An Improved Particle Swarm Optimization with Biogeography-Based Learning Strategy for Economic Dispatch Problems

Economic dispatch (ED) plays an important role in power system operation, since it can decrease the operating cost, save energy resources, and reduce environmental load. This paper presents an improved particle swarm optimization called biogeographybased learning particle swarm optimization (BLPSO) for solving the ED problems involving different equality and inequality constraints, such as power balance, prohibited operating zones, and ramp-rate limits. In the proposed BLPSO, a biogeographybased learning strategy is employed in which particles learn from each other based on the quality of their personal best positions, and thus it can provide a more efficient balance between exploration and exploitation. The proposed BLPSO is applied to solve five ED problems and compared with other optimization techniques in the literature. Experimental results demonstrate that the BLPSO is a promising approach for solving the ED problems.


Introduction
Economic dispatch (ED) is an important optimization task in power system operation and planning.The main objective of ED problems is to allocate generation among the committed generating units so as to meet the required load demand at minimum operating cost, with various physical constraints [1].The cost of power generation is high, and economic dispatch can help in saving a significant amount of revenue [2].
In the original ED problem, the cost function for each generation unit is approximately represented by a single quadratic function, and traditional approaches based on mathematical programming techniques have been utilized to solve the ED problem, including the lambda-iteration method, gradient method, Newton's method, linear programming, interior point method, and dynamic programming [3][4][5].Usually, these methods are highly sensitive to starting points and rely on the assumption that the cost function needs to be continuous and convex.However, the practical ED problems exhibit nonconvex and nonsmooth characteristics because of valve-point effects, ramp-rate limit, multifuel cost, prohibited operating zones, and so on [6].The traditional methods are not capable of efficiently solving the ED problems with these characteristics.
In the past decades, more and more researchers are turning to metaheuristic search (MS) algorithms for solving the ED problems.These methods have the ability to identify higher-quality solutions and can be grouped into three categories, as original, improved, and hybrid MS algorithms.
In this paper, an improved PSO algorithm with biogeography-based learning strategy is proposed to solve the ED problems.The main contributions of this paper are listed as follows: (1) A biogeography-based learning particle swarm optimization (BLPSO) algorithm which employs a biogeography-based learning strategy (BLS) is presented.The computational complexity of BLPSO is also analyzed.
(2) By combining the feature of ED problems, the BLPSObased economic dispatch method is developed.
(3) BLPSO is applied to solve five ED problems with various practical constraints, and the experimental results demonstrate that the proposed method can obtain promising results for ED problems.
This paper is organized as follows: Section 2 briefly introduces the formulation of ED problems.Section 3 introduces the original PSO and its three variants.In addition, a biogeography-based learning particle swarm optimization algorithm is presented in this section.Section 4 addresses the implementation of BLPSO for solving ED problems.Section 5 provides the experimental results on five test systems.Finally, the paper is concluded in Section 6.

Formulation of ED Problems
The objective of the ED problem is to minimize the fuel cost of thermal power plants for a given load demand subject to various physical constraints.
2.1.Objective Function.The traditional fuel cost or objective function of the ED problem is the quadratic fuel cost equation of the thermal generating units and is given by where N g is the total number of generating units or generators, F j P j is the cost function of the jth generating unit ($/hr), P j is the real output of the jth generating units (in MW), and a j , b j , and c j are fuel cost coefficients of the jth generator.
In some ED problems, the admission valves control the steam entering the turbine through separate nozzle groups.When the valve opens, the fuel cost will increase dramatically because of the wire drawing effect, and this makes the practical objective function have many nondifferentiable points [44].Therefore, the fuel cost function often contains many nonsmooth ripple curves due to the presence of valve-point effects.The objective function when the valve-point effect is taken into account is represented as where e j and f j are nonsmooth fuel cost coefficients of the jth generator with valve-point effects and P min j is the minimum power generation limit of the jth generator (in MW).

