Setting Underfrequency Relays In Power Systems Via Integer Programming

Setting Under-Frequency Relays in Power Systems via Integer Programming

Frida Ceja Gómez

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 Gómez 2011

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 limits for 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.

i

RÉSUMÉ Dans un réseau électrique, l’écart de la fréquence du réseau par rapport à sa valeur nominale est le reflet d’un manque d’équilibre entre la production et la consommation. Ces écarts peuvent avoir des conséquences graves et ils doivent être contrôlés et surveillés de très près. Un impact majeur dans l’opération d’un réseau électrique avec une tolérance large autour de la fréquence nominale est le risque d’endommager les alternateurs. Pour éviter cette situation, les fabricants établissent des délais pour opérer en sous-fréquence et lorsque ces délais sont dépassés, les alternateurs sont automatiquement déconnectés du réseau, ce qui entraîne une plus grande déviation de la fréquence. Le délestage sur le seuil de sous- fréquence est conçu pour protéger le réseau électrique d'événements conduisant à une baisse soudaine de la fréquence du réseau lorsque les réglages intégrés dans le système de production sont insuffisants pour ramener la fréquence à la valeur nominale. Le délestage sur le seuil de sous- fréquence consiste à déconnecter des regroupements de consommateurs lorsque la fréquence descend en dessous d’une certaine limite. Cependant, les manœuvres de délestage sont principalement conçues à partir de l'expérience du comportement du réseau électrique. Les réglages des relais de protection de premières instances sont proposés, testés et modifiés jusqu'à ce qu'une manœuvre de délestage approprié soit obtenue. Ce processus est laborieux,

non-systématique,

et

conduit

généralement

à

un

délestage

conservateur de la demande. Cette thèse présente une formulation utilisant des techniques de programmation linéaire mixte pour déterminer les réglages des relais de protection de sous-fréquence. L'objectif est de rendre la conception des manœuvres de délestage plus systématique, moins dépendante des méthodes empiriques, et moins conservatrice au point de vue du délestage.

ii

For my mother and my husband, who offered me unconditional love and support throughout the course of this thesis

iii

ACKNOWLEDGMENTS

I would like to thank Professor Galiana, who inspired me to pursue a master’s 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 the previous 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 André Dagenais, Amir Kalatari, Mustafa Momen, Salman Nazir, Kelvin Lee, Étienne 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-Québec 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.

iv

CONTENTS ABSTRACT ............................................................................................................... i RÉSUMÉ ................................................................................................................... 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 v

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

8

7.1

Conventional Method ...............................................................................34

7.2

Revised Conventional Method .................................................................39

7.3

Comparison................................................................................................46

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

vi

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..............31 Figure 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 TABLES Table 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 vii

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

viii

1 INTRODUCTION

1.1 Background The 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 underfrequency 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 1960’s 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. 1

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 underfrequency 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 2

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.

1.2 Recent Developments in the Design of UFLS Programs Currently, 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 optimum UFLS 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 3

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 the system’s 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

4

method

requires

that

relays

be

equipped

with

microcontroller

and

communications technology.

1.3 Motivation for this Thesis Power 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 underfrequency 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 does away 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)  f s , ts , d s  ; s  1,..., ns whose values define the setting of the relay: A set of frequency set points f s ; s  1,..., ns of decreasing values whose violation over corresponding time spans t s triggers the shedding of corresponding blocks of load d s ; 3) A set of constraints on the input parameters and decision variables. For example, the requirement that the frequency time trajectory following a 5

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-integer linear 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 underfrequency 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.

6

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 the frequency returns to a value higher than f within a period of time smaller than

t , then the timer resets. Figure 1 illustrates the operation of an underfrequency load shedding relay.

Figure 1: Operation of an under-frequency load shedding relay

7

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

 f s , ts , d s  ; s  1,..., ns .

Therefore, the under-frequency relay settings are entirely defined by the design variables:

f s , t s , d s ; s  1,..., ns

(2.1)

8

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,

2 H d f  Pm  Pe  P f o dt

(3.1)

In equation (3.1), Pm refers to the input mechanical power and Pe refers to the output electrical power in per unit, while H is the generator inertia constant in seconds, fo is 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.

3.1 Load Damping There 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 9

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, Pml  Df

(3.2)

Pml 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

10

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, 2 H d f  Pm  Pe  Df f o dt

(3.3)

3.2 Primary Frequency Regulation Generating 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 by 

f , where R represents R

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 a frequency deviation of  f . R is the governor droop and T is the time constant associated with the governor action.

