Hybrid Bio-Inspired Computational Heuristic Paradigm for Integrated Load Dispatch Problems Involving Stochastic Wind

: In this research work, bio-inspired computational heuristic algorithms (BCHAs) integrated with active-set algorithms (ASA) were designed to study integrated economics load dispatch problems with valve point e ﬀ ects involving stochastic wind power. These BCHAs are developed through variants of genetic algorithms based on a di ﬀ erent set of routines for reproduction operators in order to make exploration and exploitation in the entire search space for ﬁnding the global optima, while the ASA is used for rapid local reﬁnements of the results. The designed schemes are estimated on di ﬀ erent load dispatch systems consisting of a combination of thermal generating units and wind power plants with and without valve point loading e ﬀ ects. The accuracy, convergence, robustness and complexity of the proposed schemes has been examined through comparative studies based on a su ﬃ ciently large number of independent trails and their statistical observations in terms of di ﬀ erent performance indices.


Introduction
Economic load dispatch (ELD) is a fundamental issue in power plant systems, design and analysis with the aim of optimal scheduling of generated power in order to satisfy the load demand by least probable cost, while however, fulfilling the constraints on power generators [1][2][3].Generally, the electricity generation cost with thermal power plants is excessively high and suitable planning is indeed needed to minimize the cost within reasonable levels.The ELD optimization problem is in one of the difficult constraints-based optimization systems in the power sector that usually needs excessive computations because of the nature of the cost functions and inherent non-smooth properties.A number of studies have introduced a variety of optimization procedures for ELD problems with and without valve point loading effect (VPLE) based on conventional and recently introduced meta-heuristics schemes, such as Newton methods [4,5], genetic algorithms [6], biogeography-based optimization algorithms [7], teaching learning based optimization methods [8], grey-wolf optimization algorithms [9], ant lion optimization procedures [10], modified krill herd algorithms [11,12], natural updated harmony searches [13], improved differential evolution [14], mine blast algorithms [15], and crow-search algorithms [16].
An additional aim in optimal load dispatch is to decrease or reduce the emissions that are dispersed due to the procedure of electricity generation.Normally, these environmental goals are conflicting with the economical nature of the systems, i.e., the decline in emission from generating units (GUs) results Energies 2019, 12, 2568 2 of 23 in the increased rate of electricity generation and vice versa.In such circumstances, multi-objective optimization techniques are exploited for combined ELD with emission problems, such as the symbiotic organisms search optimization method [17], simulated annealing algorithms [18], multi-objective evolutionary computing [19], multi-objective biogeography-based optimization [20], flower pollination algorithms [21], modulated particle swarm optimization [22] and chaotic bat algorithms [23].
The modern trend is to exploit the renewable energy assets for economical and unpolluted generation of electric power by incorporating the electricity generation scheme by use of wind power.The significant advantages of wind energy, besides the one-time initial cost of wind plants, are that there are no costs for production of power through wind, it is more environmentally friendly than thermal power plants and its ease in expendability i.e., installation of additional wind power generating units.There are some renewed applications of ELD involving wind energy, such as the binary artificial sheep method [24], integrated imperialist competitive with sequential quadratic programming [25], fuzzy adaptive artificial physics optimization [26], unit commitment problem involving wind power [27], multi-objective evolutionary algorithm [28] and group search optimizer with multiple producers [29].All these existing procedures have their own competency, importance, applications and drawbacks in terms of precision, stability, and computing requirements.The research community has growing interest to design, explore and exploit modern stochastic solvers by using the strength of artificial intelligence procedures for applications in the diversified field of applied science and engineering, e.g., solution of stiff optimization problems arising in nanotechnology, nonlinear optics, astrophysics, atomic physics, plasma physics, electromagnetics, fluid mechanics, electric machines, piezoelectric systems, fractional order systems, bioinformatics, signal processing, controls, economic and finance [30][31][32][33][34] along with references therein.Additionally, there are many applications in which evolutionary computing paradigms are exploited through variants of genetic algorithms (GAs) based on different set of routines in the reproduction mechanism [35,36].All of these are inspiring factors for authors to investigate in evolutionary stochastic paradigms for the solution of the emerging domain of energy and power sectors [37][38][39][40] including integrated power plant systems.As per our literature survey, evolutionary computing strategies based on variants of GAs have yet not been exploited in integrated power dispatch problems, therefore, the objective of the present study is to investigate integrated bio-inspired computational heuristic algorithms (BCHAs) based on the variants of GAs aid with the active-set algorithm (ASA) for optimization of load dispatch problems.
A brief summary of innovative contributions in terms of salient features of the proposed study are listed as:

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Global search strength of GAs and its variants is exploited for the design of BCHAs by using an appropriate set of routines for reproduction operators in order to make exploration of the entire search space supported with speedy local refinements with ASA.

