Multiobjective Collective Decision Optimization Algorithm for Economic Emission Dispatch Problem

The collective decision optimization algorithm (CDOA) is a new stochastic population-based evolutionary algorithm which simulates the decision behavior of human. In this paper, amultiobjective collective decision optimization algorithm (MOCDOA) is first proposed to solve the environmental/economic dispatch (EED) problem. MOCDOA uses three novel learning strategies, that is, a leader-updating strategy based on the maximum distance of each solution in an external archive, a wise random perturbation strategy based on the sparse mark around a leader, and a geometric center-updating strategy based on an extreme point. The proposed three learning strategies benefit the improvement of the uniformity and the diversity of Pareto optimal solutions. Several experiments have been carried out on the IEEE 30-bus 6-unit test system and 10-unit test system to investigate the performance of MOCDOA. In terms of extreme solutions, compromise solution, and three metrics (SP, HV, and CM), MOCDOA is compared with other existing multiobjective optimization algorithms. It is demonstrated that MOCDOA can generate the well-distributed and the high-quality Pareto optimal solutions for the EED problem and has the potential to solve the multiobjective optimization problems of other power systems.


Introduction
The classical economic dispatch (ED) of electric power generation operating at the absolute minimum cost is one of the mathematical optimization issues attracting many researchers' interests [1,2].However, coal, natural gas, and petroleum remain the primary feedstock for power plants around the world, although some countries are planning to build several nuclear power plants and are sourcing more of electricity from wind.The serious global environmental problems are caused by burning fossil fuel to release several contaminants, such as  2 ,  2 , and   , into the atmosphere.The ED problem therefore no longer is considered alone.In this case, the environmental/economic dispatch (EED), a short-term alternative to reduce the atmospheric emissions, has received much attention.
The EED problem is a highly constrained conflicting nonlinear multiobjective optimization problem which results in not only greater economical benefits, but also less pollutant emissions.Various approaches have been reported to solve the multiobjective EED problem.Initially, some conventional optimization methods such as a linear programming method were used as the optimizing tool to optimize the EED problem [3,4].However, they are not suitable for a complex multiobjective optimization EED problem.Hence, some heuristic search techniques especially evolutionary algorithms (EAs) [5] and swarm intelligence algorithms (SIs) [6] have gotten the attention of many researchers' interests.EAs and SIs have been successfully tried to solve the EED problem, which can be mainly divided into two categories: (i) The first category regards the multiobjective EED problem as a single-objective optimization problem.
The EED problem was handled as a single-objective problem by considering the emission as a constraint in [7,8].Another technique, using the linear weighted sum method, transforms a set of objective functions into a single objective [9][10][11][12][13][14].This approach generally uses the following formula to transform two objectives into a single objective: where  is the scaling factor which combines the total fuel cost (  ) with the total emission (  ) and  is the weight factor in the range of [0, 1].Each objective is multiplied by a weight related to a given value , which usually embodies the importance of the objective.The approach should be operated for several times to get a set of Pareto optimal solutions by setting different  values.It cannot be applied to obtain the Pareto optimal solutions for the problem with a nonconvex Pareto optimal front.
(ii) The second category tackles the both objectives, i.e., fuel cost and emission simultaneously as two competing objectives.Some multiobjective evolutionary algorithms based on genetic algorithms have been applied to generate the Pareto optimal solutions of the EED problem, which include a niched Pareto genetic algorithm (NPGA) [15], a nondominated sorting genetic algorithm (NSGA) [16], a strength Pareto evolutionary algorithm (SPEA) [17], an improved genetic algorithm (IGA) [18], and a fast and elitist Multiobjective Genetic Algorithm (NSGA-II) [19,20], etc.Some other multiobjective evolutionary algorithms based on particle swarm optimization, such as an external memory based Multiobjective Particle Swarm Optimization (MOPSO) [21], a comprehensive learning particle swarm optimizer (MOCLPSO) [22], a fuzzified multiobjective particle swarm optimization (FMOPSO) [23], a multiobjective chaotic particle swarm optimization (MOCPSO) [24], a fuzzy clustering-based particle swarm optimization (FCPSO) [25], a parameter-free bare-bones multiobjective particle swarm optimization algorithm (BB-MOPSO) [26], and a cultural quantum-behaved particle swarm optimization (CMOQPSO) algorithm [27], have also been presented to solve the EED problem.In addition, differential evolutions (DEs) have also been used to solve the EED problem [28].These algorithms have achieved good results for the EED problem.
In this paper, a new heuristic evolutionary method called a multiobjective collective decision optimization algorithm (MOCDOA) is first proposed to solve the EED problem.The paper mainly has the following four contributions: (i) Proposing a MOCDOA: we adopt the collective decision optimization algorithm (CDOA) [29] to solve the multiobjective EED problem for the first time.
(ii) Designing a leader-updating strategy: considering the uniformity performance of approximate solutions, a new technique based on the maximum distance of each solution in the external archive is proposed to update a global leader.
(iii) Designing a wise random perturbation strategy: a wise random perturbation strategy based on the sparse mark around a leader is used to enhance the uniformity of the obtained Pareto optimal solutions.(iv) Designing a geometric center-updating strategy: a geometric center-updating strategy is presented to expand the diversity performance of the Pareto optimal set, which randomly selects an extreme point on the Pareto optimal solutions to replace a geometric center.
The rest of this paper is organized as follows.Section 2 formulates the environmental/economic dispatch problem.Section 3 gives a brief review of CDOA.Section 4 presents MOCDOA.Section 5 shows the application of MOCDOA to solve the EED problem.Section 6 gives the corresponding comparative results of several existing optimization methods, and Section 7 concludes.