11

1 R(1  sT )

Figure 3: Governor model

This transfer function is expressed in time domain by, T

d r f   r dt R

(3.4)

Governor action can then be included in the swing equation as shown below,

2 H d f  Pm  Pe  r  Df f o 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 time delay, and with both.

12

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.

13

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,

d f (t ) f  0  r (t )  g  d (t )  Df (t )  dt 2H

(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=0 while d (t ) is the amount of load shed at time t by 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 constant T , is governed by,

d r (t ) 1  f (t )    r (t )  dt T R 

(3.7)

In (3.6), the equivalent inertia constant is computed by,

H  i

H i Si S

(3.8)

14

where H i is the inertia constant of generator i with power base Si and S is the system power base. In addition, in (3.7) the equivalent governor droop can be calculated from the individual generator droops Ri as,

1 S  i R i Ri S

(3.9)

Equations (3.6) and (3.7) can be discretized into time steps of duration t by defining r ( nt )  rn , d (nt )  d n , and f (nt )  f n . Then, through Euler’s method, equation (3.6) in discrete form becomes,

f n  f n1  K n 1t ; n

(3.10)

where, Kn 

f0  rn  g  d n  Dfn  2H

(3.11)

while equation (3.7) in discrete form is given by,

rn  rn1 

t  f n    rn 1   T  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 15

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.

16

5 SYSTEMATIC UNDER-FREQUENCY RELAY SETTING APPROACH USING BINARY VARIABLES

Consider a power system consisting of ng generators. Such a system has

2 ng  1 possible generation loss contingencies, the j th contingency being defined by the loss of g j 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 C of nc plausible contingencies, 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 nc contingencies 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]. 17

5.1 Relay Timer Model Consider 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 relay1 disconnects a block of load d s at time step n when the frequency trajectory f n j (computed using equations (3.10)-(3.12)) violates the frequency set point f s for a length of time greater than t s . Recall that in its most general form the quantities { f s , d s , ts ; s} define the relay settings and are decision variables of the MILP. Note as well that the frequency deviations over time n for the different contingencies j, f nj ; j , n , are also decision variables to be found by the MILP. Now, for each frequency set point, f s , a timer can be systematically defined with the aid of the binary variable vsnj , which equals 0 when the frequency at time n , f 0  f n , is above the frequency set point f s and 1 when f 0  f n  f s . This binary variable can therefore be exactly defined by the linear inequalities,

f s   f o  f n j  L

 v  1 j sn

f s   f o  f n j  L

; j, s, n

(4.1)

In inequality (4.1), L is a large positive number compared to the numerator, e.g. 60 Hz. Using the binary variable vsnj , the total time spent by trajectory j below frequency f s at time step n is explicitly given by the following linear relation,

tsnj  tsj, n1  vsnj t ; j, s, n

(4.2)

1 We can assume that the load 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.

18

5.2 Relay Operation Logic As stated above, the relay logic dictates that the block of load d s be shed when the corresponding timer t snj exceeds t s . 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 usnj which is equal to 0 when the time interval t snj is less than t s and 1 when t snj  ts . Once again, using binary math, this variable can be exactly characterized by the linear inequality,

tsnj  ts tsnj  ts j  u sn  1  ; 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 t s 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 t s can be a variable in order to increase the degrees of freedom of the relay setting problem. In such a case, t s must be restricted to be greater than the minimum time required for the circuit breaker to open, t min , that is,

t s  tmin ; s

(4.4)

19

Another important consideration is that once some load has been shed in stage s , it cannot be restored at a later time during the trajectory, a requirement that takes the form,

usnj  usj,n 1 ; 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 stages s , which in mathematical terms requires that,

u

j sn

s

  usj,n 1  1; j , n

(4.6)

s

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 is then also enforced,

usnj  usj1,n ; j , s, n

(4.7)

5.3 Discrete-Time Frequency Response Including Load Shedding From equations (3.10)-(3.12), for each contingency j , the time discretized frequency trajectory must satisfy,

f nj  f n j1  K nj1t ; j , n

(4.8)

20

where, including the load shedding amount  u snj d s , s

K nj 

f0 2H j

 j  j j j  rn  g  Df n   usn d s  ; j , n s  

(4.9)

and where the primary frequency regulation is given by,

rnj  rnj1 

t  f n j j    j  rn 1  T  R 

(4.10)

Note that the load shedding term,  usnj d s , contains products of a binary s

variable usj,n and the amounts of load shed d s , both unknown decision variables. Nonetheless, the relay setting problem can still be formulated as a MILP by replacing the product usnj d s by the variable xsnj and by imposing the following equivalent linear inequalities j, s, n ,

0  xsnj  usnj

(4.11)

0  d s  xsnj  1  usnj 

(4.12)

It is easy to see from (4.11) and (4.12) that when usnj  1 then xsnj  d s , while when usnj  0 then xsnj  0 .

