hydraulic turbine models & control models for systems dynamics studies

7/27/2019 Hydraulic Turbine Models & Control Models for Systems Dynamics Studies

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turbine flow rate, m3

2Penstock area, m

Penstock length, m

is the acceleration due to gravity, m/sec

is the static head of water column, m

is the head at the turbine admission, m

is the head loss due to friction in

the conduit, m

2

produces mechanical power.damping effect which is a function of gate opening (6).

expressed as:

There is also a speed deviation

Per unit turbine power, Pm, on generator MVA base is thus

where qnl is the per unit no-load flow, accounting for turbine fixed

power losses. A+ is a proportionality factor and is assumed constant.

It is calculated using turbine MW rating and generator MVA base.

(6)urbine MW rating' (Generator MVA rating)h,(q, - q,,,)

where hr s the per unit head at the turbine at rated low and qr s the

per unit flow at rated load.

it should be noted that the per unit gate would generally beless than unity at rated load.

The parameter A defined by Equation 6 converts the gate

Figure 2. Non-Linear Model of Turbine - Non-Elastic Water Column

Expressed in per unit this relation becomes

where h and hl are the head at the turbine, and head loss

respectively n per unit, with hbase defined as the static head of the

water column above the turbine.

Tw, called water time constant or water starting time, is defined as:

qbose s chosen as the turbine flow rate with gates fully open (Gate

posltionG = 1) and head at the turbine equal to hbase. It should

be noted that the choice of base quantities s arbitrary.The system of base quantities defined above has the following

advantages:Base head (hbase) is easily identied as the total

available static head (i.e. lake head minus tailracehead).Base gate is easily understood as the maximum gate

opening.Having established base head and base gate position, the

turbine characteristics define base flow through the relationship:

q = f (gate. head)

The per unit flow rate through the turbine is given by:

In an ideal turbine, mechanical power is equal to flow timeshead with appropriate conversion factors.

The fac t that the turbine is not 100% efficient is taken intoaccount by subtracting the no load flow from the actual flow givingthe difference as the effective flow which, multiplied by head,

opening to per unit turdine power on the volt-ampere base of thegenerator and takes into account the turbine gain. It should benoted. however, hat in some stability programs,A+ s used to convert

the actual gate position to the effective gate position, i.e. A,

= 1/(G - G ) as described in (6). A separate factor s then used tocon vet the "p',wer from the turbine rated power base to that of the

generator volt-ampere base.

2.2 Linear Models

Neglecting friction losses in the penstock, a small perturbationanalysis of the relationships in Figure 2 yields the black diagram ofFigure 3.

I - 1

I I

7-AW

Figure 3. Linearized Model of Turbine - Non-Elastic Water Column

From this figure, the change in mechanical power output canbe expressed as:

4(1 - T,s)AGAP,,, = - DG0Ao

(1 + T A

where

Go = per unit gate opening qt operating point

T i = (qo-qn$TW

T2 = G0Tw/2

(7)

= per unit steady state flow rate at operatingpoint

0

Note that Go = qo



With the damping term neglected, equation 7 is similar to thecommonly used classical penstock/turbine inear transfer function

X A ,

AP, 1 - GoTwS

AG- =

G T s1+A

c

where GOTw s an approximation o the effective water starting time

for small perturbations around the operating point.

Other, more elaborate linear models have been proposed(2.3A.5). They require more detailed turbine data.

Linear models are useful for studies of control system tuningusing linear analysis tools (frequency esponse, eigenvalue etc.). Theiruse in time domain simulations should be discouraged since inaddition o being imited o small perturbations, hey do not offer anycomputational simplicity relative to the non-linear model.

( 8 ) and Te, the wave travel time is

2.3 Traveling Wave Models

While the modeling of the hydraulic effects using theassumption of inelastic water columns is adequate for short tomedik length penstocks, there is sometimes need to consider theeffects wnich cause traveling wave; .' pressure and flow due to theelasticity of the steel in he penstock and he compressibil*yof water.For long penstocks the travel time of the pressure and flow wavescan be significant.

An analysis of the partial differential equations in time andspace defining pressure and flow rate at each point in the conduitgives rise to the classical traveling wave solution which, with theboundary conditions of zero change in head at the penstock inletand the f low/gate/head relationship at the turbine, yields the blockdiagram of Figure 4.

-rL

Figure 4. Non-Linear Model of Turbine Including Water ColumnTraveling Wave Effects

Ths block diagram incorporates the traveling wave transferfunction between head and flow rate:

where:

a

DfE

'base

PK

T,=L/a

= pg (1/K + D/fE)= density of water= bulk modulus of water= internal penstock diameter= wall thickness of penstock=

= base flowYoung's modulusof pipe wall material

hbase = base head9 = acceleration due to gravityL = length of penstock

a = wave velocity=ENoting that the water time constant in the penstock

Tw =-qd4 u m a g

and expression 11 for Te a it follows that

T = ZoTe

169

(10)

(1 1)

Typical values for wave velocity are in the range of 1000 to1200 m/sec.

