Sharif University of Technology
Scientia Iranica
Transactions D: Computer Science & Engineering and Electrical Engineering www.sciencedirect.com
Optimal reactive power flow using multi-objective
mathematical programming
A. Lashkar Ara
a,∗,
A. Kazemi
b,
S. Gahramani
a,
M. Behshad
aaDepartment of Electrical Engineering, Islamic Azad University, Dezful Branch, Dezful, P.O. Box: 313, Iran
bDepartment of Electrical Engineering, Iran University of Science and Technology, Narmak, Tehran, P.O. Box 1684613114, Iran
Received 6 February 2012; revised 3 June 2012; accepted 4 July 2012
KEYWORDS
Optimal reactive power flow (ORPF);
Multi-objective optimization; MINLP;
OA/ER/AP algorithm.
Abstract This paper presents a multi-objective optimization methodology to solve the Optimal Reactive Power Flow (ORPF) problem. The ε-constraint approach is implemented for the Multi-objective Mathematical Programming (MMP) formulation. The solution procedure uses Mixed Integer Non-Linear Programming (MINLP) model due to discrete variables, such as the tap settings of transformers and the reactive power output of capacitor banks. The optimum tap settings of transformers are directly determined in terms of the admittance matrix of the network since the admittance matrix is constructed in the optimization framework as additional equality constraints. The optimization problem is modeled in General Algebraic Modeling System (GAMS) software and solved using DICOPT solver. Simulation results are implemented on the IEEE 14-, 30-, and 118-bus test systems to simultaneously optimize the total fuel cost, power losses and the system loadability as objective functions. The simulation results show that the proposed algorithm is suitable and effective for the reactive power planning.
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1. Introduction
The multi-objective Optimal Reactive Power Flow (ORPF) is an optimization problem to help the operator to ensure the reactive power planning technically and economically as a subproblem of the Optimal Power Flow (OPF). The objective of the ORPF problem is to allocate reactive power compensation devices and optimize the objective functions while satisfying physical and technical constraints on the power system. The conventional reactive power compensation devices are tap changing transformers and shunt capacitors/reactors. The ORPF problem can be formulated as a Mixed Integer Non-Linear Programming (MINLP) model since tap ratios of transformers and outputs of shunt capacitors/reactors are inherently discrete [1].
∗Corresponding author. Tel.: +98 641 6260051; fax: +98 641 6260890.
E-mail addresses:[email protected](A. Lashkar Ara),[email protected]
(A. Kazemi).
Most of the previous researches on the OPF and ORPF have been focused on intelligent methods because they can re-solve the defect of conventional optimization algorithms in global searching and handling discrete variables and the in-feasibility problem [2]. For decades, conventional optimization techniques, such as Linear Programming (LP) [3], Non-Linear Programming (NLP) [4], Quadratic Programming (QP) [5], Se-quential Quadratic Programming (SQP) [1], Newton method [6] and Interior Point Methods (IPMs) [7,8], have been developed for solving ORPF problem.
In recent years, global optimization techniques, such as Ge-netic Algorithms (GA) [9], Evolutionary Programming (EP) [10], hybrid EP [11], fuzzy [12], Evolutionary Strategies (ES) [13], Par-ticle Swarm Optimization (PSO) [14] and Differential Evolution (DE) [2], have been suggested, which have greater or less suc-cess in solving different non-linear single-objective optimiza-tion problems. In [15] the Self-Adaptive Real Coded Genetic Algorithm (SARGA) has been applied as one of the techniques to solve Optimal Reactive Power Dispatch (ORPD) problem. The ORPF problem has been investigated using Hybrid GA and IPM (HGI) [16,17], EP [18], Reactive Tabu Search (RTS) [19,20] and PSO [21,22].
Recently, several multi-objective optimization methods as listed below have been investigated for solving the ORPF problem. In [23], multi-objective ORPF has been applied to
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doi:10.1016/j.scient.2012.07.010
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minimize active power losses and payment for the provision of the reactive power service in the framework of the UK daily balancing market. OPF problem using Biogeography-Based Optimization (BBO) has been suggested to minimize three different objectives, i.e. the fuel cost, voltage profile and voltage stability improvement [24]. The multi-objective optimization has been investigated using PSO [25,26], Adaptive Immune Algorithm (AIA) [27] and GA [28–30] to solve ORPF problem.
