differetial evolution approach for optimal reactive power dispatch
TRANSCRIPT
Differential evolution approach for optimal reactive power dispatch
M. Varadarajan *, K.S. Swarup
Department of Electrical Engineering, Indian Institute of Technology, Madras, India
Received 6 June 2007; received in revised form 28 October 2007; accepted 4 December 2007
Available online 15 December 2007
Abstract
Differential evolution based optimal reactive power dispatch for real power loss minimization in power system is presented in this paper. The
proposed methodology determines control variable settings such as generator terminal voltages, tap positions and the number of shunts to be
switched, for real power loss minimization in the transmission system. The problem is formulated as a mixed integer nonlinear optimization
problem. A generic penalty function method, which does not require any penalty coefficient, is employed for constraint handling. The formulation
also checks for the feasibility of the optimal control variable setting from a voltage security point of view by using a voltage collapse proximity
indicator. The algorithm is tested on standard IEEE 14, IEEE 30, and IEEE 118-Bus test systems. To show the effectiveness of proposed method the
results are compared with Particle Swarm Optimization and a conventional optimization technique – Sequential Quadratic Programming.
# 2007 Elsevier B.V. All rights reserved.
Keywords: Optimal power flow; Reactive power dispatch; Loss minimization; Differential evolution; Penalty function
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Available online at www.sciencedirect.com
Applied Soft Computing 8 (2008) 1549–1561
1. Introduction
Global optimization of non-continuous, non-linear functions
arising from real world complex engineering problems, which
may have large number of local minima and maxima, is quite
challenging. A number of deterministic approaches based on
branch and bound and real algebraic geometry are found to be
successful in solving these problems to some extend. Of late,
stochastic and heuristic optimization techniques, such as
evolutionary algorithms (EA), have emerged as efficient tools
for global optimization. It has been applied to a number of
engineering problems in diverse fields and one such field is
power system optimization.
The power system is a complex network used for generating
and transmitting electric power. It is expected to operate with
consumption of minimal resources giving maximum security
and reliability. The optimal power flow (OPF) problem is an
important tool to help the operator achieve these goals by
providing the optimal settings of all controllable variables. The
various objectives of OPF problem are
(1) minimization of cost of generation;
* C
E
1568
doi:1
(2) m
inimization of transmission losses or optimal reactivepower dispatch;
orresponding author.
-mail address: [email protected] (M. Varadarajan).
-4946/$ – see front matter # 2007 Elsevier B.V. All rights reserved.
0.1016/j.asoc.2007.12.002
(3) m
inimization of shift in controls;(4) m
inimization of cost of VAr Investment;(5) m
aximization of social benefit.Because of its significant influence on secure and economic
operation of power systems, optimal reactive power dispatch
has received an ever-increasing interest from electric utilities.
In this paper the optimal reactive power dispatch is done, which
is a sub-problem of the OPF problem, with an objective to
reduce transmission line power losses. It is an effective method
to improve voltage level, decrease network losses and maintain
the power system running under normal conditions. All
controllable variables, such as tap ratio of transformers, output
of shunts, reactive power output of generators and static
reactive power compensators, are determined which minimizes
real power losses or other appropriate objective functions,
satisfying a given set of physical and operational constraints.
While transformer tap ratios and output of shunts have discrete
values, reactive power output of generators, bus voltage
magnitudes and angles have, on the other hand, continuous
values. Hence the reactive power dispatch optimization is a
combinatorial optimization problem has to be formulated as a
mixed integer, nonlinear problem.
A number of mathematical optimization techniques have
been proposed in literature to solve the OPF problem. For
decades, conventional optimization techniques such as linear
M. Varadarajan, K.S. Swarup / Applied Soft Computing 8 (2008) 1549–15611550
programming (LP), quadratic programming (QP), gradient
method, Newton method, and Interior Point methods have been
used for the solving optimal reactive power dispatch problem
[1–4]. LP method requires that the objective function and
constraints have linear relationship, which may lead to loss of
accuracy. The gradient and Newton methods suffer from the
difficulty in handling inequality constraints. Conventional
methods are not efficient in handling problems with discrete
variables. The combinatorial search approaches, branch and
bound and cutting plane algorithms, which are usually used to
solve the mixed integer programming model, are non-
polynomial and all suffer from the curse of dimensionality
making them unsuitable for large scale OPF problems.
In recent years global optimization techniques such as
genetic algorithms (GA), evolutionary programming (EP),
evolutionary strategies (ES), and particle swarm optimization
(PSO) have been proposed to solve the OPF problem. Ma and
co-workers [5,6] used EP for optimal reactive power dispatch.
Lee et al. [7] solved the reactive power dispatch and investment
planning problem by using a simple genetic algorithm (SGA)
combined with the successive linear programming method. The
Bender’s cut are constructed during the SGA procedure to
enhance the robustness and reliability of the algorithm. Yoshida
et al. [8] proposed an algorithm for reactive power and voltage
control considering voltage security assessment using PSO.
Zhao et al. [9] proposed a solution to the reactive power
dispatch problem with PSO using multi-agent systems.
This paper investigates the applicability of differential
evolution (DE) algorithm for reactive power dispatch to
minimize real power loss in transmission network. DE is a
simple, population based search algorithm, for global
optimization. It has demonstrated its robustness and effective-
ness in a variety of applications, such as neural network
learning and infinite impulse response (IIR) filter design
[10,11]. DE differs from other EA’s in the mutation and
recombination phases. Unlike stochastic techniques such as GA
and ES, where perturbation occurs in accordance with a random
quantity, DE uses weighted differences between solution
vectors to perturb the population. DE employs a greedy
selection process with implicit elitist features. It has a minimum
number of EA control parameters, which can be tuned
effectively. The authors in [12–14] used DE for Optimal Var
Planning.