In the block diagram of Figure4 the effect of friction head ossin the penstock is shown proportional to flow squared.

An alternative numericalmethod of time simulation of travelingwave effects is the method of characteristics solution, detailed inreference 9. An example of this solution technique is given inSection 4.4 .

The dynamics of turbine power are an almost instantaneousfunction of head across the turbine and gate or nonle openingincluding deflector effects where applicable. The head across the

turbine is a function of the hydraulic characteristics upstream of theturbine and also downstream in cases where the flow in the drafttube and/or tailrace is constrained as in the case of Francis or KaplanTurbines. In the case of Pelton (impulse) turbines, the downstreampressure is atmospheric hence he hydraulic effects are only from theconduits between the reservoir and turbine.

Ths modulor separation of effects s shown in Figure 1 with thedistinct blocks labeled "Turbine Dynamics" and "Conduit Dynamics".The model of the combined system. turbine and conduit. is shown inFigures 2 and 3 for the simple penstock/turbine system withoutelasticity effects and in Figure 4 considering elasticity effects.Particular hydraulic conduit arrangements may require specialmodeling in cases such as constrained or vented tunnels. individualpenstocks fanning out from a common pressure shaft etc. The basicmodels for conduits can be put together to describe the specificarrangement, much as the basic models of electric components areused to describe specific networks.

Examples of models for more complex hydraulic systems aregiven in Sections 2.4 to 2.6.

also written as 2.4 Non-linear Madel Including Surge Tank Effects.Non-Elastic Water Columns.

In hydro plants with long supplyconduits. t is common practiceto use a surge tank. The purpose of the surge tank is to provide some

-Z,tan h(Tes) (9b)

2 is the surge impedance of the penstock in per unit expressed as:



170

hydraulic solationof the turbine from the head deviations generatedby transients in the conduit. Many surge tanks also include an orificewhich dissipates the energy of hydraulic oscillations and producesdamping. The hydraulic model shown in Figure 5 includesrepresentationof

penstock dynamicssurge chamber dynamicstunnel dynamicspenstock, tunnel and surge chamber orifice loses

A Wd-

A H e a d a t t u r b i n e ( \

I ISur ge Tank L eve l (head)

Figure5. Non-Linear Model of Turbine Including Surge TankEffects - Non-Elastic Water Column

Flow base, head base and water time constants aredetermined as in 2.1. C,, storage constant of surge tank, is definedas:

c, = ~A s * h m e secs (14)

%lSE

2here

A = surge tank cross section area, m

Upper and ower penstock head oses are proportional o flowrate squared through loss coefficients f

Head IoSSeS in the orifice'to the surge tan are proportional othe coefficient fo times flow rate times absolute value of flow rate to

maintain directionof head loss. The same applies to head loss in theupper penstock where flow can reverse.

The head across the lower penstock s defined by the level ofthe surge tank, which can undergo low frequency oscillations (in theorder of .01 Hz) between surge tank and reservoir.

The inclusion of surge tank effects is warranted in cases wheredynamic performance is being simulated over many seconds tominutes.

p l andp

2.5 Non-Linear Model IncludingSuraeTank E f f e c t s .

Elastic Water Column in Penstock

in cases where traveling wave effects in the penstock areimportant the model of Figure 5 is modified to that of Figure6. Herethe upper penstock or tunnel is considered inelastic because thedynamic effects contributed by that system and surge chamber

involve low frequency effects, while the high frequency responsecomponents are contributed by the lower penstock which is subjectto abrupt gate or flow area changes. The difference between themodel in Figure6 and that in Figure 4 is that the head acting on thelower penstock is the surge tank level rather than the constantreservoir elevation taken as 1 pu.

I - I

q" Lss

I L

I 'I

IFigure6. Non-Linear Model of Turbine With Surge Tank Effects

and Traveling Wave Effects in Penstock

2.6 Non-linear Modeld Multiple Penstocks and Turbines

Supplied from Common Tunnel. Inelastic Water Columns

Figure7 shows a configuration where a pressure shaft or tunnelbrings water to a manifold from which penstocks fan out to severalturbines. The coupling effect of head variations at the manifold is

illustrated n the model of Figure8 for the case of three turbines andtheir penstocks with water starting times of Twl, Tw2 and Tw 3

respectively and a tunnel water starting time of Tw. The model of

Figure 8 is derived from the basic momentum equations for eachconduit and eliminating the variable head at the manifold throughuse of the continuity equation forcing the flow in the upper tunnel tobe equal to the sum of the flows in the penstocks.