In previous research works reported in the literature, the ORPF problem has been formulated as a MINLP with discrete and continuous variables; however, they have some serious problems. In some papers, the problem has been solved in two steps. In the first step, the continuous variables have been de-termined by the OPF algorithm, and in the second step, it has been tried to handle the discrete variables in the ORPF prob-lem. Therefore, the results cannot converge to the optimal batch solution with such treatment and the authors do not have the ORPF program. In some other papers, the effect of tap chang-ing transformers has been considered only on the objective functions while it has been influenced in the active and reac-tive power balance equations. Consequently, in this paper, it is implemented in the admittance matrix of the system. The Multi-objective Mathematical Programming (MMP) methods, such as generation methods, despite having significant advan-tages are less popular due to their computational complexity and the lack of widely available software [31]. Considering our present knowledge, no research work in this area has consid-ered the MMP methods for solving ORPF problem. The main contribution of this paper is to propose the
ε
-constraint method for solving the multi-objective ORPF problem based on MINLP while the discrete and continuous variables are determined simultaneously.For this application, General Algebraic Modeling System (GAMS) is used to solve the optimal model, and Matlab is used to feed parameters to the GAMS routine. The MINLP optimization problem is modeled in GAMS software and solved by using DICOPT solver [32]. In order to select the ‘‘best’’ compromised solution among the Pareto optimal solutions of multi-objective optimization problem, a fuzzy decision-making tool is adopted. In addition, the admittance matrix is formed as additional equality constraints to involve the tap ratios of transformers in the optimization problem. The total fuel cost, the active power losses and the system loadability are simultaneously optimized as objective functions in the power system while satisfying several constraints. The proposed algorithm is implemented on the IEEE 14-, 30-, and 118-bus test systems. The single- and multi-objective optimization results show that the proposed algorithm is outperformed by the other methods.
The paper is organized as follows. The ORPF problem formulation is developed in Section 2. Section 3 describes the MINLP formulation and its solution methods briefly. The multi-objective ORPF solution procedure is explained in Section 4. Section5 contains simulation results followed by conclusions.
2. Problem formulation
The multi-objective ORPF problem seeks to optimize one or more objective functions to improve the steady state performance of the power system while satisfying several equality and inequality constraints. The objective functions, constraints, and the multi-objective optimization framework of the problem are explained in the following subsections.
2.1. The objective functions
In this paper, three objective functions are considered which are the total fuel cost, the active power losses and the system loadability. These objective functions are formulated as follows:
(1) The total fuel cost:The first objective function is to minimize the total fuel cost that can be expressed as:
F1
=
NG
i=1
ai
+
biPGi+
ciPGi2(
$/
h),
(1)wherePGiis the active power output ofith generator;NGis
the total number of generators;ai,bi, andciare the fuel cost
coefficients ofith generator [33].
(2) The real power losses:The second objective function is to minimize the real power losses in transmission lines that can be defined as:
F2
=
Nl
k=1
gk(Vi2
+
Vj2−
2ViVjcosθij),
(2)wheregkis the conductance of branchkbetween busesiand
j;Nlis the number of branches;Viis the voltage magnitude
at busi;
θij
is the voltage angle difference between busesi andj[34].(3) The system loadability: The third objective function is to maximize the system loadability that can be described as [35]:
F3
=
ρ(
x,
u),
(3)and
ρ
can be obtained by assuming constant power factor at each load in the both real and reactive power balance equations as follows:PG
−
ρ
PD=
fp(x,
u),
(4)QG
−
ρ
QD=
fq(x,
y),
(5)where PG and QG are the vectors of generators real and
reactive power, respectively; PD and QD are the vectors
of loads real and reactive power, respectively; fp and fq
are the vectors of real and reactive power flow equations, respectively.
