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Optimal Setting Sor Under Frequency Load Shedding

Relays Using Mixed Integer Linear Programming

Mohammad Alizadeh1 Turaj Amraee2 Meysam Jaefari3

1 Ph.d student, Department of electrical engineering, Imam Khomeini naval Sciences University, iran, noshahr.

golmahalleh63@gmail.com 2 Assistant Professor, Department of electrical and computer engineering, K.N. Toosi University of Technology, iran,

tehran.

amraee@kntu.ac.ir 3 Assistant Professor, Department of electrical engineering, Noshirvani University of Technology, iran, babol.

m.jafari@nit.ac.ir

Abstract :After occurrence of some disturbances in power system that causes the sever imbalance between generation power and

electrical load, the power system frequency begins to decrease. To prevent power system frequency instability and stop

the frequency decay below the power system allowable frequency limitation, load shedding schemes should be utilized

by applying under frequency load shedding relays. In this paper, power system behavior is modeled as a mathematical

mixed integer programing (MIP), moreover the parameters that are required for UFLS relays setting are determined

optimally in order to minimize the load shedding amount throughout power system in compliance with certain

restrictions. The obtained results have been verified by Simulation studies.

Keywords: Under frequency load shedding, Optimization, Mixed integer program, Disturbance, Frequency

stability.

Submission date: 20, 11, 2015

Acceptance date : 16, 04, 2017

Corresponding author: Meysam Jaefari

Corresponding author’s address: shariati Ave. Elec. Eng. Dep., Noshirvani Uni. Of Tech. babol , Iran

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1. Introduction

Under frequency load shedding (UFLS) is a common

practice for electric power utilities to prevent frequency

decline in power systems after disturbances causing

dangerous imbalance between the generation and load

[1]. Depending on the size of the frequency deviation

experienced, emergency control and protection

schemes may be required to maintain power system

frequency at a safe range. The small frequency

deviations can be attenuated by the governor natural

autonomous response (primary load frequency control)

and the secondary automatic generation control system.

For larger frequency deviations, the emergency control

and protection schemes must be used to restore the

system frequency.

Different kinds of UFLS protection schemes have been

proposed including traditional, semi-adaptive and

adaptive schemes [2, 3]. The role of rate of frequency

change (df/dt) in designing effective UFLS plan has

been discussed in [1,3,4]. The main advantage of

adaptive UFLS schemes is that the load shedding is

carried out based on the severity of disturbance. A

systematic method for setting under frequency load

shedding relays has been proposed using a mixed

integer technique [5]. The ability of a regression tree

method for estimation of the frequency decline

following a generator outage is examined in [6]. In [7]

a genetic algorithm (GA) has been employed to

determine and minimize the amount of load to be shed

at each stage for UFLS relays. In [8] a computational

method is presented using the Monte-Carlo simulation

approach to calculate the settings of UFLS relays. In

[9] the Artificial Neutral Network (ANN) and Support

Vector Regression (SVR) methods have been applied

for prediction of post-contingency power system

frequency response. A method that combines the

Monte Carlo simulation approach with artificial neural

networks (ANN) to select UFLS strategies is presented

in [10].

One of the main defections of the conventional UFLS

schemes is determination of the load amount to be shed

in each step. Insufficiency of the load shedding in any

step, shifts the system state to a point with lower

frequency until the governor or the next load shedding

stage have been imported. This situation is probably

more harmful than frequency collapse and may damage

the power system generators. Oppositely, the large

amount of load shedding in each step causes the fast

velocity of power system generators and sever increase

of network frequency which finally leads to system

collapse.

In this paper, a new method is proposed for optimal

setting of UFLS relays by using mathematical and

numerical computational methods. The objective

function is to minimize the average amount of load to

be shed over a set of random generator outage.

The proposed method examines the setting of UFLS

relays through the mathematical optimization and

significantly reduces the dependence on trial-and error

methods.

The rest of paper is organized as follows: In section 2,

the fundamentals of the load-frequency control are

discussed. The parameters that are required for UFLS

relays setting are mentioned in section 3. The UFLS

optimization problem is modeled as a mixed integer

linear programming in section 4. The simulation

studies are illustrated in section 5 and finally the

conclusion is stated in section 6.

2. Fundamentals of Load-Frequency

Control

After occurrence of sever disturbances in power

system, an imbalance between generation and

consumption power will be created. This power

imbalance decreases the power system frequency.