•
The performance of the designed schemes is estimated on ELD, ELD-VPLE and ELD-VPLE-SW problems based on a combination of thermal and wind power generating units by means of accuracy, convergence and complexity operators based on the results of statistics for a sufficiently large number of independent trails.

•
The effective operation of BCHAs for integrated load dispatch scenarios, and other illustrative hallmarks for simplicity of the concept, coherent procedures with smooth implementation, robustness, expendability and stability.
The optimization procedure of BCHAs is described in Section 1; a brief overview of the system model of the integrated load dispatch system is presented in Section 2; the results with necessary interpretations are given in Section 3, while conclusions with future relevant studies are listed in Section 4.

System Model: Integrated Load Dispatch Problems
Three type of load dispatch problems are discussed in this study involving the no valve point loading effect (VPLE), with VPLE and VPLE involving stochastic wind power.
The fuel cost function for ELD with no VPLE: The total fuel cost of the power plant J 1 is modelled in this case with the help of the quadratic cost function and, it is given mathematically as: where N g represents the total number of the power plants, a i , b i , as well as, c i denote the fuel charge coefficients of ith power plant, and P i gives the current output power of the ith plant.
The fuel cost function for ELD involving VPLE: The total fuel cost of the power plant J 1 is normally modelled with the help of the quadratic term based cost function, while the valve-point effect is similarly measured through adding of the sinusoidal term.The total fuel cost function is written as follows: The coefficient e i and f i denote the fuel charge for the valve-point effect for the ith power plant.The fuel cost function for ELD with VPLE and stochastic wind power: There are numerous ways that describe the importance of the functioning and forecasting cost in the scheme comprising of both thermal generators, as well as, wind turbines.Subsequently, the instant wind speed is arbitrary at some specified time, therefore, the operator might overestimate or underestimate the wind power availability.The cost function for the wind power generator is given mathematically as: [25] J 2 = m j=1 WPCost dir.j + WPCost oe.j + WPCost ue.j , where, WPCost dir,j represents the direct cost for the generation of wind power from the jth unit in MWh, WPCost oe,j denotes the overestimation cost for the jth wind generator in MWh and WPCost ue,j is defined for the underestimation of the cost of jth wind turbine in MWh.The WPCost dir,j is directly related to the output wind power and it is given as: where, q j and w j are the constant of direct electrical energy generation and real power generated by the jth wind generator in MWh, respectively.Similarly, the WPCost oe,j can be presented as follows: C rwj denotes the charge constant for overestimation and underestimation of the jth wind generator in MW, while E(Y oe, j ) is the expected value of wind power overestimation and underestimation for the jth wind generator.
The E(Y oe, j ) is mathematically represented as follows: [ where, K j and C j are the shape and scale influence of Weibull distribution intended for the jth wind generator, respectively.The parameters, v r , v in and v out stand for wind speed, cut in and cut out speeds in m/s, respectively.An intermediate constant v 1 is defined as The wind turbine parameters w j and w r are representing the generated and rated power of the jth plant, respectively.Moreover, in (6), the symbol Γ with two parameters represent the incomplete gamma function as: while, the symbol Γ with a single parameter represents the standard gamma function as: Similarly, WPCost ue,j can be presented as follows: where, m denotes for number of wind generators, C pwj defines the cost constant of underestimation for the jth wind generator in MWh and E(Y ue, j ) represents as the estimated charge of wind underestimation intended for the jth wind generator, while E(Y ue, j ) is provided mathematically as follows [25]: Precisely, the cost function for integrated power plant systems is given as: WPCost dir.j + WPCost oe.j + WPCost ue.j , (10) where J 1 is given in Equation (2).Further necessary details of the system model for the interested readers can be seen in [25].Constraints: The entire power generation based on thermal and wind generators should be equal to P load line losses (P loss ) as follows: where N g represents the amount of power plants, m denotes the number of wind generators, P i describesthe power of the ith power plant.w j represents the generated power of the jth wind, P load defines the total load demand and P loss defines the line losses.
The losses of the transmission may be ignored for smaller transmission distance as well as for excessive load densities.However, in an enormous interrelated network wherever power is transferred above the extended distance through low load density regions, losses due to transmission are a foremost issue and distress the optimal dispatch.The mathematical relations of the losses are considered as follows: (12) where, B ij , B i0 , B 00 is defined as the line loss coefficient and N g represented the number of power plants.
The active power of for each power plant, as well as, wind generators must fulfil the following bounds: P i,min and P i,max are representing the maximum and minimum parameters of the ith power plant, respectively, while w j and w r,j denote the produced and rated power of the jth wind generator, respectively.Basically, the operational collection of the entire generators are restricted through their ramp rate confines.These limits are reflected as follows: P i and P i 0 represent the current and prior output of the ith power plant, respectively, while D i and U i define the down and up ramp rate limits, respectively.