Pen s t o c k s

Turbines

Tunnel

Figure 7. Penstock Arrargement Fanning Out From ManifoldFrom Sinale Tunnel

01

Q2

G 3 - - - - - - -

The equ iva len t r e l f and mut ua l r t a r t ing t imer , T1 , , T1 2 , T13 , tc . are der ived

f r om t he so lu t ion of h 1 , h 2 a nd h 3 a s f u n c t io n of ql , Q2 , 4 3They are basical ly the terms in the inverre of t he mat r lx be low .

Figure 8. Model for Configuration of Figure 7 - Non-ElasticWoter Column



171

2.7 Non-Linear Model of Multiple P w t o c k s andTurbiheS SuPDhd

F r o m Common Tunnel. ElasticWater Columns n Penstocks andTunnel

Figure9shows the model accounting for traveling wave effectsin the penstocks and tunnel.

1%

I202 G3=urge impedanceof indlvidual penstocks

207 Surge Impedanceof tunnel

Te,, T 9 Te? - Travel tlme of lndlvidual penstock8

T y Travel time In tunnel

Figure 9. Model for Configuration of Figure 7 - Including

Traveling Wave Effects in Penstocks and Tunnel

Ths model incorporates he single penstock model of Figure 4 and introduces the effect of the tunnel by using the same form oftransfer function between downstream head and flow, which. for thetunnel is the sum of flows in the penstocks.

Whereas the algebraic loop between flow and head of thesimple penstock can be solved in closed form, these loops in Figure 9 are best solved by iteration.

3.0 HYDRO TURBINE CONTROLS

Hydro turbines, because of their initial inverse responsecharacteristics of power to gate changes, require provision oftransient droop features in the speed controls for stable controlperformance. The term 'transient droop' implies that, for fastdeviations in frequency, the governor exhibits high regulation (low

gain) while for slow changes and in the steady state the governorexhibits the normal low regulation. (high gain).

From a linear control analysis point of view. the case of a hydroturbine generator supplying an isolated load can be represented bythe block diagram of Figure 10.

Qate Pmech ACC. Acc.

PelecSy r te m

Figure 10. Linear Model of Hydro Turbine and Speed ControlsSupplying Isolated Load

Conventional requency response and Bode plot analysis of thiscontrol system shows that a pure Droportional controller would have

proportional control gain would be limited to about 3 per unit foracceptable stabilfty which would imply an unacceptably highregulation of 33%.

i a

1

n. .10

0.01 Oe l rad / sec '

Figure 11. Bode Plot of Open Loop Function in Figure 10 withProportional Governor

K = 2 - WCROSS = 0.28 radlsec - OHARG = 45 .8 deg

K = 3 - WCROSS = 0.47 radlsec - llHARG 2 2 . 2 deg

K = 4 - WCROSS = 0.71 radlsec - BMRG = 0.0 deg

1 ,I

100 , oo30 3.0000 5.0000 7.0000 9 ~ o o o o 11.W O3.000 15.noo

,ooO 19.000

T I M E

to be tuned with a very low gain for acceptable stability yielding avery poor (high) regulation. Ths is evident from Figure 11 showing theopen loop asymptotic gain and phase angle plots and in Figure 12 showing the response to a step change in electrical load or differentvalues of proportional gain K . Ths example using a water startingtime Tw of 2 sec and inertia constant H of 4 sec, shows that a

Figure 12. Response of Mechanical Power for a Step Change inElectrical Power in System of Figure 10 withProportional Governor



172

Transient gain reduction is thus necessary to provideacceptable steady state regulation with adequate stability.

3.1 Governors - ProportionalContrd with Transient Droop

Figure 13shows the model block diagram of a typical governorin which the turbine gate is controlled by a two stage hydraulicposition selvo. The physical meaning of the parameters used in themodel s as follows:

T - Pilot valve and servo motor time constant

8 - ~ervogain

T - Main servo time constant

I - Permanent droop

- Transient droop

- Reset time or dashpot time constantRP

TR

I ' ---U Rp IPermanent Droop

Compenrr t lon

Figure 13. Model of Typical Hydro-Turbine Governor

The permanent droop determines the speed regulation understeady state conditions. It is defined as the speed drop in percent orper unit required to drive the gate from minimum to maximumopening without change in speed reference. As noted n Section 3.0.due to the peculiar dynamic characteristicsof the hydraulic turbine.it is necessary to increase the regulation under fast transientconditions n order to achieve stable speed control. Ths is achievedby he parallel ransient droop branch with washout time constant TR.