2.2. Constraints
The multi-objective ORPF problem has two sets of con-straints including equality and inequality concon-straints. These constraints can be described in the following compact form:
g
(
x,
u)
=
0,
(6)h
(
x,
u)
≤
0,
(7)where u is a set of control variables which includes active power and voltage magnitude of generator buses, voltage angle and magnitude of the swing bus, tap of transformers and the reactive power sources;xis a set of dependent variables which includes active and reactive power of the swing bus, voltage angle and reactive power of generator buses, voltage angle and magnitude of load buses, and the admittance matrix of the power system;g
(
x,
u)
is the set of equality constraints which are usually the power flow constraints for a specified operating condition; h(
x,
u)
is the set of inequality constraints which represents the limits on the control variables and the operating limits of the power system [36]. The equality and inequality constraints are explained in the following subsections.2.2.1. Equality constraints
Power flow equations with corresponding to both real and reactive power balance equations must be satisfied as defined below:
PGi
−
PDi=
Pi(x,
u),
(8)QGi
−
QDi=
Qi(x,
u),
(9)wherePGiandQGiare the generator real and reactive power at
busi, respectively;PDiandQDiare the load real and reactive
power at busi, respectively;PiandQiare the real and reactive
power flow equations at bus i, respectively [33,36]. In this paper, another set of equality constraints is added which is related to formation of the admittance matrix. The real and imaginary parts of the off diagonal elements of admittance matrix in rowiand columnjare as follows:
Yreal,ij
=
Yreal,ji= −
rij/Tij(rij2+
x2ij),i
̸=
j,
i,
j=
1, . . . ,
N,
(10) Yimag,ij=
Yimag,ji=
xij/Tij(rij2+
x2ij),i
̸=
j,
i,
j=
1, . . . ,
N,
(11) and their diagonal elements are as follows:Yreal,ii
=
N
j=1 rij/Tij2(
rij2+
x2ij), i̸=
j,
if Tij̸=
1 thenTji=
1,
j=
1, . . . ,
N,
(12) Yimag,ii=
N
j=1−
xij/Tij2(
rij2+
x2ij)+
bc,ij/2,
i
̸=
j,
if Tij̸=
1 thenTji=
1,
j=
1, . . . ,
N,
(13)whereNis the number of buses;rijandxijare the resistor and
inductance of the branch between busesiandj, respectively;Tij
is the tap ratio of the transformer between busesiandj. If there is no transformer between busesiandj, thenTijis equal to 1.
2.2.2. Inequality constraints
The inequality constraints in the multi-objective ORPF problem include:
Generation constraints: Generator voltages, real power outputs, and reactive power outputs are restricted by their lower and upper limits as follows [33,36]:
VGimin
≤
VGi≤
VGimax,
i=
1, . . . ,
NG,
(14)PGimin
≤
PGi≤
PGimax,
i=
1, . . . ,
NG,
(15)QGimin
≤
QGi≤
QGimax,
i=
1, . . . ,
NG.
(16)Security constraints:These include the constraints of voltages at load buses, transmission lines loadings, transformer tap settings and the reactive power output of capacitor banks as follows:
VLimin
≤
VLi≤
VLimax,
i=
1, . . . ,
Nd,
(17)SLi
≤
SLimax,
i=
1, . . . ,
Nl,
(18)Timin
≤
Ti≤
Timax,
i=
1, . . . ,
NT,
(19)Qcimin
≤
Qci≤
Qcimax,
i=
1, . . . ,
Nsh,
(20)whereNdis the number of load buses;NT is the number of transformers;Nshis the number of switchable VAR sources [21].
2.3. Problem statement
In general, the multi-objective ORPF problem can be math-ematically formulated by aggregating the objective functions and constraints as follows:
minimizeF1 minimizeF2 maximizeF3 subject to:g
(
x,
u)
=
0,
h(
x,
u) <
0.
(21)The multi-objective optimization problem can be converted into a single-objective optimization and then solved by using the
ε
-constraint method in Section4.3. Mixed integer non-linear programming
Generally, the MINLP formulation can be expressed in the following form [37]: Minimize/MaximizeJ
(
x,
y)
Subject to:g(
x,
y)
=
0 h(
x,
y)
≤
0 xL≤
x≤
xU,
x∈
X⊆ ℜ
n, yL≤
y≤
yU,
y∈
Yinteger,
(22)wherexis a set of continuous variables which includes control and dependent variables;yis a set of discrete variables treated as integer variables which includes tap ratios of transformers and outputs of shunt capacitors/reactors;J
(
x,
y)
is the objective function which could be the total generation cost, active power losses and system loadability; g(
x,
y)
is the set of equality constraints which are active and reactive power balance equations and the admittance matrix equations;h(
x,
y)
is the set of inequality constraints which represents the operating limits of the power system. The integer variables y can be defined in terms of 0–1 variables (i.e., binary) denoted byzas follows:y
=
yL+
z1+
2z2+
4z3+ · · · +
2N−1zN, (23)whereNis the minimum number of 0–1 variables needed and given by: N
=
1+
INT
log(
yU−
yL)
log 2
.