Mathematical Model of the EED Problem
Satisfying several equality and inequality constraints, the EED problem requires minimizing simultaneously two competing objective functions, the fuel cost and the emission.The mathematical model for the EED problem including its objective functions and constraints is described in Section 2.1 and Section 2.2 in detail.The EED problem can be mathematically formulated as where  and  denote total fuel cost and total emission, respectively.ℎ and  are the equality and inequality constraints, respectively.
. .Objective Functions . . .Fuel Cost Minimization.The total fuel cost (  ) (dollars per hour) is made up of  generator costs expressed by quadratic functions.The total fuel cost (  ) can be represented as where the vector   = ( 1 ,  2 , ⋅ ⋅ ⋅ ,   ) is the real power outputs of generators.  is the real power output of the th generator.  ,   , and   are the cost function coefficients of the th generator.
. . .Emission Minimization.The emission function can be presented as the sum of all types of the emission considered, but only the emission of nitrogen oxides   is considered in the present study.The total emission (  )(/ℎ) of nitrogen oxides   caused by the generators can be expressed as a function of generator output in (4), that is, the sum of some quadratic functions and exponential functions.1 and 3.
. .Constraints.The power dispatch constraints include one equality constraint on the power balance and several inequality constraints on generation capacity.
. . .Power Balance Constraint.The total power generation must cover the power demand   and the transmission network loss   , namely, Here, in general,   is determined by Krons loss formula [30] as where   ,  0 , and  00 are the transmission network power loss coefficients.
. . .Generation Capacity Constraints.The power output of each generator is limited by its corresponding lower and upper bounds as shown in where  min  ,  max  are the minimum and maximum generation limits of the th generator, respectively.

Collective Decision Optimization Algorithm
Collective decision optimization algorithm (CDOA) [29] is a new evolutionary method proposed by Qingyang Zhang in 2016 [29], inspired from the decision-making behavior of human such as holding a meeting.Each member of a group will express and exchange their own thoughts or plans in meeting.The best one among the resultant plans is selected as a global optimal solution.In CDOA, an individual   generates several candidate solutions on a multistep positionselected scheme guided by different individuals, such as a local leader, an others' individual, a geometric center, and a leader.The seven major steps of CDOA, described briefly next, are group generation, experience-based phase, others-based phase, group thinking-based phase, leaderbased phase, innovation-based phase, and selection.
where  is the uniform distribution and   and   are the lower and upper limits of the th dimension, respectively.
Step (experience-based phase).In CDOA, personal experience represents the local leader, i.e., the personal best position () obtained by each individual itself so far.The new position of the individual   () will be expressed as follows: where  →  0 is a random vector with each number uniformly distributed in the interval (0, 1) and () denotes the step size of the th iteration.
where  max is the maximum number of iteration.
Step (others'-based phase).In the meeting, all individuals will interact randomly.The individual   () who is better than the current member   () is randomly selected from the group.The calculation formula of updating the  0 is defined as follows: where  denotes a random integer in [1, N],  →  1 is a random vector with each number uniformly distributed in (0, 1), and  1 and  1 are the random numbers in (-1, 1) and (0, 2), respectively.