21

5.4 Constraints on the Load Shedding Variables The amount of load that may be shed at each stage d s 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,

d s  d sspec ; s

(4.13)

In this case, equation (4.9) would be given by, K nj 

f0 2H j

 j j j j spec   rn  g  Df n   usn d s  ; j , n s  

(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 nonnegative and that their sum be smaller than the total system load,

 d

s

d

(4.15)

s

5.5 Generator Under-Frequency Time/Limits The automatic load shedding strategy must satisfy the generators’ underfrequency/time limitations specified by the manufacturer. These parameters ensure 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].

22

Under-frequency range or

Maximum allowed

upper threshold (Hz)

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 tmax . To include these limitations in the relay setting problem, a new binary variable wjn is introduced equal to 1 when, after contingency j , the frequency f 0  f n j falls below the frequency threshold f  at time step n and 0 otherwise. Thus, f    f o  f n j  L

 w  1 j n

f    f o  f n j  L

; j , , n

(4.16)

Now, the total time spent below frequency threshold f  by trajectory j at time step n can be found from,

tjn  tj,n 1  wjn t; j, , n

(4.17)

Finally, the generator under-frequency/time limitations can be enforced by requiring that, 23

tjn  tmax ; j , , n

(4.18)

5.6 Other Constraints A successful load shedding strategy must yield a steady state frequency deviation f Nj within a safe range of nominal, typically plus or minus half a Hertz, that is,

 g j  x j s sN  0.5  f   1  D j  R j N

    0.5  

(4.19)

where N is the last time step in the simulation and where, as per equation (4.11) and (4.12), xsNj 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 fn j , be close to  np N  np

the steady state within some small amount  (typically 0.15 Hz), that is,

  f Nj 

1 N  f nj   np N np

(4.20)

It is also important to impose that the frequency set points f s 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, 24

min  f    f s  f max 

(4.21)

In addition, to avoid having equal frequency set points in consecutive load shedding stages, we impose the condition, f s  f s 1  f min

(4.22)

where f min is typically around .2 Hz.

5.7 MILP Formulation Thus, of all the MILP decision variables, three sets define the relay setting: (i) the frequency set points, f s ; s  1,..., ns , (ii) the time delays before shedding, t s ; s  1,..., ns , and (iii) the amount of load shed at each stage, d s ; s  1,..., 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 the unknowns 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; 25



   

Equations (4.8), (4.9), and (4.10) of the discrete time frequency response with 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.

5.8 Objective Function The 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   p j  xs. N j   s   j 

(4.23)

where the load shed for contingency j is weighted by its probability of occurrence p 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.

26

6 SIMULATION RESULTS FOR A TEST POWER SYSTEM

To illustrate the proposed method for setting UFR’s, 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.

27

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.

28

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 C to be used for the MILP formulation. However, having a large set C increases 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. 29

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 C and 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 points were 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 Points

fs

(Hz)

Load Shedding Blocks

d s

(pu)

Time delay

t s

58.2

.134

.2

57.6

.150

.2

57.2

.134

.2

(s)

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

30

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

10% 15% 25% 25% 35% 40% 50% 50%

1 5 2 1,5 1,2 2,5 2,3 1,2,5

0 0 1 1 2 2 3 3 Total

Load shedding amount (pu) 0 0 .134 .134 .284 .284 .418 .418 1.672

Table 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

31

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)

32

Figure 9: Generation loss contingencies with load shedding relay action (3)

33

7 COMPARISON OF THE PROPOSED UFLS SCHEME WITH THE CONVENTIONAL 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 steps presented 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.

7.1 Conventional Method The 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 North America 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 34

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],

L f  d (1  ) 60 d T  1  L f 1  d (1  ) 60

(6.1)

In the above equation dT refers to the total amount of load to be shed, d refers to the load damping factor, f represents 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 perunit overload is equal to 1. We can then substitute this value in equation(6.1), as well as the settling frequency of 59.5 Hz and the load damping of 2% load reduction per 1% frequency reduction.