Because of the choice of the per unit system, with maximumgate opening defined as unity, the speed limits must be defined, forconsistency, as fractions of the maximum gate opening per second.

The Bode diagram in Figure 14 gives an asymptotic plot of theinverse of the feedback path l/ hl i.e.

and the fofward function g

(For the purpose of clarity, the effects of selvomotor time constantshave been neglected n this figure. Their effect is significant only if

their poles occur before or near crossover frequency).

control loop is

The closed loop response (C.L.R.) of such a g,/h,

If gl<<l/hl. then: ghl<<l and C.L.R. is approximately = gl,

If l/hl<cg, then: glhl>>l and C.L.R. s approximately = l/hl.

0.01 rad lsec 10

Figure 14. Bode Diagram of Governor in Figure 13, Forward andInverse Feedback Function

Hence, the closed loop response may be approximated byplotting both g1 and l/h l and choosing the lowest of both gain

responses at any frequency as an approximation to the closed loopresponse at that frequency. Referring to Figure 14, the speed-regulating control loop will "see' the governor as having a gain of1/Rp in the steady state, and a 'transient' gain of l /Ri forphenomena above the l /TR frequency range. An equivalent timeconstant of 1 (QRt)sec will result from the second intersection of theg1 and l/ hl traces.

The speed regulating loop will have acceptab le stability if:

a) The transient gain, (l/Rt) does not exceed

1 H- 1.5 -Rt Tw

b) Crossover frequency, Wc , approximately equal to

1/(2HRt), occurs somewhere in the region betweenl /TR and QRt. Th's reduces phase lag contributions

from the governor.

Several authors have proposed relations for temporary droop.

For temporary droop Ref 1 1 and 12 propose values of TW/H,

Rt,and dashpot time constant, TR:

while Ref. 7 proposes the following formula

Rt =

TdH*(1.15 -(l,

1)*0.075]

All three will result in crossover frequencies that are close to 1/2Tw,and, therefore, satisfy condition (a).

Regarding the dashpot time constant, Ref 11 suggests a valueof 4 times Tw, Ref 12 proposed a TR equal to five times Tw and Ref

7 proposes



173

TR = Tw*(5 - (Tw - 1)*0.5] (20)

For crossover frequencies in the order of 1/2Tw, l /TR values in

the order of 1/5Tw will minimize 'low frequency" phase lag

contributions rom the governor.Reference 7 suggests large values of €2, the servosystem gain,

to attain improved performance characteristics. The typicalmaximum values of 5 to 10 reported n that Same paper, will minimizethe 'high frequency' phase lag contributions rom the gavemor.

3.2 Other T v p e s of Govemon

There are cases where specific governors require morecomplex representation han in Figure 13. The differences may bedue to added ime constants in hardware and also where derivativeaction is included.

Figure 15 shows an example of more complex representations.

Saeed *r - _ -

R e f e r e n c e

U

IAd d l tlonal

ServoD y n a m i c s

IFigure 15. PID Governor Including Pilot and Servo Dynamics

3.2.1 PI Govemor (KD = 0)

Neglecting the pilot servo and additional servo dynamics inFigure 15,shown in the Bode diagram of Figure 16 are the inverse ofthe intemal oop feedback path l /h ie 1 Rp. and the forward gaing, .e. KP + Kl/s. When comparing tLe resulting frequency responsecharacteristics with those of Figure 14 it is apparent that bothgovernors achieve the same objective, i.e. transient droop increase.

Tuning objectives are identical: Transient droop Rt is given byl/KP. KP/KI is equivalent to the dashpot time constant T ,and caremust be taken that crossover does not occur at frequenfies that are

close to the inverse of the smaller servomotor time constants.

3.2.2 PID Governor

The purpose of the derivative is to extend the crossoverfrequency beyond the constraints imposed on PI governors. Figure 17 shows the governor loop frequency response when the PIDgovernor is tuned according to the authors of Reference 10.

Rt = 1/KP = 0.625Tw/H

TR = KP/KI = 3.33 Tw

w/KD ' P-w

(21)

Transient gain (1/Rt) has been increased by 60% over normalPI values. Ths results in roughly the same increase in crossoverfrequency. and thereby, in governor response speed. Thedetrimental effects on stability are averted by the phase lead effectsresulting from derivative action. There is a risk, however, hat the risein magnitude due to the derivative action, compounded with thatresulting from the hydraulic system. may result in a second crossoverat higher frequencies. Due to the high phase lags at thesefrequencies, a second crossover will certainly result in governor loopinstability. Ths is the reason for the minimum limit imposed on thevalue of KP/KD.