Complexity
Step (group thinking-based phase).In the meeting, group thinking will influence the decision of each individual.In CDOA, the position of group thinking is represented by the geometric center (  ) of all individuals.  can be described as follows: Here, updating the  1 is calculated in the following formula: where  →  2 is a random vector with each number uniformly distributed in (0, 1) and  2 and  2 are the random numbers in (-1, 1) and (0, 2), respectively.
Step (leader-based phase).In CDOA, the position of the best individual in the group is represented by a leader, i.e., a global best position (  ).The next new position of th individual will be designed as follows: where  →  3 is a random vector with each number uniformly distributed in (0, 1) and  3 and  3 are the random numbers in (-1, 1) and (0, 2), respectively.The leader's mind can only be changed randomly by himself.The leader slightly changes its position by a random walk strategy in a local search space.In this phase, five neighbors are generated randomly around   , as shown in where   →   is a random vector with each number in (0, 1).The next leader   is produced in where   is the index of a minimum objective function value.
Step (innovation-based phase).To prevent a premature convergence, an innovation operator, which makes a small change among one of the dimension of each individual, is designed in CDOA.The innovation operator is similar to a mutation operator in evolutionary algorithms to improve the population diversity.The operator can be designed as follows: where 0 represents a random number in (0, 1),  is a random integer in [1, D], and  is an innovation (mutation) factor.The  is set to 0.8 in the following algorithm.
Step (selection).In the selection phase, the fitness values in   and  4 are compared to update the population  by using a greedy selection.
The procedure of CDOA again and again repeats from Step 2 to Step 7 until the terminating condition is satisfied.

Multiobjective Collective Decision Optimization Algorithm
CDOA is initially designed for single-objective optimization problems.In this paper, we extend CDOA to make it suitable for handling multiobjective optimization problems.
Our main development of CDOA will be concerned in three novel learning strategies, which include a leader-updating strategy based on the maximum distance of each solution in nondominant solution set, a wise random perturbation strategy based on the sparse mark around the leader, and a geometric center-updating strategy based on an extreme point.The proposed three learning strategies are created for improving the uniformity and diversity of the Pareto optimal solutions.In addition, several existing techniques such as a local leader-updating strategy, a nondominated approach, an external elitist archive, and a circular crowded sorting are introduced into MOCDOA.Algorithm 1 presents the pseudocode of MOCDOA.Similar to other evolutionary algorithms, MOCDOA can be divided into three processes, initialization, mutation, and selection.In the mutation phase of Algorithm 1, the main development of the three novel learning strategies is shown in the lines 11, 12, and 14, respectively, which are underlined and shown in bold.Whenever the archive goes beyond its capacity, the redundant crowded solutions will be removed from the archive in the lines 26-27 of Algorithm 1.The details of MOCDOA are described in this section.
. .Leader-Updating Strategy.In a multiobjective optimization problem, the conflicting multiple objectives make CDOA difficult to choose a global best position.To resolve this problem, MOCDOA maintains an external archive to store and update the nondominated solutions in each iteration.The leader   of each individual is selected from the external archive to improve the uniformity and diversity of the nondominated solutions.In MOCDOA, a new maximum distance () is designed to measure the sparsity of solutions.In Figure 1, the solid dots denote the solutions of the external archive.The maximum distance of the th solution (  ) is the maximum of two side-length sums of its two adjacent cuboids.The maximum distance of the th solution (  ) is calculated as follows: where  is the maximum capacity of the archive.The distances of the th solution to the upper point and the lower point are   and   shown in ( 19) and ( 20), respectively.  () ← Geometric center updating(()) % Find the geometric center   () 13: if  < 0.5 then 14: Random access to the next generation 25: ( + 1) ←  (() ∪ ()) % Update the archive 26: if |( + 1)| >  then 27: (( + 1)) % Maintain the archive, where |( + 1)| is the element number of ( + 1) 28: for  = 1 to  do 29: ( + 1) =   (  (),   ) % Update local leader 30: =  + 1 31:  ←   and stop the algorithm %Output the obtained Pareto optimal solutions Algorithm 1: Multiobjective collective decision optimization algorithm.
where  is the number of objective functions. min Since there is only one neighbor point for each extreme point in the archive, the maximum distance () of each extreme point is assigned with the distance between it and its neighbor point.In MOCDOA, the leader-updating strategy applies a roulette wheel selection method during each individual-mutating.The probability of each solution in the external archive to be selected as the leader is proportional to its maximum distance.We use Function Leader updating to select   , which is shown in the line 11 of Algorithm 1.The leader-updating strategy is illustrated as Algorithm 2.
Here, we design a concept of sparse direction () for the wise random perturbation strategy in the next Section 4.3.The   and   of the th solution of the archive are compared to record the sparse direction of the th solution   , which will be used as a sparse mark of the solution to guide the perturbation in following strategy.If   is greater than   ; that is, the gap of the th solution for its upper neighbor solution is larger than that for its lower neighbor solution; we record   = 1.Otherwise,   = −1.For example, in Figure 1,   is greater than   for the th solution, so   = 1.For the th solution,   is greater than   , such that   = −1.The calculation formula of   is designed as   () =    () Algorithm 2: Leader-updating. . .Wise Random Perturbation Strategy.To improve the uniformity performance of the external archive in MOCDOA, an individual that has the half probability to be selected is updated by using a wise random perturbation strategy around   .The perturbation method is shown in where 1 is a random number in (0, 1).The strategy makes   move to the direction of its sparse neighbor, thus generating a new individual   , which can reduce the sparsity of the external archive.We use Function Leader guiding, shown in the line 14 of Algorithm 1, to update   .
. .Geometric Center-Updating Strategy.For improving the diversity performance of the external archive in MOCDOA, we use Function Geometric center updating to randomly select   in the extreme points of the archive.Function Geometric center updating is shown in the line 12 of Algorithm 1.The calculation formula of   is designed as where 2 is a random number in (0, 1).
. .External Archive-Retaining Strategy.It is important to retain the nondominated solutions during the entire search process to obtain a good optimal solution set at the end of MOCDOA.Many scholars have used the external archive with a given maximal capacity to store and update the nondominated solutions in each iteration.At present, the maintenance strategy of an external archive is mostly adopted a more efficient nondominated sorting method called the fast nondominated sort in NSGA-II [31].Furthermore, Deb et al. proposed an approach based on a crowding distance, which is usually the average distance of two neighbor points around each solution.Once the elitist archive has reached its maximal capacity, the crowding distance is adopted to remove the extra members and thereby keep the archive in its maximum capacity.Hence, the calculation method of the crowding distance greatly affects the distribution of the external archive.However, the crowding distance is only calculated once in each iteration of NSGA-II.Several adjacent solutions with small crowding distances will be removed, which may cause that the remaining solutions are too sparse.To overcome the above drawback, a cyclic crowded sorting algorithm [32], Function Circular crowded sorting in Algorithm 3, is adopted to improve the uniformity and the diversity of the Pareto optimal solutions.

