Authors: Ramamoorthi R, Balamurugan R
DOI Link: https://doi.org/10.22214/ijraset.2023.52161
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This article presents a new evolutionary optimization approach named grey wolf optimization (GWO), which is based on the behavior of grey wolves, for the optimal operating strategy of economic load dispatch (ELD). Nonlinear characteristics of generators like ramp rate limits, valve point discontinuities and prohibited operating zones are considered in the problem. GWO method does not require any information about the gradient of the objective function, while searching for an optimum solution. The GWO algorithm concept appears to be a robust and reliable optimization algorithm is applied to the nonlinear ELD problems. The proposed algorithm is implemented and tested on two test systems having 40 Thermal generators. The results confirm the potential and effectiveness of the proposed algorithm compared to various other methods available in the literature. The outcome is very encouraging and proves that the GWO is a very effective optimization technique for solving various ELD problems.
I. INTRODUCTION
Economic load dispatch (ELD) is a process used in power systems to determine the optimal distribution of power output from various generating units to meet the system demand while minimizing the operating costs. This is achieved by optimizing the output of each generator in the system, subject to the operating constraints of the system such as transmission capacity, voltage limits, and stability limits. The ELD problem is a nonlinear optimization problem that requires solving a set of equations to determine the optimal operating point for each generator. The solution is typically obtained using mathematical optimization techniques such as linear programming; nonlinear programming but these mathematical methods are having various disadvantages such as the increased computational complexity of techniques with the size and complexity of the problem. As a result, it may be difficult or even impossible to solve largescale non convex optimization problems using these techniques. On the other hand, heuristic optimization methods can handle nonlinear and nonconvex problems effectively by searching the solution space using a population of solutions and applying evolutionary operators such as mutation, crossover, and selection etc., also ELD problems involve multiple constraints such as power balance, generator output limits, transmission capacity limits, and voltage limits.
Evolutionary algorithm can handle multiple constraints by incorporating penalty functions or constraints handling techniques into the fitness function. In recent literature various heuristic algorithms are reported to solve the ELD problems. Gaing proposed particle swarm optimization (PSO) to solve the ELD problem in power systems and compared with Genetic Algorithm. Several nonlinear characteristics of the generator such as ramp rate limits, POZs and nonsmooth cost functions were considered [1]. In [2], Firefly algorithm [FA] was used to determine optimal solution for the ELD problems FA emulates social conduct of fire flies dependent on their blazing quality. Dubey, pandit and panigrahi [3] presented modified flower pollination algorithm [MFPA] to deal the ELD problems. In the MFPA neighborhood fertilization of FPA was constrained by a scaling component and a concentrated exploitation stage was added to determine the best solutions. A continuous version of quick group search optimizer (QGSO) algorithm was proposed to realize me ELD formulation with VPL effect POZs, transformation losses and ramp rate limits [4]. Cuckoo search algorithm (CSA) was developed for solving both convex and nonconvex ELD problems [5]. It was inspired from the obligate brood parasitism of some cuckoo species by laying their eggs in the nests of other host birds of different species. Simulated annealing technique has been applied to determine the optimal generation schedule for economic dispatch problems in a power system [6]. Hybrid evolutionary programming (HEP) [14] is used to obtain best optimal solution.[7]. In [8] authors used Evolutionary programming method for solving ELD problems .
This paper presents a new swarm based optimization algorithm known as Gray wolf optimization (GWO) developed by proposed by Mirjalili et al. in 2014 [9]. This algorithm is inspired by the hunting nature of the gray wolf. To assess the effectiveness of the GWO, Two test system having different constraints are taken in to consideration and optimized for best optimal fuel cost. Obtained results are compared with the other heuristic methods in the literature.
Organization of this research article as follows: In Section 2, the formulation of the economic load dispatch problem is discussed. In Section 3, the applied Gray wolf optimization algorithm is explained, as well as its implementation process for the ELD problem. In Section 4, Simulation results are presented to compare GWO’s effectiveness to that of the original algorithm and other algorithms. Final section summarizes this research work.
D. Attacking prey (exploitation)
The grey wolves stop the hunting by attacking the prey when it stop moving. It depends on the value of a* AM is a random value in the interval [2a, 2a]. In GWO, search agents update their positions based on the location of alpha, beta, delta wolves mentioned in hunting phase and attack towards the prey.
E. Grey wolf optimization applied to ELD
The different steps of GWO algorithm for solving ELD problems are described below.
IV. CASE STUDIES AND NUMERICAL RESULTS
In order to validate the feasibility of the proposed GWO method for the ELD problems, it is employed on a relatively large system consisting of 40 generating units. The load demand used in the simulations is 10500 MW. In order to judge the efficacy of the proposed in nonlinear environment, the valvepoint effect and prohibited operating zones are considered. Data for the test system is referred from [10]. The developed algorithm is simulated and tested in MATLAB 7.1 Software on 2 GHz Pentium IV, 1 GB RAM personal computer. The population size and the maximum iteration number are taken as 50 and 500 respectively for the test systems under consideration.
Test Case 1: 40unit system without valve point loading effect and without Transmission loss.
Test Case 2: 40unit system including valve point loading effect and Transmission Loss.
A Test System 1
In this 40 unit test system Valve point effects and Transmission losses are not considered for the simulation. The results obtained by applying the GWO algorithm and other heuristic method known as variable scaling hybrid differential evolution (VSHDE) [11] are summarized in Table 1 for 40unit power system without considering the effects of valvepoint loading without transmission losses. Analyzing the data, it can be observed that the GWO method succeeds in finding a satisfactory solution.
The minimum cost obtained by the proposed GWO method was given by 121244 $/hr, which is the best cost found so far. The analysis of these comparative results demonstrates that the proposed approach shows superior performance compared to other method reported in the literature. Convergence curve for this test system is shown in figure 3.
Table 1 : Best fuel cost Simulation test results for the Test system 1
Unit 
GWO 
VHSDE 
Unit 
GWO 
VHSDE 
1 
80 
79.63 
22 
550 
550 
2 
120 
119.99 
23 
550 
550 
3 
190 
189.98 
24 
550 
550 
4 
41.24206 
36.27 
25 
550 
550 
5 
38.08678 
42 
26 
550 
550 
6 
140 
140 
27 
550 
549.99 
7 
300 
300 
28 
10.17545 
10 
8 
300 
299.98 
29 
10.17545 
10 
9 
300 
300 
30 
10.17545 
10 
10 
130 
131.97 
31 
20 
20.01 
11 
94 
94.03 
32 
20 
20.01 
12 
100.7911 
94 
33 
20 
20 
13 
171.2593 
174.03 
34 
20 
20 
14 
339.2156 
327.7 
35 
18 
18.01 
15 
337.9593 
339.51 
36 
18 
18 
16 
337.9593 
339.49 
37 
20 
20 
17 
337.9593 
350.34 
38 
25 
25.06 
18 
500 
500 
39 
25 
25 
19 
500 
500 
40 
25 
25 
20 
550 
550 
Total Cost (S/hr)