1radhec

o1 .1

Figure 16. Bode Plot of P-l Control

100

10

1

0.1

0.01 radlsec '

10

10

Figure 17. Bode Plot of PID Control

3.3 Enhanced Govemor Modd

The governor model described in Figure 18 has modelingcapabilities not frequently found in typical hydro plant models. Itsfeatures may be crit ical for the correct simulation of partial or toto1load rejections:



114

Buffering of ga te closure may produce a reduction in 4.0 EXAMPLESOF ANALYSIS AND SIMULATIONoverpressures under load rejection. It also reduces impact loadingson the gate linkage and limits the magnitude of the pressurepulsations while the gates are fully closed during the decay of loadrejection overspeed.

A sample system with all hydraulic and control parametersdefined in the Appendix serves to illustrate various aspects Of

dynamic performance and simulation approaches.

% eRef

Speed

L

Jet Deflector

MXJDOR

MXQTCR or

r R

MXGTORMXGTCRMXBGORMXBGCRGMAXGMlNRVLVCRRVLMAXMXJDORMXJDCR

Opening

0Relief Valve

Permanent DroopTemporary DroopDashpot Time ConstantPilot Valve Tlme Constant

Qate Servo Tlme ConstantMaxlmum Gate Openlng RateMaximum Gate Closing Rate

Maxlmum Buffered Gate Openlng RateMaximum Buffered Gate Closing RateMaximum Gate LimitMinimum Gate LlmltRelief Valve Closing RateMaxlmum Rellef Valve LlmltMaxlmum Jet Deflector Openlng RateMaxlmum Jet Deflector Cloalng Rate

Figure 18. Enhanced Governor Model Used In Load RejectionStudies

4.1 Govemor L o o p Stabi l i

Traveling wave effects may become a significant factor whenanalyzing governor loop stabili. F$ure 19shows mognitude and

phase lags for the classical linear hydro turbine model (Eq. 8. G = 1 ,Tw=2 sec) and for the same ideal model when traveling wave e dc ts

are considered (Te=l sec) (14). The larger magnitudes in the

traveling wave model will result in higher crossover frequencies.Higher crossover frequencies, compounded with larger phase lags,result in smaller phasemargins, and herefore essstable performancethan when assuming a lumped-parametermodel.

I. ..y

2.0

.0

L 1 oo (rad/.sec)

On. I

Figure 19. Magnitude and Phase Lag Versus Frequency ofAP /AG Function- Lumped Parameter and TravelingW 8 e Models

In some installations a relief valve is attached t o the turbinecasing providing a bypass for the flow. It is operated directly from

the governor or the gate mechanism of the turbine. The amount ofwater bypassed is sufficient to keep the total discharge through thepenstock fairly constant, hence controlling pressure rise.

Turbine flow, as used for turbine power calculations, is

determined in this case as:

These effects have usually a negligible mpact on PI controllerstability. They should not be neglected. however, when analyzing PIDcontroller. Ths is illustrated in the example in Table 1, where traveling-wave modeling is shown to have a significant impact on PIDcontroller st ab ili .

It can be shown that the per unit error in he head/f low transferfunction due to ignoring traveling wave effects is approximatelyequal to

Turbine Flow = QpnstockatawningGate.Opn.+ Rel.Vlv.Opn.

(22)

-To' s 2 (24)error .I-n long-penstock impulse turbines, rapid reductions in watervelocity are not allowed to avoid the pressure rise which wouldoccur. To minimize the speed rise following a sudden load rejection,a governor-controlled et deflector s sometimes placed between heneedle nozzle and the runner. The governor moves this deflectorrapidly into the jet, removing part or all of the power input to theturbine.

Their significance s therefore larger for long penstocks. Longpenstocks result in larger water time constants and therefore lowergovernor response speed. The larger bandwidth in PID controllers smost attractive in such conditions. Traveling wave analysis thusbecomes cfiica l.Turbine flow in this case is calculated as:

Turbine Flow = Openstock x Min(l.,DefPos./Gate Opn.) (23)4.2 Linear vs. Non-Linear Hydraulic Model (Inelastic Flow)

The advantages of nonlinear versus linear models becomen Figure 18, position limits are shown on the controller. Limitscould also be included on the jet and gate Servos.



175

i I I I I I I I I

MECHANICAL POWER

-

-

-

-

-

I -

-

-

i I I I i I I I I

apparent when both models are subjected to large excursions inturbine loading. Figure20 shows the models' responses to a relativelysmall (0.01 pu) step in gate position. Figure21 shows simulation resultsfor a larger (0.2 pu) step. For comparison purposes, the governorrepresentation was deactivated in both models and both no-loadflow and sDeed deviations were set to zero in the nonlinear model.

