A chaotic Jaya algorithm for environmental economic dispatch incorporating wind and solar power

1.   Introduction

The need for electrical energy is growing day by day with industrial growth, and it will continue to increase due to widespread industrial growth. The electricity sector is still dominated by thermal power, in particular fossil fuels such as coal, natural gas and petroleum. They are considered the main sources of harmful pollution, and they have gained much attention in the last few decades. By considering new regulations for excessively generated greenhouse gases, a combination of economic dispatch and constraints on emission has come into existence; it is called economic emission dispatch (EED). EED is a multi-objective optimization problem of power systems, where in addition to the minimization of power generation cost, the minimization of emission is also considered simultaneously. The complicated operational constraints related to the EED problem, such as the valve-point loading effects, ramp-rate limits and prohibited operating zones (POZ) make the formulation highly nonlinear, discontinuous and non-convex. The main idea of EED is to find out the best compromise solution between two objectives i.e., cost and emission. The EED problem can be solved either by considering emission as a weighted function in the objective function ^[1,2,3] or by considering emission as a constraint ^[4]. Combined EED (CEED) is another method in which the coefficient of the price penalty factor is multiplied by the emission part of the objective function ^[5,6,7]. Also, a multi-objective optimization problem can be solved by converting it into a single-objective optimization problem using the weighted sum approach. The best part of the weighted sum method is that sets of Pareto-optimal solutions can be obtained by varying the weight ^[3].

As per the literature, deterministic approaches are not found to be suitable for dealing with large-scale integrated power systems. These methods are found to be associated with the inability to escape the local minima ^[8]. Therefore, researchers have turned toward nature-inspired optimization (NIO) methods due to their ability to find near-optimal solutions more efficiently. NIO methods knit together five categories of optimization approaches, i.e., evolutionary optimization, swarm intelligence-based optimization, ecology-based optimization, physical science-based optimization and optimization methods inspired by human intelligence ^[9].

Particle swarm optimization (PSO) is a well-accepted algorithm that belongs to the family of swarm intelligence algorithms due to its easy implementation, simplicity, fast convergence and robustness. However, PSO is very sensitive to its control parameters. A fuzzy section mechanism has been implemented and utilized for the solution to the multi objective economic dispatch (MOED) problem in ^[10]. To control the inertia weight an annealing reduction technique was implemented in ^[11]. The gravitational search algorithm (GSA) is based on Newton's law of gravity. It is a memory-less algorithm that can accelerate the optimization process without sacrificing accuracy. Obtaining the solution to multi-dimensional CEED problems by using a GSA is discussed in ^[5]. The harmony search algorithm (HSA) is a derivative-free optimization method inspired by the music improvisation of the musicians. In ^[6], the chaotic patterns and virtual memory concepts are utilized for solving the CEED problems; this modification is found to be highly efficient. The sine cosine algorithm is a population-based optimization method. It uses a mathematical model to create multiple initial random candidate solutions and requires them to fluctuate toward the best solution by utilizing sine and cosine functions. It has been applied and tested on CEED problems and found to be fast and efficient ^[7].

Literature shows that with the hybridization of two methods, proper balance between exploration and exploitation can be possible and will lead to improved performance.

Differential evolution (DE) is a heuristic method that improves candidate solutions over several generations by using three operations, i.e., mutation, crossover and selection, to reach an optimal solution. DE is found to give better solutions while satisfying all operational constraints for multimodal non-convex EED problems ^[12]. However, DE is unable to map its unknown variables efficiently when the complexity and size of the system increase. In the initial phase, the solution moves toward its optima in a faster manner; however, in a later stage, it requires fine-tuning. To achieve a proper balance between exploration and exploitation, a hybrid DE/biogeography-based optimization (BBO) method ^[13] has been used; it utilizes the migration operator of BBO, along with the three operators of DE, to find better convergence and solution quality. Similarly, by simultaneously updating the particle velocity in PSO and the acceleration coefficient of the GSA, improved performance was achieved ^[14]. By applying the time-variable acceleration coefficient in PSO to explore the entire search space and a local version of DE to the exploitation phase a hybrid DE-PSO algorithm ^[15] was used to obtain the feasible solution in a fast and efficient manner for a multi-objective economic dispatch (MOED) constraint optimization problem.