In all the previous works reported in literature, inequality
constraints were handled by use of a penalty function approach,
i.e., the constraint violation is multiplied by a penalty
coefficient or parameter and added to the objective function.
Deb [15] proposed a penalty function method without penalty
coefficients to overcome the difficulty in choosing penalty
coefficients for GA based constrained optimization problems.
Although a penalty term is added to the objective function to
penalize infeasible solutions, the method differs from the way
the penalty term is defined in conventional methods and in
earlier EA implementations. This penalty parameterless
strategy is applicable only to population based approach
because it requires the population to be divided into two sets:
feasible and infeasible sets. The fitness function depends on the
feasible and infeasible solutions. Since in a conventional
optimization approach, there is only one member in each
iteration, such a penalty parameterless scheme cannot be
applied.
In this paper, the penalty parameterless scheme is applied for
reactive power optimization using DE. The method converges
to the optimum solution, successfully meeting all equality and
inequality constraints. The validity of the proposed method is
tested on standard IEEE systems. Results obtained using PSO
and a conventional optimization technique – sequential
quadratic programming (SQP) are also provided for comparing
the performance of the proposed method.
2. Optimal power flow
The optimal power flow (OPF) is a static, non-linear, and
non-convex optimization problem, which determines a set of
optimal variables from the network state, load data and system
parameters. Optimal values are computed in order to achieve a
certain goal such as generation cost or transmission line power
loss minimization subjected to equality and inequality
constraints. In general the OPF problem can be presented as
min fðx; uÞ (2.1)
s:t: gðx; uÞ ¼ 0 (2.2)
hðx; uÞ � 0 (2.3)
xmin � x � xmax (2.4)
umin � u � umax (2.5)
where, fðx; uÞ is the objective function that typically includes
total generation cost (active power dispatch) or total losses in
transmission system (reactive power dispatch). Generally,
gðx; uÞ represents the loadflow equations and hðx; uÞ represents
transmission line limits and other security limits such as voltage
security margin (VSM). The vector of dependent and control
variables are denoted by x and u respectively. In general, the
dependent vector includes bus voltage angles u, load bus
voltage magnitudes VL, slack bus real power generation
Pg;slack, and generator reactive power Qg.
x ¼ ½u;VL;Pg;slack;Qg�T (2.6)
The control variable vector consists of real power generation,
Pg (except slack bus); generator terminal voltage, Vg; trans-
former tap ratio, t; and reactive power generation or absorption
of shunt capacitor and reactors, Qsh.
u ¼ ½Pg;Vg; t;Qsh�T (2.7)
Of the control variable mentioned in Eq. (2.7) Pg and Vg are
continuous variables, while tap ratio of the tap changing
transformer and reactive power output of shunt devices, Qsh,
are discrete variables. Loss minimization is usually required
when cost minimization is the main goal, with generator active
power generation as a control variable. When all control
variables are utilized in a cost minimization, a subsequent loss
minimization will not yield further improvements. Therefore,
M. Varadarajan, K.S. Swarup / Applied S
in optimal reactive power dispatch problem, such as loss
minimization, active power generation of all generators except
slack generator is fixed during the optimization procedure.
3. Optimal reactive power dispatch
The objective function here is to minimize the active power
loss (PLOSS) in the transmission system. There are two basic
approaches to loss minimization, namely the slack bus
approach and the summation of losses on individual lines.
Sometimes it is desirable to minimize losses in a specific area
and hence, the second approach which is more generic, is used
in this work.
3.1. Objective function
Network losses, either for the whole network or for certain
sections of network, are non-separable functions of dependent
and independent variables.
min PLOSS ¼XNl
k¼1
gk½ðtkViÞ2 þ V2j � 2tkViV jcos ui j� (3.1)
where, gk is the conductance of branch k between buses i and j,
Nl the number of branches, tk tap ratio of transformer connected
in branch k, Vi is voltage magnitude at bus i, and ui j is the
voltage angle difference between buses i and j.
3.2. Constraints
The minimization of the above objective function is
subjected to a number of equality and inequality constraints.
The equality constraints are real and reactive power balance at
each node i.e. load flow equations given by
Pi � Vi
XNB
j¼1
V jðGi jcos ui j þ Bi jsin ui jÞ ¼ 0
for i ¼ 1; . . . ;NB � 1
(3.2)
Qi � Vi
XNB
j¼1
V jðGi jsin ui j � Bi jcos ui jÞ ¼ 0
for i ¼ 1; . . . ;NPQ
(3.3)
where, NB is the number of buses, NPQ the number of PQ buses,
Gi j and Bi j are real and imaginary part of (i, j)th element of bus
admittance matrix, Pi and Qi are net real and reactive power
injection at bus i. The inequality constraints on security limits
(dependent variables) are given by
Pming;slack � Pg;slack � Pmax
g;slack (3.4)
VminL;i � VL;i � Vmax
L;i i ¼ 1; . . . ;NPQ (3.5)
Qming;i � Qg;i � Qmax
g;i i ¼ 1; . . . ;NG (3.6)
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP2
l þ Q2l
q� Smax
l l ¼ 1; . . . ;Nl (3.7)
VCPIi � VCPIthreshold i ¼ 1; . . . ;NB (3.8)
The inequality constraints on control (independent) variable
limits are given by
Vming;i � Vg;i � Vmax
g;i i ¼ 1; . . . ;NPV (3.9)
Qminsh;i � Qsh;i � Qmax
sh;i i ¼ 1; . . . ;Nsh (3.10)
tmink � tk � tmax
k k ¼ 1; . . . ;NT (3.11)
where, NG, NPV, Nsh and NT are the number of generators, PV
buses, shunts and transformers respectively. Pl, Ql and Smaxl are
real, reactive and maximum apparent power flow in line l.