Implementation of MOCDOA
In this section, the proposed MOCDOA is applied for solving the EED problem with one equality constraint on the power Input: (),  Output: ( + 1) 1:  = |()| %  is the element number of () 2: while  >  do %  is maximum capacity of the archive 3: for  = 1   do 4: ().= 0 % Initialize the crowding distance of () 5: for  = 1   do %  is the number of objective functions 6: () = s((), ) % Sort  by th objective functions value 7: 1 ().=  % Set the crowding distance of extreme solutions equal to infinite 8: ().=  9: for i=2 to Nt-1 do balance.MOCDOA uses a constraint-handling mechanism to adjust an unfeasible solution in feasible search space and a fuzzy set theory to select a best compromise solution.
Experiment design and parameter setting are introduced in final subsection.
. .Constraint-Handling. Since a resulting individual is not always guaranteed to satisfy the equality constraint, a constraint-handling strategy needs to be adopted to deal with the constrained EED problem.In order to guarantee the feasibility in all solutions, a straightforward constraint treatment method, the rejecting strategy, has been applied to handle the constraints of the EED problem in [23,24,33].However, this approach produces the Pareto optimal solutions satisfying the equality constraints at the slowest pace.
In MOCDOA, Function Constrint Handling is designed to handle the equality constraint of the EED problem.By applying Function Constrint Handling, an unfeasible solution produced by MOCDOA can be modified into a feasible one.The Function Constrint Handling is described in Algorithm 4. In this model, we set  = 1 − 12.
. .Compromise Solution.After obtaining the Pareto optimal solutions, a fuzzy membership function [34] is proposed to simulate a decision-maker's preference and to extract a Pareto optimal solution as the best compromise solution.Usually, a membership function for each of the objective functions is defined by the experiences and intuitive knowledge of the decision-maker.In this work, a simple linear membership function is considered for each of the objective functions.The linear membership function is herein defined as The membership function value represents the degree of achievement of an objective function as a value between 0 and 1.   = 1 is expressed as completely satisfactory, and   = 0 is expressed as unsatisfactory.Figure 2 where  denotes the number of the objective functions and  is the number of the solutions in the final nondominated front.The best compromise solution is the one getting the highest value of   .
. .Experiment Design and Parameter Setting.In this section, the standard IEEE 30-bus 6-unit system and 10-unit system are considered to validate the performance of the proposed MOCDOA for solving the EED problem.To evaluate the optimization performance of MOCDOA, two different cases (Case1 and Case2) of the EED problem are considered for the two test systems.In Case1, we do not consider the transmission loss of power balance constraint.On the contrary, we consider the transmission losses of power balance constraint for Case2.The data of the IEEE 30-bus 6-unit system are referenced in [26] and listed in Tables 1 and 2. The power demand of the IEEE 30-bus 6-unit system is set to 2.834 MW.The data of the IEEE 30-bus 10-unit system are referenced in [35] and listed in Tables 3 and 4. The power demand of the IEEE 30-bus 10-unit system is set to 2000 MW.We use Matlab software to run MOCDOA program on a personal computer with Pentium 2.60 GHz processor and 4.00GB RAM.
By using the orthogonal experimental method, the parameters  and  are tuned and adjusted until the optimal settings are determined.In these two cases, the population size () and the maximum capacity of the archive () are set as 100 and 50, respectively.The maximum iterations are restricted to 100 and 200 for Case1 and Case2, respectively.MOCDOA is set to conduct 30 runs to collect the statistical results for all the two test systems.For comparison, MOPSO [21], NSGA-II [19], and PESA-II [36] are applied to solve the EED problem and use the above same parameters.In case of MOPSO, the inertia weight, personal learning coefficient, global learning coefficient, and number of grids per dimension are selected as 0.7, 1.4, 1.4, and 7 for all the two test systems.In case of NSGA-II and PESA-II, the crossover percentage and mutation percentage are set to 0.7.

