# Multiobjective Collective Decision Optimization Algorithm for Economic Emission Dispatch Problem

Authors: Xinlin Xu; Zhongbo Hu; Qinghua Su; Zenggang Xiong
Journal: Complexity
 (2018)

---

## Abstract

The collective decision optimization algorithm (CDOA) is a new stochastic population-based evolutionary algorithm which simulates the decision behavior of human. In this paper, a multiobjective 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.


---

## Body

## 1. 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 SO2, CO2, and NOx, 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 [9–14]. This approach generally uses the following formula to transform two objectives into a single objective:(1)minwFPG+1-wσEPGwhereσ is the scaling factor which combines the total fuel cost F(PG) with the total emission E(PG) and w is the weight factor in the range of [0,1]. Each objective is multiplied by a weight related to a given value w, 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 w 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. Section2 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.


## 2. 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 Section2.1 and Section 2.2 in detail. The EED problem can be mathematically formulated as(2)minFPG,EPGst.hPG=0gpG≤0where F and E denote total fuel cost and total emission, respectively. h and g are the equality and inequality constraints, respectively.


### 2.1. Objective Functions

#### 2.1.1. Fuel Cost Minimization

The total fuel costF(PG) (dollars per hour) is made up of n generator costs expressed by quadratic functions. The total fuel cost F(PG) can be represented as (3)FPG=∑i=1nai+biPGi+ciPGi2where the vector PG=(PG1,PG2,⋯,PGn) is the real power outputs of generators. PGi is the real power output of the ith generator. ai, bi, and ci are the cost function coefficients of the ith generator.


#### 2.1.2. Emission Minimization

The emission function can be presented as the sum of all types of the emission considered, but only the emission of nitrogen oxidesNOx is considered in the present study. The total emission E(PG)(ton/h) of nitrogen oxides NOx 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.(4)EPG=∑i=1n10-2αi+βiPGi+γiPGi2+ζieλiPGiwhere αi, βi, γi, ζi, and λi are the emission function coefficients of the ith generator. The parameters αi, βi, γi, ζi, and λi are shown in Tables 1 and 3.Table 1
Generator cost and emission coefficients in the IEEE 30-bus 6-unit system.
Unit p m i n p m a x a i b i c i α i β i γ i ζ i λ i G 1 0.05 0.5 10 200 100 4.091 - 5.554 6.490 2.0 E - 4 2.857 G 2 0.05 0.6 10 150 120 2.543 - 6.047 5.638 5.0 E - 4 3.333 G 3 0.05 1 20 180 40 4.258 - 5.094 4.586 1.0 E - 6 8.000 G 4 0.05 1.2 10 100 60 5.326 - 3.550 3.380 2.0 E - 3 2.000 G 5 0.05 1 20 180 40 4.258 - 5.094 4.586 1.0 E - 6 8.000 G 6 0.05 0.6 10 150 100 6.131 - 5.555 5.151 1.0 E - 5 6.667


### 2.2. Constraints

The power dispatch constraints include one equality constraint on the power balance and several inequality constraints on generation capacity.


#### 2.2.1. Power Balance Constraint

The total power generation must cover the power demandPD and the transmission network loss PL, namely,(5)∑i=1nPGi=PD+PLHere, in general,PL is determined by Krons loss formula [30] as(6)PL=∑i=1n∑j=1nPGiBijPGj+∑i=1nB0iPGi+B00where Bij, B0i, and B00 are the transmission network power loss coefficients.


#### 2.2.2. Generation Capacity Constraints

The power output of each generator is limited by its corresponding lower and upper bounds as shown in(7)PGimin≤PGi≤PGimax,i=1,2,⋯,n.where PGimin, PGimax are the minimum and maximum generation limits of the ith generator, respectively.


## 2.1. Objective Functions

### 2.1.1. Fuel Cost Minimization

The total fuel costF(PG) (dollars per hour) is made up of n generator costs expressed by quadratic functions. The total fuel cost F(PG) can be represented as (3)FPG=∑i=1nai+biPGi+ciPGi2where the vector PG=(PG1,PG2,⋯,PGn) is the real power outputs of generators. PGi is the real power output of the ith generator. ai, bi, and ci are the cost function coefficients of the ith generator.


### 2.1.2. Emission Minimization

The emission function can be presented as the sum of all types of the emission considered, but only the emission of nitrogen oxidesNOx is considered in the present study. The total emission E(PG)(ton/h) of nitrogen oxides NOx 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.(4)EPG=∑i=1n10-2αi+βiPGi+γiPGi2+ζieλiPGiwhere αi, βi, γi, ζi, and λi are the emission function coefficients of the ith generator. The parameters αi, βi, γi, ζi, and λi are shown in Tables 1 and 3.Table 1
Generator cost and emission coefficients in the IEEE 30-bus 6-unit system.
Unit p m i n p m a x a i b i c i α i β i γ i ζ i λ i G 1 0.05 0.5 10 200 100 4.091 - 5.554 6.490 2.0 E - 4 2.857 G 2 0.05 0.6 10 150 120 2.543 - 6.047 5.638 5.0 E - 4 3.333 G 3 0.05 1 20 180 40 4.258 - 5.094 4.586 1.0 E - 6 8.000 G 4 0.05 1.2 10 100 60 5.326 - 3.550 3.380 2.0 E - 3 2.000 G 5 0.05 1 20 180 40 4.258 - 5.094 4.586 1.0 E - 6 8.000 G 6 0.05 0.6 10 150 100 6.131 - 5.555 5.151 1.0 E - 5 6.667


## 2.1.1. Fuel Cost Minimization

The total fuel costF(PG) (dollars per hour) is made up of n generator costs expressed by quadratic functions. The total fuel cost F(PG) can be represented as (3)FPG=∑i=1nai+biPGi+ciPGi2where the vector PG=(PG1,PG2,⋯,PGn) is the real power outputs of generators. PGi is the real power output of the ith generator. ai, bi, and ci are the cost function coefficients of the ith generator.


## 2.1.2. Emission Minimization

The emission function can be presented as the sum of all types of the emission considered, but only the emission of nitrogen oxidesNOx is considered in the present study. The total emission E(PG)(ton/h) of nitrogen oxides NOx 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.(4)EPG=∑i=1n10-2αi+βiPGi+γiPGi2+ζieλiPGiwhere αi, βi, γi, ζi, and λi are the emission function coefficients of the ith generator. The parameters αi, βi, γi, ζi, and λi are shown in Tables 1 and 3.Table 1
Generator cost and emission coefficients in the IEEE 30-bus 6-unit system.
Unit p m i n p m a x a i b i c i α i β i γ i ζ i λ i G 1 0.05 0.5 10 200 100 4.091 - 5.554 6.490 2.0 E - 4 2.857 G 2 0.05 0.6 10 150 120 2.543 - 6.047 5.638 5.0 E - 4 3.333 G 3 0.05 1 20 180 40 4.258 - 5.094 4.586 1.0 E - 6 8.000 G 4 0.05 1.2 10 100 60 5.326 - 3.550 3.380 2.0 E - 3 2.000 G 5 0.05 1 20 180 40 4.258 - 5.094 4.586 1.0 E - 6 8.000 G 6 0.05 0.6 10 150 100 6.131 - 5.555 5.151 1.0 E - 5 6.667


## 2.2. Constraints

The power dispatch constraints include one equality constraint on the power balance and several inequality constraints on generation capacity.


### 2.2.1. Power Balance Constraint

The total power generation must cover the power demandPD and the transmission network loss PL, namely,(5)∑i=1nPGi=PD+PLHere, in general,PL is determined by Krons loss formula [30] as(6)PL=∑i=1n∑j=1nPGiBijPGj+∑i=1nB0iPGi+B00where Bij, B0i, and B00 are the transmission network power loss coefficients.


### 2.2.2. Generation Capacity Constraints

The power output of each generator is limited by its corresponding lower and upper bounds as shown in(7)PGimin≤PGi≤PGimax,i=1,2,⋯,n.where PGimin, PGimax are the minimum and maximum generation limits of the ith generator, respectively.


## 2.2.1. Power Balance Constraint

The total power generation must cover the power demandPD and the transmission network loss PL, namely,(5)∑i=1nPGi=PD+PLHere, in general,PL is determined by Krons loss formula [30] as(6)PL=∑i=1n∑j=1nPGiBijPGj+∑i=1nB0iPGi+B00where Bij, B0i, and B00 are the transmission network power loss coefficients.


## 2.2.2. Generation Capacity Constraints

The power output of each generator is limited by its corresponding lower and upper bounds as shown in(7)PGimin≤PGi≤PGimax,i=1,2,⋯,n.where PGimin, PGimax are the minimum and maximum generation limits of the ith generator, respectively.


