A population based heuristic algorithm for optimal relay operating times

Abstract

The RST algorithm of Mohan and Shanker [10] is a population based heuristic algorithm designed to determine the global optimal solution of nonlinear constrained optimization problems. Unlike other opti-mization techniques it does not require an initial guess value of the decision parameters, but only requires the lower and upper bounds of the decision parameters. The algorithm does not require the continuity and differ-entiability of the objective function or constraints. The algorithm has been well tested on standard benchmark problems in Mohan and Shanker [10] . In this paper, an attempt has been made to solve a problem that has its origin in electrical power systems. The problem is to compute the values of the decision variables called "Re-lays", which control the act of isolation of faulty lines from the system without disturbing the healthy lines. In other words it is required to determine the optimal relay operating times. Firstly, the generalized model is presented. Then, two models have been considered namely 3–bus system and 4–bus system. The results obtained by RST are compared with that of MATLAB Toolbox. It is found that the results obtained by RST are superior from the results of MATLAB because the later are not feasible solutions.

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ISSN 1 746-7233, England, UK
World Journal of Modelling and Simulation
Vol. 2 (2006) No. 3, pp. 167-176
A population based heuristic algorithm for optimal relay operating times∗
Kusum Deep1†, Dinesh Birlar2, R. P. Maheshwari2, H. O. Gupta2, Manoj Thakur1
1Department of Mathematics, IIT Roorkee, Roorkee 247667, India
2Department of Electrical Engineering, IIT Roorkee, Roorkee 247667, India
(Received February 16 2005, Accepted March 28 2006)
Abstract. The RST algorithm of Mohan and Shanker[10]is a population based heuristic algorithm designed
to determine the global optimal solution of nonlinear constrained optimization problems. Unlike other opti-
mization techniques it does not require an initial guess value of the decision parameters, but only requires the
lower and upper bounds of the decision parameters. The algorithm does not require the continuity and differ-
entiability of the objective function or constraints. The algorithm has been well tested on standard benchmark
problems in Mohan and Shanker[10]. In this paper, an attempt has been made to solve a problem that has its
origin in electrical power systems. The problem is to compute the values of the decision variables called “Re-
lays”, which control the act of isolation of faulty lines from the system without disturbing the healthy lines.
In other words it is required to determine the optimal relay operating times. Firstly, the generalized model
is presented. Then, two models have been considered namely 3–bus system and 4–bus system. The results
obtained by RST are compared with that of MATLAB Toolbox. It is found that the results obtained by RST
are superior from the results of MATLAB because the later are not feasible solutions.
Keywords: electric power system relays, constrained global optimization, population based heuristic algo-
rithms
1Introduction
Directional Overcurrent Relays (DOCRs) are provided in electrical power systems to isolate only the
faulty lines, in the event of the faults in the system. These relays are placed at both ends of each line. Thus,
number of directional overcurrent relays in an electrical power system is twice the number of the lines. To
maintain the continuity of supply to healthy sections and to isolate the faulty section only, relays are coordi-
nated. This ensures that minimum lines are disrupted when fault occurs. This is done in DOCRs by properly
fixing the two adjustable parameters of each relay called “settings”. The two settings of each relay are plug
setting (PS) and time dial setting (TDS). There happen to be many relays in the system depending on the size
of the system. Thus, each relay introduces two decision variables (one TDS and one PS) in the problem.
The above stated problem of coordinating each DOCR with one another in electrical power systems can
be modeled as a non-linear constrained optimization problem. The objective function to be minimized is the
sum of the operating times of all the primary relays, which are expected to operate in order to clear the faults of
their corresponding zones. The constraints are bounds on all decision variables, complexly interrelated times
of the various relays (called selectivity constraints) and restrictions on each term of the objective function to
be between certain specified limits.
∗The second author acknowledges Council for Scientific and Industrial Research (CSIR), New Delhi, India, for providing the
financial support vide grant number 9/143 (439) 2003 – EMR – I.
†Corresponding author. Tel.: +91-1332-285339; fax: +91-1332-273560.
E-mail address: kusumfma@iitr.ernet.in.
