setting underfrequency relays in power systems via integer programming
TRANSCRIPT
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Setting Under-Frequency Relays in Power
Systems via Integer Programming
Frida Ceja Gmez
A thesis submitted to the Department of Electrical Engineering
in partial fulfillment of the requirements for the degree of
Master of Engineering
McGill University
Montreal, Quebec, Canada
June 2011
Copyright Frida Ceja Gmez 2011
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ABSTRACT
The deviation of the frequency of a power system from its nominal value
is a reflection of the mismatch between generation and load. Such deviations are
serious and must be monitored and controlled very closely. One major impact of
operating outside a narrow range around the nominal frequency is that
generators can be damaged. To avoid this, manufacturers set time interval limitsfor under-frequency operation and when such limits are exceeded, the generator
trips. However, unless generator tripping is coordinated with some
accompanying load shedding, the system inability to supply its load can be
exacerbated resulting in an even worse frequency deviation.
Under-frequency load shedding (UFLS) is designed to protect the power
system from events leading to a sudden drop in system frequency, when the
primary frequency regulation built into the generation system is not enough to
bring the frequency back to nominal. Under-frequency load shedding
disconnects blocks of load when the frequency drops below given thresholds.
However, the conventional design of UFLS schemes is primarily based on
experience about the behavior of the system. Basically, trial relay settings are
proposed, tested, and revised until a successful UFLS scheme is obtained. This
process is tedious, not very systematic, and usually leads to shedding
conservative amounts of load.
This thesis presents a mixed-integer linear programming formulation of
the UFLS relay setting problem. The goal is to render the design of UFLS more
systematic, less dependent on trial and error, and less conservative in terms of
the amount of load shed.
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RSUM
Dans un rseau lectrique, lcart de la frquence du rseau par rapport
sa valeur nominale est le reflet dun manque dquilibre entre la production et la
consommation. Ces carts peuvent avoir des consquences graves et ils doivent
tre contrls et surveills de trs prs. Un impact majeur dans lopration dun
rseau lectrique avec une tolrance large autour de la frquence nominale est le
risque dendommager les alternateurs. Pour viter cette situation, les fabricants
tablissent des dlais pour oprer en sous-frquence et lorsque ces dlais sont
dpasss, les alternateurs sont automatiquement dconnects du rseau, ce qui
entrane une plus grande dviation de la frquence.
Le dlestage sur le seuil de sous- frquence est conu pour protger le
rseau lectrique d'vnements conduisant une baisse soudaine de la frquence
du rseau lorsque les rglages intgrs dans le systme de production sont
insuffisants pour ramener la frquence la valeur nominale. Le dlestage sur le
seuil de sous- frquence consiste dconnecter des regroupements de
consommateurs lorsque la frquence descend en dessous dune certaine limite.Cependant, les manuvres de dlestage sont principalement conues partir de
l'exprience du comportement du rseau lectrique. Les rglages des relais de
protection de premires instances sont proposs, tests et modifis jusqu' ce
qu'une manuvre de dlestage appropri soit obtenue. Ce processus est
laborieux, non-systmatique, et conduit gnralement un dlestage
conservateur de la demande.Cette thse prsente une formulation utilisant des techniques de
programmation linaire mixte pour dterminer les rglages des relais de
protection de sous-frquence. L'objectif est de rendre la conception des
manuvres de dlestage plus systmatique, moins dpendante des mthodes
empiriques, et moins conservatrice au point de vue du dlestage.
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For my mother and my husband, who offered me unconditional love and support
throughout the course of this thesis
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ACKNOWLEDGMENTS
I would like to thank Professor Galiana, who inspired me to pursue a
masters degree in power engineering. I really admire the passion with which he
teaches and the care that he gives to all his students. I will always be in debt with
him for his valuable guidance and advice. He makes graduate school a fun
place.
The completion of this work would not have been possible without theprevious research done by Syed Saadat Qadri, who I kindly thank for his support
and his interest in helping me.
I am very grateful to fellow students Andr Dagenais, Amir Kalatari,
Mustafa Momen, Salman Nazir, Kelvin Lee, tienne Veilleux and Da Qian Xu;
who were always willing to help me and created a beautiful friendly
environment in the power lab.
I would also like to express my gratitude to the Natural Sciences and
Engineering Research Council of Canada and to Hydro-Qubec for their financial
support.
Finally, I would like to thank my dear friends Joey, Mike, Aldo, Manar,
and Olivier for always being there for me in moments of doubt.
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CONTENTS
ABSTRACT ............................................................................................................... i
RSUM ................................................................................................................... ii
ACKNOWLEDGMENTS .................................................................................... iv
LIST OF FIGURES ............................................................................................... vii
LIST OF TABLES .................................................................................................. vii
1 INTRODUCTION ........................................................................................... 1
1.1 Background .................................................................................................. 1
1.2 Recent Developments in the Design of UFLS Programs ......................... 3
1.3 Motivation for this Thesis .......................................................................... 5
2 THE UNDER-FREQUENCY RELAY .......................................................... 7
3 POWER SYSTEM DYNAMICS ................................................................... 9
3.1 Load Damping ............................................................................................. 9
3.2 Primary Frequency Regulation ................................................................ 11
4 DISCRETE-TIME FREQUENCY RESPONSE MODEL ...................... 14
5 SYSTEMATIC UNDER-FREQUENCY RELAY SETTING
APPROACH USING BINARY VARIABLES ............................................................. 17
5.1 Relay Timer Model ................................................................................... 18
5.2 Relay Operation Logic .............................................................................. 19
5.3 Discrete-Time Frequency Response Including Load Shedding ............. 20
5.4 Constraints on the Load Shedding Variables .......................................... 22
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5.5 Generator Under-Frequency Time/Limits .............................................. 22
5.6 Other Constraints ..................................................................................... 24
5.7 MILP Formulation .................................................................................... 25
5.8 Objective Function.................................................................................... 26
6 SIMULATION RESULTS FOR A TEST POWER SYSTEM .............. 27
7 COMPARISON OF THE PROPOSED UFLS SCHEME WITH THE
CONVENTIONAL METHOD ........................................................................................ 34
7.1 Conventional Method ............................................................................... 34
7.2 Revised Conventional Method ................................................................. 39
7.3 Comparison................................................................................................ 46
8 EFFECT OF DIFFERENT PARAMETERS ON THE MILP
FORMULATION ................................................................................................................ 49
8.1 Choosing the Contingencies to Include in the MILP Formulation ....... 49
8.2 About the Number of Contingencies to Include in the MILP
Formulation 51
8.3 About the Number of Load Shedding Stages .......................................... 52
8.4 Effect of the Number of Time Steps and Step Size .................................. 54
8.5 Having Fixed Load Shedding Blocks ....................................................... 56
8.6 Having Fixed Frequency Set Points ........................................................ 59
CONCLUSIONS .................................................................................................... 61
REFERENCES ......................................................................................................... 62
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LIST OF FIGURES
Figure 1: Operation of an under-frequency load shedding relay ............................... 7
Figure 2: Effect of load damping ...................................................................................... 10
Figure 3: Governor model ................................................................................................. 12
Figure 4: Effect of governor action and time delay ...................................................... 13
Figure 5: Test power system ............................................................................................. 27
Figure 6: Generation loss contingencies without load shedding relay action ....... 29
Figure 7: Generation loss contingencies with load shedding relay action.............. 31Figure 8: Generation loss contingencies with load shedding relay action (2) ........ 32
Figure 9: Generation loss contingencies with load shedding relay action (3) ........ 33
Figure 10: Frequency trajectory for a 15% generation loss plotted with the
discrete-time frequency response model as implemented in MATLAB ................. 44
LIST OF TABLESTable 1: Manufacturer-specified generator under-frequency/time limits ............. 23
Table 2: Generation loss contingencies for test power system .................................. 28
Table 3: Relay settings obtained with the MILP model .............................................. 30
Table 4: Blocks of load shed per contingency with proposed relay settings .......... 31
Table 5: Relay settings obtained with the conventional method .............................. 38
Table 6: Modified relay settings resulting from the conventional method ............ 39
Table 7: Relay settings obtained with revised conventional method....................... 46
Table 8: Comparison of relay settings ............................................................................. 47
Table 9: Comparison of blocks of load shed per contingency with different relay
settings ................................................................................................................................... 47
Table 10: Relay settings obtained considering a different set of 3 contingencies .. 49
Table 11: Blocks of load shed per contingency for relay settings in Table 5 .......... 50
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Table 12: Relay settings obtained with a set of 4 contingencies ................................ 51
Table 13: Blocks of load shed per contingency with relay settings obtained
considering a set of 4 contingencies ................................................................................ 52
Table 14: Proposed UFLS plan with 4 load shedding stages ..................................... 53
Table 15: Blocks of load shed per contingency with relay settings having 4 load
shedding stages ................................................................................................................... 53
Table 16: Relay settings obtained with a step size of .2 s ............................................ 55
Table 17: Blocks of load shed per contingency with relay settings obtained with a
step size of .2 seconds ......................................................................................................... 55
Table 18: Relay settings obtained when having the load shedding blocks as
inputs ..................................................................................................................................... 57
Table 19: Relay settings obtained when having the load shedding blocks as
inputs (2) ............................................................................................................................... 58
Table 20: Blocks of load shed per contingency when having the load shedding
blocks as inputs .................................................................................................................... 58
Table 21: Relay settings obtained when having the frequency set points as inputs................................................................................................................................................. 59
Table 22: Block of load shed per contingency when having the frequency set
points as inputs .................................................................................................................... 60
Table 23: Relay settings obtained when having the frequency set points as inputs
(2) ............................................................................................................................................ 60
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1 INTRODUCTION
1.1BackgroundThe frequency of a power system will suffer a decline when the demand
for electricity exceeds the generation capacity. Such an event or contingency
occurs randomly due to the sudden loss of one or more generating units.
