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Numerical Modeling of Active Hydraulic Devices and Their
Significance for System Performance and Transient Protection
by
Qinfen (Katherine) Zhang
A thesis submitted in conformity with the requirements
for the degree of Doctor of Philosophy
Graduate Department of Civil Engineering
University of Toronto
© Copyright by Qinfen (Katherine) Zhang 2009
ABSTRACT
Numerical Modeling of Active Hydraulic Devices and
Their Significance to System Performance and Transient Protection
by
Qinfen (Katherine) Zhang
Graduate Department of Civil Engineering University of Toronto
2009
The thesis numerically explores the use and behavior of Active Hydraulic Devices
(AHDs), creating a new capability to simulate and control a pipe system’s transient
performance.
Automatic control valves are the first type of AHDs studied in this research. Due
to the challenges inherent in the design of a pressure relief valve (PRV), the general
principles of PRV use and selection are studied along with the system’s response to the
PRV parameters. A new application of PID (proportional, integral and derivative) control
valve is envisioned that combines a remote sensor at the upstream end of a pipeline to
create a non- or semi- reflective boundary at the downstream end. Case studies show that,
with such a boundary, the reflection and resonance of pressure waves within the pipeline
are sometimes eliminated and invariably limited.
The second type of AHDs studied in this research is the governed hydro turbine,
the most complicated hydraulic component in terms of transient analysis and
waterhammer control. A complete numerical model is developed for the turbine
installations in either urban water networks or conventional hydropower generation
systems. Using the model, transient simulations for several realistic hydro projects are
presented along with various transient control measures.
ii
Acknowledgements
My recognition and heartfelt thanks to Professor Bryan W. Karney, for his continuous
and enthusiastic supervision and financial support throughout my studies at the
University of Toronto, his influence on me not only in academic aspect but also in other
aspects of my life.
I appreciate Prof. Stanislav Pejovic for his knowledge, guidance, insight,
comments and the opportunity that he brought for turbine model application in MeS
Hydro project. I am also grateful to the team Professors of my Ph.D. committee,
including Barry Adams, Brent Sleep, Heather MacLean and Christopher Kennedy, for
their encouraging and invaluable comments on my thesis and presentations. I also thank
the external reviewer of my thesis, Dr. Anton Bergant of Litostroj Power, for the time he
allocated to reading my dissertation, his enthusiasm for this work and the invaluable
suggestions he offered for strengthening it. That a researcher and scholar of his high
calibre was involved with the evaluation of my work is an honor.
During my study at University of Toronto, I enjoyed the company, friendship and
help from Yimin Zhang, Lihong Sheng, Julia Koycheva, Andrew Colombo, Bong Seog
Jung, Bahman Nasar, Yves Filion et al. My special thanks to Andrew Colombo for his
help on my written English. It was a great and valuable experience for the transition from
a new immigrant to a professional in Canada.
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Most importantly, accomplishing a PhD during my middle life is only impossible
due to the continuous support of my family, the direct support from my dear husband Hui
(Howard) Guo and our older son Tingxi (Tim) Guo, and the understanding of my
extended families in China.
I experienced huge personal changes and emotional waves during the years that I
was perusing my PhD. In 2007, I was heartbroken for losing and finally lost my dear
mother. While on April 23rd, 2009, just one week after my final defense, I gave birth to
my second son Aaron Guo. These are the real pain and joy of our real lives in this world;
thank God for his incredible love and blessing through this journey.
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Contents
Abstract
Acknowledgments
Contents
List of Tables
List of Figures
Notation
1 Introduction……………………………………………………………………….....1
1.1 Active hydraulic devices and system transient control………………………...1
1.2 Thesis objectives……………………………………………………………….4
1.3 Thesis overview and layout……………………………………………….……4
1.4 Publications related to thesis research………………………………………….7
2 Pressure relief valve Selections and Transient Control…………………………...9
2.1 Introduction…………………………………………………………………...10
2.2 Types of PRVs and their applications…………………………………………11
2.3 PRV operation in pump and turbine systems………………………………….14
2.4 Case studies…………………………………………………………………...18
2.4.1 Brief description of the system………………………………………18
2.4.2 Case study 1: upstream control………………………………………18
2.4.3 Case study 2: downstream control…………………………………...22
2.5 Summary and conclusions…………………………………………………….24
3 Transient Simulation of Active Hydraulic Devices with
an Introduction to Automatic Control…………………………………………...36
3.1 Numerical method of pipe transient flow…………………………….………36
3.1.1 Governing equations and method of characteristics (MOC)………...37
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3.1.2 Extended MOC equations in pipe networks…………………………40
3.2 Controllers and active hydraulic devices……………………………………..43
3.2.1 Open-loop and closed-loop control………………………………….43
3.2.2 Feedback and feedforward control…………………………………..44
3.2.3 Evolution of controllers……………………………………………...46
3.2.4 Elements in a control system………………………………………...53
3.3 Generalized mathematical model for active hydraulic devices……………….55
3.3.1 Extended MOC equations……………………………………………55
3.3.2 AHD’s dynamic characteristics………………………………………56
3.3.3 PID controller equation………………………………………………57
3.4 Summary……………………………………………………………………...58
4 “Non-Reflective” Boundary Design via Remote Sensing and PID Control Valve………………………………………………………………………………...60
4.1 Introduction to “non-reflective” boundaries…………………………………..61
4.2 Transient performances with dead-ends and valve control…………………...64
4.3 Mathematical model for conventional/local PID control valve………………69
4.3.1 Extended MOC equations……………………………………………70
4.3.2 Valve discharge equation…………………………………………….70
4.3.3 PID controller equations……………………………………………..71
4.4 Consideration and mathematical model of “non-reflective” valve opening.…74
4.5 Simulations and case studies………………………………………………….79
4.5.1 Case studies with non-zero initial flow………………………………79
4.5.2 Case studies without initial flow (static initial state)………………...86
4.6 Frequency analysis and “non-reflective” boundary verification……………...88
4.6.1 System responses to pressure oscillations with various frequencies...89
4.6.2 “Non-reflective” boundary verification using hydraulic impedance method……………………………………………………………………97
4.7 Tuning PID controller………………………………………………………....99
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4.8 Summary and conclusions…………………………………………………...103
5 Fundamentals of Transients in Hydro Turbine Systems……………………….105
5.1 Hydro turbine classification…………………………………………………106
5.1.1 General classification of hydro turbines……………………………106
5.1.2 Hydro turbine classification in context of numerical modeling……108
5.2 Hydro turbine parameters……………………………………………………111
5.3 Hydro turbine characteristics………………………………………………..116
5.4 Model hill diagrams and conversion………………………………………...118
5.4.1 Francis turbines……………………………………………………..118
5.4.2 Kaplan turbines……………………………………………………..124
5.4.3 Impulse turbines…………………………………………………….125
5.4.4 Pump-turbines...…………………………………………………….127
5.5 Layout of water conveyance system in hydro systems……………………...130
5.5.1 Conventional hydropower system………………………………….130
5.5.2 Energy recovery turbines in municipal water supply system………133
5.6 Governor and control system………………………………………………..133
5.6.1 Three levels of turbine flow control………………………………..135
5.6.2 Mechanism of a speed-control-governor…………………………...137
5.7 Fundamental knowledge of synchronous turbine-generator units…………..139
5.7.1 Load distribution and unit operation with a governor……………...139
5.7.2 Startup and load acceptance………………………………………..142
5.7.3 Shutdown and load rejection……………………………………….143
5.7.4 Speed-no-load (SNL) and runaway condition…...…………………144
5.8 Features of transients caused by governed turbines…………………………145
5.8.1 Key features of turbine transient analysis…………………………..145
5.8.2 Speed variation……………………………………………………...146
5.9 Summary.……………………………………………………………………148
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6 Hydro Turbine Modeling and Model Verification……………………………..149
6.1 Mathematical model of turbine boundary condition………………………...149
6.1.1 Head balance equation without turbine-attached valves……………150
6.1.2 Head balance equation with turbine-attached valves……………….153
6.1.3 Speed change equation (torque balance equation)………………….157
6.1.4 Governor equation………………………………………………….159
6.2 Combination of basic equations for different scenarios……………………..161
6.3 Computer implementation…………………………………………………...164
6.3.1 Steady-state simulation……………………………………………..164
6.3.2 Transient simulation and flow chart of programming……………...164
6.3.3 Data input and output……………………………………………….166
6.4 Model and program verification……………………………………………..166
6.4.1 Comparison with Wylie’s simulation……………………………….166
6.4.2 Comparison with field measurement……………………………….172
6.5 Summary………………………………………………………………….....177
7 Case Study 1: Energy Recovery Hydro Turbines in Las Vegas Water Supply System……………………………………………………………………………..178
7.1 Project background and objectives…………………………………………..179
7.2 System and hydraulic characteristics………………………………………..181
7.3 Transient scenarios and worst case identification…………………………...183
7.4 Simulation and field tests on turbine load rejections………………………..185
7.4.1 Field tests…………………………………………………………...185
7.4.2 Simulation vs. field test……………………………………………..186
7.5 Effect of hydrant flow on turbine unit operation…………………………….196
7.5.1 Upstream hydrants………………………………………………….196
7.5.2 Downstream hydrants………………………………………………204
7.6 Summary and conclusions…………………………………………………...210
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8 Case Study 2: Evaluation of Transient Performance and Control Measures for Bone Creek Hydropower Project.…………………………………………..…...212
8.1 Introduction to Bone Creek project………………………………………….213
8.1.1 System description and data………………………………………...213
8.1.2 Objectives of transient analysis.……………………………………216
8.1.3 Simplifications and assumptions made in transient analysis……….217
8.2 Conventional open surge tank scheme………………………………………218
8.2.1 Transient performance of original system design…………………..218
8.2.2 Sensitivity analysis of surge tank distance………………………….225
8.2.3 Sensitivity analysis of wicket-gate closure time……………………230
8.2.4 Sensitivity analysis for surge tank design parameters.……………..236
8.3 Air-cushioned surge chamber scheme……………………………………….240
8.3.1 Design parameters & input data of simple cylinder air surge tank…241
8.3.2 Transient performance with simple cylinder air surge tank………...243
8.3.3 Sensitivity analysis of air chamber height………………………….243
8.3.4 Transient performance with a pipe-like air surge tank……………...248
8.4 Summary and conclusions…………………………………………………...253
9 Thesis Summary and Suggestions for System Transient Performance Evaluation…...…………………………………………………………………….256
9.1 Perspectives on system transient performance evaluation…………………..258
References………...…………………………………………………………………....262
Appendix Ⅰ: Turbine Characteristics Input Data Sample and Variable Index…269
Appendix Ⅱ: Turbine Characteristics Output Data Sample……………………...275
Appendix Ⅲ: Turbine Boundary Input Data Sample and Variable Index……….279
Appendix Ⅳ: Turbine Boundary Output Data Sample and Variable Index……..287
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List of Tables
Table 5.1 Classification of hydro turbines
Table 6-1 System data of Wylie’s example
Table 6-2 System data of MeS hydro project
Table 7-1 Linden turbine-generator parameters
Table 8-1 System data of Bone Creek hydro project
Table 8-2 Sensitivity of surge tank distance to transient performance
Table 8-3 Sensitivity analysis of simple tank diameter
Table 8-4 Sensitivity analysis of feeder and connector pipe diameter
Table 8-5 Sensitivity of cylinder air surge tank height to transient performance
Table 8-6 Sensitivity of air chamber pipe length
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List of Figures
Figure 1.1 Complexity of hydraulic devices
Figure 2.1 Operation of main valve and PRV
Figure 2.2 Synchronous operation of turbine and PRV
Figure 2.3 PRV installations in a simple pipeline system
Figure 2.4 System transient performance varying with the design of upstream PRV
Figure 2.5 Sensitivity analysis of upstream PRV parameters
Figure 2.6 System transient performance varying with the design of downstream PRV
Figure 2.7 Sensitivity analysis of downstream PRV parameters
Figure 3.1 MOC grid for single pipe
Figure 3.2 Generalized node with an external flow
Figure 3.3 ON/OFF control
Figure 3.4 Proportional control
Figure 3.5 Integral control derived from the area under the control error curve
Figure 3.6 Illustration of the need for derivative control
Figure 3.7 Loop of PID-control-valve system
Figure 3.8 Active hydraulic device in a pipe network
Figure 4.1 Scheme of a branched system and initial pressure head
Figure 4.2 Comparison of transient responses to dead-ends and small orifices
Figure 4.3 Traveling pulse waves and “tailored” valve reflections
Figure 4.4 Case studies: System scheme and initial steady state
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Figure 4.5 Development of steady-oscillatory flow in frictionless pipeline with fixed
valve opening at downstream, due to the upstream pressure oscillations
Figure 4.6 Responsive PID control valve vs. fixed-opening-valve in frictionless pipeline
(f = 0)
Figure 4.7 Responsive PID control valve vs. fixed-opening-valve in frictional pipeline (f
= 0.012)
Figure 4.8 Responsive PID control valve vs. fixed-opening-valve for static initial states
with frictional pipeline (f = 0.012)
Figure 4.9 System responses to forced pressure oscillation with various frequencies in a
frictionless pipeline (f = 0)
Figure 4.10 System responses to forced pressure oscillation with various frequencies in a
frictional pipeline (f = 0.012) (Fixed valve vs. responsive PID control valve)
Figure 4.11 Effect of controller parameters on system pressure control
Figure 5.1 Illustrative Hill Diagram of Francis turbine
Figure 5.2 Francis turbine characteristic curves in Suter parameters
Figure 5.3 Interpolation scheme of turbine characteristics
Figure 5.4 Illustrative characteristics of a Kaplan turbine
Figure 5.5 Illustrative characteristics of a Pelton turbine
Figure 5.6 Illustrative characteristics of a Turgo turbine with inclined nozzle
Figure 5.7 Illustrative characteristics of a pump turbine
Figure 5.8 Water-supply modes in conventional hydropower stations
xiii
Figure 6.1 Schematic of turbine boundary condition
Figure 6.2 Scheme of the turbine attached valves
Figure 6.3 Flow chart of turbine boundary condition programming
Figure 6.4 Comparisons with Wylie’s simulations
Figure 6.5 Comparison of Wylie’s turbine characteristics with other Francis turbine
characteristics
Figure 6.6 Schematic waterway of two parallel turbine units
Figure 6.7 Comparison of numerical modeling and field measurement
Figure 7.1 Energy recovery hydro turbine and system constituents
Figure 7.2 Turbine transient responses to 100% load rejection
Figure 7.3 Main pipeline pressure responses to 100% load rejection
Figure 7.4 Turbine transient responses to 50% load rejection
Figure 7.5 Main pipeline pressure responses to load rejection of Sloan Forebay turbine
unit
Figure 7.6 Sketch of South Valley Lateral System
Figure 7.7 Isolated turbine responses to 10 upstream hydrants flow
Figure 7.8 Online turbine responses to 10 upstream hydrants flow
Figure 7.9 Isolated turbine responses to single upstream hydrant flow
Figure 7.10 Online turbine responses to single upstream hydrant flow
Figure 7.11 Isolated turbine responses to 10 downstream hydrants flow
Figure 7.12 Online turbine responses to 10 downstream hydrants flow
xiv
Figure 7.13 Isolated turbine responses to single downstream hydrant flow
Figure 7.14 Online turbine responses to single downstream hydrant flow
Figure 8.1 Planned system layout with differential surge tank
Figure 8.2 Sketch of planned differential surge tank
Figure 8.3 Originally specified wicket-gate closure law
Figure 8.4 System responses during full load rejection with planned system layout and
design and original specified wicket-gate closure law (T1 = 6 s, T2 = 8 s)
Figure 8.5 Turbine speed rise during full load rejection with planned system layout and
design and wicket-gates refusal to close (runaway condition)
Figure 8.6 Envelope of max. and min. HGLs along main penstock during full load
rejection with 20 m diameter of surge tank and originally specified WG
closure law (T1 = 6 s, T2 = 8 s)
Figure 8.7 Sensitivity analysis of surge tank distance
Figure 8.8 System responses during full load rejection with a closer differential surge
tank and original specified wicket-gate closure law (T1 = 6 s, T2 = 8 s)
Figure 8.9 Turbine speed rise during full load rejection with a closer differential surge
tank and wicket-gate refusal to close (runaway condition)
Figure 8.10 Sensitivity analysis of wicket-gate closure time
Figure 8.11 System responses during full load rejection with planned differential surge
tank and extended wicket-gate closure law (T1 =11 s, T2 = 8 s)
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Figure 8.12 System responses during full load rejection with extended wicket-gate
closure law (T1 = 35 s, T2 = 15 s) and flywheel (adding inertia 28000 kg.m2)
for no surge tank scenario
Figure 8.13 Sensitivity of simple tank diameter D
Figure 8.14 Sensitivity of feeder and connector pipe diameter Dc
Figure 8.15 System layout with suggested underground air-cushioned surge tank
Figure 8.16 Sketch of simple cylinder air-cushioned surge tank
Figure 8.17 System responses during full load rejection with originally specified
wicket-gate closure (T1 = 6 s, T2 = 8 s) and simple cylinder air surge tank
(1000 m3 in volume)
Figure 8.18 Sensitivity analysis of the height of simple cylinder air surge tank
Figure 8.19 System responses to full load rejection with originally specified wicket-gate
closure (T1 = 6 s, T2 = 8 s) and a pipe-like air surge tank (500 m3 in volume)
Figure 8.20 System responses to full load rejection with extended wicket-gate closure (T1
= 18 s, T2 = 8 s) and pipe-like air surge tank (500 m3 in volume)
Figure 8.21 System responses to full load rejection with originally specified wicket-gate
closure (T1 = 6 s, T2 = 8 s) and a greater size of air surge chamber (840 m3 in
volume)
Figure 8.22 Sensitivity of pipe-air-chamber length to peak pressure during full load
rejection with originally specified wicket-gate closure (T1 = 6 s, T2 = 8 s) and a
pipe-like air surge tank (500 m3 volume)
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Notation
a – wave speed
a0 – turbine wicket gate opening.
B – pipe constant in MOC equations, gAaB =
D – internal diameter of pipe cross section
D1 – diameter of turbine runner
d – diameter of Pressure Relief Valve and bypass line
E – valve energy dissipation coefficient, a valve size parameter determined by the energy
dissipation potential of the valve
EB – energy dissipation coefficient of turbine bypass valve
EC – energy dissipation coefficient of turbine control/shut-off valve
ES – energy dissipation coefficient of turbine surge valve
e – control error or deviation of the process variable u(t) from its set point u*;
dimensionless error is defined as *)(1
utue −= .
FDR –Relative Friction Decay Rate
f – Darcy-Weisbach friction factor of a pipe
g – acceleration due to gravity
H or H(t) – instant pressure head difference across a hydraulic device
H0 – pressure head difference across a hydraulic device at initial steady state
H1 or H1(t) – instant pressure head at the inlet of a hydraulic device
xvii
H10 – initial pressure head at the inlet of a hydraulic device
H1*(t) – the dynamic set point of the pressure head at the inlet of a hydraulic device
H2 or H2(t) – instantaneous pressure head at the outlet of a hydraulic device
H20 – initial pressure head at the outlet of a hydraulic device
H2*(t) – the dynamic set point of the pressure head at the outlet of a hydraulic device
Have – average turbine head over years
HC – pressure head across the turbine control/shut-off valve
HR or HR(t) – instantaneous pressure head at reservoir
HR0 – initial pressure head at reservoir
HTail – water level in tailrace channel or downstream reservoir
Hmax – maximum turbine hydraulic head
Hmin – minimum turbine hydraulic head
HR – the design/rated turbine head
H (x, t) – piezometric head (HGL) at time t and position x.
h – relative/dimensionless head of a turbine, RH
Hh =
I – polar moment of inertia of rotating fluid and mechanical parts in the turbine-generator
unit, I =WRg2
KC – proportional gain of controller
L – pipe length
xviii
N1 – number of pipes at a junction, whose flow direction is assumed toward the node
N2 – number of pipes at a junction, whose flow direction is assumed away from the node
Ndy – number of wicket gate settings;
Nblade – number of runner blade angle settings.
n – rotational speed of turbine unit (rpm)
nE – synchronous speed of generator
ng – specific speed of a turbine or pump
ns – specific speed of a turbine
nR – rated speed or normal speed of turbine unit
nrun – runaway speed of turbine unit
n11 or – unit speed of a turbine, '1n 1'
1 HnDn =
P – rated energy produced from turbine or generator per second (kW, Hp)
PG – generator power output
PT – turbine power output
P11 or – unit power of a turbine: '1P 2/32
1
'1 HD
PP =
Q or Q(t) – instantaneous flow in pipeline or passing through a hydraulic device
Q* – dynamic set point of the flow passing through a hydraulic device
Q0 – initial steady state flow in pipe system or passing through a hydraulic device
Q1 or Q1(t) – instantaneous flow at the inlet of a hydraulic device (also called external
flow)
xix
Q2 or Q2(t) – instantaneous flow at the outlet of a hydraulic device (also called external
flow)
QBV – discharge of turbine bypass valve
QCV – discharge of turbine control/shut-off valve
QSV – discharge of turbine surge valve
QR – design/rated turbine discharge
Q11 or – unit discharge of a turbine, 1Q′ 2
11 HD
QQ =′
Qext – external flow at a junction, governed by a hydraulic device or boundary condition
R – pipe constant in MOC equations, 22gDAxfR Δ
=
Rg – gyration radius of turbine-generator rotating parts
SET1 – high-pressure set point of PRV
SET2 – low-pressure set point of PRV
T – turbine shaft torque exerted by the water
T11 or – unit turbine shaft torque, '1T 3
1
'1 HD
TT =
TR – rated turbine shaft torque exerted by the water
Tg – resistant shaft torque from the generator
TV1 – opening time period of PRV
TV2 – closure time period of PRV
xx
Td – derivative time constant of controller; dashpot time constant in turbine speed
governor
Tf – the shortest time period for closure of turbine wicket gate
Ti – integral time constant of controller
Tm – mechanical strating time of turbine-generator unit, RRRRRm TITIT ωωω == 2
Tα – turbine governor’s promptitude time constant without temporary speed droop
'αT – turbine governor’s promptitude time constant with temporary speed droop, that is,
dTTT ' δαα +=
t – elapse of time
u(t) – process variable to be controlled
u* – set point or desired value of process variable
V (x, t) – fluid velocity whose direction is defined as the same as the distance x
v – relative/dimensionless flow of a turbine, RQ
Qv =
W – weight of turbine-generator rotating parts and fluid contained
WH – Suter parameter of turbine head, )( 22 vhxWH +
=α
, in which the
coordinate tan 1
vx α−=
WB – Suter parameter of turbine torque,B )( 22 vxWB +=α
β , in which the
coordinate tan 1
vx α−=
x – distance along the centerline of a pipeline
y – dimensionless opening of turbine wicket-gates or a control valve
xxi
α – relative/dimensionless speed of a turbine, Rn
n=α
β – relative/dimensionless torque of a turbine, RT
T=β
δ – turbine governor’s temporary speed droop
σ – turbine governor’s permanent speed droop
γ– the specific weight of water
τ – dimensionless valve opening
Bτ – dimensionless opening of turbine bypass valve
Cτ – dimensionless opening of turbine control/shut-off valve
Sτ – dimensionless opening of turbine surge valve
Gη – generator efficiency
Tη – turbine efficiency
η – efficiency of a turbine-generator unit.
ω – angular speed of a turbine, 602 nπω =
Rω – rated angular speed of a turbine, 602 RR nπω =
tΔ – time step, or dimension of time in MOC grid
xΔ – dimension of the pipe distance in MOC grid
ϕ – runner blade angle for Kaplan turbines
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Chapter 1 Introduction
1.1 Active Hydraulic Devices and Transient Control
Unsteady or transient states are inevitable for any pipeline carrying water or any other
substance. Indeed, transients can be caused by routine switch on/off of a device, system
initialization (pipeline filling), accidents occurred during normal operation (e.g., power
failure, load rejection) or any other event that causes a flow change. When a pressured
pipe system experiences a transient event, both the physical components of the system
and the services provided by the system could be at risk due to the possible extreme high
pressure, unacceptable low or negative pressure, vibration of devices, or the overspeed of
any associated turbo machines. Therefore, the unsteady condition and transient flow
should be well managed and controlled to maintain the normal operation of the system.
Although transient flow is transmitted through pipelines, such flows are
almost universally initiated and modified by hydraulic devices in the system. In other
words, boundary conditions are usually the key to control system transients, and they
play crucial roles in determining both the nature and magnitude of system responses
during unsteady states. The term “Active Hydraulic Device” (AHD) is coined here to
describe a category of hydraulic component with the potential to create, manipulate and
control the system transients, typically by direct or active modification of their head-
discharge characteristic. Thus, as a class AHDs include all types of control valves,
1
hydraulic turbines, pumps and so on. Each device specifies a certain relationship between
pressure and flow, and both the flow and associated pressure are usually adjustable to
certain degree. Interestingly, a pump or turbine usually adjusts its working head and flow
through the operation of its control valve or wicket-gates, although the speed or load
variation of the hydraulic machine could also lead to certain changes of its flow-head
relationship. Thus, the control valves or wicket-gates are in this sense more active than
hydraulic machinery in terms of transient control. Moreover, even less active hydraulic
devices, such as reservoirs, surge tanks, and air chambers, are usually operated in
conjunction with any control valves used for waterhammer protection during the system
transients. Each hydraulic device has different level of complexity and activity that can
be conceptually visualized in Figure 1.1, although the placement of the device in the
diagram needs to be considered approximate only.
Activity Level
Hydro Turbines Variable Speed Pumps Fixed Speed Pumps Modulating Valves Air Chambers Water Tanks On-off Valves Reservoirs
Modeling and Operational Complexity
Figure 1.1 Complexity of hydraulic devices
2
It is not surprising that most transient events in a pipe system are created by the
actions of these devices either on purpose or by accident, while different active devices
(or strategies) are implemented in different systems to control and limit system transients.
Specifically, a pump with Variable Frequency Drive (VFD), by dynamically altering
pump speed, allows not only a more controlled start/stop of the pump, but also a more
precise match of pressure and flow to system requirements. Similarly, a governed turbine,
by regulating the unit speed, is able to change the opening of wicket-gates in response to
the demand from the power grid system. The particular strategy of Misaligned Guide
Vanes (MGV), specially designed for a high-head reversible turbine, is a typical active
control strategy, which modifies the “S-shape” turbine character by allowing pre-opening
of some guide vanes before the movement of others and mediates the instability at zero-
load operating region and reduces the difficulty of synchronization (Liu, et al., 2007). In
addition, by automatically adjusting the valve opening, control valves are able to relieve
excessive transient pressures (e.g., through pressure relief valves and air valves), and or
to at least partly sustain the desired pressure or flow rate at some point of system
(modulating valves), or even to create “non-reflective” boundaries for
eliminating/limiting reflection and resonance of pressure waves in the pipe system.
Overall, no hydraulic device is a panacea for all pipeline surge and waterhammer
problems, yet a combination of different components often provides a cost effective and
holistic solution to unsteady flow challenges. The resulting control system must be
examined thoroughly and comprehensively utilizing appropriate computer software for
transient simulation.
3
1.2 Thesis Objectives
This thesis aims to develop new capacity of transient analysis modeling and control for
several AHDs and their associated protection strategies. A generalized mathematical
model is first adapted for the hydraulic devices with PID (proportional, integral and
derivative) type of controllers since automatic control is inherently relevant to the AHDs;
the governing equations in the general model of AHDs are quite readily created for an
active control valve. In order to demonstrate the capability of an automatic control valve
for manipulating the wave reflection and resonance in a pipeline system, an attempt is
undertaken to modify the model of a PID control valve by combing a remote sensor with
the controller to create a “non-reflective” boundary condition in the pipe system.
Meanwhile, based on the general mathematical model of AHDs, with the knowledge of
hydro systems and characteristics of turbine-generators, a numerical model of governed
turbine unit is established with the intent of analyzing and controlling the system
transient performance for realistic hydro projects. More broadly, this thesis endeavors to
place the issue of transient analysis of AHDs within the larger context of comprehensive
evaluation methodology of system hydraulic transient performance and protection.
1.3 Thesis Overview and Layout
The progress of this thesis was guided by the creation of conference and journal articles
and project reports; some of the core material has been disseminated in different venues.
This dissertation is organized around these papers and reports, but effort is made to link
4
them logically using the “AHD” concept as their essential core, though readers might not
find the traditional “linear” narrative throughout the thesis.
Automatic control valves are the first type of active hydraulic devices studied
in this research. Such a valve could be either a cause of transient flow or a measure used
to protect system from detrimental transients; it is often a key element in a water system
to control the flow rate, reverse flow, pipe pressure or water level in a tank. Due to the
challenges inherent in the design of a Pressure Relief Valve (PRV), the general principles
of their use and selection are studied along with the sensitivities of a system’s response to
the PRV parameters (Chapter 2). Chapter 2 elucidates what to pay attention when
designing or applying an AHD (i.e., PRV) to protect a system from excessive
waterhammer pressures.
Chapter 3 introduces the general numerical modeling challenges associated
with AHDs. In this chapter, the extended Method of Characteristic (MOC) for transient
analysis and theory of automatic control and controllers are reviewed and a generalized
mathematical model of AHDs is described. This chapter also briefly reviews the relevant
literature, providing a foundation for the overall thesis research.
Chapter 4 is one of the core contributions of this research; it creates a new
strategy of transient pressure control using an AHD (i.e., the PID-controlled valve).
Based on the numerical model of conventional PID control valves, a new application of
PID control valve is envisioned that combines a remote sensor at the upstream of a
pipeline to create a non- or semi- reflective boundary at the downstream end. Case study
shows that, at such a boundary, the valve is able to adjust its opening automatically in
5
response to the pressure changes remotely sensed to eliminate or mediate the reflection
and resonance of pressure waves within a pipeline system.
The second type of active hydraulic device studied in this thesis is the
governed hydro-turbine, the most complicated hydraulic component in terms of transient
analysis and waterhammer control. The motivation and interest in this hydro system
research arises partly from the author’s previous background, as well as renewed interest
worldwide in hydro-related businesses (especially small hydro) motivated by growing
environmental concerns and improved hydro technologies. Meanwhile, waterhammer
has been a serious issue in many hydro systems, and numerous accidents or troublesome
behavior is associated with the operation of turbine units (Pejovic, et al., 2007). Although
there are a number of numerical models developed for transient analysis of hydro systems,
few of the published ones are particularly designed for the applications in water networks.
Chapter 5 reviews the hydro turbine and its control system, including the
treatment of turbine model characteristics. Then, Chapter 6 develops a comprehensive
numerical model for turbine installations in either conventional hydro pipeline systems or
complex municipal water networks. Following that, as case studies of transient
performance evaluation and system transient control, two realistic hydro projects are
presented, including an energy recovery hydro-turbine project in urban water supply
system (Chapter 7) and a conventional hydroelectric power project (Chapter 8).
Chapter 9 summarizes the thesis research and addresses the perspective that
system performance and transient protections should be evaluated at different technical
levels and in a more broad way.
6
1.4 Publications Related to Thesis Research
The following list comprises the journal articles and conference papers that have either
been published, or are under work for publication that are related to the thesis. As
aforementioned, the core contributions of this thesis have been disseminated in
publication format.
1. Qinfen Zhang, Bryan W. Karney and David L. McPherson. (2008) “Pressure Relief
Valves Selection and Transient Pressure Control”. August 2008, J. of AWWA. Vol.
100, No. 8, p62-69 (closely linked to Chapter 2).
2. Bryan Karney, Qinfen Zhang, Feng Wang and Stanislav Pejovic (2008). “Non-
reflective Boundary Design through Remote Sensing and PID Control Valve”.
Published and orally presented at Pressure Surges 10th International Conference,
BHR Group, Edinburgh, Scotland, May14-16, 2008 (loosely linked to Chapter 3
and 4).
3. Katherine Qinfen Zhang, Bryan Karney (2008). “Energy Recovery Hydro Turbines in
water Supply system”. Published and orally presented at HydroVision 2008,
Sacramento, California USA, July 14-18, 08 (somewhat linked to Chapter 7).
4. Stanislav Pejovic, Bryan Karney, Qinfen Zhang. (2004) “Water Column Separation in
Long Tailrace Tunnel”, Proceedings of HYDROTURBO 2004 international
conference, Brno, Czech Republic, Oct. 2004 (loosely linked to Section 6.4.2,
Chapter 6).
7
5. Zhang Qinfen, Karney Bryan. (2003) “Pipe System with Micro-Turbines:
Waterhammer Considerations”, Proceedings of FEDSM’03 4th ASME-JSME Joint
Fluids Engineering Conference, Hawaii, USA, July 2003 (loosely linked to Chapter
5, 6 and 7).
Chapters 2 and 4 are based on the published journal paper entitled “Pressure
Relief Valves Selection and Transient Pressure Control” (No.1 in the Publication List)
and conference paper “Non-reflective Boundary Design through Remote Sensing and
PID Control Valve” (No. 2 in the Publication List). The materials of other published
papers are referenced in the thesis. For these thesis-related papers, I wrote the papers and
did the majority of research and analysis behind them. My Ph.D thesis supervisor, Prof.
Bryan Karney, was naturally a significant coauthor for the publications, providing ideas
and insights, and proofreading/editing the manuscripts. Other coauthors on more limited
aspects include David L. McPherson, Stanislav Pejovic and others, they all performed the
customary duties of coauthor by way of editorial suggestions and appropriate caveats. I
have received their written permission and endorsement to include the material derived
from, or based on, our joint publications.
8
Chapter 2 Pressure Relief Valve Selection and
Transient Pressure Control
Valves are the first type of active hydraulic devices studied in this thesis, specifically
automatic control valves which are treated in depth. While by no means simple, such
valves serve as an excellent starting point for any foray into active hydraulic devices
because they are considerably less complex than a turbine control system or variable
speed pump. The operation of automatic control valves could be either a cause of
transient flow or, conversely, a measure to protect a system from the detrimental effects
of such unsteady flow; valves are key elements in water system for regulating flow rate,
preventing reverse flows, limiting pressures, or moderating tank water levels. Due to the
challenges inherent in the design of a Pressure relief valve (PRV), perhaps better referred
to as a Surge Relief Valve (SRV), the general principles of PRV/SRV use and selection
are studied in this chapter along with an exploration of the sensitivity to system response.
This chapter highlights the appropriate design and application of an AHD (i.e., SRV) to
protect a system from excessive waterhammer pressures.
This work is now published in the Journal of the American Water Works
Association (August, 2008) and was co-authored with Bryan Karney and David L.
McPherson. As the first author, I wrote the full early draft of the paper, and did all the
numerical runs and analysis. The idea of this work originated with Prof. Karney and
various editorial suggestions and practical caveats were provided by the paper’s
coauthors.
9
Abstract: A pressure relief valve (PRV) is installed for surge protection and pressure
relief in a pipeline system. The valve, which is normally closed, is designed to open
rapidly once its pressure setting is met or exceeded, thus permitting fluid discharge to
relieve pressure. A PRV’s effectiveness depends on system properties, the surge
characteristics, and the way in which the valve’s attributes and settings are configured.
This chapter illustrates the design challenges pertaining to PRVs and shows that an
appropriately designed PRV can protect some systems from extreme pressures but that
inappropriate use can actually worsen a system’s transient response. The general
principles of PRV use and selection are presented along with a sensitivity analysis of
PRV parameters. Although this understanding is essential to effective system design, the
final selection of a PRV is made through numerical simulation that evaluation of PRV
viability and cost-effectiveness in specific systems.
Key Words: Pressure relief valve (PRV), Parameters selection, Numerical simulation,
Transient performance, Sensitivity analysis, Transient pressure control.
2.1 Introduction
Transient events, in particularly rapid changes of flow rate, can cause serious problems in
pipeline systems and networks. High waterhammer pressure can permanently deform or
rupture a pipeline and its components; low pressures can collapse a pipe, inducing leaks,
causing service disruption, and contaminating water in the pipeline.
Numerous transient control strategies have been developed, including changes
within the pipe system (pipe diameter, thickness, alignment, profile or other hydraulic
10
components), wave speed reduction, optimized operation, and installation of dedicated
devices such as automatic control valves, surge tanks, and air chambers (e.g., Wylie et al.,
1993; Karney and Simpson, 2007). Automatic control valves, including pressure relief
valves (PRVs), flow- or pressure-regulating valves, air valves and check valves, are
common and often cost-effective. Depending on the type, a valve is used to control
transient conditions either by reducing the rate of flow velocity change or by discharging
or admitting air into the pipeline. When triggered by a pressure that is beyond its preset
limit, a PRV opens to allow flow. The resultant outflow causes a pressure drop and
presumably reduces the maximum pressure. Conversely, inflow through the valve
compensates for reduced flow and can limit low pressures and/or avert cavitation. A
PRV must have a low physical inertia so that it can respond rapidly to the detected
pressure fluctuations and open before the set-point is greatly exceeded (Chaudhry, 1979).
As portrayed in the case studies presented in this chapter, delays in valve opening can
compromise system transient protection.
2.2 Types of PRVs and Their Applications
Depending on the application and industry, various types of PRVs are used. A safety
valve (also referred to as an overpressure pop-off valve), usually applied in steam or gas
conveyance systems, is a spring- or weight-loaded valve that opens once the pressure in
the pipeline exceeds its set point and closes immediately when the pressure drops below
such threshold. Thus, a safety valve is either fully open or fully closed. If not activated
too frequently, a rupture disk can be used as an alternative to a safety valve. This type of
valve is not a “true” valve but rather an opening in the pipe. The opening is covered by a
11
diaphragm that ruptures and relieves pressure when the pressure set point is exceeded.
One disadvantage of a rupture disk is that it continues to discharge until it is replaced.
In liquid applications, a PRV is usually mounted on the discharge side of a pump,
hydro turbine, or main cutoff valve. It is typically a pilot-controlled throttling valve,
opened or closed either hydraulically or by a servomotor, with opening and closing rates
that can be individually set. It is distinct from a pressure-regulating valve, although both
have pilot systems. In fact, a regulating or modulating valve typically uses a feedback
device (usually, a PID type controller) to accurately sustain a pressure set point in
response to the sensed pressure or pressure difference. By contrast, a PRV is usually
triggered by an event (On/Off controller) and, once triggered, follows a predefined
opening and closing motion. More precisely, a PRV opens to release water when the
pipeline pressure at the valve inlet exceeds a high set point (referred to as SET1) and the
normal discharge of the valve is to a zone of lower pressure or to an open discharge area
such as a pump station wet well or adjacent storm sewer, pond or stream. In addition, a
PRV can open to admit water through a bypass line when the valve’s downstream
pressure in the main pipeline falls below a low-pressure set point (referred to as SET2).
If the pressure of the water source linked with the PRV is higher than that at the pipeline
downstream of the control valve, the valve opens to supply fluid, thus compensating for
the reduced flow and limiting the magnitude of the downsurge and the subsequent
upsurge reflected from the downstream pipeline. A PRV set in this mode is commonly
referred to as a pump- or valve-station bypass assembly.
PRVs have been used successfully in a wide range of hydraulic conditions and
operating scenarios to reduce the adverse transient conditions within a pipeline system.