35

1 59.5  2(1  ) 60  .485 pu d T  2 59.5 1  2(1  ) 60

(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 load shedding 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.

36

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:

37

Frequency Set Points

fs

(Hz)

Load Shedding Blocks

d s

(pu)

Time delay

t s

(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.

38

Frequency Set Points

fs

(Hz)

Load Shedding Blocks

d s

(pu)

Time delay

t s

(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.

7.2 Revised Conventional Method As 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 39

steady-state frequency to a safe range [20, p. 596,597]. The maximum generation loss Pmax given a desired steady-state frequency deviation f ss is calculated by,

Pmax    f ss

(6.3)

In the above equation,  refers to the system’s total damping factor, which can be calculated as in [20, p. 597] given the speed governor droops and the load-frequency sensitivity factor by,

  D

1 Req

(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  Req i Ri

(6.5)

In this case i=2. Note that the equivalent droop should be first converted to the system base. This is done with the use of the next formula,

Ri , Sbase 

Ri * S Sbase Si

(6.6)

40

For our test network the system base is 500 MVA (the addition of the ratings of all the units). Therefore,

Ri , Sbase 

.05*500  .25 pu 100

(6.7)

Now we can calculate the total equivalent governor droop by taking into account that we have only two machines left,

1 1   2  8 pu Req .25

(6.8)

Then, the system’s total damping factor  can be obtained,

  D

1  2  8  10 pu Req

(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,

Pmax    f ss  10 

.5 1   .08333 pu 60 12

(6.10)

Having found this value, the minimum amount of load to be shed d is given by, 41

d  P  Pmax  .5 

1 5   .4167 pu 12 12

(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 decide to 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 1   3  12 pu Req .25

(6.12)

42

  D

1  2  12  14 pu Req

Pmax    f ss  14 

.5 7   .1167 pu 60 60

d  P  Pmax  .25 

7 2   .1333 pu 60 15

(6.13)

(6.14)

(6.15)

The minimum load to be shed for a .40 pu generation loss is given by

d  P  Pmax  .40 

7 17   .2833 pu 60 60

(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 designer’s 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

43

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

44

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 into account 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,

f 

P .25  fo   60  1.071Hz  14

(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.

45

f 

The

second

P (.40  .135)  fo   60  1.135 Hz  14

block

of

load

would

then

(6.18)

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 Points

fs

(Hz)

Load Shedding Blocks

d s

(pu)

Time delay

t s

(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.

7.3 Comparison Table 8 shows the relay settings obtained with the MILP formulation as well as the ones obtained with the two variations of the conventional method.

46

Load Shedding Blocks d s (pu)

Frequency Set Points f s (Hz)

MILP

Conventional

58.2 57.6 57.2

58.5 58.1 57.2

Revised Conventional 58.4 58.0 57.5

MILP

Conventional

.134 .150 .134

.140 .160 .190

Time Delay t s (s)

Revised Conventional .135 .150 .165

.2 .2 .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 per block presented in Table 9.

No. of blocks shed

Lost generators

MILP Conventional 1 5 2 1,5 1,2 2,5 2,3 1,2,5

0 0 1 1 2 2 3 3

1 1 1 1 2 2 3 3 Total

Load shedding amount (pu)

Improved MILP Conventional Conventional 0 0 .14 0 0 .14 1 .134 .14 1 .134 .14 2 .284 .30 2 .284 .30 3 .418 .49 3 .418 .49 1.672 2.14

Improved Conventional 0 0 .135 .135 .285 .285 .45 .45 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 47

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.

48

8 EFFECT OF DIFFERENT PARAMETERS ON THE MILP FORMULATION

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.

8.1 Choosing the Contingencies to Include in the MILP Formulation The 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 previous example. Frequency Set Points

fs

(Hz)

Load Shedding Blocks

d s

(pu)

Time delay

t s

58.0

.160

.2

57.3

.118

.2

57.2

.152

.2

(s)

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. 49

% generation loss

Lost generators

No. of blocks shed

10% 15% 25% 25% 35% 40% 50% 50%

1 5 2 1,5 1,2 2,5 2,3 1,2,5

0 0 1 1 2 3 3 3 Total

Load shedding amount (pu) 0 0 .160 .160 .278 .430 .430 .430 1.88

Table 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 of generation 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.