-C

C

Table 1. Governor Loop Stability for TypicalController Tuning and Alternative Hydraulic Models

Hydraulic

Lumped

Controller Frequency Margin

Parameter rad/sec 19.5 deg

-

I I I I 1 I I I I

1 :[ 1 Parameterumped 1 r? 1 20.0deg

Wave rad/sec 18.1 deg

Wave rad/sec 13.8 deg

Traveling

Traveling 0.54

H=4sec.TW=2sec.Te=1 sec

PI Controller:KP = 2, KI = 0.25PID Controller:KP = 3.2, KI = 0.48. KD = 2.13

'z

The hydroulic system parameters were:

T, = 1.83 s8c

Go = 0.762 puAt = 1.004

The linear model fails to represent the increases in effective watertime with changes in penstock flow.

I I I I I I I i I

e-

MEC H AN IC AL POUER

4.3 Effectd Surae lank

Surge tank effects should be included in dynamic analyses ofhydro plants when the time range of interest is comparable to thesurge tank natural period. For shorter time periods. the simpler short-term model can be used.

(25)Surge Tank Natural P e r i o d = P x

Figure 21. Mechanical Power Response to .2 pu Step in GatePosition. Linear vs. Nonlinear Model

Figure 22 shows the result of simulating a 0.1 pu step loadincrease on on isolated hydro plant with and without surge tankeffects. The surge tank natural period s 3 minutes. For the normal 3to 5 sec transient stability range simulation results are almost identical.For longer simulation intervals. surge tank level starts falling, andmechanical power recovery lags behind that of the short-termmodel, which assumes an infinite surge tank.

Simulation of surge tank dynamics is necessary when the tankis small enough to be emptied by a large load increase (8).Long-

term simulations are valuable in establishing acceptable operatingprocedures that avoid such catastrophic consequences.

For plants with more complex layouts. 'high frequency"oscillations resulting from pendulum action between surge chambersand other hydraulic resonant modes may interfere with the

governor's speed-regulating loop. Dynamic simulations andfrequency response analyses representing these 'long-term" effectsare required tools in such types of analyses (81.

Figure20. Mechanical Power Response to 0.01 pu Step in GatePosition. Linear vs . Non-Linear Model



176

4.4 Traveling-Wave HydraulicSimulation where:

The results shown in Figure 22 were for the case of inelasticcondults and incompressible fluid. Taking into account the effect ofelasticity and compressibility leads to a traveling wave solutionmethod described in Figure 23. This method of calculation oftraveling wave effects is an alternative to implementation of theFigure 6 model with time delay of transport time simulation of thewave.

a - condult wave velocityg - gravitational accelerationA - cross-sectional area0 - conduit siope

Ond On equation Of motion:

SPEED D E Y I A T I f f l

ITIME

Figure 22. Response of Mechanical Power and Speed to a 0.1 puLoad Increase - With and Without Surge Tank Effects

Time TUNNEL (TUNLGTH. TUNSPD, TUNARE. TUNLOS)

Surge ChamberC onat r a in t r

Space

Tunnel InletConst raints

t PENSTOCK (PENLGTH. PENSPD. PENARE. PENL OS)

Tur b lno C ona t r r in t r

Space

Surge ChamberC onr t r a ln t r

Figure 23. Solution of Traveling Wave Effects by Method ofCharacteristics

Flows and heads along the penstock and tunnel are analyred

in terms of a continuity equation

Time-space lattices such as those shown in Figure 23 aredefined for each of the conduits, and both equations aresimuitaneously solved using the method of characteristics [9).

The accuracy of the results is proportional to the number ofsegments into which the conduit is divided. Practical application ofthese models seems to suggest that a minimum of ten segments isrequired.

Time and space increments are related by conduit wavevelocity. Time increments must be equal to or multiples of theSimulation time step. The minimum size requirements on simulationtime step may create additional computational burdens for largesystem simulations.

Simultaneous considerationoftwoor more conduits, while usinga unique simulation time step makes it impossible to T i an exactlattice on each conduit. Recognizing the problem uncertainties,particularly on conduit wave velocity, conduit lengths are adjusted

to the nearest increment.The sine term in (26) recognizes pressure rises. and therefore,

specific volume and flow reductions, resulting from reductions inelevation. This complicates the initialization process (flows along thesame conduit are not equal in the steady-state), but has negligibleeffects in simulation results. Horieontal conduits may be assumed.

The additional computational burden of programming andrunning a traveling-wave model has to be weighed against the errorcaused by the use of an inelastic model. As previously mentioned,the per-unit errors are proportional to the square of conduit traveltime times the square of the main frequency of the dynamicphenomena. As shown in Section 4.1 this per-unit difference willusually be negligible unless very long penstocks are studied or unlessgovernor bandwidth has been expanded by derivative action. Acritical case run under both assumptions assesses the difference.