Therefore for maintaining the system frequency in

secure range, some substations throughout the network

have been equipped with UFLS relays to shed the

sufficient amounts of load.

The application of UFLS relays to remove the specified

amounts of load at the predetermined frequency

thresholds is the simplest automatic load shedding

plan. UFLS relays can be optimally set to shed the

minimum amounts of load and also arrest the system

frequency decay at a safe level [11]. As the power

system frequency decreases, some amounts of load are

disconnected in discrete steps according to frequency

thresholds.

Assume that the swing equation of i-th generation unit

is expressed as follows [1, 12, 15]:

2i i

i mi ei

n

H dfp p p

f dt (1)

where Pmi is the mechanical turbine power in p.u, Pei is

the electrical power in p.u, ΔPi is the load-generation

imbalance in p.u, Hi is the inertia constant in second, fi

is the frequency in Hz and fn is the rated value of

frequency. By combining Ng swing equations, the

following expression is obtained for the total load-

generation imbalance on the COI reference using the

new base MVA, 1

gN

iiS S

as follows [15]:

(2) 2

eq

m e

cd

n

H dfP P P

f dt

where

(3) 1 1

( , ) ,g gN N

i

i i i i

e

i

qc

H f H Sf H

H S

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(4) 1 1

. .( , ) ,

g g

mi i ei i

N N

i i

m e

P S P SP P

S S

where fc is the frequency of the equivalent inertial

center and Heq is the equivalent inertial constant

[13,15].

3. UFLS Relays Setting

In an effective load shedding scheme, the minimum

load amount must be shed in different disturbances

while the system frequency is never allowed to

decrease below the secure frequency range. For

adjusting the UFLS relays, 3 parameters should be

optimally set which are:

1- Frequency threshold for each load shedding step: If

the power system frequency has decreased more than

the predetermined frequency threshold, the UFLS

relays will be operate.

2- Time delay for each load shedding step: If the power

system frequency has crossed the frequency threshold,

the UFLS relay will be operate after a certain delay

time.

3- The amounts of load to be disconnected in each load

shedding step.

4. Optimization Problem

In this paper to optimize the parameters that are

required for UFLS relays setting, a Mixed Integer

Programing (MIP) model is utilized. To achieve this

purpose, the Generalized Algebraic Modeling System

(GAMS) software is applied. GAMS is a high-level

modeling system for mathematical programming and

optimization and is specifically designed for modeling

linear, nonlinear and mixed integer optimization

problems. It consists of a language compiler and a set

of integrated high-performance solvers and is tailored

for complex and large scale modeling applications.

4.1. Discrete-Time Frequency Response

Modeling

According to (1), in a multi-machine power system,

each generator has a unique frequency response

depending on parameters such as inertia constant and

governor droop. Assuming that all generators swing

synchronously at a common frequency, as is commonly

done, an approximation to the system frequency

response can be obtained through an equivalent single-

machine swing equation expressed as follows [5]:

( )( ( ) ( ) . ( ))

2 eq

nfd f tr t g d t D f t

dt H

(5)

Where fn is the nominal frequency, Δf(t) refers to

frequency deviation from the nominal frequency

following the contingency, Δd(t) is the amount of load

to be shed at time t by applying UFLS relays, D is the

system load damping factor, and Δr(t) is the primary

frequency regulation of governor action with time

constant T that is represented by[15]:

(6) ( ) 1 ( )

( ( ))eq

d r t f tr t

dt T R

Where Req is the equivalent governor droop that is

calculated from the individual generator droops Ri as:

1

1 g

i

i

N

ieq

S

R R S

(7)

Where Ri is the inertia constant of generator i with base

power Si.

By considering the time interval Δt for simulation

points, (5) and (6) are changed to discrete forms. The

smaller Δt results more accurate answer but will also

increase the number of model variables and the

computational time.