Optimization Techniques
The optimization procedure in this study consists of two parts.In the first part, the design of the bio-inspired heuristic algorithms based on the variant of GAs through its reproduction operators is presented, along with an overview of ASA used for rapid local convergence of the results.While in the second part, the learning procedure of these optimization algorithms to three constrained ELD systems involving VPLE and wind power generators is presented.
GAs is a meta-heuristic algorithm for viable global search and introduced by Holland in the 1970s [41].GAs work through their fundamental operators of selection, crossover and mutation.These have been effectively utilized in diversified applications of constrained and unconstrained optimization problems with better control, stability, robustness and convergence.The workflow in terms of block structure for GAs is provided in Figure 1, while few potential applications of GAs in power sector can be seen in [42][43][44].The steady state optimization performance of GAs is speeded-up by the process of combination with the efficient local search method based on ASA.ASA is one of the best local search procedures for linear and nonlinear, constrained and unconstrained optimization problems.The standard working of ASA is to divide the original stiff problems to relatively non-stiff sub-problems and these sub-problems are solved with the ease of algorithms.The block structure form of the workflow of ASA is shown in Figure 1.ASA addresses effectively many optimization problems which include nonnegative matrix factorization problems [45], variation deblurring problems [46] and warehouse location problems [47].well as, wind GUs with VPLE are presented in this section.The nine variants of GA were applied initially and later on, all the results of these variants were given to ASA for further refinements.The load demand (PD) remained fixed at 10,500 MW for all three load dispatch problems.The maximum generators output powers Pmax, the minimum generators output powers Pmin and the cost coefficients in the case of 40 GUs are given in Appendix A [48].The parameter of wind GUs is given in Appendix B [25,49].

Cost function formulation:
The cost function for ELD problems with 40 GUs, i.e., Ng = 40, having quadratic cost function using Equation (1), is written as: The paramount importance of GAs and ASA has encouraged the use of memetic variants of GAs with ASA (GA-ASA) for integrated load dispatch problems.Nine different sequential computing paradigms, GA-ASA-1 to GA-ASA-9 are designed for optimization based on a different set of reproduction operators as provided in Table 1.The selection operator stochastic uniform, means that GAs move along the line in steps of equal size.The section operator reminder, means that the probability for the selection of the parent is proportional to the fractional part of its scaled value.The selection operator roulette, means that an individual is chosen randomly with a probability equal to their respective area.The crossover operator heuristic, means an offspring that lies on the line containing their parents.The crossover operator arithmetic, means the create/generate offspring that are the weighted arithmetic mean of their parents.The crossover operator scatter, means a random one point, two point or intermediate crossover between the genes of two parents to have new child.The mutation operator adaptive feasible, means randomly generated feasible directions according to the last known successful or unsuccessful generation.The workflow diagram of the proposed approach is presented in Figure 1.In this study, implementation of variants of GAs and ASA is made through the optimization toolbox of the software package, Matlab with the help of ga, gaoptimset, fmincon and optimset routines.All three load dispatch problems are solved by these functions with appropriate settings of the parameters.The pseudocode of ASA is given in Algorithm 1.