12
However, they should not be used without proper assessment of their ability to
adequately protect the system. Because a PRV is a reactionary device, many hydraulic
systems will not benefit from a PRV. For example, if the PRV is used to relieve high
pressure, the pressure relief will initially be local to the PRV itself, because its operation
obviously depends on a local discharge of fluid. Protection of the pipeline as a whole will
depend on a variety of factors that involve the complex interplay between the strength
and source of the original surge condition, the way in which the valve action modifies the
wave propagation, and the way in which these conditions interact with the system’s
strength. In some cases, a PRV may only provide quick local protection, making other
mitigation strategies more cost effective. One particular misapplication is to deploy a
PRV in the bypass mode to protect a pumped system with a rising main from an
uncontrolled shutdown (power failure). Under these conditions, the available pressure on
the suction side of the pump station may not allow sufficient flow to effectively limit the
resultant downsurge. The first step to develop a positive transient control system using a
PRV is to understand the valve’s operation and limitations. A preliminary numerical
evaluation provides the design engineer with insights into the suitability of a PRV for
controlling surge pressures.
Once a decision has been made to use a PRV, it must be carefully and
appropriately designed. There are four design considerations for a PRV. The first defines
the valve’s location. In a high-pressure relief mode of operation, the PRV should be
positioned so that the high pressure and flow can be diverted around the pressure-
sensitive areas and excess flow can be discharged to an appropriate place. The remaining
three design considerations relate to the PRV characteristics and valve parameters,
13
including the valve size (d), the high- or low-pressure set point (SET1 or SET2), and the
opening and closure time periods (TV1/TV2). Each valve parameter profoundly
influences system performance. For instance, an oversized PRV will not be cost-
effective (the system safety enjoys little improvement), while an undersized valve cannot
effectively alleviate excessive pressure surges. Using a fully dynamic transient model,
TransAM, a simplified pipeline system is analyzed in order to illustrate both the general
principals of PRV design and the sensitivity of the response to control valve parameters.
2.3 PRV Operation in Pump and Turbine Systems
To protect the system from unacceptable low pressure or excessive high pressure during
the sudden pump shutdown, a PRV can sometimes be installed on a bypass line around
the pump station. When a pump’s power supply is interrupted, the reduced flow at the
pump causes an imbalance in flow and a rapid reduction in pressure. The net effect is that
a rarefaction wave propagates into the discharge line. When the low pressure wave
reflects off a downstream boundary (e.g., water tank), the pressure will be normalized to
the free water surface level in the tank and thus reflected back into the system,
establishing pressures that are sometimes higher than those originally experienced before
the flow was reduced. After this reflected positive wave has reached the pump, and the
pump check valve has closed, the check valve effectively becomes a “dead-end”, creating
another reflection and magnification. In theory, the reflected wave doubles at a closed
valve, producing a higher transient pressure. In a pumped system, a PRV will open at the
initial low-pressure set point once power to the pump motor is cut off. The PRV, now
open in anticipation of the returning positive wave, will offer a means for releasing water
14
from the system and prevent the onset of a high transient pressure condition. If the
operating suction pressure is not positive in a pumped system, a partial or full vacuum
pressure will develop at the PRV once the valve is opened. Should a full vacuum develop,
the severity of the transient pressure may be exacerbated because of cavity collapse. In
the absence of positive pressure, particular measures (such as an air valve) might be
needed when contemplating whether a PRV is an appropriate control device.
Main Valve Movement
PRV Movement
TV1 TV2R
elative Valve O
peningTime (s)
Figure 2.1 Operation of main valve and PRV
Unlike with regular startup and shutdown, power failure is always sudden and
unpredictable. In Figure 2.1, the main valve closure is equivalent to the reduction of
pump flow due to the power failure. This kind of system (e.g., typical of certain types of
booster pumping stations) assumes a positive suction operating head. As a result, the
PRV in the bypass line opens rapidly to admit water into, or release water from, the
pipeline back to the suction side of the pump. The PRV is then gradually closed to
15
eventually shut down the system and bring it to rest; a slowly closing valve has little
effect on the transient condition. The check valve, which is typically installed at each
pump, will close upon flow reversal. However, when the check valve is closed, the
rotational moment of inertia that controls the rate of pump run-down during a power
failure will be effectively eliminated (Ruus and Karney, 1997; Chaudhry, 1979). As a
result, the energy dissipation required to slowly bring the system to rest and dampen the
adverse transient condition is left to the PRV bypass assembly.
The application of a PRV in a hydropower plant depends on many factors. It is
more commonly used in small hydro turbine installations, where large surge protection
measures such as a surge tank and air chamber are not essential. In large hydropower
plants, a PRV usually functions as a secondary surge protection measure and is not
typically a substitute for a surge tank. For a power plant isolated from the electric grid,
the most common configuration is to have the PRV in the bypass line kept partly open to
provide for the maximum anticipated rapid load changes. As shown in Figure 2.2(a),
when the turbine unit rapidly accepts a load, the PRV begins to close almost
simultaneously with the opening of the wicket gates in order to compensate the turbine
flow from the PRV bypass line. Alternatively, following a load reduction and as the
wicket gates are closing, the PRV opens to divert the turbine flow as illustrated in Figure
2.2(b); subsequently, the PRV closes slowly. Yet, when a full load rejection occurs, either
in the synchronous or isolated operation of the turbine unit, all the turbine flow is
switched to the PRV, as shown in Figure 2.2(c). In this way, a much more constant flow
velocity in the penstock is maintained during either load acceptance or load rejection.
16
Some water is wasted through the PRV operation; however, the amount of discharged
water is insignificant compared with the total cost of transient control (Chaudhry, 1979)
.
Figure 2.2 Synchronous operation of turbine and PRV
17
2.4 Case Studies
The design of a PRV to control adverse transient pressure is system specific and depends
on the physical and hydraulic conditions in the system and the nature of its response to
operation of the PRV during the transient event. Fully dynamic hydraulic transient
modeling can be used to illustrate the system’s reactions and estimate its transient
performance associated with the selection of various PRV parameters. To illustrate, a
simple system with a PRV installation at both upstream and downstream locations, is
discussed here.
2.4.1 Brief description of the system
As shown in Figure 2.3, two reservoirs are linked by a uniform pipeline with a length L =
500 m, diameter D = 1.0 m, friction factor f = 0.012, and wave speed a = 1200 m/s. The
water level is 15 m at upstream reservoir and 12 m at downstream reservoir. At each end
of the pipeline, there is a primary valve and a PRV; the PRV is situated on a short bypass
line that connects either end of pipeline to its associated reservoir. At initial steady state,
both main valves are fully opened, both PRVs are fully closed, and flow in the pipeline is
Q0 = 1.73 m3/s.
2.4.2 Case study 1: upstream control
Main-valve-1 at the upstream end is fully closed within 5 s, causing an incident
rarefaction pressure wave in the pipeline. When the pressure at the outlet of PRV-1 (i.e.,
the pressure at the outlet of main-valve-1) falls below its set point (SET2), the PRV opens
18
to admit water and avoid the downsurge at the outlet of main-valve-1, thus reducing the
subsequent upsurge reflected from the downstream reservoir.
12 m
Q
PRV-1
Main-Valve-1
15 m
Datum
PRV-2
Figure 2.3 PRV installations in a simple pipeline system
Comparison of different upstream PRV parameters. The transient pressures in
the pipeline are compared in Figure 2.4 for the following systems: one without PRV
protection, one with an appropriately designed PRV, and four with poorly designed PRVs.
Without PRV protection (Figure 2.4(a)), rapid closure of main valve would result in
approximately 260 m of maximum pressure head and -240 m of minimum pressure head
in the system. The maximum head 260 m is extremely high, and this may cause pipe
rupture or impose extra cost for a pipe of greater strength. The negative pressure head (-
240 m) is a numerical value from the simulation model and does not factor in
vaporization and cavitation. However, the modeled -240 m rarefaction wave is
completely unacceptable because of the occurrence of cavitation and subsequent cavity
collapse. Cavitation will lead to vapor cavity formation and subsequent cavity collapse
(the so-called column rejoinder effect), which can cause extremely high pressure. The
19
large negative pressures indicate that the response is unacceptable when the PRV is not
operated, and thus protection from transient pressure is necessary for this system. As
shown in Figure 2.4(b), operation of an appropriately designed PRV will limit maximum
and minimum transient pressure heads along the pipeline to within the range of 3.6 m and
22 m. However, inappropriate selection of PRV parameters, such as an undersized PRV,
a pressure set point for SET2 that is too law, a PRV opening that is too slow, and closure
that is too fast, would result in poor performance and unacceptable transient pressures, as
evident in Figure 2.4 (c), (d) (e) and (f), respectively. However, Figure 2.4 also shows
that even a poorly designed PRV provides some protection and is certainly preferable to
completely ignoring PRV deployment.
Sensitivity analysis of upstream PRV parameters. Proper selection of a PRV’s
control parameters is important, particularly because these parameters are inevitably
uncertain. For example, activation of a PRV depends on exactly how and where the
pressure is sensed and how this information is transformed by both the ongoing transient
and control valve action. Sensitivity analyses would reveal how the system’s transient
performance changes with variation of each PRV parameter, and could aid in identifying
rules for PRV design. Therefore, the sensitivities of each parameter of PRV-1 (d, SET2,
TV1, TV2) were analyzed and results are summarized in Figure 2.5.
Figure 2.5(a) indicates that an increase in valve size would improve the transient
performance, but only up to a certain limit; an increase in valve size beyond 0.5 m would
not effectively improve transient performance in this case study. Design and field
experience indicate that PRV diameters usually range from one twentieth to one third of
the main pipe diameter, with most values falling near the middle of this range. The logic
20
for achieving this type of range is evident in Figure 2.5(a), although not exactly followed.
The slight discrepancy that appears can be explained as follows: in this case study, only 3
m of pressure head existed along a 500-m length of pipe; thus the driving heads in this
system is very low. This implies that in order to discharge a certain amount of water, a
larger PRV is needed. The ultimate choice of PRV size is usually a compromise between
cost, which increases rapidly with diameter, and hydraulic control and protection, which
usually provide diminishing returns with PRV diameter.
Figure 2.5(b) illustrates that the variation in transient pressures diminishes with an
increase in the low-pressure set point (SET2). An issue is how quickly the bypass or
PRV will open immediately after a transient down-pressure wave is detected in
association with the motion of the primary valve. Because of the system configuration
presented in case study, there is no improvement in response when SET2 is beyond 15 m,
because the valve stays open once the set point is higher than this upstream reservoir
pressure. In general, the lower or more sensitive the threshold pressure for PRV
activation, the better the transient protection, because the valve will begin its response
earlier. However, there is an obvious trade-off, because a valve that activates quickly is
more prone to activate not only for important design events but also for normal or routine
transients that pose no threat to the system. A valve that opens prematurely not only
requires more frequent routine maintenance but, depending on system configuration, can
waste both energy and fluid from the system.
Figure 2.5(c) shows that the variation of opening time period (TV1) does not bear
on the maximum transient pressure in this case study; in general, the shorter the opening
time (TV1), the less severe the resulting negative pressure. When TV1 is less than 5 s, a
21
reduction in TV1 no longer changes the minimum transient pressure. When TV1 is longer
than 10 s, the negative pressure would occur in the pipeline and rapidly becomes more
severe with increasing TV1. In this case study, the appropriate opening time for the PRV
should be within 5-10 s when both ease of PRV operation and the prevention of negative
pressures in the pipeline are considered. As mentioned here, and discussed by Chaudhry
(1987), low valve inertia or highly responsive control system is required for a PRV to
achieve a response and activation. These components can increase either, or both, valve
cost and maintenance requirements. However, the timing of PRV operation and the
transient response are unique to each hydraulic system and essentially govern the needs
and costs of the control valve, the configuration of the system, and the nature of the
disturbance that actually generates the transient response.
As shown in Figure 2.5(d), slowing PRV closure (i.e., increasing TV2) results in
smaller variations in transient pressures, which benefits hydraulic control but diverts
more water during PRV operation. In addition, when TV2 is shorter than 60 s, the range
of maximum and minimum pressures becomes more sensitive to the variation of TV2. In
this case study, 60- 80 s is suggested for the PRV closure time period.
2.4.3 Case study 2: downstream control
In this case study, main-valve-2 at the downstream end is fully closed in 5 s, causing an
incident upsurge in the pipeline. When the pressure at the inlet of PRV-2 (i.e., the
pressure at the inlet of main-valve-2) exceeds its set point (SET1), the PRV opens to
discharge water into the downstream reservoir to mediate the pressure rise in the
upstream pipeline.
22
Comparison for different downstream PRV parameters. As shown in Figure
2.6, if there is no PRV protection, the rapid closure of main-valve-2 results in 232 m of
the maximum pressure and -200 m of the minimum pressure (see previous text regarding
pressure in excess of full vacuum condition) in the system. Operation of an appropriately
designed PRV will reduce the envelope of extreme pressures to between 4.2 and 25.8 m.
An undersized PRV, or the inappropriate setting of the PRV set point or valve timing,
would result in unacceptable transient conditions, shown in Figure 2.6 (c-f).
Sensitivity analysis of downstream PRV parameters. The sensitivities of each
PRV-2 parameter of (d, SET1, TV1, TV2) were analyzed and are shown in Figure 2.7;
the results are similar to those from the previous case study. Figure 2.7(a) illustrates that
the most economically efficient and hydraulically effective diameter of the PRV is
around 0.2-0.3 m, which is well within the range from one fifth to one third of the main
pipe diameter.
Figure 2.7(b) shows that the range of transient pressures becomes narrower as
SET1 is reduced. However, there is little improvement in transient control when SET1 is
set lower than 15 m, which is the water level in the upstream reservoir. In addition, a
negative or partial vacuum pressure resulting from the boundary reflection will become
more severe if SET1 is set higher than 32 m. In this case study, a proper SET1 value of
15-32 m should be selected.
Figure 2.7(c) demonstrates how a reduction in PRV opening time (TV1) will
reduce the maximum transient pressure in the system, though only slightly affecting the
minimum transient pressure in the pipeline. Curtailing TV1 to less than 20 s does not
23
change the transient response any further. In this case study, results indicate that the
system is rather insensitive to the selection of TV1 as long as TV1 is not extremely long
(e.g., longer than 60 s).
Figure 2.7(d) shows that a slow closure of PRV-2 would improve transient
pressure performance. In this case study, 60 s or longer is required for TV2 to prevent the
negative pressure from occurring in the pipeline. The trade-off is that a slower closing
time will result in more water being diverted out of the system during the PRV operation.
2.5 Summary and Conclusions
The objective of PRV design is to transform an existing hydraulic system to a new one
capable of a more acceptable transient response. In evaluating a PRV, the first step is to
qualitatively assess the PRV’s viability as a protection device. Doing this requires a
comprehensive understanding of how a PRV contributes to transient mitigation,
especially appreciation that a PRV is a reactionary device that establishes a pathway to
either introduce or relieve energy from the hydraulic system. Many designs have failed
because a PRV was not designed or deployed correctly. However, if such a device is
found to be a viable surge-control strategy, a series of interdependent design parameters
must be considered in its design. The basic principles of PRV designing and its
associated parameter selection have been described in this chapter and are summarized as
follows:
It is obvious that an undersized PRV would be insufficient to protect a system
from extreme transient pressures. There is a misunderstanding in practice,
24
that is, the bigger the valve diameter is the safer the system will be. However,
this intuition is not always valid, and there is no value in over-sizing a PRV
because, beyond a certain point, there is little further improvement in its
waterhammer performance but potential under pressure problems due to
larger flow discharge, even though its cost continues to increase.
If the pressure set point is exaggerated (i.e., too low for the low-pressure set
point or too high for the high-pressure set point), the PRV will not adequately
provide transient pressure control.
If the PRV opens too slowly or closes too quickly, it could cause dangerous
waterhammer incidents, although in general a PRV helps to alleviate extreme
waterhammer pressures.
The case studies presented in this chapter suggested common principles to follow
when selecting a PRV at preliminary design stage, but the detailed values of each
parameter will be different from case to case. For instance, the valve opening time period
TV1 is required for 5-10 s in the upstream PRV, while it could be as long as 20 s in the
downstream PRV. Therefore, to make wise and informed choices for these parameters at
final design stage, a fully dynamic hydraulic simulation is required to verify the
performance of each pipeline system.
25
Max. H
Steady State H
Pipeline
Min. H
(a) Simulation without PRV protection
Max. H
Min. H
(b) Simulation with an appropriately designed PRV
(d = 0.9 m, TV1 = 5 s, TV2 = 80 s)
Figure 2.4 System transient performance varying with upstream PRV design (Con’d)
26
Max. HSteady State H
Pipeline
Min. H
(c) Simulation with an under-sized PRV
(d = 0.2 m, TV1 = 5 s, TV2 = 80 s)
Max. H
Min. H
(d) Simulation with a too-low-pressure set point PRV
(d = 0.9 m, SET2 = 2 m, TV1 = 5 s, TV2 = 80 s)
Figure 2.4 System transient performance varying with upstream PRV design (Con’d)
27
Max. H
Min. H
(e) Simulation with a too-slow-opening PRV
(d = 0.9 m, TV1 = 20 s, TV2 = 80 s)
Max. H
Min. H
(f) Simulation with a too-fast-closure PRV
(d = 0.9 m, TV1 = 5 s, TV2 = 40 s)
Figure 2.4 System transient performance varying with upstream PRV design
(Note: Default SET2 is equivalent to SET2>=15m, i.e., the PRV is activated instantly when Main-Valve-1 closes)
28
Max. / Min. Waterhammer Pressure vs. PRV Diameter
-80
-60
-40
-20
0
20
40
60
80
0 0.2 0.4 0.6 0.8
PRV Diameter (m)
Max
. and
Min
. Pre
ssur
e (m
)
Max.
Min.
(a) PRV size (d) (TV1= 5s, TV2=80s, SET2=0)
Max./ Min. Waterhammer Pressure vs . PRV Low Pressure Set Point
-40
-30
-20
-10
0
10
20
30
40
0 5 10 15 20 25 30
PRV Low Pressure Set Point (m)
Max
. and
Min
. Pre
ssur
e (m
)
Max.
Min.
(b) PRV pressure set point (SET2) (TV1=5s, TV2=80s, d=0.9m)
Figure 2.5 Sensitivity analyses of upstream PRV parameters (Con’d)
29
Max. /Min. Waterhammer Pressure vs. PRV Opening Time Period
-30
-20
-10
0
10
20
30
0 5 10 15 20 25
PRV Opening Time Period (s)
Max
./ M
in. P
ress
ure
(m)
Max.
Min.
(c) PRV opening time period (TV1) (TV2=80s, d=0.9m, SET2=0)
Max./ Min Waterhammer Pressure vs. PRV Closure Time Period
-20
-10
0
10
20
30
40
50
0 20 40 60 80 100 120 140
PRV Closure Time Period (s)
Max
. and
Min
. Pre
ssur
e (m
)
Max.
Min.
(d) PRV closure time period (TV2) (TV1=5s, d=0.9m. SET2=0)
Figure 2.5 Sensitivity analyses of upstream PRV parameters
30
Max. H
Steady State H
Pipeline
Min. H
(a) Simulation without PRV protection
Max. H
Min. H
(b) Simulation with an appropriately designed PRV
(d = 0.5 m, TV1 = 10 s, TV2 = 70 s)
Figure 2.6 System transient performance varying with downstream PRV design (Con’d)
31
Max. H
Steady State H
Pipeline
Min. H
(c) Simulation with an under-sized PRV
(d = 0.1 m, TV1 = 10 s, TV2 = 70 s)
Max. H
Steady State H
Min. H
(d) Simulation with a too-high-pressure set point PRV
(d = 0.5 m, SET1 = 50 m, TV1 = 10 s, TV2 = 70 s)
Figure 2.6 System transient performance varying with downstream PRV design (Con’d)
32
Max. H
Min. H
(e) Simulation with a too-slow-opening PRV
(d = 0.5 m, TV1 = 60 s, TV2 = 70 s)
Max. H
Steady State H
Pipeline
Min. H
(f) Simulation with a too-fast-closure PRV
(d = 0.5 m, TV1 = 10 s, TV2 = 40 s)
Figure 2.6 System transient performance varying with downstream PRV design
(Note: Default SET1 is equivalent to SET1<=15m, i.e., the PRV is activated instantly when Main-Valve-2 closes)
33
Mi H
Max. / Min. Waterhammer Pressure vs. PRV Diameter
-100-60-202060
100140180
0 0.2 0.4 0.6 0.8
PRV Diameter (m)
Max
. and
Min
. Pre
ssur
e (m
)
Max.
Min.
(a) PRV size (d) (TV1=10s, TV2=70s. SET1=0)
Max./ Min. Waterhammer Pressure vs . PRV High Pressure Set Point
-20
0
20
40
60
80
100
0 10 20 30 40 50 60
PRV High Pressure Set Point (m)
Max
. and
Min
. Pre
ssur
e (m
)
Max.
Min.
(b) PRV pressure-set point (SET1) (TV1=10s, TV2=70s, d=0.5m)
Figure 2.7 Sensitivity analysis of downstream PRV parameters (Con’d)
34
Max. /Min. Waterhammer Pressure vs . PRV Opening Time Period
0
10
20
30
40
50
60
70
80
0 10 20 30 40 50 60
PRV Opening Time Period (s)
Max./
Min.
Pres
sure
(m)
Max.
Min.
(c) PRV opening time period (TV1) (TV2=70s. d=0.5m, SET1=0)
Max./ Min Waterhammer Pressure vs. PRV Closure Time Period
-80
-60
-40
-20
0
20
40
60
80
100
120
0 20 40 60 80 100
PRV Closure Time Period (s)
Max.
and M
in. Pr
essu
re (m
)
Max.
Min.
(d) PRV closure time period (TV2) (TV1=10s. d=0.5m, SET1=0)
Figure 2.7 Sensitivity analysis of downstream PRV parameters
35
Chapter 3 Transient Simulation of Active Hydraulic
Devices with an Introduction to Automatic Control
This chapter provides an introduction to the numerical modeling of AHDs. Since
automatic control is a defining element of AHDs, the key feature is a general
mathematical model for the transient analysis of AHDs with a PID (proportional, integral
and derivative) type controller. Familiarity with the material contained in this chapter is
important for understanding the overall thesis and is based on a literature review of
automatic control and hydraulic devices. Specifically, the Method of Characteristics
(MOC) and its implementation to pipe networks is first reviewed; this is followed by a
brief introduction of control theory as well as the evolution and variety of controllers.
Finally, a set of basic governing equations is presented for the transient simulation of a
generalized AHD in a pipe network.
3.1 Numerical Method of Pipe Transient Flow
The momentum and continuity equations governing transient flow in closed conduits are
classified as quasi-linear hyperbolic partial differential equations for which no analytical
solutions are available for a general pipe system. With respect to pipe systems and the
resolution of the governing equations, the superiority of the MOC over finite element or
finite difference methods in terms of both simulation accuracy and computational effort
has been demonstrated (e.g., Karney & McInnis, 1992).
36
3.1.1 Governing Equations and Method of Characteristics (MOC)
The momentum equation (3-1) and continuity equation (3-2) can be derived for unsteady
closed conduit flow based on certain reasonable assumptions and approximations (Wylie,
etc., 1993 and Chaudhry, 1987).
Δx Δx A C Bi-1 i i+1
PC+ C-
j+1 △t
j
j-1 x
t
Figure 3.1 MOC grid for single pipe
2)-(3 0
1)-(3 0||2
2
2
1
=+=
=++=
xt
tx
VgaHL
VVDfVgHL
As shown in Figure 3.1, x is the distance along the centerline of the conduit,
and t is the time. In the equations, H (x, t) is the piezometric head (HGL), V (x, t) is the
fluid velocity whose positive direction is defined as the same as the distance x. Both H
and V are dependent on x and t, and the subscripts represent partial derivatives. The other
constants include f = Darcy-Weisbach friction factor, a = wave speed and g = acceleration
due to gravity.
37
Introducing an unknown multiplier λ , equations (3-1) and (3-2) can be
combined into:
3)-(3 02
|| 2
21 =+⎥⎦
⎤⎢⎣
⎡++⎥⎦
⎤⎢⎣⎡ +=+=
DVfVV
gaVHgHLLL txtx
λλ
λλ
Comparing the terms in the brackets of equation (3-3) with the full derivatives of H and V:
, and txtx VdtdxV
dtdVH
dtdxH
dtdH
+=+= suggests letting 2
dtdx
gag
==λ
λ.
This substitution implies that , adtdx
ag
±=±=λ and that equation (3-3) can be
expressed as a system of two pairs of ordinary differential equations, that is, the
following two sets of compatibility equations (3-4) to (3-7). They are also referred to as
the characteristics equations because equation (3-4) is only valid along C+ characteristic
line (3-5) for the arriving pressure wave while equation (3-6) is only valid along C-
characteristic line (3-7) for the leaving wave:
⎪⎪⎩
⎪⎪⎨
⎧
−=
=++−
⎪⎪⎩
⎪⎪⎨
⎧
+=
=++
−
−
+
+
7)-(3 line sticcharacteri C
6)-(3 equation ity compatibil 02
||
5)-(3 line sticcharacteri C
4)-(3 equation ity compatibil 02
||
adtdx
DVfV
dtdV
dtdH
ag
C
adtdx
DVfV
dtdV
dtdH
ag
C
In Figure 3.1, to solve the unknown point P from the known points A and B,
the compatibility equations can be integrated along AP (C+) and BP (C-), respectively,
38
using the pipe discharge Q and cross-sectional area A instead of velocity V. Here, a linear
approximation is used for the friction term; that is,
⎪⎪⎭
⎪⎪⎬
⎫
=Δ
−−−−
=Δ
+−+−
−−−+
−−−−
++
−−
0||2
)(
0||2
)(
12
111
12
111
11
11
jjjjji
j
jjjjji
j
iiiii
iiiii
QQgDA
xfQQgAaHH
QQgDA
xfQQgAaHH
These equations are only valid if tax Δ⋅=Δ , which has to be met for the pipe
discretization and time step choice. Introducing two pipe constants gAaB = and
22gDAxfR Δ
= , the above integrating equations can be written as:
⎪⎭
⎪⎬⎫
+−+=
−−−=−−−
−−−
+++
−−−
||)(
||)(
111
111
111
111
jjjjjj
jjjjjj
iiiiii
iiiiii
QRQQQBHH
QRQQQBHH
These are further re-written as:
|
|
in which
)9-3(
8)-(3
11
11
11
11
11
11
⎪⎩
⎪⎨⎧
+=
+=
⎪⎩
⎪⎨⎧
+=
+=
+=
−=
−+
−
−+
−+
−−
−
+
jiM
ji-P
ji
jiM
ji-
ji-P
jjMM
ji
jiPP
ji
R|Q B B
R|Q B B
BQHC
BQ H C
QBC: HC
QBC: HC
Here, the subscript ‘P’ represents positive ‘+’ sign; and ‘M’ represents minus ‘-’ sign in
the characteristic equations. Therefore, starting from the initial flow state (t = 0),
equations (3-8) and (3-9) can be solved, so the transient pressure head H and flow Q at
39
any internal pipe node along the distance x (i =1, 2, 3, ….) and at any time step tjt Δ⋅=
(j =1, 2, 3,….) can be obtained step by step:
11)-(3
10)-(3
11
1
MP
MP
M
Mi
P
j
iPj
i
MP
PMMPj
i
BBCC
BCH
BHCQ
BBBCBCH
+−
=−
=−
=
++
=
++
+
Yet, only one of characteristic equations [either (3-8) or (3-9)] is available for the node at
a pipe’s upstream and downstream ends, and one boundary condition at each end has to
be provided to define either the flow Q (t, x=0 or L), or the pressure H (t, x=0 or L), or the
relation between H and Q. Thus, one characteristic equation and one or more boundary
conditions are always combined to determine the transient conditions at the pipe end.
3.1.2 Extension of MOC in Pipe Network
The MOC has been extended to network applications and the resultant solution is
algebraically simple and computationally flexible (Karney and McInnis, 1992). The
extended MOC is implemented in this research for the numerical development of
different AHDs.
In a pipe network, a junction (i.e., several pipe ends meeting together as
depicted in Figure 3.2) is the most common boundary condition and can usually be
assumed frictionless. Let the set N1 be the pipes whose flow direction is assumed toward
the node and N2 the set of pipes whose flow direction is assumed away from the node.
Qext is external flow governed by some boundary condition, with a positive magnitude
referring to flow away from the junction. If there is no external flow at the junction, Qext
40
=0. HP is assumed as the common piezometric head at a junction and the flow in
each pipe. Here the second subscript k indicates a particular pipe in the set. For any pipe
whose flow direction is assumed toward the junction, we use the C
KPQ
+ compatibility
equation (3-8), which gives:
12)-(3 1 N KBH
BC
QKK
K
K
P
P
P
P
P ∈−=
For any pipe whose flow direction is assumed away from the junction, we use C-
compatibility equation (3-9), which gives:
13)-(3 2 N KBH
BC
QKK
K
K
M
P
M
M
P ∈+−=
The continuity equation requires that the total flow is zero at the junction; that is:
14)-(3 021
11=−−∑∑
==ext
N
KP
N
kP QQQ
KK
Substituting equations (3-12) and (3-13) into equation (3-14), the following extended
MOC equation can be obtained:
15)-(3 extP QBCH ′−′=
in which
16)-(3 ; 11 1 21 2 Nk
1
N ⎟⎟⎠
⎞⎜⎜⎝
⎛+′=′⎟
⎟⎠
⎞⎜⎜⎝
⎛+=′ ∑ ∑∑ ∑
∈ ∈
−
∈ ∈ NkM
M
P
P
k NkMP k
k
k
k
kkBC
BC
BCBB
B
41
Qext
Figure 3.2 Generalized node with an external flow
It is interesting that the equation (3-15) at a junction has the same form as the
C+ compatibility equation for the downstream end of a simple pipe. In this way, any
junction within a network (with or without external flow) can be treated as equivalent to a
downstream node. Note that if only one pipe is connected to a junction (the so-called one-
degree-node) we can deduce MPMP CCCBBB or and ,or =′=′ from equation (3-16).
This indicates that the extended MOC equation is a generalized solution for any junctions
in the system, implying a higher efficiency in numerical modeling. It should be noted
that the junction without any connecting pipe (the so-called zero-degree-node) is not
allowed in the extended MOC model because 0==jj MP BB would occur in equation (3-
16). Meanwhile, a junction only permits attachment of one “device”. When more than
one device is associated with a junction, the extra devices have to be treated either by
combining them as one “device” or setting at least one characteristic section of the
connection pipe between them when the system is discretized and modeled.
42
3.2 Controllers and Active Hydraulic Devices
Active hydraulic devices are intrinsically associated with control theory and automatic
controllers. In mathematics and engineering, control theory deals with the behavior of
dynamical systems. The desired output of a system is called its reference. When one or
more output variables of a system need to follow a certain reference over time, a
controller attempts to manipulate the input to the system to obtain the desired output
effect.
3.2.1 Open-loop and Closed-loop control
There are two basic types of control, open-loop control and closed-loop control. For
example, in an automobile's cruise control system, the output variable is vehicle speed
and the input variable is the engine's throttle position which influences engine torque
output. A simple way to implement cruise control while driving is to lock the throttle
position. However, on hilly terrain, the vehicle will slow down going uphill and
accelerate going downhill. This type of control is called open-loop control because there
is no direct connection between the output of the system and its input. A drawback of
open-loop control is that it requires perfect system knowledge (i.e., it needs to know
exactly what inputs will produce the desired output). Open-loop control can be used in a
well-characterized system so that one can adequately predict which inputs will
necessarily achieve the desired output. It is usually a simple system, and the mathmtaical
relationship between inputs and outputs is clear and an analytical solution is possible.
For example, the rotational velocity of an electric motor may be sufficiently characterized
for the supplied voltage to make feedback unnecessary.
43
To avoid the shortcomings of open-loop control, feedback is introduced into a
controller to regulate the operational states of a dynamic system. The process output is
measured with sensors and acted upon by the controller; the result (i.e., control signal) is
input to the process, closing the loop. This control type is known as closed-loop control,
which offers certain key advantages over its open-loop counterpart, such as disturbance
rejection, a stabilized pocesses, reduced sensitivity to parameter variations, improved
reference tracking performance, and guaranteed performance even with model
uncertainties when the model structure does not match perfectly the real process and the
model parameters are at least partly inaccurate. For instance, using the closed-loop
control in the automobile's cruise system, the engine's throttle position can be adjusted to
reach the desired driving speeds no matter what the road condition is (Tan et al., 1999;
http://en.wikipedia.org/wiki/Controller_(control_theory)).
3.2.2 Feedback and Feedforward Control
Closed-loop control is also called feedback control since, in this form of system
regulation, the sensed process outputs are used as “feedback” to control subsequent
outputs or states of a dynamic system. The household thermostat is an example of
feedback control. The feedback controller relies on measuring the governing variable, in
this case the temperature of the house, and then adjusting the output, whether the furnace
or heater is on or off. However, feedback control usually results in intermediate periods
where the controlled variable is not at the desired set-point. With the thermostat example,
the furnace or heater would switch on only after the house temperature falls below such a
point.
44
In some systems, closed-loop and open-loop control are used simultaneously.
In such cases, the open-loop control is termed feedforward and serves to further improve
reference tracking performance. Feedforward control can avoid the slowness of feedback
control because, with feedforward control, the disturbances are measured and accounted
for before they have time to affect the system. In the thermostat example, a feedforward
system may measure the fact that a door is opened and automatically turn on the furnace
before the house can get too cold. The difficulty with feedforward control is that the
effect of the disturbances on the system must be accurately predicted and there should be
few surprise disturbances. For instance, if an unmonitored window were opened, the
feedforward-controlled thermostat might still let the house cool down.
To achieve the benefits of feedback control (controlling unknown disturbances
and not having to know exactly how a system will respond to disturbances) and the
benefits of feedforward control (responding to disturbances before they can affect the
system), there are combinations of feedback and feedforward that can be used. Dead-
time compensation and inverse response compensation are examples where feedback and
feedforward control are used together. Dead-time compensation is used to control devices
that take a long time before exhibiting any adjustment to an input change and uses an
element to determine how an instant change in the controller’s input will affect the
controlled variable (output) in the future. The controlled variable is also measured and
used in feedback control. Inverse response compensation involves controlling systems
where a change at first affects the measured variable one way, but then later affects it in
the opposite way; for example, when one eats candy, at first it provides an energy boost,
but subsequently, one may feel tired or even lethargic. In this way, it can be difficult to
45
control this type of system with feedback alone. Therefore, a predictive feedforward
element is necessary to inform of the reverse effect that a change in state might entail in
the future (Tan et al., 1999; http://en.wikipedia.org/wiki/Controller_(control_theory)).
3.2.3 Evolution of Controllers
Controller design evolves with the development of associated science and technology.
Most control systems in the past were implemented using mechanical systems or solid
state electronics. Pneumatics were often utilized to transmit information and control using
pressure. Nowadays, most systems and industries rely on computers for controller design,
which obviously renders implementation of complex control algorithms easier than with
a mechanical system.
On/Off Control. On/Off control is the simplest and most intuitively akin to
direct manual control, like the thermostat that turns the furnace on or off depending on
the ambient temperature’s departure from the set-point. It is undoubtedly the most widely
used control type for both industrial and domestic service. It has an output signal u,
which may be changed to either a maximum or a minimum value, depending on whether
the process variable is greater or less than the set-point. As shown in Figure 3.3, the
control law is described by (Tan, et al., 1999):
,0 , ,0 ,
min
max
<=>=
euueuu
where e is the control error, yre −= ; y is the controlled process variable and r is its set-
point (reference). The minimum value of the control output is usually zero (off), the
46
maximum value of the control output could be either a positive or negative constant. The
mechanism to generate On/Off control is usually a simple relay.
One well acknowledged and significant disadvantage of On/Off control is that
it oscillates around the constant set-point. This will directly affect the process variable
which also oscillates around the desired value. If, for example, On/Off control is applied
to regulating a tank’s water level with the aid of a valve which can only open or shut
completely, the On/Off controller therefore naturally opens and shuts the valve
alternatively. This can cause rapid wear and tear on the moving parts in the actuating
device and the “ringing” or oscillation phenomenon may be intolerable in certain cases.
The solution in most On/Off controllers is to establish a dead zone, or hysteresis, of about
0.5% to 2% of the full range. This dead zone straddles the set-point so that no control
action takes place when the process variable lies within the dead zone.
e
umax
umin
u
Figure 3.3 ON/OFF control
Propotional/Correspondance/Modulating Control. Except for the use of a
small dead zone to mitigate signal oscillations associated with On/Off control, one
alternative is to use a small gain for the controller when the error e is modest and use a
47
large gain when the error is large. This can be achieved with a proportional or P-
controller, a basic continuous control mode. The control rule in a P-controller is given by
(Tan, et al., 1999):
, , , ,
, ,
0min
000
0max
eeueeeeKu
eeuu
c
<=<<−+=
>=
(3-17)
where u0 is the level of control signal when there is no control error, and Kc is the
proportional gain (or sensitivity) of the controller. In fact, Kc indicates the changes in
control signal per unit change in the error signal; it is indeed an amplification and may be
adjusted by the operator. The proportional gain Kc will be positive if an increase in the
input variable requires an increase in the output variable (direct-acting control), and it
will be negative if an increase in the input variable requires a decrease in the output
variable (reverse-acting control). A typical example of a direct-acting system is
controlling the flow of cooling water. When the temperature increases, the flow must be
increased to maintain the desired temperature. Conversely, a typical example of a
reverse-acting system is controlling the flow of steam for heating. If the temperature
increases, the flow must be decreased to maintain the desired temperature. The P-
controller can also be described graphically as in Figure 3.4.
While the signal oscillations with On/Off control may be quenched with
proportional control, a new problem arises. For most systems, the P-controller will never
entirely remove errors at steady state. In other words, after the transients have died down,
there may remain a deviation between the set-point and the process variable. Since the
48
control error is given by cKuu
e 0−= , at least one of the following conditions must be true
to meet the zero control error at the steady state (Tan, et al., 1999):
1) Kc is indefintely large, which effectively reverts P-control back to On/off control,
bringing with it the issue of instablity or requiring physical impossibilities like
infinitely large valves;
2) The steady state control signal, uss, equals, the level of the control signal when there is
no control error, u0, that is, uss = u0. If so, when the error is zero, the controller only
provides the steady state control action so the system will settle back to the original
state which is probably not the new set-point desired for the system. In other words,
uss = u0 cannot be generally satisfied by all set-points. Even if u0 can be adjusted
relative to the set-point, it is still necessary to know at least the process static gain
before the adjustment can be done.
e
umax
umin
u
u0
Proportional Band
-e0 e00
Figure 3.4 Proportional control
49
PI control. To eliminate the remaining steady state error with pure P-control,
the integral or reset action can be introduced. The I-part control is able to find the correct
value for u0 automatically in response to any set-point without needing to know the
process static gain. Integral control action usually is combined with proportional control
action, although it is possible but less common than using integral action by itself. The
combination (PI-control) is favorable because of the advantages of both control actions.
The control signal in a PI-controller is given by (Tan, et al., 1999):
⎟⎟⎠
⎞⎜⎜⎝
⎛+= ∫ eedt
TKu
ic
1 (3-18)
where Ti is the integral time of the controller. The constant level of u0 found in the P-
controller has thus been replaced by the integral:
∫= edtTK
ui
c0 (3-19)
The integral of the control error is effectively proportional to the area under the curve
between the process variable and the set-point (Figure 3.5).
To better understand the capacity of a PI-controller for the elimination of
steady state error, it is assumed that the closed-loop system is stable and that a steady
state control error existed despite having a PI-controller, which implies . At steady
state, the integrator will continuously accumulate the error signal at the input and thus the
control signal u will be either rising or falling, depending on whether the error is positive
or negative. The P-part has a constant value corresponding to K
0≠sse
cess and thus will not
50
affect the analysis. If the control signal rises or falls, the process variable will also rise or
fall. This in turn means that the error e = r - y cannot be constant at steady state and thus
contradicts the assumption that the error is stationary. Thus, it is impossible to have a
non-zero steady state error when the controller has an intergral part and the closed-loop is
stable.