Optimization Constraints
2.2.1.Power Balance Constraint.The total generated power should be equal to the sum of the total system demand (P D ) and the total transmission network loss (P L ): The B coefficient method is widely utilized to calculate the total transmission network loss P L .In such a way, P L can be calculated as follows: where B ji , B j0 , and B 00 are the loss coefficients or B coefficients.It can be seen that B ji is an N g × N g matrix.

Power Generation Limits.
The power generation of each generator should be within its minimum and maximum limits: 2 Complexity where P min j and P max j are the minimum and maximum power generation limits of the jth generator.

2.2.3.
Ramp-Rate Limits.The physical limitations of starting up and shutting down of generators impose ramp-rate limits, which are modeled as follows.The increase in generation is limited by Similarly, the decrease is limited by where P 0 j is the previous output power and UR j and DR j are the up-ramp limit and the down-ramp limit of the jth generator, respectively.
Combining 6 and 7 with 5 results in the change of the effective operating or generation limits to max P min j , P 0 j − DR j ≤ P j ≤ min P max j , P 0 j + UR j 2.2.4.Prohibited Operating Zones.The prohibited operating zones (POZ) are due to steam valve operation or vibration Algorithm 2: (BLPSO).Complexity in shaft bearing.The feasible operating zones of the jth generator can be described as follows: where n j is the number of prohibited zones of the jth generator.P l j,k and P u j,k are the lower and upper power output of the kth prohibited zone of the jth generator, respectively.
Combining the equations from 2 to 9, the ED problem can be formulated as proposed by Eberhart and Kennedy [45].It is based on the swarm intelligence theory, and the fundamental idea is that the optimal solution can be found through cooperation and information sharing among individuals in the swarm.In the past decade, PSO has gained increasing popularity due to its effectiveness in performing difficult optimization tasks.
In PSO, each individual is treated as a particle in the D-dimensional space, with a position vector x i t = x i1 t , x i2 t , … , x iD t and a velocity vector v i t = v i1 t , v i2 t , … , v iD t .The particle updates its velocity and position according to the following equations: where pbest i t = pbest i1 t , pbest i2 t , … , pbest iD t is the personal best position of particle i; gbest t = gbest 1 t , gbe st 2 t , … , gbest D t is the position of the best particle in the population; w is the inertia weight; c 1 and c 2 are acceleration coefficients; and r 1 and r 2 are two random real numbers distributed uniformly within [0,1].

Comprehensive Learning Particle Swarm Optimization.
Liang et al. [46] proposed a comprehensive learning PSO (CLPSO) which uses a novel comprehensive learning strategy whereby all other particle personal best positions are used to update a particle velocity.This strategy can preserve the diversity of the swarm to discourage premature convergence.CLPSO uses the following velocity updating equation: is the learning exemplar indices for particle i, which is generated based on tournament selection procedure.The CLPSO does not introduce any complex operations to the original simple PSO framework, and the main difference from the original PSO is the velocity update equation.

Social
Leaning Particle Swarm Optimization.Cheng and Jin [47] proposed a social learning PSO (SLPSO) inspired by learning mechanisms in social learning of animals.The SLPSO is performed on a sorted swarm, and particles learn from any better particles in the current swarm.The particles learn from different particles based on the following equations: where P L i is the learning probability for particle i, p i t is a randomly generated probability that satisfies 0 ≤ p i t ≤ P L i ≤ 1, x kj t is the demonstrator of particle i in the jth dimension, x j t = ∑ N i=1 x ij /N is the mean position of the all particles in the current swarm, and ε is the social influence factor.In addition, the SLPSO adopts dimension-dependent parameter control methods to determine the three parameters, that is, the swarm size N, P L i the learning probability, and the social influence factor ε.

