We focus on solving the ED problem with nonsmooth cost functions considering the valve point loading effect for verifying the utility of the proposed framework, rECGA.
The nonsmooth cost functions were described as Equation (4). In order to examine the performance, rECGA for ED was applied to two ED problems that were adopted as test problems in the literature (Walters and Sheble, 1993; Sinha et al., 2003) for the purpose of comparison. One consists of three generation units, and the other consists of 40 generation units. The input data for the three-generator system are given by Walters and Sheble (1993), and those for the 40-generator system are given by Sinha et al.
(2003). The detailed problem parameters for the two test problems, including the lower bound and upper bound for the output of each generator as well as the coefficients for computing the cost functions, are given in Tables 6 and 7. The total system demand for the three-unit system is 850 MW, and that for the 40-unit system is 10,500 MW. It has been proven that for the three-unit system, the global optimal solution is 8234.07 (Lin et al.,
Table 7: Parameters for test case II (40-unit system) with the valve point loading effect.
a, b, c, e, and f are the cost coefficients in the fuel cost function: Fj(Pj)= ajPj2+ bjPj + cj + |ejsin(fj × (Pjmin− Pj))|.
Generator Pmin(MW) Pmax(MW) a b c e f
1 36 114 0.0069 6.73 94.705 100 0.084
2 36 114 0.0069 6.73 94.705 100 0.084
3 60 120 0.2028 7.07 309.54 100 0.084
4 80 190 0.00942 8.18 369.03 150 0.063
5 47 97 0.0114 5.35 148.89 120 0.077
6 68 140 0.01142 8.05 222.33 100 0.084
7 110 300 0.00357 8.03 287.71 200 0.042
8 135 300 0.00492 6.99 391.98 200 0.042
9 135 300 0.00573 6.6 455.76 200 0.042
10 130 300 0.00605 12.9 722.82 200 0.042
11 94 375 0.00515 12.9 635.2 200 0.042
12 94 375 0.00569 12.8 654.69 200 0.042
13 125 500 0.00421 12.5 913.4 300 0.035
14 125 500 0.00752 8.84 1,760.4 300 0.035
15 125 500 0.00708 9.15 1,728.3 300 0.035
16 125 500 0.00708 9.15 1,728.3 300 0.035
17 220 500 0.00313 7.97 647.85 300 0.035
18 220 500 0.00313 7.95 649.69 300 0.035
19 242 550 0.00313 7.97 647.83 300 0.035
20 242 550 0.00313 7.97 647.81 300 0.035
21 254 550 0.00298 6.63 785.96 300 0.035
22 254 550 0.00298 6.63 785.96 300 0.035
23 254 550 0.00284 6.66 794.53 300 0.035
24 254 550 0.00284 6.66 794.53 300 0.035
25 254 550 0.00277 7.1 801.32 300 0.035
26 254 550 0.00277 7.1 801.32 300 0.035
27 10 150 0.52124 3.33 1,055.1 120 0.077
28 10 150 0.52124 3.33 1,055.1 120 0.077
29 10 150 0.52124 3.33 1,055.1 120 0.077
30 47 97 0.0114 5.35 148.89 120 0.077
31 60 190 0.0016 6.43 222.92 150 0.063
32 60 190 0.0016 6.43 222.92 150 0.063
33 60 190 0.0016 6.43 222.92 150 0.063
34 90 200 0.0001 8.95 107.87 200 0.042
35 90 200 0.0001 8.62 116.58 200 0.042
36 90 200 0.0001 8.62 116.58 200 0.042
37 25 110 0.0161 5.88 307.45 80 0.098
38 25 110 0.0161 5.88 307.45 80 0.098
39 25 110 0.0161 5.88 307.45 80 0.098
40 242 550 0.00313 7.97 647.83 300 0.035
2002). As for the 40-unit system, the global optimal solution has not been determined.
To the best of our limited knowledge, the known best solution previously obtained by other methods is 122,252.265 as reported by Park et al. (2005) with MPSO.
The parameter settings in rECGA for ED are population size= 400, crossover prob-ability = 0.975, tournament size = 8, γ = 0.5, = 0.999, and the maximum function
Table 8: Comparison of the results obtained by various methods on the nonsmooth cost function considering the valve point loading effect. For the three-unit system, EP, CMA-ES, MPSO, and rECGA were able to find the global optimum (Lin et al., 2002).