## 3. 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 Xi generates several candidate solutions on a multistep position-selected 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, leader-based phase, innovation-based phase, and selection.Step 1 (group generation).
CDOA usesN individuals as the group or population in each generation. The group and each individual at the tth iteration are denoted by pop(t)=X1t,…,Xit,…,XNt and Xi(t)=xi1t,xi2t,⋯,xiDt, respectively, where D denotes the dimension of each generation. The group is initialized as shown in (8), so the N individuals are uniformly distributed in the feasible solution space. (8)xik~Ulxk,uxkwhere U is the uniform distribution and lxk and uxk are the lower and upper limits of the kth dimension, respectively.Step 2 (experience-based phase).
In CDOA, personal experience represents the local leader, i.e., the personal best position(Xb) obtained by each individual itself so far. The new position of the individual Xi(t) will be expressed as follows:(9)newXi0=Xit+μ0→·stept·d0,d0=Xb-Xitwhere μ0→ is a random vector with each number uniformly distributed in the interval (0, 1) and step(t) denotes the step size of the tth iteration.(10)stept=2-1.7t-1Tmax-1where Tmax is the maximum number of iteration.Step 3 (others’-based phase).
In the meeting, all individuals will interact randomly. The individualXj(t) who is better than the current member Xi(t) is randomly selected from the group. The calculation formula of updating the newXi0 is defined as follows:(11)newXi1=newXi0+μ1→·stept·d1,d1=α1·d0+β1·Xjt-Xitwhere j denotes a random integer in [1,N], μ1→ is a random vector with each number uniformly distributed in (0, 1), and α1 and β1 are the random numbers in (-1, 1) and (0, 2), respectively.Step 4 (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 (XG) of all individuals. XG can be described as follows: (12)XG=1NX1t,X2t,⋯,XNt=1N∑i=1Nxi1t,1N∑i=1Nxi2t,⋯,1N∑i=1NxiNt
Here, updating thenewXi1 is calculated in the following formula:(13)newXi2=newXi1+μ2→·stept·d2,d2=α2·d1+β2·XGt-Xitwhere μ2→ is a random vector with each number uniformly distributed in (0, 1) and α2 and β2 are the random numbers in (-1, 1) and (0, 2), respectively.Step 5 (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 (XL). The next new position of ith individual will be designed as follows:(14)newXi3=newXi2+μ3→·stept·d3,d3=α3·d2+β3·XLt-Xitwhere μ3→ is a random vector with each number uniformly distributed in (0, 1) and α3 and β3 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 aroundXL, as shown in (15)newXq=XL+Wq→,q=1,2,3,4,5where Wq→ is a random vector with each number in (0, 1). The next leader newXL is produced in (16)newXL=newXk,k=min_ObjFunnewXq,q=1,2,3,4,5where min_ObjFun is the index of a minimum objective function value.Step 6 (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:(17)newXi4=newXi3,Xi4p~Ulxp,uxp,r0<MRnewXi3,otherwisetwhere r0 represents a random number in (0, 1), p is a random integer in [1,D], and MR is an innovation (mutation) factor. The MR is set to 0.8 in the following algorithm.Step 7 (selection).
In the selection phase, the fitness values inXi and Xi4 are compared to update the population pop by using a greedy selection.The procedure of CDOA again and again repeats from Step2 to Step 7 until the terminating condition is satisfied.


## 4. 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. Algorithm1 presents the pseudocode of MOCDOA.Algorithm 1:Multiobjective collective decision optimization algorithm.
Input: input parameters ObjFun,N,D,Na,Tmax Output:  Op 1:N: size of the population, D: dimension of the population, Na: maximum capacity of the archive, Tmax: maximum number of iterations,pop(t): population in tth iteration, Xi(t): ith individual of population pop(t) in tth iteration, Ar(t): external archive to store all the updated non-dominated solutions intth iteration 2:Initialization 3: Generate an initial populationpop(0)={X1(0),⋯,XN(0)},whereXi(0)={xi1(0),⋯,xiD(0)} 4:Xb←pop(0) % Initialize the local leader Xb 5:F(pop(0))←ObjFun(pop(0)) % Evaluate the fitness of each individual 6:Ar(0)←Non_dominated(pop(0)) % Return the non-dominated solutions from pop0 7:while  t≤Tmax  do 8:Mutation 9:Ar(t)←sort(Ar(t)) % Sort Ar by objective functions value 10:for  i=1toN  do 11:XLi(t)←Leader_updating_(Ar(t)) % Find the global best of each individual XLi(t) 12:XGi(t)←Geometric_center_updating_(Ar(t)) % Find the geometric center XGi(t) 13:if  rand<0.5  then 14:newpopi(t)←Leader_guiding_(XLi(t)) % Directed search XLi(t) neighbor 15:else 16:newpopi(t)←Individual_updating[newXi0(t),newXi1(t),newXi2(t),newXi3(t),newXi4(t)] % Update the ith individual 17:Selection 18:F(newpop(t))=ObjFun(newpop(t)) % Return the fitness value of each new population 19:for  i=1 to N  do 20:if  newpopi(t)≺Xi(t)   then 21:Xi(t+1)=newpopi(t) 22:F(Xi(t+1))=F(newpopi(t)) 23:else if  newpopi(t)⊀Xi(t)andXi(t)⊀newpopi(t)   then 24:Random access to the next generation 25:Ar(t+1)←Non_dominated(pop(t)∪Ar(t)) % Update the archive 26:if  |Ar(t+1)|>Na  then 27:Circular_crowded_sorting(Ar(t+1)) % Maintain the archive, where |Ar(t+1)| is the element number of Ar(t+1) 28:for  i=1 to N  do 29:Xbi(t+1)=Local_leader_updating(Xbi(t),Xi) % Update local leader 30:t=t+1 31:Op←Art and stop the algorithm %Output the obtained Pareto optimal solutionsSimilar to other evolutionary algorithms, MOCDOA can be divided into three processes, initialization, mutation, and selection. In the mutation phase of Algorithm1, 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.


### 4.1. 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 leaderXL 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 (md) 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 ith solution (mdi) is the maximum of two side-length sums of its two adjacent cuboids. The maximum distance of the ith solution (mdi) is calculated as follows:(18)mdi=maxudi,ldii=1,…,Nawhere Na is the maximum capacity of the archive. The distances of the ith solution to the upper point and the lower point are udi and ldi shown in (19) and (20), respectively.(19)udi=∑j=1MfjXi-fjXi-1fjmax-fjmin(20)ldi=∑j=1MfjXi-fjXi+1fjmax-fjminwhere M is the number of objective functions. fjmin and fjmax are the maximum and minimum values of the jth objective function, respectively. fj(Xi) is the jth objective function value of the ith solution.Figure 1
Calculation of maximum distance and sparse direction.Since there is only one neighbor point for each extreme point in the archive, the maximum distance (md) 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 XL, which is shown in the line 11 of Algorithm 1. The leader-updating strategy is illustrated as Algorithm 2.Algorithm 2:Leader-updating.
Input:  Ar(t) Output:  XL 1:Nt=|Ar(t)| %  Nt is the element number of Ar(t) 2:for  i=1toNt  do 3:calculate mdi and li 4:for  i=1toN  do %  N is size of the population 5:%  XLi(t) is selected according to the probability based on the percentage of the mdi as a whole 6:indexi=Roulette_wheel_selection(md) 7:XLi(t)=Arindexi(t)Here, we design a concept of sparse direction (l) for the wise random perturbation strategy in the next Section 4.3. The udi and ldi of the ith solution of the archive are compared to record the sparse direction of the ith solution li, which will be used as a sparse mark of the solution to guide the perturbation in following strategy. If ldi is greater than udi; that is, the gap of the ith solution for its upper neighbor solution is larger than that for its lower neighbor solution; we record li=1. Otherwise, li=-1. For example, in Figure 1, ldA is greater than udA for the Ath solution, so lA=1. For the Bth solution, udB is greater than ldB, such that lB=-1. The calculation formula of li is designed as (21)li=1,ldi>udi-1,ldi≤udi


### 4.2. 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 aroundXL. The perturbation method is shown in (22)newpopit=XLit+r1·Arindexi+lit-XLitwhere r1 is a random number in (0, 1). The strategy makes XLi move to the direction of its sparse neighbor, thus generating a new individual newpopi, which can reduce the sparsity of the external archive. We use Function Leader_guiding, shown in the line 14 of Algorithm 1, to update Xi.


### 4.3. Geometric Center-Updating Strategy

For improving the diversity performance of the external archive in MOCDOA, we use Function Geometric_center_updating to randomly selectXG in the extreme points of the archive. Function Geometric_center_updating is shown in the line 12 of Algorithm 1. The calculation formula of XG is designed as (23)XGt=Ar1t,r2<0.5Arendt,r2≥0.5where r2 is a random number in (0, 1).


### 4.4. 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 individualXi and the old local leader Xbi in MOCDOA. If Xb dominates Xi, we keep Xb in memory. If Xb is dominated by Xi, Xi is selected as a new local leader to replace the old one. Otherwise, we randomly choice one of Xi and Xb 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. (24)Xbit+1=Xbit,Xbit≺XitXit,Xit≺XbitXbitorXit,Xbit⊀Xit∧Xit⊀Xbit


### 4.5. 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.Algorithm 3:Cycled crowding sorting.
Input:  Ar(t),Na Output:  Ar(t+1) 1:Nt=|Ar(t)| %  Nt is the element number of Ar(t) 2:while  Nt>Na  do %  Na is maximum capacity of the archive 3:for  i=1toNt  do 4:Ari(t).distance=0 % Initialize the crowding distance of Ar(t) 5:for  m=1toM  do %  M is the number of objective functions 6:Ar(t)=sort(Ar(t),m) % Sort Ar by mth objective functions value 7:Ar1(t).distance=Inf % Set the crowding distance of extreme solutions equal to infinite 8:ArNt(t).distance=Inf 9:for i=2 to Nt-1 do 10:Ari(t).distance=Ari(t).distance+Ari+1(t).distance-Ari-1(t).distanceArNt(t).distance-Ar1(t).distance 11:k=min_Ar(t).distance % Find the solution k with the smallest crowding distance 12:Ark(t)= % Delete the solution k from A 13:Nt=Nt-1 % Prepare for the next cycle 14:Ar(t+1)←Ar(t)


## 4.1. 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 leaderXL 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 (md) 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 ith solution (mdi) is the maximum of two side-length sums of its two adjacent cuboids. The maximum distance of the ith solution (mdi) is calculated as follows:(18)mdi=maxudi,ldii=1,…,Nawhere Na is the maximum capacity of the archive. The distances of the ith solution to the upper point and the lower point are udi and ldi shown in (19) and (20), respectively.(19)udi=∑j=1MfjXi-fjXi-1fjmax-fjmin(20)ldi=∑j=1MfjXi-fjXi+1fjmax-fjminwhere M is the number of objective functions. fjmin and fjmax are the maximum and minimum values of the jth objective function, respectively. fj(Xi) is the jth objective function value of the ith solution.Figure 1
Calculation of maximum distance and sparse direction.Since there is only one neighbor point for each extreme point in the archive, the maximum distance (md) 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 XL, which is shown in the line 11 of Algorithm 1. The leader-updating strategy is illustrated as Algorithm 2.Algorithm 2:Leader-updating.
Input:  Ar(t) Output:  XL 1:Nt=|Ar(t)| %  Nt is the element number of Ar(t) 2:for  i=1toNt  do 3:calculate mdi and li 4:for  i=1toN  do %  N is size of the population 5:%  XLi(t) is selected according to the probability based on the percentage of the mdi as a whole 6:indexi=Roulette_wheel_selection(md) 7:XLi(t)=Arindexi(t)Here, we design a concept of sparse direction (l) for the wise random perturbation strategy in the next Section 4.3. The udi and ldi of the ith solution of the archive are compared to record the sparse direction of the ith solution li, which will be used as a sparse mark of the solution to guide the perturbation in following strategy. If ldi is greater than udi; that is, the gap of the ith solution for its upper neighbor solution is larger than that for its lower neighbor solution; we record li=1. Otherwise, li=-1. For example, in Figure 1, ldA is greater than udA for the Ath solution, so lA=1. For the Bth solution, udB is greater than ldB, such that lB=-1. The calculation formula of li is designed as (21)li=1,ldi>udi-1,ldi≤udi


## 4.2. 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 aroundXL. The perturbation method is shown in (22)newpopit=XLit+r1·Arindexi+lit-XLitwhere r1 is a random number in (0, 1). The strategy makes XLi move to the direction of its sparse neighbor, thus generating a new individual newpopi, which can reduce the sparsity of the external archive. We use Function Leader_guiding, shown in the line 14 of Algorithm 1, to update Xi.


