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A novel disruption based symbiotic organisms search to solve economic dispatch

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Abstract

Economic dispatch (ED) is one of the vital prospects in the energy management system for determining the optimal power generation distribution among several committed power generating units. The objective of ED is to generate the electric power to meet the load demand by minimizing the total operating cost subjected to various equality and inequality constraints. This ED problem is generally solved using conventional techniques such as a gradient method by assuming the objective function as linear and convex. However, in general, the ED problem is constrained, non-convex, large scale, multimodal and highly nonlinear optimization problem. In order to overcome this drawback, various heuristic techniques have been introduced in the literature. Symbiotic organism search (SOS) is one of the emerging nature-inspired meta-heuristic algorithms. Due to simple in implementation and free from algorithm-specific control parameters, the SOS algorithm has gained a reputation among the researchers. The SOS technique mimics the biological interaction between two distinct species to survive and proliferate in the ecosystem. To evade the suboptimal solution and to enhance the exploration and exploitation of the SOS algorithm, the disruption operator is integrated with the basic SOS to form a novel disruption based SOS (DSOS) algorithm. The disruption strategy is originated from astrophysics and has the ability to shift between exploration and exploitation during the process of finding an optimum solution. To illustrate the effectiveness of the proposed method, it is applied on various test systems such as 3-unit, 5-unit, 6-unit, 13-unit, 20-unit, 38-unit, and 110-unit systems to solve different ED problems, namely, without transmission losses (TL), with transmission losses, by simultaneously considering valve point loading (VPL) effect along with TL, by simultaneously considering the effect of ramp rate limits (RRLs), prohibited operating zones (POZs) and TLs, and dynamic economic load dispatch by considering RRLs, POZs, VPL, and TL. The comparison of DSOS based simulation results with other methods reveal the efficiency and robustness of the proposed method to provide minimum total generation cost in all the cases with better convergence rate and less computational time.

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  1. Electrical and Electronics Engineering Department, S R Engineering College, Warangal, 506371, India

    B. Vedik & C. K. Shiva

  2. Indian Institute of Technology, Roorkee, Uttarakhand, 247667, India

    P. Naveen

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Appendix

Appendix

To test the efficiency and performance of the proposed DSOS technique, it is employed on 10 benchmark functions that include five unimodal and five multimodal functions [90]. These benchmark functions are summarized in Table 29, and the function details are taken from [90, 91]. The first five functions namely, f1 to f5 are unimodal functions that have a single global solution with no local optimum. Therefore, these functions determine the exploitation capability of the techniques along with the convergence rate of the algorithm. The next five functions f6 to f9 and f10 are multimodal and low dimensional functions, respectively. The multimodal functions have a considerable number of local optimum solutions. In contrast, low dimension function has a few local optimum solutions. Therefore, these multimodal and low dimensional functions determine the exploration capability in identifying the global optimum region by avoiding local optimum solutions.

Table 29 Unimodal, multimodal, and fixed low-dimensional benchmark test functions
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The proposed DSOS has been implemented on the above said benchmark functions and the results obtained are tabulated in Table 30. Further, the obtained results have been compared with recently published technique, namely, atom search algorithm (ASO) along with SA, PSO, GA, and GSA techniques. In order to show the effectiveness of the proposed DSOS, the results are also compared with the basic SOS technique. Table 30 consists of 10 columns. The first column represents the benchmark function utilized. Columns 2 to 6 signify the results obtained using SA, PSO, GSA, GA, and atom search optimization (ASO). These results are taken from [90] for comparison. The population size and number of iterations considered for these techniques are 50 and 1000, respectively. Columns 7 and 9 denote the results obtained from SOS and the proposed DSOS techniques. Here, the population size and number of iterations considered for both these algorithms are 30 and 100, respectively. Further, to show the effectiveness of the proposed DSOS compared to basic SOS, these two techniques have been executed with less number of iterations, i.e. the number of iterations has been decreased from 100 to 20. The results obtained for this case are shown in Table 30 Columns 8 and 10, respectively. It is to be noted here that the best results obtained by all the algorithms after 50 runs are shown in Table 30.

Table 30 Comparisons of best-so-for results for 50 trials obtained by different techniques
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From Table 30, it can be perceived that for unimodal functions (f1 to f5) the proposed DSOS technique provides better results not only when compared to basic SOS but also when compared to other techniques even with less number of iterations. This shows the exploitation capability of the proposed method when compared to other techniques. Further, it can also be seen that when the number of iterations has been decreased from 100 to 20, the proposed algorithm has provided the same results except for f5; on the other hand, basic SOS failed to provide. Similarly, for multimodal functions, the proposed method provides better results when compared to SA, GA, PSO, ASO and GSA techniques for all test functions from f6 to f8 and provides similar results for test functions f9 and f10. The similar results obtained for benchmark functions f9 and f10 is due to the small dimensional size of the problem and further, the number of iterations considered for SA, GA, PSO, ASO and GSA techniques is 10 times more than the DSOS technique. Similarly, when compared to basic SOS, the DSOS technique provides similar results for all the functions ranging from f6 to f10 except f7. However, when the number of iterations decreased to 20, the proposed DSOS provided better results when compared to basic SOS for all the functions. Furthermore, it can also be seen from Table 30 that the proposed method provides similar results even when the number of iterations has been decreased. This shows the exploration capability of the proposed DSOS when compared to other techniques [90].

The statistical indices, namely, the average (mean), standard deviation (std), and minimum best-so-far solution (best) are utilized to better evaluate the performance of the proposed algorithm. For all the 50 runs, minimum average of the best solutions obtained so far shows the ability of the algorithm to avoid local optimum, minimum standard deviation of the best solutions obtained so far denotes the closer the obtained solution to the average value and minimum best-so-far solution denotes the ability of the algorithm closer to the global solution [90]. The statistical results obtained by GA, SA, GSA, PSO, ASO, basic SOS, and DSOS techniques for 50 runs are given in Table 31. From this table, it can be perceived that the proposed DSOS technique provides better results when compared to all other techniques and similar results when compared to ASO and SOS for functions f8 to f10.

Table 31 Comparisons of statistical results obtained by different techniques
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Further, the convergence curve that indicates the behavior of the algorithm to attain global optimum accurately and quickly is utilized to compare basic SOS and DSOS algorithms. From Figs. 16, 17, 18, 19, 20, 21, 22, 23, 24 and 25, it can be seen that the proposed DSOS algorithm convergence quickly and find the accurate global solution when compared to the basic SOS technique. It is to be noted here that, even though DSOS has a better convergence rate than basic SOS for all the benchmark functions, the similar characteristics for f9 and f10 functions is due to low dimensional size of the problem. This shows the preeminence of the proposed method compared to the basic SOS algorithm.

Fig. 16
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Convergence curve for function f1

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Fig. 17
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Convergence curve for function f2

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Fig. 18
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Convergence curve for function f3

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Fig. 19
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Convergence curve for function f4

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Fig. 20
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Convergence curve for function f5

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Fig. 21
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Convergence curve for function f6

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Fig. 22
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Convergence curve for function f7

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Fig. 23
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Convergence curve for function f8

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Fig. 24
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Convergence curve for function f9

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Fig. 25
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Convergence curve for function f10

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Vedik, B., Naveen, P. & Shiva, C.K. A novel disruption based symbiotic organisms search to solve economic dispatch. Evol. Intel. 15, 255–290 (2022). https://doi.org/10.1007/s12065-020-00506-5

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