Figure 3.5 Integral control derived from the area under the control error curve
A PI-controller thus resolves the problem of remaining stationary error and the
problem of oscillation associated with On/Off control. A PI-controller is therefore an
effcient controller without any significant faults and is often sufficient when the control
requirements are low to modest.
PID control. Both the P- and the I-componets of a PI-controller operate on
past control errors, and do not attempt to predict future control errors. This characteristic
limits the achieveable performance of the PI-controller. The problem is more clearly
illustrated in Figure 3.6. The two curves in the figure show the time graph of the control
51
error for two different processes. At time t, the P-part, being proportional to the control
error, is the same for both cases. Assuming that the I-parts, being proportinal to the area
under the two control error curves, are also equal in the two cases. This means that a PI-
controller gives exactly the same control signal at time t for the two processes. However,
it is clear that there is a great difference between the two cases if the rate of change in the
control error is considered. For Repsone I in Figure 3.6, the control error changes rapidly
and the controller should reduce its output to avoid an overshoot in the process variable
occuring in the future. For Response II, the control error changes more sluggishly and the
controller should react strongly in order ro reduce the error more rapidly. Derivative or
rate control indeed carries out this type of compensation.
Figure 3.6 Illustration of the need for derivative control
It is not even theoretically possible to have a control action based solely on the
rate of error signal change (de/dt), since the derivative controller output would be zero if
the error is large but unchangaing. Thus, derivative control is usually combined with at
least a proportional control. The D-part of PID controller is proportional to the predicted
t t
e
I
e
IIP P
52
error at time t+Td, where Td is the derivative time of the controller. The control law for
the PID controller is described as (Tan, et al., 1999):
⎟⎟⎠
⎞⎜⎜⎝
⎛++= ∫ dt
deTedtT
eKu di
c1
(3-20)
Since set-point r is normally constant or has rather abrupt changes (to reflect
changes in circumatnces), it will thus normally not contribute to the derivative term. For
this reason, it is common parctice to apply the derivative action only to the process output
y, except when the process variable is required to track a continually changing set-point.
3.2.4 Elements in a Control System
As shown in Figure 3.7, a control loop usually consists of the following hardware
elements:
1) Sensors that monitor the ongoing process condition;
2) Data transmission system (wireless, wire, or cable transmitters) that carries the
measured data from sensors to controller and signals from controller to the final
control element;
3) Controller that compares the “process variable” received from the transmitter with the
set-point (i.e., the desired process), and then makes decisions and sends corrective
signals to the final control element;
4) Actuator is a pneumatic, hydraulic or electrically powered device that supplies force to
open or close the valve, or moves the other final control elements.
53
5) Final control element that manipulates the process. Control valves are the most
commonly used final control elements; others include wicket-gates in turbine units;
MGV (misaligned guide vanes) device at high-head pump-turbines, VFD (variable
frequency drive) on pumps; dampers; and devices that regulate (throttle) electric
energy such as silicon-controlled rectifiers.
Sensor
Control Valve
Control signal for valve operation
Pipeline
PID Controller
Figure 3.7 Loop of PID-control-valve system
AHDs are typically combined with automatic controllers, and different types of
AHDs use different controllers and actuators. For instance, an On/Off controller or P-
controller is used in pressure relief valves (PRV), check valves, and air valves; and a PID
controller is used in some regulating control valves, speed-governed hydro-turbines, and
variable speed pumps.
54
3.3 Generalized Mathematical Model for AHD Transient Simulations
A general mathematical model for a PID controlled AHD is presented next; specific
features, design functions and case studies for individual devices are discussed in the
following chapters.
3.3.1 Extended MOC Equations
Figure 3.8 shows an AHD in a pipe network. Using the extended MOC equation (3-15) to
relate the nodal heads to the flow rate across the device, the compatibility equations at
AHD’s upstream and downstream node, respectively, are written as:
AHDQ Node2 Node1 Pipe
Pipe
Q Pipe
Pipe
H1
H2
H
Figure 3.8 Active Hydraulic Device in a pipe network
55
22)-(3 21)-(3
222
111
QBCHQBCH′+′=
′−′=
in which , , and are calculated using equation (3-16). By combining the two
extended MOC equations, a head-flow relationship at the AHD is obtained:
1'B 2'B 1C ′ 2C′
23)-(3 )()( '2
'1
'2
'121 QBBCCHHH +−+=−=
in which, H = the head difference across the AHD. This equation links the device to the
reminder of whole system, since it will be resolved by a moment-by-moment simulation
in the connected pipe system starting from the initial steady state.
3.3.2 AHD’s Dynamic Characteristics
As mentioned in Chapter 1, a control system is essential to make a hydraulic device
“active”. The head-discharge relationship of an AHD can usually be dynamically
modified via its control system to change the opening of its control valve or wicket-gates.
The generalized dynamic characteristics of an active hydraulic device is expressed as:
24)-(3 0)... ,,,,( =dtdyyHQF
in which y = the opening of control valve or turbine wicket-gates and Q = the flow rate
passing through the AHD. For a control valve, equation (3-24) is the valve discharge
equation, representing the relationship between the valve resistance and valve opening;
more specifically, that is equation (4-3) in Chapter 4. For a hydro-turbine, equation (3-24)
represents a set of characteristic curves (reference to different turbine characteristics
shown in Section 5.3 Chapter 5).
56
3.3.3 PID Controller Equation
Instead of a predefined opening or closing motion for most conventional valves with
On/Off controllers, the opening of a PID control valve is automatically regulated in
response to the sensed pressure or pressure difference, which is desired to control.
Consequently, y (t) in equation (3-24) is unknown in a PID control valve and needs to be
dynamically evaluated by the characteristics of the PID controller. While for a speed-
governed hydro-turbine, the wicket-gate opening is adjusted instantaneously by tracking
the rotational speed of the turbine unit in response to the variation of power load on the
unit.
Depending on the actuator’s power source (pneumatic, electric or hydraulic)
and the internal design of a valve and controller, the equation of the PID controller could
be established corresponding to each process variable for which control is sought.
Usually, we need to establish a relationship between the PID control signal (amplified
control error) and the actuator action, which is expressed by:
25)-(3 ,...),(
)1()(0
dtdyyF
dtdeTedt
TeKtu d
t
iC
=
++−= ∫
Here, KC, Ti and Td are three parameters of PID-controller, representing the
characteristics of a controller. The function ,...),(dtdyyF represents the action of the
actuator. More specifically, equations (4-5) and (4-6) are two different actions taken by
the PID control valve for different control variables. And the turbine governor equation
57
(6-19) is another example of PID controller equation, which is to adjust wicket gate
opening for the speed control of turbine unit.
In sum, the mathematical model of a PID controlled AHD basically consists of
three governing equations, as presented above. However, it is possible to have additional
governing equations for some AHDs. These additional equations might include a torque
balance equation for a pump or hydro turbine, reflecting the variable power load or the
speed change of the turbine unit, e.g., equation (6-17) in Chapter 6
3.4 Summary
In this chapter, the Method of Characteristics (MOC) and its implementation in pipe
networks is first reviewed and then a brief introduction of control theory, as well as the
evolution and variety of controllers, is provided. Finally, a mathematical model for the
transient analysis of a general AHD, combined transient response with control theory, is
presented. AHD models must capture three specific relationships:
1. the characteristics of the hydraulic component, that is, the heart of the component,
whether valve, turbo machine or storage element;
2. the control behavior, that is, what action the device takes as a result of the hydraulic
event it creates or experiences; and
3. the wave behavior of the reminder of the system, captured in this case by the Method
of Characteristics (MOC), that is a moment-by-moment simulation of arriving and
leaving waves.
58
With the fundamental understanding of AHDs, the next chapter develops a new
active device to create a “non-reflective” boundary condition in a pipeline through a PID
control valve combined with a remote pressure sensor.
59
Chapter 4 “Non-Reflective” Boundary Design via
Remote Sensing and PID Control Valve
This chapter represents one of the core contributions of thesis research. Using the
concept of AHD, a new strategy is created to actively control the reflection and
resonance of pressure waves in a pipeline system, using a so-called “non-reflective”
boundary. The system performance is evaluated and compared for both scenarios with
and without application of this strategy, and the role of “non-reflective” or “semi-
reflective” boundary design is demonstrated.
The core results of this chapter have been published and presented at
Pressure Surges 10th International Conference (BHR Group, Edinburgh, Scotland, May
14-16, 2008), co-authored with Bryan Karney, Feng Wang and Stanislav Pejovic. As the
second author, I wrote the full early draft of the published paper, developed the
numerical model and did all the numerical runs and analysis. The initial idea of this
work originated with Prof. Karney and Stanislav Pejovic; Feng Wang helped to
incorporate the model into the program TransAM and also was involved in discussion of
the details of model “conception” during the time when he visited the University of
Toronto in 2006.
Abstract: This chapter explores a new approach to limit the surge pressures by
creating so-called “non-reflective” (or “semi-reflective”) boundary in a pipe system. This
60
boundary condition is established using the combination of a remote sensor and a control
system to operate a relief valve. In essence, the idea is to sense the pressure change at a
remote location (say, at a tee to a dead-end link) and then to use the measured data to
adjust the opening of an active control valve at the end of the line so as to eliminate or
attenuate the wave reflections at the valve and thus to control system transient pressures.
This novel idea is explored here initially by the means of numerical simulation and
shows considerable potential for transient protection, as demonstrated in case studies.
Using this model, wave reflections and resonance can be effectively eliminated for
frictionless pipelines or initially no-flow conditions and better controlled in more
realistic pipelines for a range of transient disturbances. In addition, the feature of even
order of harmonics as well as “non-reflective” boundary conditions during steady
oscillation, obtained through time domain transient analysis, are verified by hydraulic
impedance analysis in the frequency domain.
4.1 Introduction to “Non-Reflective” Boundaries
One of the most interesting aspects of transient events is that the phenomenon is
primarily created and controlled by the action of boundary devices in the system. The
actions of, or changes in, these boundaries both initiate the transient event and control its
severity, creating at operating devices a sequence of velocity and pressure fluctuations
that then propagate through the pipe system. The action or response of other devices and
components, either by design or by accident, then reflects, refracts, amplifies or
attenuates the primary pressure and velocity waves.
61
There is an intriguing connection between water hammer control and the use
of “tailored” or active boundary conditions that control wave reflections. Certainly such
“non-reflective” boundary conditions have been partially explored before and used in the
past to represent certain network junctions and components (Almeida and Koelle, 1992;
Pejović-Milić A., Pejović S, and Karney B., 2003). Indeed in a conventional sense,
various “semi-reflective” boundaries are the basis of water hammer protection, and their
role as general energy dissipation device in systems with dead-ends is extended and
explained in the following sections; however, as some devices can be sources of
resonance, special consideration must be given to the frequency domain. Moreover, the
physical, mathematical and numerical aspects of such boundaries are considered and
developed along with possible applications that move these boundary conditions toward
use and application in systems and devices.
With the development of electronic and computer technologies, dynamic PID
(proportional, integral and derivative) controllers are now being used more frequently to
maintain a desired performance in a water system; the variable to be controlled may be
turbine speed, turbine power output, pump speed, pump torque, pump discharge flow,
water level in tanks, upstream or downstream pressure at valve, and so on. This
developed control capability also provides an alternative arrangement for suppression of
hydraulic transients in pipe systems. In fact, local/conventional PID control valves, even
when modulating, have implications for transient control by maintaining a desired
pressure or flow in the system. However, the combination of remote sensing and PID
control valve could provide a broad range of waterhammer protection through designing
of “non-reflective” boundaries into a pipe system. Yet, the relevant literature to date has
62
mainly focused on the responses of transient flow to the action of PID control valves by
coupling hydraulic transient analysis and control theory (Koelle 1992, Koelle & Poll
1992, Lauria & Koelle 1996, Bounce & Morelli 1999, and Poll 2002), it is an innovation
to study how to actively control a valve in response to the remotely sensed pressure for
system waterhammer protection.
The goal of this chapter is to explore how “non-reflective” boundaries can be
designed and applied for transient protection, particularly at problematic locations like
dead-ends or cul-de-sacs in a distribution system. Though having the same role to limit
the transient pressure in pipe systems, this approach is different from the operation of a
pressure relief valve (PRV) that opens a by-pass line to release excessive flow when the
pressure in pipeline exceeds the set point. Such as an arrangement is also quite distinct
from a local/conventional pressure sustaining valve (PSV) or backpressure valve
(holding the pressure at valve inlet or outlet) that modulates the valve opening to
maintain the set point corresponding to the locally sensed pressure (Hopkins, 1998). The
key issue in the remote control is that the transformation of transient pressure waves
between the remote sensor and active control valve.
In this chapter, an example is first presented to demonstrate the possibility of
dangerous waterhammer occurring in the system due to reflections at dead-ends. The role
of a control valve to dissipate the transient energy and thus protect the system from
excessive transient pressures is illustrated. Then, the mathematical model for the
local/conventional PID control valves is addressed as a prerequisite to the solution of
remote sensing and “non-reflective” valve opening, and the key novel features in the
remote control model are discussed. After that, case studies involving a successful
63
numerical application of the remote control model are presented. Theses case studies
show the ability of “non-reflective” boundary to control the reflection of pressure wave
and potential resonance within the pipeline. Moreover, the developed “non-reflective”
boundary condition during the steady oscillation is verified using hydraulic impedance
analysis in frequency domain. Finally, using the developed simulation tool, the selection
and tuning of PID controller parameters are discussed based on sensitivity analyses.
4.2 Transient Performances with Dead-ends and Valve Control
A dead-end is often a tricky arrangement in a pipe system. Such an arrangement includes
a closed valve, which carries no discharge and can cause unexpected high pressure in the
system. When a pressure wave is transmitted into the dead-end pipe, the flow or velocity
is stopped rapidly at the end and the wave transmission also terminates there, which
induces a doubled value in local pressure and creates an increased pressure wave that
returns into the system. This is the so-called dead-end reflection. By contrast, a pressure
wave would be fully reflected with reverse sign from a constant head reservoir. In other
words, neither a reservoir nor a dead-end is intrinsically dissipative; they both reflect
waves but conserve energy. However, a partially open valve at a reservoir dissipates
energy and acts somewhat between a dead-end and a reservoir. If the size of the valve
opening is systematically adjusted, a value can be found for a given system so that the
disturbance/excitation does not reflect at all. This setting thus produces a “non-
reflective” boundary with a maximum rate of transient/oscillation energy dissipation.
64
Consider a pipeline system shown in Figure 4.1. Initially, the terminal control
valves at the end of branches are fully closed. If the pipes do not leak and have been
open to the reservoirs for some time, no flow will occur in the system and the head will
be uniformly 100 m as found in the reservoirs. However, following Boulos et al.
(Boulos, Lansey and Karney 2006), suppose the second section of the main pipeline
(from B to C) is pressurized to a uniform value of 130 m. If this initial condition is
released, two pressure waves, both 15 m in amplitude, are created and propagate into the
system. The response to the traveling pulse waves are simulated and shown in Figure 4.2,
representing the envelope of maximum and minimum transient pressures along the pipe
length from A to C and then from C to F. More specifically, in Figure 4.2 (a), the
terminal valves keep fully closed, and we see that the wave is reflected and magnified by
the dead-end due to the overlapping of incident and reflected pressure waves. In Figure
4.2 (b), the terminal valves now are opened to 10% of the full size in 1 second when the
pulse waves start to travel, which reduces the maximum pressure but causes the
unacceptable negative pressure. Figure 4.2 (c) represents the condition that the terminal
valves are opened to 0.35% of full size in 1 second, and in this case the wave energy is
largely dissipated when it arrives at this small orifice. These differences in system
response can be exploited. Indeed, even when the valve opening is chosen somewhat
arbitrarily, the transient pressures are likely to be at least partly mitigated. So, what if the
valve opening is systematically refined to eliminate the wave reflection?
65
100 m 100 m
80 m
130 m
A B C D E
F G
Figure 4.1 Scheme of a branched system and initial pressure head
(Note: Pipe length AB = BC = CD = DE = CF = DG = 1000 m, friction factor f = 0.012, and the
diameter of main pipeline D = 1 m, and the diameter of branches d = 0.5 m.)
Max. H
Min. H
Initial H
Pipeline
A B C F
(a) Transient response in the system with dead-end branches
66
Max. H
Min. H
Initial HPipeline
A B C F
(b) Transient response in the system with 10% orifices at branch ends
Max. H
Min. H
Initial H
Pipeline
A B C
(c) Transient response in the system with 0.35% orifices at branch ends
Figure 4.2 Comparison of transient responses to dead-ends and small orifices
67
(a) Sketch of system configuration
(b) Initial pressure head along the pipeline
Valve Opening=10% Valve Opening=20%
Valve Opening=30% Valve Opening=60%
Valve Opening=0%
Valve Opening=100%
(c) Transient pressure head along the pipeline after reflections from both ends
Figure 4.3 Traveling pulse waves and “tailored” valve reflections
68
Further insight is obtained by comparing the responses for different sizes of
valve opening in a single pipeline system, sketched in Figure 4.3 (a). In this system, a
uniform pipeline links two reservoirs with the same constant head; and a control valve is
installed at the right hand reservoir. A traveling pulse wave is initially created within the
middle section as described in the literature (Boulos, Lansey and Karney 2006), see in
Figure 4.3 (b). Figure 4.3 (c) shows the sign of the wave reflection shifts as the right
hand boundary shifts with the valve opening increasing from fully closed (i.e., dead-end)
to fully opened (i.e., constant-head reservoir). By systematic adjustment, a valve
opening of about 30% is found in this system to eliminate all wave reflections.
In practice, such “non-reflective” boundary would and could only be “tailored”
by an automatic control valve, which measures and dynamically adjusts in response to
the incoming pressure waves.
4.3 Mathematical Model for Local/Conventional PID Control Valve
PID control valves are usually installed at the connection of subsystems to maintain the
desired operating condition in hydraulic networks. Most probably, the subsystems were
originally designed for separate operation and then have been connected or expanded due
to the urbanization and development of water distribution networks (B. Bounce & L.
Morelli, 1999). For instance, to ensure the minimum flow demand in downstream
subsystem, an accurate and continuous control of the pressure or flow rate in the
connection pipeline is needed.
69
The mathematical model and numerical simulation of PID control valves
provides a tool to better understand the system hydraulics. Usually, there is a built-in
PID controller and sensor at a control valve, as shown in Figure 3.7. For different control
variables, the mathematical models are largely the same except for a slight difference in
the PID controller equation.
4.3.1 Extended MOC Equations
To simulate a PID control valve in a pipe network (reference to Figure 3.8), the extended
Method of Characteristics (MOC) is used to calculate the nodal heads at the valve inlet
and outlet, respectively (Karney and McInnis, 1992). The MOC equations are repeated
here:
1)-(4 111 QBCH ′−′=
2)-(4 222 QBCH ′+′=
in which , , and are calculated as shown in Chapter 3. 1'B 2'B 1C ′ 2C′
4.3.2 Valve Discharge Equation
Valve discharge equation defines the relationship between the flow passing through a
valve and the head difference across the valve. The commonly used valve discharge
equation is as follows:
)34( 21 −−= HHEQ Sτ
70
in which ES is a valve conductance parameter determined by the energy dissipation
potential of the valve, and τ is the dimensionless valve opening. For the steady flow Q0
and the corresponding head loss of H0, 2010
0 and 1HH
QES
−==τ ; and for no flow case
with the valve closed, 0=τ (Wylie, et al., 1993).
Instead of the predefined opening or closing motion of a conventional valve,
the opening of a PID control valve is adjustable in response to the sensed pressure or
pressure difference, which is desired to be controlled. That is, )(tτ is unknown for a PID
control valve and needs to be dynamically determined by the characteristics of the PID
controller.
The valve relationship, equation (4-3), is suitable for quasi-steady flow only
and might not be valid for rapid transient flow, because during transient state it may
occur that the flow direction is inconsistent with the head difference across the valve
when valve opening is significantly small. This phenomenon has similar influence as the
backlash or dead time of the valve. Future research should consider this inconsistency in
the model, but the challenge is that there is a little data on transient behavior of valves
and other components of the hydraulic system.
4.3.3 PID Controller Equations
The output signal or response of a typical parallel-structured PID controller is given as
(Tan, et al., 1999, page 9):
4)-(4 )1()(0 dt
deTedtT
eKtr d
t
iC ++= ∫
71
where e is the controller error, that is the deviation of the process variable u(t) from its
set point u*. Usually, the desired set point u* is a given constant and the dimensionless
error is defined as *)(1
utue −= . For a control valve, u(t) could be either of the inlet
pressure head H1(t), outlet pressure head H2(t), or the flow passing through the valve Q(t).
In a physical system, the values of these control variables are continuously measured by
sensors; while in numerical counterpart, the values of these variables are simulated step
by step.
Constants KC, Ti and Td are, respectively, the proportional gain, integral time
and derivative time constants of a PID-controller; they represent the characteristics of the
controller. The questions about how to select these parameters and how they affect the
speed and stability of the control process are discussed in the last section of this chapter.
The control law expressed by equation (4-4) is general for all types of PID-
controllers. It is straightforward to set the parameter Td to zero for a PI-controller;
furthermore, for a controller that has proportional part only (i.e., a P-controller), Ti is
given a large value. Besides, for a series-structured controller with given parameters, the
parameters for corresponding parallel type can always be obtained by the relationships
between these two controller structures. On the other hand, given the parameters for a
parallel type controller, it is not always possible to obtain the corresponding parameters
for the series type (Tan, et al., 1999, page 17). Therefore, equation (4-4) for parallel type
PID controllers will work for series type as well by transforming the given parameters to
those for the corresponding parallel type.
72
Depending on the power source of the actuator (pneumatic, electric or
hydraulic) and the internal design of the valve and controller, the operational equation of
the PID controller could be established corresponding for each specified process variable.
For the control of pressure at the valve inlet, H1, the output signal (the amplified error) of
the controller is set equal to the rate of valve flow reduction (Hopkins, 1998); that is,
5)-(4 )1()(0 dt
dQdtdeTedt
TeKtr d
t
iC −=++= ∫
here *)(1
1
1
HtHe −= . The hydraulic implication of the above control action can be
explained as follows: when pressure H1(t) at the inlet of the valve begins to exceed the
set point H1* ( ), the valve would open slightly to discharge the excess water
volume (
0<e
0>dtdQ ). By contrast, if the pressure H1(t) at the valve inlet begins to decay
below the set point H1* ( ), the valve would throttle and reduce the discharge
(
0>e
0<dtdQ ). With a PID controller, the automatic adjustment of the valve opening would
be smooth and continuous.
Similarly, for the pressure control at the outlet of the valve H2, the output
signal of the controller would be set equal to the rate of valve flow increase (dtdQ
+ ).
That is, the minus sign is removed from the right-hand side of equation (4-5) and one
would use *)(1
2
2
HtHe −= instead.
73
For the control of valve flow rate Q, the signal of control error is set equal to
the change of the head difference across the valve, H, and this change is based on the
initial steady state, that is,
6)-(4 ] )()([][
)()1()(
212010
00
tHtHHH
tHHdtdeTedt
TeKtr d
t
iC
−−−=
−=++= ∫
here *)(1
QtQe −= .
In sum, for a local/conventional PID control valve, two MOC equations, the
valve discharge equation and the PID controller equation constitute the mathematical
model of the boundary condition in the pipe network. Therefore, the four unknown
variables at valve boundary (τ , H1, H2, Q) can be numerically resolved using the finite
difference method.
4.4 Consideration and Model of “Non-Reflective” Valve Opening
Based on the developed mathematical model for a local PID control valve, the physical
consideration and mathematical model to search for a “non-reflective” boundary by
using remote sensing and PID control valve is elucidated in this section.
Theoretically, a remote sensor could be located either upstream or downstream
of the control valve wherever an incident pressure surge occurs, since the pressure surge
will usually propagate towards both directions. However, this research focuses on the
74
case of upstream incident surge, and the main idea is illustrated by a simple system
shown in Figure 4.4.
HR
Valve Sensor Q
L
H1 H2
f = 0
f = 0.012
H1
Figure 4.4 Case studies: System scheme and initial steady state
In this system, it is assumed that a series of pressure surges, HR(t), (say, that
represent periodic waves following a sinusoidal law) occur at an upstream reservoir due
to a certain disturbance, where a sensor is installed and linked to a PID control valve at
downstream reservoir. The pressure waves would then propagate at speed a towards the
downstream end along the pipeline. The wave would arrive at the valve inlet within L/a
seconds. To limit the wave reflection from the valve, the PID controller adjusts the valve
opening continuously and accurately to "absorb" the upcoming incident wave when it
passing through the valve (i.e., to dissipate the wave energy at the valve). As a result, at
any instant time t, the pressure at the valve inlet H1(t) should maintain the same
magnitude as the upcoming wave when it reaches the valve. In other words, the set point
75
of valve inlet pressure, H1*, would be equivalent to the “successor” of the upstream
reservoir pressure at a L/a time ahead, i.e., HR(t-L/a). Since the upcoming pressure
changes with time, the set point, H1*, must change with time as well, which is so-called
time-variable or dynamic set point, H1*(t).
The pressure at the valve inlet H1(t) will be tracked in this “non-reflective”
boundary model, correspondingly, the mathematical model for the local PID control
valve with H1 control case, including the extended MOC equations (4-1) and (4-2), the
valve discharge equation (4-3) and the PID controller equation (4-5), is applicable.
However, there is a key difference in the controller equation; in particular, the control
error e in “non-reflective” boundary model is the deviation of instant pressure head H1(t)
from its dynamic set point H1*(t), that is, )(*
)(11
1
tHtHe −= . Instead of a given constant for
the set point in the local control valve, the set point H1*(t) in this case is an unknown
variable (transmission of HR), dependent of the remote pressure waves and pipe system
features, which could significantly complicate the discretization of the governing
equations and their numerical solution. Yet, if H1*(t) could be predicted at any instant
time t, then the PID controller will send the actuator a series of “commands” (the error
signals) to adjust the valve opening continuously to achieve H1(t) = H1*(t). Therefore,
the only remaining issue is how to determine the dynamic set point, H1*(t), according to
the remotely sensed pressure HR(t)?
For a frictionless pipeline without reflections from the downstream boundary,
the upstream pressure wave HR(t) would have no change when it propagates to the front
of valve in L/a seconds. Thus, at any instant, the set point for the pressure at the valve
76
inlet is equivalent to the pressure at the remote sensor that has been recorded at prior
time of L/a. Mathematically, we have:
7)-(4 )/()(*1 aLtHtH R −=
However, for a more realistic pipeline having friction resistance, the
magnitude of a pressure wave decays somewhat as it propagates downstream. The key
and challenging question is how to transform transient pressures between the two ends of
the pipeline with friction. Since no analytic solutions are available, the set point must be
estimated based on the remotely measured pressure waves and pipe system features. In
this research, a relative Friction Decay Rate (FDR) is introduced based on the hydraulic
grade line at initial steady state, which is defined as the ratio of pressure heads at the
valve inlet (H10) and at the remote sensor (HR0):
8)-(4 0
10
RDR H
HF =
Certainly, FDR is system specific but constant for a particular system with certain initial
hydraulic condition. Using the FDR defined in equation (4-8), the magnitude of pressure
wave when it arrives at the valve could be largely estimated:
9)-(4 )/()(*1 aLtHFtH RDR −×=
Note that equations (4-8) and (4-9) are consistent with and valid for
frictionless pipelines as well. When the pipeline friction is negligible we have HR0 = H10
and FDR = 1, and thus equation (4-9) reduces to equation (4-7). Therefore, for both
frictional and frictionless pipelines, the set point in controller equation (4-5) could be
77
dynamically estimated using equations (4-8) and (4-9). Interestingly, for those cases
with an initially static state (i.e., the initial flow Q0 = 0 in the system), there is no initial
headloss along the pipeline no matter the pipeline is frictional or frictionless, so we also
have HR0 = H10 and FDR = 1, and equation (4-9) still reduces to equation (4-7).
Through this FDR compromise, we are actually using the steady state FDR to
estimate the frictional decay under transient states. Fortunately, the case studies using
this model have sufficiently shown the potential of PID control valve with remote
pressure sensor in limiting the pressure oscillations and resonance. As better pressure
control would arise from a better estimate for the dynamic set point, H1*(t); but this
value is challenging to find since there is no universally appropriate estimate for transient
pressure decay rate. These difficulties arise from the complexity of the friction term and
wave interference in the momentum equation of transient flow, which is not only
dependent of the roughness or other properties of pipe system and the hydraulic
conditions, but also relevant to the direction and frequency of the propagating waves.
Finally, it is understood that the pressure at the remote location for the L/a
second earlier, HR (t-L/a), could be always retrieved at any instant time t in both physical
and numerical systems. In the physical system, the value of HR (t-L/a) was measured and
recorded at the remote sensor, and then sent instantly to the controller by wireless or
cable transmission at any time t. Thus, at least two pressure sensors are required in the
system, one is to measure the control variable H1 (t) at the valve and another is to
measure remotely to obtain the dynamic set point H1*(t). While in the numerical
counterpart, the pressure at the remote sensor node for L/a second time ahead, HR(t-L/a),
was already simulated and stored, which could obviously be retrieved at any instant time.
78
4.5 Simulations and Case Studies
4.5.1 Case Studies with Non-Zero Initial Flow
The same system, as shown in Figure 4.4, is studied to illustrate the application of
current mathematical model. In this system, a long horizontal pipeline (with L = 5000 m,
diameter D = 1.0 m, and wave speed a = 1000 m/s) links two constant head reservoirs.
At the entrance of downstream reservoir, there is a valve with constant ES = 7 m2.5/s and
initial opening 0τ = 0.1. The upstream reservoir has 30 m water head initially (HR0 = 30
m) and downstream reservoir has a constant head of 15 m. The water level at upstream
reservoir starts to fluctuate sinusoidally, which induces transient events in the system.
The amplitude of oscillatory wave is 2 m; the cyclic period is 10 s, which equals 2L/a, so
that resonance would be expected to take place. The instant pressure at the upstream
reservoir can be described as:
)10/2sin(2)( 0 tHtH RR π+= (4-10)
In the first case study, it is assumed the pipeline is frictionless, i.e., the Darcy-
Weisbach friction factor f = 0. At initial steady state, the hydraulic head at the valve inlet
equals the reservoir head, that is, H10 = HR0 = 30 m, and the flow rate in the system and
passing through the valve is Q0 = 2.71 m3/s.
If the downstream valve remains at its initial opening 0τ = 0.1 (i.e., fixed
orifice), the oscillations of upstream pressure (i.e., incidental pressure wave, or forced
vibration) will cause hydraulic resonance and pressure amplification in the middle part of
the pipeline. Figure 4.5 shows the development of the steady-oscillatory flow in the
79
pipeline; the solid line represents the pressure oscillations at the middle-point of the
pipeline and the dashed line represents the pressure oscillations at the valve inlet. There
is a L/2a time difference (i.e., 1/8 phase difference) between these two pressure waves,
and the amplitudes of both waves are initially small and grow gradually until they finally
(within 300 s) stabilize at a resonance condition. This mode shape of the pressure waves
can be understood and explained as follows: in this system, the upstream incidental wave
will reach the valve inlet in 5 s (L/a = 5 s), and it will take 7.5 s for the first wave peak to
arrive at the valve inlet (the oscillatory period of upstream incidental wave is 10 s).
During the time period 5 to 7.5 s, the pressure at the valve inlet, indeed at any internal
pipe node, is the superposition of the incident wave and the reflected wave from the
fixed valve at downstream reservoir. Since the reflection at downstream boundary in this
case is negative, the first wave peak is reduced when it arrives at downstream valve.
However, the pressure waves with different amplitudes would continuously proceed and
be reflected. The result of superposition increases until the maximum amplitude reached
and the steady-oscillatory flow condition developed in the system. This simulation result
with fixed valve opening is also represented by the dashed lines in Figure 4.6. The value
of the steady amplitude at each position of pipeline depends on the system frequency and
resistance characteristics. As matter of fact, this phenomenon resulting from the forced
vibration of upstream pressure (with fixed downstream valve opening) is equivalent to
the responses caused by periodic valve motion at downstream end (while keeping the
upstream reservoir head constant), as summarized in Figure 8.2 on page 204 of Chaudhry
(1979).
80
To eliminate or limit the reflection and superposition of pressure waves in the
pipeline, a sensor is installed at the upstream node, and a PID controller positioned at the
downstream valve. The constants of PID controller are taken as KC = 250, Ti = 0.002 s
and Td = 0.5 s. The total simulation time is run for 800 s, and the time step is 1/100 s that
is sufficiently small to look into the detailed valve responses.
The developed remote control model for frictionless pipeline is used in this
case study to simulate the system responses to the oscillations of pressure head at
upstream reservoir. Figure 4.6 compares the system responses for the responsive PID
control valve (black/solid lines) and the conventional valve with fixed orifice (red/dashed
lines). The solid/black lines in Figure 4.6(a) show that the maximum and minimum
pressures along the pipeline remain the same as those introduced at upstream reservoir,
demonstrating no wave reflection or resonance ever occurred in the system by using
remote sensing and PID control valve. By contrast, if the valve opening is fixed at the
initial size ( 1.0=τ ), the pressure waves are reflected and superposed in the pipeline,
resulting in the increased envelope of maximum and minimum pressures along the
pipeline, see the red/dashed curves in Figure 4.6(a). In Figure 4.6(b), the black/solid
curves show the opening of PID control valve is adjusted periodically in response to the
oscillating incident pressure waves, as expected, which exactly eliminates the wave
reflections at the valve (by exciting a contrary wave) and thus remains the same mode
shape of pressure oscillation as of the upstream incident wave; while the red/dashed lines
show that the amplitude of pressure waves at the inlet of fixed orifice is reduced at the
beginning and grows until stabilized at certain value because of resistance of the valve.
81
In the second case study, the friction factor f = 0.012 is given for the pipeline.
At the initial steady state, the hydraulic grade line declines along the pipe, the pressure at
the valve inlet is H10 = 19.37 m, and the flow in the system and passing through the valve
Q0 = 1.46 m3/s. The same parameters of PID controller, total simulation time and time
step as in the first case study are used here.
To simulate the system responses to the oscillations of pressure head at
upstream reservoir, the frictional decay rate FDR defined in equation (4-8) and the
dynamic set point estimated by equation (4-9) are applied. The simulated PID control
results are compared with the corresponding conditions for fixed-opening-valve case. As
shown in Figure 4.7, the reflection is not completely eliminated because of the roughly
estimated FDR in transient state. However, the simulation results converge to a steady
oscillation flow within 400 s, and the comparison in Figure 4.7(a) shows that the
maximum and minimum pressure envelope using the PID control (solid/black curves) is
smaller than the case of fixed-opening-valve (dashed/red curves), which demonstrates
the reflections and resonance of pressure waves are constrained by using the remote
sensing and PID control valve. The smaller the initial flow in the system, the more
reduction in the magnitude of pressure amplitude envelopes, and this point can be
demonstrated by the following case study with zero initial flow. Figure 4.7(b) shows
that the opening of the PID control valve is adjusted periodically in response to the
proceeding waves, and the steady oscillation remains at the valve inlet with the same
decayed amplitude as initial steady state. Yet, for the case of fixed-opening-valve, the
amplitude of pressure waves at the valve inlet varies with time, reduced at the beginning
and increasing gradually until a steady-oscillatory flow developed over about 300 s.
82
0
10
20
30
40
50
60
0 50 100 150 200 250 300
Time (s)
Pres
sure
hea
d (m
)
H(L) H(L/2)
Figure 4.5 Development of steady-oscillatory flow in frictionless pipeline with fixed-
valve-opening at downstream, due to the upstream pressure oscillations
83
0
10
20
30
40
50
60
0 1000 2000 3000 4000 5000
Distance along pipeline (m)
Max
./Min
. pre
ssur
e he
ad (m
)
PID Valve Fixed Valve Steady State
Max. H
Min. H
S. S.
(a) Maximum and minimum pressure head envelopes
24
26
28
30
32
34
0 20 40 60 80 100 120 140 160 180 200
Time (s)
H 1(m
)
0
0.1
0.2
0.3
0.4
Tau
PID Valve Fixed Valve
H1
Tau
(b) Valve opening and inlet pressure wave varying with time
Figure 4.6 Responsive PID control valve vs. fixed-opening-valve in frictionless pipeline (f=0)
84
0
10
20
30
40
50
60
0 1000 2000 3000 4000 5000
Distance along pipeline (m)
Max
./Min
. pre
ssur
e he
ad (m
)
PID Valve Fixed Valve Steady State
Min. H
S. S.
Max. H
(a) Maximum and minimum pressure head envelopes
12
14
16
18
20
22
0 20 40 60 80 100 120 140 160 180 200
Time (s)
H1(
m)
0
0.1
0.2
0.3
0.4
Tau
PID Valve Fixed Valve
Tau
H1
(b) Valve opening and inlet pressure wave varying with time
Figure 4.7 Responsive PID control valve vs. fixed-opening-valve
in frictional pipeline (f=0.012)
85
4.5.2 Case Studies without Initial Flow (Static Initial State)
In the system sketched in Figure 4.4, if the constant heads are 30 m at both upstream and
downstream reservoirs, there is no flow in the system and the initial valve opening is of
no consequence. For both cases with frictionless and frictional pipeline, the wave
reflection and resonance can be completely eliminated and the “non-reflective” boundary
achieved. In this system, the oscillatory head causes the flow, and the flow direction
shifts as the head oscillates around the original constant level. At any specific point of
the pipeline (e.g., at the valve), the magnitude of oscillatory flow is small and the
average value with time is zero. The f value by itself seems hardly influence the wave
reflection.
Figures 4.8 shows the simulation results for the case study with friction factor
f=0.012. Without the flow (and thus without resistance from flow), the resonance would
be stabilized in around 600 s for the fixed valve opening case and the range of the
max./min. pressures are much larger than that with non-zero flow case. While with
responsive PID control valve, there is no wave reflection and resonance at all in the
system, and the oscillations remain steady at any point and at any time, as the same as
the upstream incident pressure oscillations.
86
-70-50-30-101030507090
110130
0 1000 2000 3000 4000 5000
Distance along Pipeline (m)
Max
./Min
. Pre
ssur
e H
ead
(m)
Fixed Valve (Tau=0.1) PID Control valve
Max. H
Min. H
(a) Maximum and minimum pressure head envelopes
0
4
8
12
16
20
24
28
32
36
0 10 20 30 40 50 60 70 80 90 100
Time (s)
H1 (
m)
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Q (m
3 /s)
H1
Q
(b) Inlet pressure wave and flow at the PID valve varying with time
Figure 4.8 Responsive PID control valve vs. fixed-opening-valve for static initial
states with frictional pipeline (f=0.012)
87
4.6 Frequency Analysis and “Non-Reflective” Boundary Verification
Based on the time domain transient analysis using method of characteristics (MOC), the
numerical model for “non-reflective” boundary design has been developed in previous
sections. However, for a periodic oscillation originating at a remote location, transient
analysis in the frequency domain is a more practical and efficient way to reveal the
oscillatory conditions in the fluid system.
In the frequency domain, there are two types of analyses: frequency response
analysis and free vibration analysis. They apply the basic unsteady flow equations, using
complex variables and a mathematical shorthand common in variation problems. The
former is based on fully developed steady-oscillatory forced vibrations (i.e., a persistent
excitation). The objective is to obtain the system responses to the explicit excitation
imposed on the system. The free vibration of a fluid system is caused by some temporary
excitation. The initial fluid motion is of little interest in the Free Vibration Analysis, it
aims only at the residual time-dependent oscillation. Thus, it doesn’t need to know the
excitation, but the natural frequencies of the entire system would be defined and the
vibration mode shape of the system be identified. The analysis is also useful in
evaluating system stability, in judging system performance, and identifying critical
resonance conditions.