## 4.3. Geometric Center-Updating Strategy

For improving the diversity performance of the external archive in MOCDOA, we use Function Geometric_center_updating to randomly selectXG in the extreme points of the archive. Function Geometric_center_updating is shown in the line 12 of Algorithm 1. The calculation formula of XG is designed as (23)XGt=Ar1t,r2<0.5Arendt,r2≥0.5where r2 is a random number in (0, 1).


## 4.4. 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 individualXi and the old local leader Xbi in MOCDOA. If Xb dominates Xi, we keep Xb in memory. If Xb is dominated by Xi, Xi is selected as a new local leader to replace the old one. Otherwise, we randomly choice one of Xi and Xb 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. (24)Xbit+1=Xbit,Xbit≺XitXit,Xit≺XbitXbitorXit,Xbit⊀Xit∧Xit⊀Xbit


## 4.5. 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.Algorithm 3:Cycled crowding sorting.
Input:  Ar(t),Na Output:  Ar(t+1) 1:Nt=|Ar(t)| %  Nt is the element number of Ar(t) 2:while  Nt>Na  do %  Na is maximum capacity of the archive 3:for  i=1toNt  do 4:Ari(t).distance=0 % Initialize the crowding distance of Ar(t) 5:for  m=1toM  do %  M is the number of objective functions 6:Ar(t)=sort(Ar(t),m) % Sort Ar by mth objective functions value 7:Ar1(t).distance=Inf % Set the crowding distance of extreme solutions equal to infinite 8:ArNt(t).distance=Inf 9:for i=2 to Nt-1 do 10:Ari(t).distance=Ari(t).distance+Ari+1(t).distance-Ari-1(t).distanceArNt(t).distance-Ar1(t).distance 11:k=min_Ar(t).distance % Find the solution k with the smallest crowding distance 12:Ark(t)= % Delete the solution k from A 13:Nt=Nt-1 % Prepare for the next cycle 14:Ar(t+1)←Ar(t)


## 5. Implementation of MOCDOA

In this section, the proposed MOCDOA is applied for solving the EED problem with one equality constraint on the power 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.


### 5.1. 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 Algorithm4. In this model, we set σ=1e-12.Algorithm 4:Constraint-handling.
Input:  Xi,Pmax,Pmin Output:  Xi′′ 1:Xi:ith individual,Pmax: the upper limits of generation capacity, Pmin: the lower limits of generation capacity 2:e=PL+PD-sum(Xi) % Calculate and return the error between PL+PD and the element sum of Xi 3:while  |e|>σdo % Check the feasibility of Xi 4:k=rand(1,N) % Pick a random integer between 1 to N 5:Xik′=Xik(PD+PL)/sum(Xik) % Adjust Xi to make it satisfy the constraint 6:if  Xik′>PkmaxorXik′<Pkmin  then 7:Xik′=Pkmin+rand∗Pkmax-Pkmin 8:Xik=Xik′ 9:e=PL+PD-sum(Xi) % Calculate and return the error again


### 5.2. 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(25)μij=1,fjXi≤fjminfjmax-fjXifjmax-fjmin,fjmin≤fjXi≤fjmax0,fjXi≥fjmaxThe membership function value represents the degree of achievement of an objective function as a value between 0 and 1.μij=1 is expressed as completely satisfactory, and μij=0 is expressed as unsatisfactory. Figure 2 illustrates a fuzzy-based membership function. The normalized membership function μk is calculated at the kth solution as follows:(26)μk=∑j=1Mμkj∑i=1Nt∑j=1Mμijwhere M denotes the number of the objective functions and Nt is the number of the solutions in the final nondominated front. The best compromise solution is the one getting the highest value of μk.Figure 2
The fuzzy-based membership function.


### 5.3. 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.Table 2
Transmission loss coefficients in the IEEE 30-bus 6-unit system.
0.1382 - 0.0299 0.0044 - 0.0022 - 0.0010 - 0.0008 - 0.0299 0.0487 - 0.0025 0.0004 0.0016 0.0041 B= 0.0044 - 0.0025 0.0182 - 0.0070 - 0.0066 - 0.0066 B 00=0.00098573 - 0.0022 0.0004 - 0.0070 0.0137 0.0050 0.0033 - 0.0010 0.0016 - 0.0066 0.0050 0.0109 0.0005 - 0.0008 0.0041 - 0.0066 0.0033 0.0005 0.0244 B 0 = 0.0107 0.0060 - 0.0017 0.0009 0.0002 0.0030Table 3
Generator cost and emission coefficients in the IEEE 30-bus 10-unit system.
Unit p m i n p m a x a i b i c i d i e i α i β i γ i ζ i λ i G 1 10 55 1000.403 40.5407 0.12951 33 0.0174 360.0012 - 3.9864 0.04702 0.25475 0.01234 G 2 20 80 950.606 39.5804 0.10908 25 0.0178 350.0056 - 3.9524 0.04652 0.25475 0.01234 G 3 47 120 900.705 36.5104 0.12511 32 0.0162 330.0056 - 3.9023 0.04652 0.25163 0.01215 G 4 20 130 800.705 39.5104 0.12111 30 0.0168 330.0056 - 3.9023 0.04652 0.25163 0.01215 G 5 50 160 756.799 38.5390 0.15247 30 0.0148 13.8593 0.3277 0.00420 0.24970 0.01200 G 6 70 240 451.325 46.1592 0.10587 20 0.0163 13.8593 0.3277 0.00420 0.24970 0.01200 G 7 60 300 1243.531 38.3055 0.03546 20 0.0152 40.2669 - 0.5455 0.00680 0.24800 0.01290 G 8 70 340 1049.998 40.3965 0.02803 30 0.0128 40.2669 - 0.5455 0.00680 0.24990 0.01203 G 9 135 470 1658.569 36.3278 0.02111 60 0.0136 42.8955 - 0.5112 0.00460 0.25470 0.01234 G 10 150 470 1356.659 38.2704 0.01799 40 0.0141 42.8955 - 0.5112 0.00460 0.25470 0.01234Table 4
Generator cost and emission coefficients in the IEEE 30-bus 10-unit system.
0.000049 0.000014 0.000015 0.000015 0.000016 0.000017 0.000017 0.000018 0.000019 0.000020 0.000014 0.000045 0.000016 0.000016 0.000017 0.000015 0.000015 0.000016 0.000018 0.000018 B= 0.000015 0.000016 0.000039 0.000010 0.000012 0.000012 0.000014 0.000014 0.000016 0.000016 0.000015 0.000016 0.000010 0.000040 0.000014 0.000010 0.000011 0.000012 0.000014 0.000015 0.000016 0.000017 0.000012 0.000014 0.000035 0.000011 0.000013 0.000013 0.000015 0.000016 0.000017 0.000015 0.000012 0.000010 0.000011 0.000036 0.000012 0.000012 0.000014 0.000015 0.000017 0.000015 0.000014 0.000011 0.000013 0.000012 0.000038 0.000016 0.000016 0.000018 0.000018 0.000016 0.000014 0.000012 0.000013 0.000012 0.000016 0.000040 0.000015 0.000016 0.000019 0.000018 0.000016 0.000014 0.000015 0.000014 0.000016 0.000015 0.000042 0.000019 0.000020 0.000018 0.000016 0.000015 0.000016 0.000015 0.000018 0.000016 0.000019 0.000044 B0= 0  B 00 = 0By using the orthogonal experimental method, the parametersN and Na are tuned and adjusted until the optimal settings are determined. In these two cases, the population size (N) and the maximum capacity of the archive (Na) 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.


## 5.1. 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 Algorithm4. In this model, we set σ=1e-12.Algorithm 4:Constraint-handling.
Input:  Xi,Pmax,Pmin Output:  Xi′′ 1:Xi:ith individual,Pmax: the upper limits of generation capacity, Pmin: the lower limits of generation capacity 2:e=PL+PD-sum(Xi) % Calculate and return the error between PL+PD and the element sum of Xi 3:while  |e|>σdo % Check the feasibility of Xi 4:k=rand(1,N) % Pick a random integer between 1 to N 5:Xik′=Xik(PD+PL)/sum(Xik) % Adjust Xi to make it satisfy the constraint 6:if  Xik′>PkmaxorXik′<Pkmin  then 7:Xik′=Pkmin+rand∗Pkmax-Pkmin 8:Xik=Xik′ 9:e=PL+PD-sum(Xi) % Calculate and return the error again


## 5.2. 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(25)μij=1,fjXi≤fjminfjmax-fjXifjmax-fjmin,fjmin≤fjXi≤fjmax0,fjXi≥fjmaxThe membership function value represents the degree of achievement of an objective function as a value between 0 and 1.μij=1 is expressed as completely satisfactory, and μij=0 is expressed as unsatisfactory. Figure 2 illustrates a fuzzy-based membership function. The normalized membership function μk is calculated at the kth solution as follows:(26)μk=∑j=1Mμkj∑i=1Nt∑j=1Mμijwhere M denotes the number of the objective functions and Nt is the number of the solutions in the final nondominated front. The best compromise solution is the one getting the highest value of μk.Figure 2
The fuzzy-based membership function.