Inputs:
The best individual of nine variants of GAs for each ELD, ELD-VPLE and ELD-VPLE-SW in the case involving 40 generation units.Mathematically represented as: The refined weights by ASA represented as: Initialize the values of random assignments, constraints and parameters of the ASA.

Refinements
Refine the values of the decision variables at each iteration with ASA using the fmincong routine with algorithm active-set in the MATLAB optimization toolbox.

End Storage
Store the values of decision variables for ELD, ELD-VPLE and ELD-VPLE-SW along with their costs, time, function count for current execution of ASA.Statistics: Repeat the steps from initialization to storage for 100 trials for all nine variants of GA, i.e., GA-1 to GA-9 to generate a dataset of GA-ASA-1 to GA-ASA-9 results for comparative analysis of performance.
Table 1.The functions invoke to design the variants of the proposed optimization solvers based on genetic algorithms (GAs) supported with the active-set algorithm (ASA).

Results and Discussion
The numerical experimentation of the all nine design schemes for three load dispatch problems based on 40 generation units (40-GUs) involving no VPLE, with VPLE and combined thermal, as well as, wind GUs with VPLE are presented in this section.The nine variants of GA were applied initially and later on, all the results of these variants were given to ASA for further refinements.The load demand (PD) remained fixed at 10,500 MW for all three load dispatch problems.The maximum generators output powers Pmax, the minimum generators output powers Pmin and the cost coefficients in the case of 40 GUs are given in Appendix A [48].The parameter of wind GUs is given in Appendix B [25,49].
Cost function formulation: The cost function for ELD problems with 40 GUs, i.e., N g = 40, having quadratic cost function using Equation (1), is written as: where the values of Pmin, Pmax and cost coefficients vectors a, b and c are given in Table A1 of Appendix A. The constraints associated with the problem are written as: Similarly, the cost function for ELD problems with VPLE (ELD-VPLE) for 40 GUs, i.e., N g = 40, is written as: where the values of P min , P max along with cost coefficients vectors a, b, c, e and f are given in Table A1 of Appendix A. The constraints associated with the problem are given in Equation ( 16).
The cost function for ELD problems involving VPLE by considering stochastic wind availability (ELD-VPLE-SW), in the case of 3 wind GUs, i.e., m = 3, and 37 thermal GUs, Ng = 37, are given as: . sin( f i (P i.min − P i )) WPCost dir.j + WPCost oe.j + WPCost ue.j (18) where the values of P min , P max and cost coefficients vectors a, b, c, e and f are given in Table A1 of Appendix A, while the parameter of wind generating units is given in Table A2 of Appendix B. The constraints associated with the problem are given in Equation ( 16).
The design nine variants of GA were applied to solve ELD, ELD-VPLE and ELD-VPLE-SW problems using the cost function given in Equations ( 15), ( 17) and (18), respectively, while satisfying the constraints given in Equation (16).The learning curves of GA-1 along its fitness value and output power are shown graphically in Figure 2a,c,e for ELD, ELD-VPLE and ELD-VPLE-SW problems, respectively.The global best weights of GA-1 for all three load dispatch problems are given to ASA for further refinements and respective results of GA-ASA-1 in Figure 2b,d,f.It can be seen that by the process of combination, a significant improvement in the values of the cost function was observed for all three load dispatch problems.Accordingly, the results of all nine variants of GA and GA-ASA were determined.The results of in terms of costs, time consumed, generation (Gen) executed, and fitness function evaluated (FE) are given in Table 2 for ELD, ELD-VPLE and ELD-VPLE-SW problems, while the results of output power Pi of GA and GA-ASA for all three load dispatch problems ELD, ELD-VPLE and ELD-VPLE-SW are listed in Tables A3-A5, respectively, of Appendix C. For ELD problems without considering VPLE, the minimum cost was