Wolf Search Algorithm for Solving Optimal Reactive Power Dispatch Problem

Indonesian Journal of Electrical Engineering and Informatics (IJEEI)
Vol. 3, No. 1, March 2015, pp. 7~15
ISSN: 2089-3272  7
Received November 18, 2014; Revised January 4, 2015; Accepted January 18, 2015
Wolf Search Algorithm for Solving Optimal Reactive
Power Dispatch Problem
K. Lenin*, B. Ravindhranath Reddy, M. Surya Kalavathi
Jawaharlal Nehru Technological University Kukatpally, Hyderabad 500 085, India
email: gklenin@gmail.com*
Abstract
This paper presents a new bio-inspired heuristic optimization algorithm called the Wolf Search
Algorithm (WSA) for solving the multi-objective reactive power dispatch problem. Wolf Search algorithm is
a new bio – inspired heuristic algorithm which based on wolf preying behaviour. The way wolves search for
food and survive by avoiding their enemies has been imitated to formulate the algorithm for solving the
reactive power dispatches. And the speciality of wolf is possessing both individual local searching ability
and autonomous flocking movement and this special property has been utilized to formulate the search
algorithm. The proposed (WSA) algorithm has been tested on standard IEEE 30 bus test system and
simulation results shows clearly about the good performance of the proposed algorithm.
Keywords: modal analysis, optimal reactive power, transmission loss, wolf search algorithm
1. Introduction
Optimal reactive power dispatch problem is subject to number of uncertainties and at
least in the best case to uncertainty parameters given in the demand and about the availability
equivalent amount of shunt reactive power compensators. Optimal reactive power dispatch
plays a major role for the operation of power systems, and it should be carried out in a proper
manner, such that system reliability is not got affected. The main objective of the optimal
reactive power dispatch is to maintain the level of voltage and reactive power flow within the
specified limits under various operating conditions and network configurations. By utilizing a
number of control tools such as switching of shunt reactive power sources, changing generator
voltages or by adjusting transformer tap-settings the reactive power dispatch can be done. By
doing optimal adjustment of these controls in different levels, the redistribution of the reactive
power would minimize transmission losses. This procedure forms an optimal reactive power
dispatch problem and it has a major influence on secure and economic operation of power
systems. Various mathematical techniques like the gradient method [1, 2] Newton method
[3]and linear programming [4-7] have been adopted to solve the optimal reactive power dispatch
problem. Both the gradient and Newton methods has the difficulty in handling inequality
constraints. If linear programming is applied then the input- output function has to be expressed
as a set of linear functions which mostly lead to loss of accuracy. The problem of voltage
stability and collapse play a major role in power system planning and operation [8]. Enhancing
the voltage stability, voltage magnitudes within the limits alone will not be a reliable indicator to
indicate that, how far an operating point is from the collapse point. The reactive power support
and voltage problems are internally related to each other. This paper formulates by combining
both the real power loss minimization and maximization of static voltage stability margin (SVSM)
as the objectives. Global optimization has received extensive research attention, and a great
number of methods have been applied to solve this problem. Evolutionary algorithms such as
genetic algorithm have been already proposed to solve the reactive power flow problem [9, 10].
Evolutionary algorithm is a heuristic approach used for minimization problems by utilizing
nonlinear and non-differentiable continuous space functions. In [11], by using Genetic algorithm
optimal reactive power flow has been solved, and the main aspect considered is network
security maximization. In [12] is proposed to improve the voltage stability index by using Hybrid
differential evolution algorithm. In [13] Biogeography Based algorithm proposed to solve the
reactive power dispatch problem. In [14] a fuzzy based method is used to solve the optimal
reactive power scheduling method and it minimizes real power loss and maximizes Voltage
Stability Margin. In [15] an improved evolutionary programming is used to solve the optimal