Biogeography-Based Learning Particle Swarm
Optimization.In this paper, a biogeography-based learning particle swarm optimization (BLPSO) which employs a new biogeography-based learning strategy (BLS) [48] is proposed for the ED problems.[46] and biogeography-based optimization [49,50].It has two characteristics: (1) Each particle updates itself by using the combination of its own personal best position and personal best positions of all other particles, which is similar to the comprehensive learning strategy of CLPSO.This updating method enables the diversity of the swarm to be preserved to discourage premature convergence.
(2) The migration operator of biogeography-based optimization is used to generate the learning exemplar for each particle, in which a ranking technique is employed to make particles learn more from particles with high-quality personal best positions.This can provide a more effecient balance between exploration and exploitation for the new PSO algorithm.
In BLS, each particle updates its velocity and position according to the following equations: where pbest τ i t = pbest τ i 1 ,1 t , pbest τ i 2 ,2 t , … , pbest τ i D ,D t is the learning exemplar for particle i; is the learning exemplar indices for particle i, which is generated by the biogeographic migration.
In the biogeographic migration, all particles are firstly sorted based on the value of their pbest from best to worst and assigned with ranking values.For a minimization problem, assume where s 1 is the subscript of the particle with the best pbest, s 2 is the subscript of the particle with the second best pbest, and s N is the subscript of the particle with the worst pbest; N is the population size.Then, the rankings of particles are assigned as below:   Second, immigration and emigration rates are assigned for all particles.The immigration and emigration rates for all particles can be calculated as follows: According to 20, the solution x s 1 with the best pbest s 1 will have the lowest immigration rate and highest emigration rate; and the solution x s N with the worst pbest s N will have the highest immigration rate and lowest emigration rate.
Third, the biogeography-based exemplar indices τ i = τ i 1 , τ i 2 , … , τ i D for particle i can be generated based on the biogeography-based exemplar generation method, see Algorithm 1.

Procedures of BLPSO.
Using the BLS, the procedures of BLPSO can be outlined in Algorithm 2. It can be seen from Algorithm 2 that the structure of BLPSO is as simple as the classic PSO.
In addition, based on lines 10-13 in Algorithm 2, it can be seen that Algorithm 1 is executed to generate new learning exemplar indices τ i only when there is a stagnation for G generations, which is used to save computational cost of the BLPSO.In other words, if new learning exemplar indices τ i are generated for all particles in each generation, Algorithm 2 will be executed too frequently, and this may cost a large computational time, which is inappropriate for real-world ED problems.

Remarks
(1) Complexity Analysis.The computational costs of the original BLPSO algorithm involve the initialization (T ini ), biogeography-based exemplar generation method (T bio ), velocity and position update (T upd ), and evaluation (T eva ) for each particle.Assume D is the dimensionality of the optimization problem, N is the population size, and maxF ES is maximum number of functional evaluations allowed for the algorithm.
The total computational complexity of BLPSO is In general, the population size N is often set to be proportional to the problem dimension D (i.e., N = kD) [47].Thus, the complexity of BLPSO is

Complexity
(2) Compared with Previous Hybrid PSO/BBO Algorithms.Several hybrid PSO/BBO algorithms have been proposed in the literature.For example, Guo et al. [51] presented a biogeography-based particle swarm optimization with fuzzy elitism (BPSO-FE) for constrained engineering problems.In this BPSO-FE algorithm, the whole population is split into several subgroups, and BBO is employed to search within each subgroup while PSO for the global search.Mo and Xu [52] applied the position updating strategy of PSO to increase the diversity of population in BBO and develop a biogeography particle swarm optimization algorithm (BPSO) to optimize the paths in path network.However, there are some differences between BLPSO and them.First, the hybrid strategies of BLPSO, BPSO-FE, and BPSO are different.In BLPSO, the biogeography-based migration is used to generate the learning exemplar for each particle; while in BPSO-FE and BPSO, the biogeography-based migration is used as search operator.Second, the application areas of BLPSO, BPSO-FE, and BPSO are different.BLPSO is presented for ED problems, while BPSO-FE and BPSO are proposed for classical engineering optimization problems and robot path planning, respectively.