achieved by GA-4 and the worst cost was achieved by GA-8, while no noticeable difference in time, generation (Gen) and function evaluated (FE) were observed (see data presented in Table 2).However, in the process of sequential computing the nine variants of GA-ASA, all nine algorithms converged to the same minimum cost.This is understandable given the ELD problems based on smooth/convex cost functions with unique local minima (see the results listed in Table 2).Apart from the comparison of nine variants of GA and GA-ASA with each other, a detailed analysis of the proposed results in both cases of ELD-VPLE for 40GUs system and integrated power plant systems, i.e., ELD-VPLE-SW for 40GUs system with 3 wind units, is made with reported results.In the case of ELD-VPLE-SW, the results of reported solutions with Hybrid imperialist competitive-sequential quadratic programming (HIC-SQP) [25], PWTED1 [50], DWTED1 [50] and best compromise [50] are listed in Table 3.The best results were reported in HIC-SQP [25] for ELD-VPLE-SW based on the integrated load dispatch problem.Similarly, was the case for ELD-VPLE reported solutions for evolutionary programming aided with sequential quadratic Apart from the comparison of nine variants of GA and GA-ASA with each other, a detailed analysis of the proposed results in both cases of ELD-VPLE for 40GUs system and integrated power plant systems, i.e., ELD-VPLE-SW for 40GUs system with 3 wind units, is made with reported Energies 2019, 12, 2568 10 of 23 results.In the case of ELD-VPLE-SW, the results of reported solutions with Hybrid imperialist competitive-sequential quadratic programming (HIC-SQP) [25], PWTED1 [50], DWTED1 [50] and best compromise [50] are listed in Table 3.The best results were reported in HIC-SQP [25] for ELD-VPLE-SW based on the integrated load dispatch problem.Similarly, was the case for ELD-VPLE reported solutions for evolutionary programming aided with sequential quadratic programming (EP-SQP) [51], HIC-SQP [25], ant colony optimization (ACO) [52], biogeography-based optimization (BBO) [53], differential evolution aided with BBO (DE-BBO) [53], bacterial foraging optimization combined with Nelder-Mead (BF-NM) [54], new particle swarm optimization supported with local random searches (NPSO-LRS) [55] and real coded genetic algorithms (RCGA) [56].The minimum cost achieved by HIC-SQP, PWTED1 and DWTED1 methods are listed in Table 3 for ELD-VPLE-SW, while the minimum costs of ES-SQP, HIC-SQP, ACO, BBO, DE-BBO, BF-NM, NPSO-LRS and RCGA for ELD-VPLE problems are also presented in Table 3.In the case of ELD-VPLE, the reported and our sequential computing algorithms have close resemblance with standard solutions, however none of the variants of GAs, GA-1 to GA-9 and their memetic computing techniques, i.e., GA-ASA-1 to GA-ASA-9, give the best solution reported so far for ELD-VPLE problems.Whereas, the significance of the proposed algorithms was evidently seen in the case of ELD-VPLE-SW problems based on the stiff cost function as defined in Equation ( 18), involving the calculation of incomplete gamma functions for each evaluation of the objective function.For example, the result achieved by the integrated computing approach GA-ASA-2 was 127,345.345$/h,which was better than the best reported optimization solutions for ELD-VPLE-SW problems in [25].Additionally, it was observed that the proposed results of all nine variants of GA were poorer than in the reported results.However, after performance with ASA, the results of all nine variants of GA-ASA improved considerably, and even better than the reported results of recently applied algorithms based on HIC-SQP, PWTEDI, DWTEDI and best compromise.The analysis on multiple run of algorithms: The performance analysis on the basis of multiple runs for all nine variants of GA and GA-ASA were carried out to solve the optimization problems based on ELD, ELD-VPLE and ELD-VPLE-SW systems which consisted of 40 Gus, including both thermal and wind power plants.