(24)Here the INT function truncates its real argument to an integer value. Therefore,
y
∈
Y= {
0,
1}
q,
andynow is a vector ofq0–1 variables.
Several different methods have been developed to solve the MINLP problem in the literature. Some of the most important algorithms, in chronological order of development, are Generalized Benders Decomposition (GBD), Branch and Bound (BB), Outer Approximation (OA), Feasibility Approach (FA), Outer Approximation with Equality Relaxation (OA/ER), Outer Approximation with Equality Relaxation and Augmented Penalty (OA/ER/AP), Generalized Outer Approximation (GOA), and Generalized Cross Decomposition (GCD). In this paper, the OA/ER/AP algorithm is used to solve the ORPF problem. It has been implemented in the DICOPT solver by GAMS [38] and explained in detail in the following.
The OA/ER/AP algorithm, DICOPT, solves a series of NLP and Mixed Integer Linear Programming (MILP) subproblems
called primal and master subproblems, respectively. It starts by relaxing the integer variables as continuous variables and solving the relaxed NLP problem. If the yield solution is an in-teger solution, then the search terminates immediately. Oth-erwise, it continues with a sequence of primal subproblems by fixing the integer variables, and master subproblems gen-erated by augmented penalty function. The primal subproblem is solved to find the upper bound as well as the Lagrange multi-pliers for fixed integer variables that are identified by the mas-ter subproblem at each imas-teration. The masmas-ter subproblem also provides a lower bound on the objective function for the case of convex and minimization problems. As iterations proceed, the lower bound increases monotonically due to the accumula-tion of linear approximaaccumula-tions. The iteraaccumula-tions are continued until there is an increase in the optimal value of the feasible primal subproblems for consecutive iterations. In the case of non-convex problems, the valid bound can not be obtained by the master subproblem. Therefore, the termination criterion can be viewed only as a heuristic approach. We refer the readers to [38] for implementation and technical details about the DI-COPT solver.
4. Multi-objective mathematical programming
The MMP problems which simultaneously optimize two or more conflicting objectives subject to certain constraints are solved using prevalent techniques, such as the
ε
-constraint strategy. Existing methods generally cannot identify a single optimal solution that simultaneously optimizes all the objective functions. In contrast to the optimal solution, the decision maker is looking for the ‘‘most preferred’’ solution among Pareto optimal solutions. The Pareto optimal (or efficient, non-dominated, non-inferior) solution is a feasible solution that other feasible solutions cannot be improved in all the objective functions simultaneously. Theε
-constraint method to generate Pareto optimal solutions and the fuzzy decision-making tool to select the most preferred solution are described in the following subsections.4.1. The
ε
-constraint methodAlthough the other objective functions are used as con-straints, the
ε
-constraint method is applied for optimizing one objective function as follows [39]:minF1
(
x),
subject toF2
(
x)
≤
e2 F3(
x)
≤
e3· · ·
Fp(x)
≤
ep,(25) where pis the number of objective functions. For the range of every objective function calculated from the payoff table, at leastp
−
1 objective function are considered. The payoff table constructed by the calculation of the individual optima of the objective functions is applied in order to manage this method. The optimum value of theith objective function is shown by F∗i and the value of other objective functions is indicated by
F1i
, . . . ,
Fii−1,
Fii+1, . . . ,
Fpi; therefore, theith row of the payoff table is calculated. This way is carried out for all rows of the payoff table as follows:Φ
=
F1∗(
x∗1)
· · ·
Fi(
x∗1)
· · ·
Fp(x ∗ 1)
...
...
...
F1(
x∗i)
· · ·
F ∗ i(
x ∗ i)
· · ·
Fp(x ∗ i)
...
...
...
F1(
x∗p)· · ·
Fi(
x∗p)· · ·
Fp∗(
x ∗ p)
.