In this section, oscillation is introduced through various harmonics that might
occur in the pipe system. Then, the system responses to the forced vibrations (upstream
pressure oscillations) with different frequencies and the applicability of developed “non-
reflective” model are checked. After that, the steady “non-reflective” boundary
88
conditions for the frictionless pipeline system, obtained by the traditional MOC, would
be verified using the method of hydraulic impedance in the frequency domain.
4.6.1. System Responses to Pressure Oscillations with Various Frequencies
Unexpected resonance could be destructive in practical hydraulic systems. The
consequences of resonance in fluid systems range from objectionable operating
conditions, such as instability, noise, and vibration, to fatal damage of system elements
overstressed during severe pressure oscillations. Thus, the phenomenon of hydraulic
resonance should be predicted and prevented.
In this case study, the incident pressure oscillation at upstream reservoir is one
type of forced excitation. The fundamental period of pipeline system T0 = 4L/a = 20 s
(i.e., natural frequency is 1/20 Hz) and the given period of forced excitation T = 10 s (the
forcing frequency is 1/10 Hz), and the system responses to this forced excitation are
shown in Figure 4.6(a) and Figure 4.7(a) for frictionless and frictional pipeline,
respectively. For a fixed orifice at downstream reservoir, the system responses
demonstrate the characters of second harmonics, and the maximum amplitude of
pressure oscillation, occurred at the middle point of the pipeline, is about 12 and 9 times
as large as the incident pressure oscillation, respectively, for frictionless and frictional
cases. Now, what if we change the frequency of the forced excitation? Could we adjust
downstream valve opening to eliminate the potential resonance in the system using the
developed “non-reflective” boundary design?
89
(a) No resonance (T0/T = 1)
(b) 2nd order of harmonics (T0/T = 2)
(c) No resonance (T0/T = 3)
Figure 4.9 System responses to forced pressure oscillation with various frequencies
in a frictionless pipeline (f=0) (Con’d)
90
(d) 4th order of harmonics (T0/T = 4)
(e) No resonance (T0/T = 5)
(f) 6th order of harmonics (T0/T = 6)
Figure 4.9 System responses to forced pressure oscillation with various frequencies
in a frictionless pipeline (f=0) (Con’d)
91
(g) 8th order of harmonics (T0/T = 8)
(h) 10th order of harmonics (T0/T =10)
(i) 20th order of harmonics (T0/T =20)
Figure 4.9 System responses to forced pressure oscillation with various frequencies
in a frictionless pipeline (f=0)
92
Frictionless pipeline system. In this system, for the fixed valve opening at downstream
reservoir, even order of harmonics exist, as shown in Figure 4.9 (b), (d) and (f)-(i). The
even harmonics indicates that the reflective characteristic at downstream orifice is
similar to a reservoir (negative reflection) (Wylie, et al., 1993 and Chaudhry, 1987). This
can be verified by comparing the hydraulic impedance of the fixed valve with the
characteristic impedance of the system. Hydraulic impedance in a fluid system is defined
as the ratio of the complex head to the complex discharge at a particular point in the
system (Wylie, et al., 1993. Pg301-302).
For the frictionless pipeline system as shown in Figure 4.4 (reservoir-pipeline-
orifice at reservoir), the characteristic impedance of fluid system is calculated as follows:
11)-(4 s/m 130m /411416.3m/s 9.81
m/s 1000 2222 ≈
××==
gAaZC
While for the fixed orifice or valve, the valve equation (4-3) can be written as:
12)-(4 0HEQ Sτ=
in which 0H is the head drop across the valve for the mean flow Q . From the initial
steady state, we have: . /m 711.2 m, 15 ,/m 7 0.1, 30
2.5 sQHsES ====τ So the
hydraulic impedance of this fixed valve:
13)-(4 s/m 112 20 ≈==QH
QH
ZV
VV
93
here HV and QV are the complex head and flow at the oscillatory valve, respectively.
Thus, we have , this is the condition that even harmonics occurs in the system.
On the contrary, if , the odd harmonics could occur, which indicates the orifice
would provide a response similar to a dead-end. If we adjust the valve opening to make
, then the orifice becomes “non-reflective”.
CV ZZ <
CV ZZ >
CV ZZ =
When the cyclic period of the forced vibration is given as fractional a part of
the fundamental period (T0 = 4L/a =20 s), the different orders of harmonics and different
mode shapes of pressure waves would occur in the pipe system. However, the frequency
change doesn’t affect the amplitude of each harmonics if the amplitude of incidental
pressure oscillation remains the same. The order of harmonics equals the system
fundamental period T0 divided by the period of forced vibration T. If we give even
number of T0 /T (i.e., T = 10 s, 5 s, 3.33 s, 2.5 s, 2 s and 1 s, as shown in Figure 4.9),
excessive energy influx to the system during oscillatory flow leads to resonance. In an
ideal lossless system (for this frictionless pipeline system the only energy dissipation
occurs at the downstream valve), there is generally no energy transmission in steady-
oscillatory flow, although alternating energy conversion between kinetic energy and
pressure energy may occur. With terminal wave reflection, steady-oscillatory motion
shows a combination of forward and reflected waves that results in a standing wave.
Within the standing-wave pattern, energy is converted from pressure energy to kinetic
energy, then back to pressure energy, and so on. If we give odd number of T0 /T (as
shown in Figure 4.9 (a), (c) and (e) for T = 20 s, 6.67 s and 4 s), the mode shape of the
pressure waves is quite different from the even harmonics, the energy dissipated
gradually and resonance does not occur in the system.
94
It is not surprising to find the resonance with different orders of harmonics
and amplification of pressure head can be completely eliminated in frictionless pipeline
by designing a “non-reflective” boundary. The simulated results are all the same as the
black/dotted line in Figure 4.6 (a), no matter how the period of the forced vibration T
changes. However, it is noticed that for the higher frequency forced oscillations (smaller
T), the PID integral and derivative parameters (Ti and Td) require smaller values to obtain
the precise adjustment of “non-reflective” valve opening.
Frictional pipeline system with non-zero initial flow. For the frictional pipeline case,
we have a similar finding regarding even harmonics when we change the period of the
forced vibration. The application of remote sensor and PID control valve cannot
completely eliminate the reflections and resonance if the initial flow is not zero, but the
amplitude of the pressure waves is significantly reduced for each harmonics, as shown
Figure 4.10. In this Figure, the different responses to the forced oscillations at the
upstream end are compared for the system with a fixed-opening-valve and “non-
reflective” boundary condition at the downstream end of pipeline.
95
Figure 4.10 System responses to forced pressure oscillation with various
frequencies in a frictional pipeline (f=0.012) (Fixed valve vs. responsive PID control
valve)
96
4.6.2. “Non-Reflective” Boundary Verification using Hydraulic Impedance Method
From the viewpoint of frequency domain, the automatically adjustable PID valve creates
an artificial excitation and the consequence of this designed valve-oscillation would
exactly cancel out the effect of incidental pressure oscillation at the upstream reservoir.
For the frictionless pipeline system, we have verified the condition of a “non-reflective”
boundary, that is, , for the developed steady-oscillatory flow (i.e., after 300 s of
PID valve adjustment when the amplitude of the pressure waves in the pipeline
stabilized). This steady valve-oscillating condition has been obtained from the numerical
simulation in the time domain, which uses the “non-reflective” boundary model
developed in the previous section.
CV ZZ =
For the frictionless pipeline system, the characteristic impedance has been
calculated as equation (4-11). For an oscillating valve, the hydraulic impedance at the
upstream side of the valve can be calculated by
14)-(4 22 00
V
V
V
VV Q
THQH
QH Z
τ−==
We already know /m 711.2 m, 15 0.1, 30 sQH ===τ , and here HV, QV and are the
complex head, flow and opening at the oscillatory valve, respectively.
VT
From numerical simulation in the time domain, we found the maximum valve
opening 107.0( max =τ , amplitude = 0.007) corresponding to the minimum valve flow
(Qmin = 2.695 m3/s, amplitude = 0.016 m3/s) and minimum valve inlet head (H1min = 28.0
m, amplitude = 2.0 m). On the contrary, the minimum valve opening 094.0 ( min =τ ,
97
amplitude = 0.006) corresponds to the maximum flow (Qmax = 2.726 m3/s, amplitude =
0.015 m3/s) and maximum valve inlet head (H1max = 32.0 m, amplitude = 2.0 m). In other
words, we have complex hydraulic values:
10
2sin 2.0 10
)/( 2sin 0.2 ⎥⎦⎤
⎢⎣⎡ +=⎥⎦
⎤⎢⎣⎡ +
= πππ taLtHV (4-15)
10
2sin 0155.0 10
)/( 2sin 0155.0 ⎥⎦⎤
⎢⎣⎡ +=⎥⎦
⎤⎢⎣⎡ +
= πππ taLtQV (4-16)
10
2sin 0065.0 ⎥⎦⎤
⎢⎣⎡=
tTVπ (4-17)
It is noticed that there is a phase difference of 10/)/2( aLπ between the oscillation of
valve opening (TV) and the oscillations of valve flow (QV) and inlet head (HV). So, we
can also calculate 129≈=V
VV Q
HZ s/m2, or 13722 00 ≈−=
V
VV Q
THQH Z
τ s/m2 (from
equations (4-16) and (4-17), we know TV and QV have opposite sign; the numerical error
is acceptable due to only 3 digitals of the simulation results recorded). Therefore, the
“non-reflective” boundary condition CV ZZ ≈ =130 s/m2 has been verified.
In the case without initial flow but the pipeline with friction, as shown in
Figure 4.8, we obtained the same pressure and flow oscillations at the valve as described
in equations (4-15) and (4-16), and thus V
VV Q
HZ = ≈ ZC =130 s/m2 can be verified.
Moreover, by changing the amplitude of the incident pressure waves at the upstream
reservoir, the amplitude of induced flow oscillations would change proportionally, so the
98
value of V
VV Q
HZ = remains near 130 and “non-reflective” boundary condition would be
always achieved for static initial state cases.
To further verify this law of “non-reflective” boundary condition, the wave
speed of pipeline is reduced to 500 m/s, so the system characteristic impedance also
reduces to s/m 65 2≈=gAaZC . The “non-reflective” boundary condition, , could
be verified as well by the corresponding numerical simulation results in the time domain.
CV ZZ =
4.7 Tuning PID Controller
The final tuning of the parameters (KC, Ti and Td) for a PID controller would be
important during the commissioning stage of the system. Similar to the trial and error
method used for a physical system, the numerical model could provide a tool for
preliminary selection of these parameters. To better understand how the variation of
each parameter affects on the system control results, a sort of sensitivity analysis for
three controller parameters is performed here, using the same system shown in Figure 4.4
with frictionless pipeline as aforementioned, and the comparative results are summarized
in Figure 4.11.
Proper selection of controller parameters means finding a compromise
between the requirement for fast control and the need of stable control. More
specifically, with increases in the proportional gain (KC), the speed of control increases
but the stability of control reduces. Figure 4.11 (a) shows that given the same simulation
time period (200 s), the reduction of KC (slower control) enlarges the maximum and
99
minimum pressure envelope. Clearly, KC must be greater than a certain value to
effectively control the system. In this case study, a KC of about 50, or greater, is needed.
The tendencies for the variation of integral time (Ti) are opposite to KC. With
Ti increases, the speed of control reduces while the stability of control increases. Figure
4.11 (b) shows that the maximum and minimum pressure envelope expands with the
increase of integral time Ti (slower control), and Ti ≤ 0.001 s is required in this system.
The derivative part produces both faster and more stable control when Td
increases. However, this is only true up to a certain limit and if the signal is sufficiently
free of noise (calculation error is a kind of noise in numerical system). If Td rises above
this limit it will result in reduced stability of control. As we know that the function of the
derivative part is to estimate the change in the control a time Td ahead. This estimate will
naturally be poor for large values of Td. Another consideration is the noise and other
disturbances. The noise is amplified to a greater extent when Td increases, and thus it is
often the noise that sets the upper limit for the magnitude of Td. The above theoretical
analysis could be verified by the numerical simulations in Figure 4.11 (c), which shows
that the control stability is better when Td = 0.5 s (dashed lines) than that when Td =
0.005 s (dot-dash lines). However, when Td = 5 s the stability of control is poor.
In addition, the simulations also show that for high frequency oscillatory flow
(e.g., 1 s of the period of incidental pressure wave), the smaller values for both the
integral time constant Ti and derivative time constant Td are required.
The stability and speed of control process are associated with the parameters
of controller. The mathematical model and numerical simulation are useful for selection
100
and tuning of the controller parameters, which could save time and cost in
commissioning of the physical system.
0
10
20
30
40
50
60
0 1000 2000 3000 4000 5000
Distance along pipeline (m)
Max
./Min
. pre
ssur
e he
ad (m
)
Kc=250
Kc=50
Kc=5
S.S.
Max. H
Min. H
S. S. H
(a) Max./Min. pressure envelopes expanding with reduction of proportional gain KC
(Ti = 0.001 s, Td = 0.5 s)
Figure 4.11 Effect of controller parameters on system pressure control (Con’d)
101
20
22
24
26
28
30
32
34
36
38
40
0 1000 2000 3000 4000 5000
Distance along pipeline (m)
Max
./Min
. pre
ssur
e he
ad (m
)Ti=0.001
Ti=0.01
Ti=0.02
S.S.
Max. H
Min. H
S. S. H
(b) Max./Min. pressure envelopes expanding with increase of integral time Ti
(KC = 250, Td = 0.5 s)
20
22
24
26
28
30
32
34
36
38
40
0 1000 2000 3000 4000 5000
Distance along pipeline (m)
Max
./Min
. pre
ssur
e he
ad (m
)
Td=0.5
Td=0.005
Td=5
S.S.
Max. H
Min. H
S. S. H
(c) Max./Min. pressure envelopes varying with different derivative time Td
(KC = 250, Ti = 0.02 s)
Figure 4.11 Effect of controller parameters on system pressure control
102
4.8 Summary and Conclusions
Dead-end branches are a common component in pipe networks; of such locations, dead-
end reflection may cause unexpected high pressures when system experiences transients,
which should and can be avoided by using a suitable dissipative valve.
The creation of “non-reflective” (or “semi-reflective”) boundaries through
remote sensing and PID control valve is a new concept to more accurately limit dead-end
reflection and resonance in pipelines. This idea is explored here by the means of
numerical simulation, and a considerable potential for transient protection has been
demonstrated in the case studies. Using the model developed in this paper, the wave
reflection and resonance could be eliminated for frictionless pipelines or the pipelines
with static initial states; while for the pipeline with some friction, the pressure waves’
reflection at the valve and superposition within the pipeline can be effectively limited.
Moreover, the theory of hydraulic impedance in the frequency domain, regarding the
condition of even order of harmonics and “non-reflective” boundary conditions, has been
verified by the numerical simulations using the transient analysis model developed in the
time domain.
However, complications and challenges may arise when the model of “non-
reflective” (or “semi-reflective”) boundary is applied in real systems. First, the model is
developed based on a single pipeline system with the remote sensor at upstream end and
PID control valve at downstream end. It is feasible to apply the model for a branched
pipeline in a complex system with remote sensor at the junction and a terminal valve at
the branch-end, but further application to a pipe loop or an arbitrary pipeline with other
103
components between the two ends would be significantly challenging, since the primary
pressure wave would be reflected, refracted, or attenuated by those components and thus
the theoretical estimate of pressure set point become almost impossible. Besides, in real
systems, the uncertainties (or frequency-relevance) in the magnitudes of waves speed,
pipe friction factor and pressure decay rate at transient state may cause significant over-
or under-estimate of pressure set point and thus require for careful system calibrations in
advance. The upstream pressure disturbance may not be a steady oscillation as in the
case studies, and then the reflections at the upstream reservoir would further complicate
the “non-reflective” valve opening control. Thus, at present the design of “non-
reflective” boundary, though promising, calls for further research as well as cooperation
with the device manufacturers.
104
Chapter 5 Fundamentals of Transients in Hydro
Turbine Systems
This chapter supplies the required background for the second type of AHD studied in this
thesis, a governed hydro turbine. A hydro turbine is the most complicated hydraulic
component in terms of transient analysis and waterhammer control. Motivation and
interest in hydro system research arises partly from the author’s previous background, as
well as a global reawakening of interest in the hydro industry (especially small scale
facilities). Unavoidably, waterhammer remains a serious issue in many hydro systems
where accidents and disruptions are caused by imperfect operation of turbine units
(Pejovic et al, 2007). Although there are several numerical models developed for hydro
system transient analysis, none of those published is directly applicable to a water network.
To better understand the transient flow associated with turbine operation, the
fundamental knowledge of turbine and hydro system, including turbine classification,
turbine characteristics, governor and control equipment, and the common arrangements of
hydro systems, are reviewed in this chapter.
105
5.1 Hydro Turbine Classification
Although turbine classifications are similar throughout the world, the specific terminology
of different turbine types and models may vary from one region to another.
5.1.1 General Classification of Hydro Turbines
Hydraulic turbines are classified, with respect to their hydraulic action or energy
conversion, into two major categories: reaction turbines and impulse turbines, as shown in
Table 5.1. The reaction turbines include Francis and Propeller types; in these types, flow
takes place under pressure in a closed chamber and only a part of the available water
energy is converted into kinetic energy at the entrance to the runner, with a substantial part
remaining capable of doing work on the runner through pressure. Some propeller-type
turbines have fixed runner blades, i.e., the angle of runner blade ϕ is fixed at a near optimal
position for the design operating condition. Another family of propeller turbines, the
Kaplan turbines, has adjustable blades; the blade angle ϕ can be adjusted within a certain
range to achieve higher efficiencies over a wide spectrum of flow conditions. Both the
runner blade angle and wicket gate position are regulated through governor control system
by a cam mechanism (a designed relationship between wicket gate servo stroke and runner
blade servo stroke). Kaplan turbines are typically used on run-of-the-river hydroelectric
plants with relatively high flow rates but low heads (Tong, et al., 1997 and Liu, et al.,
1997).
106
Table 5.1 Classification of Hydro Turbines
Turbines
Impulse Turbines (small ns)
Pelton Turgo
Reaction TurbinesFrancis Propeller (with fixed runner blades) Kaplan (with adjustable runner blades)
In an impulse turbine, a jet of water issuing from a nozzle impinges on the
vanes/buckets of the turbine wheel/runner (usually only one or two vanes/buckets at a
time), which is exposed to atmospheric pressure. All of the available potential energy of
the water is first converted into kinetic energy with the help of a nozzle. The impulse
turbine is a logical choice for high-head installations and it is also suitable for many lower
head sites if the discharge is relatively small. Because of this, it is the turbine of choice for
many small-scale hydropower plants. The primary advantages of the impulse turbine are
its simplicity and ease of maintenance. One of its disadvantages is that the impeller must be
set above the highest level of the downstream pool. This requirement implies that
run-of-the-river plants would tend to waste available head in dry seasons when the river
discharge is low and the downstream pool level depressed.
According to the main flow direction within the runner passage, hydro turbines
can be classified as types of axial-flow, cross-flow, radial-flow (Thompson) etc.; they also
be classified as vertical, horizontal (Bulb, Pit) and inclined types according to the
107
arrangement of the main shaft.
A pump-turbine is a special type of turbo machine used in pump-storage hydro
plants. All pump-turbines are reversible; that is, when water enters the spiral case at the
periphery and flows inward to the machine, it acts as a turbine; when water enters at the
center (or eye) and flows outward, it acts as a pump. The direction of rotation is, of course,
opposite in the two cases. Depending on the design pressure heads, pump-turbines may be
of the Francis, mixed-flow, or axial-flow type. A pump-turbine is connected to a motor
generator which acts as either a motor or generator depending on the direction of rotation
(Franzini, 1997).
Some pump-turbines are speed-adjustable either in pumping or generating mode
because, with variable speed, it is possible to operate the unit at or near the optimum speed
for the particular head and output and thus obtain a higher efficiency than with a
single-speed unit.
5.1.2 Hydro Turbine Classification in the Context of Numerical Modeling
To describe a turbine in computer code, it is necessary to precisely define which turbine
parameters are varying and which are fixed. In the numerical model developed in this
thesis, turbines are classified as:
1) ‘Impulse’ (equivalent in most ways to an external discharge valve);
108
2) ‘Fixed WG’ (with fixed wicket gates and fixed runner blades);
3) ‘Francis’ or ‘WG’ (wicket gates adjustable but runner blades fixed);
4) ‘Kaplan’ (both wicket gates and runner blades are adjustable).
Impulse Turbines. The flow of an impulse turbine is a function of head and nozzle
opening only. In other words, the variation of turbine speed has no direct influence on the
turbine flow. Some impulse turbines (Pelton) may have dual flow control systems, i.e., a
deflecting board and a needle valve. To avoid excessive acceleration of the runner during a
load rejection, it is necessary to block its water supply; the deflector servomotor shifts the
deflector to the required deflecting position for a short period of time (2-3 seconds or more)
while the discharge through the nozzle has not been changed. Simultaneously, the nozzle
servomotor moves the needle to the required position. The full travel of the needle may
take 30-40 seconds or longer. During movement of the needle, the deflector is withdrawn
gradually from the jet and at the end of its motion has no effect on the flow. In other words,
under load-rejection, the deflecting board does not change the flow rate but makes the jet
alter its direction to temporarily relieve the force impacting on the Pelton wheel. Therefore,
there is no need to incorporate the deflecting board into a waterhammer analysis. In the
current version of TransAM, the impulse turbine is simplified and treated as a valve
discharging to the atmosphere. That is, the turbine governor is not included in the model
and the turbine characteristics are not required; however, the nozzle characteristics have to
109
be provided for numerical simulation and the setting of the nozzle or needle valve
operation determined by the servomotor control system.
‘Fixed WG’. In this thesis, the variable Ndy represents the number of wicket gate
settings; and Nblade is the number of runner blade angle settings. For turbines with fixed
wicket gates and fixed blades, Ndy=1 and Nblade=1. An inversely operating pump would
fit this description and there is only one characteristic curve for this type of turbine. Also
for this type of turbine, an upstream control valve is usually necessary to govern inflow
into the turbine; the valve can be considered as either a turbine-attached-valve (see in a
later section) or a separate component in the system. In both cases, no governor is involved
in the turbine simulation.
Francis or ‘WG’. This type of turbines includes Francis turbines and propeller
turbines with fixed runner blades. The position of wicket gates can be adjusted but the
runner blades are fixed (i.e., Nblade=1). There are Ndy sets of characteristic curves
corresponding to different wicket gate openings (Ndy>1). The range and intervals of
wicket gate opening are dependent on the available model hill diagram. During turbine
operation, wicket gate setting is typically controlled by a speed-control-governor and
adjusted by a servomotor system. In this case, a governor equation has to be included in the
turbine simulation.
110
Kaplan Turbine. For a Kaplan turbine, both the wicket gate opening and runner
blade angle are adjustable (Ndy>1 and Nblade>1). A governor controls both the runner
blade angle and wicket-gate opening through a designed cam mechanism. In the current
simulation (adapted to TransAM) version, the Kaplan model has been considered but not
fully implemented.
5.2 Hydro Turbine Parameters
The terminologies commonly used in hydro engineering are reviewed in this section. They
include both dimensional and dimensionless turbine parameters.
1) Turbine head (H): the hydraulic head differential between the high- and low- pressure
reference sections of the turbine unit (in feet, meters, etc.). Hydraulic head is defined
as the energy available for per unit weight of water at a section, including the elevation,
water pressure and kinetic energies.
Net head is the hydraulic head differential between points just in front of, and behind,
the turbine unit (say, from the inlet of the spiral case to the outlet of draft tube).
The maximum and minimum turbine hydraulic head (Hmax and Hmin) establishes the
allowable head range for turbine operating conditions.
The average hydraulic head (Have) is an average of available head over the years. The
111
Design/Rated head (HR) is determined based on the range of available net head, under
which the rated turbine speed and runner size are selected such that the best overall
efficiency would be achieved over the operating range of turbine flow.
2) Turbine Discharge (Q): the flow rate through the turbine (m3 3/s, ft /s, etc.).
Design/Rated discharge (QR) is the turbine flow required to produce the rated power at
the design head.
3) Turbine Speed (n): the rotational speed of turbine unit, typically in rpm.
Rated speed or normal speed (nR) is the turbine rotation speed under normal operating
conditions; it is typically the speed of which the unit enjoys its best efficiency.
Synchronous speed (nE) is the speed of the generator when regulated by the frequency
of the power grid system to which the unit is connected, that is, pfnE 60= . Here, f
is the frequency of the a.c. supply system in Hz (50-60 Hz), the permitted variation of
frequency +/- 0.2 Hz for a big system, and +/- 0.5 Hz for a small system; p is the
number of pairs of generator poles. For a unit having a common shaft connecting the
turbine and generator directly, the synchronous speed is the rated speed of the turbine
and unit.
Runaway speed (nrun) is the highest speed associated with a particular position of
needle valve(s), wicket gates and/or runner blades, after all transient waves have been
112
dissipated, when the generator is disconnected from its load and network under a
specific turbine head. At steady state, for a particular head and opening position, a
turbine’s runaway speed can be tested and measured on the model. However, the
runaway speed during transient states will often surpass its steady counterpart because
of waterhammer effects. Runaway speed is not the same as the speed rise during the
closure of wicket gates (particularly for fast closures) when load is rejected.
4) Power (P): the rated energy produced from turbine or generator per second (kW, Hp).
Turbine output is the waterpower transferred to the generator or motor through the
main shaft. The rated turbine output depends on the rated flow, rated head and the
corresponding efficiency ; that is, Tη
1)-(5 81.9 RRTT HQP η=
The rated output of a turbine-generator unit is obtained using the efficiency of
generator multiplying with the rated turbine output to obtain Gη
2)-(5 81.9 RRTGTGG HQPP ηηη ==
5) Efficiency (η): the ratio of output to input power.
3)-(5 QHPT
T γη = Turbine efficiency:
ωTPT = ; T Here, γis the specific weight of water. Turbine output can be expressed as
113
30nπω =is the shaft torque exerted by the water; , the angular speed of the turbine.
The efficiency of a turbine-generator unit can be expressed as:
4)-(5 QHPG
GT γηηη ==
6) Specific Speed: Specific speed is defined differently for a turbine and a pump:
5b)-(5
5a)-(5 or
4/3
4/34/5
HQn
n
HQn
nH
Pnn
g
gs
=
==For turbines For pumps
Here, P is the turbine power at maximum efficiency; Q is the pump flow at maximum
efficiency. The specific speed n or ns g represents the speed of a series of geometrically
homologous turbines (or pumps) at H =1 m and P=1 kW (or flow Q=1 m3/s), which is
unrelated to the machine size. Specific speed is an important turbine parameter, largely
reflecting a turbine’s performance and typically indicative of turbine type or model.
7) Unit Parameters: Unit parameters are defined for standard-sized runner (i.e., throat
runner diameter D1=1 m) under unit net head (i.e., H =1 m), and used to describe
turbine model characteristics in hill diagrams. They are sometimes also referred as
specific parameters.
114
6)-(5 2
11 HD
QQ =′Unit discharge: Q or 11
7)-(5 1'1 H
nDn =Unit speed: n or 11
8)-(5 2/321
'1 HD
PP =Unit Power: P or 11
9)-(5 31
'1 HD
TT =Unit Torque: T or 11
8) Suter Parameters: Based on the following four dimensionless turbine parameters:
10)-(5 ,,,RRRR n
nTT
QQv
HHh ==== αβ
the independent variable x and two dependent variables W and WH B regarding turbine
head and torque (i.e., the Suter parameters) are expressed as follows:
B
13)-(5 )(
12)-(5 )(
11)-(5 tan
22
22
1
vxW
vhxW
vx
B
H
+=
+=
= −
αβ
α
α
The above definition of x is consistent with pump characteristics used in the simulation
115
model TransAM. The relationships between W ~x and WH B ~x are the turbine’s head
and torque characteristics in Suter parameters.
5.3 Hydro Turbine Characteristics
To simulate transient flow through turbines, the relationships between the net head,
discharge and/or other turbine parameters have to be specified. The flow rate through a
turbine depends upon various parameters. For an impulse turbine, the discharge through
the turbine Q is related to the net head H and the opening of nozzle-needle-valve y only, so
the relationship of its flow and head is as simple as a valve. However, for a Francis turbine,
the discharge of the turbine Q is related to the net head H, rotation speed n and wicket gate
opening y (the variable y represents the dimensionless opening of turbine in this thesis); the
flow through a Kaplan turbine also depends upon the runner blade angle ϕ. The curves
representing the relationships between these parameters are called turbine characteristics,
with combined model characteristic curves (i.e., hill diagram in unit parameters) defining
the relationships among various turbine parameters for a series of homologous turbines.
The prototype turbine characteristics, also known as turbine performance curves, are all
converted from model hill diagram. They include curves of efficiency vs. output (i.e., η –
P), output or efficiency vs. discharge vs. (i.e., P – Q or η-Q ), output or efficiency vs. head
(i.e., P–H or η – H), and discharge, output or efficiency vs. speed (i.e., Q–n, P–n or η –n).
116
Except for large and important units, turbine designers and manufacturers usually
do not conduct load rejection tests for model turbines (Standard laboratory for model tests
cannot do transient tests); that is, the turbine characteristics are usually obtained from
steady-state cavitation free model tests, from which the prototype turbine’s performance
curves would be plotted. However, an actual turbine cannot avoid experiencing hydraulic
transients, and the effect of waterhammer does influence the head–discharge relationship
and thus, to some extent, all other characteristic curves. Particularly, in no-load conditions,
the runaway speed at steady state cannot exceed a certain value at a particular net head and
gate opening. During a transient, the prototype speed at no-load condition would exceed
the steady-state runaway speed for a short duration. To account for this, the curves are
extrapolated for speed values beyond the steady-state runaway value by assuming they
follow the same trend observed for inferior speeds (Chaudhry, 1987).
In addition, because of the assumption of equal efficiencies in the homologous
relationship between the model and prototype, the prototype efficiency should be modified
after conversion of turbine parameters from model to prototype. For medium and large
turbine units, the prototype efficiency is usually higher than that determined by model tests
at the same operating point because of scale effects and the limitations of manufacturing
technologies for small-size models. Various empirical formulas for the normal operating
zone have been proposed for efficiency correction (IEC 193, 1965).
117
5.4 Model Hill Diagrams and Conversion
There are four quantities associated with turbo-machine characteristics: the pressure head
H, the discharge Q, the rotational speed n, and the shaft torque T (or efficiency η ). Two of
these may be considered independent. For instance, if Q and n are given, H and T (or η )
can be determined from the turbine characteristics. The shaft torque T and efficiency η can
be converted from each other in the range of the normal operating zone based on the shaft
power balance equation; that is, for a pump and for a turbine. ηγω QHT = ηγω QHT =
As aforementioned, the available turbine characteristics are usually expressed in
unit parameters and obtained from model tests. For simulation, the model relations have to
be converted for the prototype turbine. However, the prototype turbine’s relationships
between H, Q, n and T (or η ) are difficult to handle in numerical simulation because they
all may cross zero and change sign during a transient. Thus, the Suter parameters, WH and
WB, are used to describe a turbine’s characteristics in transient simulation; it is particularly
necessary for reversible pump-turbines and turbines operated in pump mode.
B
5.4.1 Francis Turbines
Figure 5.1 is an illustrative hill diagram for a typical Francis turbine, which is usually
furnished by the turbine designer or manufacturer. At each wicket gate setting (e.g., y = y1),
a series of values are read from the Figure; and then the following dimensionless
variables can be calculated with known rated turbine parameters:
'1
'1 , Qn
118
16)-(5
15)-(5
14)-(5 1
'1
21
21
'1
2'1
2221
2
2
2'1
21
2
RRR
RR
RR
R
R
R
R
RR
R
R
R
R
RR
vhhvT
HQTQH
TT
hQQ
HDHH
QHDQ
QQv
nHnD
nn
nDn
HHHh
αηηη
αωγ
ωηγβ
α
=⎟⎟⎠
⎞⎜⎜⎝
⎛===
⎟⎟⎠
⎞⎜⎜⎝
⎛===
⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛==
2According to the above equations (5-14) to (5-16), the dimensionless variables h, βand v
are proportional to α2 (i.e., ). Therefore, no matter what the relative
speed α is, the ratios of and are independent of α. For
convenience, α =1 (i.e., n=n
22 ,, αβαα ∝∝∝ vh
)/( 22 v+αβ)/( 22 vh +α
R) is assumed in the data conversion.
y2
y1
'1Q
η1n′
Figure 5.1 Illustrative hill diagram of Francis turbine
119
The steps to convert a hill diagram (Figure 5.1) to Suter parameter curves (Figure
5.2) are summarized as follows:
(1) For a given opening (e.g., y = y ), read one set of from the model hill diagram; '1
'1 , Qn1
22 vWB +
=α
β
vx α1tan −=
y=0 .2 .4 .6 .8 1.0 22 v
hWH +=α
vx α1tan −=
y=0
.2
.4
.6
.8
1.0
Figure 5.2 Francis turbine characteristic curves in Suter parameters
(2) With α =1, calculate h, ν, β from equations (5-14) to (5-16), respectively;
(3) Calculate x, WH and WB from equations (5-11) to (5-13), respectively;
~x and W ~x at opening y(4) Repeat the above steps (1)-(3), two curves of WH B 1 can be
plotted;
(5) Repeat the above steps (1) – (4) for each of wicket gate opening to plot a set of curves
of WH ~x and WB ~x at each wicket gate opening.
120
If the model characteristic curves are in terms of unit power instead of unit flow,
equations (5-15) and (5-16) would be replaced by following (5-15a) and (5-16a),
respectively; and the same procedure as above described would be followed to plot curves
of WH ~x and WB ~x at each wicket gate opening.
16a)-(5
15a)-(5 /
2/3'1
2/321
'1
2/321
'1
21
'1
2/321
αωωωωωβ
ηγγηηγ
hPT
HDT
PHDT
PTT
TT
hPQ
DHHQ
PHDQ
HPQQv
RR
R
RRRR
R
R
RRR
⎟⎟⎠
⎞⎜⎜⎝
⎛=====
⎟⎟⎠
⎞⎜⎜⎝
⎛====
In the case of prototype turbine characteristics, represented by the head H vs. Q
and turbine shaft torque T (or power P) vs. Q, the data conversion to Suter parameters can
be calculated directly from prototype variables, which is even more straightforward.
Interpolation and Extrapolation of Turbine Characteristics. Since the
characteristic curves in Suter parameters are usually built with intervals ∆x=3ºand ∆y
=0.1-0.2, it is necessary to interpolate the values of WH and WB at a specified instantaneous
operating point (x, y) for turbine simulation. Experience points to linear interpolation as the
most reliable method. In this study, the concept of shape function is implemented, which is
actually a two-step linear interpolation but more straightforward and conceptually clear.
B
121
x1 x2 x
y
WH
(WB)
y2
y1
A1 A2
A4A3
P (x, y)
Figure 5.3 Interpolation scheme of turbine characteristics
Assuming that Suter parameters WH(x, y) and WB(x, y) are related to independent
variables x and y as:
B
18)-(5 ),(
17)-(5 ),(
321022
321022
xybybxbbv
yxW
xyayaxaav
hyxW
B
H
+++=+
=
+++=+
=
αβ
α
, a , a , aHere, coordinates (x, y) represent turbine instant operating point (P); a0 1 2 3 and b , b0 1,
b , b are interpolation constants. To obtain the values of WH(x, y) and W2 3 B(x, y), the
interpolation constants need to be solved first. In fact, the Gaussian Elimination Method is
employed to find these interpolation constants based on four nearby known operating
points A
B
1, A , A and A2 3 4 (see in Figure 5.3). Specifically, the interpolation procedures are
described as follows:
122
1) Determine the instantaneous operation point P (x, y). The coordinate x can be
calculated according to the instantaneous turbine speed n and discharge Q, and the
coordinate y is the gate opening;
2) Find four nearest known points around P (x, y). From data tables of Suter parameter
characteristics, the points A (x , y (x1 1 1), A2 2, y (x1), A3 1, y (x , y) and A2 4 2 2) and their
corresponding known values of WH(1,1), WH(2,1), WH(1,2) and WH(2,2), and WB(1,1),
W
B
BB(2,1), WB(1,2) and WBB(2,2) can be determined; B
3) Substitute coordinates (x1, y ), (x , y1 2 1), (x1, y ) and (x , y ) and values of WH2 2 2 (1,1),
WH(2,1), WH(1,2) and WH(2,2) into equation (5-17) to obtain the appropriate equation
(5-19) and solve it for constants a , a , a and a3; 0 1 2 similarly, substitute the values of the
coordinates and WB at the four points into equation (5-18) to obtain the determinant
equation (5-20) and solve it for constants b
B
, b , b , and b ; 0 1 2 3
, a4) Finally, substitute the coordinates of instant operating point (x, y) and constants a0 1,
a and a2 3 or b , b0 1, b and b2 3 back into equation (5-17) or (5-18) to obtain the
instantaneous turbine head or torque.
123
19)-(5
)2,2(
)2,1(
)1,2(
)1,1(
22322210
21322110
12312210
11312110
⎪⎪⎪⎪
⎩
⎪⎪⎪⎪
⎨
⎧
=+++
=+++
=+++
=+++
H
H
H
H
Wyxayaxaa
Wyxayaxaa
Wyxayaxaa
Wyxayaxaa
20)-(5
)2,2(
)2,1(
)1,2(
)1,1(b
22322210
21322110
12312210
11312110
⎪⎪⎪⎪
⎩
⎪⎪⎪⎪
⎨
⎧
=+++
=+++
=+++
=+++
B
B
B
B
Wyxbybxbb
Wyxbybxbb
Wyxbybxbb
Wyxbybxb
When the operating point P(x, y) is beyond the range of the characteristic curves,
extrapolation is necessary to obtain the values of WH(x, y) and WB(x, y). B
5.4.2 Kaplan Turbines
Figure 5.4 shows the hill diagram of a Kaplan turbine, depicting the optimum relationship
between the runner blade position (blade angle ϕ) and wicket gate opening (a0). The
design of a Kaplan turbine provides a stepless control of both wicket gate opening and
blade position to achieve the best turbine efficiencies for a wide discharge range.
For a Kaplan turbine, using the same procedure as that for a Francis turbine, two
sets of curves WH ~x and WB ~x can be obtained for the series of wicket gate openings (y , 1
124
y y2, 3 ….) and for the series of runner blade angles (ϕ1, ϕ2, ϕ3 …), respectively. The values of
WH and WB for an instant operating point (x, y,ϕ) have to be interpolated or extrapolated by
two sets of curves, that is, WH (x, y), WB (x, y) and WH (x, ϕ), WB (x, ϕ).
Figure 5.4 Illustrative characteristics of a Kaplan turbine
(Revised based on the original source: Small hydropower series no. 4, p40 Fig.20)
5.4.3 Impulse Turbines
Figures 5.5 and 5.6 show the features of hill diagrams of Pelton and Turgon turbines.
Firstly, with variation of the efficiency increases or declines rapidly. Secondly, with
variation of the turbine discharge by means of the nozzle needle, the efficiency remains
high over a wide range of operating conditions. Thirdly, at constant nozzle opening, the
value remains constant irrespective of the value of .
'1n
'1Q '
1n
Since an Impluse turbine is simulated as a valve, there is no need to convert the hill
125
diagram, but the nozzle characteristics (the relationship between the flow and nozzle
opening) should be specified by the manufacturer. However, the turbine characteritics
must be used to simulate the speed rise and runaway condition at load rejections.