## 5.3. 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.Table 2
Transmission loss coefficients in the IEEE 30-bus 6-unit system.
0.1382 - 0.0299 0.0044 - 0.0022 - 0.0010 - 0.0008 - 0.0299 0.0487 - 0.0025 0.0004 0.0016 0.0041 B= 0.0044 - 0.0025 0.0182 - 0.0070 - 0.0066 - 0.0066 B 00=0.00098573 - 0.0022 0.0004 - 0.0070 0.0137 0.0050 0.0033 - 0.0010 0.0016 - 0.0066 0.0050 0.0109 0.0005 - 0.0008 0.0041 - 0.0066 0.0033 0.0005 0.0244 B 0 = 0.0107 0.0060 - 0.0017 0.0009 0.0002 0.0030Table 3
Generator cost and emission coefficients in the IEEE 30-bus 10-unit system.
Unit p m i n p m a x a i b i c i d i e i α i β i γ i ζ i λ i G 1 10 55 1000.403 40.5407 0.12951 33 0.0174 360.0012 - 3.9864 0.04702 0.25475 0.01234 G 2 20 80 950.606 39.5804 0.10908 25 0.0178 350.0056 - 3.9524 0.04652 0.25475 0.01234 G 3 47 120 900.705 36.5104 0.12511 32 0.0162 330.0056 - 3.9023 0.04652 0.25163 0.01215 G 4 20 130 800.705 39.5104 0.12111 30 0.0168 330.0056 - 3.9023 0.04652 0.25163 0.01215 G 5 50 160 756.799 38.5390 0.15247 30 0.0148 13.8593 0.3277 0.00420 0.24970 0.01200 G 6 70 240 451.325 46.1592 0.10587 20 0.0163 13.8593 0.3277 0.00420 0.24970 0.01200 G 7 60 300 1243.531 38.3055 0.03546 20 0.0152 40.2669 - 0.5455 0.00680 0.24800 0.01290 G 8 70 340 1049.998 40.3965 0.02803 30 0.0128 40.2669 - 0.5455 0.00680 0.24990 0.01203 G 9 135 470 1658.569 36.3278 0.02111 60 0.0136 42.8955 - 0.5112 0.00460 0.25470 0.01234 G 10 150 470 1356.659 38.2704 0.01799 40 0.0141 42.8955 - 0.5112 0.00460 0.25470 0.01234Table 4
Generator cost and emission coefficients in the IEEE 30-bus 10-unit system.
0.000049 0.000014 0.000015 0.000015 0.000016 0.000017 0.000017 0.000018 0.000019 0.000020 0.000014 0.000045 0.000016 0.000016 0.000017 0.000015 0.000015 0.000016 0.000018 0.000018 B= 0.000015 0.000016 0.000039 0.000010 0.000012 0.000012 0.000014 0.000014 0.000016 0.000016 0.000015 0.000016 0.000010 0.000040 0.000014 0.000010 0.000011 0.000012 0.000014 0.000015 0.000016 0.000017 0.000012 0.000014 0.000035 0.000011 0.000013 0.000013 0.000015 0.000016 0.000017 0.000015 0.000012 0.000010 0.000011 0.000036 0.000012 0.000012 0.000014 0.000015 0.000017 0.000015 0.000014 0.000011 0.000013 0.000012 0.000038 0.000016 0.000016 0.000018 0.000018 0.000016 0.000014 0.000012 0.000013 0.000012 0.000016 0.000040 0.000015 0.000016 0.000019 0.000018 0.000016 0.000014 0.000015 0.000014 0.000016 0.000015 0.000042 0.000019 0.000020 0.000018 0.000016 0.000015 0.000016 0.000015 0.000018 0.000016 0.000019 0.000044 B0= 0  B 00 = 0By using the orthogonal experimental method, the parametersN and Na are tuned and adjusted until the optimal settings are determined. In these two cases, the population size (N) and the maximum capacity of the archive (Na) 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.


## 6. 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.


### 6.1. Experimental Analysis of Three 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:(27)minf1=4x12+4x22minf2=x1-52+4x2-52st.0<x1<5,0<x2<3The parameters including N=40, D=2, Na=35, and Tmax=30 are provided. The following four algorithms are compared for this test:(1) The first one is a multiobjective CDOA (origin MOCDOA 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 center-updating strategy into the origin MOCDOA.The Pareto optimal solutions obtained by the four algorithms are depicted on Figure3. According to the picture, we can get two results:Figure 3
Comparisons of original MOCDOA and adding strategies MOCDOA for test1.(1) As can be seen from Figure3, 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 Figure3, 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.


### 6.2. IEEE 30-Bus 6-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.


#### 6.2.1. 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 Table5. 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 $/h for Case1 and Case2, respectively. The optimal values of CDOA with the emission as only objective function are 0.194203 ton/h and 0.194179 ton/h for Case1 and Case2, respectively. Next, the results of the multiobjective extreme solutions and the compromise solutions are discussed. The best results in all the tables are highlighted in bold for each case.Table 5
IEEE 30-bus 6-unit system best solutions for cost and emission optimized individually.
C a s e 1 C a s e 2 Best cost Best emission Best cost Best emission G 1 0.109712 0.406081 0.120952 0.410946 G2 0.299772 0.459067 0.286307 0.463662 G3 0.524300 0.537945 0.583597 0.544409 G4 1.016191 0.382950 0.992842 0.390386 G5 0.524308 0.537930 0.523967 0.544445 G6 0.359717 0.510024 0.351894 0.515483 Fuel cost ($/h) 600.111408 638.273933 605.998370 646.207369 Emission (ton/h) 0.222145 0.194203 0.220730 0.194179Figure 4
IEEE 30-bus 6-unit system convergence of cost and emission objective functions onCase1 and Case2.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 Figure5, 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.Table 6
IEEE 30-bus 6-unit system best solutions for cost with nine algorithms onCase1.
MOCDOA LP MOSST NSGA NPGA SPEA NSGA-II FCPSO BB-MOPSO G 1 0.110009 0.1500 0.1125 0.1567 0.1080 0.1062 0.1059 0.1070 0.1090 G 2 0.299387 0.3000 0.3020 0.2870 0.3284 0.2897 0.3177 0.2897 0.3005 G 3 0.521287 0.5500 0.5311 0.4671 0.5386 0.5289 0.5216 0.525 0.5234 G 4 1.019763 1.0500 1.0208 1.0467 1.0067 1.0025 1.0146 1.015 1.0170 G 5 0.523732 0.4600 0.5311 0.5037 0.4949 0.5402 0.5159 0.5300 0.5238 G 6 0.359821 0.3500 0.3625 0.3729 0.3574 0.3664 0.3583 0.3673 0.3603 Emission 0.222376 0.22330 0.22220 0.22282 0.22116 0.2215 0.22188 0.2223 0.22220 Fuel cost 600.112351 606.314 605.889 600.572 600.259 600.15 600.155 600.1315 600.112 ∑ i = 1 N P G i - P D 0 0.026 0.2026 1 e - 04 0 1 e - 04 0 0 0 FEs 10,000 - - 100,000 100,000 100,000 10,000 20,000 10,000Table 7
IEEE 30-bus 6-unit system best solutions for emission with nine algorithms onCase1.
MOCDOA LP MOSST NSGA NPGA SPEA NSGA-II FCPSO BB-MOPSO G 1 0.404380 0.4000 0.4095 0.4394 0.4002 0.4116 0.4074 0.4097 0.4071 G 2 0.461118 0.5500 0.4626 0.4511 0.4474 0.4532 0.4577 0.4550 0.4591 G 3 0.537554 0.4500 0.5426 0.5105 0.5166 0.5329 0.5389 0.5363 0.5374 G 4 0.382199 0.4000 0.3884 0.3871 0.3688 0.3832 0.3837 0.3842 0.3838 G 5 0.538118 0.5500 0.5427 0.5553 0.5751 0.5383 0.5352 0.5348 0.5369 G 6 0.510631 0.5000 0.5152 0.4905 0.5259 0.5148 0.5110 0.5140 0.5098 Fuel cost 638.327501 639.600 644.112 639.231 639.182 638.51 638.269 638.3577 638.262 Emission 0.194203 0.19424 0.19418 0.19436 0.19433 0.1942 0.19420 0.1942 0.194203 ∑ i = 1 N ‍ P G i - P D 0 0.016 0.027 1 e - 04 0 0 1 e - 04 0 0 FEs 10,000 - - 100,000 100,000 100,000 10,000 20,000 10,000Table 8
IEEE 30-bus 6-unit system best compromise solutions with six algorithms onCase1.
G 1 G 2 G 3 G 4 G 5 G 6 Fuel cost Emission ASD MOCDOA 0.269923 0.372134 0.529157 0.699728 0.546824 0.416234 609.656267 0.200942 0.759457 BB-MOPSO 0.2595 0.3698 0.5351 0.6919 0.5500 0.4277 609.747 0.20083 0.7555 NSGA 0.2571 0.3774 0.5381 0.6872 0.5404 0.4337 610.067 0.20060 0.7551 NPGA 0.2696 0.3673 0.5594 0.6496 0.5396 0.4486 612.127 0.19941 0.7491 SPEA 0.2785 0.3764 0.5300 0.6931 0.5406 0.4153 610.254 0.20055 0.7527 FCPSO 0.3193 0.3934 0.5359 0.5921 0.5457 0.447 619.998 0.19715 0.7267Figure 5
IEEE 30-bus 6-unit system Pareto front using MOCDOA onCase1.From Table6, 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.ForCase2, 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.Table 9
IEEE 30-bus 6-unit system best solutions for cost with six algorithms onCase2.
G 1 G 2 G 3 G 4 G 5 G 6 Fuel cost Emission FEs MOCDOA 0.120436 0.287096 0.580268 0.992202 0.525923 0.353730 605.999549 0.220643 20,000 NSGA 0.1168 0.3165 0.5441 0.9447 0.5498 0.3964 608.245 0.21664 100,000 NPGA 0.1245 0.2792 0.6284 1.0264 0.4693 0.3993 608.147 0.22364 100,000 SPEA 0.1086 0.3056 0.5818 0.9846 0.5288 0.3584 607.807 0.22015 100,000 NSGA-II 0.1182 0.3148 0.5910 0.9710 0.5172 0.3548 607.801 0.21891 10,000 FCPSO 0.1130 0.3145 0.5826 0.9860 0.5264 0.3450 607.7862 0.2201 20,000Table 10
IEEE 30-bus 6-unit system best solutions for emission with six algorithms onCase2.
G 1 G 2 G 3 G 4 G 5 G 6 Fuel cost Emission FEs MOCDOA 0.410372 0.466870 0.543728 0.390095 0.542564 0.515742 646.334746 0.194179 20,000 NSGA 0.4113 0.4591 0.5117 0.3724 0.5810 0.5304 647.251 0.19432 100,000 NPGA 0.3923 0.4700 0.5565 0.3695 0.5599 0.5163 645.984 0.19424 100,000 SPEA 0.4043 0.4525 0.5525 0.4079 0.5468 0.5005 642.603 0.19422 100,000 NSGA-II 0.4141 0.4602 0.5429 0.4011 0.5422 0.5045 644.133 0.19419 10,000 FCPSO 0.4063 0.4286 0.5510 0.4084 0.5432 0.4974 642.8964 0.1942 20,000From Tables9 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.