VCPIi the voltage collapse proximity indicator at bus i, Vg;i and
VL;i are bus voltage magnitude at generator and load bus i,
respectively, Qg;i the reactive power generation at bus i, Qsh;i the
shunt reactive power at bus i, tk the tap ratio of transformer k
and Pg;slack the real power generation at slack bus. VminL;i , Vmax
L;i ,
Vming;i , Vmax
g;i , Pming;slack, Pmax
g;slack, tmink , tmax
k , Qming;i , Qmax
g;i , Qminsh;i , and
Qmaxsh;i , are minimum and maximum limits of the corresponding
variables, respectively.
4. Differential evolution
Differential evolution is a simple population based,
stochastic parallel search evolutionary algorithm for global
optimization. DE is capable of handling non-differentiable,
non-linear, and multi-modal objective functions. In DE, the
population consists of real valued vectors with dimension D
that equals the number of design parameters/control variables.
The size of the population is adjusted by the parameter NP. The
population of a DE algorithm is randomly initialized within the
initial parameter bounds. The optimization process is con-
ducted by means of three main operations: mutation, crossover
and selection. In each generation, each individual of the current
population becomes a target vector. For each target vector, the
mutation operation produces a mutant vector, by adding the
weighted difference between two randomly chosen vectors to a
third vector. The crossover operation generates a new vector,
called trial vector, by mixing the parameters of the mutant
vector with those of the target vector. If the trial vector obtains a
better fitness value than the target vector, then trial vector
replaces the target vector in the next generation. The
evolutionary operators are described below.
4.1. Initialization
In DE, a solution or an individual i, at generation G is a
multidimensional vector xGi ¼ ðxi;1; . . . ; xi;DÞ. The population is
initialized by randomly generating individuals as
xGi;k ¼ xkmin
þ rand½0; 1� � ðxkmax� xkmin
Þ i2 ½1;NP�;
k2 ½1;D�(4.1)
oft Computing 8 (2008) 1549–1561 1551
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where, NP is the population size, D is the solution’s dimension
i.e. number of control variables. Each variable k in a solution
vector i in the generation G is initialized within its boundaries
xkminand xkmax .
4.2. Mutation
DE does not use a predefined probability density function to
generate perturbing fluctuations. It relies upon the population
itself to perturb the vector parameter. For every i2 ½1; . . . ;NP�the weighted difference of two randomly chosen population
vectors, xr2and xr3
, is added to another randomly selected
population member, xr1, to build a mutated vector vi.
vi ¼ xGr1þ SðxG
r2� xG
r3Þ (4.2)
In Eq. (4.2), i; r1; r2 and r3 are mutually different indices from
the current generation. S is the user defined parameter, called
step size, which is typically chosen from the range [0, 2]. If vi is
found outside variable limit, it will then be fixed to the violated
upper or lower limit.
4.3. Crossover
The next task after mutation is crossover, to increase the
diversity of the perturbed parameter vectors. A trial vector ui is
created incorporating the mutated vector vi and the target vector
xi:
ui ¼ uGþ1i;k ¼ vi;k if randk;i � CR or k ¼ Irand
xGi;k if randk;i >CR and k 6¼ Irand
�(4.3)
where randk;i 2 ½0; 1� and Irand is chosen randomly from the
interval ½1; . . . ;D� once for each vector to ensure that at least
one vector component originates from the mutated vector vi.
Eq. (4.3) is applied for every vector component i2 ½1; . . . ;NP�,k2 ½1; . . . ;D�. CR is the DE control parameter, called the
crossover rate, and is a user defined parameter within range
[0, 1]. Trial parameter with randomly chosen index, Irand, is
taken from mutant vector to ensure that the trial vector does not
duplicate xi.
4.4. Selection
DE determines survivors by pairwise comparison i.e. a form
of tournament selection. If the trial vector ui has an equal or
better objective function value than that of its target vector xi, it
replaces the target vector in the next generation. Otherwise,
target retains its place in the population for at least one more
generation. By comparing each trial vector with the target
vector from which it inherits parameters, DE more tightly
integrates recombination and selection. All solutions in the
population have the same chance of being selected as parents.
xGþ1i ¼ uGþ1
i if f ðuGþ1i Þ � f ðxG
i ÞxG
i otherwise
�(4.4)
By using this selection procedure, all individuals of the next
generation are as good as, or better than the individuals of the
current population.
4.5. Stopping criteria
The stopping criteria depends on the type of problem. The
iterative procedure can be terminated when any of the following
criteria is met, (i) an acceptable solution has been reached, (ii) a
state with no further improvement in solution is reached, (iii)
control variables has converged to a steady value or (iv) a
predefined maximum number of iterations have been com-
pleted.