 ISSN: 2089-3272
IJEEI Vol. 3, No. 1, March 2015 : 7 – 15
8
reactive power dispatch problem. In [16] the optimal reactive power flow problem is solved by
integrating a genetic algorithm with a nonlinear interior point method. In [17] a standard
algorithm is used to solve ac-dc optimal reactive power flow model with the generator capability
limits. In [18] proposed a two-step approach to evaluate Reactive power reserves with respect
to operating constraints and voltage stability. In [19] a programming based proposed approach
used to solve the optimal reactive power dispatch problem. In [20] is presented a probabilistic
algorithm for optimal reactive power provision in hybrid electricity markets with uncertain loads.
This research paper proposes a new bio-inspired heuristic search optimization algorithm, the
Wolf Search Algorithm (WSA), for solving the optimal reactive power dispatch problem and this
algorithm is based on wolf preying behaviour activity. Algorithm possesses both individual local
searching ability and autonomous flocking movement. Wolf hunts independently by
remembering its own trait and it will merge with its peer when the peer is in better position. The
swarming behaviour of WSA has more advantage than that of algorithms like PSO [21], Fish
[22] and Firefly [23]. WSA functions as multiple leaders swarming from multiple directions to
reach the best solution, rather than searching as a single flock. How the wolf jumps far out of its
hunter’s visual range to avoid being trapped like that algorithm design will jump away from the
local optimal solution. The wolves in the nature have best memory capability for they can hide
food in caches; also they sense and track down a prey from distances of miles away. They
themselves do set markers in their territory in various methods like by urinating at the borders.
Researcher Sebastian Vetter and his team, from the University of Vienna have been studying
the high level of observational spatial memory in the wolf. Main assumption is that the wolves
are functioning as searching agents in the WSA optimization algorithm are empowered by
memory caches that can able to store the previously visited various positions. The proposed
algorithm WSA been evaluated in standard IEEE 30 bus test system & the simulation results
shows that our proposed approach outperforms all reported algorithms in minimization of real
power loss and voltage stability index .
2. Voltage Stability Evaluation
2.1. Modal Analysis for Voltage Stability Evaluation
Modal analysis is one among best methods for voltage stability enhancement in power
systems. The steady state system power flow equations are given by.
∆P
∆Q
J θ J
J θ J
(1)
Where
∆P = Incremental change in bus real power.
∆Q = Incremental change in bus reactive
Power injection
∆θ = incremental change in bus voltage angle.
∆V = Incremental change in bus voltage Magnitude
Jpθ , JPV , JQθ , JQV jacobian matrix are the sub-matrixes of the System voltage
stability is affected by both P and Q.
To reduce (1), let ∆P = 0 , then:
∆Q J J θJ θ J ∆V J ∆V (2)
∆V J ∆Q (3)
Where,
J J J θJ θ JPV (4)
J is called the reduced Jacobian matrix of the system.

IJEEI ISSN: 2089-3272 
Wolf Search Algorithm for Solving Optimal Reactive Power Dispatch Problem (K. Lenin)
9
2.2. Modes of Voltage Instability
Voltage Stability characteristics of the system have been identified by computing the
Eigen values and Eigen vectors.
Let,
J ξ˄η (5)
Where,
ξ = right eigenvector matrix of JR
η = left eigenvector matrix of JR
∧= diagonal eigenvalue matrix of JR and
J ξ˄ η (6)
From (5) and (8), we have:
∆V ξ˄ η∆Q (7)
Or,
∆V ∑
ξ η
λ
∆Q (8)
Where ξi is the ith column right eigenvector, and η is the ith row left eigenvector of JR. λi is the
ith Eigen value of JR.
The ith modal reactive power variation is:
∆Q K ξ (9)
Where,
K ∑ ξ 1 (10)
Where
ξji is the jth element of ξi
The corresponding ith modal voltage variation is:
∆V 1 λ⁄ ∆Q (11)
If | λi | =0 then the ith modal voltage will collapse.
In (10), let ∆Q = ek where ek has all its elements zero except the kth one being 1.
Then:
∆V ∑
ƞ ξ
λ
(12)
ƞ
k th element of ƞ
V –Q sensitivity at bus k
∑
ƞ ξ
λ
∑
λ
(13)
3. Problem Formulation
The objectives of the reactive power dispatch problem is to minimize the system real
power loss and maximize the static voltage stability margins (SVSM).
3.1. Minimization of Real Power Loss
Minimization of the real power loss (Ploss) in transmission lines is mathematically
stated as follows.