Published by World Academic Press, World Academic Union

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K. Deep & D. Birlar et al: A population based heuristic algorithm
In this paper, the use of the well tested RST algorithm of Mohan and Shanker[10]has been exhibited to
determine the global optimal solution of the above stated problem. The results are compared with the results
obtained from MATLAB Toolbox.
In section 2, the previous attempts in this direction are mentioned. In section 3, the RST algorithm of
Mohan and Shanker (1994) has been described. In section 4, the mathematical model of the problem has been
described. In section 5, the numerical results are discussed. Finally, in section 6 conclusions are drawn.
2Literature review
The use of optimization techniques in relay coordination was first suggested in 1988[12]. A survey of all
coordination philosophies used by various researchers in the past has been presented recently by authors of
this paper [4]. The optimal coordination problem of DOCRs is a non-linear optimization problem[11]. Dimen-
sion of the problem becomes very large for the modern interconnected power systems. Due to this problem
and complexities of non-linear programming techniques, most of the researchers have solved the problem in
linear environment by presuming the values of decision variables (all plug settings), which make the prob-
lem non-linear. This presumption is made based on the expert engineer’s experience and wisdom. Thus, the
coordination problem reduces to a linear programming problem[12]. However, linear approaches can’t ensure
correct settings of the relays[9]. They cannot consider all possible operating conditions of the system. The
results obtained may be trapped on local optimum relay settings[14]. Non-linear methods can produce optimal
resultsbyoptimizingall settingsofrelays andthus,avoidundesired trippingofthose relays,whicharenot sup-
posed to operate for a fault under consideration. First optimization attemptted in this area used simplex-based
linear approach for optimizing TDS settings for presumed PS settings and Generalized Reduced Gradient
non-linear technique to optimize the PS settings for already optimized TDS[? ]. This procedure was iterated
till convergence was achieved. Ref. [7] used Sparse Dual Revised Simplex method of linear programming
suggested by [8] to optimize TDS settings for pre-assumed non-linear PS settings. Ref. [9] applied Simplex
and Rosenbrock-Hillclimb methods to optimize TDS and PS settings respectively, in a similar way, as used by
Ref. [12]. These approaches were further followed by simplex-based approaches with more and more sophis-
tications about finer aspects of the relays[2, 6, 13, 15, 16]. Recently, authors of this paper also made non-linear
attempts by making use of “MATLAB Toolbox”[5]and “Numeric Algorithm Group” Sequential Quadratic
Programming routines[3].
3The RST algorithm
In this section, we briefly discuss the RST algorithm and its additional features as to why the method is
advantageous as compared to other optimization methods.
The RST algorithm is a heuristic algorithm which can be used to determine the global optimal solution
of a nonlinear constrained optimization problem of the type:
Minimize
f(x1,x2,...,xn)
i.e. f is nonlinear real valued function in n variables.
Subject to:
gj(x1,x2,··· ,xn) ≥ 0,
ai≤ xi≤ bi,
for j = 1,2,3,··· ,m,
for i = 1,2,3,··· ,n.
and
The RST algorithm developed by Mohan and Shanker [1994] works in two phases. In the first phase
the objective function is evaluated at a number of randomly generated feasible solutions, while in the second
phase, at each iteration these solutions are manipulated by local searches (using quadratic approximation)
to yield possible candidate for global optima. As the iterations proceed the sample of solutions converge to
the global optimal solution. The Algorithm has been well tested on a number of benchmark test problems in
Mohan and Shanker[10]. The computational steps of RST are:
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World Journal of Modelling and Simulation, Vol. 2 (2006) No. 3, pp. 167-176169
(1) Choose a sample of NBIG, n-dimensional random feasible solutions and evaluate the objective func-
tion at each of them. Store in an NBIG by (n + 1) array A. Set ITERATION =1.
(2) Out of these feasible solutions, determine M and L as the feasible solutions with greatest and least
function values f(M) and f(L) respectively. If stopping criteria, |f(M)−f(L)
message that L is the global minimum solution. Otherwise go to step 3.