Q11
n11
a0 = const. η= const.
ηmax
Figure 5.5 Illustrative characteristics of a Pelton turbine
126
a0 = const. η= const.
n11
ηmax
Q11
Figure 5.6 Illustrative characteristics of a Turgo turbine with inclined nozzle
5.4.4 Pump Turbines
The discharge and torque characteristics of a typical pump turbine are shown in Figure 5.7
(a) and (b). There are different characteristic curves for various gate openings and they
overlap with each other. On these charts, it is possible to find more than one value of unit
flow or unit torque at a given gate opening and a given unit speed, as illustrated in Figure
5.7 (c). The multi-valued area (so-called S-characteristic in the runaway zones) results in a
difficulty in application of these curves to transient simulation. For reversible machines
with the low specific speed, such multi-valued nature is pronounced as the unit’s operation
passes through this region, alternately experiencing positive and negative discharges as the
wicket gates close after load rejection. To overcome the challenge, various representations
have been used in numerical modelling and each presents certain merits and limitations
(Chaudhry, 1987 and Pejovic, et al., 1983).
The head WH and torque WB characteristics are defined as single-valued in the
entire area covered by the polar angle (Suter, 1966; Thorley, et al., 1966):
B
)/(tan 1 vx α−=
22)-(5 )()(
21)-(5 )()(
22
22
vSIGNxW
vh
hSIGNxW
B
H
+=
+=
αβ
β
α
When using the Suter curves, difficulties arise from an uneven distribution of
127
curves at the wicket gate opening. At small openings, the curves space themselves out
further, and for a definite angle x when the wicket gates are being closed, the values of WH
and WB become indefinitely great. Consequently, the selection of interpolation and
extrapolation methods of these characteristics has at least some influence on the accuracy
of transient calculation results.
B
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-80 -60 -40 -20 0 20 40 60 80
Unit Speed n11 (rpm)
Uni
t Flo
w Q
11 (c
ms)
TAU=0.00
TAU=1.50
TAU=3.00
TAU=6.00
TAU=8.80
TAU=11.9
TAU=14.9
TAU=17.9
TAU=20.9
TAU=23.8
TAU=26.8
TAU=29.9
TAU=32.9
(a) Unit Discharge
Figure 5.7 Characteristics of a pump turbine (Con’d)
128
-2500
-2000
-1500
-1000
-500
0
500
1000
1500
2000
2500
-80 -60 -40 -20 0 20 40 60 80
Unit Speed n11 (rpm)
Uni
t Tor
que
T11
(N.m
)
TAU=0.00
TAU=1.50
TAU=3.00
TAU=6.00
TAU=8.80
TAU=11.9
TAU=14.9
TAU=17.9
TAU=20.9
TAU=23.8
TAU=26.8
TAU=29.9
TAU=32.9
(b) Unit Torque
Unit Speed n11
Unit D
ischarge Q11
(c) Multivalued characteristics
Figure 5.7 Illustrative characteristics of a pump turbine
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5.5 Water Conveyance System Layout in Hydro Systems
5.5.1. Conventional hydropower system layout
In Figure 5.8, several water supply modes in conventional hydropower stations are
sketched; the “+” sign represents the necessary valve or gate; “×” indicates that the valve
may or may not be there depending on the situation and design. Mode (a) with separated
penstocks is suitable for the cases with short distance between the reservoirs/surge tank
and powerhouse. Modes (b) and (c) with combined water-supply pipes are economical for
long penstock cases, but the risk of hydraulic transients in the common water supply
pipeline could be severe in the case where all units simultaneously reject their power loads
and it is necessary to install a shut-off valve upstream of each turbine unit.
In this study, each turbine unit (not a hydropower station) is simulated as an
individual boundary condition (coded as BCTYPE=’TURB’), although several turbine
units are often arranged in parallel in a hydro station (as shown in Figure 5.8). This
numerical treatment is adopted for three reasons: firstly, each turbine unit might be
operated under different conditions or regulated in different ways; secondly, the pipeline
from the upstream junction to the turbine entrance can be long and pipes’ properties often
change along their route, and thus they usually cannot be neglected; finally, it is more
convenient in numerical modeling for each individual turbine unit than for a group of units.
In many conventional hydropower projects, the downstream conduit system
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comprises a short draft tube, followed by a free-surface flow tunnel or an open channel. In
this case, the downstream conduit is typically negligible for transient analysis, and the
turbine unit is called a “one-node” turbine boundary condition (i.e., coded as a nodal
dependence number NDN=1). However, the pressurized tunnel/pipeline, even with a surge
tank, is common for those underground powerhouses and for turbine installations within
multi-use systems where the discharge water from the turbines may be directed to urban
utilities, industry users or irrigation systems.
Theoretically, the developed turbine model is applicable to any water system, no
matter how complex the system is, because the numerical models of almost all other
hydraulic devices have been previously developed in the simulation program TransAM.
For instance, the forebay or reservoir is a Constant Head Reservoir; the surge tank may be
a Linear Reservoir or a Tank With Overflow or an Air Chamber; and the valve distant from
the turbine can be a valve-in-line.
131
T
Reservoir
/ Forebay
T
T
T
A B
(a) Separated Water-supply Pipes
(b) Grouped Water-supply Pipes
(c) Combined Water-supply Pipes
Figure 5.8 Water-supply modes in conventional hydropower stations
Surge Tank
T
T
T
T
A B
Surge Tank T
T
T
A BT
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5.5.2 Energy recovery turbines in municipal water supply system
Unlike in a conventional hydropower system, there are three major features governing the
layout of energy recovery turbine units in municipal water supply systems: 1) there is
usually a long pipeline and pipe loops/networks downstream of the turbine, so transient
analysis should be carried out both upstream and downstream; 2) topographic variation is
often more significant along the pipeline, so the highest point should be closely scrutinized
to prevent vacuum pressure and cavitation; and 3) many different hydraulic devices in the
system (such as pumps, hydrants, etc.) could interact with the hydro turbine units.
5.6 Governor and Control System
The power load of a turbine-generator unit varies constantly with demand fluctuations and
the turbine governor and control system is designed to equate the dynamic power produced
by the water to the power required by the electrical system in order to ensure a stable power
supply frequency (either a large or isolated network). Also, it governs the turbine when the
unit starts up, is brought on line (synchronized), and shuts down normally or in an
emergency.
The governor control system for hydro turbines is basically a feedback regulation
system which senses the speed and power of the generating unit, or the water level of the
forebay of the hydroelectric installation, and takes a particular action for operating the
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discharge/load controlling devices in accordance with the deviation of the actual control
variable from the reference/set point. An isolated turbine unit should use a feedback
control system to adjust its speed and power output. For the small and grid-connected units,
water level controllers may be used.
The governing system comprises of the control section and the mechanical
hydraulic actuation section. The control section may be mechanical (PI type controller),
analogue electronic or digital (PID type controller). The present trend is toward use of
digital governing control systems in hydroelectric units. The advantages of a
microprocessor-based system over the earlier analogue governors (based on solid state
electronic circuitry) include higher reliability, self diagnostic features, modular design,
flexibility of changing control functions via software, stability of set parameters, reduced
wiring and easy remote control through optical fiber cables.
The turbine control actuator can be hydraulic, mechanical (motor) or a load
actuator. A load actuator is used in micro hydro range (through adjusting the shunt/bypass
of the load bank), while mechanical (motor operated) actuators may be used for units up to
1000 kW in size. Overall, hydraulically controlled actuators (pressure oil system with oil
servomotor) are most commonly used. The actuator system compares the desired actuator
position with its instant position. In most hydroelectric units it requires positioning of the
wicket gates in reaction turbines, runner blades in Kaplan turbines and needles in Pelton
turbines (2008 MNREG, India).
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5.6.1 Three Levels of Turbine Flow Control
The output of turbine power is controlled by adjustment of turbine flow discharge; there
are three levels of turbine flow control.
1) Speed-control governor
A speed-control governor is the most popular control device. During normal operation, it
continually adjusts the gate opening (and/or runner blade angle) according to the variations
of power load exerting on the turbine shaft to maintain the unit speed at the
rated/synchronous speed. In addition, it operates to ensure a smooth shutdown or start-up
of the turbine unit.
Large hydroelectric installations are usually equipped with hydraulic-mechanical
or hydraulic-electric governors. Depending upon the criteria for the corrective action, the
hydraulic governors may be classified into three types: a) temporary-droop governors; b)
accelerometric governors; and c) proportional-integral-derivate (PID) governors. In a
temporary-droop governor, the corrective action of the governor is proportional to the
speed deviation (α =N/NR) and its integral with time. In an accelerometric governor, the
speed change is proportional to the derivative of speed deviation ( dtdα ). And PID
governor is the sum of actions proportional to the speed deviation (α), its derivative
( dtdα ), and the integral of speed deviation with time. The PID was introduced in the
early seventies and is now being used extensively.
135
2) Non-speed-control governor
Small hydroelectric installations generally have little effect on the frequency of the power
grid and thus can have governors without speed regulation for cost savings.
Non-speed-control governors may be either hydraulically or electrically-operated to
control the gate opening (and turbine discharge) based on measuring forebay water levels.
Their function is to bring the turbine to near synchronous speed for start-up, to regulate
load after synchronous speed has been achieved and to shut down the unit during both
normal and emergency conditions.
3) Inlet control valve
In cases where load regulation is not required (with small or micro capacity sizes), the
governor is not needed but a main control valve or shut-off valve must be installed and able
to shut down the unit under normal and emergency conditions.
In a large turbine system, the main control valve is also equipped with a governor
(dual control system). The control valve could be a butterfly, a deadweight butterfly, a gate,
ball, or rotary valve. It is installed immediately upstream of the turbine entrance to shut off
the flow when the turbine water passage is under inspection, overhaul or standing idle. In
addition, it is also used to shut off the water flow in the emergency of wicket gate failure
during load rejection in order to protect the turbine-generator unit from a protracted
runaway speed condition.
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5.6.2 Mechanism of a Speed-Control-Governor
With respect to the mechanism of a speed-controller, two basic concepts require
clarification: (1) the turbine speed change is controlled only by the net torque acting on the
shaft of turbine-generator; and (2) the speed change, acting through the governor, controls
the main servomotor. The main servomotor is the governing system device that adjusts the
energy input the turbine. This mechanism might be a needle valve, wicket gates, runner
blades, or a combination of them and other elements.
In a turbine unit, if friction is negligible, we have the following torque balance
equation (speed change equation):
23)-(5 gTTdtdI −=ω
2Here, I =WRg , is the polar moment of inertia of rotating fluid and mechanical parts in the
turbine-generator unit, in which W is the weight of turbine-generator rotating parts and
fluid contained and Rg is the gyration radius; ω is the angular speed of the unit
( 60/2 nπω = where n is in rpm); T is the torque produced by water flowing through the
unit; T is the resistant torque from the generator. g
With a rise in the electrical load, Tg increases and the unit slows down, the
governor would activate the gate opening to augment the flow discharge, allowing the
hydraulic torque T to increase until a new torque equilibrium is achieved. The reverse logic
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holds when electric load (and T ) decreases. g
The dynamic characteristics of a governing system represent the variation of
adjustable parameter (e.g., the speed n) with time during the governing process. The
stability of a governing system is essential; besides, the good quality of governing system
also requires minimizing the amplitude and duration of the deviation of turbine speed from
the rated speed within the smallest number of oscillations.
The static characteristics of a governing system indicate the relationship between
the adjustable parameters, usually the speed n, and the output power Pg during steady state.
There are two types of static characteristics. One is the no-droop governing system, in
which the speed of turbine is unrelated to the load Pg; no matter the load value, the speed
keeps at the rated value nR. The second governing system relationship allows for a small
difference between the new steady state and the old one, i.e., there is a permanent speed
droop σ. Usually σ is between 0 to 0.08, which is a relative value and defined as
Rnnn /)( minmax −=σ , nmax is the speed at no-load condition, nmin is the speed at rated load
condition, and nR is the rated speed. Most units in the network adopt a drooped governing
system because it facilitates a fixed load; those units that provide the network as a whole
with frequency control use a no-droop governing system.
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5.7 Fundamental Knowledge of Synchronous Turbine-Generator Units
Synchronous units (on-line units) mean that the turbine-generators feed their output to a
large electricity grid. In the modern hydro industry, most turbine units are linked to
electricity networks. Therefore, it is necessary to understand the fundamentals and basic
features of synchronous units.
5.7.1 Load Distribution and Unit Operation with a Governor
In a large interconnected system, there are usually many tie lines between power stations,
and each power station would adjust its output to cover local load and tie line loads, which
can be explained as the load borrowed from somewhere else. The tie lines do not start and
end at generating stations, thus it is difficult to control the amount of load for each tie line.
The frequency of the power system must remain stable to ensure the quality of
power supply. The question is how to operate a unit at a plant to maintain this master
frequency. “Frequency Signaling” is used to operate the plant or the unit. A clock
reflecting the speed of the unit at the plant is compared with a master clock reflecting the
system frequency and the difference shows whether the unit’s speed is too high or too low.
The output of the plant/unit is then changed according to the given frequency error.
The governor usually enables straightforward synchronization, particularly for
the units that carry peak loads. The speed-drop characteristic is also desirable to keep
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several turbines working in parallel; otherwise the load carried by each unit cannot be fixed
when the new balance is being reached while the system experiences load changes. In fact,
the permanent speed drop causes a slight speed alteration as the load changes; however,
this small speed-drop could be re-adjusted after the load variation, which inevitably
requires expensive electrical equipment. However, the final allowable master frequency
fluctuation is only 0.1-0.4%, and thus can be neglected for hydraulic transient analyses.
Therefore, the assumption of synchronous speed for on-line turbine units is reasonable and
the torque balance (or speed change) equation is invalid for the units connected to a large
power network.
For unit speed fluctuations, the governor would adjust the wicket gate opening
slightly and accordingly to maintain the constant master frequency, while, for a larger load
variation of on-line units, the adjustment involves both the governor function and load
re-distribution among units in the power system. Load re-distribution is planned by the
dispatcher who determines the proportion of total load to be carried by each individual
plant or unit. A massive unplanned load change (e.g., a large power generation plant is
accidentally shut down) may cause too low a frequency of the local power network and
thus automatically shut down other generating units connected to the network as a
protective measure. This action would cascade down to the neighboring and connected
power networks (because the local load demand is transferred to the neighboring power
140
networks) if there were no isolation measures taken among the interconnected networks, a
unfolding of events experienced during the August 2003 North American Blackout.
Close frequency control and proper load distribution among units and plants are
requirements which cannot be simultaneously met by a governor because they are
contradictory. It is therefore practical to maintain that feature of the governor which
assures the best distribution of the load and realizes as constant a frequency as possible via
auxiliary corrective means that can be applied gradually enough so as not to offset the
already established proper load distribution. This indicates that for a significant load
change of a particular unit, it is the only way that the turbine gate is opened/closed at a
prescribed rate or by a time function to the position at which turbine output will be equal to
the required final output. In other words, the gate opening cannot be adjusted by the
governor for a required significant load change (Chaudhry, 1987).
In this thesis, the above recommendations are followed when the unit is connected
to a large electricity network. However, further discussion from the literature (Zaruba,
1993) is summarized below:
a) During the time when the generator is connected to a power network, the moment of
inertia considered in the speed change equation is not constant and is defined by the
relation Ic = I + fI (t). Here, I is the constant inertia of the turbine unit, and the
function fI (t) represents the effect of the power network.
141
b) The resistant torque Tg of the generator may be dependent on the turbine speed, but
also on the connected power network. The effect of the network may change with time.
The resistant torque employed in the speed change equation is defined by the relation,
T = (n-ng E) f (t) + fn c(t). The functions f (t) and fn c(t) represent the effect of the
connected power network (the effect of self-regulation and the moment of loading),
(n-nE) is the speed deviation from the synchronous speed.
c) With above two corrections, the control regime of the turbine blades angle and
wicket-gate opening is not set in advance but is defined by the function of turbine
speed governor, i.e. the governor equation.
5.7.2 Startup and Load Acceptance
To start up a unit, the wicket gates are first turned on to an opening at which the generated
power is sufficient to offset static friction. This gate opening is maintained until the unit
speed reaches about 60% of the rated speed; then the wicket gates are closed to the position
of speed-no-load and during a short period of time the unit is running at the synchronous
speed and ready for load acceptance. Since no significant flow change occurs during the
synchronizing process (unit start-up), pressure surges are only concerned during load
acceptance; that is, when the turbine opens from the speed-no-load condition
(corresponding to zero load at speed-no-load opening and synchronous speed) to the
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normal operation condition (corresponding to the target load at final opening and
synchronous speed).
5.7.3 Shutdown and Load Rejection
There are two types of load relief responses. One is associated with the normal shutdown
of the unit and the other is load rejection. Load rejection is a sudden unexpected load
cut-off, which is a frequent occurrence in hydro units that results from the power failure,
device malfunction or some other accident; load rejection actually serves as a kind of
protection for the system and units.
In normal unit shutdown, the gates are first closed to a gate opening
corresponding to speed-no-load condition. In this nil efficiency operating condition, the
turbine is still connected to the line and running at its synchronous speed; from this
opening, the gates are slowly closed to zero and the unit is taken out of grid/service. The
speed-no-load opening can be found from turbine characteristic curves (zero-efficiency or
zero net torque point), and it varies from 2-3% at high head plants to up to 35-40% at low
head propeller wheels.
However, when load rejection occurs, the unit is instantly disconnected from the
system. Then, the unit has to be quickly closed to prevent excessive runaway condition,
which may induce severe draft tube pressure drops.
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5.7.4 Speed-No-Load (SNL) and Runaway Condition
The speed-no-load (SNL) gate is the lowest gate opening at which the turbine maintains
synchronous speed and carries no (zero) load. In this condition, the turbine power is just
sufficient to overcome the turbogenerator windage and friction losses at synchronous
speed, so the net output is zero. The value of SNL opening varies with the net head, since
lower head needs greater flow (greater gate opening) to offset the losses. SNL gate opening
with corresponding flow and head (zero-efficiency or zero net torque point) are usually
provided by turbine performance curves.
To keep the unit running at synchronous speed when the gate opening is less than
the SNL opening, an outside power source must be supplied, a procedure known as
motoring the unit.
When a load rejection occurs, the wicket gates are supposed to be closed quickly
to prevent extended periods of high over speed. However, if the closure fails, such as when
there is a malfunction of the governor, the imbalanced torque induces an acceleration of
the unit shaft. With the turbine speed growing, friction loss escalates until finally the
available hydraulic power is fully dissipated by friction losses in the machinery; the net
output of the turbine is zero, and the acceleration is reduced to zero with the speed reaching
its maximum value. As aforementioned, this is defined as runaway condition and the
maximum rotational speed is called runaway speed.
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The runaway speed range varies according to unit type. Typical runaway speeds
for Kaplan turbines are 2.5 to 3 times normal speed, whereas with Impulse and Francis
turbines they are rarely over twice normal. Runaway speed values are measured during the
steady-state model tests of the turbine. In operation, a unit cannot exceed the runaway
speed for a particular net head and gate opening. Turbo-generating machinery must be
designed to be strong enough to withstand runaway speed, but with a highly reduced safety
factor. However, as mentioned in Section 5.3, the prototype speed may exceed the
steady-state runaway speed for a short duration in transient states since the instantaneous
head is governed by waterhammer effects. Therefore, the turbine steady state characteristic
curves have to be extended to obtain the transient runaway speed. The same concepts and
observations about runaway condition hold for both isolated turbines and the turbines
originally interconnected.
5.8 Features of Transients Caused by Governed Turbines
5.8.1 Key Features of Turbine Transient Analysis
Compared to pumps, the modeling of a governed turbine includes two new features that
further complicate the problem: first, the governor produces a change in wicket-gate
position or runner blade angle in response to a speed change, therefore an equation is
needed to describe the dynamic response of the governor. Second, turbine characteristic
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curves are different for every gate position and, thus, additional characteristic curve data
must be available for the numerical solution. For a pump, only one curve is needed for each
of the head and torque characteristics, respectively (a pump with variable characteristics –
other than pump speed – is highly exceptional); for a turbine, two families of curves
represent H and T characteristics with different gate positions have to be provided.
5.8.2 Speed Variation
A change in turbine operating condition results in transient flow in the turbine and in the
associated hydraulic system. This change may be due to a load adjustment, a unit start-up,
or a unit stoppage, either normally or by accident. Among other things, these adjustments
cause an instantaneous imbalance between the power absorbed by the generator and that
produced by the water; the resulting shaft force changes the rotation speed of a
disconnected unit. For reaction types of turbines, such as Kaplan and Francis turbines, the
turbine speed exerts a considerable influence on the turbine flow, and the speed changes
should therefore be taken into consideration in transient analysis. However, the speed
variation is small and can be neglected if a unit keeps on line and the power of this unit is
only a small portion of network power, since the speed has to correspond to the
synchronous speed whenever its load increases or reduces.
When we calculate the waterhammer caused by load rejection or load variation,
the speed variation must be taken into account for the following two cases:
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(a) The unit of turbine-generator is operating stand-alone or in a small power system;
whether for the case of load rejection or load variation, the speed change has an influence
on the discharge and efficiency for most turbines types. The selection of criteria to
determine whether a power system is large or small is somewhat complicated. In general,
any unit in a system that would be supplying 40% or more of the total capacity should be
designed as an isolated unit. The relationship between system capacity and load change
should also be compared. Any load change of 10% or more of the system capacity should
be analyzed to determine its effect on frequency (Gordon, 1961).
(b) If the unit is initially on-line (connected to a large network), when the load is rejected
it would be disconnected from the power network immediately and thus become isolated.
The maximum allowable speed rise is approximately 45-65%. During load
rejection, reducing the duration of regulation (the closure time of wicket gates or control
valves) can decrease the speed rise but will bring about an excess waterhammer (pressure
rise), so that the reasonable closing time becomes a compromise between the pressure rise
and speed control, which is called a Guaranteed Regulation Calculation.
Is it necessary to calculate the speed change of an impulse turbine for the purpose
of waterhammer analysis? The answer is no, since the speed change has no influence on
the turbine discharge. But for the purpose of speed control, the torque equation is suitable
for any type of turbine, though naturally each turbine has its own characteristics curves.
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5.9 Summary
In this chapter, the classification of turbines, their key parameters, and the characteristics
of some typical turbines are reviewed. Following this, hill diagram conversions for Francis
turbine are introduced. Then, the layouts of a water conveyance system for conventional
electricity dedicated hydropower and energy recovery turbine systems are briefly
discussed. And the hydraulic control devices and their working mechanisms are
summarized. Finally, the basics of synchronous units and different scenarios of transient
simulations, as well as the features of turbine transient analysis, are introduced.
The focus of this overall review is to establish the relevance of the system’s
characteristics to the transient numerical modeling, which is essential for an understanding
of the numerical model of hydro turbine and transient analysis in the next chapter.
The challenges for hydro turbine modeling arise from several factors: (1) turbine
operation interacts with hydraulic, mechanical and electrical systems; (2) turbine
characteristics have more interdependent parameters, such as the turbine head, flow, speed,
efficiency, wicket-gate opening and even the blade angle for Kaplan turbines; and (3) in
some parts of the water passage, such as, in the runner and draft tube, the flow is fully
three-dimensional and might be mixed with air. Despite above challenges, reasonable
accuracy in prediction of transient peak values is required to prevent waterhammer
problems and failures.
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Chapter 6 Hydro Turbine Modeling and Model
Verification
This chapter is another core contribution of the thesis. It develops a numerical model and
computer program for transient analysis of hydro turbine units, which is incorporated
into TransAM, a transient analysis model specifically designed for complex networks. To
date, few commercially available transient analysis models for the turbine installation
allow for the existence of a complex network. In this model, the turbine units could be
either isolated or interconnected; and with or without attached valves. The simulation
results are compared a published example (Wylie, et al., 1993) and also verified by a set
of field data measured during the test of large hydro plant. The developed model has
become a useful tool to evaluate transient performance and to find effective measures to
transient mediation and control in the systems with installation of turbine units (as shown
in case studies of Chapters 7 and 8).
6.1 Mathematical Model of Turbine Boundary Condition
A general mathematical model for active hydraulic devices has been developed in
Chapter 3. However, hydro turbine units have their own features and variables and thus
their governing equations are developed in this section.
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6.1.1 Head Balance Equation without Turbine-Attached Valves
For the configuration of turbine system as shown in Figure 6.1, the turbine upstream
node is often set at spiral case inlet and the downstream node at draft tube exit. There is
an energy balance (or head balance) equation between these two nodes:
1)-(6 0)2
()2
( 22
22
221
21
11 =−+−+= HgA
QHgAQHF
Here, H is the difference of hydraulic head (i.e., the energy per unit of weight of water)
between the turbine upstream and downstream nodes, the so-called turbine net head. H1
and H2 are instantaneous piezometric heads, including the nodal elevation and pressure
heads. To account for the kinetic energy head, the nodal external flow (Q1 or Q2) and
pipe cross-sectional area (A1 or A2) at upstream and downstream node, respectively, are
used to calculate the velocity head at the nodes.
H
U/S Node
Pipe
Q
T D/S Node
Pipe
Pipe
Q1 Q2Pipe
Figure 6.1 Schematic of turbine boundary condition
150
As described in Chapter 3, taking the turbine upstream and downstream nodes as
the junctions in a pipe network, and H1 and H2 can be calculated from the extended MOC
equation, that is,
QBC H
QBC H
3)-(6 )(
2)-(6
2'2
'22
1'1
'11
−−=
−=
Here, the constants at the turbine upstream node and the constants
at the downstream node can be calculated by equation (3-16). It has been
noted that the sign of external flow Q
11 and CB ′′
22 and CB ′′
1 is positive and of Q2 is negative, and their values
are equivalent if water losses are negligible at the turbine boundary condition. With the
turbine flow Q, we have the flow continuity equations at nodes A and B:
4)-(6 21 QQQ ==
This relation explicitly excludes any small capacitance or water storage within the
turbine, which is usually an excellent approximation except when considerable air is
present. If air or other capacitance effects are desired to be included, minor medication of
the solution procedure can be undertaken.
The draft tube that connects the turbine to the downstream reservoir can
sometimes be very short (e.g., straight cone draft tube); indeed the tailrace could be
non-pressurized, so the transient effect in water passage of turbine downstream would be
151
negligible. In this case, the water level in tailrace channel (or downstream reservoir),
HTail, is directly used in head balance equation (6-1), which results in:
1a)-(6 0)2
()2
( 22
22
21
21
11 =−+−+= HgAQH
gAQHF Tail
This configuration of turbine connection to system is called one-node-turbine boundary
condition, since the turbine unit is associated with only one node at the end of system
pipeline, and the variable NDN=1 in TransAM program. A NDN value of 2, for the
two-node-turbine boundary conditions, is typical in multi-use hydro systems, energy
recovery urban water systems, or many conventional hydropower plants especially with
underground powerhouses.
Based on equations (6-1) to (6-4), it is straightforward to obtain a general form
of the head balance equation, with respect to the turbine flow Q and net head H:
5)-(6 0),(1 =HQF
Furthermore, substituting the turbine flow vQQ R= and head characteristic
relationship into this equation, a general form of
head balance equation in dimensionless variables α, v and y can be obtained:
))(( 321022 xyayaxaavHH R ++++= α
6)-(6 0),,(1 =yvF α
The above equation is referred to the first basic/governing equation of the turbine
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boundary condition. In this equation, both the turbine gate opening y and the speed α
become explicit unknown variables though they are implicit in equation (6-5). Moreover,
the turbine characteristics have been implied in this equation.
6.1.2 Head Balance Equation with Turbine-Attached Valves
In hydro turbine systems, a main control valve (i.e., a shut-off or control valve) is often
mounted upstream of the turbine, either inside the powerhouse or immediately outside it.
The pipe between the valve and spiral-case entrance is typically only 2-3 m long, which
can create numerical challenges. To avoid too short a pipe section, and thus too small a
time step, it might be more appropriate to combine the control valve within the turbine
boundary condition rather than treating it as an individual hydraulic device.
In some reactive turbine installations, a Surge/Pressure Relief Valve (SRV or
PRV) is used to mediate waterhammer pressures during load rejections, together with, or
instead of, a surge tank. Turbine SRVs are also regulated by the turbine governor; their
function and working mechanism is discussed in Section 2.3 of Chapter 2. This type of
SRV could be combined with turbine boundary condition in numerical modeling.
These turbine-combined valves are also referred to “attached valves” in this
thesis. They occur more often for the small/micro hydro turbines in urban water systems
than for the large conventional hydropower plants. One reason is that a dedicated surge
tank is often economically inefficient for small/micro turbines in urban water systems;
153
moreover, the connection pipes between the valve and turbine are often relatively short
for the urban long pipeline systems.
In the current code, a maximum of three attached valves can be combined within
each turbine unit, as shown in Figure 6.2, including an upstream main control valve
(TvalTYP=’TCV’), a bypass valve (TvalTYP=’Bypass’), and a surge relief valve
(TvalTYP =’Surge’). There is little difference between the ‘Bypass’ valve and ‘Surge’
valve in modeling, since they both have multiple settings for different parameters.
The “degree” of a node is defined as the number of pipes connected to the
node/junction. Each hydraulic device (i.e., a boundary condition in pipe system) is
considered to be associated with one or two nodes; a zero degree of node (an unattached
device) is not permitted. Therefore, depending on the distance from the turbine, the main
control valve can be taken as either a ‘valve-in-line’ or a turbine-attached valve. As a
separate b/c –‘valve-in-line’, there has to be a connection pipe between the valve and the
turbine, so its simulation has already been completed in TransAM. As for the solution
of “TCV”, the valve head loss has to be accounted for in the head balance equation. In
addition, the nodal continuity equations can be used in the flow relationship between Q,
QSV, QBV and QCV (see Figure 6.2) when the surge valve and bypass valve are present in
the system. The detailed description of turbine attached valves can be found in Appendix
Ⅲ.
154
If an attached turbine main control valve is included, the additional head loss HC
across the ‘TCV’ has to be taken into account in the head balance equations. Thus
equation (6-1) turns to (6-7), and (6-1a) turns to (6-7a):
7a)-(6 0)2
()2
( 22
22
21
21
11 =−−+−+= CTail HHgAQ
HgAQ
HF
7)-(6 0)2
()2
( 22
22
221
21
11 =−−+−+= CHHgA
QHgAQHF
The continuity equation (with the usual assumptions) is applied in the scheme of Figure
6.2, and we have:
8)-(6 21 QQQQQQ BVSVCV =++==
11)-(6
10)-(6
9)-(6
HEQ
HEQ
HEQ
BBBV
SSSV
CCCCV
τ
τ
τ
=
=
=
As indicated in Figure 6.2, HC represents the head loss across the control valve ‘TCV’.
The subscripts C, S, B in equation from (6-8) to (6-11) indicate different valve types,
such as, QCV is the flow through the control valve, QSV is the flow through surge valve,
and QBV is the flow through bypass valve. As usual, τ represents the relative valve
opening, E represents valve resistance coefficient. For bypass valve and surge valve,
valve opening and coefficient can be set flexibly; for example, setting EB = 0 indicates
155
there is no bypass valve present; setting 0=Bτ if the bypass valve is present but fully
closed. However, a logical variable is necessary to identify whether ‘TCV’ is present or
not, and then to decide which head balance equation to use in the model.
HGL
H1
H2
HC
H
Bypass Valve
Surge Valve
QSV
QBV
‘TCV’
QCVQ
U/S Node Q1 Q2 D/S Node
Figure 6.2 Scheme of the turbine attached valves
Substituting equation (6-9) to (6-11) into equation (6-8), the headloss across the
control valve can be derived:
156
12)-(6 )(2
⎥⎦
⎤⎢⎣
⎡ ++=
CC
BBSSRC E
HEEvQHτ
ττ
Further substituting equation (6-8) to (6-12) into the head balance equation (6-7) or
(6-7a), the general form of turbine Q and H relationship, F1 (Q, H) = 0, can be derived as
well for the turbine boundary condition with attached valves. Therefore, the head balance
equation, as the first governing equation, always implies a relationship between turbine
flow Q and head H no matter ‘TCV’ present or not, no matter with or without bypass
valves.
6.1.3 Speed Change Equation (Torque Balance Equation)
The torque balance equation has been described in Section 5.6.2 of Chapter 5.
Multiplyingω at both sides of equation (5-23), we have:
13)-(6 dtdIPT gωωω =−
In above equation, Pg is the power absorbed by the generator, reflecting the variable
power demand. During normal operation, the variation of Pg with time is typically
planned or can be predicted in advance. Divided by the product of RRT ω at rated
operation condition, equation (6-13) can be further deduced to
14)-(6 RRRRR
g
RR dtd
TI
TP
TT R ω
ωω
ωωω ω
ω
⋅=−
157
Using dimensionless turbine performance parameters βα , as defined in Chapter 5, the
equation (6-14) can be further transformed to:
15)-(6 dtdT
PP
mR
g ααβα =−
or
16)-(6 α
βα
R
gm P
PdtdT −=
Here, RRRRRm TITIT ωωω == 2 is the mechanical starting time. Integrating (6-16) over
∆t, the second basic equation for a turbine boundary condition can be derived:
( ) 17)-(6 01 0 =−Δ
−− ααα
βt
TPP
mg
R
where represents the value of dimensionless turbine speed at the previous time-step;
β = (α
0α
2 + v2) (b0 + b1 x+ b2 y + b3 x y) represents the torque characteristics of the turbine,
so equation (6-17) can be expressed in dimensionless variables using the general form of
18)-(6 0),,(2 =yvF α .
The speed-change equation is invalid for interconnected units, since in this
case the inertia of associated power network has to be taken into account. Instead, the
constant synchronous speed can be assumed for the interconnected turbine units.
158
6.1.4 Governor Equation
As introduced in Chapter 5, the function of a governor is at all times to supply the turbine
with a discharge which, multiplied by instantaneous net head and the instantaneous
efficiencies of turbine and generator, would match the power demand on the generator
shaft. Specifically, the objectives of governing are to minimize speed variations caused
by load changes and to provide adequate stability and a quick return to normal speed.
When the load changes, the unit will tend to speed up or slow down. The governor has no
way of knowing of load change before hand, but it detects the speed changes and acts to
compensate for the speed change by opening or closing the turbine gates. The action of
the governor is always a correcting action; there must be a deviation from normal speed
(a speed error) detected by governor to produce correcting action to compensate for this
error.
The equation of a PID speed-control governor can be derived (Wylie, et al.,
1993):
( ) ( ) 19)-(6 011'2
2
=+−+−++dtdTy
dtdyT
dtydTT dd
αασαα
Where y = dimensionless turbine gate opening;α = dimensionless turbine speed as
defined in Chapter 5; Td = dashpot time constant; Tα = promptitude time constant without
temporary speed droop; , promptitude time constant with temporary dTTT ' δαα +=
159
speed droop; δ = temporary speed droop; σ = permanent speed droop.
Generally, , the above equation reduces to dTT <<α
20)-(6 0)1()1( =+−+−+dtdTy
dtdyT dd
αασδ
Let dy/dt=Z, the rate of the movement of turbine gate opening. To discrete over Δt, we
have:
21)-(6 0
tyy
ZΔ−
=
22)-(6
000
2
2
tZtyy
tZZ
dtdZ
dtyd
Δ⎟⎠⎞
⎜⎝⎛ −
Δ−
=Δ−
==
Substituting the above two equations into the governor equation (6-20) and integrating
over Δt, we have:
( ) ( ) ( ) ( ) 23)-(6 011 1 00
'
00 =−+−
Δ+−
Δ+−+⎟
⎠
⎞⎜⎝
⎛ −Δ−
ααασ
αααα
α
TTTty
TTtyy
TTT
Ztyy
ddd
Here, subscript ‘0’ represents the values at the previous time step; and the dimensionless
opening y represents the main servomotor position that may control a needle valve,
wicket gates, runner blades or a combination of these or other elements. For a Kaplan
turbine, both the wicket gate opening and the runner blade angle have been obtained once
y is solved if the cam relationship between runner blade servo stroke and wicket gate
160
servo stroke has been designed at the specific net head; otherwise, the runner blade angle
at the instant operating condition has to be found from the set of turbine characteristic
curves of WH (x, ϕ) and WB (x, ϕ).
Equation (6-23) applies for mechanical, electric or digital governing system, as
long as the coefficients in the equation are properly identified. For different type of
governor, the governor equation may changes. Yet, the third basic equation for a turbine
boundary condition regarding its governor can always be generalized as
24)-(6 0),(3 =yF α
6.2 Combination of Basic Equations for Different Scenarios
The head balance equation, the speed change equation and the governor equation
discussed above are the three basic equations, which govern a turbine boundary condition.
They are applied in different combinations according to the specific problem and
operation conditions. Several common scenarios and the corresponding equation sets are
summarized as follows.
1) Isolated Units
Isolated unit refers to the unit operating stand-alone or supplying its power generation to
a small isolated power system. A speed control governor is essential for an isolated unit.
161
Head balance equation F1 (α, v, y)=0, speed change equation F2 (α, v, y)=0, and governor
equation F3 (α, y)=0 constitute a set of governing equations for transient analysis of an
isolated turbine unit. Therefore, three unknowns gate opening y, turbine speedα and
turbine flow v can be obtained.
For isolated units without governors (usually only run-of–river installations with
small capacity), the servomotor stroke of turbine gates are usually set at full opening
position, or set manually to match the site flow condition, that is, y = y (t) is known. If
there is a governor installed but it is out of function, then the gate position remains at the
opening of initial state during the transients, that is, y = y0 is known. Obviously, the
governor equation is no longer valid under these two scenarios. So, the head balance
equation F1 (α, v) = 0 and speed change equation F2 (α, v) = 0 are applied to solve two
unknowns of turbine speed α and turbine flow v during the transient states.
As aforementioned, any turbine units would be disconnected when load
rejection happens. So, for transient analysis of a load rejection, the units are always
treated as isolated ones.
2) Interconnected Units
Whenever the demand changes in a large electricity system, there is a load re-distribution
among tie line loads. In other words, the load variation is assumed not only by a specific
unit but by all connected units, and the load Pg applied on the specific unit remains
162
around the dynamical power produced by the turbine, i.e., Tωηg = γQHηηg = Pg during
adjustment of the gate opening. Therefore, the normal speed change equation no longer
exists for an interconnected unit.
If the permanent speed droop σ and the dynamic variation of unit speed during
the process of adjustment are negligible, the speed of connected turbine can be
considered as constant and it is the synchronous speed which depending on system
frequency (n = nE). α =1 is known; and the governor equation is reduced to F3 (y)=0
which means the gate opening y is actually set directly.
The speed regulation governor is not as necessary for those units carrying on
block load (also called as base load) in the daily load diagram, particularly for those units
with extremely long penstocks. In cases with extremely long penstocks, a longer time
period for both opening and closing of gate is necessary to limit the waterhammer when
the unit rejects or accepts any load, which would cause a greater over speed. Therefore, it
is appropriate to let the unit only produce a constant load within the grid to avoid
frequently adjusting flows, with consequent savings to the governing system. With
constant speed (n=nR), α =1 is known; without the governor, the turbine opening has to
be set automatically or manually, i.e., y=y (t) is prescribed, which is applied into the first
basic equation: F1 (v, y)=0, then the turbine flow v can be obtained step by step.
163
6.3 Computer Implementation
A computer program for the solution of a turbine boundary condition was developed in
FORTRAN90 based on the mathematical model developed above. Except for some minor
additional and modifications to a few of the related codes, only a few new subroutines
had to be associated with turbine boundary condition. These include:
Turbdat.f90 — To read the input data and se up the initial conditions.
