# An Improved Symbiosis Particle Swarm Optimization for Solving Economic Load Dispatch Problem

Authors: Jianxia Zhang; Jianxin Zhang; Feng Zhang; Minglu Chi; Linbin Wan
Journal: Journal of Electrical and Computer Engineering
 (2021)

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## Abstract

To realize the sustainable development of social economy, energy conservation and emission reduction has become one of the problems that must be considered in the current power system. Under the electric power market system, the economic load dispatch problem not only is important but also has practical significance and broad application prospects. In order to minimize the costs of electric-power generation, the load capacity should be reasonably assigned among many different generating sets. In this paper, an improved symbiosis particle swarm optimization algorithm was proposed, aiming at providing a better solution to this problem. First of all, a mathematical model was established with certain constraints, which successfully converted the practical problem into a mathematical one. Then, to balance the global optimization and local search capability, an improved symbiosis particle swarm optimization algorithm with mutualistic symbiosis strategy in nature was presented. The improved symbiosis particle swarm optimization algorithm consisted of three swarms inspired by the proverb “two heads are better than one,” and its specific analysis was through the standard test functions. At last, the economic load dispatch problem could be optimized by the proposed improved symbiosis particle swarm optimization algorithm. In addition, two different kinds of practical examples were also adopted for algorithm evaluation. From the simulation results, it can be seen clearly that the costs of electric-power generation gained were the lowest compared with the results of particle swarm optimization algorithm, improved chaos particle swarm optimization algorithm, and symbiotic organisms search algorithm, well demonstrating the effectiveness of the improved symbiosis particle swarm optimization algorithm in solving the economic load dispatch problem.


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## Body

## 1. Introduction

With the rapid development of market system and the establishment of resource-saving society, the economic load dispatch (ELD) has become an important and practical optimization problem [1–3]. It can be described as the following definition. It refers to the fact that the generator cost of the whole power system increases least, which is under the different constraint conditions [4]. The different constraint conditions include satisfying the user electric requirement, reducing the loss of power system, being in the set range of active power generation, and so on [5]. At present, the ELD problem affects the safety and economic benefit of the whole power system. How to reduce the generator cost of the whole power system has become the wide focus of attention of domestic and foreign scholars [6–8].The traditional approaches for solving the ELD problem include the Lagrange method [9], the equal incremental principle [10], the sequential method [11], the point-by-point method [12], and so on. These traditional approaches cannot solve the ELD problem well, because this problem has characteristics of very strong non-linearity and different various results. In recent years, with the constant development and extending application of intelligent optimization algorithms, how to solve the ELD problem using the intelligent optimization algorithms has become a hot topic for study [13]. In the year of 2011, a differential artificial bee colony algorithm on solving the ELD problem was proposed by Xiong et al. which was inspired by differential evolution algorithm. This algorithm not only improved the search behaviors of employed bees, but also added an effective repair procedure [14]. An improved chaos particle swarm optimization (ICPSO) algorithm had been presented by Han et al. to solve the ELD problem for thermal power plants in 2015. The ICPSO algorithm was based on the chaos particle swarm optimization (CPSO) algorithm. In order to increase the diversity of particles in the swarm, the crossover operation in genetic algorithm was adopted and the generation of inertia weight used the parabola and chaotic sequence [15]. In 2016, a novel hybrid algorithm of differential evolution (DE) and particle swarm optimization (PSO) for solving the ELD problem was described by Parouha and Das. It was abbreviated as “DPD,” because it used DE-PSO-DE on a population composed of three swarms. It has better convergence and efficiency [16]. Together, a new approach to economic dispatch problem using a hybrid algorithm named ACO-ABC-HS was put forward by Sen and Mathur in the year of 2016, which combined the framework of ant colony optimization (ACO), artificial bee colony (ABC), and harmonic search (HS). It could find out the optimum results satisfying minimizing the costs of generation [17]. In 2019, a new phasor particle swarm optimization (PPSO) algorithm was put forward by Gholamghasemi et al. to solve the non-convex economic load dispatch problems. It is efficient and reliable [18]. In 2020, a Marine Predators Algorithm (MPA) was presented by Faramarzi et al., which has got better solutions with low computational cost compared with other algorithms [19]. An improved equilibrium optimization algorithm (IEOA) was described by Shaheen et al. in 2020. It can be used to optimize the multi-objective function of total active power loss minimization [20]. Besides, a multiobjective salp swarm optimizer (MSSO) was presented to enhance the performance of distribution systems. It was more advantageous in large-scale 118-node radial system [21].In this paper, an improved symbiosis particle swarm optimization (ISPSO) algorithm was proposed. It mainly focused on solving the ELD problem. It is based on the principle of mutualistic symbiosis in nature. The mutualistic symbiosis refers to the fact that the biological populations can gain benefits and have mutual influence on each other. The ISPSO algorithm consists of three swarms, which was inspired by the proverb “two heads are better than one.” The two swarms carried on iteration optimization through the PSO of linear descending inertia weight. The third swarm used the position information of the above two swarms to self-update. The specific remaining structure of this paper is organized as follows. In the second part, a mathematical model of ELD problem was established with certain constraint conditions. The concrete ISPSO algorithm was explained and analyzed in the third part. In the fourth part, the two different practical examples were used to do the experiments, which were compared with the PSO algorithm, the ICPSO algorithm, and the symbiotic organisms search (SOS) algorithm. Lastly, the conclusion is presented.