Rising power demand with minimum pollution constraints can be achieved through the integration of renewable energy sources (RESs) in the existing power network. However, the integration of RESs further complicates the problem due to their stochastic nature. RESs such as wind and solar power have maximum power generation limits that are variable and change with time. The uncertainty associated with RESs is a serious factor that must be considered for power generation planning for a longer time frame. The uncertainty function associated with RESs can be modeled by using the beta, log-normal, or Weibull probability distribution function (PDF). The solution to the EED problem for wind-thermal systems, as obtained by using a Weibull PDF is presented in ^[9,16,17]. Recently, in ^[18,19,20] solutions for EED were presented as a result of incorporating solar power. The binary flower pollination algorithm ^[18] has been applied to solve the CEED problem by incorporating solar power. Risk probability concepts were utilized to attain a better solar share of photovoltaic units and this reduced the total cost of the hybrid system. Impact analysis with a focus on the total operating cost and reduction in emission level of a solar-wind-thermal system was carried out by using a hybrid teaching-learning-based optimization (TLBO)-PSO algorithm in ^[19]. A new constrained multi-objective extremal optimization algorithm that has advanced constrained handling capability was proposed for the solution of the EED problem incorporating variable wind and solar power ^[20]. The lognormal PDF for calculating solar power, as well as the Weibull PDF for the calculation of wind power are utilized here.

Dynamic EED (DEED) is an extension of the EED problem, where the scheduling of committed generator units is carried out over the scheduled time period. Here, the ramping constraints of generators are also taken into consideration ^[3]. It is much more complex to solve than the classical EED problem due to the application of much more variable and operational constraints. In 2006, a PSO-based goal attainment method was used to solve the five-unit DEED problem. The multi-objective problem has been converted to single objective optimization goal attainment and then solved by using PSO ^[21]. PSO with avoidance of worst locations (AWL) and gradually increasing directed neighborhoods (GIDN) has been used to solve the 10-unit DEED problem. In the aforementioned study, a weighted sum approach was applied to convert the multi-objective problem into a single objective problem. The simulation results demonstrated that the performance of PSO with GIDN topology and AWL performs best ^[22]. The non-dominated sorting genetic algorithm II (NSGA II) was applied to solve 10 DEED problems with non-smooth cost and emission functions ^[23]. In their study, the NSGA simulation results were found to be better than those for the classical approach. To improve the computational efficiency of the bacterial foraging algorithm (BFA), a BFA with a crossover operation and parameter automation strategy has been used to solve the 10-unit DEED problem ^[24]. Finally, the fuzzy selection mechanism was adopted to find non-dominating solutions. In this previous study, the simulation results show that an improved BFA performs better than the classical BSA and NSGA-II. To avoid entrapment in local optima, TLBO phase angle-based mechanisms have been proposed and applied to solve 5-, 10- and a large-scale 120-unit DEED problem ^[25]. Simulation results demonstrate that ϴ-TLBO was able to provide high-quality well-distributed solutions in a single run. Variants of DE like multi-objective neural networks evolved with DE ^[26] to generate the Pareto front and the efficient fitness-based DE ^[27] which has a double mutation strategy, random mutation factor and crossover rate with learning ability have been proposed to solve DEED problem. In ^[28], a new enhanced harmony search algorithm was used to solve the DEED problem that utilizes (i) three arbitrary distance bandwidths to enhance global and local search capability and (ii) consideration of both the best and worst memory vectors in the second half of generation to enhance solution quality and avoid premature convergence.

