An Elitist Multi-Objective Particle Swarm Optimization Algorithm for Sustainable Dynamic Economic Emission Dispatch Integrating Wind Farms

In recent years, wind energy has been widely used as an alternative energy source as it is a clean energy with a low running cost. However, the high penetration of wind power (WP) in power networks has created major challenges due to their intermittency. In this study, an elitist multi-objective evolutionary algorithm called non-dominated sorting particle swarm optimization (NSPSO) is proposed to solve the dynamic economic emission dispatch (DEED) problem with WP. The proposed optimization technique referred to as NSPSO uses the non-dominated sorting principle to rank the non-dominated solutions. A crowding distance calculation is added at the end of all iterations of the algorithm. In this study, WP is represented by a chance-constraint which describes the probability that the power balance cannot be met. The uncertainty of WP is described by the Weibull distribution function. In this study, the chance constraint DEED problem is converted into a deterministic problem. Then, the NSPSO is applied to simultaneously minimize the total generation cost and emission of harmful gases. To proof the performance of the proposed method, the ten-unit and forty-unit systems—including wind farms—are used. Simulation results obtained by the NSPSO method are compared with other optimization techniques that were presented recently in the literature. Moreover, the impact of the penetration ratio of WP is investigated.


Research Background
In recent years, power system operators have been advised to use non-conventional energy sources such as wind energy and solar energy. Although renewable energy sources have positive environmental impacts, their exact output power is not evident to predict. For this reason, these sources are mostly integrated with conventional sources, such as thermal units, to meet the balance between load demand and power generation. However, mismanagement of thermal generating units leads to high operation cost and unacceptable emission level. Moreover, high penetration of renewable energy, especially wind energy, into the power system caused major challenges due to their intermittent outputs. As countermeasures, decision makers in the power sector should use an optimal power dispatch under wind power (WP) uncertainty for reducing both operational cost and emission. Due to the dynamic characteristic of today's network loads, it is necessary to schedule the generation units according to power demand variations. To achieve the aforementioned tasks, the dynamic economic emission dispatch (DEED) problem incorporating wind energy has become a key issue for power system operators. The purpose of the DEED problem is to minimize simultaneously the total fuel cost and total emission by finding the power production of thermal power plants according to the predicted its solution. Hagh et al. [24] used an exchange market algorithm method for solving the stochastic economic emission dispatch problem, where the fuel cost, penalty costs of WP and emission were combined in one objective function. Unfortunately, most of the aforementioned methods did not necessarily provide the real Pareto front and the compromise solution which are frequently requested for the decision making. In addition, the solution programs need to run several times to get the non-dominated solutions. In consequence, several multi-objective optimization algorithms such as NSGA-II (non-dominated sorting genetic algorithm-II), MOPSO (multi-objective PSO) and MODE (multi-objective DE) were used for solving this kind of problem [25]. These algorithms are mainly based on the non-dominance principle and may provide the Pareto front in a single run. However, some works demonstrated that MOPSO-based algorithms provide frequently the more accurate Pareto front [25]. In fact, MOPSO adopts the non-dominated sorting principles to improve the solution diversity. In addition, PSO and its variants use the experience acquired during the search for the optimum solution in order to best guide the search process. Unlike GA and DE, PSO does not have mutation and crossover operators, but it emulates the social behavior of organisms, which enables the PSO-based techniques to efficiently provide the local solutions. Table 1. List of some techniques used for dispatch problems.

Techniques Dispatch Problems
Dynamic programming [4] Static economic dispatch problem without valve-point loading effects (VPLE) constraints Interior point method [6] Nonlinear optimal power flow Particle swarm optimization (PSO) [8] Static economic dispatch with VPLE constraints Artificial bee colony (ABC) [9] Genetic algorithm [7] Dynamic economic dispatch problem with VPLE constraints Bacterial foraging [10] Dynamic economic emission dispatch (DEED) problem with VPLE constraints Simulated annealing [11] Differential evolution (DE) [12] Here-and-now approach [14] Static economic dispatch problem including wind power and without VPLE Stochastic optimization technique [15] Static economic emission dispatch (SEED) problem considering wind power Chance-constraint programming [16] SEED problem considering wind power Chance-constraint programming [17] DEED problem considering wind power Chance-constraint programming [18] Static economic dispatch considering wind power and without VPLE Scenario-based approach [20] DEED problem considering wind power Scenario-based approach [21] Reactive power dispatch considering renewable energy sources and with uncertainties in loads.