This is shown in Figure 24, where the hydro plant described inthe Appendix cTw = 1.83 s. Te = 0.42 s) is subjected to a 0.2 pu

increase in load under isolated conditions. Except for some transienthigh frequency effects, the difference between the elastic and

inelastic solutions is negligible.There are times, however, when traveling wave analysis is

essential. The analysis of overpressures and pressure pulsations dueto total load rejection s generally carried out with this type of tool.A closed or almost closed gate gives rise to poorly attenuatedtraveling waves of pressure.

Figure 25 shows a total gate closure simulation for the systemdescribed in the Appendix. For gate positions at or near total closure,the inelastic simulations of scroll case head and penstock flows areno longer applicable, and are replaced by an algebraic, steady-state solution of the penstock. Surge chamber levels and tunnel flowsare not affected by these high-frequency effects. Frequency controlis not affected either since turbine power is practically zero at thesesmall gate openings.

The effect of buffering the gate closure is shown inFigure 26 forthe same total load rejection simulation as in Figure 25, but with amaximum buffered closing rate of 4.05 pu/sec. applied after gateopening is less than 0.15 pu. Overpressures and pressure pulsationsore significantly reduced. at the expense of a larger overspeed.

Figure 27 shows the same total load rejection as in Figure 25,but including simulotion of a relief valve with a -0.01 closing rate.Both overpressures and overspeed are significantly improved by reliefvalve operation.

Figure 28 simulates total load rejection including jet deflectoraction with a -0.5 closing rate. While gate closing rate has beenreduced to a tenth of Rs value in Figure25 , reducing overpressures.the jet deflector manages to control speed.



177

It is recognized that specific applications may require thedevelopment of special models including effects suchasdeadbands.hysteresis. etc. One of the objectives of the paper is to Present thebasic physicsOf hydraulic urbines and thekcontrok that,in the State of the afl, the development Of code for a particularmodel is routine once the physics are well defined.

5.1 Transient Stabilitv Studier

5.0 CONCLUSIONS AND RECOMMENDANONS

This has presented a number of different models forhydraulic turbinesand for their speed controllers. The models vary incomplexity, and are meant to be used for the study of power systemproblems of different types. General recommendotions or their use

are given below.

UITH TRAVELING ~ A V E FFECTS

T I M E

igure 26. Speed and Head Response to a Full Load Rejection

With Gate Buffering

I : L I I ' 19.000TIME

-1 .60 ,lmm!0000 5!woo 7.0000 9 . ~ 0 0 l'.ooo 13,ooo 15.u00 l7.000

Figure 25. Speed and Head Response to a Full Load RejectionWith and Without Traveling Wave Effects T I M E

Figure27. Speed, Head and Flow Response to a Full Load

Rejection With Relief Valve Operotion



178

-

L-- OPEN IN^ -

TU R BI N E H EAD

..

, - - - OfFLfcroR FIOU -* c . OPEN IN^ PEED ~ ~ y l ~ r l o N * - - . _ _

. - - -_ .- - - _ _-..S

Q ~

-

---&(

---_--_-Lb . ,/? -_

/I , I .$

y,:."I :,/ ' 1 o

'3%'I '

-

In studies of small isolated power systems, the governor and

turbine characteristics play an important part in the response of thesystem frequency to disturbances. Here the action of the govemorspeed regulation and the response of the turbine must be includedin the model. The effects of gate posit ion and speed limits can be

significant in such cases, and should thus correspond to those inservice in the modeled plants.

APPENDIX

Sample System (Figures 5,6 and 18)

Rated

Rated Turbine Power:

Rated Turbine Flow:

Rated Turbine Head:

Gate Position at Rated Cond:

NoLoad Flow, qNL:

MVA:

Permanent Droop. R :

Temporor) Droop, Rt:

Dashpot Time Constant. TR:

5 2 Pilot Time Constant, T :m - P

+ Servo Time Constant, T :2 9' 0f 4 fi Maximum Gate Opening Rate, MXGTOR: 0.1 pu./s

2 5 Maximum Gate Closing Rate, MXGTCR: -0.1 pu./S- - f "

m - 4 7 7I O 0

- 7 7 Maximum Gate Limit, GMAX: 1. pu.

0. pu.s

Minimum Gate Limit, GMIN._.-

100. MVA

90.94 MW

7 1.43 m3/s

138.9 m

0.90pu.

4.3 m3/s

0.05 pu.

0.45 u.

8. s

0.02 s

0.5 s

T I M E

Figure 28. Speed, Head and Flow Response to a Full loadRejection with Jet Deflector Operation

The turbine model of Section 2.1 coupled with a governormodel chosen from that of Section 3.1 or Section 3.2 as appropriateis recommended for use in transient stability programs. A simplelinear turbine model s not recommended since ts parameters wouldhave to be adjusted as a function of operating conditions and theaccuracy of representationwould be affected by the magnitudeofthe perturbations.