5. Constraint handling
The most common approach in the EA to handle constraints
is to use penalties. The basic approach is to define the fitness
values of an individual by extending the domain of the objective
function.
5.1. Penalty function based on penalty coefficients
In this method of constraint handling, in minimization
problems, the fitness function FðxÞ is defined as the sum of the
objective function f ðxÞ and a penalty term which depends on
the constraint violation hhðxÞi.
FðxÞ ¼ f ðxÞ þXn
j¼1
R jhh jðxÞi2 (5.1)
where hi gives the absolute value of the operand if the operand
is negative and returns a zero if the operand is positive. The
parameter R j is the penalty coefficient of the jth inequality
constraint and it is user defined parameter. For reactive power
dispatch optimization problem, equality constraints given by
(3.2) and (3.3) are met by the load flow solution, while (3.9)–
(3.11) are enforced during the population coding and (3.8) is
considered outside the optimization loop. Hence effectively, the
inequality constraints to be handled here are (3.4)–(3.7). In
penalty function method, this is incorporated by modifying the
objective function as given below.
F ¼ f þ R1ðPg;slack � Plimg;slackÞ
2 þX
i2NPQ
R2ðVi � V limi Þ
2
þX
i2NG
R3ðQgi � Qlimgi Þ
2 þXi2Nl
R4ðjSlj � Smaxl Þ2 (5.2)
where, R1, R2, R3, and R4 are penalty coefficients associated
with real power generation at slack bus, voltage magnitude,
reactive power generation, and apparent line flow limit viola-
tions respectively. Plimg;slack, V lim
i , and Qlimgi can be expressed in
general form as
xlimi ¼
xmaxi if xi > xmax
i
xmini if xi < xmin
i
xi otherwise
8<: (5.3)
M. Varadarajan, K.S. Swarup / Applied Soft Computing 8 (2008) 1549–1561 1553
Since the order of magnitude violation is different for different
constraints, it is difficult to find a unique value for R1, R2, R3,
and R4. These can be fixed only by trial and error method and
problem dependent.
5.2. Effect of penalty coefficients
Theoretically, the penalty should be kept just above the limit
below which infeasible solutions are optimal. In most problems
the exact location of the boundary between the feasible and
infeasible regions is unknown and hence minimum penalty rule
is not easy to implement in practice. To highlight the influence
of the choice of penalty coefficients on the objective function
value, simulation studies were carried out on various test
systems using DE algorithm with different values of penalty
coefficients as explained above. The results obtained for IEEE
14-bus system [8] are listed in Table 1. The table shows the
results for three different cases with same initial population. It
can be observed that the PLOSS value is greatly dependent on the
choice of penalty coefficients.
5.3. Penalty function based on feasibility
In this scheme, employed in this paper, the composite fitness
function for any x is given as follows
FðxÞ ¼ f ðxÞ if x is feasible
f max þ CVðxÞ otherwise
�(5.4)
Here, f max is the objective function value of the worst feasible
solution in the population. In situation where none of the
solutions in a population are feasible, f max is defined. Hence,
such situations are handled by artificially inserting the base case
solution into the population. CV(x) is the overall constraint
violation of solution x. It is calculated as follows.
CVðxÞ ¼ max ð0;Pg;slack � Pmaxg;slack;P
ming;slack � Pg;slackÞ
þXNPQ
i¼1
max ð0;Vi � Vmaxi ;Vmin
i � ViÞ
þXNG
i¼1
max ð0;Qgi � Qmaxgi ;Qmin
gi � QgiÞ
þXNl
l¼1
max ðjSlj � Smaxl Þ
(5.5)
All feasible solutions have zero constraint violation and all
infeasible solutions are evaluated according to their constraint
Table 1
Effect of penalty coefficients on PLOSS
Coefficients Case 1 Case 2 Case 3
R1 500 100 200
R2 1000 2000 3000
R3 100 200 300
R4 100 100 100
PLOSS (MW) 13.42 13.29 13.31
violations alone. Hence, both the objective function value and
constraint violation are not combined in any solution in the
population. Thus there is no need to have any penalty coeffi-
cient R for this approach. The advantages of this scheme as
compared to the usual penalty parameter based scheme are (i)
The tedious process of choosing a suitable penalty coefficient R
can be avoided, the inappropriate choice of which will affect the
final solution and (ii) there is no need to evaluate the objective
function value for individuals with constraint violation, which
reduces the computation time. The following criteria are
enforced while selecting the individuals for the next generation.
(1) A
ny feasible solution is preferred to any infeasible solution.(2) A
mong two feasible solutions, the one having betterobjective function value is preferred.
(3) A
mong two infeasible solutions, the one having smallerconstraint violation is preferred.
6. Differential evolution approach to optimal reactive
power dispatch
The control variables selected for reactive power dispatch
problem are: the generator voltages, tap ratio of tap changing
transformers and output of shunts. Among the control variables,
the generator voltages are continuous, whereas the transformer
tap ratios and the outputs of shunts are discrete. But tap ratio of
transformers and output of shunts depend upon the tap position
and the number of shunts switched. Hence, the generator
voltage Vg, tap position (integer), and the number of shunts to
be switched (integer) are selected as control variables for
optimization problem.