121244

121253

21 
550 
550 
B. Test system 2
The simulation results obtained by proposed GWO method for 40 unit test system considering transmission losses and valve point loading effects are compared with GAAPI [12],SDE [10],TLBO [13] in Table 2. From Table 2, it was observed that GWO outperforms other optimization methods. In this case, the GWO obtained total cost 2267.8 $/hr, 560.3 $/hr, 217 $/hr lesser than the GAAPI, SDE, and TLBO algorithms respectively. Convergence curve for this test system is shown in figure 3. The simulation results clearly suggest that GWO produces feasible solutions. To judge the superiority and robustness,
Table 2 : Best fuel cost Simulation test results for the Test system 1
Unit 
GAAPI 
SDE 
TLBO 
GWO 
Unit 
GAAPI 
SDE 
TLBO 
GWO 
1 
114 
110.06 
114 
114 
22 
550 
550 
522.1852 
523.2794 
2 
114 
112.41 
114 
114 
23 
550 
550 
526.1804 
523.2794 
3 
120 
120 
120 
120 
24 
550 
528.16 
521.1967 
523.2794 
4 
190 
188.72 
182.4448 
179.7331 
25 
550 
524.16 
525.801 
523.2794 
5 
97 
85.91 
90.6923 
87.7999 
26 
550 
539.1 
526.0022 
541.3818 
6 
140 
140 
140 
140 
27 
11.44 
10 
13.0804 
10 
7 
300 
250.19 
300 
300 
28 
11.56 
10.37 
11.0397 
10 
8 
300 
290.68 
296.0682 
300 
29 
11.42 
10 
12.9373 
10 
9 
300 
300 
288.8518 
300 
30 
97 
96.1 
89.7412 
87.7999 
10 
205.25 
282.01 
281.952 
279.5997 
31 
190 
185.85 
190 
190 
11 
226.3 
180.82 
238.1293 
243.5997 
32 
190 
189.54 
190 
190 
12 
204.72 
168.74 
251.012 
243.5997 
33 
190 
189.96 
190 
190 
13 
346.48 
469.96 
483.1175 
484.0392 
34 
200 
199.9 
200 
200 
14 
434.32 
484.17 
481.9042 
484.0392 
35 
200 
196.25 
200 
200 
15 
431.34 
487.73 
488.2883 
484.0392 
36 
200 
185.85 
164.7435 
164.7998 
16 
440.22 
482.3 
396.3448 
484.0392 
37 
110 
109.72 
110 
110 
17 
500 
499.64 
494.2577 
489.2794 
38 
110 
110 
110 
110 
18 
500 
411.32 
408.3826 
489.2794 
39 
110 
95.71 
110 
110 
19 
550 
510.47 
510.5206 
511.2794 
40 
550 
532.47 
547.9677 
511.2794 
20 
550 
542.04 
521.2217 
511.2794 
Total cost ($/hr) 
139865 
138157.5 
137814.2 
137597.2 
21 
550 
544.81 
540.57 
433.5196 
Total Loss MW 
1045.06 
974.43 
1002.63 
1021.504 
In this work, an efficient metaheuristic algorithm named GWO is proposed to solve the ELD problem taking the valve point loading effects, prohibited operating zone, ramp rate limits into consideration. Two case studies are employed to demonstrate the applicability of the GWO method. The benefit of the proposed GWO is that it does not impose any convexity limitations on the generating unit characteristics. Numerical results show that the GWO method has superior features, advantages over other algorithms in terms of robustness, avoids premature convergence, simple applicability and stable convergence characteristic. Although, the proposed algorithm is applied to solve ELD problems in the current study, it seems from its unique feature that GWO has the potential to solve many other optimization problems in the field of power system planning and operation.
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Copyright © 2023 Ramamoorthi R, Balamurugan R. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Paper Id : IJRASET52161
Publish Date : 20230513
ISSN : 23219653
Publisher Name : IJRASET
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