Considering the ever-increasing power demand, fossil fuel costs and environmental legislation (e.g., Kyoto protocol) have forced expansion of the use of RESs. Hence, hybrid power generating systems come into existence. For this, a collective cost function of conventional thermal power generators with RESs and emission function needs to be investigated to analyze its impact on environmental and economic factors. The power generation by RES like wind and solar both is uncertain and variable. Therefore, direct cost, overestimation and underestimation costs are considered in the modeling. However, the uncertainty of RES leads to more complications in the formulation of the DEED problem ^[29,30,31]. A hybrid flower pollination algorithm that combines flower pollination algorithm (FPA) and DE is used to solve the DEED problem of a six-unit wind-thermal system ^[29]. Here fuzzy selection was used to find better trade-off solutions. Whale optimization algorithm (WOA) which is inspired by the hunting strategy of humpback whales ^[30], DE with ensemble selection method ^[31] was used to handle the DEED problem with wind integration. Substantial saving in cost and emission is reported in ^[32] using electric vehicles and a multi-objective evolutionary approach known as the exchange market algorithm. The membrane optimization algorithm is employed for the solution of the combined cost-emission optimization problem and produces Pareto solutions and recommendations for the best solution which is superior to reported results ^[33]. In reference ^[34] Equilibrium optimization was used for profit maximization as well as reduction in pollution content. It was tested on a hybrid thermal-wind-PV system.

PV systems and wind turbines both are dependent on climate change and hence neither system is capable of delivering enough electricity reliably and efficiently. However, a Battery storage system with its integration in a suitable size helps to improve power quality, suppresses power due to renewable energy resources and also helps to reduce the mean cost of energy ^[35,36].

According to the no-free-lunch (NFL) theorem, no algorithm can solve all types of optimization problems and there is still a chance to get a better solution by a new algorithm ^[37]. Also, literature shows that chaotic sequences with heuristic optimization have been used together to get improved performance ^[38,39].

Mostly the reported methods used to solve CEED/DEED problems have some limitations as trapping to local optima, slow convergence and complexity due to more control parameters. JAYA algorithm is selected due to the fewer control parameters to tune and easier implementation. Chaotic map with JAYA is used to avoid trapping to local optima. The main contribution in this work is summarized as follows:

• An analytical objective function model is developed for a hybrid thermal-wind-solar system. It includes the collective cost function of three types of power generating units, operational constraints, uncertainty of wind, solar system and emission function due to thermal units.

• Population-based JAYA algorithm and JAYA algorithm embedded with a chaotic map are implemented to investigate the hybrid thermal-wind-solar for CEED and DEED problems.

• The impact of wind integration, with both solar and wind integration is investigated with a 10-thermal unit non-convex system and analyzed for single and bi-objective optimization under fixed load demand.

• The competence and robustness of the proposed methodology are confirmed with the reported results.

• The impact of wind integration, both solar and wind integration is investigated with 10-thermal units analyzed for single and bi-objective optimization for 24 hours with 24 intervals of 1 hour each.

Problem formulation is presented in Section 2. A brief introduction of the JAYA algorithm, chaos maps and step-by-step implementation process of the Ch-JAYA algorithm for the solution of the EED problem are discussed in section 3. The simulation results and discussions are presented in section 4 and finally, concluding remarks are drawn in section 5.

2.   The mathematical formulation of the DEED problem

The objective of the DEED problem is to find out the optimal generation schedule over the period in such a manner that costs associated with power generation and emission are minimized simultaneously. The total cost (TC) of power generation can be symbolically represented by,

The first part of Eq (1) represents the fuel cost of the thermal power generating unit, and it is given by,

The second part of Eq (1) is the cost due to wind power. The system operator has to deal with either for deficit or more than scheduled power generation by wind farms due to the stochastic nature of wind power. The deficit in wind power can be fulfilled by maintaining a sufficient amount of spinning reserve (SR) and it is considered as an overestimation and the cost corresponding to SR is added to power generation. On the other hand, the generation of more power than scheduled power by wind farm system operators has to bear the penalty called underestimation cost. Therefore, wind power generation includes three costs: direct cost, overestimation cost/reserve cost and underestimation cost/penalty cost ^{[3,20,29,30,31]}.