#### 6.2.2. 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 to evaluate the solution quality for multiobjective optimization problems [39].(i)
Uniformity. Maintaining uniform distribution.(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:(28)SP=1Ar-1∑i=1Ard¯-di2,di=minqj∈Ar∧qj≠qi∑k=1mfkqi-fkqjwhere di refers to the Euclidean distance of two consecutive solutions in the external archive and d¯ is the mean of all di. 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. Table 11 illustrates 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 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.Table 11
IEEE 30-bus 6-unit system statistical results of the SP onCase2.
Best Worst Median Average Std MOCDOA 0.003844 0.007615 0.005605 0.005636 0.000850 MOPSO 0.012307 0.023796 0.018037 0.018255 0.002701 NSGA-II 0.015961 0.023478 0.020096 0.020139 0.002054 PESA-II 0.014728 0.056674 0.023672 0.026567 0.008761Figure 6
IEEE 30-bus 6-unit system Pareto fronts and compromise solution for the four algorithms onCase2.
(a)
MOCDOA (b)
MOPSO (c)
NSGA-II (d)
PESA-IIA 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:(29)HV=⋃i=1Arviwhere a hypercube volume vi is calculated with a reference point wr and a solution Xi∈Ar as the diagonal corners of a hypercube. For HV, a higher value is better. The reference point wr 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.Table 12
IEEE 30-bus 6-unit system statistical results of the HV onCase2.
Best Worst Median Average Std MOCDOA 1.174740 1.172396 1.173860 1.173755 0.000630 MOPSO 1.162428 1.155299 1.159138 1.158993 0.002064 NSGA-II 1.167940 1.146496 1.162377 1.160582 0.005742 PESA-II 1.158865 1.120439 1.151565 1.148653 0.009073To 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:(30)CMAr1,Ar2=x2∈Ar2,∃x1∈Ar1:x1≺x2Ar2where Ar1 and Ar2 are two solution sets of two different algorithms. CM(Ar1,Ar2)=1 indicates that all solutions in Ar2 are dominated by the solutions in Ar1. This shows that Ar1 is closer to the true Pareto optimal front than Ar2. CM(Ar1,Ar2)=0 represents that none solution in Ar2 is covered by Ar1. Table 13 shows 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 onCase2.
MOCDOA MOPSO NSGA-II PESA-II CM(MOCDOA,∗) - 0.6 0.18 0.16 CM(MOPSO,∗) 0 - 0.6 0.1 CM(NSGA-II,∗) 0.04 0.38 - 0.16 CM(PESA-II,∗) 0.02 0.36 0.12 -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 inCase2. 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 perturbation strategy in Section 4.2. 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.


#### 6.2.3. 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. Tables14 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.Table 14
IEEE 30-bus 6-unit system statistical results of the three objectives onCase1.
Best Worst Median Average Std Fuel cost 600.112572 600.277057 600.138081 600.159425 0.041390 Emission 0.194203 0.194325 0.194232 0.194243 3.33 e - 05 ASD 0.759457 0.747940 0.754129 0.753532 0.002742Table 15
IEEE 30-bus 6-unit system statistical results of the three objectives onCase2.
Best Worst Median Average Std Fuel cost 605.999549 606.1162352 606.015211 606.024775 0.026281 Emission 0.194179 0.194249 0.194188 0.194197 1.98 e - 05 ASD 0.758042 0.744444 0.754744 0.753989 0.003077Figure 7
IEEE 30-bus 6-unit system statistical results obtained by MOCDOA onCase1 and Case2.It can easily be seen from Figure7 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, 3.33e-05, and 0.002742 for Case1. In Table 15, the standard deviations of the three objective are 0.026281, 1.98e-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.


### 6.3. IEEE 30-Bus 10-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 onCase1 and Case2.The fuel cost and the emission are minimized individually by the basic CDOA in all cases. Table16 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 $/h on Case1 and Case2, respectively. The minimum values to consider the emission as only objective function are 3651.072701 ton/h and 3933.012596 ton/h on Case1 and Case2, respectively. Multiobjective results obtained by optimizing the cost and the emission simultaneously are discussed below.Table 16
IEEE 30-bus 10-unit system best solutions for cost and emission optimized individually.
C a s e 1 C a s e 2 Best cost Best emission Best cost Best emission G 1 54.354899 54.969212 54.237557 54.992582 G 2 77.475920 76.928036 79.941879 78.938898 G 3 88.276828 78.916625 104.852031 80.557478 G 4 81.509714 78.967157 99.757723 82.288308 G 5 66.071942 160 84.152371 159.974756 G 6 71.847842 240 87.902733 239.943905 G 7 287.674673 275.155451 298.512528 289.201904 G 8 332.788181 276.363682 338.327682 296.527420 G 9 470 379.528659 469.615571 400.653771 G 10 470 379.171177 469.619537 398.609472 Fuel cost ($/h) 106183.951158 111870.335739 111521.601406 116381.181212 Emission (ton/h) 4278.459561 3651.072701 4545.826580 3933.012596Figure 8
IEEE 30-bus 10-unit system convergence of cost and emission objective functions onCase1 and Case2.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 Tables17, 18, and 19, respectively. Figure 9 shows the Pareto fronts of MOCDOA, MOPSO, NSGA-II, and PESA-II.Table 17
IEEE 30-bus 10-unit system best solutions for cost with four algorithms onCase1 and Case2.
C a s e 1 C a s e 2 MOCDOA MOPSO NSGA-II PESA-II MOCDOA MOPSO NSGA-II PESA-II G 1 54.158870 54.134753 53.691308 54.803688 54.127113 52.863198 51.755735 54.997131 G 2 78.422533 78.139386 75.399985 77.335058 79.207632 78.825931 73.834838 79.995928 G 3 97.804234 110.593988 84.188022 89.476491 94.923188 104.094617 93.469134 112.435619 G 4 93.483470 92.354559 87.079197 62.613846 92.962807 102.294750 93.444728 85.764687 G 5 76.419930 80.279634 110.049490 67.144314 81.761262 97.001150 111.290861 91.517523 G 6 73.121641 70.016616 115.870843 86.785823 110.961706 95.871712 139.548089 82.535812 G 7 280.934498 279.361011 278.237057 289.140313 299.253109 294.030349 293.422964 299.966549 G 8 311.644912 321.711273 309.757782 335.411589 336.963929 325.450671 316.810101 339.967105 G 9 466.664170 446.519442 437.948030 468.245354 467.138652 467.865756 456.774987 469.961342 G 10 467.345738 466.889336 447.778284 469.043519 469.426001 468.141081 454.933764 469.969235 Emission 4252.190307 4260.635634 4001.839787 4249.562117 4463.735501 4487.984281 4294.006682 4545.689410 Fuel cost 106264.834496 106376.240273 107024.921918 106288.960020 111647.243250 111665.652452 112339.243763 111543.388926Table 18
IEEE 30-bus 10-unit system best solutions for emission with four algorithms onCase1 and Case2.
C a s e 1 C a s e 2 MOCDOA MOPSO NSGA-II PESA-II MOCDOA MOPSO NSGA-II PESA-II G 1 52.532519 52.852002 52.991134 54.222507 53.96756 54.236469 53.956584 53.908881 G 2 72.435520 69.597818 75.182766 75.066663 72.905941 78.981985 78.700245 77.049995 G 3 82.379093 71.767701 75.899072 84.396494 91.018244 76.975535 84.717066 84.158908 G 4 84.050261 81.122486 75.723913 84.227764 86.648978 77.977286 84.825929 80.928031 G 5 152.491324 158.672750 146.318487 151.316167 159.951634 159.519513 154.470253 151.279898 G 6 238.073874 238.166505 203.771856 230.429877 238.723255 237.781338 180.056057 235.222332 G 7 274.238437 264.847460 268.891862 271.546877 276.910571 276.088787 299.343585 290.055655 G 8 265.509554 286.431874 305.379732 277.395716 298.133654 300.031806 307.019564 304.265163 G 9 404.020356 391.188744 397.523330 384.398490 389.569294 405.729260 425.923887 403.498611 G 10 374.269061 385.352658 398.317844 386.999443 413.796996 414.886666 414.052021 401.556885 Fuel cost 111555.483102 111696.091418 110017.658015 111208.638485 116322.893631 116209.080846 114211.583909 115939.573531 Emission 3680.578627 3667.508603 3728.840966 3681.510679 3950.764391 3945.408627 4048.348353 3960.706739Table 19
IEEE 30-bus 10-unit system best compromise solutions with four algorithms onCase1 and Case2.
C a s e 1 C a s e 2 MOCDOA MOPSO NSGA-II PESA-II MOCDOA MOPSO NSGA-II PESA-II G 1 51.811253 54.146837 51.986438 54.500735 53.019793 54.564607 54.852490 54.399261 G 2 74.716733 79.720887 73.179620 76.228593 75.439733 73.581243 76.037006 77.493806 G 3 81.292214 83.454045 84.019834 82.014741 89.556375 88.832532 88.950481 87.999806 G 4 82.075395 75.651603 82.096182 81.748678 86.966333 87.819192 89.313517 87.373982 G 5 121.509751 144.184062 131.099548 121.142989 137.307583 128.411293 136.436222 120.078071 G 6 147.511266 148.565014 147.290760 139.813680 147.649522 144.095165 155.293274 142.652156 G 7 280.003081 284.922355 287.204027 288.236121 289.584082 298.618120 296.733338 297.184408 G 8 307.750415 293.335856 295.380918 299.231441 322.652284 321.940573 318.147618 330.804592 G 9 432.468854 407.299855 424.290718 421.127067 442.611979 443.421947 434.291352 445.836158 G 10 420.861036 428.719484 423.451949 435.955950 439.648299 443.427277 433.995612 441.036231 Fuel cost 107838.048975 108390.120075 108052.518200 107665.729718 112729.353799 113274.171282 113135.673565 112556.796831 Emission 3891.545095 3827.895782 3867.175411 3900.324575 4180.868813 4208.176152 4152.140421 4231.601876 ASD 0.6810 0.6279 0.5842 0.6581 0.6802 0.6566 0.5730 0.6471Figure 9
IEEE 30-bus 10-unit system Pareto fronts and compromise solution for the four algorithms onCase2.
(a)
MOCDOA (b)
MOPSO (c)
NSGA-II (d)
PESA-IIFrom Table17, 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. It is quite evident that the proposed MOCDOA performs better than the other three algorithms. Combining the above results, we can see that MOCDOA is more efficient than almost the other three compared algorithms.