Turbintss.f90 — To initialize the steady state of turbine boundary condition.
Turb.f90 — Core subroutine of turbine b/c, the solution to the set of basic equations.
ShapeWHH.f90 — To interpolate the turbine head characteristics.
ShapeWBB.f90 —To interpolate the turbine torque characteristics.
6.3.1 Steady-State Simulation
In modeling, the input initial data from the input file are often inconsistent with each
other, so it is necessary to start simulation with initial steady state to eliminate the “false
transient conditions”.
6.3.2 Transient Simulation and Flow Chart of Programming
The transient simulation follows the procedure charted in Figure 6.3.
164
Figure 6.3 Flow chart of turbine boundary condition programming
With Generator?
Isolated Unit? Synchronous Speed (α=1)
Governed Unit?
Basic equations: F1 (α, v, y)=0 F2 (α, v, y)=0 F3 (α, y)=0 (α=1 and remove F2 (α, v, y)=0 for Interconnected Units)
The opening of wicket-gate is prescribed, and/or the flow controlled by shut-off valve or other valves. Basic Equations: y=y(t)
F1(α, v, y)=0 F2 (α, v, y)=0 (α=1 and remove F2 (α, v, y)=0for Interconnected Units)
No
Yes
Yes
Inverse Pump
No
Yes
Solve the Eqs. by Newton’s Method
(No Governor, or Governor Malfunctioned)
Valve-Controlled Unit
165
6.3.3 Data Input and Output
A new interface was created to effectively input system data and to visualize output of
the simulation results. For further development of the turbine boundary condition, a
sample set of data for a turbine boundary condition and the definition and description of
each input variable is provided in Appendix Ⅲ. A general summary of turbine boundary
condition is includes in the output data table. A sample output data abstracted from the
data file of *.BND and the definition of each variable in the time series are listed in
Appendix Ⅳ. The output data can also be found and reflected in Debug and Graphic
files.
6.4 Model and Program Verification
6.4.1 Comparison with Wylie’s Simulation
An example from literature (Wylie, et al., 1993) is used to verify the model and computer
program of turbine boundary condition developed in this thesis. This is a
reservoir-pipe-turbine system, in which a turbine unit is located at the downstream end of
the penstock. The draft tube is short and thus can be ignored, so it is a one-node turbine
boundary condition, which means the turbine downstream pressure head is constant. The
turbine unit is assumed to be an isolated one, and a PID speed control governor is used to
adjust the wicket gate opening during load rejection and variation, but the maximum rate
166
of wicket gate closure is limited by the shortest closure time period Tf = 6.5 s. The
turbine and system data are listed in Table 6-1.
Table 6-1 System data of Wylie’s example
Pipeline Turbine Unit Governor
L = 411 ft HR = 269 ft Td = 3.7 s
A = 254 ft2 NR = 200 rpm s 325.0=αT
a = 4100 ft/s QR = 4025 ft3/s Tf = 6.5 s
TR = 3.03(106) ft-lb 18.0=δ
WR g 2 =35.3(106) lb-ft2 0.0=σ
Three cases are simulated. The first case is that the unit initially generating 61.7
MW of power and then rejected within 0.1 s; the second case is that the same initial load
reduces to 44.8 MW in 1 s; and the third case is that the same initial load increases to 81
MW. The simulated responses of turbine dimensionless variables to above three load
variations are showed graphically in Figures 6.4, 6.5 and 6.6, respectively, in which the
solid lines represent simulations of this thesis; and the dashed lines represent the
simulations from Wylie’s classic book, of which the program and data files are directly
cited from the appended disk of the book. By the way, other case studies do not show this
obvious difference between these two interpolation methods.
167
The comparison in Figure 6.4 shows that two simulations are generally quite
close to each other. However, the difference is noticed in the case of load rejection, as
shown in Figure 6.4 (a), for the simulation of turbine head at the range of wicket-gate
opening from y = 0.4 to y = 0.0. The turbine head, which is actually the pressure head
at the end of penstock in this case, analyzed by this model, has obvious “waves” while
Wylie’s simulation is quite flat. This difference is believed to arise largely from the
slightly different interpolation methods used for turbine characteristics and the particular
turbine characteristics. Figure 6.5 (a) shows that the slope of this particular turbine
characteristic curves WH (x, y) change with coordinates x when wicket-gate opening y is
small (less than 0.5). This feature requires a special treatment of interpolation for the
large gaps of WH values between the wicket-gate openings y; in Wylie’s simulation,
linear interpolation is used twice each time, between the openings y first and then
between the coordinates x; and the values of WH at certain range of small openings and
small coordinates are replaced by the values of coordinates x when they are interpolated
between the openings y. While in this developed model, the concept of shape function is
implemented for interpolation of WH between the openings y and coordinates x, which
causes the numerical instability in the range of small openings for this particular example.
However, this feature of WH curves is not necessarily found, or very rarely found, in
many other Francis turbine characteristic curves. Figure 6.5 (b) shows a typical Francis
turbine’s WH curves generated by the program developed in Chapter 5, they keep a
steady slope with coordinates x though big gaps exist between different openings. So far,
168
this numerical instability has never been found in the simulations of actual hydro projects
using the model developed in this thesis. However, certainly the interpolation method
could be further improved, particularly for each individual hydro turbine characteristics
and especially for the range of small openings.
(a) Responses to Load Rejection (61.7- 0 MW)
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 1 2 3 4 5 6 7 8 9 10
Time (s)
Dimensionless Turbine
Variables
y
v
h
ALPH
BETA
Figure 6.4 Comparisons with Wylie’s Simulations (con’d)
169
(b) Responses to Load Reduction (67.1- 44.8 MW)
0
0.2
0.4
0.6
0.8
1
1.2
0 2 4 6 8 10 12 14 16 18 20 22 24 26
Time (s)
Dimensionless Turbine Variables
y
v
h
ALPH
BETA
(c) Responses to Load Increase (61.7- 81.0 MW)
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5 6 7 8 9 10
Time (s)
Dimensionless Turbine
Variables
y
v
h
ALPH
BETA
Figure 6.4 Comparisons with Wylie’s Simulations
170
(a) Wylie's_Francis_Turbine_Characteristic_Curve
0
1
2
3
4
5
6
7
8
9
10
-9 -6 -3 0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78 81 84 87 90
x=arctg(v/a) [deg]
WH
(x,y
)
y= 0.0 0.1 0.2 0.3
0.4
0.5
(b) MeS_Francis_Turbine_Characteristic_Curve
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
-9 -6 -3 0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78 81 84 87 90
X=arctg(v/a) [deg]
WH
(x, y
)
y=0.13y=0.22y=0.39y=0.48y=0.57y=0.65y=0.74y=0.83y=0.91y=1.0
Figure 6.5 Comparison of Wylie’s turbine characteristics with other Francis turbine
characteristics
171
6.4.2. Comparison with Field Measurement
The MeS hydropower plant in Iran has eight 250 MW units, as shown in Figure 6.6,
every two-units sharing a common penstock (Pipe #1) and a common tailrace tunnel
(Pipe #6). The units are submerged 13 m below the tailwater to prevent cavitation, but
the submersion is still insufficient based on the research (Pejovic, et al. 2004). This plant
has been built and operated for several years. However, during the simultaneous load
rejection of two units, there is a risk of water column separation and subsequent rejoinder
in the draft tube, the event that could potentially cause serious damage on both the
machine and water conveyance system (Pejovic, et al. 2004). Therefore, field
investigation and numerical simulation were carried out in 2004 to find an effective way
to prevent the excessive low pressures occurring in the draft tube.
To further verify the numerical model developed in this chapter, one set of data
measured during the site test at this power plant is used to compare with the simulation
results. As shown in Figure 6.7, this is the case that one unit rejects its full load while
another unit is closed. The (blue) thinner lines in the figure represent measurement and
the (red) thicker lines are from simulation. The greater time period of measured pressure
fluctuation is caused by the air chamber effects of the void in the draft tube; the bigger
the void, the greater the wavelength. There are two pressure taps set at turbine
downstream, one is positioned at a cross-section of the draft tube cone (3.2 m below the
turbine spiral case); another is positioned at the sluice gate where is near the draft tube
172
exit. The calculated pressure surges at the inlet of spiral case and outlet of draft tube are
initially calibrated to agree with the measurement. The calculated pressure at the
measured section of draft tube cone is retrieved based on the head balance equation from
the simulated pressure at the draft tube outlet (sluice gate) by considering the vertical
variations of both elevation and diameter. As expected, the simulated minimum pressure
values at the draft tube cone are lower than the measured ones for both initial steady and
transient states, because the role of vortex core void cannot be mathematically simulated
by one-dimensional model and thus the simulated pressure represents the average value
on the cross-section; despite this, the simulation agrees quite well with measurement
values.
173
368-372 m
222-235mUnit #3
Unit #4
Pipe #1 Pipe #2
Pipe # 3Pipe #6
Pipe #4
Pipe #5
Figure 6.6 Schematic waterway of two parallel turbine units in MeS plant
Table 6-2 System data of MeS Hydro Project
Pipe #1 Pipe #2/3 Pipe #4/5 Pipe #6 Turbine Unit Governor
L = 255.4 m L = 73.6 m L = 76.6 m L = 393.9 m HR = 142 m Td = 3.7 s
D = 9.5 m D = 5.81 m D = 7.315 m D = 11.0 m NR=187.5 rpm s 325.0=αT
a = 1200 m/s a = 1200 m/s a = 600 m/s a = 600 m/s QR = 191.5 m3/s Tf = 20 s
f = 0.03348 f = 0.02485 f = 0.01454 f = 0.02264 PR = 250 MW 18.0=δ
WRg2=7.1*107 N.m2 0.0=σ
174
(a) Turbine Speed
80
90
100
110
120
130
140
150
160
50 60 70 80 90 100 110
Time (s
Spee
d (%
)
-10
0
10
20
30
40
50
60
70
80
90
Ope
ning
(%)
Figure 6.7 Comparison of numerical modeling and field measurement (con’d)
(Note: measurement; simulation)
(b) Penstock Pressure
162
164
166
168
170
172
174
176
178
180
182
184
50 60 70 80 90 100 110
Time (s)
Pens
tock
Pre
ssur
e (m
)
)
WG Opening
175
(c) Pressure at Sluice Gate
0
2
4
6
8
10
12
14
16
18
20
22
50 60 70 80 90 100 110
Time (s)
Flap
Gat
e Pr
essu
re (m
)
(d) Pressure at Draft Tube Cone
-4
-2
0
2
4
6
8
10
12
14
16
18
20
22
50 60 70 80 90 100 110
Time (s)
Dra
ft Tu
be C
one
Pres
sure
(m)
Figure 6.7 Comparison of numerical modeling and field measurement
(Note: measurement; simulation)
176
6.5 Summary
This chapter demonstrates an application of AHD model to a system containing a hydro
turbine. A transient analysis model, including the head balance equation derived from
extended MOC equations, the speed change equation (i.e., the torque balance equation of
turbine shaft) and the governor equation (i.e., the relationship of PID speed controller
and corrective actions), has been developed for hydro turbine units. The numerical
solution and computer program have been coded to facilitate model implementation in
either pipe networks or conventional hydropower plants.
The applications of three basic equations for both isolated and interconnected
units, and for different operating scenarios are discussed in detail, and the computer
program and its input and output data are introduced as well for advanced TransAM
users.
Finally, the model and program have been verified by both Wylie’s simulation
example and a field test. Next, the developed turbine model and program are applied in
the new case studies presented in the next two chapters.
177
Chapter 7 Case Study 1: Energy Recovery Hydro
Turbines in Las Vegas Water Supply System
This chapter presents one application of the developed turbine model for the transient
analysis of small hydro turbines installed in Las Vegas water supply system for the
purpose of energy recovery in the place of the more traditional pressure reducing valves.
The transient analysis was carried between 2002 and 2004 during the feasibility study
and design stages of the project. The project has now been put in operation since 2006,
and field tests of load rejections were performed during system commissioning; field
data were kindly provided by the design engineer, Kinetrics Inc. Given that the original
purpose of the analysis was to predict transient pressures conservatively, and that the
results presented here are essentially “as is” without subsequent calibration, and taking
into account the complexity of the network system, the agreement shown in this chapter
between the numerical simulation and field test measurement for turbine transient
responses on load rejections is considered acceptable. Overall, this study provides
some additional verification of the developed turbine model, and to provide an indication
of the accuracy that might be expected when predicting transients in topologically
complex systems with active hydraulic devices.
This chapter also explores a novel application, specifically that of the effect of
178
fire-fighting flow on the operation of turbine unit. Such a flow disturbance would
obviously not be considered normal in the context of a more typical turbine application.
7.1 Project Background and Objectives
Small or micro hydro turbines can be applied not only in dedicated hydroelectric plants
but also in multiple-use systems and for power recovery purposes. In fact, installation of
micro-turbines in many water conveyance and irrigation systems is gradually becoming a
popular alternative to the wasteful dissipation of energy in devices such as pressure
reducing valves and head loss chambers. The use of small or micro turbines in zones with
a significant topographic difference not only produces renewable energy, but also can
avoid the use of stronger (and thus more expensive) downstream pipes and may even
reduce downstream water leakage. Moreover, extensive (and sometimes oversized)
storage tanks in urban water supply systems may have a valuable hydraulic potential first
for energy storage and then for subsequent electricity production during periods of high
electrical demand, thus allowing the municipal system to sometimes function somewhat
as a pumped storage scheme.
An important issue in any turbine installation is that of system hydraulics. In
particular, waterhammer in systems with turbines can cause numerous problems,
especially in systems with long pressurized pipelines. These concerns originate largely
from the complexity of turbine operation whose performance is influenced by the
179
operation of both the governor and the generator, from the properties of the water
conveyance network, and from the properties of the power grid connected to the unit.
Yet, even allowing for this complexity, at least a conservative estimate is required in
prediction and calculation of transient performance to avoid waterhammer problems and
associated system failures.
To take advantage of the flow rates (to feed the urban water distribution system)
and pressure differences (mostly from topographical difference) traditionally dissipated
by pressure reducing valves, four vertical Francis turbines were proposed to be installed
individually at Horizon Ridge, Bermuda, New Sloan (Linden) and Sloan Forebay ROFC
(Rate of Flow Control) stations in the Las Vegas water supply system. The system
hydraulics associated with the proposed turbine installations becomes a new issue for the
water supply system. Because of the complexity of long pipeline profiles, multiple
hydraulic devices and pipe connections in the system, the most vulnerable points to
waterhammer are not necessarily at upstream and downstream nodes of turbine (thus not
at the points which have typically been a focus of waterhammer studies in conventional
hydropower stations).
The developed turbine model was applied to analyze the proposed turbines and
associated pipe systems. The main objectives were to identify and predict the most severe
transient pressures in the pipeline under all possible conditions related to turbine
operations; and to identify possible mitigation techniques that could be implemented to
180
alleviate transient pressures.
Based on the turbine model characteristics provided by the manufacturer and
associated water system data converted from a supplied EPANET data file, the full load
rejection was identified as the most severe transient event. The corresponding numerical
simulation results provided designers with conservative estimates of the required pipe
pressure ratings and the minimum pipe internal pressures. Finally the key points in the
system to which more attention should be paid during design and operation were
identified.
7.2 System and Hydraulic Characteristics
The scheme summarized in Figure 7.1 shows the main constituents in one water district
of the water supply system. The Zone L1 is a part of water distribution network. The peak
daily water demand of this zone (approximately 22,000 gpm of flow) is supplied through
Linden ROFC station (Rate of Flow Control) from Pump Station A (shortened as PS A) to
the Reservoir S and/or through the 42-inch mains from the Reservoir S to the PS B.
During the summer months, some pumps at the PS B also can feed water to the zone L1,
typically pumping 13,500 gpm while 16,000 gpm is being taken from the ROFC station.
Because of the topographic difference between PS A and the distribution zone L1 the
pressure head across the ROFC can range from 165 to 240 feet, which was originally
dissipated by pressure reducing valves. To take advantage of this pressure head and the
flows that feed the downstream network, a Francis turbine manufactured by CHC
181
(Canadian Hydro Components Ltd.) has been installed within one valve train of ROFC
vault. The parameters of the turbine unit are summarized in Table 7-1.
Table 7-1 Linden turbine-generator parameters
Dth
(mm)
N11
(rpm)
Q11
(cms)
NR
(rpm)
QR
(cms)
HR
(m)
Turbine
Efficiency
(%)
Turbine
Output
(kW)
Generator
Nameplate
(kW)
385 57.8 0.764 1200 0.92 65.5 91.4 558 522
Reservoir S
Proposed
Turbine
in ROFC
30’’(68’ long)
96’’(80’ long)
36’’ (80’ long)
42’’ Mains (41,200’ long)
Feed to Zone L2
2 Spherical Surge Tanks
PS B
PS A
PS A Discharge
Pipeline 78’’
(Lower-elevation Distribution Zone L1)
(Higher-elevation Water Supplier)
Hydrant
Figure 7.1 Linden Energy Recovery Hydro Turbine and System Constituents
182
The total length of mains is approximately 41,200 ft, including the 11,000 ft
portion of the line from Reservoir S to the ROFC and the 30,200 ft line between the
ROFC and PS B. The inner diameter of most of these pipes is 42 inches, and some are 60
inches. The friction factor for the Hazen-William’s formula was taken as 135
(equivalent Darcy-Weisbach friction factor f = 0.012), and the wave speed is 3500 ft/s; a
sensitivity analysis was performed and demonstrated that the transient response was only
mildly dependent on these specific values.
7.3 Transient Scenarios and Worst Case Identification
There are three potentially unfavorable situations that require special attention when
turbine units are installed and operated in a water supply system.
1) The first and most severe situation is that any control valve or the wicket-gate of
the turbine closes essentially instantly because of an extreme accident, such as the
sudden failure of the control ring or some other event not necessarily related to
turbine load rejection. This situation is termed “Instantaneous Closure” since the
flow at the turbine would be cut-off to zero in a negligible time. Not surprisingly,
the numerical simulation of “Instantaneous Closure” shows it would cause severe
negative and positive pressures within the connected pipe system.
However, this event can be considered highly unlikely for this application; in fact,
considerable effort is made to prevent such an occurrence. Specifically, through
183
careful design of the wicket-gate control ring actuator, this event is routinely made
extremely improbable. Protecting against such an event would be expensive and
would almost certainly include surge protection equipment what may never
subsequently be used.
2) The second, and more realistic, unfavorable transient event is caused by turbine
load rejection. The load rejection of a low- or medium-specific-speed Francis
turbine causes a rapid flow reduction or choking as the turbine accelerates to
runaway speed, and thus causes the receiving and supplying distribution system to
go through a phase of transient re-adjustment, as conditions adjust to the new flow
through the turbine. When load is rejected from large turbine-generator units, the
wicket-gate is usually governed to close gradually after load rejection, thus
reducing the time period (and intensity) of runaway speed (though the wicket-gate
closure can cause additional pressure surges in the system). For large machines that
cannot tolerate excessive runaway speeds, the calculation of wicket gate responses
is indispensable, as is the prediction of both the associated waterhammer and the
unit speed-rise.
However, the small turbine-generators considered in this project can be designed to
tolerate full runaway speeds for relatively long time periods (e.g., 60 minutes), and
thus the pressure surges in the system associated with only the load rejection itself
become the main transient event of concern. The 600-second (10-minute) time
184
period for the wicket-gate closure planned for this project is effectively unbounded
relative to the time constants of the hydraulic system, and thus the results are quite
similar to the full runaway situation. In other words, a controlled load rejection
(with wicket gate closure) and a full runaway condition (without wicket gate
closure) are effectively equivalent in this system. Thus, in summary, analysis and
prediction of the transient response associated with a load rejection and full
runaway condition is the major objective.
3) If there are hydrants nearby the turbine unit, either upstream or downstream, could
the in emergency use have significant impact on operation of the turbine unit?
How should the turbine unit be operated properly as a protection if this happens?
In addition, water column separation may become a concern because of the long
pressured downstream pipelines in a water distribution system. These topics, as one
of special issues for energy recovery turbines in water supply system, are addressed
as well in this chapter.
7.4 Simulation and Field Tests on Turbine Load Rejections
7.4.1 Field Tests
The load rejection tests were carried out on site during the commissioning of the turbine
systems; these tests permit an evaluation of the turbine’s transient performance (e.g., the
185
max. speed rise, the transient pressures, the turbine unit stresses and etc.). However, the
tests were no originally intended to verify the transient analysis model, and indeed the
data records are somewhat lacking for model verification. At the Linden
turbine-generator unit, the load was rejected at 50% and 100% wicket gate positions.
During the field tests, almost full flow is in the parallel piping path (train #1). The data
were recorded in steps of 1 or 2 (or 5) seconds. Since the rate of wicket gate closure is
at 6% per minute (full closure in 15 minutes), there is a long period of turbine runaway
condition with slowly decreasing flow before the system becomes to a final steady state.
7.4.2 Simulation vs. Field Test
Figure 7.2 shows the transient response of turbine parameters when the load is rejected
initially from the rated operation condition (100% wicket gate opening) and with
wicket-gate closure in 900 s. As expected, the transient responses are mild. This mild
response is achieved in several ways: (1) a flow of 0.924 m3/s (almost half of total system
flow) is continuously passed through the paralleled pressure reducing valve and bypass
piping line down to the main pipelines; and (2) the wicket gate closes extremely slowly,
so the discharge reduction is basically from the speed rise of the turbine at the initial stage
rather than from wicket gate closure, and the speed rise restricts the turbine flow from
initially 0.97 m3/s to 0.75 m3/s in the first 5 seconds.
In this system, the main pipelines are at the turbine downstream, so the
186
simulation of associated downstream pressure drop and subsequently pressure rise from
wave reflection are the major interest of the project. The comparison of turbine flow –
Figure 7.2 (a), speed rise – Figure 7.2 (b), upstream pressure rise – Figure 7.2 (c), and
downstream pressure drop – Figure 7.2 (d), shows a reasonable agreement between the
field test measurement and numerical simulation. Obviously, though, the simulation
results are generally somewhat more conservative, as indeed they were intended to be.
For instance, the maximum speed rise in Figure 7.2 (b) is 2070 rpm from simulation
while it is 2060 rpm from field test (the only two peak speed data recorded); the
simulated pressure at the turbine downstream in Figure 7.2 (d) shows the lowest pressure
drop to 68.6 m (more conservative peak pressure drop) and a slower transient energy
dissipation process (showing two pressure drop waves), while it has only one pressure
wave (demonstrating a fast transient energy dissipation process) with the lowest drop to
70.5 m from field test.
The observed discrepancies almost certainly arise because, in a more realistic
pipeline system, there are many more energy dissipation mechanisms than could be
captured in the numerical model. For example, small branch pipes are neglected in the
model yet they enhance energy dissipation in the system; this also holds for leaks which
cause water and energy losses from the distribution system; additional friction headlosses
occur at the pipe junctions, and many other mechanisms. The efficient dissipation
associated with waterhammer in complex networks has been commonly observed in other
187
studies.
The upstream pressure seems to rise gradually; this is because Pump Station A is
continuously supplying the same amount flow to the system, which results in the
accumulation of water in the double spherical surge tank, a process that occurs as the
turbine flow is gradually reduced.
An alternate reason for the noted difference between the simulation and field
measurement might be the complexity of flow conditions inside the turbine water passage
(three-dimensional and two-phases) or from incorrect input values of system parameters.
This difference would include, but not necessarily be limited to, variations in wave
speeds, pipe friction factors, device settings and system demands. It is challenging to
obtain the accurate values of these parameters from a realistic system, because many
factors could influence these parameters. For instance, a small amount of air in the
pipeline could significantly affect the value of wave speed and thus change the pressure
wave patterns and dissipation rate. A careful calibration of system parameters would be
required to minimize the differences between numerical simulation and field
measurement. However, from the perspective of engineering design practice, it is most
important to conservatively predict the peak pressure heads in the system and the peak
speed rise of the turbine unit. For this purpose, the differences shown in Figure 7.2 are
acceptable.
188
Figure 7.3 shows the transient pressure responses in the associated downstream
mains when the turbine unit experiences the full load rejection (i.e., the 100% initial
wicket gate opening). The maximum HGL are well below the pipe ratings; and the
minimum HGLs are generally above the pipeline profiles. Thus, no additional measures
are required for transient mitigation and control.
Figure 7.4 shows a comparison of numerical simulation and test data for the
transient responses of turbine parameters to the 50% load rejection (i.e., at the 50% of
initial wicket gate opening). The same trends and agreement are demonstrated as in
Figure 7.2 (100% load rejection case), except the transient response is milder for this
case.
189
(a) Turbine Flow Reduction During 100% Load Rejection
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 5 10 15 20 25 30 35 40 45 50
Time (s)
Turb
ine
Flow
(m3/
s)
From Field Test
From Simulation
(b) Turbine Speed Rise During 100% Load Rejection
1100
1300
1500
1700
1900
2100
0 5 10 15 20 25 30 35 40 45 50
Time (s)
Turb
ine
Spee
d R
ise
(rpm
)
From Field Test
From Simulation
Figure 7.2 Turbine Transient Responses to 100% Load Rejection (con’d)
190
(c) Turbine Upstream Pressure Rise During 100% Load Rejection
120
122
124
126
128
130
132
0 5 10 15 20 25 30 35 40 45 50
Time (s)
Turb
ine
Ups
trea
m P
ress
ure
Hea
d (m
)
From Field Test
From Simulation
(d) Turbine Downstream Pressure VariationDuring 100% Load Rejection
60
64
68
72
76
80
0 5 10 15 20 25 30 35 40 45 50
Time (s)
Turb
ine
Dow
nstr
eam
Pre
ssur
e H
ead
(m)
From Field Test
From Simulation
Figure 7.2 Turbine Transient Responses to 100% Load Rejection
191
(a) Distance from ROFC (Turbine) to PS B
(b) Distance from ROFC (Turbine) to Reservoir S
Pipe Ratings
Pipe Ratings
Pipe Profile
Pipe Profile
Max. H
Max. H
Min. H
Min. H
S.S. H
Figure 7.3 Main Pipeline Pressure Responses to 100% Load Rejection
192
(a) Turbine Flow Reduction During 50% Load Rejection
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Time (s)
Turb
ine
Flow
(m3/
s)From Field TestFrom Simulation
(b) Turbine Speed Rise During 50% Load Rejection
1000
1200
1400
1600
1800
2000
0 10 20 30 40 50 60 70 80 90 100
Time (s)
Turb
ine
Spee
d R
ise
(rpm
)
From Field Test
From Simulation
Figure 7.4 Turbine Transient Responses to 50% Load Rejection (con’d)
193
(c) Turbine Upstream Pressure Rise During 50% Load Rejection
120
122
124
126
128
130
132
134
136
138
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Time (s)
Turb
ine
Ups
trea
m P
ress
ure
Hea
d (m
)
From Field Test
From Simulation
(d) Turbine Downstream Pressure Variation During 50% Load Rejection
60
64
68
72
76
80
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Time (s)
Turb
ine
Dow
nstr
eam
Pre
ssur
e H
ead
(m)
From Field Test
From Simulation
Figure 7.4 Turbine Transient Responses to 50% Load Rejection
194
Figure 7.5 shows a simulation result for one main pipeline of SNWA water
supply system due to the load rejection of hydro turbine unit installed at New Sloan
Forebay (the detailed description and data of this turbine system are not provided here).
The maximum pressure rise at the pipe sections around the lowest position of the pipeline
(Las Vegas Wash) did exceed the existing pipe rating. So, the topographic variations in
the water supply system may create transient concerns due to the installation of hydro
turbine units.
Figure 7.5 Main Pipeline Pressure Responses to Load Rejection of Sloan Forebay
Turbine Unit
Distance from RMR to SF Turbine along 78'' New EVL Pipeline (ft)
Pipe Ratings
Pipe Profile
Max. HGL
S.S. HGL
Min. HGL
Las Vegas Wash
195
7.5 Effect of Hydrant Flow on Turbine Unit Operation
Hydrants are indispensable devices in water supply and distribution systems. Depending
on the population, the size and layout of a water network, the required fire flow rate
ranges from 1000 gpm to 3500 gpm (i.e., 65-220 l/s), the design duration 4-10 hours.
The operation of hydrant and water demand for the fire fighting can cause significant
transient surges in the system. If the hydrants are located close to turbine units, they
could induce undesirable impacts on the turbine operation. In this section, the effect of
fire fighting flow on the operation of turbine unit is explored; and the behavioral
responses of the turbine to the hydrant demands are graphically shown in the following
case studies with the hypothetical hydrants nearby the turbine units.
7.5.1. Upstream Hydrants
In the system sketched in Figure 7.6, since the downstream end of the turbine at Horizon
ROFC station is directly linked to a water tank, there is no possibility of hydrants
existing in its downstream pipeline. One to ten (intentionally extreme) hydrants are
assumed to locate at 2000 ft upstream of the turbine, where is marked as a star in Figure
7.6. The total flow in that pipeline is around 190 mgd (7.4 m3/s), and the turbine rated
flow is 33.75 mgd (1.48 m3/s). Initially, the turbine unit is operating at the rated condition
either isolated or connected to a large electricity system, and the hydrant flow is suddenly
initiated. The fire fighting demand would initially induce a rapid flow increase in the
196
associated pipeline and turbine, and thus result in a subsequent pressure drop at the
turbine, as well as a series of turbine behavioral variations, including the changes of gate
opening (automatically regulated by the governor) and rotational speed for an isolated
turbine unit.
Figure 7.7 shows the responses of an isolated operating turbine unit to 50 mgd
(2.2 m3/s) of upstream fire flow, which is equivalent to the extreme case of 10 hydrants
operating instantaneously (opened within 1 s). In this and the following figures, the
dimensionless turbine parameters v, h, ALPH (α), BETA (β) are defined as in equation
(5-10), Chapter 5; and they are turbine’s relative flow, head, speed and torque,
respectively; and y is the dimensionless wicket gate opening. The downsurge arrives at
the turbine in 0.5 –1.5 s from the hydrants, and then the turbine upstream head drops 60
feet relative to the initial head. This reduction of available waterpower causes the turbine
to rotate at a lower speed (α). Meanwhile, to balance its power load (resistance on the
turbine shaft), the turbine governor tries to open the wicket gates wider to pass more flow.
However, in this case even at full gate opening the available waterpower is still not
sufficient to meet the requirement of power load, so the rotation of the unit has to slow
down and finally would switch off automatically.
Figure 7.8 shows the responses of an online operating turbine unit to 50 mgd
(2.2 m3/s) of upstream fire flow. Since the online unit would transfer/distribute its power
load to the interconnected generator units, the unit speed and gate opening would keep at
197
their initial values. However, its flow (v) and dynamic torque (β) would keep at a lower
level because of the reduction in net head. Therefore, the turbine unit would finally reach
a new balanced condition with a lower power output.
Figures 7.9 and 7.10 demonstrate the turbine responses to single hydrant flow
(0.22 m3/s), the variation patterns of turbine behavior are the same as previously but the
variation ranges are naturally much milder, e.g., the speed of isolated turbine goes down
to 90% of the rated value at 16 s. This speed drop is much milder compared with the case
of 10 hydrants, in which the speed is lowered to 22% of the rated speed at 16 s. However,
in power system applications, to avoid tripping of underfrequency relays, the allowable
speed deviation of isolated turbine unit (i.e., the acceptable system frequency excursion)
is around 2-3% only (Gordon, 1985); so even a single hydrant flow would lead to the
stoppage of turbine unit operation.
Finally, attention should be paid to the pressure drop at the turbine downstream.
The sudden reduction of flow at the first initiation of hydrant operation would result in a
pressure drop at turbine downstream as well, which may cause the local pressure to drop
below atmospheric pressure for a short time period.
198
PT
Bermuda R.
WL=2422.1’
To 2420 Zone
To 2300 Zone
84’’ 90’’
Bermuda ROFC & Turbine
T
108’’
Horizon Ridge R. WL=2365’
West-end SVL East-end Point
South Valley R.
WL=2530’
Parkway R.
WL=2495’
Black Mountain R.
WL=2220’
Horizon Ridge ROFC & Turbine Burkholder R. &
River Mountains PS
Warm Springs R.WL=2422’
Hydrant
Figure 7.6 Sketch of South Valley Lateral System
199
Upstream hydrant flow 2.2 cms Isolated turbine unit
0
0. 2
0. 4
0. 6
0. 8
1
1. 2
1. 4
0 2 4 6 8 10 12 14 16Time (s)
Turb
ine
dim
ensi
onle
ss v
aria
bles
y
v
h
ALPH
BETA
Upstream hydrant flow 2.2 cms Isolated turbine unit
-200
20406080
100120140160180200
0 1 2 3 4 5
Time (s)
Turb
ine
upst
ream
and
dow
nstre
am h
ead
(ft)
Upstream head
Downstream head
Figure 7.7 Isolated turbine responses to 10 upstream hydrants flow
200
Upstream hydrant flow 2.2 cms Online turbine unit
0
0. 2
0. 4
0. 6
0. 8
1
1. 2
1. 4
0 1 2 3 4 5Time (s)
Turb
ine
dim
ensi
onle
ss v
aria
bles
y
v
h
ALPH
BETA
Upstream hydrant flow 2.2 cmsOnline turbine unit
-20
0
20
40
60
80
100
120
140
160
180
200
0 1 2 3 4 5
Time (s)
Turb
ine
upst
ream
and
dow
nstre
am h
ead
(ft)
Upstream Head
Downstream Head
Figure 7.8 Online turbine responses to 10 upstream hydrants flow
201
Upstream hydrant flow 0.22 cms Isolated turbine unit
0. 7
0. 8
0. 9
1. 0
1. 1
1. 2
0 2 4 6 8 10 12 14 16 18 20
Time (s)
Turb
ine
dim
ensi
onle
ss v
aria
bles
y
v
h
ALPH
BETA
Upstream hydrant flow 0.22 cms Isolated turbine unit
-20
0
20
40
60
80
100
120
140
160
180
200
0 1 2 3 4 5
Time (s)
Turb
ine
upst
ream
and
dow
nstre
am h
ead
(ft)
Upstream HeadDownstream Head
Figure 7.9 Isolated turbine responses to single upstream hydrant flow
202
Upstream hydrant flow 0.22 cms Online turbine unit
0.7
0.8
0.8
0.9
0.9
1.0
1.0
1.1
1.1
1.2
1.2
0 1 2 3 4 5
Time (s)
Turb
ine
dim
ensi
onle
ss v
aria
bles
y
v
h
ALPH
BETA
Upstream hydrant flow 0.22 cms Online turbine unit
-20
0
20
40
60
80
100
120
140
160
180
200
0 1 2 3 4 5Time (s)
Turb
ine
upst
ream
and
dow
nstre
am h
ead
(ft)
Upstream HeadDownstream Head
Figure 7.10 Online turbine responses to single upstream hydrant flow
203
7.5.2. Downstream Hydrants
In the system schemed in Figure 7.1 (Linden hydro turbine system), it is assumed that
hydrants are installed on the 42-inch mains at the turbine downstream, the location is
about 2900 ft of distance away from the turbine (see the star in Figure 7.1). The turbine
rated flow is 0.92 m3/s, which is also the total initial flow in the system for this case
study.
Figure 7.11 shows the responses of an isolated operating turbine unit to 1.0 m3/s
of downstream fire demand, which is equivalent to the case of 5 hydrants operating
instantaneously (opened within 1 s). Such a demand is unavailable in this system; thus,
then what would happen with this over demand? Since the location of the hydrant is not
far away from the turbine unit, this sudden flow is similar to a sudden opening of a valve,
which will cause an immediate pressure drop and subsequently a series of pressure waves
at turbine downstream pipeline. Even in this case of unrealistically over demand at
turbine downstream pipeline, the turbine upstream pressure is only slightly disturbed, and
the fluctuations in the turbine head would cause the variations in the turbine flow, speed
and wicket gate opening, which are gradually attenuated over time. The maximum speed
deviation is within the range of 82.5% to 115.5% of normal speed, and the main concern
is the negative pressure from the first down surge at the turbine downstream.
Figure 7.12 shows the responses of an online operating turbine unit to 1.0 m3/s of
204
downstream fire flow. Since the online unit would transfer/distribute its load to the
interconnected power system, so the turbine speed and wicket gate opening remain at
their initial values. The simulation shows that the turbine flow fluctuates with the turbine
head.
Figures 7.13 and 7.14 demonstrate the turbine responses to a smaller hydrant
flow (0.22 m3/s, equivalent of single typical hydrant); the variation patterns of turbine
behavior are similar to the previous 10 hydrants’ flow case, but the variations are much
milder. However, the transient waves would not be attenuated with time in this case
study; this might be a coincidence of resonance that depends on the characteristics of
pipelines.
The above discussions regarding the interaction between the hydrant flow and
turbine unit operation are based on certain hypothetical situations, particularly about the
speed change of isolated unit; the purpose is to reveal the potential significance of the
hydrant flow to the turbine operation. In any case, it is only prudent to place turbines as
far as possible from actively operating hydraulic devices to reduce the potential impacts
on the turbine operation. Under certain circumstance even the demand from a single
hydrant could cause instability of turbine operation.
205
Downstream hydrant flow 1.0 cmsIsolated turbine unit
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
0 5 10 15 20 25 30 35 40 45 50
Time (s)
Turb
ine
dim
ensi
onle
ssva
riabl
esy
v
h
ALPH
BETA
Downstream hydrant flow 1.0 cmsIsolated turbine unit
-200
-100
0
100
200
300
400
500
600
0 5 10 15 20 25 30 35 40 45 50
Time (s)
Turb
ine
upst
ream
and
dow
nstre
am h
ead
(ft)
Upstream HDownstream H
Figure 7.11 Isolated turbine responses to 5 downstream hydrants flow
206
Downstream hydrant flow 1.0 cms Online turbine unit
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0 5 10 15 20 25 30 35 40 45 50
Time (s)
Turb
ine
dim
ensi
onle
ss v
aria
bles
y
v
h
ALPH
BETA
Downstream hydrant flow 1.0 cms Online turbine unit
0
100
200
300
400
500
600
0 5 10 15 20 25 30 35 40 45 50
Time (s)
Turb
ine
upst
ream
and
dow
nstre
amhe
ad (f
t)
Upstream H
Downstream H
Figure 7.12 Online turbine responses to 5 downstream hydrants flow
207
Downstream hydrant flow 0.22 cmsIsolated turbine unit
0.0
0.5
1.0
1.5
2.0
2.5
0 10 20 30 40 50
Time (s)
Turb
ine
dim
ensi
onle
ssva
riabl
esy
v
h
ALPH
BETA
Downstream hydrant flow 0.22 cmsIsolated turbine unit
0
100
200
300
400
500
600
0 5 10 15 20 25 30 35 40 45 50
Time (s)
Turb
ine
upst
ream
and
dow
nstre
am h
ead
(ft)
Downstream HUpstream H
Figure 7.13 Isolated turbine responses to single downstream hydrant flow
208
Downstream hydrant flow 0.22 cms Online turbine unit
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0 5 10 15 20 25 30 35 40 45 50
Time (s)
Turb
ine
dim
ensi
onle
ssva
riabl
es
y
v
h
ALPH
BETA
Downstream hydrant flow 0.22 cmsOnline turbine unit
0
100
200
300
400
500
600
0 5 10 15 20 25 30 35 40 45 50
Time (s)
Turb
ine
upst
ream
and
dow
nstre
amhe
ad (f
t)
Upstream HDownstream H
Figure 7.14 Online turbine responses to single downstream hydrant flow
209
7.6 Summary and Conclusions
This chapter presents an application of energy recovery hydro turbines in an urban water
supply system. The main points obtained from this project are summarized below:
(a) During load rejection of a turbine unit, the net pressure head at the lowest point of
upstream pipeline could exceed the existing pipe rating, and thus could lead to pipe
rapture and flooding. Therefore, it may be necessary to raise the pipe rating for
installations using hydro turbines.