Generating units cannot operate for an extended period of time in under-frequency conditions, since the mechanical resonance will damage the turbine
blades. For this reason, the manufacturers set under-frequency/time limitations
that if violated will cause the unit to trip. This means that if the frequency is not
promptly returned to its nominal value by either generation regulation action
(primary frequency regulation) or by automatic load shedding, more generating
units will trip and the system frequency will continue to drop.
A local shortage of generation will also cause interconnected systems to
supply extra power to meet the load. This action might overload the connecting
tie-lines and make them trip as well, thus exacerbating the system degradation.
Under-frequency load shedding (UFLS) has been widely used since the
1960s as the last resort to protect power systems from total blackouts following
contingencies that lead to a significant decline in frequency.
The implementation of UFLS plans dates back to 1965, when a severe
blackout in the Northeast region of the United States left more than 30 million
people without electricity for up to 13 hours [1]. The severity of this event
prompted the North American Electric Reliability Council (NERC) to
recommend the implementation and coordination of UFLS plans in each region
of the United States.
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Each region belonging to the NERC jurisdiction has different rules
regarding the total amount of load to be shed and the frequency thresholds that
must be respected by their UFLS scheme. For example, the North East Power
Coordinating Council (NPCC) says that [2]:
The goal of the program is to arrest the system frequency decline and to
return the frequency to at least 58.5 Hz in ten seconds or less and to at least 59.5
Hz in thirty seconds or less, for a generation deficiency of up to 25% of the load.
Over the past few years, UFLS programs have proved to be successful in
maintaining system stability when disturbances that cause dangerous under-frequency conditions occur. One of these events occurred in the region controlled
by the Western Electricity Coordinating Council (WECC) on August 10, 1996 [2].
A generation outage caused the separation of the region in four islands. The
automatic UFLS plan was used on each island to arrest the frequency decline,
which avoided a complete system collapse. However, it was noted that more
than enough load was shed, which led to some problems with generator
voltages.
Another significant under-frequency event occurred in Italy on September
28, 2003. The Italian grid was separated from the rest of the continent because
some transmission lines tripped. This caused a deficit in active power, which led
to a frequency decline that caused generators to trip, resulting in a general
blackout. According to [3], the automatic UFLS program was not properly
designed for the loss of imported power and did not arrest the frequency decline.About 60 million people were affected by this blackout for more than 3 hours.
On November 4 2006, the inadequate planning of the disconnection of a
power line in Germany so that a ship could cross the Ems River safely caused the
European transmission grid to split into three areas. The western area was the
most affected, with a 22% power imbalance that caused the frequency to drop to
49 Hz [4]. About 15 million households were affected by the power outage, but
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the system was restored and a complete blackout was prevented by the fast
action of the automatic UFLS scheme.
The historical under-frequency events presented in this section show the
importance of UFLS plans in preventing blackouts.
1.2Recent Developments in the Design of UFLS ProgramsCurrently, UFLS plans are designed by performing an iterative series of
dynamic performance simulations, the results of which are combined with
historical data, heuristics and practical experience to settle on the relay settings
[2].
In the last decades several studies on the optimal tuning of UFLS relays
have been conducted. In [5] an optimization algorithm for the design of an UFLS
scheme is presented, in which the objective function is divided in a dynamic and
a static part. The dynamic part consists of the integral of the deviation from
nominal frequency and the static part is the total load shedding. The optimumUFLS must not lead to a system frequency below a minimum value and must
satisfy constraints on the load shedding amounts and time delays. The
optimization problem is solved using the gradient projection method with the
partial derivatives expressed using analytic approximations. The resulting UFLS
solution corresponds to a global minimum that is highly dependent on the initial
guess used. This means that to obtain good relay settings several initially guesses
must be tried. Also, this method considers the frequency set points that trigger
load shedding as pre-defined values, whereas the method presented in this thesis
treats them as decision variables.
Adaptive schemes have also been studied, as in [6-8], the goal of which is
to react better to a broader range of contingencies than the conventional method.
In [7], a six-step procedure to find adaptive relay settings is proposed. First, a
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dynamic model of the system under study is constructed to simulate a set of
generation outage scenarios.The results are then used to find the rate of change
of the frequency loci for each contingency, which then define the parameters for
the first load shedding stage. The subsequent load shedding stages are also set
adaptively, based on the operation of the previous stage. Once all the stages are
planned, the scheme is tested to verify if some parameters must be adjusted. The
proposed ULFS scheme has an improved performance for large disturbances,
and a response similar to the conventional scheme for small disturbances.
In [7], it is shown that the frequency gradient is a reliable indicator of thesystems generation deficiency only if considering other system parameters such
as voltage profile, system loading and load characteristics. The authors therefore
use a gradient curve that is a linear function of both the frequency gradient and
of the aforementioned parameters. This technique resulted in less load shedding
than the conventional approach.
One of the latest innovations in UFLS schemes is to include the use of
SCADA systems to modify the relay settings in real time. In [9], a SCADA-based
scheme is proposed in which the magnitude of the disturbance is estimated by
computing the mean system frequency. This is achieved by collecting, comparing
and analysing current and past system data. From the disturbance magnitude, it
is determined whether or not the system requires load shedding. If required,
there are two possible conditions: (i) When the mean system frequency is above
the allowed value, the frequency might nonetheless be low for a particular
generator, so local frequency monitoring and a delayed load shedding scheme
are imposed; (ii) When the mean system frequency is below the allowed value, a
pre-calculated amount of load is shed with the minimum possible delay. This
method proved to maintain the frequency of all the generators within a safe
range, while shedding less load than some traditional schemes. However, the
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method requires that relays be equipped with microcontroller and
communications technology.