Aims and Contributions
Within this context, this paper presents an elitist multi-objective method to solve the DEED for the wind-thermal system. In the optimization process, all cited operating constraints are considered and the stochastic characteristic of the WP is represented by a chance-constraint which describes the probability that the power balance cannot be met. The stochastic constraint is incorporated in the system constraints in order to mitigate the penalty costs of WP. The resolution of the problem is carried out mainly in two steps. Firstly, the stochastic problem is converted into a deterministic problem by representing the random characteristic of the wind speed by the Weibull PDF. Then, the problem is solved by an elitist multi-objective evolutionary algorithm. The proposed optimization technique, called non-dominated PSO algorithm (NSPSO), uses a crowding distance comparison at the end of iterations of the classic PSO in order to facilitate the convergence to the real Pareto front. The Pareto front is generated in one run of the solution program. The proposed method is tested on the ten-unit and forty-unit systems including the wind energy source.

Chance-Constrained DEED Problem
In the literature, the DEED problem was considered as a bi-objective optimization problem. It aims to minimize simultaneously the total fuel cost and total emission by finding the power production of thermal units according to the predicted load demands. The resolution of the DEED problem can be accomplished by solving the SEED problem over a certain period of time subdivided into smaller time intervals. The total fuel cost and emission over time horizon of T hours are described, respectively [2], by Equations (1) and (2). In this study, T = 24 h.
The two objective functions will be minimized subject to the constraints given in Equations (3)-(7). Inequality (3) is the chance-constraint describing the stochastic characteristic of WP. It represents the probability to meet the system load requirement at time t. Real power losses P t loss are calculated using the B-loss formula [2]. In practice, the power generation of each unit i during two successive time periods is confronted by its RRLs defined by Inequalities (4) and (5). The POZ constraints are described as given in Equation (6). Maximum and minimum generations of both thermal units and the wind energy source are stated in Equations (7) and (8), respectively.

Probability Model of WP Output
The high penetration of WP into power networks has created major challenges due to the intermittency of the wind speed. From the literature review, it is found that the wind speed is widely described by two-parameter Weibull PDF [17]. The Weibull PDF and cumulative distribution function are given, respectively, in Equations (9) and (10) [17].
Parameters c and k in Equation (9) depend on the wind farm location. Mostly, they are in the range of (5.0, 20.0) and (1.0, 3.0), respectively. Figure 1 depicts the impact of parameters c and k on the Weibull PDF. It can be seen that the curve shape is influenced by the value of parameter k. It is noteworthy also that if c increases, the curves move toward higher wind speed. In this paper, WP output as a function of the wind speed is described by the following equation.
The cumulative distribution function (CDF) of the random P w may be calculated as follows.
Thus, constraint (3) can be modified as follows. Pr