Reserve cost/overestimation cost of wind power is given by,

The penalty cost/underestimation cost of wind power is given by,

In this work, the Weibull PDF is used for wind speed distribution as the wind speed is uncertain and irregular. The Weibull PDF is represented by,

The corresponding cumulative distribution function (CDF) can be represented by,

For each wind power generating unit, the power output at a given wing speed can be expressed by using ^[29,30,37],

The probability of wind power is 0 to ${\mathrm{P}}_{\mathrm{w}𝓇}$, and it can be calculated by using,

Wind power in the range ${v}_{in} < v < {v}_{r}$ is given by ^[37],

where

A typical Weibull PDF with a shape factor of 2 and scale factors of 5 and 10 is shown in Figure 1. The third part of Eq (1) represents the cost associated with solar power. It also has three cost components including direct cost, reserve cost and penalty cost ^[40].

Reserve cost associated with solar power generation that is derived from the overestimation of solar power and it can be represented by using ^[40],

Penalty cost associated with solar power generation is derived from the underestimation of solar power and it can be represented by using ^[40],

The solar irradiation (${G}_{pv}$) to energy conversion function of solar PV generators can be represented as ^[40],

The output of a solar power plant depends on irradiation at a particular location which can be modeled by Beta, Weibull, or Lognormal distribution. Here, Weibull PDF is used and it is represented by using ^[40],

The cumulative distribution function (CDF) of Eq (18) can be represented by using,

As per the transformation of the random variable, linear transformation is carried out with solar irradiation (${G}_{pv})$ random variable, and it can be represented by using ^[40],

Solar power probability for the piecewise function can be represented by using,

where

The second-order transformation is accomplished with solar irradiation (${G}_{s})$, and it can be represented by using,

The total emission (TE) from various pollutants can be symbolically represented as follows:

minimize

where

The multi-objective optimization problem was converted into a single-objective optimization problem using the weighted sum approach ^[3]. The objective of the DEED problem can be written as,

minimize

Subjected to the following operational constraints.

2.1.   Operational constraints

These constraints are expressed as,

2.2.   Ranking approach

To aggregate two conflicting objectives (cost and emission), the fuzzy-min ranking method is used. Linear membership function ${\mu }_{i, r}({i}^{th}$ solution of ${r}^{th}$ objective function) is described for each objective function ${F}_{i}$ in Eq (36) and also in Figure 2.

For ${i}^{th}$ solution, the rank is defined as ^[24],

The solution with maximum membership value (${\mu }_{r})$ is considered as the best compromise solution (BCS).

3.   Chaotic JAYA algorithm

3.1.   Jaya algorithm

The Jaya algorithm is one of the simple and powerful optimization methods proposed by Rao ^[41]. The basic idea behind the Jaya algorithm is to obtain a solution for a specified optimization problem that avoids the worst solution and moves toward the best one. It is a population-based evolutionary algorithm and does not require any algorithm-specific parameter to tune for its convergence.

Let us consider an objective function $f\left(X\right)$, where X is a d-dimensional variable and the population size is p. Let the best and worst values of the objective function produced by the candidate solution be $f\left({X}_{best}\right)$ and $f\left({X}_{worst}\right)$, respectively ^[39]. Then, the ${j}^{th}$ element of the ${i}^{th}$ solution is updated by using,

where ${𝓇}_{1, ij}$ and ${𝓇}_{2, i, j}$ are the two random numbers in the range [0, 1].

The second term of Eq (38) helps to move solutions towards the best solution and the third term helps to escape away from the worst solution. All the improved objective function values at the end of every iteration are transferred to successive iterations. Hence, the algorithm can achieve victory by attending to the global best solution. This process carries forward the victorious members of the population through the iterations. Therefore, the algorithm has been named as Jaya which means victory in the Sanskrit and Hindi languages.