(3) From the current array A choose three distinct feasible solutions R1= L, R2and R3randomly and
determine a new feasible solution P as the point of minima of the quadratic curve passing through R1, R2and
R3by the formula:
?(R2
(4) Find f(P). If f(P) < f(M) , increment ITERATION and go to step 4, else go to step 3.
(5) Replace M by P in the array A and go to step 2.
The essential features of the RST Algorithm are:
• It attempts to determine the global optimal solution rather than the local optimal solution;
• It does not require an initial guess value of the decision parameters, but instead requires only the lower
and upper bounds of the decision parameters;
• It does not assume the continuity and differentiability of the objective function or the constraints, and
hence can be used for non differentiable objective functions as well;
• It is easy to program and has a wide applicability.
f(L)
| < ε, is satisfied, stop with the
P = 0.5
2− R2
(R2− R3)f(R1) + (R3− R1)f(R2) + (R1− R2)f(R3)
3)f(R1) + (R2
3− R2
1)f(R2) + (R2
1− R2
2)f(R3)
?
.
4 General model of the problem
In this section, firstly generalized relay coordination problem model is presented. Then two specific
models are presented.
4.1Generalized model
The operating time (T) of a DOCR is non-linear function of the relay settings (TDS and PS) and the fault
current (I) seen by the relay. Therefore, Relay operating-time equation for a directional overcurrent relay is
given by a non-linear equation as under:
T =
α ∗ TDS
I
?
PS ∗ CTpri rating
?β
− γ
(1)
Only TDS and PS are unknown variables in above equation. These are the “decision variables” of the
problem. α, β and γ are the constants representing the behaviour of the characteristic of the system in a
mathematicalway,inwhichoperatingtimeoftheDOCRvariesandaregivenas0.14,0.02and1.0respectively
as per[1]. Value of CTpri rating depends upon the number of turns in the equipment CT (Current Transformer).
CT is used to reduce the level of the current so that relay can withstand it. With each relay one “Current
Transformer” is used and thus, CTpri rating is known in the problem. Valueof I (Fault current passing through
the relay) is also known, as it is a system dependent parameter and continuously measured by measuring
instruments.
Number of constraints for systems of bigger sizes will be dependent upon the number of lines in the
system. Details of the number of lines in few larger systems are given in Table I. In practice, in electrical
engineering, power systems may be of even bigger sizes and there are other types of relays also besides
DOCRs. Coordinating DOCRs with other types of relays generates even larger number of constraints than
shown in Table 1. It is evident from Table 1 that simultaneous optimization of both the settings (TDS and PS)
of each DOCR of the system is a complex problem.
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K. Deep & D. Birlar et al: A population based heuristic algorithm
Table 1. No. of lines, relays, decision variables, selectivity constraints and constraints imposing restrictions on each term
of objective function for large power systems
Power systems
IEEE 57-bus
78
156
312
680
IEEE 14-bus
20
40
80
184
IEEE 30-bus
41
82
164
390
IEEE 64-bus
88
176
352
900
IEEE 118-bus
179
358
766
2036
No. of lines
No. of DOCRs (relays)
No. of decision variables
No. of selectivity constraints
Constraints imposing
restrictions on each
term of objective function
160332 6247041532
4.1.1Objective function
The relay,which is supposed to operate first to clear the fault, is called primary relay.A faultclose to relay
is known as the close-in fault for the relay and a fault at the other end of the line is known as a far-bus fault
for this relay. The locations of these faults are shown by crosses (A to H) in Fig. 2. Conventionally, objective
function in coordination studies is constituted as the summation of operating-times of all primary relays,
responding to clear all close-in and far-bus faults. OBJ, as under, gives objective function of the problem.
OBJ =
Ncl
?
i=1
Ti
pri cl in+
Nfar
?
j=1
Tj
pri far bus
(2)
where,
N clis number of relays responding for close-in fault;
Nfaris number of relays responding for far-bus fault;
Tpri cl inis primary relay operating-time for close-in fault;
Tpri far busprimary relay operating-time for far-bus fault.
4.1.2Constraints
Bounds on variables TDSs (Time dial setting of each relay)
TDSi
min≤ TDSi≤ TDSi
max,i = 1,2,··· ,Ncl
where,TDSi
Bounds on variables PSs (Plug setting of each relay)
minislowerlimitandTDSi
maxisupperlimitofTDSi.Theselimitsare0.05and1.1,respectively.