 ISSN: 2089-3272
10
P ∑ g θ
,
(14)
Where n is the number of transmission lines, gk is the conductance of branch k, Vi and Vj are
voltage magnitude at bus i and bus j, and θij is the voltage angle difference between bus i and
bus j.
3.2. Minimization of Voltage Deviation
Minimization of the voltage deviation magnitudes (VD) at load buses is mathematically
stated as follows.
Minimize VD = ∑ |V 1.0| (15)
Where nl is the number of load busses and Vk is the voltage magnitude at bus k.
3.3. System Constraints
Objective functions are subjected to these constraints shown below.
Load flow equality constraints:
P – P V ∑
G cos θ
B sin θ
0, i 1,2 … . , nb (16)
Q Q V ∑
G cos θ
B sin θ
0, i 1,2 … . , nb (17)
Where, nb is the number of buses, PG and QG are the real and reactive power of the generator,
PD and QD are the real and reactive load of the generator, and Gij and Bij are the mutual
conductance and susceptance between bus i and bus j.
Generator bus voltage (VGi) inequality constraint:
V V V , i ∈ ng (18)
Load bus voltage (VLi) inequality constraint:
V V V , i ∈ nl (19)
Switchable reactive power compensations (QCi) inequality constraint:
Q Q Q , i ∈ nc (20)
Reactive power generation (QGi) inequality constraint:
Q Q Q , i ∈ ng (21)
Transformers tap setting (Ti) inequality constraint:
T T T , i ∈ nt (22)
Transmission line flow (SLi) inequality constraint:
S S , i ∈ nl (23)
Where, nc, ng and nt are numbers of the switchable reactive power sources, generators and
transformers.

IJEEI ISSN: 2089-3272 
11
4. Wolf Search Algorithm
Wolves are social predators that hunt in packs and uses stealth when hunting prey
together. In behaviour of ants it utilizes pheromones to communicate with their peers to know
about food source. WSA [24] also do this kind of communication, which decreases the run time
of the search. Wolves are unique, partially cooperative characteristics and usually move in a
group in coupled formation, but have tendency to take down the prey individually. WSA naturally
balances scouting the problem space in random groups and individual. During hunting, wolves
will group themselves as they approach their prey. This peculiar characteristic prompts the
searching agents in WSA to move for a better position, like the same way wolves continuously
change their positions for better ones. When hunting, wolves search for prey and also keenly
watch the threats from hunters or other animals like tigers etc. Each wolf in the pack chooses its
own way & position continuously moving to a better state for the prey and also for threats in all
directions. When wolves’ bumping into their enemies it is well equipped with a threat probability
and it dashes a great distance away from its present position. The same way in WSA avoids the
deadlock of getting trapped in local optimal solution. The direction and distance the wolf moving
away from a threat are random, and is similar to mutation and crossover in Genetic algorithm
.Wolves have very high sense of smell and it can easily locate prey by scent. Similarly, in the
WSA each wolf has a sensing distance that creates visual distance. This visual distance is
applied to search the global optimum and in moving to a better position and for jumping out of
visual range. In search mode, the wolves are move in Brownian motion (BM), which imitates the
random drifting of particles suspended in fluid.
Basic logics of wolf search
There are three rules that act as basic logics of the Wolf Search Algorithm (WSA)
Rule 1: Each wolf has visual area as a fixed one and with a radius defined by v for X as
a set of continuous possible solutions. Each wolf can sense companions who are all appear
within its visual circle. The footstep expanse by which the wolf moves at a time is normally
smaller than its visual distance.
Rule 2: The fitness of the objective function represents the wolf’s current position. If
there is more options the wolf will chose the best terrain inhabited by another wolf from the
given options. If not, the wolf will continue to move randomly in BM.
Rule 3: if the wolf will sense an enemy then the wolf will immediately escape to a
random position far from the threat and beyond its visual range.
WSA implementation in based on the fitness of the objective function and it reflects the
quality of a terrain position which will eventually lead to food.
Wolf often changes in position in search of food and also to safeguard form the
enemies. Wolf trust with other wolves in movement because they never prey each other. The
movement done by one wolf will be watched by other wolves simultaneously and they position
themselves in chance of finding food also with care of them by continuously moving. If the
current wolf’s location is greater the distance of the companion location, then that new location
will be less attractive one even though the new position may be good one. Wolf’s willingness to
move is decreased means, and then that movement will obey the inverse square law. The
formula is r , where Io is the origin of food and r is the distance between the food or we
can denote that distance between the new terrain and the wolf.
This is the similar formula in the firefly algorithm, for the calculation of attractiveness.
The incentive formula for the wolf search by using absorption coffeicient and gaussian equation,
can be written as:
β r β e (24)
Normally all the wolves want to move better position based on colonized by their peers
position and it depends on many factors like visual distance and how the initial wolf covers the
area. Wolf will visualize the other wolves location each other i.e. it will compare the range of
distance and set by itself in best position for preying and also from enemies. The movement can
be written as:
x i x i β e x j x i escape (25)