3.2.   Chaotic JAYA algorithm

The search process of the Jaya algorithm is governed by the two uniformly distributed random numbers ${𝓇}_{1, ij}$ and ${𝓇}_{2, i, j}.$ The Jaya algorithm was found to saturate prematurely for practical real-life problems where the objective functions have non-convex and discontinuous nature and there are probabilistic variables. From the literature, it was found that the random numbers generated using chaotic sequences enhance the population diversity and the global search capability of evolutionary algorithms ^[6,24,36,37] thus avoiding convergence to the local optimum solution. Different chaotic maps were used to generate a sequence of chaotic random numbers introduced to replace two random numbers. The ${j}^{th}$ element of the ${i}^{th}$ modified solution vector in the ${(n+1)}^{th}$ iteration will be computed by using

where ${Ch}^{n+1}$ is the random number generated by using a chaotic map as explained in the next section. The solution strategy used for the optimization process using the Chaotic Jaya algorithm is presented using the flowchart in Figure 3.

3.3.   Chaotic map embedded with JAYA algorithm

Chaos theory is a branch of mathematics that deals with nonlinear dynamic systems and chaos systems are found to be highly sensitive to the initial condition. Chaos helps to improve the performance of population-based metaheuristic algorithms. The ten chaos maps ^[36,37] which are embedded with the Jaya algorithm are listed below. For n = 1, ${\mathcal{X}}_{n} = rand$.

3.1.1.   Chebyshev map

3.1.2.   Circle map

3.1.3.   Gauss/mouse map

3.1.4.   Iterative map

here, the range of the map is (-1, 1).

3.1.5.   Logistic map

3.1.6.   Piecewise map

here $\mathcal{P}$ is the control parameter considered as 0.4.

3.1.7.   Sine map

3.1.8.   Singer map

3.1.9.   Sinusoidal map

3.1.10.   Tent map

3.4.   Implementation of chaotic JAYA algorithm for environmental economic dispatch

To verify the effectiveness of the Ch-JAYA algorithm for the solution of the EED problem, first define the experimental data like the number of power generating units as the dimension of the problem, cost coefficients, emission cost coefficients, min-max limit of power generating units, operational constriction, population size and maximum iteration as stopping criteria.

Step 1: Initialize the population randomly within upper and lower power generation limits as below:

where $𝓇\in \left[\mathrm{0, 1}\right]$ is a random number and $N$ is the number of power-generating units.

Step 2: Calculate the total cost for each candidate solution using Eq (1), check for all associated operational constraints by using the Eqs (31−35), identify the best and worst solutions and preserve them.

Step 3: This process indicates the modification process of the algorithm. Modification of each power generating unit has been carried out as per Eq (39).

This step helps to move a solution towards the best solution and away from the worst one. The chaotic term $Ch,$ acts as a scaling factor to ensure good diversification during the optimization process. The new best solution ${P}_{best}^{new}$ and the new worst solution ${P}_{worst}^{new}$ are preserved for use in the next iteration.

Step 4: Calculate the total cost for a modified solution after each iteration. All operational constraints given by Eqs (31−35) are checked for violations if any. The violations are used to convert the constrained optimization problem into an unconstraint problem by using the penalty function approach. Thus, a feasible solution gets a better fitness as compared to an infeasible solution.

Step 5: If the updated solution is found to be better, then replace the modified solutions with previous solutions otherwise retain the previous one. The best solutions are stored when the stopping criterion is reached.

Step 6: Similarly, compute all Pareto optimal solutions and rank them based on $fuzzy\_min$ to get the best possible solution by using Eqs (36) and (37).

4.   Case studies, results and discussion

The proposed approach for the solution of the EED problem with the integration of RESs has been implemented using an improved version of the Jaya algorithm. The proposed Ch-Jaya algorithm is tested on two standard power system test cases, representing static/dynamic cases respectively, with different complexity levels as shown in Table 1. Both the test cases have a non-convex, multimodal and stochastic objective function, where RESs uncertainty is modeled by using the random variables. In addition to these, Test Case II has a discontinuous objective function and the optimization variables are dynamically coupled in successive intervals through the ramp rate limits.