PSj
min≤ PSj≤ PSj
max,j = 1,2,··· ,Ncl
where, PSj
Limits on primary operation times: This constraint imposes constraint on each term of objective function
to lie between 0.05 and 1.0.
Selectivity constraints for all relay pairs:
minis lower limit and PSj
maxis upper limit of PSj. These are 1.25 and 1.50, respectively.
Tbackup− Tprimary− CTI ≥ 0
where, Tbackupis operating time of backup relay and Tprimaryis operating time of primary relay. Value of
CTI is 0.3.
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World Journal of Modelling and Simulation, Vol. 2 (2006) No. 3, pp. 167-176171
4.2 Model-I
In the model-I shown in Fig. 1 there are three lines i.e. this is a 3-bus system. So, number of DOCRs in the
system is 6 and number of decision variables, whose optimal set of solution is to be found, is 12 (two settings
with each DOCR). The number of terms in the objective function for problem is also 12 (twice the number of
DOCRs). Thus, the number of constraints involving the restrictions on each term of objective function is 24
(for upper and lower limits). The number of selectivity constraints is dependent upon the system data of the
model, and it is 8 for the considered model-I.
Fig. 1. Model-I (A typical 3-Bus system)
Fig. 2. Model-II (A typical 4-Bus system)
4.2.1 Objective function
The relay, which is supposed to operate first to clear the fault, is called primary relay. A fault close to
relay is known as the close-in fault for the relay and a fault at the other end of the line is known as a far-bus
fault for this relay.
Conventionally, objective function in coordination studies is constituted as the summation of operating-
times of all primary relays, responding to clear all close-in and far-bus faults. The objective function repre-
sented by OBJ is as under
OBJ =
6
?
i=1
Ti
pri cl in+
6
?
j=1
Tj
pri far bus
where,
Nclis number of relays responding for close-in fault;
Nfaris number of relays responding for far-bus fault;
Tpri cl inis primary relay operating-time for close-in fault;
Tpri far busprimary relay operating-time for far-bus fault.
Here,
Ti
pri cl in=
0.14 ∗ TDSi
ai
PSi∗ bi
??0.02
− 1
;
Ti
pri far bus=
0.14 ∗ TDSj
ci
PSi∗ di
??0.02
− 1
.
The values of constants ai, bi, ciand diare given in the Table 2.
4.2.2Constraints
Constraints for the model will be as under:
Bounds on variables TDSs (Time dial setting of each relay)
TDSi
min≤ TDSi≤ TDSi
max
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K. Deep & D. Birlar et al: A population based heuristic algorithm
where, i = 1,2,··· ,Ncl,TDSi
and 1.1, respectively.
Bounds on variables PSs (Plug setting of each relay)
minis lower limit and TDSi
maxis upper limit of TDSi. These limits are 0.05
PSj
min≤ PSj≤ PSj
max
where, j = 1,2,··· ,Ncl, PSj
respectively.
Limits on primary operation times: This constraint imposes constraint on each term of objective function
to lie between 0.05 and 1.0.
Selectivity constraints for all relay pairs:
minis lower limit and PSj
maxis upper limit of PSj. These are 1.25 and 1.50,
Tbackup− Tprimary− CTI ≥ 0
Tbackupis operating time of backup relay and Tprimaryis operating time of primary relay. Value of CTI is
0.3. Here,
Ti
backup=
0.14 ∗ TDSp
ei
PSp∗ fi
??0.02
− 1
;
and
Ti
primary=
0.14 ∗ TDSq
gi
PSq∗ hi
??0.02
− 1
.
The values of constants ei, fi, giand hiare given in the Table 3.