1.3Motivation for this ThesisPower systems have recently undergone significant changes due to the
proliferation of wind power and to the introduction of electricity markets, both
of which introduce new sources of uncertainty. As a result, the operational set
points of the power system not only deviate from their typical values but are
harder to predict. This requires that the measures in place to ensure the security
of power systems be reassessed and refined, in particular the setting of under-frequency relays, which is the motivation for the work presented here.
Also, the literature review presented above shows that there is no
standard systematic approach regarding the setting of under-frequency relays
(UFR) and that heuristics, experience, and trial and error still play a major role.
Therefore, this thesis describes the development of an under-frequency
load shedding scheme based on mixed-integer linear programming [10] that doesaway with trial and error methods and minimizes the amount of load shed. The
main steps of the proposed optimization-based UFR setting approach are:
1) The system frequency behaviour versus time following a contingency is
estimated by a discrete-time equivalent swing equation model;
2) The UFR model is characterized by three sets of design parameters (or
optimization decision variables) , , ; 1,...,s s sf t d s ns whose values define the
setting of the relay: A set of frequency set points ; 1, ...,sf s ns of decreasing
values whose violation over corresponding time spans st triggers the shedding
of corresponding blocks of load sd ;
3) A set of constraints on the input parameters and decision variables. For
example, the requirement that the frequency time trajectory following a
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contingency return to a safe frequency range within specified time limits. These
safety requirements consider that each generator is subject to a set of
manufacturer-defined under-frequency thresholds and corresponding time
limitations[11];
4) The selection of a suitable objective function whose minimization with
respect to the UFR decision variables defines the relay setting. In this thesis, we
minimize the expected load shed over a set of random contingencies with known
probabilities;
5) The UFR setting optimization problem is formulated as a mixed-integerlinear program (MILP), and solved and solved without resorting to trial and error by
very efficient commercially available software [12].
The next sections of this thesis present some basic concepts about under-
frequency relays and power system dynamics. This is followed by the
development of the proposed MILP formulation to set under-frequency relays.Finally, a case study is presented and motivation for further work is provided.
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2 THE UNDER-FREQUENCY RELAY
The purpose of under-frequency load shedding relays is to detect an
under-frequency condition in the power system and disconnect some of the load
to prevent the system from becoming unstable. The bus frequency is monitored
at every substation and if the bus frequency goes below a certain set point f, a
timer is activated. When the timer reaches a preset value t , the circuit breaker
receives a trip signal that disconnects a local block of load d . Note that if thefrequency returns to a value higher than fwithin a period of time smaller than
t , then the timer resets. Figure 1 illustrates the operation of an under-
frequency load shedding relay.
Figure 1: Operation of an under-frequency load shedding relay
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Most under-frequency load shedding schemes have more than one load
shedding stage (typically 3 to 5 load shedding stages [2]). An under-frequency
relay can then be designed to take action for more than one load shedding stage
or different relays can be used to react to different stages. Regardless of the way
in which the UFLS scheme with ns load shedding stages is implemented, the
scheme must specify ns sets of the three variables , , ; 1,...,s s sf t d s ns .
Therefore, the under-frequency relay settings are entirely defined by the design
variables:
;, , 1,...,s s sf t d s ns (2.1)
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3 POWER SYSTEM DYNAMICS
A generating unit is a rotating mechanical system, in which the rotation of
a turbine shaft caused by some input mechanical power is transformed into
electrical power. A mechanical torque is generated by the input mechanical
power while an opposing electrical torque is caused by the load connected to the
generator. If there is a change in the generation or demand, this imbalance will
be reflected on the turbine speed, which results in a fluctuation of the system
frequency.
The swing equation of a generating unit defines the relationship between
the active power imbalance and its frequency response. This equation in its
simplest form is given by,
2m e
o
H d fP P P
f dt
(3.1)
In equation (3.1), Pmrefers to the input mechanical power and Perefers to
the output electrical power in per unit, while His the generator inertia constant
in seconds, fois the nominal frequency in Hz, and f is the frequency deviation
from nominal in Hz.
There are however many other parameters that significantly affect the
frequency trajectory of a generating unit. These are discussed in the following
subsections.
3.1Load DampingThere are different kinds of electric loads in a power system. Resistive
loads do not modify their power consumption when there are frequency
fluctuations, but this is not the case of motor loads. Motor speeds vary according
to the frequency of the input power supply. A motor load reduces its active
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power consumption when there is a decline in the system frequency. The
dependence of power consumption on frequency for motor loads is defined by
the relation,
mlP D f (3.2)
mlP refers to the change in active power consumed by motor loads and
f is the frequency deviation from nominal. The damping constant D is defined
as the percent change in load for a one percent change in frequency. For example,
a damping constant of 2% indicates that a 1% change in frequency would cause a
2% change in load.
The sensitivity of loads to frequency changes should be included in the
swing equation to accurately reflect the frequency response following a
contingency. Figure 2 shows the frequency trajectory after a 25% generation loss
contingency with and without considering load damping.
Figure 2: Effect of load damping
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It can be seen that the frequency decline is more severe when not
considering the effect of load frequency dependence. Therefore, if the relay
settings are found with the use of a model that does not include load damping,
the resulting UFLS plan will be too conservative and may shed excess load. This
reflects the importance of including load damping in the swing equation as
follows,
2m e
o
H d fP P D f
f dt
(3.3)
3.2Primary Frequency RegulationGenerating units in a synchronous power system are equipped with speed
governors that are responsible to oppose changes in frequency by modifying its
active power generation. This means that for a frequency deviation f the
governor would change the units power generation byf
R
, where Rrepresents
the governor droop or frequency regulation constant of the generating unit in
Hz/MW. Typical governor droop values are between 3 and 6 Hz for the loss of
the rated power.
There are time constants associated with the opening of valves that range
from 5 to 10 seconds. The operation of a speed governor can be modeled as
shown in Figure 3, where r is the primary frequency regulation due to afrequency deviation of f . R is the governor droop and T is the time constant
associated with the governor action.
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1
(1 )R sT
Figure 3: Governor model
This transfer function is expressed in time domain by,
d r fT r
dt R
(3.4)
Governor action can then be included in the swing equation as shown
below,
2m e
o
H d fP P r D f
f dt
(3.5)
Figure 4 shows the system frequency trajectory following a contingency of
15% generation loss without governor action, with governor action but no timedelay, and with both.
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Figure 4: Effect of governor action and time delay
It can be seen that the frequency goes back close to the nominal value
when considering primary frequency regulation, whereas the frequency reaches
an unsustainable level without regulation. Also, it can be seen that if the time
delay associated with governor action is not considered, the frequency reaches
steady state without oscillations. These oscillations might however activate some
load shedding stages, which means that ignoring them would lead to
inappropriate relay settings.
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4 DISCRETE-TIME FREQUENCY RESPONSE MODEL
In a multi-machine power system with ng generators, depending on
parameters such as inertia constant and governor droop, each generator has a
unique frequency response to a contingency as shown in the previous section.
However, by assuming, as is commonly done, that all generators swing
synchronously at a common frequency f , an approximation to the system
frequency response can be obtained through an equivalent single-machine swing
equation expressed by,
0( )
( ) ( ) ( )2
fd f tr t g d t D f t
dt H
(3.6)
In equation (3.6), f refers to the system frequency deviation from
nominal at time t following the loss of some amount of generation g at t=0while ( )d t is the amount of load shed at time tby under-frequency relaying
action. In addition, D is the system load damping factor while ( )r t , the primary
frequency regulation with governor action and time constantT , is governed by,
( ) 1 ( )( )
d r t f t r t
dt T R
(3.7)
In (3.6), the equivalent inertia constant is computed by,
i i
i
H SH
S (3.8)
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where iH is the inertia constant of generator i with power base iS and Sis the
system power base. In addition, in (3.7) the equivalent governor droop can be
calculated from the individual generator droops iR as,
1 i
i i
S
R R S (3.9)
Equations (3.6) and (3.7) can be discretized into time steps of duration t
by defining ( ) nr n t r , ( ) nd n t d , and ( ) nf n t f . Then, through
Eulers method, equation (3.6) in discrete form becomes,
1 1 ;n n nf f K t n (3.10)
where,
02
n n n n
fK r g d D f
H (3.11)
while equation (3.7) in discrete form is given by,
1 1nn n n
ftr r rT R
(3.12)
The initial conditions for equations (3.10) and (3.12) are zero since prior to
a contingency there is no frequency deviation or primary frequency regulation.