 ISSN: 2089-3272
12
Where, escape ( ) is a function that calculates a random position to jump to with a constraint of
minimum length; v, x is the wolf, which represents a candidate solution; and x(j) is the peer with
a better position as represented by the value of the fitness function. The second term of the
above equation represents the change in value or gain achieved by progressing to the new
position. r is the distance between the wolf and its peer with the better location.
There are three types of preying that takes place in sequence,
1) Preying initiatively
Wolf feed on prey it represents the optimization function as objective. By using the
visual boundary wolf will have step by step movement on constantly seeing the prey and it will
have random movement from the current step to forward or backward depending on the prey
position. If it thinks particular position as best one then it will omit other wolves movements.
Then it will move in own direction.
2) Prey passively
In passive mode the wolf will compare the position with its peers and will improve the
current position. Wolf will move to passive mode when its own movement does not find food or
insecurity for its movement.
3) Escape
Wolves normally have enemies in nature and threat will be there always. If any threat is
found, it will relocate very quickly form the current position to new position which will be normally
greater distance than that of the normal visual range. This can be written in equation as:
if moving
x i x i α ∙ r ∙ rand prey
x i x i α ∙ s ∙ escape escape
(26)
Where x(i) is the wolf’s location; a is the velocity; v is the visual distance; rand() is a random
function whose mean value distributed in [-1,1], s is the step size, which must be smaller than v;
and escape() is a custom function that randomly generates a position greater than v and less
than half of the solution boundary.
Wolf algorithm for solving optimal reactive power dispatch problem:
Step 1: Objective function f(x), x =(x1,x2,..xd)T
Step 2: Initialize the population, xi(i=1,2,..,W)
Step 3: initialize parameters
r = radius of the visual range
s = step size by which a wolf moves at a time
α = velocity factor of wolf
pa = a user-defined threshold [0-1], determines how often foe appears
Step 4: WHILE (t<generations and also for stopping criteria is not met)
step5: FOR i=1: W // each wolf
step6: Prey new food initiatively ();
step7: Generation of new location ();
step8: To check whether the next location suggested by the random number generator
is new one .
step8: If not, repeat generating random location.
Step9:IF(dist(xi,xj) < r and xj is better as f(xi)<f(xj)) xi moves towards xj // xj is a better
than xi
Step 10: ELSE IF
xi = Prey new food passively ();
Step 11: END IF
Generation of new location ();
IF (rand ()>pa)
xi = xi + rand() + v; wolf escape to a new position.
END IF
END FOR
END WHILE