## 6.1. Experimental Analysis of Three 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:(27)minf1=4x12+4x22minf2=x1-52+4x2-52st.0<x1<5,0<x2<3The parameters including N=40, D=2, Na=35, and Tmax=30 are provided. The following four algorithms are compared for this test:(1) The first one is a multiobjective CDOA (origin MOCDOA 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 center-updating strategy into the origin MOCDOA.The Pareto optimal solutions obtained by the four algorithms are depicted on Figure3. According to the picture, we can get two results:Figure 3
Comparisons of original MOCDOA and adding strategies MOCDOA for test1.(1) As can be seen from Figure3, 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 Figure3, 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.


## 6.2. IEEE 30-Bus 6-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.


### 6.2.1. 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 Table5. 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 $/h for Case1 and Case2, respectively. The optimal values of CDOA with the emission as only objective function are 0.194203 ton/h and 0.194179 ton/h for Case1 and Case2, respectively. Next, the results of the multiobjective extreme solutions and the compromise solutions are discussed. The best results in all the tables are highlighted in bold for each case.Table 5
IEEE 30-bus 6-unit system best solutions for cost and emission optimized individually.
C a s e 1 C a s e 2 Best cost Best emission Best cost Best emission G 1 0.109712 0.406081 0.120952 0.410946 G2 0.299772 0.459067 0.286307 0.463662 G3 0.524300 0.537945 0.583597 0.544409 G4 1.016191 0.382950 0.992842 0.390386 G5 0.524308 0.537930 0.523967 0.544445 G6 0.359717 0.510024 0.351894 0.515483 Fuel cost ($/h) 600.111408 638.273933 605.998370 646.207369 Emission (ton/h) 0.222145 0.194203 0.220730 0.194179Figure 4
IEEE 30-bus 6-unit system convergence of cost and emission objective functions onCase1 and Case2.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 Figure5, 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.Table 6
IEEE 30-bus 6-unit system best solutions for cost with nine algorithms onCase1.
MOCDOA LP MOSST NSGA NPGA SPEA NSGA-II FCPSO BB-MOPSO G 1 0.110009 0.1500 0.1125 0.1567 0.1080 0.1062 0.1059 0.1070 0.1090 G 2 0.299387 0.3000 0.3020 0.2870 0.3284 0.2897 0.3177 0.2897 0.3005 G 3 0.521287 0.5500 0.5311 0.4671 0.5386 0.5289 0.5216 0.525 0.5234 G 4 1.019763 1.0500 1.0208 1.0467 1.0067 1.0025 1.0146 1.015 1.0170 G 5 0.523732 0.4600 0.5311 0.5037 0.4949 0.5402 0.5159 0.5300 0.5238 G 6 0.359821 0.3500 0.3625 0.3729 0.3574 0.3664 0.3583 0.3673 0.3603 Emission 0.222376 0.22330 0.22220 0.22282 0.22116 0.2215 0.22188 0.2223 0.22220 Fuel cost 600.112351 606.314 605.889 600.572 600.259 600.15 600.155 600.1315 600.112 ∑ i = 1 N P G i - P D 0 0.026 0.2026 1 e - 04 0 1 e - 04 0 0 0 FEs 10,000 - - 100,000 100,000 100,000 10,000 20,000 10,000Table 7
IEEE 30-bus 6-unit system best solutions for emission with nine algorithms onCase1.
MOCDOA LP MOSST NSGA NPGA SPEA NSGA-II FCPSO BB-MOPSO G 1 0.404380 0.4000 0.4095 0.4394 0.4002 0.4116 0.4074 0.4097 0.4071 G 2 0.461118 0.5500 0.4626 0.4511 0.4474 0.4532 0.4577 0.4550 0.4591 G 3 0.537554 0.4500 0.5426 0.5105 0.5166 0.5329 0.5389 0.5363 0.5374 G 4 0.382199 0.4000 0.3884 0.3871 0.3688 0.3832 0.3837 0.3842 0.3838 G 5 0.538118 0.5500 0.5427 0.5553 0.5751 0.5383 0.5352 0.5348 0.5369 G 6 0.510631 0.5000 0.5152 0.4905 0.5259 0.5148 0.5110 0.5140 0.5098 Fuel cost 638.327501 639.600 644.112 639.231 639.182 638.51 638.269 638.3577 638.262 Emission 0.194203 0.19424 0.19418 0.19436 0.19433 0.1942 0.19420 0.1942 0.194203 ∑ i = 1 N ‍ P G i - P D 0 0.016 0.027 1 e - 04 0 0 1 e - 04 0 0 FEs 10,000 - - 100,000 100,000 100,000 10,000 20,000 10,000Table 8
IEEE 30-bus 6-unit system best compromise solutions with six algorithms onCase1.
G 1 G 2 G 3 G 4 G 5 G 6 Fuel cost Emission ASD MOCDOA 0.269923 0.372134 0.529157 0.699728 0.546824 0.416234 609.656267 0.200942 0.759457 BB-MOPSO 0.2595 0.3698 0.5351 0.6919 0.5500 0.4277 609.747 0.20083 0.7555 NSGA 0.2571 0.3774 0.5381 0.6872 0.5404 0.4337 610.067 0.20060 0.7551 NPGA 0.2696 0.3673 0.5594 0.6496 0.5396 0.4486 612.127 0.19941 0.7491 SPEA 0.2785 0.3764 0.5300 0.6931 0.5406 0.4153 610.254 0.20055 0.7527 FCPSO 0.3193 0.3934 0.5359 0.5921 0.5457 0.447 619.998 0.19715 0.7267Figure 5
IEEE 30-bus 6-unit system Pareto front using MOCDOA onCase1.From Table6, 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.ForCase2, 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.Table 9
IEEE 30-bus 6-unit system best solutions for cost with six algorithms onCase2.
G 1 G 2 G 3 G 4 G 5 G 6 Fuel cost Emission FEs MOCDOA 0.120436 0.287096 0.580268 0.992202 0.525923 0.353730 605.999549 0.220643 20,000 NSGA 0.1168 0.3165 0.5441 0.9447 0.5498 0.3964 608.245 0.21664 100,000 NPGA 0.1245 0.2792 0.6284 1.0264 0.4693 0.3993 608.147 0.22364 100,000 SPEA 0.1086 0.3056 0.5818 0.9846 0.5288 0.3584 607.807 0.22015 100,000 NSGA-II 0.1182 0.3148 0.5910 0.9710 0.5172 0.3548 607.801 0.21891 10,000 FCPSO 0.1130 0.3145 0.5826 0.9860 0.5264 0.3450 607.7862 0.2201 20,000Table 10
IEEE 30-bus 6-unit system best solutions for emission with six algorithms onCase2.
G 1 G 2 G 3 G 4 G 5 G 6 Fuel cost Emission FEs MOCDOA 0.410372 0.466870 0.543728 0.390095 0.542564 0.515742 646.334746 0.194179 20,000 NSGA 0.4113 0.4591 0.5117 0.3724 0.5810 0.5304 647.251 0.19432 100,000 NPGA 0.3923 0.4700 0.5565 0.3695 0.5599 0.5163 645.984 0.19424 100,000 SPEA 0.4043 0.4525 0.5525 0.4079 0.5468 0.5005 642.603 0.19422 100,000 NSGA-II 0.4141 0.4602 0.5429 0.4011 0.5422 0.5045 644.133 0.19419 10,000 FCPSO 0.4063 0.4286 0.5510 0.4084 0.5432 0.4974 642.8964 0.1942 20,000From Tables9 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.


### 6.2.2. 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 to evaluate the solution quality for multiobjective optimization problems [39].(i)
Uniformity. Maintaining uniform distribution.(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:(28)SP=1Ar-1∑i=1Ard¯-di2,di=minqj∈Ar∧qj≠qi∑k=1mfkqi-fkqjwhere di refers to the Euclidean distance of two consecutive solutions in the external archive and d¯ is the mean of all di. 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. Table 11 illustrates 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 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.Table 11
IEEE 30-bus 6-unit system statistical results of the SP onCase2.
Best Worst Median Average Std MOCDOA 0.003844 0.007615 0.005605 0.005636 0.000850 MOPSO 0.012307 0.023796 0.018037 0.018255 0.002701 NSGA-II 0.015961 0.023478 0.020096 0.020139 0.002054 PESA-II 0.014728 0.056674 0.023672 0.026567 0.008761Figure 6
IEEE 30-bus 6-unit system Pareto fronts and compromise solution for the four algorithms onCase2.
(a)
MOCDOA (b)
MOPSO (c)
NSGA-II (d)
PESA-IIA 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:(29)HV=⋃i=1Arviwhere a hypercube volume vi is calculated with a reference point wr and a solution Xi∈Ar as the diagonal corners of a hypercube. For HV, a higher value is better. The reference point wr 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.Table 12
IEEE 30-bus 6-unit system statistical results of the HV onCase2.
Best Worst Median Average Std MOCDOA 1.174740 1.172396 1.173860 1.173755 0.000630 MOPSO 1.162428 1.155299 1.159138 1.158993 0.002064 NSGA-II 1.167940 1.146496 1.162377 1.160582 0.005742 PESA-II 1.158865 1.120439 1.151565 1.148653 0.009073To 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:(30)CMAr1,Ar2=x2∈Ar2,∃x1∈Ar1:x1≺x2Ar2where Ar1 and Ar2 are two solution sets of two different algorithms. CM(Ar1,Ar2)=1 indicates that all solutions in Ar2 are dominated by the solutions in Ar1. This shows that Ar1 is closer to the true Pareto optimal front than Ar2. CM(Ar1,Ar2)=0 represents that none solution in Ar2 is covered by Ar1. Table 13 shows 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 onCase2.
MOCDOA MOPSO NSGA-II PESA-II CM(MOCDOA,∗) - 0.6 0.18 0.16 CM(MOPSO,∗) 0 - 0.6 0.1 CM(NSGA-II,∗) 0.04 0.38 - 0.16 CM(PESA-II,∗) 0.02 0.36 0.12 -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 inCase2. 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 perturbation strategy in Section 4.2. 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.