The accuracy of the discrete-time frequency response is dependent on the
size of the integration step t , with smaller integration steps resulting in a better
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approximation but in a greater number of decision variables. Also, when
simulating the effect of a contingency, it is important to use enough time steps to
allow the frequency to reach steady state, typically within 20 s.
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5 SYSTEMATIC UNDER-FREQUENCY RELAY SETTING APPROACHUSING BINARY VARIABLES
Consider a power system consisting of nggenerators. Such a system has
2 1ng possible generation loss contingencies, the thj contingency being defined
by the loss of jg from the total pre-contingency generation level. Since for large
ng the number of possible contingencies is very large, the under-frequency relay
settings are traditionally based on the worst credible contingency [2]. This
approach can however lead to a conservative strategy, shedding more load than
necessary when milder contingencies materialize. On the other hand, if the relays
are set based on an average contingency, the resulting scheme may not protect
the system adequately against more severe contingencies.
Hence, in this thesis we base the relay settings on a set Cof ncplausiblecontingencies, each with a known probability of occurring. Methods for choosing
such an appropriate set of representative contingencies have been suggested in
[13, 14].
As shown below, the under-frequency relay setting problem can then be
formulated as a mixed integer linear program with an appropriate objective
function to be minimized subject to the following conditions: (i) the power
system frequency response versus time to each of the nccontingencies including
the response of the load shedding relays; (ii) respecting the under-frequency time
limitations of the generators as specified by the manufacturers.
The following subsections formulate the under-frequency relay setting
problem as a mixed integer linear program (MILP), which is an extension of the
MILP model developed in [15].
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5.1Relay Timer ModelConsider an UFLS scheme with a specified number ns of load shedding
stages. When contingency j C occurs, for each load shedding stage s , the
under-frequency relay1disconnects a block of loadsd at time step n when the
frequency trajectory jnf (computed using equations (3.10)-(3.12)) violates the
frequency set pointsf for a length of time greater than st . Recall that in its most
general form the quantities { , , ; }s s sf d t s define the relay settings and are
decision variables of the MILP. Note as well that the frequency deviations over
time nfor the different contingenciesj, ; ,jnf j n , are also decision variables to be
found by the MILP.
Now, for each frequency set point, sf , a timer can be systematically
defined with the aid of the binary variable jsnv , which equals 0 when the
frequency at time n , 0 nf f , is above the frequency set point sf and 1 when
0 n sf f f . This binary variable can therefore be exactly defined by the linear
inequalities,
1 ; , ,
j j
s o n s o nj
sn
f f f f f fv j s n
L L
(4.1)
In inequality (4.1), L is a large positive number compared to the
numerator, e.g. 60 Hz. Using the binary variablej
snv , the total time spent by
trajectory j below frequency sfat time step n is explicitly given by the following
linear relation,
, 1 ; , ,
j j j
sn s n snt t v t j s n (4.2)
1We can assume that the lo ad shedding strategy is implemented by a single relay that sends instructions to various load
shedding breakers or, alternatively, that the strategy is implemented by many relays, each responsible for one load shedding stage.
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5.2Relay Operation LogicAs stated above, the relay logic dictates that the block of load sd be shed
when the corresponding timersn
t exceeds st . Note that at this stage these
three quantities are unknown decision variables. The logic behind this relay
operation can be modeled by a new load shedding binary variable jsnu which is
equal to 0 when the time interval jsnt is less than st and 1 when
j
sn st t . Once
again, using binary math, this variable can be exactly characterized by the linearinequality,
1 ; , ,j j
jsn s sn ssn
t t t t u j s n
L L
(4.3)
where the parameter L is a sufficiently large positive number such as the
maximum expected duration of any frequency transient, e.g. 20 s.
Note that in the conventional way of setting relays, the amount of time st
that the frequency is allowed to remain under a frequency set point is usually
fixed to around 0.2 s. However, in this MILP formulation st can be a variable in
order to increase the degrees of freedom of the relay setting problem. In such a
case, st must be restricted to be greater than the minimum time required for the
circuit breaker to open, int , that is,
min ;st t s (4.4)
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Another important consideration is that once some load has been shed in
stages , it cannot be restored at a later time during the trajectory, a requirement
that takes the form,
, 1; , ,
j j
sn s nu u j s n (4.5)
In addition, to avoid different stages from shedding load simultaneously,
at most one load shedding occurs at any time n over all possible stagess , which
in mathematical terms requires that,
, 1 1; ,j j
sn s n
s s
u u j n (4.6)
Finally, if a load shedding priority exists in which the higher the index s
the lower the priority of the corresponding load block, the following condition isthen also enforced,
1,; , ,j j
sn s nu u j s n (4.7)
5.3Discrete-Time Frequency Response Including Load SheddingFrom equations (3.10)-(3.12), for each contingency j , the time discretized
frequency trajectory must satisfy,
1 1 ; ,j j j
n n nf f K t j n (4.8)
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where, including the load shedding amount jsn ss
u d ,
0 ; ,2
j j j j j
n n n sn sjs
fK r g D f u d j n
H
(4.9)
and where the primary frequency regulation is given by,
1 1
jj j jn
n n nj
ftr r r
T R
(4.10)
Note that the load shedding term, jsn ss
u d , contains products of a binary
variable,
j
s nu and the amounts of load shed
sd , both unknown decision
variables. Nonetheless, the relay setting problem can still be formulated as a
MILP by replacing the product jsn su d by the variable
j
snx and by imposing the
following equivalent linear inequalities , ,j s n ,
0 jsn snx u (4.11)
0 1 js sn snd x u (4.12)
It is easy to see from (4.11) and (4.12) that when 1jsnu then j
sn sd , while
when 0jsn
u then 0jsn
x .
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5.4 Constraints on the Load Shedding VariablesThe amount of load that may be shed at each stage sd has, so far, been
treated as a continuous variable between zero and the total pre-contingency load.
However, load shedding could be restricted to a set of pre-specified blocks,
;specs sd d s (4.13)
In this case, equation (4.9) would be given by,
0 ; ,2
j j j j j spec
n n n sn sjs
fK r g D f u d j n
H
(4.14)
in which case it would not be necessary to include equations (4.11) and (4.12).
On the other hand, if the load shedding blocks are continuous variables,
the conditions on the load amounts shed over the stages is that they be non-
negative and that their sum be smaller than the total system load,
s
s
d d (4.15)
5.5Generator Under-Frequency Time/LimitsThe automatic load shedding strategy must satisfy the generators under-
frequency/time limitations specified by the manufacturer. These parametersensure that the time that the frequency remains under a given threshold does not
result in generator damage. The following table shows an example of these
limits[11].