50

8.2 About the Number of Contingencies to Include in the MILP Formulation 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. All other 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 Points

fs

(Hz)

Load Shedding Blocks

d s

(pu)

Time delay

t s

58.4

.135

.2

57.7

.160

.2

57.3

.123

.2

(s)

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.

51

% generation loss

Lost generators

No. of blocks shed

10% 15% 25% 25% 35% 40% 50% 50%

1 5 2 1,5 1,2 2,5 2,3 1,2,5

0 0 1 1 2 2 3 3 Total

Load shedding amount (pu) 0 0 .135 .135 .295 .295 .418 .418 1.696

Table 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 relay settings 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.

8.3 About the Number of Load Shedding Stages The 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. 52

Frequency Set Points

fs

(Hz)

Load Shedding Blocks

d s

Time delay

t s

(pu)

58.4

.134

.2

58.1

.100

.2

57.4

.050

.2

57.2

.134

.2

(s)

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

10% 15% 25% 25% 35% 40% 50% 50%

1 5 2 1,5 1,2 2,5 2,3 1,2,5

0 0 1 1 2 3 4 4 Total

Load shedding amount (pu) 0 0 .134 .134 .234 .284 .418 .418 1.622

Table 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 th load shedding stage. All other contingencies have the same amount of load shed with both UFLS plans. 53

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.

8.4 Effect of the Number of Time Steps and Step Size The 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 of time 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.

54

Frequency Set Points

fs

(Hz)

Load Shedding Blocks

d s

Time delay

t s

(pu)

58.5

.134

.2

57.8

.150

.2

57.7

.134

.2

(s)

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 a generation 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

10% 15% 25% 25% 35% 40% 50% 50%

1 5 2 1,5 1,2 2,5 2,3 1,2,5

0 0 1 1 2 3 3 3 Total

Load shedding amount (pu) 0 0 .134 .134 .284 .418 .418 .418 1.622

Table 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 to minimize 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.

55

8.5 Having Fixed Load Shedding Blocks All previous examples treat the load shedding blocks as continuous unknown variables. There are however certain constraints on the load shedding blocks that may not allow to shed the appropriate blocks sizes or to shed them in the right order. Under-frequency load shedding relays are placed at distributions or transmission stations where major load feeders are located. Load shedding is performed by automatically opening a circuit breaker that disconnects the load at the feeder. This means that the load that can be shed by a given relay depends on the amount of load connected to the feeders that can be controlled by the relay. Also, most UFLS plans must follow a specific priority order in which the blocks must be shed. Load shedding priority has a significant impact on the success of the UFLS plan because even if the optimal block sizes can be shed, the imposed order might lead to excessive load shedding. For example, consider a power system in which for a generation loss of 22% it is required to shed a minimum amount of 18% load to have the system frequency settle at 59.5 Hz. Now suppose that we have 3 load shedding blocks of 10%, 15%, and 20% and that they must be shed in that specific order. Without priority load shedding, we could choose to shed the block of 20% which would allow the system frequency to recover without shedding excess load. However, if the priority order must be followed, the total load shed will be 25% which is more than the actual generation loss. For these reasons it is possible that when designing an UFLS scheme, the load shedding blocks are already predefined and we are only interested in finding the appropriate frequency set points. The proposed MILP formulation can be easily adapted to accept the load shedding blocks as inputs by adding equation (4.13) and modifying equation (4.14) as shown in Section 5.4. 56

Table 18 shows the relay settings obtained with the MILP formulation when providing the load shedding blocks as inputs. The load shedding blocks were chosen to be the ones found with the complete model in Section 5 in order to see if specifying the blocks had a significant change in the results. The highest allowed frequency set point was reduced from 59 to 58.5, since previous knowledge of the system shows that a higher value would cause unnecessary load shedding. All other parameters remained the same as in the previous examples. The simulation time was less than 30 seconds.

Frequency Set Points

fs

(Hz)

Predefined Load Shedding Blocks

d s

Time delay

t s

(s)

(pu)

58.2

.134

.2

57.5

.150

.2

57.2

.134

.2

Table 18: Relay settings obtained when having the load shedding blocks as inputs

The resulting relay settings are very similar to the ones obtained when having the load shedding blocks as variables. The frequency set point of the second load shedding stage changed from 57.6 to 57.5. This however does not change the response of the system frequency during the target contingencies and the loads shed per contingency are identical to the results shown in Table 4. Table 19 shows the relay settings obtained when specifying a different set of load shedding blocks.

