application of gravitational search algorithm for optimal...
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Application of Gravitational Search Algorithm for Optimal Reactive Power Dispatch Problem
Serhat Duman Department of Electrical Education, Technical Education Faculty, Duzce University, Duzce, 81620 TURKEY
Yusuf Sonmez Department of Electrical
Technology, Gazi Vocational Collage, Gazi University, Ankara,
06760 TURKEY [email protected]
Nuran Yorukeren Department of Electrical
Engineering, Engineering Faculty, Kocaeli University, Izmit, 41040
TURKEY [email protected]
Ugur Guvenc Department of Electrical and
Electronic Engineering, Faculty of Technology, Duzce University,
Duzce, 81620 TURKEY [email protected]
Abstract— In this paper, Gravitational Search Algorithm (GSA) is applied to solve the optimal reactive power dispatch (ORPD) problem. The ORPD problem is formulated as a nonlinear constrained single-objective optimization problem where the real power loss and the bus voltage deviations are to be minimized separately. In order to evaluate the proposed algorithm, it has been tested on IEEE 30 bus system consisting 6 generator and compared other algorithms reported those before in literature. Results show that GSA is more efficient than others for solution of single-objective ORPD problem.
Keywords-gravitational search algorithm, optimal reactive power dispatch, power systems, optimization
I. INTRODUCTION In recent years the optimal reactive power dispatch (ORPD)
problem has received great attention as a result of the improvement on economy and security of power system operation. Solutions of ORPD problem aim to minimize object functions such as fuel cost, power system loses, etc. while satisfying a number of constraints like limits of bus voltages, tip settings of transformers, reactive and active power of power resources and transmission lines and a number of controllable variables [1, 2].
In the literature, many methods for solving the ORPD problem have been done up to now. At the beginning, several classical methods such as gradient based [3], interior point [4], linear programming [5] and quadratic programming [6] have been successfully used in order to solve the ORPD problem. However, these methods have some disadvantages in the process of solving the complex ORPD problem. Drawbacks of these algorithms can be declared insecure convergence properties, long execution time, and algorithmic complexity. Besides, the solution can be trapped in local minima [1,7]. In order to overcome these disadvantages, researches have successfully applied evolutionary and heuristic algorithms such
as Genetic Algorithm (GA) [2], Differential Evolution (DE) [8] and Particle Swarm Optimization (PSO) [9]. It is reported in those that evolutionary or heuristic algorithms are more efficient than classical algorithms for solving the RPD problem. Gravitational Search Algorithm (GSA) is a new meta-heuristic and population based search algorithm based on Newton’s law of gravity and law of motion and it proposed firstly by Rashedi et al. in 2009 [10]. GSA has many advantages which are reported in [10] such as, adaptive learning rate, memory-less algorithm and, good and fast convergence. Besides, in [10] authors have compared the GSA with PSO, Central Force Optimization (CFO) and Real Genetic Algorithm (RGA) using 23 different benchmark functions and they have reported that GSA is more powerful than other algorithms. Due to all of these advantages, Duman et al. presented gravitational search algorithm to solve the economic dispatch with valve point effects for different test systems [14]. In this paper, GSA is applied for solving the ORPD problem. In the process of solving, ORPD problem is formulated as a nonlinear constrained single-objective optimization problem where the real power loss and the bus voltage deviations are to be minimized respectively. Simulations have been done using MATLAB program. The proposed algorithm is tested on IEEE 30-bus system for evolution of effectiveness of it. Results obtained from GSA are compared results reported those in [1]. Results show that proposed algorithm is more effective and powerful than other algorithms in solution of ORPD problem.
II. FORMULATION OF ORPD PROBLEM The objective of the ORPD problem is to minimize one or
more objective functions while satisfying a number of constraints such as load flow, generator bus voltages, load bus voltages, switchable reactive power compensations, reactive power generation, transformer tap setting and transmission line flow. In this paper two objective functions are minimized separately as single objective. Objective functions minimized
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in this paper and constraints are formulated taking from [1, 11] and shown as follows.
A. Minimization of Real Power Loss It is aimed in this objective that minimizing of the real
power loss (Ploss) in transmission lines of a power system. This is mathematically stated as follows.
Minimize ∑==
−+=n
jikk
ijjijikloss VVVVgP),(
1
22 )cos2( θ (1)
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.
B. Minimization of Voltage Deviation It is aimed in this objective that minimizing of the
deviations in voltage magnitudes (VD) at load buses. This is mathematically stated as follows.
Minimize ∑=
−=nl
kkVVD
10.1 (2)
where nl is the number of load busses and Vk is the voltage magnitude at bus k.
C. System Constraints In the minimization process of objective functions, some
problem constraints which one is equality and others are inequality had to be met. Objective functions are subjected to these constraints shown below.