Implementation of BLPSO for ED Problems
When solving the ED problems using BLPSO, the following three important issues should be considered: initialization of population, constraint handling, and stopping criterion.

Initialization of Population.
In BLPSO, each individual of the population is a solution of an ED problem.If there are N g units that must be operated to provide power to load, then the current position of the ith particle can be given by where N is the population size, j is index of the generating unit, and P ij is the generation power output of the jth generating unit in the ith particle.

Constraint
Handling.One of the most important issues in solving ED problems is how to handle the quality and inequality constraints.There are four types of constraints in the ELD problems: power generation limits, ramp-rate limits, prohibited operating zone, and power balance constraint.For power generation limit and ramp-rate limit constraints, the following strategy is employed: For prohibited operating zone constraints, if P ij is located in the kth prohibited operating zone, that is, P l j,k ≤ P ij ≤ P u j,k , it is truncated to the closest boundary of the kth prohibited operating zone as follows: 9 Complexity where P l j,k and P u j,k denote the lower and the upper bounds of prohibited operation zone k of generator j, respectively.
For the power balance constraint, a repaired operator together with a common penalty is employed [28].The repaired operator is shown in Algorithm 3, and the objective function becomes where K is the penalty coefficient and the penalty term ∑ N g j=1 P j − P D − P L is the measure of violation of the equality constraint.

Stopping Criterion.
The BLPSO algorithm will be terminated if the maximum number of functional evaluations ma xFES is reached.

Results and Discussion
To test the effectiveness of the proposed BLPSO algorithm, five different test systems of varying computational difficulty levels have been solved using BLPSO.The results obtained by BLPSO are compared with two PSO algorithms, comprehensive learning PSO (CLPSO) [46] and social leaning PSO (SLPSO) [47].The results are also compared with several techniques reported in the literature whose abbreviations are listed in Table 1.
To compare the performance of the BLPSO, 50 independent trial runs are made, and the statistical results including the minimum, mean, maximum fuel cost, and standard deviation, as well as average run time, are tabulated for each test system.The parameters of BLPSO are set as follows: population size N = 40, inertia weight w linearly decreases from 0.9 to 0.2, acceleration coefficient c = 1 496, and refreshing gap G = 5.The parameters of CLPSO and SLPSO are set as those recommended in their original papers.The maximum number of functional evaluations maxFES is set as 10,000; 50,000; 50,000; 50,000; and 200,000 for the five test systems, respectively.The programs are implemented in MATLAB language on a personal computer with a 3.2 GHz processor and 8 GB RAM.
5.1.Test System 1.This is a small system comprising 6 generators and meeting a load demand of 1263 MW and includes transmission loss, POZ, and ramp-rate limits.The system data are taken from [8,53] and listed in Table S1.Table 2 presents the optimal generation values and fuel cost obtained by BLPSO.The obtained optimal cost is 15447.34$/hr.It can be seen that the generation values satisfy the generation limit constraints and do not fall in the POZs.
Table 3 shows the comparison of the statistical results of different algorithms.In the table, the results obtained by BLPSO are compared with CLPSO, SLPSO, NPSO-LRS [54], MTS [55], TS [55], SA [55], GAAPI [56], HCRO-DE [42], DE [57], MABC [31], CBA [58], RDPSO [26], IRDPSO [26], and ST-IRDPSO [26].It can be seen that the minimum and mean fuel costs obtained by BLPSO are similar to SLPSO and less than all the other methods with the exceptions of  10 Complexity HCRO-DE [42].In addition, the smaller value of standard deviation indicates that BLPSO is consistent.It is also important to note that the BLPSO is very efficient according to the average computational time (0.50 s), which is less than most of other methods.Figure 1 presents the convergence characteristics obtained by CLPSO, SLPSO, and BLPSO.From Figure 1, SLPSO has the fastest convergence speed, and BLPSO has the second.Both BLPSO and SLPSO can converge to the optimal cost after about 6000 functional evaluations.
5.2.Test System 2. This test system consists of 15 generators meeting a load demand of 2630 MW and includes transmission loss, POZ, and ramp-rate limits.The system data are taken from [8,59] and listed in Table S2.Table 4 presents the optimal generations and the costs obtained.The optimal cost obtained by BLPSO is 32587.33$/hr, and the generations satisfy the generation limit constraints.Table 5 shows the comparison of the statistical results of the BLPSO and other algorithms, including CLPSO, SLPSO, CCPSO [60], HBMO [59], CIHBMO [59], FA [15], MsEBBO [35], DEPSO [40], SWT-PSO [61], IA [62], IODPSO-G [63], and IODPSO-L [63].The minimum and mean fuel costs obtained by BLPSO are the least of all methods with the exceptions of HBMO [59].The average computation time of BLPSO (2.85 s) is also very small.The convergence characteristics obtained by CLPSO, SLPSO, and BLPSO are plotted in Figure 2. SLPSO has the fastest convergence speed in the beginning, but it is surpassed by BLPSO and CLPSO in the end.Only BLPSO can get the optimal cost in this case.5.3.Test System 3. This test system consists of 20 generators supplying a demand of 2500 MW.Transmission losses are included in this system.The cost coefficient and B coefficient data are taken from [22,64] and listed in Table S3.Table 6 presents the optimal generation values and fuel cost obtained by the BLPSO.The optimal obtained fuel cost is 62456.58$/hr.It is seen that the all the generation limit constraints are satisfied.
Table 7 shows the comparison of the statistical results of different algorithms.In the table, the results obtained by BLPSO are compared with CLPSO, SLPSO, EHNN [65], λ-iteration [64], HM [64], GSO [66], CQGSO [66], BBO [22], BSA [22], and CBA [58].It can be seen that the best fuel cost obtained by BLPSO is the least of all methods, and the mean fuel cost is the least of all methods with the sole exception of CBA [58].In addition, the standard deviation and average computation time of BLPSO are both very small.Again, the BLPSO is efficient for this case.The convergence characteristics obtained by CLPSO, SLPSO, and BLPSO are plotted in Figure 3.