6.1. Treatment of control variables
In its basic form, DE algorithm can handle only continuous
variables. However, reactive power source installations and tap
position of tap changing transformers are discrete variables in
the reactive power dispatch problem. In this paper, DE has been
extended to handle mixed integer variables, by the proper
treatment of control variables as explained below. For integer
variables the value is rounded off to the nearest integer value of
the variable.
xi ¼xi for continuous variables
b xi c for integer variables
�(6.1)
The b x c function gives the nearest integer less than or equal to
x. A typical individual xi can be represented as
xi ¼ ½V1g ; . . . ;VNPV
g ; n1t ; . . . ; nNT
t ; n1Qsh; . . . ; nNsh
Qsh� (6.2)
where nt is the number of tap positions in a tap changing
transformer and nQshis the number of shunt reactive power
devices available at a particular bus.
Initial generator terminal voltages, which are continuous
variables, are generated randomly between upper and lower
limits of the voltage specification values. The value is then
modified in the search procedure, within the specified limits.
Table 2
Description of test systems
IEEE 14 IEEE 30 IEEE 118
No. of buses NB 14 30 118
No. of generators NG 5 6 54
M. Varadarajan, K.S. Swarup / Applied Soft Computing 8 (2008) 1549–15611554
Transformer tap positions, which are integer, are initially
generated randomly between the minimum and maximum tap
positions. The value is then modified in the search procedure
among existing set of integer tap positions. Based on the tap
position, the corresponding tap ratio is calculated as follows.
tk ¼ tmink þ nk�Dtk (6.3)
where, nk is the number of tap positions and Dtk is the step size.
Using this tap ratio, the corresponding admittance of the
transformer is determined for the load flow calculation. Reactive
power compensation devices at a bus, which are again integer in
nature, are initialized randomly from the integer set generated
between 0 to the number of existing equipment at the bus. This
value is also modified in the search procedure, always limiting it
to be within the integer set of existing shunt devices.
6.2. Termination criteria
The iterative procedure can be terminated when any of the
following criteria is met, i.e., an acceptable solution has been
reached, a state with no further improvement in solution is
reached (stall generation), control variables has converged to a
steady value or a predefined number of iterations have been
completed. In most of the cases, it is not easy to test whether the
obtained solution is the most acceptable one. Also, the lack of
further improvement in the solution or convergence of control
variables need not necessarily translate to achievement of the
global solution. A commonly used approach is to run the
iterations to a fixed maximum number of generations which is
dependent on the problem under consideration. Usually the
number of maximum generations is fixed by a trial and error
process. In this work, combination of maximum number of
generations and stall generation limit is used as the termination
criteria.
6.3. Algorithm
DE is employed to find the best control variable setting
starting from a randomly generated initial population. At the end
of each generation, the best individuals, based on the fitness
value, are stored. The VCPI at the bus k obtained from Eq. (A.1),
will vary from zero to one, with zero indicating a voltage stable
condition and one indicating a voltage collapse [16]. The VCPI
value for the best individual is compared with the threshold value
and if the value is less than the threshold value, it indicates a
voltage secure condition. The threshold value is fixed by
conducting off-line study on the system for different operating
conditions. Evaluation of the voltage security, independent of the
OPF algorithm simplifies the optimization procedure. The details
of the proposed algorithm is as follows.
No. of transformers NT 3 4 9No. of shunts Nsh 2 2 12
No. of branches Nl 20 41 186
Step 1. GNo. of equality constraints 28 60 236
enerate an initial population randomly within the
control variable bounds.
No. of inequality constraints 65 125 566 Step 2. F No. of control variables 10 12 75No. of discrete variables 5 6 21
or each individual in the population, run power flow
algorithm such as Newton Raphson method, to find
the operating points.
Step 3. E
valuate the fitness of the individuals according toEq. (5.4).
Step 4. P
erform mutation and crossover operation asdescribed in Sections 4.2 and 4.3.
Step 5. S
elect the individuals for the next generation as givenin Section 4.4.
Step 6. S
tore the best individual of the current generation.Step 7. R
epeat Steps 2–5 till the termination criteria is met.Step 8. S
elect the control variable setting corresponding to theoverall best individual.
Step 9. D
etermine VCPI at each bus for the selected controlvariable setting and check whether it is less than
threshold value.
Step 10. I
f the solution is acceptable, output the best individualand its objective value. Otherwise, take the settings
corresponding to the next best individual and repeat
the Step 8.
7. Simulation results
The proposed DE approach for optimal reactive power
dispatch algorithm is tested on standard IEEE 14-bus [8], IEEE
30 [17], and IEEE 118 [18] bus test systems. Table 2 gives the
details of the test systems. A comparative study with PSO,
employing a constriction coefficient [19–22], was done to
verify the performance of the proposed algorithm. The DE and
PSO algorithm was implemented using MATLAB15.3 running
on Pentium IV PC. DE and PSO parameters used for the
simulation are summarized in Table 3. Number of individuals in
a population for each test system is decided by experimentation.
To validate and compare the results obtained by DE
algorithm, the dispatch problem is also solved by SQP
technique, using Matlab Optimization Toolbox [23], assuming
all the variables to be of continuous. The results of DE and PSO,
which follow, are the best solutions over 30 independent trails.
7.1. IEEE 14-bus system
The modified IEEE 14-bus system data and initial operating
conditions of the system is given in [8]. For IEEE 14-bus system
shown in Fig. 1, there are 14 buses, out of which 5 are generator
buses. Bus 1 is the slack bus, 2, 3, 6 and 8 are taken as PV
generator buses and the rest are PQ load buses. The network has
20 branches, 17 of which are transmission lines and 3 are tap
Table 3
Simulation parameters
DE PSO
Population size 30 Population size 30
Max. no. of generations 200 Max. no. of generations 200
Step size (F) 0.6 C1, C2 2.05
Crossover rate (CR) 0.8 vmin 0.4
– – vmax 0.9
– – x 0.7298
Fig. 2. Performance characteristics of IEEE 14-bus system.