Implementation of the Non-Dominated Sorting PSO Algorithm
The PSO algorithm emulates the social behavior of organisms [26]. In the PSO algorithm, the i-th individual, called particle, is represented at each iteration k by its position X k i = X k i1 , . . . , X k in and its velocity V k i = V k i1 , . . . , V k in . From iteration k to the next iteration (k + 1), position and velocity are updated as given in the two following equations.
where w, c 1 and c 2 are the PSO parameters; r 1 and r 2 are random numbers in the range [0, 1]; pbest k i and gbest k are the best solution of the i-th particle and the best solution in the overall population at the k-th iteration, respectively.
In order to adopt the PSO algorithm for multi-objective problems, several modifications of the original PSO have been developed in the literature [27][28][29][30]. In this study, the non-dominated sorting concept suggested by Deb et al. [28] for the non-dominated sorting genetic algorithm is incorporated in the classical PSO algorithm.
At each generation t, an offspring population Q t is generated from the parent population P t . The two populations are combined in one population R t as in Equation (16). Then, the obtained population is sorted into different non-domination levels F j as given in Equation (17). This process is shown in Figure 2 and it is detailed in [28].
where r is the number of fronts.
Sustainability 2020, 12, x FOR PEER REVIEW 6 of 15 (15) where w, c1 and c2 are the PSO parameters; r1 and r2 are random numbers in the range [0, 1]; k i pbest and k gbest are the best solution of the i-th particle and the best solution in the overall population at the k-th iteration, respectively. In order to adopt the PSO algorithm for multi-objective problems, several modifications of the original PSO have been developed in the literature [27][28][29][30]. In this study, the non-dominated sorting concept suggested by Deb et al. [28] for the non-dominated sorting genetic algorithm is incorporated in the classical PSO algorithm.
At each generation t, an offspring population Qt is generated from the parent population Pt. The two populations are combined in one population Rt as in Equation (16). Then, the obtained population is sorted into different non-domination levels Fj as given in Equation (17). This process is shown in Figure 2 and it is detailed in [28].
where r is the number of fronts.

Results and Discussion
The performance of the proposed method for solving the DEED of a wind-thermal system is verified by using two test systems which are the standard ten-unit 39-bus New England power system and the forty-unit system. The single line diagram of the first system is shown in Figure 3. All aforementioned constraints are considered for both systems. Generator cost and emission coefficients of system 1 are shown in Table A1 in the Appendix. Generation limits and RRLs of all units of this system are given in Table A2. All data of system 2 are taken from [24,25]. The hourly variation of the load is given in Table A3.

Results and Discussion
The performance of the proposed method for solving the DEED of a wind-thermal system is verified by using two test systems which are the standard ten-unit 39-bus New England power system and the forty-unit system. The single line diagram of the first system is shown in Figure 3. All aforementioned constraints are considered for both systems. Generator cost and emission coefficients of system 1 are shown in Table A1 in the Appendix. Generation limits and RRLs of all units of this system are given in Table A2. All data of system 2 are taken from [24,25]. The hourly variation of the load is given in Table A3.
Three cases are considered in this section.
(i) SEED problem without wind power.
(ii) DEED problem without wind power.
(iii) DEED problem with wind power. Three cases are considered in this section.
(i) SEED problem without wind power.
(ii) DEED problem without wind power.
(iii) DEED problem with wind power.

Case 1: SEED Problem without Wind Power
To prove the superiority of the proposed technique, the fuel cost and the emission are minimized for the forty-unit system with VPLE. In this case, the total load is 10,500 MW. The optimization results are shown in Table A4 in the Appendix. As seen in Table A4, the minimum fuel cost and emission provided by NSPSO are USD 121,153/h and 389,953 ton/h, respectively. It is clear that these values are better than those obtained by using the classical PSO algorithm. In addition, it can be verified easily that all constraints are satisfied.

Case 2: DEED Problem without Wind Power
In this case, the performance of the proposed optimization algorithm NSPSO is tested on the DEED problem without incorporation of WP. Initially, the NSPSO is applied to the ten-unit system with constant load Pld = 1700 MW and a comparison with the classical PSO is provided. The convergence and the Pareto set of solutions of both algorithms are shown in Figures 4 and 5, respectively. From Figures 4 and 5, it is clear that the proposed NSPSO provides the best results compared to the PSO algorithm. It is noticed that the minimum cost and the minimum emission achieved by NSPSO for the static problem are 9.9334 × USD 10 4 /h and 1.1158 × 10 4 ton/h while are 9.9555 × USD 10 4 /h and 1.1233 × 10 4 ton/h for the PSO algorithm, respectively. From Figure 5, it is obvious that fuel cost and emission are conflicting objective functions. In other words, if a power system operator or decision maker wants lower production cost, more emissions of harmful gases will be emitted and vice versa.