The analysis on the precision and reliability were performed through a hundred independent trails of each variant of GA and GA-ASAs in order to optimize all three load dispatch problems.The results for GA-1 and GA-ASA-1 in terms of best cost against the number of runs of the algorithms for ELD, ELD-VPLE and ELD-VPLE-SW systems are shown in Figure 3a,b, respectively, while the histogram plots of GA-1 to solve the ELD, ELD-VPLE, and ELD-VPLE-SW are shown in Figure 3c-e, respectively, and respective histogram plots for GA-ASA-1 algorithms are plotted in Figure 3f-h.Accordingly, the best cost against number of runs along with their histogram studies were conducted for all three load dispatch problems for GA-2 and GA-3 as well as GA-ASA-2 and GA-ASA-3.Similarly, the results of the cost against the number of runs are shown in Figure 4 for GA-4 and GA-ASA-4, while in Figure 5 for GA-8 and GA-ASA-8.From Figure 3, a small variation in the values of GA-1 was observed for all the load dispatch models while such small oscillations were also evident in solving ELD-VPLE and ELD-VPLE-SW systems by GA-ASA-1.However, no variations were seen in ELD problems optimized with GA-ASA-1.The results in Figure 3 also showed that the same trend of GA-1 and GA-ASA-1 were followed by GA-2 to GA-3 and GA-ASA-2 to GA-ASA-3, respectively, for all three ELD, ELD-VPLE, ELD-VPLE-SW power generation systems.Accordingly, the similar behavior of the results is evidently seen from the rest of the illustrations presented in Figures 4 and 5.
Complexity Analysis: The complexity analysis for all nine variants of GA and GA-ASAs are presented in terms of time consume, generation (Gen)/iteration executed and cost function evaluated /counted (FCs) for all three load dispatch systems.The result of complexity operators based on the mean along with its standard deviations (STD) magnitudes are presented in Table 4 for ELD, ELD-VPLE, ELD-VPLE-SW power generating systems in each case of GA and GA-ASAs.Regarding the ELD problem without VPLE, cost, time, Gen and FCs were 147,196 ± 4373, 92 ± 9, 174 ± 14 and 27,126 ± 6754 for GA, while for GA-ASA values of cost, time, Gen, FCs were 118,660 ± 0, 94 ± 12, 196 ± 7, and 28,036 ± 144.The cost, time, Gen and FCs were 152,000 ± 4000, 94 ± 10, 199 ± 20 and 29,000 ± 4000 for GA for ELD by considering VPLE, while for GA-ASA values of cost, time, Gen, FCs were 125,000 ± 1000, 95 ± 9, 428 ± 91 and 47,534 ± 7378.For ELD problems based on VPLE-SW, cost, time, Gen and FCs were 163,788.43 ± 4772.70, 98.37 ± 10.65, 173.60 ± 41.13 and 26,190.00± 222.00 for GA, while for GA-ASA the values of cost, time, Gen, FCs were 129,676.74± 897.77, 110.05 ± 11.23, 424.41 ± 83.55, and 47,214.99± 7472.99.The time based complexity analysis of the proposed variants GA-1 to GA-9, as well as, GA-ASA-1 to GA-ASA-9 is dependent on the specification of the machine on which optimization algorithms are executed.Thus, for better processing platforms, the computing time of optimization of the decision variable is reduced and vice versa.Similarly, varied computational requirement are associated with single generation/cycle of meta-heuristic paradigm based on GAs.Therefore, generations/iterations are also not effective for measurement of the complexity.To overcome these issues, the number of fitness function evaluated during the process of optimization of decision variables has been used as a measure for the analysis of the complexity which is a machine independent gauge.The complexity of the variants is given on the basis of time, iterations and FCs in the current study.These values are used for comparison whenever the same problems are addressed with counterpart meta-heuristic methodologies.The reported values of complexity in terms of time in seconds for the execution of a single generation/iteration were given 0.0597 for HIC-SQP [25] for ELD-VPLE-SW problems based on 37 thermal and three wind turbines based generated units.The CPU time per iteration of HIC-SQP [25] was better than reported PWTED1 [50], DWTED1 [50] and best compromise [50].The similarly calculated values of complexity measure of proposed variants also provided the consumed time in close vicinity of the reported results.