Experimental Results and Analysis
The first experimental results are to verify the effectiveness of the three learning strategies proposed in this paper.The second experimental results are obtained on the IEEE 30 bus 6-unit system.Further, the last experimental results are gained on the IEEE 30-bus 10-unit system.
. .Experimental Analysis of ree Learning Strategies.In order to verify that the uniformity and the diversity of the obtained Pareto optimal solutions are improved with three learning strategies, one type of multiobjective unconstrained test function is used.The function is as follows: The parameters including  = 40,  = 2,  = 35, and   = 30 are provided.The following four algorithms are compared for this test: (1) The first one is a multiobjective CDOA (origin MOC-DOA for short) obtained by transforming single-objective CDOA directly.
(2) The second one is an origin MOCDOA + strategy1, which is to add the leader-updating strategy into the origin MOCDOA.
(3) The third one is an origin MOCDOA + strategy1 + strategy2, which is to add the leader-updating strategy and the wise random perturbation strategy into the origin MOCDOA.
(4) The fourth one is an origin MOCDOA + strategy1 + strategy2 + strategy3, which is the algorithm proposed in this paper.It is to add the leader-updating strategy, the wise random perturbation strategy, and the geometric centerupdating strategy into the origin MOCDOA.
The Pareto optimal solutions obtained by the four algorithms are depicted on Figure 3.According to the picture, we can get two results: (1) As can be seen from Figure 3, a set of nondominant solutions can be found in all four methods.The Pareto front of the origin MOCDOA + strategy1 is more uniform than that of the origin MOCDOA, and the Pareto front of the origin MOCDOA + strategy1 + strategy2 is more uniform than that of the origin MOCDOA + strategy1.Furthermore, the origin MOCDOA + strategy1 and the origin MOCDOA + strategy1 + strategy2 have no multiple nondominant solutions converging in a small region.This shows that the strategy1 (the leader-updating strategy) and the strategy2 (the wise random perturbation strategy) can improve the uniformity of Pareto optimal solutions.
(2) As indicated by the first graph, the third graph, and the fourth graph in Figure 3, the coverage of the extreme solutions of the Pareto front for the origin MOCDOA + strategy1 + strategy2 marked by two green diamonds is more widespread than that of the origin MOCDOA marked by two blue squares.The coverage of the extreme solutions of the Pareto front for the origin MOCDOA + strategy1 + strategy2 + strategy3 marked by two red circles is more widespread than that of the origin MOCDOA + strategy1 + strategy2.Based on the above, one can draw a conclusion that the strategy3 (the geometric center-updating strategy) can increase the diversity of the Pareto optimal solutions, and combining the strategy3 with the other two strategies does not destroy the diversity.
. .IEEE -Bus -Unit System.All the experiments in this section are carried out for IEEE 30-bus 6-unit system.The first experiment is performed to evaluate the MOCDOA's performance by comparing the results of the extreme solutions and the compromise solutions in all cases for MOCDOA and other algorithms.The second experiment is carried out to evaluate the solution quality by comparing three metrics, i.e., SP, HV, and CM for four algorithms.The third experiment is implemented to analyze the robustness of MOCDOA by comparing the statistical results for the solutions of the minimal fuel cost, the minimal emission, and the ASD of compromise solution.