### 6.2.3. 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. Tables14 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.Table 14
IEEE 30-bus 6-unit system statistical results of the three objectives onCase1.
Best Worst Median Average Std Fuel cost 600.112572 600.277057 600.138081 600.159425 0.041390 Emission 0.194203 0.194325 0.194232 0.194243 3.33 e - 05 ASD 0.759457 0.747940 0.754129 0.753532 0.002742Table 15
IEEE 30-bus 6-unit system statistical results of the three objectives onCase2.
Best Worst Median Average Std Fuel cost 605.999549 606.1162352 606.015211 606.024775 0.026281 Emission 0.194179 0.194249 0.194188 0.194197 1.98 e - 05 ASD 0.758042 0.744444 0.754744 0.753989 0.003077Figure 7
IEEE 30-bus 6-unit system statistical results obtained by MOCDOA onCase1 and Case2.It can easily be seen from Figure7 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, 3.33e-05, and 0.002742 for Case1. In Table 15, the standard deviations of the three objective are 0.026281, 1.98e-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.


## 6.2.1. 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 Table5. 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 $/h for Case1 and Case2, respectively. The optimal values of CDOA with the emission as only objective function are 0.194203 ton/h and 0.194179 ton/h for Case1 and Case2, respectively. Next, the results of the multiobjective extreme solutions and the compromise solutions are discussed. The best results in all the tables are highlighted in bold for each case.Table 5
IEEE 30-bus 6-unit system best solutions for cost and emission optimized individually.
C a s e 1 C a s e 2 Best cost Best emission Best cost Best emission G 1 0.109712 0.406081 0.120952 0.410946 G2 0.299772 0.459067 0.286307 0.463662 G3 0.524300 0.537945 0.583597 0.544409 G4 1.016191 0.382950 0.992842 0.390386 G5 0.524308 0.537930 0.523967 0.544445 G6 0.359717 0.510024 0.351894 0.515483 Fuel cost ($/h) 600.111408 638.273933 605.998370 646.207369 Emission (ton/h) 0.222145 0.194203 0.220730 0.194179Figure 4
IEEE 30-bus 6-unit system convergence of cost and emission objective functions onCase1 and Case2.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 Figure5, 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.Table 6
IEEE 30-bus 6-unit system best solutions for cost with nine algorithms onCase1.
MOCDOA LP MOSST NSGA NPGA SPEA NSGA-II FCPSO BB-MOPSO G 1 0.110009 0.1500 0.1125 0.1567 0.1080 0.1062 0.1059 0.1070 0.1090 G 2 0.299387 0.3000 0.3020 0.2870 0.3284 0.2897 0.3177 0.2897 0.3005 G 3 0.521287 0.5500 0.5311 0.4671 0.5386 0.5289 0.5216 0.525 0.5234 G 4 1.019763 1.0500 1.0208 1.0467 1.0067 1.0025 1.0146 1.015 1.0170 G 5 0.523732 0.4600 0.5311 0.5037 0.4949 0.5402 0.5159 0.5300 0.5238 G 6 0.359821 0.3500 0.3625 0.3729 0.3574 0.3664 0.3583 0.3673 0.3603 Emission 0.222376 0.22330 0.22220 0.22282 0.22116 0.2215 0.22188 0.2223 0.22220 Fuel cost 600.112351 606.314 605.889 600.572 600.259 600.15 600.155 600.1315 600.112 ∑ i = 1 N P G i - P D 0 0.026 0.2026 1 e - 04 0 1 e - 04 0 0 0 FEs 10,000 - - 100,000 100,000 100,000 10,000 20,000 10,000Table 7
IEEE 30-bus 6-unit system best solutions for emission with nine algorithms onCase1.
MOCDOA LP MOSST NSGA NPGA SPEA NSGA-II FCPSO BB-MOPSO G 1 0.404380 0.4000 0.4095 0.4394 0.4002 0.4116 0.4074 0.4097 0.4071 G 2 0.461118 0.5500 0.4626 0.4511 0.4474 0.4532 0.4577 0.4550 0.4591 G 3 0.537554 0.4500 0.5426 0.5105 0.5166 0.5329 0.5389 0.5363 0.5374 G 4 0.382199 0.4000 0.3884 0.3871 0.3688 0.3832 0.3837 0.3842 0.3838 G 5 0.538118 0.5500 0.5427 0.5553 0.5751 0.5383 0.5352 0.5348 0.5369 G 6 0.510631 0.5000 0.5152 0.4905 0.5259 0.5148 0.5110 0.5140 0.5098 Fuel cost 638.327501 639.600 644.112 639.231 639.182 638.51 638.269 638.3577 638.262 Emission 0.194203 0.19424 0.19418 0.19436 0.19433 0.1942 0.19420 0.1942 0.194203 ∑ i = 1 N ‍ P G i - P D 0 0.016 0.027 1 e - 04 0 0 1 e - 04 0 0 FEs 10,000 - - 100,000 100,000 100,000 10,000 20,000 10,000Table 8
IEEE 30-bus 6-unit system best compromise solutions with six algorithms onCase1.
G 1 G 2 G 3 G 4 G 5 G 6 Fuel cost Emission ASD MOCDOA 0.269923 0.372134 0.529157 0.699728 0.546824 0.416234 609.656267 0.200942 0.759457 BB-MOPSO 0.2595 0.3698 0.5351 0.6919 0.5500 0.4277 609.747 0.20083 0.7555 NSGA 0.2571 0.3774 0.5381 0.6872 0.5404 0.4337 610.067 0.20060 0.7551 NPGA 0.2696 0.3673 0.5594 0.6496 0.5396 0.4486 612.127 0.19941 0.7491 SPEA 0.2785 0.3764 0.5300 0.6931 0.5406 0.4153 610.254 0.20055 0.7527 FCPSO 0.3193 0.3934 0.5359 0.5921 0.5457 0.447 619.998 0.19715 0.7267Figure 5
IEEE 30-bus 6-unit system Pareto front using MOCDOA onCase1.From Table6, 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.ForCase2, 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.Table 9
IEEE 30-bus 6-unit system best solutions for cost with six algorithms onCase2.
G 1 G 2 G 3 G 4 G 5 G 6 Fuel cost Emission FEs MOCDOA 0.120436 0.287096 0.580268 0.992202 0.525923 0.353730 605.999549 0.220643 20,000 NSGA 0.1168 0.3165 0.5441 0.9447 0.5498 0.3964 608.245 0.21664 100,000 NPGA 0.1245 0.2792 0.6284 1.0264 0.4693 0.3993 608.147 0.22364 100,000 SPEA 0.1086 0.3056 0.5818 0.9846 0.5288 0.3584 607.807 0.22015 100,000 NSGA-II 0.1182 0.3148 0.5910 0.9710 0.5172 0.3548 607.801 0.21891 10,000 FCPSO 0.1130 0.3145 0.5826 0.9860 0.5264 0.3450 607.7862 0.2201 20,000Table 10
IEEE 30-bus 6-unit system best solutions for emission with six algorithms onCase2.
G 1 G 2 G 3 G 4 G 5 G 6 Fuel cost Emission FEs MOCDOA 0.410372 0.466870 0.543728 0.390095 0.542564 0.515742 646.334746 0.194179 20,000 NSGA 0.4113 0.4591 0.5117 0.3724 0.5810 0.5304 647.251 0.19432 100,000 NPGA 0.3923 0.4700 0.5565 0.3695 0.5599 0.5163 645.984 0.19424 100,000 SPEA 0.4043 0.4525 0.5525 0.4079 0.5468 0.5005 642.603 0.19422 100,000 NSGA-II 0.4141 0.4602 0.5429 0.4011 0.5422 0.5045 644.133 0.19419 10,000 FCPSO 0.4063 0.4286 0.5510 0.4084 0.5432 0.4974 642.8964 0.1942 20,000From Tables9 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.


## 6.2.2. 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 to evaluate the solution quality for multiobjective optimization problems [39].(i)
Uniformity. Maintaining uniform distribution.(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:(28)SP=1Ar-1∑i=1Ard¯-di2,di=minqj∈Ar∧qj≠qi∑k=1mfkqi-fkqjwhere di refers to the Euclidean distance of two consecutive solutions in the external archive and d¯ is the mean of all di. 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. Table 11 illustrates 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 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.Table 11
IEEE 30-bus 6-unit system statistical results of the SP onCase2.
Best Worst Median Average Std MOCDOA 0.003844 0.007615 0.005605 0.005636 0.000850 MOPSO 0.012307 0.023796 0.018037 0.018255 0.002701 NSGA-II 0.015961 0.023478 0.020096 0.020139 0.002054 PESA-II 0.014728 0.056674 0.023672 0.026567 0.008761Figure 6
IEEE 30-bus 6-unit system Pareto fronts and compromise solution for the four algorithms onCase2.
(a)
MOCDOA (b)
MOPSO (c)
NSGA-II (d)
PESA-IIA 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:(29)HV=⋃i=1Arviwhere a hypercube volume vi is calculated with a reference point wr and a solution Xi∈Ar as the diagonal corners of a hypercube. For HV, a higher value is better. The reference point wr 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.Table 12
IEEE 30-bus 6-unit system statistical results of the HV onCase2.
Best Worst Median Average Std MOCDOA 1.174740 1.172396 1.173860 1.173755 0.000630 MOPSO 1.162428 1.155299 1.159138 1.158993 0.002064 NSGA-II 1.167940 1.146496 1.162377 1.160582 0.005742 PESA-II 1.158865 1.120439 1.151565 1.148653 0.009073To 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:(30)CMAr1,Ar2=x2∈Ar2,∃x1∈Ar1:x1≺x2Ar2where Ar1 and Ar2 are two solution sets of two different algorithms. CM(Ar1,Ar2)=1 indicates that all solutions in Ar2 are dominated by the solutions in Ar1. This shows that Ar1 is closer to the true Pareto optimal front than Ar2. CM(Ar1,Ar2)=0 represents that none solution in Ar2 is covered by Ar1. Table 13 shows 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 onCase2.
MOCDOA MOPSO NSGA-II PESA-II CM(MOCDOA,∗) - 0.6 0.18 0.16 CM(MOPSO,∗) 0 - 0.6 0.1 CM(NSGA-II,∗) 0.04 0.38 - 0.16 CM(PESA-II,∗) 0.02 0.36 0.12 -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 inCase2. 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 perturbation strategy in Section 4.2. 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.