(b) Compared with conventional hydropower projects, there are often more topographical
variations along the pipeline profiles in water supply systems. The pressure drop
caused by load rejection could be a concern at the highest position in the downstream
pipeline of the turbine unit, which may cause water column separation and subsequent
rejoinder.
(c) The potential interaction with active flow adjustments may cause serious interruption
of turbine unit operation, so the location of turbine installation should be isolated as
much as is reasonable from any sudden system disturbance.
(d) However, water storage tanks in the system may function as surge tanks, there is
seldom any waterhammer concern for the turbine installation near a water tank.
Appropriate sites for energy recovery turbine installation must clearly take into
210
account transient conditions. Topological complexity, such as associated with small
branched pipes in water networks, may dissipate transient energy and can provide
additional waterhammer protection.
Overall, there is considerable potential for energy recovery in urban water
supply and distribution systems, and hydro installations are often efficient ways of
generating clean and sustainable energy. Simulation tools, such as the developed in this
thesis, allow potential transient problems caused by turbine operation to be predicted and
avoided.
211
Chapter 8 Case Study 2: Transient Performance
Evaluation and Control Measures for Bone Creek
Hydropower Project
This chapter presents a case study of waterhammer analysis and control for the Bone
Creek hydroelectric power project in Western Canada. This is a typical hydropower
system with a long penstock, and is currently at design stage. Since there is no ideal
topographically and geologically feasible location for construction of open surge tank in
the proximity of the powerhouse, the surge tank was provisionally designed to be over
1000 m upstream of the powerhouse, connected by a branch pipeline bifurcated from the
penstock. Transient analysis shows the simultaneous full load rejection of two turbine
units would cause excessive high and low (negative) transient pressures in the penstock.
Therefore, various hydraulic system schemes are re-considered and simulated to seek for
a design plan with acceptable system transient performance; meanwhile, sensitivity
analyses of several important design parameters are performed to find the most
economical solution.
This chapter originated from the submitted consulting work for Bone Creek
Hydro Project; the thesis author carried out the numerical analyses and drafted the report
under the supervision of Prof. Karney.
212
8.1 Introduction to Bone Creek Project
8.1.1 System description and data
As sketched in Figure 8.1, this hydroelectric facility is proposed to have a 5957 m long
buried steel penstock from the intake to the powerhouse (Pipe No. 1 and 2). A
bifurcation (Node No. 2) is located 559 m upstream from the powerhouse (at the ground
elevation of 781 m) and a surge tower is tentatively planned at ground elevation of 873.5
m and distance of 578 m upstream from the bifurcation along a buried branch pipeline
(so-called surge pipeline, Pipe No. 3).
As shown in Figure 8.2, the originally proposed surge tower is a differential
surge tank with an internal 1.6 m diameter riser and a 10 m diameter external tank. The
top of the tank is at elevation (short as “EL”) 889.46 m, the top of the internal riser is at
EL 887 m, and the tank bottom is at EL 871 m. The 1.6 m diameter of riser is directly
connected with the sloped surge pipeline at EL 867 m below the tank bottom.
The upstream reservoir water level (short as “WL”) is at EL 882 m and
tailwater level is at EL 695 m; the centerline EL at the intake is 873 m. The system design
flow is 14.5 m3/s (for two units operating at full power load), and the expected head loss
at design flow is 41.4 m during normal operation. So, the steady state WL at the location
of surge tank is around 845 m, in somewhere of the surge pipeline as shown in Figure 8.2,
which is lower than the bottom of the surge tank.
Two turbine-generator units are designed for this facility, both of a double-runner
Francis turbine type directly connected to a generator through a horizontal shaft. Each
unit has a rated flow of 7.25 m3/s, a rated speed of 900 rpm and a rated power of 9,000
213
kW. The runner throat diameter is 0.7 m and tT he inertia of Turbine-Generator rotating
assembly is 10,700 kg.m2 per unit. The originally specified wicket-gate closure law is
shown in Figure 8.3 and entails a closure with first 70% closure in 6 seconds (T = 6 s)
and the last 30% over 8 seconds (T = 8 s).
1
2
Other dimensions around the powerhouse include the branched pipe (Pipe No.
4 or 5) to each turbine unit is 1.4 m in diameter, the center line EL at spiral case inlet is
694.52 m, the EL at draft tube exit and tailrace bottom is 691.52 m.
Figure 8.1 Planned system layout with differential surge tank
(Note: The drawing is directly copied from TransAM interface; the red digitals represent the Node
No./Identifiers and the black digitals represent the Pipe No./Identifiers)
214
Table 8-1 System data of Bone Creek Hydro Project
Pipe #1 Pipe #2 Pipe #3 Pipe #4-7 Turbine Unit Governor
L = 5398 m L = 559 m L = 578 m L = 10 m HR = 145.6 m Td = 3.7 s
D = 1.98 m D = 1.98 m D = 1.6 m D4-5 = 1.4.m D6-7 = 2.0 m
NR= 900 rpm s 325.0=αT
a = 1000 m/s a = 1000 m/s a = 1000 m/s a = 1000 m/s QR = 7.25 m3/s Tf = 20 s
f = 0.012 f = 0.012 f = 0.012 f = 0.012 PR = 9.0 MW 18.0=δ
WRg2=1.07*105N.m2 0.0=σ
Figure 8.2 Sketch of planned differential surge tank
215
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10 12 14 16
WG Closure Time T1+T2 (second)
WG
Ope
ning
Figure 8.3 Originally specified wicket-gate closure law
8.1.2 Objectives of Transient Analysis
A number of requirements for system transient performance need to be met under all
potential transient scenarios during turbine unit operations. In particular, the maximum
peak pressure rise (at spiral case inlet) needs to be no greater than 50% of the static head
(i.e., 280.5 m net head or 975.5 m EL) and the minimum pressure heads anywhere in the
system should be at least 2 m to avoid the possibility of negative pressures. The peak
head is expected around 880 m EL at the bifurcation and 882 m EL at the surge tank. The
maximum speed of turbine unit is not to exceed 150% (later relaxed to 160%) of normal
speed (i.e., 1350 rpm or 1440 rpm). The allowable runaway speed, given by the turbine
manufacturer, is 1570 rpm. Runaway speed occurs when the power load is rejected but
the turbine gates do not close due to system malfunction; and the maximum runaway
216
speed is also called full runaway speed, associated with full load rejection. The event of
simultaneous full load rejection of both units has been recognized as the most demanding
or unfavorable transient scenario during the operation, so the discussions in this case
study refer almost exclusively to this event. In addition, the requirements are interlinked
in that it is easier to meet the pressure rise constraint if the speed requirement is relaxed
through a slower closure of the wicket gates.
Transient analysis shows that the tentatively planned surge tank and system layout
would not automatically achieve all of the specified constraints, in particular, after full
load rejection with the originally specified wicket-gate closure (also shown in Figure 8.4).
The objective of the case study is to suggest possible changes in the system layout, the
surge tank scheme, or wicket-gate closure law that will control the transient response and
to allow the system to perform well under emergency conditions.
8.1.3 Simplifications and Assumptions Made in Transient Analysis
The representation of devices, system characteristics and constraints was made on the
basis of data provided and the standard hydraulic assumptions in modelling turbine
systems. The standard assumptions include valid algebraic head loss relations for friction,
constant wave speed, and dominantly one-dimensional (1-D) transient flow.
Because the actual turbine characteristic data was unavailable at the time of the
analysis (for confidentiality issues), the double-runner Francis turbine unit is modeled as
a single regular Francis turbine and the corresponding model characteristics for two
different-sized turbines from other projects are tested during simulation; the most
conservative simulation results turned out to be for the larger-sized of these two units,
and it is these results that are presented in this chapter.
217
The above approach may naturally cause some slight inaccuracy in the simulation
results; especially for the turbine runaway speed (since the model efficiency is more
sensitive than the head characteristics). However, the approach is believed to reasonably
reflect the system hydraulic features and to be able to capture the peak transient pressures
and speed rise caused by the load rejection of turbine units. Moreover, sensitivity studies
help to provide a control on the nature of the assumptions and their influence on the
modelling exercise.
8.2 Conventional Open Surge Tank Scheme
8.2.1 Transient Performance for Original System Design
Based on the planned system and differential surge tank scheme (Figures 8.1 and 8.2),
numerical simulations are performed using the developed simulation code for two units
simultaneously rejecting their full loads, and the results are summarized in this section.
As shown in Figure 8.4 (c), the turbine head goes up to 270 m maximum with the
flow cut-off by the originally specified wicket-gate closure law. As shown in Figure 8.4
(a) and (b), the maximum pressure head occurs at the inlet of turbine spiral case, nearly
965 m in EL; and the minimum pressure head is negative (about -22 m in theoretical
value), existing at upstream sections of the main penstock and thus cavitations would be
caused in the main pipeline. In addition, the peak pressure at the pipe bifurcation node is
around 920 m as shown in Figure 8.4(d); and the peak water level in the differential surge
tank is 884 m as shown in Figure 8.4(e).
218
As shown in Figure 8.4(f), the maximum speed rise is about 1290 rpm with
the originally specified wicket-gate closure, which is below the allowable maximum
speed 1350 rpm (or 1440 rpm).
Yet, the simulated full runaway speed is 1780 rpm, as shown in Figure 8.5,
which is higher than the specified allowable value or 1570 rpm given by the manufacturer.
However, interesting, the speed rise is below the allowable value if the turbine
characteristics (Hill Diagram) of a smaller size is used in simulation. Because the
runaway speed is mainly associated with turbine characteristics, it is suggested to consult
with the turbine manufacturer for the allowable runaway speed. In author’s view, there is
justification to down play concerns relating to runaway speed, particularly since this
scenario is highly unlikely in modern hydropower plants, equipped with advanced
automatic control and alarm system. With the appropriate mechanical design, the main
control valve at the penstock (butterfly or spherical valve) can be automatically closed if
the wicket-gates were to malfunction. The modern alarm system also ensures the
emergency reactions in the shortest time period, including possible manually intervention
of system closure in the emergency. One option that was briefly considered was to,
increase in the inertia of rotating parts; but this approach won’t significantly reduce the
runaway speed. This can be understood as follows: when a unit’s load is rejected, the
major resistance torque would be removed from the turbine shaft, but the dynamic torque
remains due to the water that is continuously arriving as the wicket-gate or the main
control valve fail to act to cut off the inflow. This huge unbalanced torque would not only
accelerate the turbine unit, but the runaway speed that would be achieved would be
largely dictated by the incoming flow.
219
The negative pressure predicted to occur at the upstream pipe sections is the major
concern for the current system layout. This negative pressure essentially results from
wave reflections from the upstream reservoir: a rapid high-pressure rise originates from
the downstream gate closure at the turbine unit, and this is turned into a negative wave
when it reflects off the upstream reservoir. Thus, the real issue is the significant inertia of
the water in the pipe between the powerhouse and the surge tank location. The 1157 m
long distance from the surge tank to the powerhouse, and the associated inertia of water
in it, reduces the ability of the originally proposed tank to mitigate transient events.
Moreover, because of the long distance of the surge tank and the associated inertia
of water it represented, the transient performance cannot be significantly improved by
increasing the size of surge tank only. If the original location of surge tank is retained, the
simulation shows increasing the surge tank diameter even from 10 m to 20 m while
lowering its bottom from 871 m EL to 860 m EL results in nearly identical peak upsurge
pressures and as well negative pressures. Figure 8.6 shows the envelope of maximum
and minimum pressures for this larger-sized tank, along the main pipeline, during the full
load rejection with the originally specified wicket-gate closure law.
220
(a) Envelope of max. and min. HGLs along main penstock
(b) Envelope of max. and min. HGLs from upstream reservoir to surge tank
Figure 8.4 System Responses during full load rejection with planned system layout
and design and originally specified wicket-gate closure law (T1=6 s, T2= 8 s) (Con’d)
221
(c) Pressure variations at spiral case inlet
(d) Pressure variations at bifurcation node
Figure 8.4 System Responses during full load rejection with planned system layout
and design and originally specified wicket-gate closure law (T1=6 s, T2= 8 s) (Con’d)
222
(e) Water level and flow at surge tank
(f) Turbine speed rise
Figure 8.4 System Responses during full load rejection with planned system layout
and design and originally specified wicket-gate closure law (T1=6 s, T2= 8 s)
223
Figure 8.5 Turbine speed rise during full load rejection with planned system layout
and design and wicket-gates refusal to close (runaway condition)
Figure 8.6 Envelope of max. and min. HGLs along main penstock during full load
rejection with 20 m diameter of surge tank and originally specified WG closure law
(T1=6 s, T2= 8 s)
224
8.2.2 Sensitivity Analysis of Surge Tank Distance
In order to better understand the influence of the distance between the differential surge
tank and the powerhouse, a sensitivity study was briefly undertaken. It is realized that
this location of the tank may be constrained by geologic and topographic conditions on
the site, but understanding this variable is key not only to the design of the surge tank but
to its possible substitution with a suitably sized air chamber.
Thus, the possibility is considered to install the surge tank closer to the
transient source (i.e., turbine unit) in order to reduce the peak pressure rise and eliminate
the negative pressures in the system. The question is how the reduced distance affects the
transient performance caused by load rejection. A sensitivity study is performed for this
project and the result is shown in Table 8-2 and Figure 8.7.
In Table 8-2 and Figure 8.7, the distance refers to the pipeline length L from
the powerhouse to the surge tank, that is, L = L2 + L3; and L2 = L3 is assumed in this study.
The maximum and minimum pressures (Hmax and Hmin) represent for the peak pressure
heads of upsurge and downsurge within the whole system following full load rejection.
The maximum unit speed is the peak turbine overspeed under the same scenario.
As shown in Table 8-2 and Figure 8.7, the distance L significantly affects the
transient response. In order to reach the desired transient performance during the full load
rejection, the surge tank would need to be installed within 200 m of the powerhouse;
further shortening this distance does not provide additional protection for the given
wicket gate closure law.
225
Table 8-2 Sensitivity of surge tank distance to transient performance
L (m) Hmax (m) Hmin (m) Max unit speed (rpm)
1157 270 -21.9 1290
600 219 -8.4 1225
400 199 -1 1197
200 184 3 1168
100 184 3 1157
150
170
190
210
230
250
270
290
0 200 400 600 800 1000 1200
Distance from Powerhouse to Surge Tank (m)
Max
imum
pre
ssur
e he
ad in
sys
tem
(m)
-25
-20
-15
-10
-5
0
5
Min
imum
pre
ssur
e he
ad in
sys
tem
(m)
Hmax (m) Hmin (m)
Figure 8.7 Sensitivity analysis of surge tank distance
226
As shown in Figure 8.8, the hydraulic pressures would be within the desired
range if the surge tank moved to the location within 200 m range from the powerhouse.
Yet, the full runaway speed would remain at 1780 rpm when the wicket gates refuse to
close (yet the flow is chocked by over-speeding runner), as shown in Figure 8.9. As
aforementioned, the speed rise at full runaway condition is mainly affected by the turbine
characteristics and turbine head; it is hardly reduced by changing system layout for this
particular high head turbine unit.
(a) Envelope of max. and min. HGLs along main penstock
Figure 8.8 System responses during full load rejection with a closer differential
surge tank and originally specified wicket-gate closure law (T1=6 s, T2= 8 s) (Con’d)
227
(b) Pressure variations at spiral case inlet
(c) Pressure at bifurcation node
Figure 8.8 System responses during full load rejection with a closer differential
surge tank and originally specified wicket-gate closure law (T1=6 s, T2= 8 s) (Con’d)
228
(d) Water level and flow at surge tank
(e) Turbine speed rise
Figure 8.8 System responses during full load rejection with a closer differential
surge tank and originally specified wicket-gate closure law (T1=6 s, T2= 8 s)
229
Figure 8.9 Turbine speed rise during full load rejection with a closer differential
surge tank and wicket-gate refusal to close (runaway condition)
8.2.3 Sensitivity Analysis of Wicket-gate Closure Time
Slowing down the closure of wicket-gates will reduce the peak pressures of original
upsurge at the turbine and thus largely eliminate the negative pressures reflected from the
upstream reservoir; however, this will also increase the maximum speed rise. However,
the transient simulations show that the maximum turbine speeds under the given closure
law of wicket-gates are well below the allowable maximum speed 1440 rpm (the relaxed
speed constraint), so it is possible to control both the transient pressures and turbine
speeds within desirable range by extending the wicket-gate closure time.
Due to the insensitivity of the second time period (T2) of wicket-gate closure,
T2 = 8 s is kept constant while changing T1 from 5 s to 15 s. The sensitivity study is based
230
on the original system layout with (1) the planned differential surge tank, (2) simple surge
tank without differential features, or (3) no surge tank in the place (certainly neither the
bifurcated pipeline). The simulation results are compared in Figure 8.10 for these three
scenarios.
The effects are only slightly different when using a simple surge tank or a
differential surge tank in this project. For both scenarios, it is found T1 needs to be
increased to 11 s so that the negative pressure can be eliminated and the speed rises are
still tolerable. The transient performance is shown in Figure 8.11 for the planned
differential surge tank and extended wicket-gate closure time T1 =11 s.
However, without a surge tank in the system, it is impossible to meet both
requirements of pressure head and speed rise during the transient by only adjusting the
speed of wicket-gate closure. Additional simulations show that further increasing T1 up
to 35 s (and T2 = 15 s), the maximum pressure head could be reduced to the desirable
value (278 m) with no negative pressures predicated in the system, but the speed would
rise to 1846 rpm. To reduce the maximum speed to below 1440 rpm, a flywheel with
inertia of 28000 kg.m2 would need to be added, which is infeasible due to its excessive
size and cost. The system transient performance under this scenario is shown in Figure
8.12.
231
-120
-100
-80
-60
-40
-20
0
20
5 6 7 8 9 10 11 12 13 14 15
WG Closure Time T1 (s)
Hm
in (m
)
Differential Surge Tank
Simple Surge Tank
No Surge Tank
(a) Min. pressure head in main pipeline vs. wicket-gate closure time T1
200
250
300
350
400
450
500
550
5 6 7 8 9 10 11 12 13 14 15
WG Closure Time T1 (s)
Hm
ax (m
)
Differential Surge Tank
Simple Surge Tank
No Surge Tank
(b) Max. pressure head in system vs. wicket-gate closure time T1
Figure 8.10 Sensitivity analysis of wicket-gate closure time (Con’d)
232
1200
1300
1400
1500
1600
1700
1800
5 6 7 8 9 10 11 12 13 14 15
WG Closure Time T1 (s)
Max
Spe
ed (
rpm
) Differential Surge Tank
Simple Surge Tank
No Surge Tank
(c) Max. speed rise of turbine unit vs. wicket-gate closure time T1
Figure 8.10 Sensitivity analysis of wicket-gate closure time
(a) Envelope of max. and min. HGLs along main penstock
Figure 8.11 System responses during full load rejection with designed differential
surge tank and extended wicket-gate closure law (T1 = 11 s, T2 = 8 s) (Con’d)
233
(b) Pressure variations at spiral case inlet
(c) Turbine speed rise
Figure 8.11 System responses during full load rejection with designed differential
surge tank and extended wicket-gate closure law (T1 = 11 s, T2 = 8 s)
234
(a) Envelope of maximum and minimum HGLs
(b) Turbine speed rise
Figure 8.12 System responses during full load rejection with extended wicket-gate
closure law (T1 = 35 s, T2 = 15 s) and flywheel (adding inertia 28000 kg.m2) for no
surge tank scenario
235
8.2.4 Sensitivity Analysis for Surge Tank Design Parameters
Sensitivity of diameter D of surge tank: For simple surge tank, the diameter change will
not cause an observable difference for either the maximum waterhammer pressure or the
overspeed, but the water level in the tanks does change with tank D and thus the tank
height could be reduced by increasing the tank diameter. The sensitivity analysis results
are shown in Table 8.3 and Figure 8.13.
With a differential surge tank, the diameter change will not cause an
observable difference for either the maximum waterhammer pressure or the overspeed,
and the variation of water level in the surge tank is also very slight. This is because the
current differential model is actually "a simple tank plus an overflow riser", and the
overflow water won't enter back into the big tank. This simplification maybe a concern
for small diameter of tank (e.g., D = 8 m or 6 m), the overflow would stabilize the water
level oscillations in the tank.
236
Table 8-3 Sensitivity Analysis of Simple Tank Diameter
Tank Dia
D
Tank Cross- Section Area
Turbine Max H
Over- speed
Max. HGL at Spiral Inlet
HGL @ Tee WL @ Tank
Tank Min.Height
(m) (m2) (m) (rpm) (m) (m) (m) (m)
10 78.54 226.6 1439 921.11 904.4 890.5 5.5
12 113.10 226.6 1439 921.11 904.4 889.0 4.0
14 153.94 226.6 1439 921.11 904.4 888.1 3.1
16 201.06 226.6 1439 921.11 904.4 888.4 2.4
18 254.47 226.6 1439 921.11 904.4 886.9 1.9
20 314.16 226.6 1439 921.11 904.4 886.6 1.6
Notes:
1. WG closure time T1=13 s, T2=18.33 s;
2. Feeder and connector pipe diameter d = 5 ft = 1.524 m, A=1.824 m2;
3. The resistance coefficient at the tank bottom orifice is defined as , and the coefficient Rin = Rout = 0.005 is assumed;
2/ QhR f=
4. Tank height needs to add the safety margin.
0.0
1.0
2.0
3.0
4.0
5.0
6.0
10 12 14 16 18 20
Tank Diameter (m)
Net
Hei
ght o
f Tan
k (m
)
Figure 8.13 Sensitivity of simple tank diameter D
237
Sensitivity of the connector pipe diameter Dc: When the connector pipe
diameter is reduced, the associated magnitude of resistance coefficient is increased as
square rate of the pipe/orifice area. Moreover, the resulting maximum waterhammer
pressure and overspeed do increase with the decrease of connector pipe diameter Dc. For
differential surge tank, when Dc reduces to 4 ft from current 5.25 ft, the maximum
pressure head increases from 212 to 216 m; moreover, overspeed increases from 1437 to
1440 rpm. Interestingly, the magnitude of resistance coefficient itself is insensitive to the
system transient performance, while the magnitude of connector/orifice area does result
in a changed transient performance. Various simulation results are summarized in Table
8-4 and Figure 8.14.
Table 8-4 Sensitivity Analysis of Feeder and Connector Pipe Diameter
Pipe Dia D
Pipe Dia D
Pipe Cross-Section Area
ResistanceCoefficient R
TurbineMax. H Over-Speed
HGL @ Tee
WL @ Tank
(ft) (m) (m2) (m) (rpm) (m) (m)
3 0.914 0.657 0.042 291.7 1549 974.7 888.0
4 1.219 1.167 0.013 251.5 1484 930.9 889.7
5 1.524 1.824 0.005 226.6 1439 904.4 890.5
6 1.829 2.627 0.002 210.9 1408 892.2 890.8
Notes:
1. WG closure time T1=13 s, T2=18.33 s;
2. Simple Surge Tank with D = 10 m;
3. The resistance coefficient at the tank bottom orifice is defined as (Rin = Rout), which is inversely proportional to the square of connector pipe area, plus 10% of additional head loss in the connector pipe;
2/ QhR f=
4. The connector pipe has the same diameter as the feeder pipeline
238
887.0
887.5
888.0
888.5
889.0
889.5
890.0
890.5
891.0
3.0 3.5 4.0 4.5 5.0 5.5 6.0
Pipe Diameter (feet)
Wat
er L
evel
in T
ank
(m)
200
210
220
230
240
250
260
270
280
290
300
Max
.pre
ssu
re h
ead
at p
ower
hou
se(m
)
WL @ tank (m) Turbine Hmax (m)
(a) Tank water level/max. pressure head at PH vs. pipe diameter Dc
800
820
840
860
880
900
920
940
960
980
1000
3.0 3.5 4.0 4.5 5.0 5.5 6.0
Pipe Diameter (feet)
Pres
sure
hea
d at
Tee
(m)
1400
1420
1440
1460
1480
1500
1520
1540
1560
Turb
ine
unit
spee
d ris
e (r
pm)
HGL @ Tee (m) Speed Rise (rpm)
(b) Pressure at Tee and turbine speed rise vs. pipe diameter Dc
Figure 8.14 Sensitivity of feeder and connector pipe diameter Dc
239
8.3 Air-Cushioned Surge Tank Scheme
With the advance of underground engineering technology, more and more hydro projects
with high-head and long-headrace-tunnel are being constructed (e.g., most pump-storage
hydro projects). For this type of hydro project, the topographic condition often doesn’t
permit easy excavation of a large conventional open surge tank. Therefore, an innovation
in recent hydropower plant design is to replace the conventional open surge tank by an
air-cushioned chamber, especially when the geological condition is favorable and thus
there is no concern with air leakage or specific water treatment issues. In many cases, it
saves significantly on the initial investment compared with the conventional surge tank,
though the air pressure in the chamber needs to be well monitored and controlled during
operation, and a long time period may be necessary to compensate for the compressed air
subsequent to the system drainage and overhaul. In addition, the required stable cross-
sectional area is sometimes larger than that of conventional surge tank though the total
volume of chamber is less, so a long-hall-shaped tunnel with arched-cover is often
constructed. Such an arched chamber also helps stabilize the rock structurally while still
providing the required cross-sectional area. In other words, the detailed design of an air
surge tank must be optimized to maximize the economical and technical benefits (Bergh
Christensen 1982; Broch E. 1984, Gu Zhaoqi 1986, Zhang Jian et al. 2000, Liu Deyou et
al. 2000; Fan Boqin et al. 2005)
In the Bone Creek project, due to the topographical constraint on the
conventional open surge tank, the use of an air-cushioned surge tank near the powerhouse
was explored. As shown in Figure 8.15, an air-cushioned surge tank can be designed and
240
installed at the main penstock line as close as possible to the powerhouse (L1 = 5882 m,
L2 = 75 m, D1 = D2 = 1.98 m are assumed in the simulations), and the surge pipeline is
certainly eliminated. A simple cylindrical tank as sketched in Figure 8.16 is first
simulated; then a more realistic pipe-like air chamber is explored. As is typical in this
kind of system, the upper part is filled with compressed air gas and the lower part stores
the water that is exchanged with the pipeline as the pressure changes. Taking advantage
of air cushion allows the peak pressure of hydraulic surge to be suppressed during
transient events.
Figure 8.15 System layout with suggested underground air-cushioned surge tank
(Note: The drawing is directly copied from TransAM interface; the red digitals represent
the Node No./Identifiers and the black digitals represent the Pipe No./Identifiers)
8.3.1 Design Parameters and Input Data for Simple Cylinder Air Surge Tank
Figure 8.16 helps define the key input parameters for simulation of an air-cushioned
surge tank include:
1) The diameter of the tank D=10 m and its height h=12 m, although the key issue is that
the tank has a total volume of about 1000 cubic meters (air and water combined);
241
2) The elevation of tank bottom is 700 m;
3) The net hydraulic head loss resistance (R, defined as ) for inflow and
outflow are assumed around 0.0001 s
2RQhf =
2/m5, but this would vary depending on the
relative size of connection to the main penstock line;
4) The local free atmosphere pressure = 9.55 m WC based on local elevation of the
project;
5) The polytropic exponent of the air gas, n = 1.6. Researches in China and Norway show
that n = 1.2-1.4 may not be sufficiently safe for prediction of the maximum gas
pressure and maximum water level in the tank (Zhang Jian, et al, 2004).
6) The initial water depth in the chamber Z0 = 4 m, so the height of air-chamber l0 = 8 m;
7) There are two ways to input the initial absolute air pressure P0 in the chamber. One
way is to input the number of P0 directly; another way is to input the static hydraulic
head (same as the WL at upstream reservoir 882 m) and then the software program
will automatically calculate the initial pressure P0 in the air chamber.
Figure 8.16 Sketch of simple cylinder air-cushioned surge tank
242
8.3.2 Transient Performance with Simple Cylinder Air Surge Tank
Based on the system model (Figure 8.15) and the above design of air-cushioned surge
tank (other system data are the same in Table 8-1), the transient event of full load
rejection is simulated, and the results are shown in Figure 8.17. The maximum peak
pressure at the spiralcase inlet is 960 m (Figure 8.17 (b)), and the maximum speed is
1142 rpm with normal closure of wicket-gate (Figure 8.17 (f)). Due to its better location
(shorter wave reflection time and reduced inertia of the intervening water), the concerns
with negative pressures are eliminated. Compared with the conventional surge tank
under the same condition, the peak of the pressure waves is reduced, though it is more
persistent. It is found that the transient waves have not declined in 600 s but they have
declined in 200 s for the corresponding case with open surge tank (Figures 8.4 and 8.8).
Another feature of air-cushioned tank is that the slope of pressure waves is related to the
water level, the peak pressure waves are sharper than the downsurge waves (Fan Boqin,
2005).
8.3.3 Sensitivity Analysis of Air Chamber Height
Table 8-5 and Figure 8.18 show the sensitivity of the height of air-cushioned surge tank
to the system transient performance while the initial water depth in the chamber, Z0 ,
keeps at 4 m; both the maximum turbine speed and maximum pressure head at the inlet of
spiralcase increase rapidly with the decrease of tank height. To control the peak pressure
below EL 975.5 m (i.e., the maximum head below 280.5 m), the minimum required
height of air-cushioned surge tank for this project would be between 11 to 12 m with as
per design described above.
243
(a) Envelope of max. and min.. HGLs along main penstock
(b) Pressure variations at spiralcase inlet
Figure 8.17 System responses during full load rejection with originally specified wicket-gate closure (T1=6 s, T2= 8 s) and simple cylinder air surge tank (1000 m3 in
volume) (Con’d)
244
(c) Water level and flow at air-cushioned surge tank
(d) Nodal pressure head and air gas pressure head at air-cushioned surge tank
Figure 8.17 System responses during full load rejection with originally specified wicket-gate closure (T1=6 s, T2= 8 s) and simple cylinder air surge tank (1000 m3 in
volume) (Con’d)
245
(e) Air volume in air-cushioned surge tank
(f) Turbine speed rise
Figure 8.17 System responses during full load rejection with originally specified
wicket-gate closure (T1=6 s, T2= 8 s) and simple cylinder air surge tank (1000 m3 in volume)
246
Table 8-5 Sensitivity of cylinder air surge tank height to transient performance
Tank Height (m) Max. Speed (rpm) Max. Pressure Head (m)
8 1156 319
10 1146 284
12 1142 265
14 1140 252
16 1139 243
18 1138 237
230
250
270
290
310
330
350
8 10 12 14 16 18
Air Chamber Height (m)
Max
imum
Pre
ssur
e H
ead
(m)
1130
1135
1140
1145
1150
1155
1160
Max
imum
Tur
bine
Spe
ed (r
pm)
Max. Pressure Head
Max. Turbine Speed
Figure 8.18 Sensitivity analysis of the height of simple cylinder air surge tank
247
8.3.4 Transient Performance with a Pipe-like Air Surge Tank
From a construction and cost perspective, a more feasible plan for this system is to install
a pipe-like chamber, parallel to the penstock, starting from the bottom of the last steep
section near the powerhouse (75 m upstream from powerhouse and the tank bottom
elevation is 700 m). The air chamber pipe would use the same material and size as the
adjacent penstock (i.e., 1.98 m in diameter), and would be tentatively designed for 160 m
long sloping upwards at 16.1 degree (assuming 40% of volume is initially water-
occupied), so the total volume is around 500 m3 (a value that could be later optimized
through simulation). The pipe-like air chamber has hemispherical heads at each end and
an outlet pipe in the bottom end, and the outlet pipe diameter is about 1200 mm (or 1980
mm) with a 45-degree bend connecting to the penstock. An access manhole of 600 mm
and make-up compressed air supply will be set up at near the top of the air chamber.
If the originally specified wicket-gate closure law is retained, the above
dimension of air chamber is insufficient to control the system transient response within
the desirable range. As shown in Figure 8.19, there is no negative pressure head in the
system but the peak waterhammer pressure (1020 m EL) at the spiralcase inlet is
considered too high. Moreover, this high-pressure surge is hardly to be mediated by
extending the wicket-gate closure time. As shown in Figure 8.20, even if the closure time
T1 extends to 18 s (and T2 = 8 s), the peak pressure is again 1014 m EL, though the
overspeed 1402 rpm is acceptable for the relaxed criterion.
248
(a) Envelope of maximum and minimum HGLs along main penstock
(b) Water level and flow at air-cushioned surge tank
Figure 8.19 System responses to full load rejection with originally specified wicket-gate closure (T1=6 s, T2= 8 s) and a pipe-like air surge tank (500 m3 in volume)
249
(a) Envelope of maximum and minimum HGLs along main penstock
(b) Turbine speed rise
Figure 8.20 System responses to full load rejection with extended wicket-gate closure (T1= 18 s, T2= 8 s) and pipe-like air surge tank (500 m3 in volume)
250
(a) Envelope of max. and min. HGLs along main penstock
(b) Turbine speed rise Figure 8.21 System responses to full load rejection with originally specified wicket-
gate closure (T1=6 s, T2= 8 s) and a greater size of air surge chamber (840 m3 in volume)
251
However, as Figure 8.21 shows, the peak pressure is reduced to 974 m EL
when the volume of air chamber increases to 840 m3 (say either by increasing the pipe
diameter from 1.98 m to 2.6 m with a constant length of 160 m; by increasing pipe length
from 160 m to 280 m keeping the 1.98 m in diameter; or by using a pipe with 250 m
length with 2.07 m diameter).
Table 8-6 and Figure 8.22 show the peak waterhammer pressure at the inlet of
spiralcase reduces with the increase of the total length of air-chamber-pipe when using
the same diameter 1.98 m. All these analysis results are based on the originally specified
wicket-gate closure law and 40% of initial water level in the pipe-like air chamber.
Simulation also shows that if the total chamber volume and initial water level
are constant, there is no difference in the results when the pipe-like air chamber is
simulated as an actual slopped pipe-like chamber (with shorter but greater cross-section
due to the slopped angle) or as an equivalent vertical air chamber.
Table 8-6 Sensitivity of air-chamber-pipe length
Air-chamber-pipe length L (m) Max. HGL (m)
160 1020
200 998
240 984
250 981
280 974
252
Peak Pressure Head vs. length of Air-chamber-pipe
970
980
990
1000
1010
1020
150 170 190 210 230 250 270 290
length of Air-chamber-pipe L (m)
Pea
k P
ress
ure
Hea
d E
L (m
)
Figure 8.22 Sensitivity of length of air-chamber-pipe to peak pressure during full load rejection with originally specified wicket-gate closure (T1=6 s, T2= 8s) and a
pipe-like air surge tank (500 m3 volume)
8.4 Summary and Conclusions
This chapter documents the search for feasible solutions to the excessive transient
pressures and speed rise associated with two units experiencing simultaneous full load
rejection. More specifically, systematic simulations and sensitivity analysis of important
system parameters were performed. The main conclusions are summarized as follows:
1. For the open surge tank scheme to be a viable way meeting all the imposed
constraints of head change and speed rise, the surge tank should be constructed
within 200 m distance from the powerhouse if both the planned surge tank scheme
and originally specified wicket-gate closure law were maintained. Unfortunately,
such a location is infeasible due to the topographic constraints.
253
2. The simulations show that increasing the size of the surge tank could not improve
system transient performance effectively and sufficiently if both the planned surge
tank location and wicket-gate closure law were remained.
3. However, more positively, adjusting the speed of the wicket-gate closure does
significantly reduce the peak of original upsurge and reduces/eliminates the extent
and duration of the possible negative pressure in the system. Although there is a
slight increase in the speed rise of the turbine unit, this speed rise is still tolerable.
To be more specific, a staged closure curve is quite feasible: if the first 70% of the
wicket-gate opening is closed linearly in 11 s, and the final 30% closed in 8 s, then
both the transient pressure and turbine speed rise can be kept within the desired range
for the current system layout and surge tank design. Moreover, the transient
response, both in terms of head rise and speed rise, is insensitive to the duration of
the second stage closure; variations of up to 50% have no visible influence on surge
conditions.
4. There is no significant difference between the effects of differential surge tank and
simple surge tank; but the transient response would be much worse if there is no
surge tank in the system.
5. For the case with no surge tank, the peak pressure rise can be constrained by
extending wicket-gate closure time to 35 s for the closure of the first 70% of the
wicket-gate opening; but the peak speed would be too high. Moreover, increasing the
inertia of Turbine-Generator rotating parts by adding a flywheel is an ineffective way
to reduce the peak speed rise to the desired range.
254
6. When the overspeed constraint is relaxed to 160% of normal speed (1440 rpm), the
wicket gate close time can be extended to 14 s for the first 70% of the full gate
opening and 18.7 s for the remaining 30% of opening, and thus the currently planned
system layout and open surge tank scheme (either simple or differential surge tank)
would be sufficient to meet the constraints of the transient pressures. This probably is
the most efficient way to eliminate the negative pressure and excessive high
pressures in the system during full load rejection.
7. Another effective way to improve the system’s transient performance is to replace
the differential surge tank with an air-cushioned surge tank nearby the powerhouse.
Such an air-cushioned surge tank could be installed underground nearby the
powerhouse provided the geological conditions are favorable, or could be configured
as a suitable length of pipe. To be effective in meeting the hydraulic constraints, a
proposed pipe-like air chamber needs approximately 840 m3 in volume (the pipe with
250 m length and 2.1 m in diameter is suggested). Such a dimension would be
associated with the originally specified wicket-gate closure law. Sensitivity studies
have shown that the exact shape for configuring the air chamber volume is of less
consequence. Moreover, as long as the connecting pipe is of adequate size, the
nature of the connecting pipe to the air chamber is of strictly secondary importance
compared to the volume of air provided. Moreover, given the fact that many of the
air chamber’s requirements are symmetrical in that they involve both load acceptance
and load rejection (upsurge and downsurge), there is little advantage to arranging a
significant variation between inflow and outflow orifice characteristics.
255
Chapter 9 Thesis Summary and Suggestions for System
Transient Performance Evaluation
This thesis focuses on the transient analysis associated with a general class of hydraulic
components, herein called active hydraulic devices (AHDs), and on several specific
realizations of these devices. AHDs are of strategic importance in the context of a
hydraulic system, as they are either the source of transient disturbances, or a critical
strategy for surge control and migration.
In particular, a new strategy using an AHD was explored which seems to
eliminate/mitigate wave reflection and potential resonance in pipelines in order to ensure
adequate transient control and protection. The application of AHDs entails a profound
effect on system behavior and, in turn, response to such devices is influenced by the
details of system design and investment as well as its operation and management. In fact,
it is emphasized that there is no ‘safe’ territory for AHDs and no simple way to achieve
better transient performance; rather, one needs to understand the nature of a system’s
response with or without AHDs. Moreover, AHDs play an important role not only in
transient management but also relative to overall system control and behavior.
Automatic control valves are the first type of active hydraulic devices studied
in this research. Such a valve could be either a cause of transient flow or a measure to
protect a system from its detrimental effects; it is often a key element in a water system
used to control the flow rate, reverse flow, pipe pressure or water level in a tank. Due to
256
the challenges inherent in the design of a pressure relief valve (PRV), the general
principles of PRV use and selection are studied along with the sensitivities of a system’s
response to the PRV parameters. Then, a numerical model for conventional PID
(proportional, integral and derivative) control valves is studied. Based on the model, a
new application of PID control valves is envisioned that combines a remote sensor
upstream in a pipeline to create a non- or semi- reflective boundary at the downstream
end. The case study shows that, at such a boundary, the valve is able to adjust its opening
automatically in response to the remotely sensed pressure changes in order to eliminate or
mediate the reflection and resonance of pressure waves within the conduit.