## 2. Establishing a Mathematical Model of Economic Load Dispatch Problem

The ELD problem can be represented as a nonlinear programming subject with inequality constraints and boundary constraints. With satisfying the several different conditions, the minimizing value of fitness function should be found out.


### 2.1. Different Constraint Conditions

The constraint conditions have two forms that include equality and inequality constraints. In this paper, the equality constraints refer to the fact that the obtained parameter values satisfy the balance of load power. The particular expression is as shown in the following formula:(1)∑i=1nPi=P,where n is the total number of generators, Pi represents the active power of the ith generator, and P is the total load power of whole system.In this paper, the inequality constraints mean that the active power of unit is less than or equal to the upper limit but greater than or equal to the lower limit, as is shown in the following formula:(2)Pi_min≤Pi≤Pi_max,i=1,2,…,n,where Pi_min is the minimizing active power of the ith generator and Pi_max represents the maximizing active power of the ith generator.


### 2.2. Description of the Fitness Function

The fitness function in this paper is established with satisfying constraints (1) and (2). Furthermore, the valve-point effect of practical generator is also considered. The valve-point effect is that there will be a kind of phenomenon, which is named the wire drawing. It appears when the turbine inlet valve suddenly opened, which produces a pulse on the consumption characteristic curve of generator [22]. It can be expressed as the following formula:(3)Ei=eisinfiPi−Pi_min,where ei and fi both express the consumption characteristic parameters, which are the constants.The consumption characteristic curve of generator can be expressed using the quadratic functions of active power, which is as shown in the following formula:(4)FPi=aiPi2+biPi+ci,where ai, bi, and ci are constants.Considering the above various factors, the fitness function of ELD problem can be expressed as the following formula:(5)min fitnessPi=∑i=1naiPi2+biPi+ci+eisinfiPi−Pi_min.


## 2.1. Different Constraint Conditions

The constraint conditions have two forms that include equality and inequality constraints. In this paper, the equality constraints refer to the fact that the obtained parameter values satisfy the balance of load power. The particular expression is as shown in the following formula:(1)∑i=1nPi=P,where n is the total number of generators, Pi represents the active power of the ith generator, and P is the total load power of whole system.In this paper, the inequality constraints mean that the active power of unit is less than or equal to the upper limit but greater than or equal to the lower limit, as is shown in the following formula:(2)Pi_min≤Pi≤Pi_max,i=1,2,…,n,where Pi_min is the minimizing active power of the ith generator and Pi_max represents the maximizing active power of the ith generator.


## 2.2. Description of the Fitness Function

The fitness function in this paper is established with satisfying constraints (1) and (2). Furthermore, the valve-point effect of practical generator is also considered. The valve-point effect is that there will be a kind of phenomenon, which is named the wire drawing. It appears when the turbine inlet valve suddenly opened, which produces a pulse on the consumption characteristic curve of generator [22]. It can be expressed as the following formula:(3)Ei=eisinfiPi−Pi_min,where ei and fi both express the consumption characteristic parameters, which are the constants.The consumption characteristic curve of generator can be expressed using the quadratic functions of active power, which is as shown in the following formula:(4)FPi=aiPi2+biPi+ci,where ai, bi, and ci are constants.Considering the above various factors, the fitness function of ELD problem can be expressed as the following formula:(5)min fitnessPi=∑i=1naiPi2+biPi+ci+eisinfiPi−Pi_min.


## 3. An Improved Symbiosis Particle Swarm Optimization Algorithm

### 3.1. Overview of Particle Swarm Optimization

The PSO algorithm was studied by Drs. Eberhart and Kennedy in 1995 [23]. It was originated from the research of predatory behavior of birds, which is similar to other optimal algorithms based on swarm intelligence. The PSO algorithm has less parameters and better structure. Nor is this all; its iterative rules are relatively simple. It has a good application and promotion in various fields. In the PSO algorithm, there are two different variables that are velocity and position. Each particle can update its own velocity and position through the optimal position of itself and overall situation. The expression in detail is as shown in the following formula:(6)vi,jt+1=wvi,jt+c1r1pi,j−xi,jt+c2r2pg,j−xi,jtxi,jt+1=xi,jt+vi,jt+1,where vi,j represents the velocity of the ith particle, j=1,2,…,d, d is the dimension of searching space, xi,j expresses the position of the ith particle. t is the current iteration, w indicates the inertia weight, c1 and c2 are the learning factors, r1 and r2 show the random number uniformly distributed between 0 and 1, pi,j expresses the current optimal position of a particle, and pg,j represents the global optimal position.