Table 2. Values of constants ai,bi,ciand difor Model-I
Ti
pri cl in
ai
9.4600
26.9100
8.8100
37.6800
17.9300
14.3500
Ti
pri far bus
ci
100.6300
14.0800
136.2300
12.0700
19.2000
25.9000
TDSi
TDS1
TDS2
TDS3
TDS4
TDS5
TDS6
bi
TDSj
TDS2
TDS1
TDS4
TDS3
TDS6
TDS5
di
2.0600
2.0600
2.2300
2.2300
0.8000
0.8000
2.0600
2.0600
2.2300
2.2300
0.8000
0.8000
Table 3. Values of constants ei,fi,giand hifor Model-I
Ti
backup
ei
14.0800
12.0700
25.9000
14.3500
9.4600
8.8100
19.2000
17.9300
Ti
primary
gi
14.0800
12.0700
25.9000
14.3500
9.4600
8.8100
19.2000
17.9300
p
5
6
4
2
5
6
2
4
fi
q
1
3
5
6
1
3
6
5
hi
0.800
0.800
2.2300
0.800
0.800
0.800
2.0600
2.2300
2.0600
2.2300
0.8000
2.0600
2.0600
2.2300
0.8000
0.8000
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4.3 Model-II
In the model shown in Fig. 2, there are four lines and so the number of DOCRs is 8. The number of
decision variables is 16 (two settings with each DOCR). The number of terms in the objective function for
problem is 16 (twice the number of DOCRs). Hence, the number of constraints involving the restrictions on
each term of objective function is 32 (for upper and lower limits). The number of selectivity constraints is
dependent upon the model, and it is 9.
4.3.1Objective function and constraints
For the coordination problem of model-II, value of each of Ncland Nfaris 8 (equal to number of relays
or twice the lines). Accordingly, there are 16 decision variables (two for each relay) in this problem i.e. TDS
1to TDS8and PS1to PS8.
The objective function and constraints for the model will be of same form as in the case of Model-I
problem (with Ncl= 8) described in section 4.2.2.
The values of constants ai,bi,ci,di,ei,fi,giand hifor Model-II are given in Table 4 and Table 5.
Table 4. Values of constants ai,bi,ciand difor Model II
Ti
pri cl in
ai
20.3200
88.8500
13.6100
116.8100
116.7000
16.67000
71.7000
19.2700
Ti
pri far bus
ci
23.7500
12.4800
31.9200
10.3800
12.0700
31.9200
11.0000
18.9100
TDSi
TDS1
TDS2
TDS3
TDS4
TDS5
TDS6
TDS7
TDS8
bi
TDSj
TDS2
TDS1
TDS4
TDS3
TDS6
TDS5
TDS8
TDS7
di
0.4800
0.4800
1.1789
1.1789
1.5259
1.5259
1.2018
1.2018
0.4800
0.4800
1.1789
1.1789
1.5259
1.5259
1.2018
1.2018
Table 5. Values of constants ei,fi,giand hifor Model-I
Ti
backup
ei
20.3200
12.4800
13.6100
10.3800
1.1600
12.0700
16.6700
11.0000
19.2700
Ti
primary
gi
20.3200
12.4800
13.6100
10.3800
116.8100
12.0700
16.6700
11.0000
19.2700
p
5
5
7
7
1
2
2
4
4
fi
q
1
1
3
3
4
6
6
8
8
hi
1.5259
1.5259
1.2018
1.2018
0.4800
0.4800
0.4800
1.1789
1.1789
0.4800
0.4800
1.1789
1.1789
1.1789
1.1789
1.5259
1.2018
1.2018
5 Discussion of results
In this section, the numerical results for Model-I and Model-II are presented by solving the correspond-
ing optimization problems using two approaches viz. RST algorithm described in section 3 and MATLAB
optimization toolbox. RST algorithm and MATLAB Toolbox are compared for the considered two relay co-
ordination problem models – model-I (Fig. 1) and model-II (Fig. 2).
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K. Deep & D. Birlar et al: A population based heuristic algorithm
Model-I Results: Table 6 shows a comparison of the optimal values of the decision variables and the
optimal objective function value. Table 7 shows the value of 8 selectivity constraints.