(26)The compromised values of thejth column of the payoff table are used in order to obtain the total (qj
+
1) grid pointsfor thejth objective function. The range of thejth objective function is obtained among its minimum and maximum values. In addition, the deviation of objective function toqjintervals
using (qj
−
1) intermediate equidistant grid points gives(
q2+
1
)
×
(
q3+
1)
× · · · ×
(
qp+
1)
which are total number ofoptimization subproblems. The values ofqi will be helpful in
controlling the density of the Pareto optimal set representation. Furthermore, the cost of higher computational time should be noted. In this paper, 4-interval is the best selection for the objective function. Also, a trade off between the computation time and the Pareto optimal set cannot be ignored [31]. Three objective functions F1, F2 and F3 described in Eqs. (1), (2), (3), respectively, are considered regarding the relation with the power system operation and its MMP problem. Therefore, the optimization subproblems are formulated in the following form: minF1
(
x)
subject toF2(
x)
≤
e2,i F3(
x)
≥
e3,j. (27) e2,i=
max(
F2)
−
max(
F2)
−
min(
F2)
q2
×
i,
i=
0,
1, . . . ,
q2 (28) e3,j=
min(
F3)
+
max(
F3)
−
min(
F3)
q3
×
j,
j=
0,
1, . . . ,
q3 (29)where max
(
·
)
and min(
·
)
represent maximum and minimum values of the individual objective function, respectively. MINLP is the common solution for the optimization subproblems applied by the mentioned constraints of the MMP problem to obtain Pareto optimal solutions.4.2. The fuzzy decision-making tool
The Pareto optimal solutions are obtained by solving the optimization subproblems. According to the best compromise solution among the Pareto optimal solutions, the optimal settings are chosen by the decision maker. For this purpose, the fuzzy decision maker calculates a linear membership function (
µi
) for minimized objective functions as follows:µ
n i i=1,2=
1 Fin≤
min(
Fi) max(
Fi)−
Fin max(
Fi)
−
min(
Fi) min(
Fi)≤
Fin≤
max(
Fi) 0 Fin≥
max(
Fi)
(30)and for maximized objective function it is expressed as:
µ
n i i=3=
0 Fin≤
min(
Fi) Fin−
min(
Fi)max
(
Fi)−
min(
Fi) min(
Fi)≤
Fn
i
≤
max(
Fi)1 Fin
≥
max(
Fi)(31)
whereFinand
µ
ni are the value of theith objective function in thenth Pareto optimal solution and its membership function, respectively. In terms of evaluating the optimal degree of the Pareto optimal solutions, the membership functions are used. The most preferred degree of the Pareto optimal solutions can be defined as follows:µopt
=
sup n∈M p
i=1wi
·
µ
n i M
n=1 p
i=1wi
·
µ
n i,
(32)where:
wi
≥
0,
p
i=1wi
=
1,
(33)here
wi
is the weight value of theith objective function andMis the number of Pareto optimal solutions. For a large value ofwi
it is possible to favorFi over other objective functions. Basedon the importance of the economical and technical aspects the weight values
wi
can be selected by the power system dispatcher. Therefore, as the best Pareto optimal solution, the optimal settings are obtained by the proposed algorithm based on the adopted weighting factors.5. Case studies
The effectiveness of the proposed algorithm is investigated on the IEEE 14-, 30-, and 118-bus test systems. The multi-objective ORPF problem is solved to simultaneously optimize the total fuel cost, active power losses and system loadability as objective functions, using MINLP as the solution procedure. The proposed algorithm is done for both single- and multi-objective optimization problems in the test systems. The various combinations of objective functions in multi-objective ORPF can be expressed by the following frames:
Case 1: minimizing the total fuel cost and active power losses;
Case 2: minimizing the total fuel cost and maximizing the system loadability;
Case 3: minimizing active power losses and maximizing the system loadability;
Case 4: minimizing the total fuel cost and active power losses and maximizing the system loadability.
The simulation studies were carried out in Matlab 7.6 and GAMS 23.0 softwares and executed on a 2.66-GHz Pentium-IV PC with 2 GB RAM. The MINLP optimization problem is modeled in GAMS software and solved using DICOPT solver. The DICOPT solver is based on the extensions of the outer approximation algorithm for the equality relaxation strategy. It iteratively invokes the MINOS5 and XA10.0 solvers for non-linear (NLP) and Mixed-Integer-Programming (MIP) solutions, respectively [32,40]. The NLP solution is obtained by MINOS5 by extremizing an augmented Lagrangian function [41] using the reduced gradient algorithm. XA10.0 uses the primal/dual simplex method to obtain the linear programming solution, combined with the ‘‘branch-and-bound’’ method to obtain the MIP solution.