. . . Comparison of Extreme Solutions and Compromise Solutions.
Initially, the basic CDOA is implemented to optimize the fuel cost and the emission individually in order to explore the extreme points of the trade-off surface in all cases.The obtained best results are given in Table 5.The convergence of the fuel cost and the emission objectives for Case1 and Case2 are shown in Figure 4. CDOA with the cost as only objective function obtains the optimal values 600.111408 $/h and 605.998370 $/h for Case1 and Case2, respectively.The optimal values of CDOA with the emission as only objective function are 0.194203 ton/h and 0.194179  ton/h for Case1 and Case2, respectively.Next, the results of the multiobjective extreme solutions and the compromise solutions are discussed.The best results in all the tables are highlighted in bold for each case.
For Case 1, an experiment is performed to search for the extreme solutions and the compromise solution on the Pareto optimal set.The distribution of the nondominated solutions in Pareto front is displayed in Figure 5, which indicates clearly that these found solutions are almost well distributed on the entire Pareto front of Case1.The best results for the fuel cost and the emission (the extreme points on the Pareto front) obtained by MOCDOA are compared with those obtained by linear programming(LP) [4], multiobjective stochastic search technique(MOSST) [37], NSGA [16], NPGA [16], SPEA [38], NSGA-II [20], FCPSO [25], and BB-MOPSO [26] in Tables 6 and 7.The average satisfactory degree (ASD) [26] of the decision-maker for the compromise solution of MOCDOA is then calculated.The results of the compromise solutions and ASDs of MOCDOA, BB-MOPSO, NSGA, NPGA, SPEA, and FCPSO are shown in Table 8.Indeed, we use ASDs to compare the compromise solutions of the different algorithms.From Table 6, it is quite evident that the proposed MOCDOA performs better than LP, MOSST, NSGA, NPGA, SPEA, NSGA-II, and FCPSO and almost as the same as BB-MOPSO in terms of the minimum fuel cost.Moreover, MOCDOA outperforms NSGA, NPGA, SPEA, and FCPSO and as the same as NSGA-II and BB-MOPSO in terms of the lowest fitness function evaluations (FEs).MOCDOA performs better than LP, MOSST, NSGA, and SPEA and almost as the same as NPGA, NSGA-II, FCPSO, and BB-MOPSO in terms of the error of equality constraint to obtain a Pareto front equal to 0. In Table 7, the minimum emission and the error of equality constraint in MOCDOA are equal to 0.194203 ton/h and 0, respectively, by using 10,000 FEs.MOCDOA has the lowest error of equality constraint and the minimal FEs for all the compared algorithms.Although MOSST has the minimum emission equal to 0.19418 ton/h, its error of equality constraint is the largest one equal to 0.027.MOCDOA's emission is close to the minimum emission equal to 0.1942 ton/h while the condition of zero error of equality constraint is satisfied.In addition, MOCDOA provides a higher ASD of compromise solution than those of the other five algorithms as shown in Table 8.As seen from the above discussions, MOCDOA is more efficient than almost all the other compared algorithms.
For Case2, this problem has been solved by using NSGA and NPGA in [16], SPEA in [38], NSGA-II in [20], and FCPSO in [25].The minimum fuel cost and the minimum emission for MOCDOA and these five algorithms are presented in Tables 9 and 10.
From Tables 9 and 10, MOCDOA obtains the minimum fuel cost and the minimum emission equal to 605.999549 $/h and 0.194179 ton/h, respectively, by using 20,000 FEs, which is lower than those of other five algorithms.Except NSGA-II with the lowest FEs 10,000, MOCDOA outperforms the other 4 algorithms in terms of FEs.Thus, the above results illustrate the stronger competitiveness of MOCDOA than other algorithms for the best solutions and the less computational time.
. . .Comparison of Solution Quality.Unlike single-objective optimization problems, the evaluation of solution quality of a multiobjective optimization problem is substantially more complex.The following criteria are generally considered Complexity 13 to evaluate the solution quality for multiobjective optimization problems [39].
(ii) Diversity.Maximum the distribution extent of the obtained nondominated set.
(iii) Convergence.Minimum the distance of the obtained Pareto optimal set and the true Pareto optimal front.
To evaluate MOCDOA's performance on the solution quality, three well-known algorithms including MOPSO [21], NSGA-II [19], and PESA-II [36] are selected and compared with MOCDOA.Here, we set all algorithms to have the same population size, archive size and maximum iteration.The constraint-handling strategy proposed in Section 5.2 is adopted into the four algorithms.The results of different algorithms are compared in terms of the above three criteria.
For comparing the uniformity of Pareto optimal solutions, the spacing metric (SP) [40] is adopted to measure the uniformity of the obtained nondominated solutions.The calculation of the SP is as follows: where   refers to the Euclidean distance of two consecutive solutions in the external archive and  is the mean of all   .The smaller the SP value is, the more uniform the distribution of solutions on the obtained Pareto front is.SP=0 represents that all solutions of the obtained Pareto front are equidistantly spaced.6.It can be deduced from Figure 6 that the distribution of MOCDOA shows an advantage over the other three algorithms.The above discussions confirm that MOCDOA has a better uniformity performance than the other three algorithms.