M. Varadarajan, K.S. Swarup / Applied Soft Computing 8 (2008) 1549–1561 1555
changing transformers. It is assumed that capacitor compensa-
tion is available at buses 9 and 14. Totally, there are nine control
variables which consists of four PV generator voltages, three
tap changing transformers with 20 discrete steps of 0.01 p.u.
each and two shunt compensation capacitor banks with three
discrete steps of 0.06 p.u. each.
Fig. 2 gives the performance of the optimization technique in
terms of PLOSS with DE and PSO for the best run out of 30 trials.
It can be observed that PLOSS reduces over the evolutions and
converge to a minimum value. From the base case value of
13.49 MW, the PLOSS was reduced to 13.239 MW with DE.
The iterative procedure is terminated when there is no change in
the result for 40 consecutive generations or when 200
generations are reached, whichever occurs first. To understand
and study how evolution is going on, information about average
value, standard deviation and variance at each generation were
observed. The Fig. 3 shows this information for the proposed
algorithm.
To verify the performance of DE, the results are compared
with PSO and a conventional optimization technique – SQP.
Table 4 shows the minimum value of PLOSS in MW obtained
by different methods. Between DE and PSO approaches, DE
performance is better as it obtained the optimum solution
with less number of generations and function evaluations.
Since SQP method assumes all variables as continuous, after
optimization procedure, load flow program is used to find the
actual PLOSS, with the discrete variables are adjusted to the
nearest possible value.
Fig. 1. Network diagram of IEEE 14-bus system.
A good optimization results in convergence of all control
variables to a steady value. Fig. 4 shows the variation of the
continuous control variable, Vg, with respect to the number of
generations. All generator voltages settle to a steady value by
30 generations. Fig. 5 shows the variation of the discrete control
variables – tap position and capacitor bank switching. It can be
observed that all discrete control variables also converge well
before 30 generations.
Fig. 6 shows the effect of optimum control variable setting
on static voltage security in terms of VCPI. At all buses, the
VCPI value is less than the threshold value for this system,
which is 0.2065 as obtained from off-line studies. Table 5 gives
the details of the control variables and PLOSS obtained with
different optimization techniques.
In order to verify the robustness of the proposed
methodology simulation is carried out for 30 independent
runs with different initial population. For each run, the final
solution and cpu time were observed. The important statistical
details are listed in Table 6. It can be seen that DE algorithm
Fig. 3. Statistics of DE in each generation for IEEE 14-bus system.
Table 4
PLOSS before and after optimization for IEEE 14-bus system
Compared item Base case DE PSO SQP
PLOSS (MW) 13.49 13.239 13.250 13.246
No. of iterations – 63 80 9
No. of function evaluations – 1890 2400 316
Fig. 6. VCPI for IEEE 14-bus system.
M. Varadarajan, K.S. Swarup / Applied Soft Computing 8 (2008) 1549–15611556
more robust than PSO and faster. To ensure a near optimum
solution for any random trial, the standard deviation for
multiple runs should be very low, which is satisfied better by
DE, when compared to PSO.
7.2. IEEE 30-bus system
In this section, results obtained for IEEE 30-bus system is
presented. The system data giving branch parameters and loads
are available in [17]. The network consists of 41 branches, six
generator buses and 24 load buses. Four branches 6–9, 6–10, 4–
12 and 27–28 have tap changing transformers with 20 discrete
Fig. 4. Convergence of control variable Vg for IEEE 14-bus system.
Fig. 5. Convergence of discrete control variables for IEEE 14-bus system.
steps of 0.01 p.u. each. The buses with possible reactive power
source installations are 10 and 24. The available reactive
powers of capacitor banks are within the interval 0 to 30 MVAr
in discrete steps of 1 MVAr. All bus voltages are required to be
maintained within the range of 0.95–1.1 p.u. Voltages of PQ
buses 26 (V26 ¼ 0:932 p.u.), 29 (V29 ¼0.940 p.u.) and 30
(V30 ¼ 0:928 p.u.) violates the lower limit in the base case.
Fig. 7 shows the convergence characteristics for the best
solution. It can be seen that PLOSS is reduced to a minimum
value of 5.011 MW from the base case loss of 5.66 MW. Fig. 8
shows the information about the average value, standard
deviation and variance of the population at each generation.
The PLOSS values before and after optimization obtained with
various methods are given in Table 7.
As in the case of IEEE 14-bus system, all control variables
converge to a steady value. Fig. 9 shows the variation of the
control variable, Vg, with respect to the number of generations.
All generator voltages settle to a steady value by 40
generations. Fig. 10 shows the variation of the discrete control
variables – tap position and capacitor bank switching. It can be
observed that all discrete control variables also converges to
steady value. Table 8 shows the control variable setting and
PLOSS obtained by different methods.