In all cases, the reactive powers of capacitor banks are within the range of 0–30 MVAr and the lower and upper limits of the transformer tap settings and voltage magnitude are considered within the interval 0.9–1.1 p.u, respectively. The transformer tap settings and the reactive powers of capacitor banks are discrete variables with the change step of 0.01 p.u. In this paper, the control and dependent variables are simultaneously utilized in the optimization framework in contrast to the other methods which only use the control variables [1,14,15].
5.1. IEEE 14-bus test system
The IEEE 14-bus test system consists of five generator buses (the bus 1 is the slack bus and buses 2, 3, 6 and 8 are PV buses with continuous operating values), 9 load buses and 20 branches, in which 3 branches (4-7, 4-9 and 5-6) are tap changing transformers. In addition, buses 9 and 14
Table 1: Results of single-objective optimization in IEEE 14-bus test system.
Parameters F1 F2 F3 Vg2 1.051 1.056 1.046 Vg3 1.041 1.047 1.021 Vg6 1.06 1.06 1.06 Vg8 1.06 1.021 1.06 T4-7 1 1.0700 1.03 T4-9 0.95 0.90 0.90 T5-6 0.98 0.98 0.98 Qc9 0.30 0.3 0.28 Qc14 0.07 0.07 0.11
Total fuel cost ($/h) 17270.327 18170.323 30700.100
Ploss(MW) 1.5952 1.0252 6.5716 Loadability index 1 1 1.558 Qloss(MVAr) 10.9466 9.7365 7.3115 Execution time (s) 0.352812 0.355180 0.326678
Table 2: Results of multi-objective optimization in IEEE 14-bus test system.
Parameters Case 1 Case 2 Case 3 Case 4
(F1&F2) (F1&F3) (F2&F3) (F1&F2&F3)
Vg2 1.053 1.047 1.05 1.048 Vg3 1.043 1.026 1.029 1.032 Vg6 1.06 1.06 1.06 1.06 Vg8 1.022 1.06 1.045 1.036 T4-7 1.05 1.02 1.05 1.03 T4-9 0.90 0.90 0.90 0.90 T5-6 0.98 0.98 0.98 0.98 Qc9 0.30 0.24 0.3 0.3 Qc14 0.07 0.1 0.1 0.09
Total fuel cost ($/h) 17518.663 25271.910 27675.854 21932.977
Ploss(MW) 1.1929 4.6605 4.4841 3.2453 Loadability index 1 1.362 1.428 1.223 Qloss(MVAr) 9.9148 1.1550 0.7234 3.3488 Execution time (s) 3.061191 2.493990 2.538650 3.269690
are selected as shunt compensation buses [14]. The results of single- and multi-objective optimization problems are obtained and presented in Tables 1 and 2, respectively. As can be observed inTable 1, the single-objective ORPF is done for every three function, separately. By minimizing the total fuel cost objective function, the total fuel cost, active power losses and the system loadability are equal to 17270.327 $/h, 1.5952 MW and 1 while they are equal to 18170.323 $/h, 1.0252 MW and 1 by minimizing active power losses, respectively.
In order to study the conflict among the objective functions, the multi-objective optimization is performed in four cases. In each case, the best Pareto optimal solution is selected among the Pareto optimal solutions based on adopted weight factors in the fuzzy decision-making process. In all the cases, the same weight values are assigned to the objective functions. The results of multi-objective optimization by the proposed algorithm are shown in Table 2. Some interesting cases observed from the table are considered here. In case 1, the total fuel cost increases as much as 248.336 S/h while active power losses decrease to 0.4023 MW with respect to single-objective optimization of the total fuel cost. In case 4, the best compromise solution for the total fuel cost, active power losses and the system loadability are equal to 21932.977 $/h, 3.2453 MW and 1.223, respectively.