5.4.
Test System 4. This test system consists of 38 generators, and the demand of this system is 6000 MW.The system data are taken from [28,41] and listed in Table S4.Table 8 presents the optimal generation values and cost obtained by 11 Complexity BLPSO.The optimal cost is 9417208.19$/hr.It is seen that the generations satisfy the generation limit constraints.
The results obtained by BLPSO are compared with those obtained by CLPSO, SLPSO, New-PSO [67], PSO-TVAC [67], HS [68], HHS [68], BBO [41], DE/BBO [41], MsEBBO [35], and IDE [28], as shown in Table 9.It can be seen that the minimum and mean fuel costs obtained by BLPSO are the least of all the methods.The average computation time of BLPSO is 2.89 s, smaller than all methods, with the exception of CLPSO.The convergence characteristics obtained by CLPSO, SLPSO, and BLPSO are plotted in Figure 4.

Test System 5.
In order to study the performance of the BLPSO on high-dimensional ED problems, a large system with 110 generators is considered.The demand of this system is 15,000 MW, and the system data are taken from [69,70] and listed in Table S5.Table 10 presents the optimal generation values and cost obtained by BLPSO.The optimal cost is 197988.16$/hr.Table 11 shows the comparison of the statistical results of BLPSO and other algorithms, including CLPSO, SLPSO, SAB [71], SAF [71], SA [71], ORCCRO [72], BBO [72], DE/BBO [72], and OIWO [69].The minimum, mean, and maximum fuel costs obtained by BLPSO are the least of all the methods.Meanwhile, the smaller value of standard deviation indicates that BLPSO is consistent.The average computation time of BLPSO is also very small compared with other methods.The convergence characteristics obtained by CLPSO, SLPSO, and BLPSO are presented in Figure 5.