Load flow equality constraints:
0sin
cos
1=
⎥⎥⎦
⎤
⎢⎢⎣
⎡
+−− ∑
=
nb
j ijij
ijijjiDiGi B
GVVPP
θθ
, i=1,2,…,nb (3)
0cos
sin
1
=⎥⎥⎦
⎤
⎢⎢⎣
⎡
+−− ∑
=
nb
j ijij
ijijjiDiGi B
GVVQQ
θθ
, i=1,2,…,nb (4)
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:
maxminGiGiGi VVV ≤≤ , ngi ∈ (5)
Load bus voltage (VLi) inequality constraint:
maxminLiLiLi VVV ≤≤ , nli ∈ (6)
Switchable reactive power compensations (QCi) inequality constraint:
maxminCiCiCi QQQ ≤≤ , nci ∈ (7)
Reactive power generation (QGi) inequality constraint:
maxminGiGiGi QQQ ≤≤ , ngi ∈ (8)
Transformers tap setting (Ti) inequality constraint:
maxminiii TTT ≤≤ , nti ∈ (9)
Transmission line flow (SLi) inequality constraint:
maxLiLi SS ≤ , nli ∈ (10)
where, nc, ng and nt are numbers of the switchable reactive power sources, generators and transformers.
During the simulation process, all constraints satisfied as explained below [11].
• The load flow equality constraints are satisfied by power flow algorithm.
• The generator bus voltage (VGi), the transformer tap setting (Ti) and the Switchable reactive power compensations (QCi) are optimization variables and they are self-restricted between the minimum and maximum value by the GSA algorithm.
• The limits on active power generation at the slack bus (PGs), load bus voltages (VLi) and reactive power generation (QGi), transmission line flow (SLi) are state variables. They are restricted by adding a penalty function to the objective functions.
III. GRAVITATIONAL SEARCH ALGORITHM GSA is the new meta-heuristic optimization algorithm
motivated by the Newton’s laws of gravity and motion. GSA was firstly produced by Rashedi et al. in 2009 [GSA]. According to this algorithm, agents are considered as objects and their performance is measured by their masses. Every object attracts every other object with gravitational force. GSA algorithm can be explained following steps [10, 12, 13].
A. Step 1: Initialization
When it is assumed that there is a system with N (dimension of the search space) masses, position of the ith mass
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is described as follows. At first, the positions of masses are fixed randomly.
),...,( 21 niiii xxxX = , Ni ,...1= (11)
where, xid is the position of the ith mass in dth dimension.
B. Step 2: Fitness Evaluation of All Agents In this step, for all agents, best and worst fitness are
computed at each epoch described as follows.
)(min)(},..,1{
tfittbest jNj∈= (12)
)(max)(},..,1{
tfittworst jNj∈= (13)
where fitj(t) is the fitness of the jth agent of t time, best(t) and worst(t) are best (minimum) and worst (maximum) fitness of all agents.
C. Step 3:Compute the Gravitational Constant (G(t)) In this step, the gravitational constant at t time (G(t)) is
computed as follows.
)exp()( 0 TtGtG α−= (14)
where G0 is the initial value of the gravitational constant chosen randomly, α is a constant, t is the current epoch and T is the total iteration number.
D. Step 4:Update the Gravitational and Inertial Masses In this step, the gravitational and inertial masses are
updated as follows.
)()()()(
)(tworsttbesttworsttfit
tmg ii −
−= (15)
where fiti(t) is the fitness of the ith agent of t time.
∑=
= N
jj
ii
tmg
tmgtMg
1)(
)()( (16)
where Mgi(t) is the mass of the ith agent of t time.
E. Step 5:Calculate the Total Force In this step, the total force acting on the ith agent (Fi
d(t)) is calculated as follows.
∑≠∈
=ikbestjj
dijj
di tFrandtF )()( (17)
where randj is a random number between interval [0,1] and kbest is the set of first K agents with the best fitness value and biggest mass.
The force acting on the ith mass (Mi(t)) from the jth mass (Mj(t)) at the specific t time is described according to the gravitational theory as follows.
))()(()(
)()()()( txtx
tRtMtM
tGtF di
dj
ij
jidij −
+×
=ε
(18)
where Rij(t) is the Euclidian distance between ith and jth agents (
2)(),( tXtX ji ) and ε is the small constant.
F. Step 6:Calculate the Acceleration and Velocity In this step, the acceleration (ai
d(t)) and velocity (vid(t)) of
the ith agent at t time in dth dimension are calculated through law of gravity and law of motion as follows.
)(
)()(tMg
tFta di
did
i = (19)
)()()1( tatvrandtv di
dii
di +×=+ (20)
where randi is the random number between interval [0,1].
G. Step 7:Update the Position of the Agents In this step the next position of the ith agents in dth
(xid(t+1)) dimension are updated as follows.
)1()()1( ++=+ tvtxtx di
di
di (21)
H. Step 8:Repeat In this step, steps from 2 to 7 are repeated until the
iterations reach the criteria. In the final iteration, the algorithm returns the value of positions of the corresponding agent at specified dimensions. This value is the global solution of the optimization problem also.
All these steps explained above describes how the GSA works. Besides, the principle diagram of the GSA is illustrated in Fig. 1.