The following cases are utilized for simplification:

(8) ( . ) ( )f n t f n

(9) ( . ) ( ) d n t d n

(10) ( . ) ( )r n t r n

Here the modified Euler method is used to solve the

discretized system frequency response. In modified

Euler method, the arithmetic average of the slopes at

t(n) and t(n+1) time steps are considered. The

discretized system frequency response is solved using

modified Euler method as follows [15]:

( ) ( )2

n

eq

fRHS t P t

H (11)

1

( ) ( )( 1) ( ) ,

n

n

t

t

n nn nf f RHS t f

(12)

The integral is approximated by trapezoidal rule as

given below:

1 1( 1) ( ) , ,

2n n n n

n n t f t ft

f f RHS RHS

(13)

Where

(( ) )( ) ( ) ( )1

( , ) .2 eq

n n n s nRHS t f r g d D fH

(14)

( )( 1)

1 ( ) ( )n

n n nt f

r r rT R

(15)

The initial condition for the above equations is zero

because there is no frequency deviation before the

occurrence of disturbance and the system frequency

has lied in steady state (i.e. Δf0=0).

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4.2. Binary Variables

If the freq(n,j) in time step n and contingency j

falls below the set point f(s), binary variable v(s,n,j)

will take 1, otherwise will take zero. This constraint

can be mathematically stated as[5]:

(16) ( ) ( ( , )) ( ) ( ( , ))

( , , ) 1f s freq n j f s freq n j

v s n jL L

where

(17) ( , ) 0 ( , )freq n j f f n j

and L is a large number. For each load shedding

step, a timer is defined to calculate the total time spent

by frequency trajectory j below set point f(s). The timer

for each load shedding step can be defined as:

(18) 1

1

1

( , , ) . ( , , )

n n

n

timer s n j t v s n j

If the time spent in each load shedding step has

exceeded the specified delay time ts, the binary variable

u(s,n,j) takes 1 and otherwise takes zero. This

constraint can be described as:

( , , ) ( ) ( , , ) ( )( , , ) 1

timer s n j t s timer s n j t su s n j

L L

(19) u(s,n,j) may have multiple value of 1 in consecutive

time steps. If u(s,n,j) has been considered as UFLS

relays actuator, these relays will operate several times

in each load shedding step. However, UFLS relays

operate once in each load shedding step, so binary

variable z(s,n,j) has been defined. z(s,n,j) takes 1 once

the binary variable u(s,n,j) takes 1 for the first time in

each load shedding step:

(20) ( , , ) ( , , ) ( , 1, )z s n j u s n j u s n j

The binary variable z(s,n,j) stimulates the UFLS relays

to shed the specified amounts of load in each load

shedding step. When freq(n,j) falls below the frequency

threshold f(s), The binary variable z(s,n,j) takes 1 after

specified time delay and activates the UFLS relays to

shed the predetermined amounts of load.

4.3. Optimization Model

The objective is defined as the minimum load

disconnection:

1 1

min ( ). ( , , )

js

s n j

nn N

P j LS s n j

(21)

Where

0 ( , , ) ( , , )LS s n j z s n j (22)

( ) 1 ( , , ) ( , , )d s s n j LS s n jz (23)

( , , ) ( ) 1 , ,( )LS s n j d s s n jz (24)

The load to be shed in each contingency is weighted by

the corresponding occurrence probability P(j) and

Δd(s) is the amount of load to be shed at step s. If the

contingencies are considered equally important, all of

the weights will be equal. Utilizing (22-24), the

optimization model is defined as a mixed integer linear

program and so the executing time considerably will be

declined.

The objective function is constrained to the

followings:

1- The aim of load shedding scheme is to prevent

the power system frequency more descending from the

minimum allowable frequency after every

disturbances. In most cases, the frequency of 47.5 Hz is

considered as minimum allowable frequency for the 50

Hz nominal frequency [14]. This constraint can be

stated as:

(25) , : ( , ) 47.5n j freq n j

2- The load shedding scheme begins when the power

system frequency is below 49.5 Hz. This can be

expressed as:

(26) : ( ) 49.5s f s

3- To avoid the interference between the UFLS relays

operation in different load shedding steps, there should

be a frequency distance greater than 0.1 Hz between

the set points f(s). This constraint state as:

(27) : ( ) ( 1) 0.1s f s f s

4- To avoid errors caused by frequency fluctuations,

the delay time for every load shedding step should be

greater than 0.2 seconds.

(28) : ( ) 0.2s t s

5. Simulations

In this paper, to implement the proposed method of

setting UFLS relays, the GAMS software has been

used. The typical values utilized in optimization

problem are given in Table. 1:

Table. 1. Parameters value used in optimization problem

Value definition parameter 0.1 Simulation time interval (sec) ∆t

60 number of simulation steps N

2 Number of load shedding steps ns

5 Number of disturbances nj

1 Load damping factor D

5 Governor time constant (sec) T

The equivalent inertia constant Heq and governor drop

Req following each contingency have been shown in

Table. 2 [5].