IEP MPSO
Generator GA (pop= 20) EP (par= 20) CMA-ES rECGA
1 300 300.23 300.26 300.27 300.27 300.27
2 400 400 400 400 400 400
3 150 149.77 149.74 149.73 149.73 149.73
TP 850 850 850 850 850 850
TC 8237.6 8234.09 8234.07 8234.07 8234.07 8234.07
Table 9: Comparison of the results obtained by various methods on the nonsmooth cost function considering the valve point loading effect. For the 40-unit system, rECGA was able to find the best solution.
Minimum Cost Method
123,488.3 CEP
122,679.7 FEP
122,647.6 MFEP
122,624.35 IFEP
122,252.27 MPSO
122,160.19 CMA-ES
121,462.3591 rECGA
evaluations is 200,000. One hundred independent trials were conducted for each prob-lem to collect statistically significant results. The obtained results for the three-unit system are shown in Table 8 and are compared to those obtained by IEP (Park et al., 1998), EP (Yang et al., 1996), MPSO (Park et al., 2005), and CMA-ES (Hansen, 2006).1 The results for this small ED problem indicate that rECGA was able to find the global optimal solution determined by Lin et al. (2002).
In the case of the 40-unit system, the results are compared with those obtained by using other methods given by Sinha et al. (2003) such as classical EP (CEP), fast EP (FEP), modified FEP (MFEP), and improved FEP (IFEP). The results for MPSO provided in Park et al. (2005) and obtained by CMA-ES (Hansen, 2006) are also included. The minimum costs, that is, the best solutions, achieved by these methods are presented in Table 9.
We can see that the best solution delivered by rECGA is 121,462.3591, which is better than the known, published best solution, 122,252.27, presented by Park et al. (2005) and that obtained by CMA-ES, 122,160.19. For the purpose of access and verification, the generation outputs (the values of the decision variables) and the corresponding cost (the objective values) of the solution found by rECGA are given in Table 10.
Because of the stochastic nature of evolutionary computation methods, to avoid reporting the results of a “lucky shot,” comparison of the experimental results in a statistical manner has to be conducted. First of all, Table 11 shows the range of the results in the 100 trials obtained by CEP, FEP, MFEP, IFEP, MPSO, CMA-ES, and rECGA, where the listed results except for those of rECGA and CMA-ES are given elsewhere
1We downloaded the source code and conducted the experiments for CMA-ES.
Table 10: The generator outputs and the costs of the best solution obtained by rECGA.
Generator Pmin(MW) Pmax(MW) Output Cost
1 36 114 110.80098 925.11565
Total generation and total cost 10,500 121,462.3591
(Sinha et al., 2003; Park et al., 2005). As we observe from Table 11, the distribution of the rECGA results may be considered better than those obtained by the other methods.
Furthermore, to more carefully and accurately compare the performance of rECGA and MPSO (Park et al., 2005) on the 40-unit problem, the t-test was conducted for the statistical significance of the obtained experimental results. Since the actual results of the 100 trials for MPSO are not available, in order to get a fair performance comparison and capability assessment, we set up two conditions under which the t-test was conducted.
Table11:Comparisonofmethodsonrelativefrequencyofconvergenceinthecostrange. Rangeofcost 126.5–126.0–125.5–125.0–124.5–124.0–123.5–123.0–122.5–122.0–121.5–121.0– Method∞126.5126.0125.5125.0124.5124.0123.5123.0122.5122.0121.5 CEP104—16224242———— FEP6—42102026246——— MFEP—————14265010——— IFEP——2—44185022——— MPSO————————5347—— CMA-ES91612191924451—— rECGA—————————2971
Table 12: The t-test for the experimental results obtained by rECGA and MPSO under condition 1, where the rECGA dataset contains the actual results, and the MPSO dataset contains 47 values of 122,252.265 and 53 values of 122,750.
rECGA MPSO
mean 121,777.649963 122,516.06455 t-value 27.8068829451749
p-value 2.2645299161711E–55
Table 13: The t-test for the experimental results obtained by rECGA and MPSO under condition 2, where the rECGA dataset contains the actual results, and the MPSO dataset contains 47 values of 122,252.265 and 53 values of 122,500.