Test Case I: Four test example cases are created from the 10-thermal unit non-convex system with a load of 2000 MW ^[6]. The data is appended in Table A1 and Table A2. In Test Case I(A), the transmission losses are included in the model, which creates additional complexity in the equality constraint. Test Case I(B) is selected for result validation; it is similar to Test Case I(A) but the losses have been neglected here. In Test Case I(C), the second and third thermal generators in I(B) are replaced by wind power units. Test Case I(D) is constructed by adding one additional solar PV system to Test Case I(C); the data for the wind and solar units is listed in Table A.3 and Figure 4.

System I(A) is included in the study for benchmarking the proposed Ch-Jaya algorithm with previously reported results.

Test Case II: Test Case II(A) has all the complexities described earlier in this section and presented in Table 1. The limits, coefficients and ramp rates are given in Table A.4, and hourly demand variation is listed in Table A.5. Case II(B) is created to study the impact of RESs; hence thermal units eight and nine are replaced by wind power units and rest of the data is similar to Case IIA. Test Case II(C) has one additional solar system. The data for wind and solar units is the same as listed in Table A3 and Figure 4.

The physical representation of the problem is shown in Figure 5. The programs have been written in MATLAB R2013a and executed on an Intel Core i7 processor with a 3.40 GHz computer with 2 GB RAM.

4.1.   Setting of population size

In metaheuristic optimization methods, the population size must be set such that the best solution can be obtained within the least possible computational time. Studies were conducted on Test Case I by varying the population size from 10 to 100 with stopping criteria of 100 iterations. Based on the statistical analysis of results presented in Table 2, using 30 trials it is observed that a population size of 50 is optimal for this problem. Similarly, for Test Case II, the best population size is found to be 100.

4.2.   Selection of the best map

All chaotic maps described in Section 3.3 are embedded one by one with the Jaya algorithm and their effects were investigated on Test case IA. The statistical results of trials conducted with different chaotic maps are presented in Table 3. It is observed that the results for all the chaotic maps were almost similar, but with the lowest standard deviation, the tent map was found to be the most consistent as compared to the other maps. Therefore, for further analysis 'tent map' is used in the Ch-Jaya algorithm. However, the computational time gets increased as compared to the analysis carried out using the JAYA algorithm alone. The characteristics of the tent map along with other maps are presented in Figure 6. The convergence behavior of the Ch-Jaya algorithm is found to be superior to the Jaya algorithm as shown in Figure 7.

4.3.   Validation of results of Ch-Jaya for single/multi-objective static/dynamic test cases

The results of cost and emission optimization, for single objective cases, are compared and validated with previously published results of parallel hurricane optimization algorithm (PHOA) ^[43], DE ^[12] and chaotic improved harmony search CIHSA ^[6] in Table 4. The least cost solution obtained by the Ch-Jaya algorithm is ＄ 111497.6312/hr which is better than the other methods while all operational constraints are also satisfied. Table 4, also shows that the best emission 3932.2426 $\mathrm{l}\mathrm{b}/\mathrm{h}\mathrm{r}$ is also obtained by using Ch-JAYA. These values are shown in bold.

The results of optimization of the bi-objective model given in Eq (30) are compared in Table 5 with GSA ^[5], MODE ^[12], NSGAII ^[12], enhanced multi-objective cultural algorithm (EMOCA) ^[44], flower pollination algorithm (FPA) ^[45] and CIHSA ^[6]. The results are comparable; the best cost ＄ 113246.5991/hr is found by Ch-JAYA while the lowest emission 3932.44734 lb/$\mathrm{h}\mathrm{r}$ is reported by CIHSA ^[6].

The optimal generation schedule for Test Case I(B), I(C) and I(D) is presented in Table 6 separately for cost and emission minimization. Furthermore, the statistical comparison for cost and emission obtained by Ch-JAYA and JAYA alone are compared in Table 6 (A) for Test Case I and in Table 7(A) for Test Case II. Here, it is observed that the performance of Ch-JAYA is better than JAYA in terms of either cost or emission minimization for all different cases considered for the analysis.

