Setting Underfrequency Relays in Power Systems via Integer Programming

8/13/2019 Setting Underfrequency Relays in Power Systems via Integer Programming

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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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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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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.


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.


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

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Setting Underfrequency Relays in Power Systems via Integer Programming