5.2 Smal l Signal Stabilii

In small signal stability studies, it is the effect of the governorsand turbines on the damping of low frequency inter-area modeswhich is of concern. These effects can be modeled adequately bylinearizing the non-linear turbine and governor models about theappropriate operating point. Fixed time constant lineaked modelsof the turbine without adjustment for operating point are notrecommended.

Linear models are also used for guidance in speed controltuning using linear control analysis techniques. The most criiicalcondition for such studies of governor adjustments would be with theunit supplying an isolated load at maximum output.

5.3 SrJecial ADdications

In special circumstances, additional complexity must beincluded in the turbine and governor models to study the detailedresponse of the plant to disturbances, or to study the effect of theunits on long-term dynamics. Other instances requiring additional

complexity are studies of interactions between turbine hydraulicdynamics including draft tube pulsations and electromechanicalpower oscillations. In such cases, the model must correspond asclosely as possible to the actual tubine and controls that exist a t theplant. Detailed modelingofspecial controls, such as those discussedin Section 3.3, and the penstock, ncluding traveling wave effects andsurge tank dynamics, as in Sections 2.3 and 2.4, may be required.

Lake Head,

Tail Head,

Penstock Length

Penstock Cross Section

Penstock Wave Velocity:

Penstock Head Loss Coeff.. fpl:

Tunnel Length:

Tunnel Cross Section:

Tunnel Wave Velocity:

Tunnel Head Loss Coeff., fp2:

Surge Chamber C. Section:

SCh. Orifice Head Loss C.. fo:

Turbine Damping:

1.

2.

3.

4.

5.

6.

7.

REFERENCES

307.0 m

166.4 m

465m

15.2 m2

1100 m/s

0.0003042

3850 m

38.5 m2

1200 m/s

0.00101 12 m/(m3/s)2

78.5 In2

0.0040751 m/(m3/s)2

0.5 pu/pu

IEEECommittee Report, "Dynamic Modelsfor Steam and HydroTurbines in Power System Studies', IEEE Trans. Vol. [email protected].

J.L.Woodward,"Hydraulic-Turbine Transfer Function for Use inGoverning Studies", Proc. IEE, Vol. 115, pp. 424-426, March 1968.

J.R.Smith et al..'Assessment of Hydroturbine Models for PowerSystem Studies", Proc. IEE, Vol. 130.Pt.C, No. 1, January 1983.

P.W.Agnew,"TheGoverning of Francis Turbines'. Water Power,pp . 119-127, April 1974.

R.Oldenburger. J.Donelson, 'Dynamic Response of aHydroelectric Plant", Trans. AIEE, Vol. 81, Pt. 111, pp. 403-418,1962.

J.M.Undriil and J.L.Woodward, 'Non-Linear Hydro GoverningModel and Improved Calculation for Determining TemporaryDroop'. IEEE Trans., Vol. PAS-86, No.4, pp.443-453, April 1967.

P.L.Dandeno,P.Kundur and J.P.Bayne, 'Hydraulic Unit DynamicPerformance Under Normal and Islanding Conditions - Analysisand Validation", IEEE Trans., Vol. PAS-97,No. 6, pp. 2134-2143.November/December 1978.



179

8. J.M.Undrill and W.Stmuss. 'Influence of hydro plont desiQnonregulating and resewe response capacity'. IEEETrans., Vd. PAS74, P. 1192-1200,uly/August 1974.

9. V.L.Streeter and E.B.Wylie, Fluid Mechanics' (McGrw-Hill,NewYork, 1975).

S . Hagihara, H. Yokota. K. Goda, K. Ixrobe. 'Stability of aHydraulic Turbine Generating Unit Controlled by PID Governof,iEEE Trans., Vol. PAS98No6. p. 2294-2298, ov/Dec 1979.

L. M. Hovey, 'Optimum Adjustment of Hydro Governors onManitoba Hydro System'. AEE Trans., Vol. 81, Part 111. pp. 581-587.Dec 1962.

F. R. Schleif and A. B. Wilbor, The Coordination of HydraulicTurbine Governors for Power System Operation', IEEETrans. V d .PAS85, p. 7W758, uly 1966.

L. K.Kirchmayer, Economic Controlof InterconnectedSystems.Vol. 11, Chapter I., John Wiley and Sons Inc. 1959.

C. K. Sanathanan. 'Accurate Low Order Model for HydraulicTurbine-Penstock', IEEE Trans., Vol. EC-2, No. 2.pp. 196200,June 1987.

10.

11 .

12.

13.

14.

hydraulic turbine models & control models for systems dynamics studies

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