rECGA MPSO
mean 121,777.649963 122,383.56455
t-value 39.4214198098397
p-value 9.0857670116394E–91
Based on the data given in Table 11, the first condition is that the MPSO results contain 47 values of 122,252.265, which is the optimum reported for MPSO (Park et al., 2005), and 53 values of 122,750, which is the mean value of 122,500 and 123,000. Table 12 demonstrates the t-test results for condition 1. Given the p-value: 2.26× 10−55, which is smaller than the commonly used statistically significant levels, such as .05 (5%), .01 (1%), or .001 (0.1%), we can conclude that the performance of rECGA on the 40-unit ED problem is statistically significantly better than that of MPSO on the same problem. For condition 2, the MPSO results contain 47 values of 122,252.265, which is the optimum reported for MPSO (Park et al., 2005), and 53 values of 122,500, which is the best value in the range from 122,500 to 123,000. The t-test results under condition 2 are presented in Table 13. Due to the change of the standard deviation, the p-value becomes 9.09× 10-91. The small p-value prevents us from accepting the null hypothesis, which is interpreted as that the performance of rECGA and MPSO on the problem is equivalent.
According to the experimental results, we know that the proposed algorithm, rECGA (ECGA+SoD), performs well on the two ED problems. Particularly, for the 40-unit ED problem, we improved the known best solution from 122,252.265 (Park et al., 2005) to 121,462.3591. Moreover, from Tables 11, 12, and 13, we observe that rECGA statistically significantly outperformed MPSO on the 40-unit ED problem. Therefore, rECGA is capable of solving ED problems effectively.
7 Summary and Conclusions
In this study, we proposed an adaptive discretization method, SoD, to enable EDAs designed for handling discrete variables to tackle real-parameter optimization. SoD was described in detail with its procedure, effect, and usage. In order to show the utility of SoD, an ECGA was employed as an optimization engine, and SoD was used as a variable-type interface. By combining ECGA and SoD, the real-coded ECGA (rECGA) were applied to solve a set of benchmark functions and two ED problems. The results on benchmark functions indicated that SoD was better than two well-known discretization methods, FHH and FWH. The results on the ED problems demonstrated that rECGA successfully achieved the global optimal solution of the three-unit ED problem and
was able to obtain the solutions better than the known best solution obtained by other methods and reported in the literature for the 40-unit ED problem.
The outcome of this study indicates that it is not only possible but also practical to employ an optimization method designed for handling discrete variables to tackle prob-lems consisting of continuous variables, as long as an appropriate interface is adopted.
Although many researchers in the field of evolutionary computation do not consider variable-type transformation to be an issue, in practice, except for some limited cases, most algorithms designed for discrete variables do not perform well on continuous problems and vice versa. By comparing the real-coded ECGA to the algorithms specifi-cally designed for handling continuous variables, such as particle swarm optimization (MPSO), evolutionary programming (IFEP, MFEP, FEP, CEP), and evolution strategies (CMA-ES), this paper provides the experimental results to serve as the proof of prin-ciple for transforming the variable type while retaining the capability of the back end optimization algorithm.
Given the nature of discretization, although good results were obtained in this study, we believe that the proposed framework might not be suitable for applications that require very high precision. As we can see in Table 5 in Section 5.2, rECGA seems to perform as well as the state of the art method, CMA-ES. However, for the benchmark, since there is a predefined accuracy level for each function (1× 10−6,1× 10−2, and 1× 10−1), the experimental results in terms of the number of solved functions should be appropriately interpreted instead of being considered to demonstrate that rECGA can perform very well under any situations. For applications that do not require very high precision, such as the control of robots, evolutionary arts, or machine learning, the proposed discretization technique may come in handy for handling real-parameter problems with the discrete-type optimization algorithm that is already in use.
Finally, future work of this study includes applying rECGA to handle other impor-tant problems as well as developing different integrations of optimization algorithms and transforming techniques. Moreover, theoretical understandings for the quality of the transforming techniques, such as SoD, FHH, and FWH, as well as for the interaction between the engine and the interface, should also be considered.
Acknowledgments
The work was supported in part by the National Science Council of Taiwan under Grant NSC-96-2221-E-009-196. The authors are grateful to the National Center for High-Performance Computing for computer time and facilities.
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