Model-II Results: Table 8 shows a comparison of the optimal values of the decision variables and the
optimal objective function value. Table 9 shows the value of 9 selectivity constraints. The Tables show that
Table 6. Decision variables and objective function optimal values for Model-I (Fig. 1) found by RST algorithm and
MATLAB
Decision VariablesValue by MATLAB Toolbox
TDS1
0.05000000000000
TDS2
0.19764667103805
TDS3
0.05000000000000
TDS4
0.20903171197914
TDS5
0.18120523612269
TDS6
0.18067552227651
PS1
1.25000000000000
PS2
1.50000000000000
PS3
1.25000000000000
PS4
1.50000000000000
PS5
1.50000000000000
PS6
1.50000000000000
Optimal Value of objective function4.78065070474491
Value by RST Algorithm
0.050062
0.210730
0.050021
0.218827
0.188140
0.195378
1.251234
1.353436
1.250000
1.381769
1.374343
1.250186
4.83542701927544
Table 7. Value of each selectivity constraint for Model-I (Fig. 1)
Constraint
No.
1
2
3
4
5
6
7
8
Constraint values in second
MATLAB Toolbox Allows tolerance 1.0e-08 in constraints
0.00000000000000
0.00000000000001
-0.00000000000002
0.09008111400397
0.04221324579101
0.02117707464442
-0.00000000000000
0.10042972713526
RST Algorithm
0.00051468116110
0.00013627506057
0.00050754672332
0.08576410572325
0.03879991683310
0.01422080195843
0.00050469099876
0.09584778802172
both methods have found acceptable feasible solutions and there is close agreement in the optimal values of
the objective functions. The minor difference in the optimal values is because MATLAB Toolbox allows some
tolerance (1.0e-08 considered in this paper) in satisfying the constraints, while RST technique attempts the
exact satisfaction of constraints without permitting any violation. Thus, the solution found by RST algorithm
is better in quality. Hence solution obtained by RST is superior to the solution obtained by MATLAB Toolbox.
6 Conclusion
In this paper, electrical engineering power system DOCR coordination problem has been solved by mod-
eling it as constrained non-linear optimization problem. Using two approaches, namely the MATLAB Toolbox
and the RST algorithm, two model problems have been solved and results by both the approaches are com-
pared for both the models.
MATLAB Toolbox finds the constrained minimum of a scalar function of several variables. The main
limitation with MATLAB Toolbox is that the function to be minimized and the constraints both must be con-
tinuous and differentiable. Moreover, it gives only local solutions and finds the constrained minimum starting
at an initial estimate of variables. Giving proper initial guess is also a limitation with the MATLAB Toolbox.
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World Journal of Modelling and Simulation, Vol. 2 (2006) No. 3, pp. 167-176175
Table 8. Decision variables and objective function optimal values for Model-II (Fig. 2), found by RST algorithm and
MATLAB Toolbox
Decision variables Value by MATLAB Toolbox
TDS1
0.05000000000000
TDS2
0.21216878984503
TDS3
0.05000000000000
TDS4
0.15157616149642
TDS5
0.12640045603507
TDS6
0.05000000000000
TDS7
0.13378620535474
TDS8
0.05000000000000
PS1
1.27332726543809
PS2
1.50000000000000
PS3
1.25000000000000
PS4
1.50000000000000
PS5
1.50000000000000
PS6
1.25000000000000
PS7
1.50000000000000
PS8
1.25000000000000
Optimal Value of objective function3.66974578594832
Value by RST Algorithm
0.050026
0.224205
0.050007
0.158685
0.136654
0.050017
0.138768
0.050038
1.291010
1.264540
1.250000
1.346004
1.266912
1.251230
1.393723
1.250774
3.70501831276266
Table 9. Value of each selectivity constraint for Model-II (Fig. 2)
Constraint
No.
1
2
3
4
5
6
7
8
9
Constraint values in second
MATLAB Toolbox (Tolerance 1.0e-08)
-0.00000000000001
0.10031292167693
0.00000000000001
0.04984067522577
-0.00000000000000
0.02586832094882
-0.00000000000000
0.09761599131427
0.00000000000001
RST Algorithm
0.00026841303138
0.09025829810599
0.00002999426841
0.04703765072395
0.0075951409511
0.02297110952710
0.00011738337735
0.09016411367773
0.00003275789839
It allows some tolerance in satisfying the constraints towards constraints violation. The RST algorithm has
definite advantages over the conventional techniques as discussed in section 3.
References
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