Conclusion
This paper has presented a biogeography-based learning particle swarm optimization (BLPSO) for solving the economic dispatch (ED) problems, which is nonlinear, nonconvex, and discontinuous in nature, with numerous equality and inequality constraints.In the BLPSO, a biogeography-based learning strategy is used to generate the learning exemplar for each particle, in which particles learn more from highquality particles.The biogeography-based learning strategy can provide a more effective balance between exploration and exploitation for the BLPSO.
The BLPSO was applied to five test systems with various constraints such as power balance, POZs, and ramp-rate limits.Transmission losses have also been included in some systems.The experimental results show that the fuel costs obtained by BLPSO are either comparable or lower than those reported by other methods.The application to a 110unit system shows that the BLPSO is also capable of handling high-dimensional ED problems.
In the future, we are planning to extend the BLPSO to solve other more complicated ED problems, such as dynamical ED problems and environmental ED problems.We are also interested in applying the BLPSO to other optimization problems in energy field such as solar photovoltaic modeling [73].

Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.12 Complexity
The complexity of initialization, velocity and position update, and evaluation are O D , O 2D , and O D , respectively.The computational costs of biogeographybased exemplar generation method T bio include population sorting O N • log N , ranking assignment O N , immigration and emigration rate assignment O 2N , and migration operator O N • D .Therefore,

10 3. Particle Swarm Optimization and Its Three Variants
Ng j=1 P ij ≤ P D + P L Then 3 Randomly select a component k from the set I; 4 While P ik = P max Ng j=1 P ij ≤ P D − P L to P ik , such as P ik = min P ik + w, P max Ng j=1 P ij ≤ P D + P L 11 Randomly select a component k from the set I; 12 While P ik = P min Ng j=1 P ij ≤ P D − P L to P ik , such as P ik = max P ik + w, P min 1 Initialize a set I = 1, 2, … , D 2 If ∑ i 5 Exclude k from I, and let the new set be I′; 6 Randomly select a component k′ from I′; 7 k = k′, I = I′; 8 End While 9 Add an amount w = ∑ i ; 10 Else If ∑ i 13 Exclude k from I, and let the new set be I′; 14 Randomly select a component k′ from I′; 15 k = k′, I = I′; 16 End While 17 Add an amount w = ∑ i ; 18 End If Algorithm 3: (the repair operator for power balance constraint).4 Complexity where pbest τ i t = pbest τ i 1 ,1 t , pbest τ i 2 ,2 t , … , pbest τ i D ,D t is the learning exemplar for particle i and

Table 1 :
Table of abbreviations.

Table 2 :
Optimal generations and cost obtained by BLPSO for test system 1 (6-unit system, P D = 1263 MW).

Table 3 :
Comparison of fuel costs and statistical results for test system 1 (6-unit system, P D = 1263 MW).

Table 4 :
Optimal generations and cost obtained by BLPSO for test system 2 (15-unit system, P D = 2630 MW).

Table 5 :
Comparison of fuel costs and statistical results for test system 2 (15-unit system, P D = 2630 MW).

Table 6 :
Optimal generations and cost obtained by the BLPSO for test system 3 (20-unit system, P D = 2500 MW).

Table 7 :
Comparison of fuel costs and statistical results for test system 3 (20-unit system, P D = 2500 MW).

Table 8 :
Optimal generations and cost obtained by the BLPSO for test system 4 (38-unit system, P D = 6000 MW).

Table 9 :
Comparison of fuel costs and statistical results for test system 4 (38-unit system, P D = 6000 MW).

Table 10 :
Optimal generations and cost obtained by the CBA for test system 5 (110-unit system, P D = 5000 MW).

Table 11 :
Comparison of fuel costs and statistical results for test system 5 (110-unit system, P D = 15,000 MW).means the data are not available in the literature. NA