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Under-frequency range or
upper threshold (Hz)
Maximum allowed
time below threshold
(s)
60.5-59.5 Safe continuous
operation
59.5 30
58.5 15
57.5 1
56.5 0
Table 1: Manufacturer-specified generator under-frequency/time limits
Suppose that among all generators, there are n
critical frequency
thresholds f, together with the corresponding manufacturer specified maximum
allowed times axt
. To include these limitations in the relay setting problem, a
new binary variable jnw is introduced equal to 1 when, after contingency j , the
frequency 0 jnf f falls below the frequency threshold f at time step n and 0
otherwise. Thus,
1 ; , ,
j j
o n o nj
n
f f f f f fw j n
L L
(4.16)
Now, the total time spent below frequency threshold fby trajectory j at
time step n can be found from,
, 1 ; , ,j j j
n n nt t w t j n (4.17)
Finally, the generator under-frequency/time limitations can be enforced
by requiring that,
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max ; , ,jn
t t j n
(4.18)
5.6Other ConstraintsA successful load shedding strategy must yield a steady state frequency
deviation jf within a safe range of nominal, typically plus or minus half a
Hertz, that is,
0.5 0.51
j j
sNj s
N
j
g x
f
DR
(4.19)
where N is the last time step in the simulation and where, as per equation (4.11)
and (4.12), jsNx is the total amount of load shed during trajectory j up to time N .
The condition imposed by (4.19) does not however exclude the possibility
of an oscillatory frequency being close to the steady-state at the end of the
simulation. To avoid this, we therefore impose that the average frequency over
the last np time steps (typically between 5 and 10 steps),1 N j
n
N np
fnp
, be close to
the steady state within some small amount (typically 0.15 Hz), that is,
1 Nj jN n
N np
f fnp
(4.20)
It is also important to impose that the frequency set points sf be larger
than the lowest permissible generator under-frequency limit and smaller than
some maximum value, so as to avoid shedding load too early. These constraints
can be expressed by,
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axmin
sf f f
(4.21)
In addition, to avoid having equal frequency set points in consecutive
load shedding stages, we impose the condition,
1 mins sf f f (4.22)
where inf is typically around .2 Hz.
5.7MILP FormulationThus, of all the MILP decision variables, three sets define the relay setting:
(i) the frequency set points, ; 1, ...,sf s ns , (ii) the time delays before shedding,
; 1, ...,st s ns , and (iii) the amount of load shed at each stage, ; 1, ...,sd s ns .
Note that if the load shedding block sizes are a priori fixed then the latter become
known constants and there are only two sets of decision variables.
The above formulation shows that the UFLS problem consists of a large
number of variables and complicated constraints. This explains the extreme
difficulty of finding a feasible solution, let alone an optimal one, using heuristics
or trial and error methods.
However, since the UFLS problem constraints are linear while theunknowns are either continuous or binary, it is possible to use commercially
available mixed-integer linear programming software [12] to find a feasible and
optimal solution. The relay setting problem can then be formulated as a MILP
subject to:
Equations (4.1) and (4.2) of the relay timer model; Equations (4.3), (4.4), (4.5), (4.6), and (4.7) of the relay operation logic;
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Equations (4.8), (4.9), and (4.10) of the discrete time frequency responsewith the non-linearity expressed using equations
Error! Reference source not found.and (4.12); Equation (4.15) for the load shedding variables; Equations (4.16), (4.17), and (4.18) of the generator under-frequency/time
limits; Equations (4.19) and (4.20) to force the frequency to return to a safe value; Equations (4.21) and (4.22) to obtain suitable frequency set points.
5.8Objective FunctionThe relay settings protect the system against all nc contingencies taken
into account by the MILP. However since not all contingencies have the same
probability of occurring, a suitable objective function to be minimized is the
expected load shed over all contingencies,
.min j j
s N
j s
p x
(4.23)
where the load shed for contingency j is weighted by its probability of
occurrence j . If the probabilities of the contingencies are not known, they can
be estimated by a constant equal to 1 over the number of contingencies.
The next chapter presents an analysis of the simulation results obtained by
implementing the MILP formulation.
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6 SIMULATION RESULTS FOR A TEST POWER SYSTEM
To illustrate the proposed method for setting UFRs, consider the lossless
power system with the per-unit pre-contingency generation and bus loads
shown in Fig. 1. All units are rated at 100MVA with an inertia constant of 4 s and
a governor droop of 5%.
Figure 5: Test power system
The UFLS scheme is designed to protect the system against the set of 8
contingencies indicated in Table 2. The last two columns show the equivalent
system inertia constant and governor droop following the contingency.
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Contingency Generation Loss(%)
Lost Units Heq
(s) Req
(Hz/500MW
1 10 g1 3.2 3.75
2 15 g5 3.2 3.75
3 25 g2 3.2 3.75
4 25 g1, g5 2.4 5
5 35 g1, g2 2.4 5
6 40 g2, g5 2.4 5
7 50 g2, g3 2.4 5
8 50 g1,g2,g5 1.6 7.5
Table 2: Generation loss contingencies for test power system
Fig. 2 shows the system frequency trajectory for 3 of the 8 possible
contingencies without load shedding action. The horizontal lines represent the 4
manufacturer specified generator frequency thresholds defined in Table 1. It can
be seen that for a generation loss of 15% the frequency stabilizes at 59.5 Hz,
which falls within the desired range of .5 Hz. Moreover, the trajectory respects
the generator time/frequency thresholds. In contrast, the loss of 25 and 50% of
the generation leads to violations in both the steady-state frequency and in the
maximum allowed time below the generator frequency thresholds.
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Figure 6: Generation loss contingencies without load shedding relay action
Thus, for the proposed UFLS strategy to be effective, it should not shed
any load for contingencies with a generation loss of 15% or less, but it should
take some load shedding action for all other contingencies.
It is possible to include all 8 contingencies from Table 2 in the set Cto be
used for the MILP formulation. However, having a large set Cincreases the
number of variables and simulation time. Experiments suggest that it is sufficient
to select a smaller set of representative contingencies to protect the system
against all the target contingencies. For this test system such a set consisted of 3
out of the 8 target contingencies, consisting of the mildest one requiring some
load shedding, the most severe, and an intermediate one.
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In general, if the resulting relay settings do not protect the system against
some intermediate contingencies not in set C, the most severe of these uncovered
contingencies is added to Cand new relays settings are computed. The selection
of contingencies to be included in the MILP formulation will be discussed in
more detail later.
Table 4 shows the relay settings obtained with the proposed MILP
formulation for a set C consisting of contingencies 3, 6, and 8 from Table 3. It was
assumed that all contingencies had equal probability of occurrence. The
simulation used a governor time constant of 5 s while the frequency set pointswere constrained to lie between 57.2 and 59.5 Hz. A step size of .1 s was used and
the time delays before shedding were fixed at .2 s. The steady state frequency
was constrained to settle between 60.5 Hz and 59.5 Hz. Three load shedding
stages were assumed. The minimum load shed solution was found in
approximately 1 minute with a personal computer having a Dual Core AMD
OpteronTM 2.39 GHz processor.
Frequency Set
Pointssf (Hz)
Load Shedding
Blockssd (pu)
Time delay
st (s)
58.2 .134 .2
57.6 .150 .2
57.2 .134 .2
Table 3: Relay settings obtained with the MILP model
These relay settings bring the system frequency back to the safe region
when the contingencies that require load shedding occur (contingencies 3-8),
while shedding no load for the contingencies that do not require it (contingencies
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1-2). Table 5 shows the number of blocks shed and amount of load shed for each
of the 8 target contingencies.
% generation loss Lost generators No. of blocks shed Load sheddingamount (pu)
10% 1 0 015% 5 0 025% 2 1 .13425% 1,5 1 .13435% 1,2 2 .28440% 2,5 2 .284
50% 2,3 3 .41850% 1,2,5 3 .418
Total 1.672Table 4: Blocks of load shed per contingency with proposed relay settings
Fig. 7 shows the system frequency trajectory for the 3 contingencies shown
in Fig. 6, but now with load shedding action.
Figure 7: Generation loss contingencies with load shedding relay action
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Fig. 8 and Fig. 9 show the system frequency trajectory for the 5 other
target contingencies with load shedding action. These figures along with Fig. 7
show that there are no overshoots outside the safety range and that the
frequency does not settle above 60 Hz for any case, which indicates that the load
shed is not excessive. The generator frequency/time limits are also respected in
all cases.