For case I(B), the cost ＄ 106170.3974/hr and emission 3650.7423 $\mathrm{l}\mathrm{b}/\mathrm{h}\mathrm{r}$, both computed by Ch-JAYA are superior to PHOA ^[43]. The minimum cost obtained by the Ch-JAYA algorithm for Test Case IIA using dynamic scheduling ＄ 2357135.0653, is shown to be better than the MBDE ^[27] algorithm ＄ 2482843.7918 in Table 7. Thus, the superior global search capability of Ch-Jaya is shown for the more complex Test Case II. Further analysis of Tables 6 and 7 for the impact of RESs is presented in the next section.

4.4.   Effect of RESs integration on cost and emission (single objective analysis)

From Table 6, it can be observed that as compared to the thermal system (Test Case IB), the reduction in cost with wind integration is found to be ＄ 2922.0829/hr (2.75% per hour) and with wind-solar integration is ＄ 4666.3705/hr (4.39% per hour). When two wind farms were replaced with thermal units in Test Case I(C), the greenhouse emission was reduced to 2873.4340 lb/hr (21.29%) and by integrating wind and solar PV systems (Test Case I(D)) the emission is reduced to 2434.4654 lb/hr (33.31%) as compared to emission released by the thermal system alone. So, it is concluded that optimization of "emission only" results in a greater reduction in emission as compared to reduction in cost for "cost only" optimization cases.

Similarly, the best cost solutions from Table 7 can be compared for Test Case II(A), II(B) and II(C) respectively. It is observed that the optimal cost of generation in a day for the three test cases is found to be ＄ 2357135.0653, ＄ 2275684.9410 and ＄ 2200281.3668 respectively. Comparing the results, it is observed that there is a reduction of ＄ 81450.1243 (3.45% per day) in total cost due to the integration of two wind power units in Case II(B). For the hybrid wind-solar PV-thermal system, IIC, the cost saving is ＄ 156853.6985 (6.65% per day).

From Table 7 it is also observed that greenhouse emission is reduced from 270810.7558 to 256591.0774 $\mathrm{l}\mathrm{b}$ in Test Case II(B) and to 230769.1362 $\mathrm{l}\mathrm{b}$ for Test Case II(C), which amounts to a reduction of approximately 5.25% (due to replacement of thermal units by wind units) and 14.78% per day (when one solar unit is added) respectively.

4.5.   Effect of RES integration for static/dynamic test cases (bi-objective analysis)

The impact of RES integration is shown in Table 8 by comparing the results of Test Case I(B), I(C) and I(D) considering the bi-objective model. It is observed that a reduction of ＄ 2989.2133/hr (2.77% per hour) in cost and ＄ 5141.307/hr (4.76% per hour) in emission is achieved by wind integration. When both wind and solar PV systems are integrated with the thermal system the reduction in cost and emission content was found to be ＄ 5141.307/hr (4.77%) and 1229.4157 lb/hr (31.71%) respectively as compared to the original thermal system.

For Test Case II, it is observed that the total cost is reduced by ＄ 32832.8159 (1.37% per day) in Test Case II(B) and ＄ 153443.0925 (6.40% per day) in Test Case IIC due to wind and wind-solar PV integration respectively. The emission content is reduced by 14696.425 lb (5.28%) for Test Case II(B) and 37400.0829 lb (13.44%) for Test Case II(C).

4.6.   Impact analysis of RES integration with different optimization goals

The percentage reduction in total cost and emission due to RES integration for single and bi-objective goals for static/dynamic test cases—Test Cases I and II, respectively—has been summarized and shown as a stacked bar chart in Figure 8. The results in Figure 8 clearly show that after the integration of RES, the percentage reduction in emission for both test cases is higher as compared to the percentage reduction in total cost. This is because the uncertainty cost of RES, in terms of reserve and penalty costs, is included in the model. The reduction in emission in Test Case II is found to be lesser as compared to Test Case I for the same conditions/goals. This is due to the dynamic ramp-rate constraints in the DEED problem in Test Case II. These constraints limit the ramping down of the thermal generation leading to reduced scheduling of RES. Hence, emission reduction is lesser as compared to the static conditions in Test Case I. The bi-objective optimization succeeds in reducing both, cost and emission and presents a reasonably good percentage reduction in both objectives for all the tested cases.