IV. SIMULATION RESULTS Proposed approach has been applied to solve ORPD
problem. In order to demonstrate the efficiency and robustness of proposed GSA approach based on Newtonian physical law
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of gravity and law of motion which is tested on standard IEEE 30-bus test system shown in Fig. 2 and the system data are given in [15].
Figure 1. The principle diagram of the GSA [10].
The test system has six generators at the buses 1, 2, 5, 8, 11 and 13 and four transformers with off-nominal tap ratio at lines 6-9, 6-10, 4-12, and 28-27 and, hence, the number of the optimized control variables is 10 in this problem.
Figure 2. Single line diagram of IEEE 30-bus test system
The minimum voltage magnitude limits at all buses are 0.95 pu and the maximum limits are 1.1 pu for generator buses
2, 5, 8, 11, and 13, and 1.05 pu for the remaining buses including the reference bus 1. The minimum and maximum limits of the transformers tapping are 0.9 and 1.1 pu respectively [1]. The proposed approach has been applied to solve ORPD problem for different objective functions. G is set using in Eq. (14), where G0 is set to 100 and α is set to 10, and T is the total number of iterations. Maximum iteration numbers are 200 for all case studies.
The optimum control parameter settings of proposed approach are given in Table 1. The best power loss and best voltage deviations obtained from proposed approach are 4.616657 MW and 0.106498 respectively. GSA is less by 9.772763%, 25.785365% compared to previously report best results 5.1167 MW, 0.1435 respectively. Fig. 3 and Fig. 4 show the convergence of GSA for minimum power loss and voltage deviations solutions respectively. The results obtained from proposed algorithm have been compared other methods in the literature. The results of this comparison are given in Table 2 and Table 3. The results in Tables 2 and 3 show that the reactive dispatch and voltage deviations solutions specified by the proposed GSA approach lead to lower active power loss and voltage deviations than that by the ref. [1] simulation results, which confirms that the proposed approach is well capable of specification the optimum solutions.
TABLE I. BEST CONTROL VARIABLES SETTINGS FOR DIFFERENT TEST CASES OF PROPOSED APPROACH
Control variables settings
Case 1: Power Loss
Case 2: Voltage deviations
VG1 1.049998 0.995371 VG2 1.024637 0.950069 VG5 1.025120 1.043033 VG8 1.026482 1.021292 VG11 1.037116 1.100000 VG13 0.985646 1.062669 T6-9 1.063478 0.905907 T6-10 1.083046 1.035611 T4-12 1.100000 1.038107 T27-28 1.039730 0.925607
Power loss (MW) 4.616657 6.371609 Voltage deviations 0.836338 0.106498
TABLE II. COPMARISON OF THE SIMULATION RESULTS FOR POWER LOSS
Control variables settings
GSA Individual
optimization [1]
Multi-objective EA [1]
As single objective
[1] VG1 1.049998 1.050 1.050 1.045 VG2 1.024637 1.041 1.045 1.042 VG5 1.025120 1.018 1.024 1.020 VG8 1.026482 1.017 1.025 1.022 VG11 1.037116 1.084 1.073 1.057 VG13 0.985646 1.079 1.088 1.061 T6-9 1.063478 1.002 1.053 1.074 T6-10 1.083046 0.951 0.921 0.931 T4-12 1.100000 0.990 1.014 1.019 T27-28 1.039730 0.940 0.964 0.966
Power loss (MW) 4.616657 5.1167 5.1168 5.1630
Voltage deviations
(p.u.) 0.836338 0.7438 0.6291 0.3142
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Figure 3. Convergence of GSA for power loss
TABLE III. COPMARISON OF THE SIMULATION RESULTS FOR VOLTAGE DEVIATIONS
Control variables settings
GSA Individual
optimization [1]
Multi-objective
EA[1]
As single objective
[1] VG1 0.995371 1.009 1.016 1.021 VG2 0.950069 1.006 1.012 1.021 VG5 1.043033 1.021 1.018 1.021 VG8 1.021292 0.998 1.003 1.002 VG11 1.100000 1.066 1.061 1.025 VG13 1.062669 1.051 1.034 1.030 T6-9 0.905907 1.093 1.090 1.045 T6-10 1.035611 0.904 0.907 0.909 T4-12 1.038107 1.002 0.970 0.964 T27-28 0.925607 0.941 0.943 0.941
Power loss (MW) 6.371609 5.8889 5.6882 5.6474
Voltage deviations
(p.u.) 0.106498 0.1435 0.1442 0.1446
Figure 4. Convergence of GSA for voltage deviations
V. CONCULUSION In this paper, one of the recently developed stochastic
algorithms is the gravitational search algorithm has been demonstrated and applied to solve optimal reactive power dispatch problem. The problem has been formulated as a constrained optimization problem. Different objective functions have been considered to minimize real power loss, to enhance the voltage profile. The proposed approach is applied to optimal reactive power dispatch problem on the IEEE 30- bus power system. The simulation results indicate the effectiveness and robustness of the proposed algorithm to solve optimal reactive power dispatch problem in test system. The GSA approach can reveal higher quality solution for the different objective functions in this paper.
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