57

Frequency Set Points

fs

(Hz)

Predefined Load

Time delay

Shedding Blocks

d s

t s

(s)

(pu)

58.1

.150

.2

57.4

.150

.2

57.2

.130

.2

Table 19: Relay settings obtained when having the load shedding blocks as inputs (2)

These relay settings also protect the system against all the target contingencies. The blocks of load shed per contingency are shown below.

% generation loss

Lost generators

No. of blocks shed

10% 15% 25% 25% 35% 40% 50% 50%

1 5 2 1,5 1,2 2,5 2,3 1,2,5

0 0 1 1 2 2 3 3 Total

Load shedding amount (pu) 0 0 .150 .150 .300 .300 .430 .430 1.760

Table 20: Blocks of load shed per contingency when having the load shedding blocks as inputs

It can be seen that the total load shed is larger, but this is only because the predefined blocks are larger than the ones found with the model. These relay settings still shed the minimum amount of load possible for the given load shedding blocks. The results presented in this section show that the model provides appropriate frequency set points that lead to a successful UFLS when the load shedding blocks are inputs.

58

8.6 Having Fixed Frequency Set Points Although more unlikely, there might be certain situations in which the UFLS must adhere to a given set of frequency set points and only suitable load shedding blocks must be found. It is possible to adapt the MILP formulation proposed in this thesis to receive the frequency set points as inputs by removing equations (4.21) and (4.22), at the same time as providing the set points as parameters. The next table shows the relay settings found when providing the frequency set points as inputs. The frequency set points were chosen to be the ones obtained with the complete model to observe if there were any changes on the resulting load shedding blocks. All parameters used remained unchanged.

Predefined Frequency Set Points

fs

(Hz)

Load Shedding Blocks

d s

(pu)

Time delay

t s

58.2

.136

.2

57.6

.147

.2

57.2

.134

.2

(s)

Table 21: Relay settings obtained when having the frequency set points as inputs

It can be seen that the load shedding blocks obtained are very similar to the ones obtained with the complete mode, showing the consistency of the model. The table below shows the blocks of lead shed per contingency.

59

% generation loss

Lost generators

No. of blocks shed

10% 15% 25% 25% 35% 40% 50% 50%

1 5 2 1,5 1,2 2,5 2,3 1,2,5

0 0 1 1 2 2 3 3 Total

Load shedding amount (pu) 0 0 .136 .136 .283 .283 .417 .417 1.672

Table 22: Block of load shed per contingency when having the frequency set points as inputs

The overall load shed is the same as with the relay settings originally obtained. Also, the same number of blocks is shed per contingency. The following table shows the relay settings obtained when having as input a different set of frequency set points.

Predefined Frequency Set Points

fs

(Hz)

Load Shedding Blocks

d s

(pu)

Time delay

t s

58.4

.134

.2

57.8

.150

.2

57.2

.134

.2

(s)

Table 23: Relay settings obtained when having the frequency set points as inputs (2)

The resulting relay settings also protect the system during all the target contingencies. The blocks of load shed per contingency are the same as shown in Table 4, since the load shedding block are of the same size. We can therefore conclude that the proposed MILP formulation can be used to find appropriate load shedding blocks when specifying the frequency set points. 60

CONCLUSIONS

This thesis presents a new systematic method based on mixed-integer linear programming to find the settings for under-frequency load shedding relays. The mathematical formulation of the method is explained as well as the rationale behind the choice of objective function and constraints. The proposed approach is tested on a small power system illustrating how the MILP-based relay settings obtained protect the system during all target contingencies without shedding excess load. The resulting frequency trajectories also respect the generator frequency/time limits specified by the manufacturer. The proposed scheme is compared to a conventional approach and it is shown that it results in smaller amounts of load shed. The effect of modifying certain parameters on the results obtained with the MILP formulation is also studied. It is shown that the proposed method can be modified to receive either the load shedding blocks or the frequency set points as inputs. The MILP model presented here can readily be extended to include unit generation capacity and ramp limits as in [21], as well as frequency deviation models for individual generators. Further research should be carried out to evaluate the possible advantages of adjusting the relay settings to the demand level, as well as to identify among the set of plausible contingencies a smaller but sufficient subset to include in the MILP problem.

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