The second type of active hydraulic device studied in this research is the
governed hydro-turbine, the most complicated hydraulic component in terms of transient
analysis and waterhammer control. Although there are a number of numerical models
developed for analyzing transient flows cuased by turbine operations, to the author’s
knowledge, previously published studies were not directly applicable to employment in a
water network. In this thesis, a complete numerical model is developed for turbine
installations in either urban water networks for energy recovery purposes or
conventional/pumped-storage hydropower generation systems. Two case studies are
undertaken and presented for realistic hydro projects.
The numerical models developed in this research represent an enhanced ability
to simulate and even control a pipe system’s transient performance. It is essential to
evaluate the transient performance and control efficacy for a pipe system design and to
identify and discard the inappropriate design schemes based on suitable design criteria
(e.g., the acceptable range of max. and min. pressures).
257
9.1 Perspectives on System Transient Performance Evaluation
Transient flow is unavoidable in any dynamic hydraulic system, and threats to system
integrity not only arise from pressure surges but from potential resonance and auto-
oscillations initiated by transient pressure or flow fluctuations in poorly designed and
tested systems. Furthermore, even though less visible, excessive pressure drop
experienced during transient variations is also crucial. While pressure extremes are well
appreciated as a source of trouble in systems, difficulties persist from system overload
and even pipe burst is not rare if careful prediction and system design are neglected.
However, decreased pressure, especially during vacuum conditions (as discussed in the
case study of Chapter 8) can cause a series of detrimental phenomena, such as air release
from solution, water vaporization, water column separation and cavitation. A pipe could
collapse because of negative pressure or burst due to high pressure from water column
rejoinder following separation; air and contaminants could be drawn into a pipeline
through any breach of system integrity (transient intrusion) and water quality risks being
compromised by possible pollutant intrusion for municipal water supply systems; while
the ingress of air could in turn markedly degrade transient pressures.
Overall, transient control and protection has proven to be a complex task in
water systems, and there are many helpful techniques for controlling or influencing
system response. These can be categorized according to three basic approaches: 1) to
enhance the strength of the system or change the pipeline profile; 2) to control the source
of transients (such as avoiding check valve slam, moderating rapid valve opening); and 3)
to suppress the transient using various hydraulic devices (such as PRVs, bypass lines, air
valves, surge tanks, and air chambers). Some devices may require lower initial
258
investment but higher maintenance cost (or pose an elevated risk of malfunction) during
the operation stage. The question is how to combine these techniques to form an
integrated strategy for system transient control and protection? What trade-offs and
benefits are expected to arise from the application of these devices? Which design
scheme or solution is the more cost-effective and energy-efficient for a specific system or
particular issue?
Measures of waterhammer mitigation should be ultimately refined and
scrutinized by technical evaluations (maximum/minimum transient pressure, probability
and safety performance), evaluation at the economic level (cost-effectiveness, including
risk cost due to transient failure), and considerations at the environmental level (energy-
efficiency and greenhouse gas emissions).
Ultimately, as part of future work, the assessment of system transient
performance should not only be based on the traditional deterministic transient analysis
(using the numerical models developed in the thesis) but also on a stochastic model and
risk analysis, because randomness is the nature of transient flow due to the variability of
initial conditions (hydrological variations) and the uncertainties in system parameters and
variables. In fact, the impetus for a shift from the traditional deterministic approach to a
transient stochastic analysis in hydropower facilities is three-fold:
Firstly, there is a need for adequate hydraulic performance assessment; transient
performance should be evaluated in a more informative and comprehensive way based on
the spectrum of pressures that systems experience. Beyond just knowing the extreme
values of transient pressures, an estimate of the likelihood of their occurrence would be
enormously useful. This requires statistical characterization and simulation of probability
259
distributions representing transient pressures, rather than the more limited knowledge of
max./min. pressure obtained from deterministic analysis under worst-case scenarios.
Secondly, there is a structural design motivation. In recent decades, there has
been an obvious discrepancy between the popular reliability-based design approach and
traditional deterministic waterhammer evaluation. Excessive pressure unleashed by
transient events is one of the random loads endured by a hydraulic structure, and its
stochastic information is essential to the reliability-based design of such constructions, in
which the normative loads, partial load factors and safety factors are derived from the
probability characteristics of each load. Furthermore, there is an obvious discrepancy
between the current structural design methodology and traditional deterministic transient
load analysis. Indeed, the advanced hydraulic structural design based on reliability theory
calls for a stochastic analysis of transient flow and an analysis of the statistical features of
transient pressures in pipelines and other hydraulic facilities; however, the pressure
surges as loads used in hydraulic structural design are still analyzed using a deterministic
approach by calculating the maximum values under the worst case scenarios. Therefore,
the design guidelines or regulations about transient loads in pressure pipe systems need to
reflect the observations of stochastic transient and risk analysis. For this reason, statistical
data associated with transient analysis in conventional hydro systems was collected
during my previous studies in China (Zhang et al. 1997, 1998, 2000s).
Thirdly, stochastic transient analysis can provide useful insights and contribute
to life cycle risk, economic and environmental assessment studies. Life cycle assessment
is a systematic methodology for evaluating the environmental impacts of different design
options. The life cycle monetary and environmental expenses are associated with the risk
260
of system failure, from which no system is completely immune. Even for a well-designed
system with appropriate waterhammer provision, the probability of failure due to
transient events is never absent because of potentially wide hydrological variations,
component failure, possible nefarious actions and other inherent uncertainties. Expected
damages due to transient failure (i.e., the product of an incident’s risk of occurrence and
its corresponding damage) must be taken into account in the life cycle monetary cost and
environmental burden of a system. In fact, only when the probabilistic characteristics of
extreme pressures are obtained from a stochastic transient model can risk analysis be
performed and integrated within the life cycle analysis (LCA) framework for holistic
system assessment.
At the economic and environmental levels, LCA can be implemented to assess
the monetary cost and energy consumption of competitive alternatives during the
protracted service period. Some design schemes may require lower initial investment but
result in high maintenance and reparation costs during the operation stage (Zhang et al.,
2007). Reconsidering the case study of Bone Creek Hydro Project presented in Chapter 8,
various technically feasible alternatives, such as the air-cushioned chamber deployed near
to the powerhouse, or the turbine-generator units manufactured according to an upgraded-
standard that are able to withstand higher overspeed during load rejection (with extended
wicket-gate closure time), could be assembled to eliminate the negative pressures in the
penstock upstream sections and reduce excessive high pressure to within acceptable
range, but which alternative is the best solution? To answer this question, the life cycle
cost including the risk cost, as well as the energy used and greenhouse gases generated
through the lifespan of each alternative solution should be evaluated and compared.
261
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APPENDICES:
AppendixⅠ: Turbine Characteristics Input Data Sample and Variable Index Turbine Characteristics Input Data Sample: | No. of turbines (NTURB) 1 | TURBINE-TYPE QR HR NR PR ETR Runner Dia. | (cms) (m) (rpm) (KW) (%) (m) 'Francis' 191.5 142.0 187.5 250000 0.889 4.5 | No. of turbine wicket gate sets (Ndy) 10 | The full opening a00(mm) 23.0 | GATE:3mm | No. Points (NPP) a0(mm) 7 3 | N11 Q11 EFFY | (r/min) (m^3/s) (%) 68.81 0.1582 47.193 71.46 0.1567 42.275 73.97 0.1548 37.363 76.42 0.1535 30.857 85.04 0.1407 0.000 84.05 0.1420 0.086 82.58 0.1443 5.173 | GATE:5mm | No. Points (NPP) a0(mm) 7 5 | N11 Q11 EFFY | (r/min) (m^3/s) (%) 68.91 0.1981 56.090 71.40 0.1958 54.124 73.98 0.1943 50.944 76.43 0.1915 46.841 88.31 0.1774 7.021 88.84 0.1767 4.213 90.37 0.1718 0.000
269
| GATE:9mm | No. Points (NPP) a0(mm) 10 9 | N11 Q11 EFFY | (r/min) (m^3/s) (%) 68.85 0.3680 79.151 71.42 0.3636 79.872 73.98 0.3595 78.986 76.49 0.3534 76.914 64.09 0.3717 78.877 81.06 0.3471 72.030 90.31 0.3153 50.363 100.19 0.2858 14.472 103.53 0.2698 0.000 110.68 0.2114 0.000 | GATE:11mm | No. Points (NPP) a0(mm) 10 11 | N11 Q11 EFFY | (r/min) (m^3/s) (%) 68.87 0.4587 84.601 71.46 0.4532 85.307 73.93 0.4496 85.709 76.40 0.4429 84.643 64.10 0.4671 83.574 81.01 0.4346 82.295 91.43 0.3925 66.861 100.45 0.3707 34.993 110.06 0.3067 0.000 115.36 0.2460 0.000 | GATE:13mm | No. Points (NPP) a0(mm) 10 13 | N11 Q11 EFFY | (r/min) (m^3/s) (%) 68.82 0.5329 88.033 71.45 0.5255 88.833 73.94 0.5191 88.268 76.45 0.5144 87.652 64.13 0.5382 87.226 81.04 0.5033 85.594 90.10 0.5005 77.221 100.40 0.4737 60.334 109.84 0.4205 17.679
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120.44 0.3034 0.000 | GATE:15mm | No. Points (NPP) a0(mm) 10 15 | N11 Q11 EFFY | (r/min) (m^3/s) (%) 68.91 0.6235 91.389 71.40 0.6165 91.244 73.88 0.6094 90.631 76.45 0.6041 90.210 63.99 0.6286 89.980 81.00 0.5893 87.944 90.51 0.5756 81.369 100.15 0.5548 66.185 109.78 0.4862 29.537 122.38 0.3954 0.000 | GATE:17mm | No. Points (NPP) a0(mm) 12 17 | N11 Q11 EFFY | (r/min) (m^3/s) (%) 68.85 0.7259 92.719 71.48 0.7216 92.959 72.51 0.7193 93.391 73.98 0.7177 93.187 76.39 0.7087 92.311 80.94 0.6966 89.988 63.98 0.7295 90.868 90.10 0.6623 80.822 100.26 0.6362 67.261 110.21 0.6005 48.442 120.61 0.5036 0.000 125.18 0.4767 0.000 | GATE:19mm | No. Points (NPP) a0(mm) 11 19 | N11 Q11 EFFY | (r/min) (m^3/s) (%) 68.82 0.7873 91.825 71.44 0.7833 92.210 73.93 0.7795 92.721 76.44 0.7770 92.683 64.10 0.7881 91.121
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81.06 0.7614 91.678 90.42 0.7127 80.934 100.22 0.6833 67.873 110.37 0.6466 49.790 119.92 0.5522 1.524 126.96 0.5082 0.000 | GATE:21mm | No. Points (NPP) a0(mmm) 14 21 | N11 Q11 EFFY | (r/min) (m^3/s) (%) 64.07 0.8517 89.998 68.81 0.8440 90.682 71.40 0.8396 91.048 73.92 0.8337 91.502 76.41 0.8313 91.567 81.05 0.8223 90.789 90.14 0.7899 82.723 100.23 0.7522 68.686 110.43 0.7065 48.724 120.18 0.6672 23.940 121.47 0.6272 4.861 122.33 0.6190 0.653 123.26 0.6103 0.000 130.33 0.5591 0.000 | GATE:23mm | No. Points (NPP) a0(mm) 14 23 | N11 Q11 EFFY | (r/min) (m^3/s) (%) 68.92 0.9017 89.415 71.32 0.8984 89.596 73.94 0.8935 89.692 76.50 0.8859 89.311 64.08 0.9099 88.024 80.94 0.8762 88.195 124.68 0.6557 0.000 123.57 0.6640 0.684 122.83 0.6722 4.854 90.05 0.8588 81.104 100.40 0.8195 67.687 110.36 0.7642 47.491 120.27 0.7221 23.537 132.19 0.6194 0.000
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Turbine Characteristics Input Variable Index:
KTCHAR—Turbine Characteristics (INTEGER) The number of sets of different head
and torque characteristics needed for the simulation. KTCHAR actually is the number of
different turbines in system, each of them with different characteristics. For the turbines
with the same head and torque characteristics, only one set of data is needed to input, but
the variable TCODE in *.DAT file should be specified to determine the order of sets in
TURB*.CHR for each turbine unit. The maximum number of KTCHAR is 40 as many as
the maximum turbines in a system.
TURBTYPE —Turbine Type Specifier (CHARACTER) Current version of TransAM
can deal with three types of turbines, that is, 1) ‘Fixed WG’ (no wicket gates or the wicket
gates fixed); 2) ‘Francis’ or ‘WG’ (turbines with wicket gates, Francis turbine is a typical
example); 3) ‘Kaplan’ (turbines with wicket gates and blades); and 4) ‘Pump-turbine’
(reversed turbine used in pump-storage hydropower plant). As for the Impulse turbine, it
is treated as an equivalent of BCTYPE=’Valve’. Actually, only Kaplan turbine needs the
exact input of ‘Kaplan’, which will infer other inputs for information of its blades.
In subroutine of solution of turbine boundary conditions, the type of turbine is determined
automatically by number of blade angle settings (NBlade) and the number of wicket-gate
settings (Ndy), and both of them are input from the turbine characteristics file
TURB*.CHR in BNDCND.f90.
QR—Rated Turbine Flow (REAL)
HR—Rated Turbine Head (REAL)
NR—Rated Turbine Speed (INTEGER)
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ETR—Rated Turbine Efficiency (REAL)
D1—Diameter of Turbine Runner (REAL)
NBlade—Turbine Blades Angle Settings (INTEGER) Nblade>1 (up to 10) for Kaplan
turbine; NBlade=1 for other types of turbines which can be determined automatically by
program. Therefore, this input is needed only for Kaplan turbines.
Ndy—Turbine Wicket-gate Settings (INTEGER) The number of wicket-gate settings
between fully closed position y=0 to fully open position y=1.0, it is variable between (6-
15) depending on the available model test data, more data available higher accuracy in
modeling.
a00—Absolute Value of Full Wicket-gate Opening (REAL) Unit in mm
NPP—Input Data Points for Each Wicket-gate Position (INTEGER)
a0—Absolute Value of Each Wicket-gate Opening (REAL) Unit in mm
N11—Unit Speed of Model Turbine (REAL) Definition is found in chapter 2.
Q11—Unit Flow of Model Turbine (REAL) Definition can be found in chapter 2
EFFY—Efficiency of Model Turbine (REAL) The efficiency should be modified from
model to prototype.
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AppendixⅡ: Turbine Characteristics Output Data Sample
------------------------------------------------------------------------- | Total No. of Turbine Characteristics-KTCHAR
1 | TURBINE No. 1 (Starting with first turbine characteristics) | Blade Angle set No.1 (Nblade=1 for Francis turbine) | NBlade Ndy Ndx (Ndx – Number of points at each gate and same for all gates) 1 10 89 | Wicket-gate set No. 1
0.1304348 (This is relative wicket-gate opening, used in interpolation)
| Head-WH characteristic (89 points of Suter Head Parameter from x =–180˚to 180˚) 1.570 1.592 1.614 1.637 1.661 1.685 1.710 1.735 1.762 1.789 1.817 1.846 1.875 1.906 1.937 1.969 2.003 2.037 2.073 2.109 2.147 2.186 2.226 2.267 2.310 2.354 2.400 2.447 2.495 2.545 2.597 2.650 2.705 2.761 2.819 2.879 2.940 3.002 3.066 3.131 3.196 3.262 3.328 3.394 3.459 3.521 3.581 3.636 3.685 3.725 3.755 3.771 3.769 3.745 3.694 3.611 3.489 3.322 3.103 2.826 2.488 2.087 1.625 1.107 0.659 0.287 -0.096 -0.478 -0.846 -1.191 -1.504 -1.778 -2.013 -2.206 -2.361 -2.479 -2.565 -2.624 -2.659 -2.675 -2.674 -2.661 -2.637 -2.606 -2.568 -2.526 -2.481 -2.434 -2.386 | Torque-WB characteristic (89 points of Suter Torque Parameter from x =–180˚to
180˚) 0.481 0.487 0.494 0.501 0.508 0.515 0.523 0.530 0.538 0.546 0.555 0.563 0.572 0.581 0.590 0.600 0.610 0.620 0.631 0.641 0.652 0.664 0.676 0.688 0.700 0.713 0.726 0.740 0.754 0.769 0.783 0.799 0.814 0.830 0.847 0.864 0.881 0.898 0.916 0.934 0.952 0.969 0.987 1.004 1.021 1.036 1.050 1.063 1.073 1.080 1.083 1.081 1.073 1.058 1.034 0.999 0.952 0.890 0.811 0.715 0.600 0.466 0.313 0.145 -0.007 -0.241 -0.478 -0.709 -0.928 -1.129 -1.306 -1.458 -1.583 -1.682 -1.757 -1.810 -1.843 -1.860 -1.864 -1.857 -1.841 -1.819 -1.791 -1.760 -1.725 -1.689 -1.652 -1.615 -1.577 | Wicket-gate set No. 2 0.2173913 | Head-WH characteristic 1.158 1.174 1.191 1.208 1.225 1.243 1.262 1.281 1.300 1.320 1.341 1.362 1.384 1.407 1.430 1.454 1.479 1.504 1.531 1.558 1.586 1.615 1.644 1.675 1.707 1.740 1.774 1.809 1.845 1.882 1.921 1.960 2.001 2.044 2.087 2.132 2.177 2.224 2.272 2.321 2.370 2.420 2.470 2.520 2.570 2.617 2.663 2.705 2.743 2.775 2.799 2.813 2.813 2.797 2.760 2.698 2.607 2.481 2.314 2.103 1.844 1.536 1.179 0.791 0.532 0.342 0.143 -0.057 -0.251 -0.435
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-0.603 -0.753 -0.882 -0.991 -1.079 -1.148 -1.201 -1.239 -1.264 -1.279 -1.285 -1.284 -1.278 -1.266 -1.252 -1.234 -1.215 -1.195 -1.173 | Torque-WB characteristic 0.388 0.393 0.398 0.404 0.410 0.415 0.422 0.428 0.434 0.441 0.447 0.454 0.461 0.469 0.476 0.484 0.492 0.500 0.509 0.517 0.526 0.536 0.545 0.555 0.565 0.576 0.586 0.598 0.609 0.621 0.633 0.645 0.658 0.671 0.684 0.698 0.712 0.726 0.741 0.755 0.770 0.785 0.799 0.813 0.827 0.840 0.851 0.862 0.870 0.876 0.879 0.878 0.871 0.859 0.839 0.810 0.771 0.719 0.653 0.572 0.475 0.362 0.233 0.089 -0.013 -0.064 -0.116 -0.167 -0.215 -0.258 -0.297 -0.329 -0.356 -0.377 -0.392 -0.403 -0.409 -0.412 -0.412 -0.410 -0.406 -0.401 -0.394 -0.387 -0.379 -0.371 -0.362 -0.354 -0.345 | Wicket-gate set No. 3 0.3913043 | Head-WH characteristic 0.648 0.657 0.666 0.676 0.686 0.696 0.706 0.717 0.728 0.740 0.751 0.764 0.776 0.789 0.802 0.816 0.830 0.845 0.860 0.875 0.891 0.908 0.925 0.943 0.961 0.980 1.000 1.020 1.041 1.063 1.085 1.109 1.133 1.158 1.183 1.210 1.237 1.265 1.294 1.323 1.353 1.384 1.415 1.445 1.476 1.505 1.534 1.560 1.584 1.603 1.617 1.624 1.621 1.606 1.575 1.524 1.449 1.344 1.205 1.027 0.820 0.653 0.532 0.433 0.376 0.312 0.244 0.173 0.101 0.032 -0.033 -0.093 -0.146 -0.193 -0.232 -0.265 -0.292 -0.314 -0.330 -0.343 -0.352 -0.358 -0.361 -0.363 -0.362 -0.361 -0.358 -0.355 -0.351 | Torque-WB characteristic 0.438 0.444 0.450 0.456 0.463 0.470 0.477 0.484 0.491 0.498 0.506 0.514 0.522 0.531 0.539 0.548 0.557 0.567 0.577 0.587 0.597 0.608 0.619 0.631 0.642 0.654 0.667 0.680 0.693 0.707 0.721 0.736 0.751 0.767 0.783 0.799 0.816 0.833 0.850 0.868 0.886 0.903 0.921 0.938 0.955 0.971 0.985 0.997 1.007 1.013 1.015 1.011 0.999 0.978 0.945 0.897 0.831 0.744 0.633 0.494 0.338 0.195 0.078 0.005 0.003 0.000 -0.002 -0.004 -0.007 -0.009 -0.011 -0.012 -0.014 -0.015 -0.016 -0.016 -0.017 -0.017 -0.017 -0.017 -0.017 -0.017 -0.017 -0.016 -0.016 -0.016 -0.016 -0.015 -0.015 | Wicket-gate set No. 4 0.4782609 | Head-WH characteristic 0.430 0.436 0.442 0.449 0.456 0.462 0.470 0.477 0.484 0.492 0.500 0.509 0.517 0.526 0.535 0.545 0.554 0.564 0.575 0.586 0.597 0.608 0.620 0.633 0.645 0.659 0.673 0.687 0.702 0.717 0.733 0.750 0.767 0.785 0.804 0.823 0.843 0.864 0.885 0.907 0.930 0.953 0.977 1.001 1.025 1.049 1.072 1.094 1.115 1.133 1.148 1.157 1.159 1.152 1.132 1.097 1.041 0.960 0.848 0.714 0.600 0.509 0.423 0.376 0.340 0.298 0.251 0.202 0.152 0.103 0.055 0.011 -0.029 -0.064 -0.095 -0.121 -0.143 -0.162 -0.176 -0.188 -0.197 -0.204 -0.209 -0.212 -0.214 -0.215 -0.215 -0.215 -0.213
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| Torque-WB characteristic 0.383 0.389 0.394 0.400 0.406 0.412 0.418 0.424 0.431 0.438 0.444 0.452 0.459 0.467 0.474 0.482 0.491 0.499 0.508 0.517 0.527 0.536 0.546 0.557 0.568 0.579 0.590 0.602 0.615 0.627 0.641 0.654 0.668 0.683 0.698 0.713 0.729 0.745 0.762 0.779 0.796 0.813 0.830 0.847 0.864 0.880 0.894 0.907 0.918 0.925 0.928 0.925 0.913 0.892 0.858 0.807 0.736 0.640 0.515 0.376 0.249 0.137 0.037 0.004 0.002 0.000 -0.002 -0.004 -0.006 -0.008 -0.009 -0.011 -0.012 -0.013 -0.013 -0.014 -0.014 -0.014 -0.015 -0.015 -0.014 -0.014 -0.014 -0.014 -0.014 -0.013 -0.013 -0.013 -0.013 | Wicket-gate set No. 5 0.5652174 | Head-WH characteristic 0.375 0.380 0.386 0.391 0.397 0.403 0.409 0.416 0.423 0.429 0.437 0.444 0.451 0.459 0.467 0.476 0.484 0.493 0.502 0.512 0.522 0.532 0.542 0.553 0.565 0.577 0.589 0.602 0.615 0.629 0.643 0.658 0.674 0.690 0.706 0.724 0.742 0.760 0.780 0.800 0.820 0.841 0.862 0.884 0.906 0.928 0.949 0.969 0.987 1.003 1.015 1.022 1.021 1.010 0.985 0.942 0.876 0.781 0.674 0.572 0.458 0.392 0.355 0.326 0.289 0.248 0.203 0.155 0.107 0.060 0.016 -0.025 -0.061 -0.092 -0.119 -0.142 -0.160 -0.175 -0.187 -0.196 -0.202 -0.207 -0.210 -0.211 -0.212 -0.212 -0.211 -0.209 -0.207 | Torque-WB characteristic 0.393 0.398 0.404 0.410 0.416 0.422 0.429 0.435 0.442 0.449 0.456 0.463 0.471 0.479 0.487 0.495 0.504 0.513 0.522 0.532 0.541 0.552 0.562 0.573 0.584 0.596 0.608 0.621 0.634 0.647 0.661 0.675 0.690 0.705 0.721 0.737 0.754 0.771 0.789 0.807 0.825 0.844 0.862 0.880 0.898 0.915 0.931 0.944 0.955 0.961 0.962 0.956 0.940 0.911 0.865 0.797 0.702 0.574 0.432 0.299 0.168 0.068 0.020 0.002 -0.017 -0.038 -0.058 -0.077 -0.095 -0.110 -0.124 -0.135 -0.144 -0.150 -0.155 -0.157 -0.158 -0.158 -0.157 -0.155 -0.153 -0.150 -0.147 -0.143 -0.140 -0.136 -0.133 -0.129 -0.126 | Wicket-gate set No. 6 0.6521739 | Head-WH characteristic 0.296 0.301 0.305 0.310 0.314 0.319 0.324 0.330 0.335 0.340 0.346 0.352 0.358 0.365 0.371 0.378 0.385 0.392 0.400 0.407 0.415 0.424 0.433 0.442 0.451 0.461 0.471 0.482 0.493 0.504 0.516 0.528 0.541 0.555 0.569 0.584 0.599 0.615 0.631 0.648 0.666 0.684 0.703 0.722 0.741 0.761 0.780 0.798 0.814 0.829 0.839 0.845 0.843 0.831 0.804 0.757 0.685 0.604 0.532 0.445 0.386 0.350 0.312 0.266 0.214 0.157 0.096 0.035 -0.024 -0.080 -0.131 -0.175 -0.213 -0.245 -0.270 -0.290 -0.304 -0.314 -0.321 -0.325 -0.327 -0.326 -0.324 -0.321 -0.317 -0.313 -0.308 -0.302 -0.297 | Torque-WB characteristic 0.363 0.368 0.373 0.379 0.384 0.390 0.396 0.402 0.409 0.415
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0.422 0.429 0.436 0.444 0.451 0.459 0.468 0.476 0.485 0.494 0.503 0.513 0.523 0.534 0.545 0.556 0.567 0.580 0.592 0.605 0.619 0.633 0.647 0.662 0.678 0.694 0.711 0.728 0.746 0.764 0.782 0.801 0.820 0.838 0.857 0.874 0.891 0.905 0.916 0.923 0.923 0.916 0.896 0.861 0.805 0.722 0.602 0.462 0.337 0.216 0.114 0.043 0.007 -0.032 -0.073 -0.116 -0.157 -0.196 -0.232 -0.263 -0.288 -0.308 -0.323 -0.333 -0.338 -0.341 -0.340 -0.337 -0.332 -0.326 -0.319 -0.311 -0.303 -0.295 -0.287 -0.279 -0.271 -0.263 -0.255
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Appendix Ⅲ: Turbine Boundary Input Data Sample and Variable Index
A Sample of Input Data for Turbine Boundary Condition: ------------------------------------------------------------------------------------------ | DEVICE 4 IS A TURBINE UNIT | BCTYPE NDN BCOUT 'TURB' 2 'OUTPUT' | NLBC(1) NLBC(2) (Node list) 05 06 | TSID 'No.4 TURBINE' | Zav(Htail0) Period(tail) AMP(tail) | NValT Isyn Igov 0 1 2 | TTYPE TCODE ‘Francis’ 1 | TPHA0 TPHAmax TPHAmin | NR QR HR PR N0 Q0 WRR Tg | (rpm) (cms) (m) (Kw) (rpm) (cms) (Kg-m2) (s) 187.5 191.5 142.0 250000 187.5 191.5 72.5E+5 20 | NTy 4 | yG (I) I=1,NTy 0.904 0.904 0.715 0.0 | TyG(I) I=1,NTy 0.0 100.2 104.0 130.2 | NTP 3 | PG(1) PG(2) 250000 250000 0.0 | TPG(1) TPG(2) 0.0 100.0 100.1 ------------------------------------------------------------------------------------
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Input Variable Index for Turbine Boundary Condition:
1) General Data for Turbine Boundary Condition
BCTYPE (CHARACTER) The type of boundary condition, each turbine unit is
designed by ‘TURB’, and in one system can have up to 40 turbine units.
NDN (INTEGER) The number of nodes associated with a turbine boundary
condition. Usually, the turbine upstream is always linked to a penstock, in which
the pressure rise is the concern, but the turbine downstream depends. NDN=2 is set
for an in-line turbine unit, that is, the turbine discharges water to a pressured
conduit or pressured system; NDN=1 for an one-node (external) turbine unit, that is,
its short draft tube can be ignored, and the turbine downstream is considered to be
directly linked with tailrace or reservoir.
BCOUT (CHARACTER) Set BCOUT to ‘OUTPUT’ if time history results are
required. These will be logged every IPRINT time steps in *.BND file.
NLBC (INTEGER) This defines the point of attachment of the turbine unit to the
pipe network. For external turbines only a one-node identifier is required which is
the upstream end of the turbine. The in-line turbines are always defined by
upstream node and downstream node. The first node input in the list is assumed to
be the upstream side and the later the downstream side.
TSID—Turbine Unit Identifier (CHARACTER) Each turbine may be ‘named’
for easier reference in the output results. A string (enclosed by a pair of single
quotes) of up to 35 characters may be used.
NvalT—Total Number of Valves Attached to a Turbine Unit (INTEGER)
280
In current version of TransAM, the maximum 3 valves can be attached to one
turbine unit, that is, 1 Turbine Control Valve (‘TCV’), 1 Bypass Valve and 1 Surge
Valve. If a turbine has installed more than 3 valves or has more complicated valves,
we always can deal with any valve easily as a separate valve boundary condition by
adding one characteristic pipe section at each side of the valve.
Isyn—Identifier of Synchronous or Isolated Turbine Unit. (INTEGER) Isyn=0
for synchronous units, and Isyn=1 for isolated units.
2) Sinusoidal Tail Head For one-node (external) turbines, the downstream/tail water
head can be a constant or varies as a sine wave function because of the wind or other
influences. (Reference to P147, TransAM Manual)
Htail0 = (REAL) the mean piezometric downstream/tail head
Ptail = (REAL) the time required for a complete wave cycle
AMPtail = (REAL) the amplitude of the pressure head variation
3) Input Data for Turbine Attached Valves –These valves are typically used to limit
water hammer as auxiliary flow control devices. They are activated under emergency
situations, for example, when the wicket gate failed to be closed. They also function
as an additional means when the primary flow control system (wicket-gate, pin valve,
etc.) is not efficient enough. They are quite similar to those valves in pump station
(reference to P149 -153, TransAM Manual), and the first or last character ‘T’ in these
variables is used to indicate that they belong to turbine unit.
TValTYP—Type Specifier of Turbine Attached Valves (CHARACTER)
Following 3 types of valves have been included in modeling of turbine boundary
281
condition, any other more complicated valve can be treated as a separate boundary
condition.
‘TCV’—Turbine Flow Control Valve, which is installed at upstream side of
turbine. This valve follows the input Tau-curve.
‘BYPASS’—Turbine bypass valve, which is used as an additional device to
relieve turbine flow through a bypass pipe if without wicket gates or they
cannot to be closed effectively under load rejection.
‘SURGE’—Surge relief valve, which can be activated by time, upsurge at
upstream or downsurge at downstream of turbine. The valve executes its
opening when a setpoint is exceeded at first time, and then remains (fully) open,
i.e., the subsequent violations of a setpoint have no effect on the behavior of the
valve.
TES—Valve discharge Constant (REAL) The discharge through a valve is
expressed as: HTESQV τ)(=
TTFAC—Reverse Flow Factor (REAL) In current version, we only take the
default value TTFAC=0.0 for any turbine-attached valve, which means the valve will
permit flow only in the usual direction. TTFAC is a dimensionless parameter.
TTAU0—Initial Valve Setting (REAL) TAU0 is the value of τ at the beginning of
the valve motion; it is a dimensionless relative valve opening, from 0.0 to 1.0.
TTAUF—Final Valve Setting (REAL) TAUF is the value of τ at the conclusion of
the valve motion; it is a dimensionless relative valve opening, from 0.0 to 1.0.
282
TTV1—Segment 1 Valve Motion Duration (REAL) In seconds
TTV2—Valve motion Duration (REAL) In seconds
TSET0—Valve Setpoint (REAL)
TSET1—Valve Setpoint (REAL)
TSET2—Valve Setpoint (REAL)
NSTAUT—Starting Tau Index (INTEGER)
NENDTAUT—Ending Tau Index (INTEGER)
MMT—Number of Closure Points (INTEGER)
TAUTT—Tau Curve Points (REAL)
4) Turbine Data
TTYPE—Turbine Type Specifier (CHARACTER) Current TransAM can deal with
four types of turbines, that is, 1) ‘Fixed WG’ (no wicket gates or the wicket gates
fixed); 2) ‘Francis’ or ‘WG’ (turbines with wicket gates, Francis turbine is a typical
example); 3) ‘Kaplan’ (turbines with wicket gates and blades); and 4)’Pump-turbine’
(reversed turbine used in pump-storage hydropower plant). As for the Impulse
turbine, it is treated as an equivalent of BCTYPE=’Valve’. Actually, except for
pump-turbines, other types of turbine could be determined automatically by number
of blade angle settings (NBlade) and the number of wicket gate settings (Ndy), both
of them are input from the turbine characteristic data file in BCBND.f90.
TCODE—Turbine Type Identifier (INTEGER) An integer value acts as an index
describing the ordinal position of the particular turbine’s characteristics in the data
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file TURB*.CHR. In other words, if the characteristics for a turbine are third data set
in TURB*.CHR, then Tcode for that turbine equals 3. The manner in which the
turbine characteristics are obtained for a given turbine and how these are assembled
into the file TURB*.CHR are explained in Chapter 4. Each turbine can have its own
specific characteristics in the file TURB*.CHR. If all turbines in the system could be
modeled using the same head and torque data, only a single set of characteristics
need be used and TCODE would equal 1 for all turbines. (The value of KTCHAR in
the file TURB*.CHR would be 1)
TPHA0—Initial Blades Angle Setting (REAL) Only for Kaplan turbines
TPHAmax—The Maximum Blades Angle Setting (REAL) Only for Kaplan
turbines
TPHAmin—The Minimum Blades Angle Setting (REAL) Only for Kaplan
turbines
TQR—Rated Turbine Flow (REAL)
THR—Rated Turbine Head (REAL)
TNR—Rated Turbine Speed (REAL) In rpm.
TQ0—Initial Turbine Flow (REAL)
TN0—Initial Turbine Speed (REAL)
TWRR—Turbine-Generator Unit Inertia (REAL) (Kg-m2 or lb-ft2) It is a
product of weight times the radius of gyration.
TPR—Rated Turbine Output Power (REAL)
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TTg— Wicket-gate Closure Time at the Maximum Rate (REAL) In seconds, it is
actually the shortest time period to close the wicket-gate from full opening to full
closing.
5) Turbine Governor
Igov—Identifier of Turbine Governor Status. (INTEGER)
Igov=0 for turbine governor in normal operation.
Igov=1 for malfunctioned governor, the consequence is that the wicket- gate
failed to be closed under load rejection, and may cause turbine full runaway
speed if no other means to cut off the upstream coming flow.
Under this circumstance, the main valve at turbine upstream side should be
activated to control the flow rate of turbine, so this scenario may be combined
with an attached ‘CTV’.
Igov=2 for the turbine without any governor (or the governor malfunctioned) but
the wicket-gate is closed as a prescribed rate. This status may seldom occur in
practical applications; however, it is useful to analyze some extreme situations,
for instance, to simulate the scenario of sudden wicket- gate closing by setting
0.1s from full opening to full closing.
Following Governor Data only for Igov=0
GOVTYP—Governor Type Specifier (CHARACTER) GOVTYP=’PID’ for
proportional, differential, and integral governor; GOVTYP=’TEMP’ for
temporary-droop governor; and GOVTYP=’ACCE’ for accelerometric governor.
However, in current version, only GOVTYP=’PID’ is defined and solved. The
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governor equation of ‘PID’ governor is referenced to P163 in Wylie’s book.
Following parameters are used in its equation.
SIGMA—Permanent Speed Drop (REAL)
DELTA—Temporary Speed Drop (REAL)
TAL—Promptitude Time Constant (REAL)
TD—Dashpot Time Constant (REAL)
6) Variation of Wicket gate Opening for Igov=2 Situation
NTY—Number of Points in y-curve (INTEGER)
YG—Sequence of Wicket gate Openings in y-curve (REAL)
TYG—Sequence of Time corresponding to YG in y-curve (REAL)
Wicket gate opening y is a dimensionless variable.
7) Load Variation
NTP—Number of Points in P-curve (INTEGER)
PG—Sequence of turbine loads in P-curve (REAL)
TPG—Sequence of Time corresponding to PG in P-curve (REAL)
The turbine load P is defined as the output power of turbine, which varies with the
load of unit. This set of data need to be input under any situation. If no load changes,
P-curve can be input as a straight line parallel to time axial.
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Appendix Ⅳ: Turbine Boundary Output Data Sample and Variable Index
A Sample of Output Data for Turbine Boundary Condition: ------------------------------------------------------------------------------------- |=======DEVICE No. 4 ==========| TYPE IS in-line turbine INCIDENT NODES: 05 06 OUTPUT in-line turbine No.4 TURBINE The Number of Valves with this Turbine = 0 Synchrounous Speed Specifier = 1 Governor Specifier = 2 DISCHARGE (m3/s) TURBINE HEAD (m ) Rated Initial Rated Initial 191.50 191.50 142.000 130.581 turbine No. SPEED (rpm) SPECIFIC RATED EFF. POWER WRR Rated Initial SPEED (fraction) Rated (kW) (kg-m2) No.4 187. 187. 190.49 0.930 250000.0 7250000. TIME Ty v hh ALPH BETA Qturb Hturb Nturb Hsuc Hdis
(s) (m3/s) (m ) (rpm) (m ) (m ) ======================================================================== 0.000 0.904 1.000 0.920 1.000 1.000 191.500 130.581 187.500 369.825 227.827 0.123 0.904 1.009 0.934 1.000 0.928 193.212 132.595 187.500 363.034 230.439 0.246 0.904 1.015 0.944 1.000 0.944 194.421 134.027 187.500 366.386 232.359 0.368 0.904 1.016 0.945 1.000 0.947 ……. Hsuc-NELEV HDIS-NELEV2 TAUC Hc Qc TAUB QB TAUS QS
(m ) (m ) (m ) (m3/s) (m3/s) (m3/s) ======================================================================== 160.825 18.827 0.000 0.000 --- 0.000 0.000 0.000 0.000 154.034 21.439 0.000 0.000 --- 0.000 0.000 0.000 0.000 157.386 23.359 0.000 0.000 --- 0.000 0.000 0.000 0.000 --------------------------------------------------------------------------------------------------
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Turbine Boundary Condition Output Variable Index:
Ty — Relative wicket-gate opening
v — Relative/dimensionless turbine flow
hh — Relative/dimensionless turbine head
ALPH — Relative/dimensionless turbine speed
BETA — Relative/dimensionless turbine torque
Qturb —Absolute turbine flow
Hturb —Absolute turbine head
Nturb —Absolute turbine speed
Hsuc — Piezometric pressure head at turbine upstream node
Hdis — Piezometric pressure head at turbine downstream node
Hsuc-NELEV — Pressure head at turbine upstream node
Hdis-NELEV2 — Pressure head at turbine downstream node
TAUC — Relative opening of ‘TCV’
Hc — Head loss across ‘TCV’
Qc — Flow rate passing through ‘TCV’
TAUB — Relative opening of ‘Bypass Valve’
QB — Flow rate pass through ‘Bypass Valve’
TAUS — Relative opening of ‘Surge Valve’
QS — Flow rate pass through ‘Surge Valve’
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