A performance metric of the convergence and the diversity of the Pareto optimal solutions, hypervolume (HV) [41], was proposed by Zitler and Thiele.The metric calculates the volume covered by all the solutions of a nondominated set and a given reference point.The HV is defined as follows: where a hypercube volume V  is calculated with a reference point   and a solution   ∈  as the diagonal corners of a hypercube.For HV, a higher value is better.The reference point   will affect the calculation of HV.In our experiments, a same reference point is used for all the algorithms.The statistical results of HV are compared in Table 12 among four different algorithms.From Table 12, it can be seen that the proposed MOCDOA obtains the largest HV values, which means that MOCDOA has a better convergence and a diversity performance than these of MOPSO, NSGA-II, and PESA-II.
To evaluate the quality of the obtained Pareto optimal solutions of the optimized problem with the unknown true Pareto front, C-metric (CM) [42] is quite often used.It can be used to show the dominance relationship between two different algorithms.CM is defined as follows: where  1 and  2 are two solution sets of two different algorithms.( 1 ,  2 ) = 1 indicates that all solutions in  2 are dominated by the solutions in  1 .This shows that  1 is closer to the true Pareto optimal front than  2 .From the above analysis, it can be concluded that MOCDOA has a better performance of uniformity, diversity and convergence than those of the other algorithms in Case2.More specifically, the better uniformity performance of MOCDOA attributes to the common efforts of the leader-updating strategy in Section 4.1 and the wise random . . .Robustness Analysis.In order to further investigate the robustness of MOCDOA for the EED problem, 30 trials for each case are performed to obtain the statistical results for the solutions of three objectives, that is, the minimal fuel cost, the minimal emission, and the ASD of compromise solution.Tables 14 and 15 list the statistical results of the solutions of the three objectives, respectively.Figure 7 depicts the box and whiskers plots of the solutions of the three objectives for two cases.
It can easily be seen from Figure 7 that the solutions of each trial remain close to the best obtained values for both cases.In Table 14, the standard deviations of the three objectives are 0.041390, 3.33 − 05, and 0.002742 for Case1.In Table 15, the standard deviations of the three objective are 0.026281, 1.98 − 05, and 0.003077 for Case2.From Tables 14  and 15, it is clear that the standard deviations of the solutions for the three objectives are small.This illustrates that the proposed MOCDOA provides the high-quality solutions and has a strong robustness for solving the constrained EED problem.
. .IEEE -Bus -Unit System.In this section, two experiments are carried out for IEEE 30-bus 10-unit system.The first experiment is to minimize the fuel cost and emission objectives by using basic CDOA individually.The second experiment is to compare the best solutions of the fuel cost, the best solutions of the emission, and the compromise solutions for the four algorithms on Case1 and Case2.
The fuel cost and the emission are minimized individually by the basic CDOA in all cases.Table 16 shows the best solutions.As in the above case, bold values in all tables represent the best results obtained for each case.The convergence of the fuel cost and the emission objectives for Case1 and Case2 is shown in Figure 8.The minimum values to consider the cost as only objective function are 106183.951158$/h and 111521.601406$/h on Case1 and Case2, respectively.The minimum values to consider the emission as only objective function are 3651.072701ton/h and 3933.012596ton/h on Case1 and Case2, respectively.Multiobjective results obtained by optimizing the cost and the emission simultaneously are discussed below.
In order to express how competitive the proposed algorithm is, it is compared with MOPSO, NSGA-II, and PESA-II.For fair comparison, 30 independent optimization runs have been carried out.The beat fuel cost, the best emission solutions, and the compromise solutions are given in Tables 17, 18, and 19, respectively.Figure 9 shows the Pareto fronts of MOCDOA, MOPSO, NSGA-II, and PESA-II.
From Table 17, the cost value of MOCDOA is smaller than those of the other three algorithms in Case1.In case2, the cost value of PESA-II is the smallest in those of the four algorithms, and the value of MOCDOA is only worse than that of PESA-II.From Table 18, the best emission value of MOPSO is the smallest in those of the four algorithms for Case1 and Case2.The best emission value of MOCDOA is worse than that of MOPSO and is better than those of the other two algorithms.As can be seen from Table 19, the ASD of MOCDOA is the best in those of the four algorithms; that is, the satisfaction of MOCDOA is the best in those of the four algorithms.Moreover, the EED problem is a multiobjective problem, and the results from Table 19 are the final results of the IEEE 30-bus 10-unit system in the two cases.It can be seen from Figure 9 that the Pareto front of MOCDOA is more homogeneous than those of the other three algorithms.exhibit that MOCDOA has a good compromise solution and highly diverse Pareto optimal solutions in the lossless and loss-considered cases.Compared with other well-known algorithms for three metrics, SP, HV, and CM, MOCDOA reveals its superior characteristics and strong robust in the EED problem.It can be concluded that MOCDOA has the potential to be applied for solving some other multiobjective power system optimization problems.