IJEEI ISSN: 2089-3272 
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5. Simulation Results
The accurateness of the proposed WSA method is demonstrated by testing it on
standard IEEE-30 bus system. The IEEE-30 bus system has 6 generator buses, 24 load buses
and 41 transmission lines of which four branches are (6-9), (6-10), (4-12) and (28-27) - are with
the tap setting transformers. The lower voltage magnitude limits at all buses are 0.95 p.u. and
the upper limits are 1.1 for all the PV buses and 1.05 p.u. for all the PQ buses and the reference
bus. The simulation results have been presented in Tables 1, 2, 3 &4. And in the Table 5 shows
the proposed algorithm powerfully reduces the real power losses when compared to other given
algorithms. The optimal values of the control variables along with the minimum loss obtained
are given in Table 1. Corresponding to this control variable setting, it was found that there are
no limit violations in any of the state variables.
Table 1. Results of WSA – ORPD optimal control variables
Control variables Variable setting
V1
V2
V5
V8
V11
V13
T11
T12
T15
T36
Qc10
Qc12
Qc15
Qc17
Qc20
Qc23
Qc24
Qc29
Real power loss
SVSM
1.041
1.042
1.041
1.031
1.002
1.040
1.00
1.00
1.02
1.01
3
3
4
0
3
4
4
3
4.3209
0.2462
Table 2. Results of WSA -Voltage Stability Control Reactive Power Dispatch Optimal Control
Variables
Control Variables Variable Setting
V1
V2
V5
V8
V11
V13
T11
T12
T15
T36
Qc10
Qc12
Qc15
Qc17
Qc20
Qc23
Qc24
Qc29
Real power loss
SVSM
1.043
1.043
1.042
1.031
1.006
1.034
0.090
0.090
0.090
0.090
4
4
3
3
0
3
2
3
4.9870
0.2471
ORPD together with voltage stability constraint problem was handled in this case as a
multi-objective optimization problem where both power loss and maximum voltage stability
margin of the system were optimized simultaneously. Table 2 indicates the optimal values of
these control variables. Also it is found that there are no limit violations of the state variables. It
indicates the voltage stability index has increased from 0.2462 to 0.2471, an advance in the

 ISSN: 2089-3272
14
system voltage stability. To determine the voltage security of the system, contingency analysis
was conducted using the control variable setting obtained in case 1 and case 2. The Eigen
values equivalents to the four critical contingencies are given in Table 3. From this result it is
observed that the Eigen value has been improved considerably for all contingencies in the
second case.
Table 3. Voltage Stability under Contingency State
Sl.No Contingency ORPD
Setting
VSCRPD
Setting
1 28-27 0.1410 0.1427
2 4-12 0.1658 0.1668
3 1-3 0.1774 0.1784
4 2-4 0.2032 0.2047
Table 4. Limit Violation Checking Of State Variables
State
variables
limits
ORPD VSCRPD
Lower upper
Q1 -20 152 1.3422 -1.3269
Q2 -20 61 8.9900 9.8232
Q5 -15 49.92 25.920 26.001
Q8 -10 63.52 38.8200 40.802
Q11 -15 42 2.9300 5.002
Q13 -15 48 8.1025 6.033
V3 0.95 1.05 1.0372 1.0392
V4 0.95 1.05 1.0307 1.0328
V6 0.95 1.05 1.0282 1.0298
V7 0.95 1.05 1.0101 1.0152
V9 0.95 1.05 1.0462 1.0412
V10 0.95 1.05 1.0482 1.0498
V12 0.95 1.05 1.0400 1.0466
V14 0.95 1.05 1.0474 1.0443
V15 0.95 1.05 1.0457 1.0413
V16 0.95 1.05 1.0426 1.0405
V17 0.95 1.05 1.0382 1.0396
V18 0.95 1.05 1.0392 1.0400
V19 0.95 1.05 1.0381 1.0394
V20 0.95 1.05 1.0112 1.0194
V21 0.95 1.05 1.0435 1.0243
V22 0.95 1.05 1.0448 1.0396
V23 0.95 1.05 1.0472 1.0372
V24 0.95 1.05 1.0484 1.0372
V25 0.95 1.05 1.0142 1.0192
V26 0.95 1.05 1.0494 1.0422
V27 0.95 1.05 1.0472 1.0452
V28 0.95 1.05 1.0243 1.0283
V29 0.95 1.05 1.0439 1.0419
V30 0.95 1.05 1.0418 1.0397
Table 5. Comparison of Real Power Loss
Method Minimum loss
Evolutionary programming [25] 5.0159
Genetic algorithm [26] 4.665
Real coded GA with Lindex as SVSM [27] 4.568
Real coded genetic algorithm [28] 4.5015
Proposed WSA method 4.3209
6. Conclusion
In this paper, the WSA has been successfully implemented to solve optimal reactive
power dispatch (ORPD) problem. The main advantages of WSA when applied to the ORPD
problem is optimization of different type of objective function, i.e real coded of both continuous
and discrete control variables, and without difficulty in handling nonlinear constraints. The
proposed WSA algorithm has been tested on the IEEE 30-bus system. Simulation Results
clearly show the good performance of the proposed algorithm in reducing the real power loss
and enhancing the voltage profiles within the limits.

IJEEI ISSN: 2089-3272 
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