### 3.2. Description of Improved Symbiosis Particle Swarm Optimization Algorithm

The PSO algorithm has an advantage of simple easy operation. However, like other algorithms, it also has the inevitable disadvantage, such as easily trapping in local optimum. The SOS algorithm has slow convergence speed in the later stage. In order to improve the performance when the PSO algorithm and SOS algorithm solve complex problems, an ISPSO algorithm is proposed in this paper. The ISPSO algorithm is based on the strategy of mutualistic symbiosis in nature, which makes the particles support each other during the whole iterative process. What is more, the strategy is advantageous to creeping towards the optimal individual. It can make the whole swarm prevent the phenomenon of evolutionary stagnation. In this paper, inspired by the proverb “two heads are better than one,” the ISPSO algorithm consists of three swarms. The concrete strategies are as shown in Figure1.Figure 1
Concrete strategies.From Figure1, it can be seen that swarm-1 and swarm-2 execute the iterative optimization separately. Swarm-3 updates the velocity by the fitness value and velocity obtaining from the above two swarms. At last, the global best value is chosen from the three fitness values.The descending inertia weight linearly is used by the particles in swarm-1 and swarm-2 [24]. The expression of inertia weight is as shown in formula (7). The updating of velocity and position is as formula (6).(7)w=wmax−wmax−wminMt,where wmax represents the maximum value of inertia weight, wmin is the minimum value of inertia weight, and M expresses the maximum number of iterations.The velocity of particles in swarm-3 is updated by the fitness value and velocity obtained from swarm-1 and swarm-2. The specific expression is as shown in the following formula:(8)vi,jt+1=wλλ1v1i,jt+λλ2v2i,jt+vi,jt+c1r1pi,j−xi,jt+c2r2pg,j−xi,jtxi,jt+1=xi,jt+vi,jt+1,where λ1 is the fitness value of swarm-1. Similarly, λ2 is the fitness value of swarm-2. λ=λ1+λ2 represents the sum of two fitness values. v1i,j expresses the velocity of swarm-1. v2i,j expresses the velocity of swarm-2.