4.7.   Optimal power sharing and constraint handling capability

The full optimal generation schedule obtained by Ch-JAYA under the three different optimization goals is available for test case I(A) in Tables 4 and 5, respectively. Tables 6 and 8 give the same for test cases I(B), I(C) and I(D). Similarly, Table 9 presents the optimal schedules for hybrid Test Case-II under the dynamic condition. From all these tables it can be seen that all operational constraints are fully satisfied by the proposed Ch-JAYA.

The optimal dynamic power sharing between solar, wind and thermal systems obtained by Ch-JAYA for best cost and best emission model for Test Case II(C) is shown in Figure 9(a, b). Similarly, the optimal power sharing under the bi-objective optimization of contradictory objectives is presented in Table 9. Here, all operational constraints are also fully satisfied.

4.8.   Trade-off solutions computed by Ch-Jaya

In bi-objective optimization of contradictory objectives, there are many competing solutions and the decision maker selects the best suitable solution based on case-specific constraints set by economic or environmental limitations and guidelines. Figures 10 and 11 present the multiple trade-off solutions for Case I and Case II obtained by Ch-Jaya. It can be seen that Ch-Jaya has produced solutions that cover the full spread of cost-emission solution space, between the two extreme points marked by the best cost and best emission solutions.

5.   Conclusions

A modified JAYA algorithm is developed with different chaos maps for solving a non-convex, mixed integer, multimodal and stochastic problem with practical constraints. The effect of the integration of the uncertain nature of wind and solar PV systems on the optimal scheduling of two complex test systems is modeled using a probabilistic cost function, employing single/multi-objective models. The performance of the proposed method is validated with published results for static/dynamic operating conditions, non-linear, discontinuous objective functions with multi-period, time-coupled constraints. The major findings are summarised as:

• The Jaya algorithm is an efficient population-based evolutionary algorithm that is free from convexity assumptions and any user-controlled program-specific tuning parameters.

• The results show that due to the integration of chaotic maps, the proposed Ch-JAYA has a superior convergence. The proposed Ch-JAYA algorithm is capable of producing feasible and credible results while handling complex and practical constraints.

• The bi-objective optimization succeeds in reducing both, cost and emission and presents a reasonably good percentage reduction in both objectives for all the tested cases.

• The effect of RES integration was investigated with single and bi-objective optimization goals and it was observed that the percentage reduction in emission for both the test cases is higher as compared to the percentage reduction in total cost.

• Results show that RES integration reduces the cost by about 2-4% but results in emission curtailment in the range of 20-33%.

• Considering simulation results under different test conditions it is observed that the Ch-JAYAalgorithm can provide credible and superior quality results and handle associated complex constraints as well as probabilistic functions in an efficient manner while satisfying all operational constraints.

• The fuzzy-min ranking approach is utilized to get the best solution for satisfying cost and emission.

• Pareto optimal solutions obtained under different test conditions provide various power scheduling options to GENCOs and the ISO can select scheduling options for minimizing either (i) total power generation cost, (ii) emissions or (iii) both simultaneously, to gain profit while protecting the environment.

• Power generation from wind turbine and solar PV systems is highly.

• This work may be extended with battery storage to improve power quality, suppress power fluctuation due to renewable energy resources and also to enhance supply security.

Acknowledgments

This research work was supported by "WOOSONG UNIVERSITY's (Daejeon, Republic of Korea) Academic Research Funding-2023".

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Conflict of interest

Surender Reddy Salkuti is an editorial board member for AIMS Energy and was not involved in the editorial review or the decision to publish this article. All authors declare that there are no competing interests.

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