Figure 8: Generation loss contingencies with load shedding relay action (2)
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Figure 9: Generation loss contingencies with load shedding relay action (3)
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7 COMPARISON OF THE PROPOSED UFLS SCHEME WITH THECONVENTIONAL METHOD
The following sections presents two design approaches based on the
conventional methodology to obtain a successful UFLS plan for the test system
shown in the appendix. These procedures were obtained and complemented
from different sources, such as books and journal papers. The main stepspresented were taken from [16] and [17] with some additions found in newer
sources like [18, p. 381-388] and [2]. The final section of this document compares
the resulting UFLS schemes with the one obtained using the MILP formulation.
7.1Conventional MethodThe next method to obtain the under-frequency relay settings consists of 4
steps and is based on [16] and [17]. Each of the steps is discussed below.
Step 1: Selection of the percent overload level that the load shedding program is to
protect and the value at which the system frequency must settle for that overload level.
Regardless of the design method, the first step in the design of an UFLS
plan is to choose the maximum overload for which it will provide coverage. This
selection is arbitrary and varies by region. According to [2], UFLS plans in NorthAmerica and Europe are designed for a maximum generation loss that ranges
between 25% and 70%.
Since the UFLS plan obtained using the MILP formulation was designed
to protect for a 50% generation loss, we designed the scheme using this method
to protect for the same amount of generation loss in order to have a fair
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comparison of both schemes. Also, we enforced the system frequency to return to
within .5 Hz of the nominal frequency (60 Hz). Note that the worst case scenario
of a 50% generation loss occurs when the system loses 3 generating units.
Therefore, the scheme must be designed taking into consideration how the
absence of these units affects the system parameters.
Step 2: Determination of the total required load to be shed.
To calculate the total amount of load to be shed for a given maximum
generation loss, we use the heuristic formula proposed in [16],
(1 )1 60
1 (1 )60
T
L fd
Ldf
d
(6.1)
In the above equation Td refers to the total amount of load to be shed, d
refers to the load damping factor, frepresents the minimum allowed settling
frequency, and L refers to the per-unit overload. The per-unit overload is the
ratio of the generation loss over the remaining generation. It is important to note
that equation(6.1) leads to a conservative value of the total load to be shed.
Since we wish to protect for a maximum generation loss of 50%, the per-
unit overload is equal to 1. We can then substitute this value in equation(6.1), aswell as the settling frequency of 59.5 Hz and the load damping of 2% load
reduction per 1% frequency reduction.
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1 59.52(1 )
2 60 .48559.5
1 2(1 )60
Td pu
(6.2)
Therefore, the UFLS plan must be designed to shed a total load of .49 pu,
rounding up to give a small margin for error.
Step 3: Selection of the number of load shedding stages and the load be shed per
stage
It has been found that most systems need between 3 and 5 load shedding
stages to shed an amount of load that is close to the minimum required [16, 19],
but the choice is arbitrary. Since the MILP formulation was set to find a UFLS
program of 3 load shedding stages, we will also use 3 load shedding stages for
this method.
Once we have the total load to shed we must split it among all the loadshedding stages. It is preferable to shed smaller blocks on the first stages and
increase the size of blocks towards the final stage [16-18]. This way less load is
shed for milder contingencies. Sometimes, the load shedding blocks are defined
by the way in which substation and feeders are arranged.
For our test system, we need to divide .49 into 3 blocks while trying to
make the first block smaller than the second one, and this one smaller than the
third one. Following this standard, we could choose a trial set of load shedding
blocks as: .14, .16, and .19 pu. Once we obtain the frequency set points, this
choice should be tested and if it does not prove suitable it should be revised.
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Step 4: Calculation of the relay frequency settings
This step selects the frequency set points. The choice is mostly arbitrary,
but based on some general rules combined with knowledge of the system. In [16,
17, 19], it is suggested that the first frequency set-point should be set at the
highest possible setting allowed by the relay since shedding load early can limit
the maximum frequency deviation. However, it is not recommended to select the
first frequency set-point at values higher than 59.5 Hz to avoid shedding load for
mild contingencies that recover without taking any measure. Therefore, we
choose our first frequency set-point at 59 Hz.The lowest set point should be above the minimum allowed frequency to
avoid the tripping of generators. These generator time limitations are given by
the manufacturer and we will use the same ones that were input to the MILP
formulation model. In this case, the system frequency should not remain for
below 57 Hz more than a second. Hence, the lowest frequency set points should
be above this value. We give an error margin of .2 Hz, since this is a typical delay
that the relays take to actually shed some load. The lowest frequency set point is
then chosen to be 57.2 Hz.
As for the remaining frequency set-points, it is suggested in [16] to
distribute them between the highest and lowest one in the same way as it was
done with the load shedding blocks. A trial set of frequency thresholds is
selected and the scheme is tested. If the is not successful, there is an iteration of
steps 3 and 4 until a suitable strategy is found. For our trial UFLS plan, we
choose the second frequency set-point at 58.2Hz. Then based on the above steps
we have our trial relay settings as:
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Frequency Set
Points sf (Hz)
Load Shedding
Blocks sd (pu)
Time delay
st (s)
59.0 .14 .2
58.2 .16 .2
57.2 .19 .2
Table 5: Relay settings obtained with the conventional method
After testing this UFLS scheme with the discrete-time frequency model
implemented in MATLAB, it was seen that the scheme protects for all the
contingencies. However, the scheme sheds more load than necessary for some
contingencies. The worst case observed was with contingency 1, in which the
frequency settled above 60 Hz.
In order to improve the relay settings, we could try reducing the block
sizes, but doing so might make the scheme ineffective for some contingencies.Therefore, we would have to try reducing them and repeat the simulation until
the scheme gets better, which shows how cumbersome the conventional method
can result.
Another modification to reduce the load shed for mild contingencies
would be to bring the first frequency set-point down. This could also could
problems, since then the second set-point would be too close to the first and
might be activated before the first block of load is shed, which would lead to
even more unnecessary shedding. We could then try reducing the first two
frequency set-points in order to leave some coordinating margin between them.
The third set-point is far enough, so it will not be revised.
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Frequency Set
Points sf (Hz)
Load Shedding
Blocks sd (pu)
Time delay
st (s)
58.5 .14 .2
58.1 .16 .2
57.2 .19 .2
Table 6: Modified relay settings resulting from the conventional method
After testing this UFLS, it was found that it still protects the system in all
contingencies, but now it does not shed any load for the mildest contingency. It
can be seen that the conventional method yields an effective load shedding
scheme, but it can become tedious for a larger system. Also, it sheds a larger
amount of load than required because the relay settings are selected using
conservative estimates.
7.2Revised Conventional MethodAs it was seen in the previous section, the calculations used in the
conventional method do not include the governor droop. Also, the last two steps
rely on heuristics and trial and error to find the appropriate relay settings. This
section shows the difference made by including the governor droop and it
presents some calculations that can be used to aid in the search for the load
shedding blocks and frequency set points.
Step 2: Determination of the total required load to be shed.