5.2. IEEE 30-bus test system
The single- and multi-objective optimization problems are performed on IEEE 30-bus test system considering the total fuel cost, active power losses and the system loadability as objective
Table 3: Results of single-objective optimization in IEEE 30-bus test system. Parameters F1 F2 F3 Vg1 1.06 1.06 1.06 Vg2 1.041 1.054 1.04 Vg5 1.014 1.036 0.999 Vg8 1.022 1.043 1.005 Vg11 1.06 1.06 1.06 Vg13 1.06 1.06 1.06 T6-9 1.07 1.08 1.05 T6-10 0.9 0.90 0.93 T4-12 0.96 0.96 0.99 T27-28 0.95 0.96 0.94 Qc10 0.30 0.30 0.30 Qc24 0.12 0.12 0.22
Total fuel cost ($/h) 801.757 967.798 1404.717
Ploss(MW) 9.3099 3.1567 14.6164 Loadability index 1 1 1.483 Qloss(MVAr) 6.5988 15.1985 28.4186 Execution time (s) 0.917260 1.014921 0.701415
Table 4: Results of multi-objective optimization in IEEE 30-bus test system.
Parameters Case 1 Case 2 Case 3 Case 4
(F1&F2) (F1&F3) (F2&F3) (F1&F2&F3)
Vg1 1.06 1.06 1.06 1.06 Vg2 1.047 1.039 1.043 1.039 Vg5 1.025 1.004 1.013 1.008 Vg8 1.035 1.014 1.017 1.019 Vg11 1.06 1.06 1.058 1.06 Vg13 1.06 1.06 1.06 1.06 T6-9 1.07 1.08 1.09 1.08 T6-10 0.9 0.9 0.9 0.9 T4-12 0.95 0.98 0.98 0.97 T27-28 0.95 0.94 0.94 0.94 Qc10 0.3 0.3 0.3 0.3 Qc24 0.12 0.18 0.19 0.15
Total fuel cost ($/h) 848.454 1153.473 1284.676 1009.281
Ploss(MW) 4.9664 13.5455 10.9197 12.3012 Loadability index 1 1.314 1.371 1.193 Qloss(MVAr) 8.9805 25.3956 16.6266 19.7588 Execution time (s) 7.005369 6.370397 5.831507 7.419780
functions. The test system contains 41 branches, 6 generators and 2 capacitor banks. Four branches 6-9, 6-10, 4-12 and 27-28 are under load tap setting transformer branches [42]. The buses with possible reactive power source putting in places are 10 and 24. The results of single- and multi-objective optimization problems are shown inTables 3and4, respectively. As observed in Table 3, the total fuel cost, active power losses and the system loadability are equal to 801.757 $/h, 9.3099 MW and 1, respectively, by optimizing the total fuel cost objective function. The system loadability and the total fuel cost functions increase to 1.483 and 1404.717 $/h by maximizing the system loadability. An interesting case deduced fromTable 4is in the following aspect. In case 2, the total fuel cost is near to its optimal value while the system loadability is receded from its optimal value compared to that of single-objective optimization of the system loadability.
As there is no previous literature specifically examining the multi objective ORPF on the considered objective functions, the results can not be compared to that of the other methods. However, as it can be inferred from the literature [1,14,15,22] in the single-objective optimization, the results obtained by the proposed algorithm are better than that of the other methods. Besides, as mentioned in the Section 4.1, the
ε
-constrained multi-objective approach will convert the MMP problem to the series of single-objective optimization subproblem;Table 5: Results obtained by some methods in IEEE 14- and 30-bus test system.
IEEE 14-bus test system
IEEE 30-bus test system Ploss (MW) Execution time (s) Ploss (MW) Execution time (s) DE [1] 13.239 8.172 5.011 13.647 SQP [1] 13.246 – 5.043 – PSO [14,22] 13.322 16.5 5.0921 3.720 SARGA [15] 13.216 54 – – RTS [14] 13.236 19.5 – – EP [15] 13.346 72 – – IPM [22] – – 5.101 0.636 Proposed algorithm 1.0252 0.355180 3.1567 1.014921
therefore, based on the better solutions of the single-objective optimization using DICOPT solver, it can be claimed that the proposed algorithm can reveal better solutions in the multi-objective optimization, too. By minimizing active power losses in the 14- and 30-bus test systems, some interesting methods are compared and tabulated inTable 5.