Table 5
Values of control variables (p.u.) and PLOSS before and after optimization for
IEEE 14-bus system
Variable Base case DE PSO SQP PSO [8]
Vg2 1.0450 1.0449 1.0443 1.0442 1.0463
Vg3 1.0100 1.0416 1.0138 1.0124 1.0165
Vg6 1.0700 1.1000 1.1000 1.1000 1.1000
Vg8 1.0900 1.1000 1.0882 1.1000 1.1000
T4�7 0.9467 1.0600 1.0700 1.0586 0.9400
T4�9 0.9524 1.0400 1.0400 1.0634 0.9300
T5�6 0.9091 1.1000 1.0000 1.0781 0.9700
QC9 0.1800 0.1800 0.1800 0.1751 0.1800
QC14 0.1800 0.0600 0.0600 0.0632 0.0600
PLOSS (MW) 13.49 13.239 13.250 13.246 13.32
Table 6
Statistical details for IEEE 14- bus system
Compared item DE PSO
PLOSS– best (MW) 13.239 13.250
PLOSS– worst (MW) 13.275 13.402
PLOSS – average (MW) 13.250 13.352
Standard deviation 0.0161 0.0640
Average no. of iterations 62 74
Average CPU time (s) 8.172 9.283
Table 7
PLOSS before and after optimization for IEEE 30-bus system
Compared item Base case DE PSO SQP
PLOSS (MW) 5.66 5.011 5.116 5.043
No. of iterations – 66 70 36
No. of function Evaluations – 1980 2100 2465
M. Varadarajan, K.S. Swarup / Applied Soft Computing 8 (2008) 1549–1561 1557
The bus voltage profile before and after optimization is
shown in Fig. 11. Voltages at all buses, including the buses 26,
29 and 30, are now within the required range of 0.95–1.1 p.u.
Fig. 12 shows the effect of optimum control variable setting on
static voltage security in terms of VCPI. At all buses, the VCPI
value is less than the threshold value for this system. which is
0.1121 as obtained from off-line studies.
Statistical details for IEEE 30-bus system for 30 indepen-
dent runs are shown in Table 9. It can be observed that
Fig. 7. Performance characteristics for IEEE 30-bus system.
Fig. 8. Statistics of DE in each generation for IEEE 30-bus system.
performance and robustness of DE algorithm are better than
PSO. DE obtains the optimum value in less number of function
evaluations. The standard deviation for 30 runs is very low in
the case of DE when compared to PSO, which ensures a near
optimum solution for any random trial.
7.3. IEEE 118 bus system
In this section performance of DE based optimal reactive
power dispatch was evaluated on IEEE 118 bus system with
simulation parameters given in Table 3 and the network data
Fig. 10. Convergence of discrete control variables for IEEE 30-bus system.
Fig. 9. Convergence of control variable Vg for IEEE 30-bus system.
Table 8
Values of control variable (p.u.) and PLOSS before and after optimization for IEEE 30-bus system
Variable Base case DE PSO SQP PSO [24] IPM [24]
Vg1 1.0500 1.0500 1.0500 1.0500 1.0178 1.1000
Vg2 1.0220 1.0446 0.9679 1.0467 1.0246 1.0541
Vg5 1.0000 1.0247 1.0262 1.0386 1.0247 1.1000
Vg8 1.0000 1.0265 1.0267 1.0293 1.0142 1.0335
Vg11 1.0000 1.1000 1.1000 1.0837 1.0172 1.1000
Vg13 1.0000 1.1000 1.1000 1.1000 0.9961 1.0149
T6�9 1.0000 1.0000 0.9700 1.0222 1.0969 0.9933
T6�10 1.0000 1.1000 1.1000 1.0453 0.9251 1.0593
T4�12 1.0000 1.0800 1.0600 1.0686 1.0005 1.0088
T27�28 1.0000 0.9200 0.9200 1.0819 1.0071 0.9971
QC10 0.1000 0.2600 0.3000 0.2974 0.1537 0.1525
QC24 0.1000 0.1000 0.1000 0.0999 0.0622 0.0893
PLOSS (MW) 5.66 5.011 5.116 5.043 5.092 5.101
M. Varadarajan, K.S. Swarup / Applied Soft Computing 8 (2008) 1549–15611558
given in [18]. The network consists of 186 branches, 54 generator
buses and 64 load buses. Nine branches have tap changing
transformers with 20 discrete steps of 0.01 p.u. each. There are
12 reactive power source installations. The available reactive
Fig. 12. VCPI at each bus for IEEE 30-bus system.
Fig. 11. Voltage profile for IEEE 30-bus system before and after optimization.
powers of capacitor banks are within the interval (0–30) MVAr in
discrete steps of 1 MVAr. All bus voltages are required to be
maintained within the range of 0.95–1.1 p.u. Voltages of PQ
buses 53 and 118 violates the lower limit in the base case.
Fig. 13 shows the real power loss variation against
generations. It can be seen that PLOSS is reduced to a minimum
value of 128.318 MW from a base case loss of 132.45 MW.
Fig. 14 shows evolution of the algorithm in terms of average
value, standard deviation and variance of population at each
generation. It is also found that all control variables converge to
a steady value by the time termination criteria is satisfied. Table
10 lists the minimum PLOSS obtained by using different
methods namely DE, PSO [24] and IPM [24].
Statistical details for IEEE 118 bus system is as follows:
best, worst, and average PLOSS obtained for 30 simulations are
128.318, 129.579, and 129.0817 MW, respectively. The
standard deviation of PLOSS is 0.345 MW. Average cpu time
taken is 42.1556 s with an average of 193 iterations.
7.4. Effect of initial population and population size
To study the effect of initial population on the performance
of the algorithm, simulation is carried out on test systems with
Fig. 13. Performance characteristics of IEEE 118-bus system using DE.