The minimum load required to be shed can be obtained by calculating the
maximum generation loss from which the system can recover and bring the
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steady-state frequency to a safe range [20, p. 596,597]. The maximum generation
loss axP given a desired steady-state frequency deviation ssf is calculated by,
max ssP f (6.3)
In the above equation, refers to the systems total damping factor,
which can be calculated as in [20, p. 597] given the speed governor droops and
the load-frequency sensitivity factor by,
1
eq
DR
(6.4)
In this case, we have a load-frequency sensitivity factor D of 2. All the
machines have a governor droop Ri of 5% and are rated at 100 MVA. Since the
contingency we are considering is the loss of 3 units, the equivalent system
droop must be found as follows,
1 1
ieq iR R (6.5)
In this case i=2.Note that the equivalent droop should be first convertedto the system base. This is done with the use of the next formula,
,
*i Sbasei Sbase
i
R SR
S (6.6)
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For our test network the system base is 500 MVA (the addition of the
ratings of all the units). Therefore,
,
.05*500.25
100i SbaseR pu (6.7)
Now we can calculate the total equivalent governor droop by taking into
account that we have only two machines left,
1 12 8
.25equ
R (6.8)
Then, the systems total damping factor can be obtained,
12 8 10
eq
D puR
(6.9)
Since we want the steady-state frequency to be within .5 Hz of the
nominal frequency, we can use this value as the maximum allowed frequency
deviation in order to find the maximum allowed power mismatch,
max
.5 110 .08333
60 12ssP f pu (6.10)
Having found this value, the minimum amount of load to be shed d is
given by,
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max
1 5.5 .4167
12 12d P P pu (6.11)
According to these calculations, the total load that must be shed when
there is a 50% generation loss is about 42% of the total load. It is a common
practice to revise the result of the calculations performed in this step in order to
leave some margin for error in the calculations. In this case, we could decide to
make shed a total load of .45 pu.
Step 3: Selection of the number of load shedding stages and the load to be shed per
stage
For a simple system like this one, we could find the load shedding blocks
in a more analytical way by targeting a given mild generation loss with the first
load shedding stage and a more severe one with the second one.
Taking a look at the possible generation loss contingencies we can decideto only shed one block of load for all contingencies below 25% generation loss
and the two first blocks for all contingencies of less than 40% generation loss.
Then we could use the procedure used in step 2 to find the minimum load we
must shed for each targeted generation loss and make these values the trial set of
load shedding blocks.
For a generation loss of .25 pu to return to a range within .5 Hz of the
nominal frequency, the minimum load to be shed is calculated as follows,
1 13 12
.25equ
R (6.12)
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12 12 14
eq
D puR
(6.13)
max
.5 714 .1167
60 60ssP f pu (6.14)
max
7 2.25 .1333
60 15d P P pu (6.15)
The minimum load to be shed for a .40 pu generation loss is given by
max
7 17.40 .2833
60 60d P P pu (6.16)
Based on the previous calculations, we can choose the first load shedding
block to be .135 pu load and the second block to be .15 pu load. Subtracting this
amount from the total load that we must shed (found to be .45 pu) we end up
with the third load shedding block as .165 pu load.
Step 4: Calculation of the relay frequency settings
Selecting the first frequency set-point is a tough decision to make because,
as explained before, it depends on the designers criterion and it is based on
experience and knowledge of the system. A logical requirement to impose is that
the first frequency set point should be below the lowest frequency at which the
system can recover without any load shedding or equipment damage.
It is hard to find this frequency, since most sources [16, 17, 19] use simple
linear models that do not incorporate governor time constants and sometimes
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even exclude governor action completely. A more detailed dynamic simulation
model could be then used to find this frequency.
For the current test system, we use the discrete-time frequency model
implemented in MATLAB in order to find this frequency. After testing for the
mildest contingencies, it was found that when losing generator 5 (.15 pu
generation loss) the system frequency goes back to 59.5 Hz, which is within the
acceptable range. When the next worse contingency occurs the system frequency
does not recover to an acceptable value, so we use the loss of generator 5 as the
contingency to define the highest frequency set-point.
Figure 10: Frequency trajectory for a 15% generation loss plotted with the discrete-time
frequency response model as implemented in MATLAB
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From the figure above it can be seen that the frequency trajectory reaches
a frequency of 58.5 before going back up to the allowed range. Therefore, the
highest frequency set point should be below this value. We will choose a value of
58.4 Hz.
The rate of change of frequency can be used to estimate the consecutive
frequency set points [16-18], once the first one has been selected. This is done to
avoid the overlap of successive load shedding steps.
In the previous section, we decided to target contingencies smaller than
25% generation loss with the first load shedding block. It is important to take intoaccount that there is a delay associated with the load shedding action which is
typically of about .2 s [2]. Therefore, to find the second frequency set point we
can calculate the rate of change of frequency after the targeted contingency. In
this case it is the 25% generation loss. The rate of change of frequency is found
by,
.25
60 1.07114oP
f f Hz
(6.17)
Now, considering that the relay takes .2s to take action, the first block of
load would not be shed until the system frequency reaches
58.4 (.2 1.071) 58.4 .2142 58.19Hz . Therefore, the second frequency set
point should be below this value. We can choose it to be 58.0 Hz to leave a
margin for error.The same process can be done to obtain third frequency set point. We can
calculate the rate of change of frequency when there is a 40% generation loss and
the first block of load has already been shed.
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(.40 .135)60 1.135
14o
Pf f Hz
(6.18)
The second block of load would then be shed at
58.0 (.2 1.135) 58.0 .227 57.77Hz . Therefore, the third frequency set point
should be below this value. We can choose it to be at 57.5 Hz. The final relay
settings obtained with this revised conventional method are shown in the next
table.
Frequency Set
Pointssf (Hz)
Load Shedding
Blockssd (pu)
Time delay
st (s)
58.4 .135 .2
58.0 .150 .2
57.5 .165 .2
Table 7: Relay settings obtained with revised conventional method
After running a simulation for all contingencies, it was found that this
UFLS successfully protects the system in all contingencies. It was found that in
all cases the frequency settled below 60 Hz, which shows that we are not
shedding excess load that causes overshoots. The downside of this method is that
it requires much more calculations than the original method. Also, since the
designer still needs to make many arbitrary decisions, for some systems the last
steps would need to be repeated until a successful UFLS plan is obtained.
7.3ComparisonTable 8 shows the relay settings obtained with the MILP formulation as
well as the ones obtained with the two variations of the conventional method.
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Frequency Set Points sf (Hz) Load Shedding Blocks sd (pu) TimeDelay
s
t (s)
MILP Conventional RevisedConventional
MILP Conventional RevisedConventional
58.2 58.5 58.4 .134 .140 .135 .257.6 58.1 58.0 .150 .160 .150 .257.2 57.2 57.5 .134 .190 .165 .2
Table 8: Comparison of relay settings
The performance of the UFLS schemes above can be assessed by the
comparison of the number of blocks shed and the load shedding amount perblock presented in Table 9.
Lostgenerators
No. of blocks shed Load shedding amount (pu)
MILP Conventional ImprovedConventional
MILP Conventional ImprovedConventional
1 0 1 0 0 .14 0
5 0 1 0 0 .14 02 1 1 1 .134 .14 .1351,5 1 1 1 .134 .14 .1351,2 2 2 2 .284 .30 .2852,5 2 2 2 .284 .30 .2852,3 3 3 3 .418 .49 .45
1,2,5 3 3 3 .418 .49 .45Total 1.672 2.14 1.74
Table 9: Comparison of blocks of load shed per contingency with different relay settings
It can be seen that the relay settings obtained with the MILP formulation
shed less load than the conventional approaches. This also shows that common
practice of having larger load shedding blocks for the subsequent load shedding
stages can lead to unnecessary load shedding. In this case for example, the relay
settings obtained with the MILP formulation have the third load shedding block
smaller than the second load shedding block and yet they protect for all the
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contingencies while shedding the minimum amount of load as possible with 3
load shedding stages.
Another important advantage of the MILP formulation is that the designer
does not need to perform iterative trial and error calculations to obtain the relay
settings, which can be a tedious and time consuming process.
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8 EFFECT OF DIFFERENT PARAMETERS ON THE MILPFORMULATION
The previous sections show the results obtained with the MILP
formulation using the optimal parameters and inputs for the given test power
system. However, it is important to explore how changing some of these
parameters alters the resulting UFLS scheme.