5.3. IEEE 118-bus test system
In order to test and validate the robustness of the proposed algorithm, simulations are carried out for ORPF problem in the IEEE 118-bus test system. The network consists of 186 branches, 54 generator buses and 12 capacitor banks. Nine branches 8-5, 26-25, 30-17, 38-37, 63-59, 64-61, 65-66, 68-69, and 81-80 are tap changing transformers [43]. Simulation results of single-and multi-objective optimization problems are tabulated in
Table 6. By minimizing the total fuel cost objective function, the total fuel cost, the active power losses and the system loadability are equal to 130114.429 $/h, 84.0059 MW and 1, respectively, while they are equal to 166538.201 $/h, 9.1995 MW and 1 when active power losses are minimized and they are equal to 342023.466 $/h, 207.1314 MW and 1.985 when the system loadability is maximized.
By minimizing active power losses, the active power losses and the execution time achieved by the proposed algorithm are equal to 9.1995 MW and 85.5661 s while they are equal to 116.81 MW and 185 s by the HGI method [17], respectively. As can be derived from the results, the proposed algorithm gives the best performance in comparison with the HGI method from the yield active power losses and the execution time aspects.
6. Conclusions
The reactive power control problem is formulated as single- and multi-objective optimization frameworks. The multi-objective ORPF problem is solved to simultaneously optimize the total fuel cost, active power losses and the system loadability as objective functions using the
ε
-constraint method for generating the Pareto optimal solutions. The best compromise solution is selected among the Pareto optimal solutions based on the adopted weighting factors using the fuzzy decision-making tool.The single- and multi-objective optimization problems are performed on IEEE 14-, 30-, and 118-bus test systems. The settings of reactive power compensators, such as tap changing transformers and shunt capacitors/reactors, are discrete variables while the others are continuous variables. The optimization problem is modeled as a MINLP in GAMS software and solved by DICOPT solver due to discrete variables.
Table 6: Results of single- and multi-objective optimization in IEEE 118-bus system. Total fuel cost ($/h)
Ploss(MW) Loadability index Qloss(MVAr) Execution time (s)
F1 130114.429 84.0059 1 910.2411 66.391384 F2 166538.201 9.1995 1 1260.7557 85.566133 F3 342023.464 207.1314 1.985 168.4968 54.382507 Case 1 (F1&F2) 134665.941 31.2014 1 1234.5345 333.580969 Case 2 (F1&F3) 269234.189 151.1829 1.788 550.4149 336.410529 Case 3 (F2&F3) 294611.840 81.1647 1.756 996.5298 362.130577
Case 4 (F1&F2&F3) 197994.299 95.6985 1.394 837.2031 508.840801
The simulation results show that the proposed algorithm is outperformed by the other methods in the single-objective optimization and hence leading to better solutions in the multi-objective optimization since the admittance matrix is directly constructed in the optimization framework as additional equality constraints. Therefore, it is expected that the proposed algorithm will be preferred and play a more active role in the reactive power planning.
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http://www.ee.washington.edu/research/pstca(accessed 21.01.08). Afshin Lashkar Arawas born in Tehran, Iran, in 1973. He received his B.Sc. degree in electrical engineering from the Islamic Azad University, Dezful Branch, Dezful, Iran in 1995. He received his M.Sc. degree in electrical engineering from the University of Mazandaran, Babol, Iran in 2001. He received his Ph.D. degree in electrical engineering from Iran University of Science and Technology (IUST), Tehran, Iran in 2011. He is currently a Faculty member of Islamic Azad University, Dezful Branch, Dezful, Iran as well as a member of IEEE Power and Energy Society (IEEE-PES). His current research interests include analysis, operation and control of power systems, and FACTS controllers.
Ahad Kazemiwas born in Tehran, Iran, in 1951. He received his M.Sc. degree in electrical engineering from Oklahoma State University, USA in 1979. He is currently an associate professor in electrical engineering department of Iran University of Science and Technology, Tehran, Iran. His research interests are reactive power control, power system dynamics, stability and control and FACTS devices.
Sodabe Gahramaniwas born in Tabriz, Iran, in 1983. He received his B.Sc. and M.Sc. degrees in electronic and electrical engineering from the Islamic Azad University, Tabriz branch and Dezful branch, Iran in 2008 and 2011, respectively. Her research interests are power system analysis and FACTS devices.
Mohammad Behshadwas born in Dezful, Iran, in 1984. He received his B.Sc. and M.Sc. degrees in electrical engineering from the Islamic Azad University, Dezful branch, Dezful, Iran in 2006 and 2009, respectively. His research interests are power system analysis and FACTS devices.