Case 1: SEED Problem without Wind Power
To prove the superiority of the proposed technique, the fuel cost and the emission are minimized for the forty-unit system with VPLE. In this case, the total load is 10,500 MW. The optimization results are shown in Table A4 in the Appendix. As seen in Table A4, the minimum fuel cost and emission provided by NSPSO are USD 121,153/h and 389,953 ton/h, respectively. It is clear that these values are better than those obtained by using the classical PSO algorithm. In addition, it can be verified easily that all constraints are satisfied.

Case 2: DEED Problem without Wind Power
In this case, the performance of the proposed optimization algorithm NSPSO is tested on the DEED problem without incorporation of WP. Initially, the NSPSO is applied to the ten-unit system with constant load P ld = 1700 MW and a comparison with the classical PSO is provided. The convergence and the Pareto set of solutions of both algorithms are shown in Figures 4 and 5, respectively. From Figures 4 and 5, it is clear that the proposed NSPSO provides the best results compared to the PSO algorithm. It is noticed that the minimum cost and the minimum emission achieved by NSPSO for the static problem are 9.9334 × USD 10 4 /h and 1.1158 × 10 4 ton/h while are 9.9555 × USD 10 4 /h and 1.1233 × 10 4 ton/h for the PSO algorithm, respectively. From Figure 5, it is obvious that fuel cost and emission are conflicting objective functions. In other words, if a power system operator or decision maker wants lower production cost, more emissions of harmful gases will be emitted and vice versa.
To further study the effectiveness of the NSPSO, it is executed for the classical DEED problem and obtained results are compared with other optimization techniques, such as improved bacterial foraging algorithm (IBFA) and non-dominated sorting genetic algorithm (NSGA-II), used in other research works for solving the same problem. The comparison results shown in Table 2 confirm that the NSPSO outperforms these techniques despite POZ constraints not being taken into account in some of them. The minimum total cost and minimum total emission achieved by the proposed technique are USD 2,474,472.8 and 293,416.3 ton, respectively.  To further study the effectiveness of the NSPSO, it is executed for the classical DEED problem and obtained results are compared with other optimization techniques, such as improved bacterial foraging algorithm (IBFA) and non-dominated sorting genetic algorithm (NSGA-II), used in other research works for solving the same problem. The comparison results shown in Table 2 confirm that the NSPSO outperforms these techniques despite POZ constraints not being taken into account in some of them. The minimum total cost and minimum total emission achieved by the proposed technique are USD 2,474,472.8 and 293,416.3 ton, respectively. 2.5168 × 10 6 3.1740 × 10 5

Case 3: DEED with Wind Power
In this case, the effect of the incorporation of wind energy into the power system is studied through the DEED problem. The problem is solved by using the proposed method for various values of the threshold tolerance. Due to the space limit, just the optimum solutions for α = 0.25 are presented. Tables 3-5 show the optimum solutions for the dynamic economic dispatch, dynamic emission dispatch and the DEED compromise solution, respectively. It is worth noting that the compromise solution is provided using a fuzzy-based approach described in [2]. It is clear from these tables that the optimum solutions respect the required constraints such as the generation limits and the ramp rate limits. Nevertheless, the total cost and emission are reduced, respectively, from USD  To further study the effectiveness of the NSPSO, it is executed for the classical DEED problem and obtained results are compared with other optimization techniques, such as improved bacterial foraging algorithm (IBFA) and non-dominated sorting genetic algorithm (NSGA-II), used in other research works for solving the same problem. The comparison results shown in Table 2 confirm that the NSPSO outperforms these techniques despite POZ constraints not being taken into account in some of them. The minimum total cost and minimum total emission achieved by the proposed technique are USD 2,474,472.8 and 293,416.3 ton, respectively. 2.5168 × 10 6 3.1740 × 10 5