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Table. 2. Equivalent parameters

Req Heq Disturbance magnitude (p.u) 3.75 3.2 0.1

3.75 3.2 0.15

3.75 3.2 0.25

5 2.4 0.35

5 2.4 0.4

The proposed method has been simulated three

different scenarios, including equal and different

probability for each disturbance.

5.1. Disturbances with Equal Probabilities

The power system frequency trajectories after

occurring different disturbances without considering

any load shedding scheme have been shown in Fig. 1.

Fig. 1. Power system frequency response after 5

disturbances without considering any load shedding

scheme

As shown in Fig.1, after occurring disturbances of

0.35 and 0.4 p.u, power system frequency has been

declined below 47.5 HZ, therefore UFLS relays should

be installed throughout the network to shed some

amounts of load to prevent the frequency instability.

To evaluate the proposed MINLP optimization

problem considering load shedding scheme, three cases

including different set of disturbances have been

considered which are listed in Table. 3.

Table. 3. Simulation cases

Study

case

Number of

disturbances

Disturbances

magnitude (p.u)

LS steps

Case 1 3 disturbances 0.1 - 0.25- 0.35 2 steps

Case 2 3 disturbances 0.15- 0.25-0.35 2 steps Case 3 4 disturbances 0.1-0.15-0.35-0.4 2 steps

By solving the MINLP optimization problem in

GAMS software, the UFLS relays parameters

including frequency thresholds, time delay and the load

amount to be shed in each step are illustrated in Table.4

for three cases.

Table. 4. UFLS relay setting

Load to be

shed (p.u) Time

delay(sec) Frequency

threshold(HZ) LS

step case

0.049 0.2 49.01 1 1

0.128 0.2 47.96 2

0.056 0.2 48.55 1 2

0.127 0.2 47.96 2

0 0.2 48.485 1 3

0.214 0.2 48.385 2

Fig. 2-4 show the power system frequency trajectories

for all three cases applying UFLS relays.

Fig. 2. Frequency trajectories for case 1

Fig. 3. Frequency trajectories for case 2

Fig. 4. Frequency trajectories for case 3

Analyzing the results of Fig. 2-4 considering equal

occurrence probability for disturbances, the power

system frequency decreases to 47.5 HZ bound in sever

disturbances and then increase, therefore the simulation

results verified the MIP optimization model for UFLS

relays setting.

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5.2. Disturbances with Different

Probabilities

In this section, the MINLP optimization is repeated

considering different probabilities for disturbances that

are mentioned in table. 5.

Table. 5. Disturbances weighting factor

Disturbances

magnitude (p.u)

Weighting

factor

0.1 10

0.15 9

0.25 5

0.35 2

0.4 1

The optimization results for UFLS relays setting

including three cases considering different probabilities

for each disturbance are illustrated in Table. 6.

Table. 6. UFLS relay setting

Load to be

shed (p.u) Time

delay(sec) Frequency

threshold(HZ) LS

step case

0.043 0.2 49.01 1 1

0.134 0.2 47.96 2

0.055 0.2 48.31 1 2

0.128 0.2 47.96 2

0.19 0.2 48.38 1 3

0.047 0.2 47.59 2

The power system frequency trajectories for three

different cases considering different probability are

illustrated in Fig. 5-7.

Fig. 5. Frequency trajectories for case 1

Fig. 6. Frequency trajectories for case 2

Fig. 7. Frequency trajectories for case 3

As illustrated in Fig. 5-7, the power system

frequency decrease to 47.5 HZ bound in sever

disturbances and then increase for three cases

considering different occurrence probabilities. Also

comparing the results of table 4 and 6 including

disturbances with equal and different occurrence

probabilities respectively, the frequency thresholds in

cases with different probabilities often is less than

equal probabilities.

6. Conclusions

In this paper, UFLS relays setting problem is modeled

as a mixed integer linear programming to optimize the

UFLS relays setting for preventing the power system

frequency instability. The proposed model is executed

for three cases including various set of disturbances

considering equal and different probabilities for each

disturbance. As a result, 3 main parameters that are

needed for UFLS relays setting are resulted optimally

by meeting of all constraints and limitations. The

simulation results have verified the UFLS relays

setting optimization model.

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