𝑗
and  max  are the maximum and minimum values of the th objective function, respectively.  (  ) is the th objective function value of the th solution.

Figure 1 :
Figure 1: Calculation of maximum distance and sparse direction.

Figure 4 :
Figure 4: IEEE 30-bus 6-unit system convergence of cost and emission objective functions on Case1 and Case2.

Figure 6 :
Figure 6: IEEE 30-bus 6-unit system Pareto fronts and compromise solution for the four algorithms on Case2.

Figure 8 :
Figure 8: IEEE 30-bus 10-unit system convergence of cost and emission objective functions on Case1 and Case2.

Table 1 :
Generator cost and emission coefficients in the IEEE 30-bus 6-unit system.(  +     +    2  ) +        ) (4) where   ,   ,   ,   , and   are the emission function coefficients of the th generator.The parameters   ,   ,   ,   , and   are shown in Tables . .Local Leader-Updating Strategy.The new personal best position, i.e., the new local leader, is updated according to the non dominated relationship between the current individual   and the old local leader   in MOCDOA.If  dominates   , we keep  in memory.If  is dominated by   ,   is selected as a new local leader to replace the old one.Otherwise, we randomly choice one of   and  as the new local leader.The local leader-updating shown in (24) is performed by Function Local leader updating in the line 29 of Algorithm 1.

Table 2 :
Transmission loss coefficients in the IEEE 30-bus 6-unit system.

Table 3 :
Generator cost and emission coefficients in the IEEE 30-bus 10-unit system.

Table 4 :
Generator cost and emission coefficients in the IEEE 30-bus 10-unit system.

Table 6 :
IEEE 30-bus 6-unit system best solutions for cost with nine algorithms on Case1.

Table 7 :
IEEE 30-bus 6-unit system best solutions for emission with nine algorithms on Case1.

Table 8 :
IEEE 30-bus 6-unit system best compromise solutions with six algorithms on Case1.

Table 9 :
IEEE 30-bus 6-unit system best solutions for cost with six algorithms on Case2.

Table 10 :
IEEE 30-bus 6-unit system best solutions for emission with six algorithms on Case2.

Table 11 :
IEEE 30-bus 6-unit system statistical results of the SP on Case2.
Table 11illustrates the comparison results of SP for different algorithms.It can be seen from this table that the average performance of MOCDOA is far better than those of the other algorithms and the standard deviation of MOCDOA is smallest.In order to compare intuitively the uniformity of the solutions obtained by MOCDOA and the other algorithms, the Pareto fronts of MOCDOA, MOPSO, NSGA-II, and PESA-II are depicted together, as shown in Figure

Table 12 :
IEEE 30-bus 6-unit system statistical results of the HV on Case2.,  2 ) = 0 represents that none solution in  2 is covered by  1 .Table13shows the comparison results of CM produced by the best solutions of different algorithms.From this table, it can be seen that none solution in MOCDOA is dominated by that of MOPSO, and near 4% and 2% solutions of MOCDOA are dominated by those of NSGA-II and PESA-II, respectively.Moreover, MOPSO, NSGA-II, and PESA-II have 60%, 18%, and 16% solutions to be dominated by MOCDOA, respectively.Thus, MOCDOA has a better convergence performance than the three algorithms.

Table 13 :
IEEE 30-bus 6-unit system statistical results of the CM on Case2.

Table 14 :
IEEE 30-bus 6-unit system statistical results of the three objectives on Case1.

Table 15 :
IEEE 30-bus 6-unit system statistical results of the three objectives on Case2.The better diversity performance of MOCDOA attributes to the effort of the geometric center-updating strategy in Section 4.3.The better convergence performance of MOCDOA attributes to the effort of a lot of random variables in the original CDOA, which results in the fact that the population of MOCDOA is able to escape the local Pareto front.

Table 16 :
IEEE 30-bus 10-unit system best solutions for cost and emission optimized individually.