Table 9
Statistical details for IEEE 30-bus system
Compared item DE PSO
PLOSS—best (MW) 5.011 5.116
PLOSS—worst (MW) 5.022 5.218
PLOSS—average (MW) 5.013 5.1254
Standard deviation 0.0026 0.0291
Average no. of iterations 66 69
Average CPU time (s) 13.647 16.420
Fig. 14. Statistics of DE at each generation for IEEE 118-bus system.
Fig. 15. Effect of initial population on PLOSS for IEEE 30-bus system.
Fig. 16. Effect of population size on PLOSS for IEEE 30-bus system.
Table 10
PLOSS before and after optimization for IEEE 118-bus system
Base case DE PSO [24] IPM [24]
PLOSS (MW) 132.45 128.318 131.908 132.110
M. Varadarajan, K.S. Swarup / Applied Soft Computing 8 (2008) 1549–1561 1559
a constant population size. Fig. 15 shows the effect of initial
population on the final solution for four trials in the case of
IEEE 30-bus system. It can be observed that even though the
convergence is different for different initial population, the
algorithm converges to the optimum solution. It is found that
the algorithm is insensitive to the initial population i.e. starting
points for the optimization process.
It is also essential to study the effect of population size on the
optimization procedure. Even though a number of heuristic
relations are available for finding the population size, there is no
hard and fast rule which can be universally adopted. In most
cases, the population size is fixed by trial and error method.
Fig. 16 shows the effect of population size on the objective
value, with the same initial population for IEEE 30-bus system.
As the population size increases a faster convergence to the
optimum solution can be obtained at a cost of increased
computation time. To achieve a compromise between
convergence to the optimal solution and reduced computation
time, a population size of 30 was used for all the test cases.
8. Conclusion
A differential evolution algorithm based OPF for reactive
power dispatch and voltage control in power system planning
and operation studies is proposed. The problem is formulated
as a mixed integer nonlinear optimization problem. Compared
to PSO, DE has fewer control parameters (population size, step
size and crossover rate). Further, the penalty parameterless
technique of handling inequality constraints, effectively
eliminates the trial and error method of assigning penalty
coefficients and also makes the process system independent.
The proposed DE approach has been evaluated on IEEE 14,
IEEE 30, and IEEE 118-bus systems and the results were
compared with that obtained using PSO and SQP. DE was
found to be more robust as it gave minimum standard deviation
among the solutions obtained from multiple random trials.
In each case, the security of the system was considered,
while optimizing the control variables for real power loss
minimization.
M. Varadarajan, K.S. Swarup / Applied Soft Computing 8 (2008) 1549–15611560
Appendix A. Voltage collapse proximity indicator
Using the voltage magnitude and voltage angle information,
voltage collapse proximity indicator at each bus [16] is
calculated as follows.
VCPIk ¼����1�
PNm¼1; 6¼ kV̄
0m
V̄k
���� (A.1)
and V̄0m is given by
V̄0m ¼
YkmPNj¼1; 6¼ kYk j
V̄m (A.2)
where, V̄k voltage phasor at bus k, V̄m voltage phasor at bus m,
Ykm admittance between buses k and m, N number of buses.
The VCPI at the bus k obtained from Eq. (A.1), will vary
from 0 to 1, with zero indicating a voltage stable condition and
one indicating a voltage collapse.
Appendix B. Particle swarm optimization
PSO was developed through simulation of simplified social
methods and is basically simulation of the social behavior of a
flock of birds in two-dimension space. Let x and v represent a
particle position and its corresponding velocity in a search
space. The best previous position of a particle is recorded and
represented as pBest. The index of the best particle among all
the particles in the group is represented as gBest [19].
Constriction function is used to ensure the convergence of PSO
[20]. The modified velocity of each particle can be calculated
using the current velocity and the distance from pBest and
gBest as
vkþ1i ¼ xðviv
ki þ C1 � randðÞð pBest � xk
i Þ þ C2 � randðÞ
� ðgBest � xki ÞÞ
(B.1)
From the above equation, a certain velocity that gradually gets
close to pBest and gBest can be calculated. The particle velocity
is limited by some maximum value vmax . This parameter
determines the fitness with which regions are to be searched
between the present position and the target position and
enhances the local exploration of the problem space. The
current position can be modified by the following equation.
xkþ1i ¼ xk
i þ vkþ1i (B.2)
where vki is the current velocity of particle i at iteration k, vkþ1
i is
the modified velocity of particle i, rand is the uniformly
distributed random number between 0 and 1, xki is the current
position of particle i at iteration k, xkþ1i is the modified position
of particle i, vi is the inertia weight factor of particle i, x is the
constriction factor, and C1;C2 are acceleration constant.
The constriction factor x is a function of C1 and C2 as given
below
x ¼ 2
j2� C �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC2 � 4Cp
j(B.3)
where C ¼ C1 þ C2 and C> 4. Usually C1 and C2 are both set
to be 2.05 and x set to be 0.729.
Suitable selection of inertia weight v provides a balance
between global and local explorations. In general, the inertia
weight v is usually be set as decreasing linearly from vmax to
vmin , according to the following equation
v ¼ vmax �vmax � vmin
itermax
� iter (B.4)
where, itermax is the maximum number of generations and iter
is the current generation. Empirical studies have shown that
PSO performs well when vmax ¼ 0:9 and vmin ¼ 0:4 [25,21].
A general strategy for setting v, C1, and C2, to guarantee the
convergence of the particles is given in [22].
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