8.1Choosing the Contingencies to Include in the MILP FormulationThe relay settings in the previous section were obtained using only 3 out
of the 8 target contingencies (contingencies 4, 6, and 8 from Table 2). However,
the contingencies to include in the MILP formulation must be carefully chosen
since not all subsets of 3 contingencies lead to optimal relay settings. To show
this, the relay settings obtained considering contingencies 4, 5, and 8 from Table
2 are shown below. All other parameters are the same ones used for the previousexample.
Frequency Set
Pointssf (Hz)
Load Shedding
Blockssd (pu)
Time delay
st (s)
58.0 .160 .2
57.3 .118 .2
57.2 .152 .2
Table 10: Relay settings obtained considering a different set of 3 contingencies
The relay settings from Table 10 also protect the system from all 8 target
contingencies, but the load shedding amounts differ from the settings presented
before. Table 11 shows the number of load shedding blocks and amounts of load
shed with these relay settings.
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% generation loss Lost generators No. of blocks shed Load sheddingamount (pu)
10% 1 0 015% 5 0 025% 2 1 .16025% 1,5 1 .16035% 1,2 2 .27840% 2,5 3 .43050% 2,3 3 .43050% 1,2,5 3 .430
Total 1.88Table 11: Blocks of load shed per contingency for relay settings in Table 5
It can be seen that both the amount of load shed per contingency and the
total amount of load shed over all contingencies is significantly larger that with
the relay settings from Table 3. This is caused by the way in which the total load
to be shed was distributed among the 3 load blocks. Note how with these relay
settings 3 blocks of load are shed for a generation loss of 40%, whereas only 2
blocks of load are shed with the previous relay settings.
Other subsets of 3 contingencies were used with the proposed MILP
formulation to explore the effect of choosing different contingencies, but the set
used to obtain the relay settings from Table 3 resulted in the best UFLS scheme in
terms of minimizing the amount of load shed.
It was found that considering contingencies that are far apart from each
other works better than considering contingencies close to each other in terms ofgeneration loss and severity. This implies that having a mild contingency, a
severe contingency, and an intermediate one in the set of considered
contingencies yields a successful UFLS scheme. Further research is required to
find a more systematic way of selecting the appropriate contingencies.
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8.2About the Number of Contingencies to Include in the MILPFormulation
It is desirable to obtain a successful UFLS plan considering the smallest
number possible of target contingencies in order to reduce the computation time.
We have presented two appropriate relay settings obtained with a set consisting
of 3 contingencies, but it is also possible to obtain good relay settings considering
more contingencies. Table 12 shows the relay settings obtained with the MILP
formulation with a set consisting of contingencies 4, 5, 6, and 8 from Table 2. Allother parameters are the same that were used to obtain the previous relays
settings. The running time of the model was approximately 3 minutes.
Frequency Set
Pointssf (Hz)
Load Shedding
Blockssd (pu)
Time delay
st (s)
58.4 .135 .257.7 .160 .2
57.3 .123 .2
Table 12: Relay settings obtained with a set of 4 contingencies
Table 13 shows the number of blocks and amounts of load shed for each
contingency with these relay settings, which also protect the system against all
the target contingencies.
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% generation loss Lost generators No. of blocks shed Load sheddingamount (pu)
10% 1 0 015% 5 0 025% 2 1 .13525% 1,5 1 .13535% 1,2 2 .29540% 2,5 2 .29550% 2,3 3 .41850% 1,2,5 3 .418
Total 1.696Table 13: Blocks of load shed per contingency with relay settings obtained considering a set of 4
contingencies
It can be seen that this scheme sheds less load than the one obtained with
a set consisting of contingencies 4, 5, and 8; but it sheds slightly more load than
the scheme obtained with the set of contingencies 4, 6, and 8. The number of
blocks of load shed is the same as with the first relay settings presented in this
thesis, but the block sizes are more conservative.
It was observed that considering 4 contingencies in the set provides relaysettings that result in less load shedding than the ones obtained with most other
subsets of 3 contingencies. However, if the right set of 3 contingencies is chosen,
the resulting relay settings are better than or as good as when considering 4
contingencies.
8.3About the Number of Load Shedding StagesThe UFLS schemes presented so far were designed with 3 load shedding
stages. The purpose of this section is to explore if adding load shedding stages
can aid in minimizing the load shed for each contingency. Table 9 presents the
relay settings obtained with the set of contingencies 4, 6, and 8. All other
parameters remained the same, except that now the model was set to obtain 4
load shedding stages instead of 3.
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Frequency Set
Points sf (Hz)
Load Shedding
Blocks sd (pu)
Time delay
st (s)
58.4 .134 .2
58.1 .100 .2
57.4 .050 .2
57.2 .134 .2
Table 14: Proposed UFLS plan with 4 load shedding stages
These relay settings successfully protect the system during all target
contingencies. The table below show the blocks of load shed for each
contingency.
% generation loss Lost generators No. of blocks shed Load sheddingamount (pu)
10% 1 0 015% 5 0 025% 2 1 .13425% 1,5 1 .13435% 1,2 2 .23440% 2,5 3 .28450% 2,3 4 .41850% 1,2,5 4 .418
Total 1.622Table 15: Blocks of load shed per contingency with relay settings having 4 load shedding stages
Table 15 shows that the overall total load shed with the UFLS scheme
having 4 load shedding stage is less than the load shed with 3 load shedding
stages. However, the amount of load shed during contingency 3 was the only one
reduced with the addition of the 4 thload shedding stage. All other contingencies
have the same amount of load shed with both UFLS plans.
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The optimal solution was found in approximately 15 minutes for the UFLS
plan with 4 load shedding stages, but in only took 1 minute to obtain a solution
for the UFLS plan with 3 load shedding stages. We can conclude that the benefit
obtained from having 4 load shedding stages is not as significant as the increase
in computation time.
8.4Effect of the Number of Time Steps and Step SizeThe number of time steps and the size of the time step t have a
significant effect in the accuracy of the discrete-time frequency response model
used in the MILP formulation. It is obvious that smaller time steps will result in a
better approximation of the frequency trajectory following a contingency, but it
also increases the number of variables and therefore the simulation time. This is
because enough time steps must be used to allow the frequency to reach steady
state (between 15 and 20 s), so the smaller the time step the larger the number oftime steps required.
All the previous relay settings were found using a step size of .1 s. The
simulation time with this resolution was approximately 1 minute and only 165
steps are required to obtain satisfactory relay settings. It is however interesting to
see what happens when using a larger step size. This is because even if the relay
settings are computed offline, it may not be practical to have such a small step
size. The following results were obtained with a step size of .2 seconds and 100
time steps. All other parameters remained as specified at the beginning of
Chapter 6.
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Frequency Set
Pointssf (Hz)
Load Shedding
Blockssd (pu)
Time delay
st (s)
58.5 .134 .2
57.8 .150 .2
57.7 .134 .2
Table 16: Relay settings obtained with a step size of .2 s
The resulting relay settings protect the system against all contingencies
and the simulation time was halved, but a larger amount of load is shed for ageneration loss of 40% compared to using the same model with a step size of .1.
This is shown in Table 17.
% generation loss Lost generators No. of blocks shed Load sheddingamount (pu)
10% 1 0 015% 5 0 025% 2 1 .13425% 1,5 1 .13435% 1,2 2 .28440% 2,5 3 .41850% 2,3 3 .41850% 1,2,5 3 .418
Total 1.622Table 17: Blocks of load shed per contingency with relay settings obtained with a step size of .2 seconds
It can be concluded that it is better to use a smaller step size in order tominimize the load shed for all contingencies. However, if we have a very large
system and the simulation time must be reduced, a larger step size still provides
reasonable results.
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8.5Having Fixed Load Shedding BlocksAll p