Case 3: DEED with Wind Power
In this case, the effect of the incorporation of wind energy into the power system is studied through the DEED problem. The problem is solved by using the proposed method for various values of the threshold tolerance. Due to the space limit, just the optimum solutions for α = 0.25 are presented. Tables 3-5 show the optimum solutions for the dynamic economic dispatch, dynamic emission dispatch and the DEED compromise solution, respectively. It is worth noting that the compromise solution is provided using a fuzzy-based approach described in [2]. It is clear from these tables that the optimum solutions respect the required constraints such as the generation limits and the ramp rate limits. Nevertheless, the total cost and emission are reduced, respectively, from USD

Case 3: DEED with Wind Power
In this case, the effect of the incorporation of wind energy into the power system is studied through the DEED problem. The problem is solved by using the proposed method for various values of the threshold tolerance. Due to the space limit, just the optimum solutions for α = 0.25 are presented. Tables 3-5 show the optimum solutions for the dynamic economic dispatch, dynamic emission dispatch and the DEED compromise solution, respectively. It is worth noting that the compromise solution is provided using a fuzzy-based approach described in [2]. It is clear from these tables that the optimum solutions respect the required constraints such as the generation limits and the ramp rate limits. Nevertheless, the total cost and emission are reduced, respectively, from USD 2,474,472.8 and 293,416.3 ton (Table 2) to USD 2,433,467.20 (Table 3) and 283,538.16 ton (Table 4), when WP is considered.  It is clear from Tables 3 and 4 that the production cost is USD 2,466,582.70 for dynamic economic dispatch and it is increased to USD 2,552,118.86 for dynamic emission dispatch, while emission is 331,251.40 ton under dynamic economic dispatch, and it decreases to 283,538.16 ton under dynamic emission dispatch. It is worth noting that the minimization of emission under economic dispatch is not considered and economic aspects are not considered under emission dispatch. To avoid the above-mentioned conflicts, the compromise solution given in Table 5 may be selected as the optimum solution.   Table 6 shows the effect of the threshold probability α on the minimum production cost, the minimum emission and compromise solution of the DEED problem incorporating wind energy. It is clear that the minimums of the two objective functions decrease as the probability α, that power balance described by Equation (13) cannot be met, increases, because the larger α signifies using more WP and reducing the demand of thermal energy. The penetration ratio of WP is investigated also in this study. The maximum value of WP penetration is given as follows.
where P D is the total demand power and η is the ratio of WP penetration. Table 7 shows the effect of WP penetration ratios on the total fuel cost and the total emission for total demand power equal to 1500 MW. It is clear that if the ratio increases, fuel cost and emission decrease due to the reduction in outputs of thermal units.

Conclusions
In this study, a PSO-based multi-objective optimization technique is proposed to solve the DEED problem incorporating wind energy. To avoid the penalty costs corresponding to the overestimation and underestimation of the wind farm output, the uncertain characteristic of the wind power is represented by a chance-constraint in the DEED model. The latter describes the probability that energy balance cannot be met. In order to adopt the PSO algorithm for the multi-objective DEED problem, the non-dominated sorting concept is incorporated in the classical PSO method. The proposed method referred to as NSPSO is successfully tested for the static economic emission dispatch problem. Then, its effectiveness for solving the stochastic DEED problem is evaluated. The fuel cost with VPLE and emission is minimized simultaneously where all constraints cited in the problem formulation are considered. Simulation results show that NSPSO outperforms other optimization techniques used for solving the same problem. Moreover, the effect of the penetration ratio on the objective functions is studied.
It can be concluded that the proposed algorithm can provide a variety of solutions for the decision makers in a single run. The NSPSO algorithm has the ability to optimize simultaneously more than two objective functions. Therefore, other functions can be added to the problem, e.g., total losses and voltage deviation.  Probability density function (PDF) Wind speed in m/s V and P W Wind speed and wind power random variables k and c Shape and scale factors of the Weibull distribution function, respectively v in , v out and v r Cut-in, cut-out and rated wind speeds in m/s, respectively w r Rated wind power output in MW