## 6.2.3. 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. Tables14 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.Table 14
IEEE 30-bus 6-unit system statistical results of the three objectives onCase1.
Best Worst Median Average Std Fuel cost 600.112572 600.277057 600.138081 600.159425 0.041390 Emission 0.194203 0.194325 0.194232 0.194243 3.33 e - 05 ASD 0.759457 0.747940 0.754129 0.753532 0.002742Table 15
IEEE 30-bus 6-unit system statistical results of the three objectives onCase2.
Best Worst Median Average Std Fuel cost 605.999549 606.1162352 606.015211 606.024775 0.026281 Emission 0.194179 0.194249 0.194188 0.194197 1.98 e - 05 ASD 0.758042 0.744444 0.754744 0.753989 0.003077Figure 7
IEEE 30-bus 6-unit system statistical results obtained by MOCDOA onCase1 and Case2.It can easily be seen from Figure7 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, 3.33e-05, and 0.002742 for Case1. In Table 15, the standard deviations of the three objective are 0.026281, 1.98e-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.


## 6.3. IEEE 30-Bus 10-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 onCase1 and Case2.The fuel cost and the emission are minimized individually by the basic CDOA in all cases. Table16 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 $/h on Case1 and Case2, respectively. The minimum values to consider the emission as only objective function are 3651.072701 ton/h and 3933.012596 ton/h on Case1 and Case2, respectively. Multiobjective results obtained by optimizing the cost and the emission simultaneously are discussed below.Table 16
IEEE 30-bus 10-unit system best solutions for cost and emission optimized individually.
C a s e 1 C a s e 2 Best cost Best emission Best cost Best emission G 1 54.354899 54.969212 54.237557 54.992582 G 2 77.475920 76.928036 79.941879 78.938898 G 3 88.276828 78.916625 104.852031 80.557478 G 4 81.509714 78.967157 99.757723 82.288308 G 5 66.071942 160 84.152371 159.974756 G 6 71.847842 240 87.902733 239.943905 G 7 287.674673 275.155451 298.512528 289.201904 G 8 332.788181 276.363682 338.327682 296.527420 G 9 470 379.528659 469.615571 400.653771 G 10 470 379.171177 469.619537 398.609472 Fuel cost ($/h) 106183.951158 111870.335739 111521.601406 116381.181212 Emission (ton/h) 4278.459561 3651.072701 4545.826580 3933.012596Figure 8
IEEE 30-bus 10-unit system convergence of cost and emission objective functions onCase1 and Case2.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 Tables17, 18, and 19, respectively. Figure 9 shows the Pareto fronts of MOCDOA, MOPSO, NSGA-II, and PESA-II.Table 17
IEEE 30-bus 10-unit system best solutions for cost with four algorithms onCase1 and Case2.
C a s e 1 C a s e 2 MOCDOA MOPSO NSGA-II PESA-II MOCDOA MOPSO NSGA-II PESA-II G 1 54.158870 54.134753 53.691308 54.803688 54.127113 52.863198 51.755735 54.997131 G 2 78.422533 78.139386 75.399985 77.335058 79.207632 78.825931 73.834838 79.995928 G 3 97.804234 110.593988 84.188022 89.476491 94.923188 104.094617 93.469134 112.435619 G 4 93.483470 92.354559 87.079197 62.613846 92.962807 102.294750 93.444728 85.764687 G 5 76.419930 80.279634 110.049490 67.144314 81.761262 97.001150 111.290861 91.517523 G 6 73.121641 70.016616 115.870843 86.785823 110.961706 95.871712 139.548089 82.535812 G 7 280.934498 279.361011 278.237057 289.140313 299.253109 294.030349 293.422964 299.966549 G 8 311.644912 321.711273 309.757782 335.411589 336.963929 325.450671 316.810101 339.967105 G 9 466.664170 446.519442 437.948030 468.245354 467.138652 467.865756 456.774987 469.961342 G 10 467.345738 466.889336 447.778284 469.043519 469.426001 468.141081 454.933764 469.969235 Emission 4252.190307 4260.635634 4001.839787 4249.562117 4463.735501 4487.984281 4294.006682 4545.689410 Fuel cost 106264.834496 106376.240273 107024.921918 106288.960020 111647.243250 111665.652452 112339.243763 111543.388926Table 18
IEEE 30-bus 10-unit system best solutions for emission with four algorithms onCase1 and Case2.
C a s e 1 C a s e 2 MOCDOA MOPSO NSGA-II PESA-II MOCDOA MOPSO NSGA-II PESA-II G 1 52.532519 52.852002 52.991134 54.222507 53.96756 54.236469 53.956584 53.908881 G 2 72.435520 69.597818 75.182766 75.066663 72.905941 78.981985 78.700245 77.049995 G 3 82.379093 71.767701 75.899072 84.396494 91.018244 76.975535 84.717066 84.158908 G 4 84.050261 81.122486 75.723913 84.227764 86.648978 77.977286 84.825929 80.928031 G 5 152.491324 158.672750 146.318487 151.316167 159.951634 159.519513 154.470253 151.279898 G 6 238.073874 238.166505 203.771856 230.429877 238.723255 237.781338 180.056057 235.222332 G 7 274.238437 264.847460 268.891862 271.546877 276.910571 276.088787 299.343585 290.055655 G 8 265.509554 286.431874 305.379732 277.395716 298.133654 300.031806 307.019564 304.265163 G 9 404.020356 391.188744 397.523330 384.398490 389.569294 405.729260 425.923887 403.498611 G 10 374.269061 385.352658 398.317844 386.999443 413.796996 414.886666 414.052021 401.556885 Fuel cost 111555.483102 111696.091418 110017.658015 111208.638485 116322.893631 116209.080846 114211.583909 115939.573531 Emission 3680.578627 3667.508603 3728.840966 3681.510679 3950.764391 3945.408627 4048.348353 3960.706739Table 19
IEEE 30-bus 10-unit system best compromise solutions with four algorithms onCase1 and Case2.
C a s e 1 C a s e 2 MOCDOA MOPSO NSGA-II PESA-II MOCDOA MOPSO NSGA-II PESA-II G 1 51.811253 54.146837 51.986438 54.500735 53.019793 54.564607 54.852490 54.399261 G 2 74.716733 79.720887 73.179620 76.228593 75.439733 73.581243 76.037006 77.493806 G 3 81.292214 83.454045 84.019834 82.014741 89.556375 88.832532 88.950481 87.999806 G 4 82.075395 75.651603 82.096182 81.748678 86.966333 87.819192 89.313517 87.373982 G 5 121.509751 144.184062 131.099548 121.142989 137.307583 128.411293 136.436222 120.078071 G 6 147.511266 148.565014 147.290760 139.813680 147.649522 144.095165 155.293274 142.652156 G 7 280.003081 284.922355 287.204027 288.236121 289.584082 298.618120 296.733338 297.184408 G 8 307.750415 293.335856 295.380918 299.231441 322.652284 321.940573 318.147618 330.804592 G 9 432.468854 407.299855 424.290718 421.127067 442.611979 443.421947 434.291352 445.836158 G 10 420.861036 428.719484 423.451949 435.955950 439.648299 443.427277 433.995612 441.036231 Fuel cost 107838.048975 108390.120075 108052.518200 107665.729718 112729.353799 113274.171282 113135.673565 112556.796831 Emission 3891.545095 3827.895782 3867.175411 3900.324575 4180.868813 4208.176152 4152.140421 4231.601876 ASD 0.6810 0.6279 0.5842 0.6581 0.6802 0.6566 0.5730 0.6471Figure 9
IEEE 30-bus 10-unit system Pareto fronts and compromise solution for the four algorithms onCase2.
(a)
MOCDOA (b)
MOPSO (c)
NSGA-II (d)
PESA-IIFrom Table17, 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. It is quite evident that the proposed MOCDOA performs better than the other three algorithms. Combining the above results, we can see that MOCDOA is more efficient than almost the other three compared algorithms.


## 7. Conclusions

A novel algorithm MOCDOA presented in this paper is successfully employed to solve the environmental/economic power dispatch (EED) optimization problem with constrains. The algorithm extends the single-objective collective decision optimization algorithm to solve the multiobjective optimization problems for the first time. In order to enhance the uniformity of the obtained Pareto optimal solutions, two learning strategies are adopted, which are a leader-updating strategy based on maximum distance to measure the sparse degree of solutions and a wise random perturbation strategy based on the sparse mark to search the area around a leader. Furthermore, an extreme point of the nondominated solutions in MOCDOA is picked at random to replace a geometric center in CDOA, which can improve the diversity of Pareto optimal solutions. In addition, a new constraint-handling strategy to deal with the equality constraint of the EED problem and an established cyclic crowded sorting method to retain the external archive are designed in MOCDOA. The IEEE 30-bus 6-unit system and 10-unit system are used to investigate the effectiveness of the proposed MOCDOA. The results of MOCDOA’s extreme solutions and compromise solutions are compared with those existing results obtained by other eight established algorithms. The compared results 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.



---

*Source: 1027193-2018-11-13.xml*