Conclusions
The conclusions are summarized as follows: • Bio-inspired computational heuristics is exploited for solving effectively the integrated power plants systems based on thermal and wind generating units.The proposed BCHAs were based

Conclusions
The conclusions are summarized as follows: • Bio-inspired computational heuristics is exploited for solving effectively the integrated power plants systems based on thermal and wind generating units.The proposed BCHAs were based on nine variants of GAs and were designed by using different sets of reproduction operators and each global search method was aided with ASA for rapid local convergence.

•
The performance of proposed schemes was examined for solving ELD, ELD-VPLE, ELD-VPLE-SW problems based on 40 generating units with a fix load demand of 10,500 MW.It was found that all nine integrated approaches were viable solvers with reasonable accuracy.Additionally, sequential computing schemes gave relatively better results than standalone approaches.

•
Regarding the smooth cost function based ELD problem, there was no difference in the performance for each integrated methodology, while in the variants of Gas, the best cost was achieved using GA-4, i.e., 131,173.36$/hr,while in case of GA-ASA, the best minimum cost was achieved through GA-9, i.e., 118,600$/hr.

•
Regarding the non-convex cost function based ELD-VPLE problem, the minimum cost was achieved using GA-ASA-1, i.e., 137,359$/hr., while the results of combined optimization approaches were relatively better than GA variants.However the best minimum cost was achieved by standalone GA-6 is 122,175$/hr.

•
The scenario of the integrated power plant system was represented with ELD-VPLE-SW.The most effective optimization solver was GA-1 in terms of accuracy and convergence for the standalone scheme, while for sequential computing schemes, the results of GA-ASA-8 were found to be superior.

•
The validation and verification for the performance of each optimization solver was established from 100 independent trails to solve all three load dispatch problems by using the detailed analysis through statistical operators, convergence curves, as well as, histogram illustrations.
Some potential research directions are briefly narrated as:

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The presented nine variant of GAs aided ASA, can be a good alternative to be explored or exploited in future for the unit commitment problem in the energy sector.

•
The application of proposed optimization algorithms can be explored for a variety of integrated load dispatch problems based on wind, solar, hydel, and biomass, generating units for dynamic and static requirements.

•
The newly introduced optimization solvers, including fractional order partial swarm optimization algorithms, fireworks algorithms, moth-flame algorithms, backtracking search optimization algorithms and differential search optimization algorithms can give quality solutions for problems arising in integrated power plant systems.

Appendix A
The parameters of thermal 40GUs based load dispatch systems are provided in Table A1.

Appendix B
The parameters of wind based generating units are provided in Table A2.
Table A2.The parameters of wind based generating units.

Appendix C
The results in terms of power output of GA and GA-ASA for ELD, ELD-VPLE and ELD-VPLE-SW problems are provided in Tables A3-A5, respectively.

Figure 1 .
Figure 1.Graphical overview of the proposed design schemes for solving integrated economic load dispatch problems involving stochastic wind.

Figure 1 .
Figure 1.Graphical overview of the proposed design schemes for solving integrated economic load dispatch problems involving stochastic wind.
Energies 2019, 12, 2568 9 of 23 (a) The results of GA-1 with no VPLE (b) The results of GA-ASA-1 with no VPLE (c) The results of GA-1 with VPLE (d) The results of GA-ASA-1 with VPLE (e) The results of GA-1 for VPLE-SW (f) The results of GA-ASA-1 for VPLE-SW

Figure 2 .
Figure 2. The adaptation of parameters of ELD systems based on 40 generating unit without considering VPLE, with considering VPLE and VPLE-SW.

Figure 2 .
Figure 2. The adaptation of parameters of ELD systems based on 40 generating unit without considering VPLE, with considering VPLE and VPLE-SW.

1 Figure 3 .
Figure 3.The comparison of results on the basis of 100 independent runs of GA-1 and GA-ASA-1 for all three load dispatch problems.

Figure 3 . 23 (Figure 4 .
Figure 3.The comparison of results on the basis of 100 independent runs of GA-1 and GA-ASA-1 for all three load dispatch problems.

Figure 4 . 8 Figure 5 .
Figure 4.The comparison of results on the basis of 100 independent runs of GA-4 and GA-ASA-4 for all three load dispatch problems.

Figure 5 .
Figure 5.The comparison of results on the basis of 100 independent runs of GA-8 and GA-ASA-8 for all three load dispatch problems.

Table 2 .
The results of GA and GA-ASA for ELD problems based on 40 GUs without considering VPLE, considering VPLE and integration of wind units.

Table 3 .
The comparison of reported solutions in case of 40 thermal generating without VPLE and with VPLE.

Table 4 .
The comparison of results for ELD problems without considering VPLE, with considering VPLE and VPLE-SW through statistical performance indices.

Table A1 .
The parameters of 40GUs system based ELD problems.

Table A3 .
The results in terms of power output of GA and GA-ASA for ELD problems based on 40 GUs without considering VPLE.

Table A4 .
The results in terms of power output of GA and GA-ASA for ELD problem based on 40 Gus by considering VPLE.

Table A5 .
Results in terms of power output of GA and GA-ASA for ELD problem based on 40 GUs by considering VPLE-SW.