### 3.3. Analysis of Improved Symbiosis Particle Swarm Optimization Algorithm

In order to further explain the proposed ISPSO algorithm, the six standard test functions are used as an example [25, 26]. Through the platform MATLAB, the convergence, robustness, and time complexity are discussed. The minimum values of six standard test functions are all zero. The expression of standard test functions is as shown in the following formula:(9)fSphere=∑i=1Dxi2,xi∈−100,100D,fRastrigin=∑i=1Dxi2−10cos2πxi+10,xi∈−10,10D,fGirewank=14000∑i=1Dxi2−∏i=1Dcosxii+1,xi∈−50,50D,fSchwefel=∑i=1Dxi+∏i=1Dxi,xi∈−10,10D,fElliptic=∑i=1D106i−1/D−1xi2,xi∈−100,100D,fRosenbrock=∑i=1D−1100xi2−xi+12+xi−12,xi∈−100,100D,where D indicates the number of independent variables.The parameters of ISPSO algorithm are as shown in the following formula:(10)N=30,D=10,c1=c2=2,wmax=0.9,wmin=0.4,M=1000.The parameters of PSO algorithm are as shown in the following formula:(11)N=30,D=10,c1=c2=2,w=0.5,M=1000.The parameters of ICPSO algorithm are as shown in the following formula:(12)N=30,D=10,c1=c2=2,wmax=0.9,wmin=0.4,MaxC=10,M=1000.The parameters of SOS algorithm are as shown in the following formula:(13)N=30,D=10,M=1000.The convergent curves of six standard test functions can be described as in Figure2.Figure 2
Convergent curves.From Figure2, it can be seen that the proposed ISPSO algorithm has global convergence properties for the six standard test functions. What is more, the convergence is superior to PSO algorithm, ICPSO algorithm, and SOS algorithm. In its searching later stages, the proposed ISPSO algorithm performs the fastest convergence, because of the strategy of mutualistic symbiosis in nature. It improves the diversity of population. Besides, every particle performs global search well, avoiding getting into a local optimum.In this paper, the robustness refers to the proportion reaching the threshold specified when the algorithm has the fifty operating independently. The threshold settings of six standard test functions are shown in Table1. The calculation of success rate is as the following formula:(14)Sr=50−n50×100%,where Sr is the probability of success and n represents the number greater than the threshold value.Table 1
Threshold settings of six standard test functions.
Standard functionsfSpherefRastriginfGirewankfSchwefelfEllipticfRosenbrockThresholds1.0e−0031.0e+0011.0e−0031.0e−0011.0e+0011.0e+001The analysis of robustness is shown in Table2.Table 2
Analysis of robustness.
AlgorithmsfSphere (%)fRastrigin (%)fGirewank (%)fSchwefel (%)fElliptic (%)fRosenbrock (%)ISPSO100921006882100PSO506276443694ICPSO10090100607696SOS10010092668098From Table2, it can be obviously seen that, for the standard test functions Sphere, Girewank, and Rosenbrock, the proposed ISPSO algorithm can be at the threshold of a hundred percent. Regarding it as a whole, compared with the PSO algorithm, the ICPSO algorithm, and the SOS algorithm, the success rate of reaching the threshold is higher by obtaining the proposed ISPSO algorithm.In theory, the computational complexity of the proposed ISPSO algorithm, PSO algorithm, ICPSO algorithm, and SOS algorithm is as shown in the following formula:(15)MISPSO=3ON×D×M≈ON×D×M,MPSO=ON×D×M,MICPSO=ON×D×M,MSOS=ON×D×M.From formula (15), it can be seen that the computational complexity of the proposed ISPSO algorithm and PSO algorithm is in the same order of magnitude theoretically.Table3 lists the run times of six standard test functions obtained by the proposed ISPSO algorithm, PSO algorithm, ICPSO algorithm, and SOS algorithm. The calculation method uses the command statement (tic, toc) of MATLAB.Table 3
Run times of two algorithms.
AlgorithmsfSpherefRastriginfGirewankfSchwefelfEllipticfRosenbrockISPSO4S5S6S6S5S4SPSO1S1S1S1S1S1SICPSO1S1S1S1S1S1SSOS2S2S3S2S3S2SAs seen in Table3, it can be known that the run times obtained by the proposed ISPSO algorithm are in the same order, compared with the PSO algorithm, the ICPSO algorithm, and the SOS algorithm. The strategy of introduction has not increased the computational complexity of the algorithm, which is consistent with the theoretical analysis.


### 3.4. An Economic Load Dispatch Problem Based on Improved Symbiosis Particle Swarm Optimization Algorithm

The ELD problem can be solved by the proposed ISPSO algorithm, the detailed description of which is as follows:Figure 3
Basic flowchart.(1)
The velocity and position of all particles in the three swarms can be initialized by formula (16), which is shown as follows specifically:(16)vi,j=randN,1∗Pi_max−Pi_min20,xi,j=randN,1∗Pi_max−Pi_min+Pi_min,whererandN,1 represents a random number between 0 and 1.(2)
With the above subjected functions (1) and (2), every particle calculates the objective function values by its own different iteration strategy, which is according to formula (5).(3)
The velocity and position of all particles in the three swarms can be updated through the proposed ISPSO algorithm in this paper. The particles of swarm-1 and swarm-2 are updated according to formulas (6) and (7), while the particles of swarm-3 according to formulas (7) and (8).(4)
The local optimal values outputted from every swarm are compared with each other. Then, the global optimal value is got.The basic flowchart of ELD problem solved by the proposed ISPSO algorithm is as shown in Figure3.


## 3.1. Overview of Particle Swarm Optimization

The PSO algorithm was studied by Drs. Eberhart and Kennedy in 1995 [23]. It was originated from the research of predatory behavior of birds, which is similar to other optimal algorithms based on swarm intelligence. The PSO algorithm has less parameters and better structure. Nor is this all; its iterative rules are relatively simple. It has a good application and promotion in various fields. In the PSO algorithm, there are two different variables that are velocity and position. Each particle can update its own velocity and position through the optimal position of itself and overall situation. The expression in detail is as shown in the following formula:(6)vi,jt+1=wvi,jt+c1r1pi,j−xi,jt+c2r2pg,j−xi,jtxi,jt+1=xi,jt+vi,jt+1,where vi,j represents the velocity of the ith particle, j=1,2,…,d, d is the dimension of searching space, xi,j expresses the position of the ith particle. t is the current iteration, w indicates the inertia weight, c1 and c2 are the learning factors, r1 and r2 show the random number uniformly distributed between 0 and 1, pi,j expresses the current optimal position of a particle, and pg,j represents the global optimal position.


## 3.2. Description of Improved Symbiosis Particle Swarm Optimization Algorithm

The PSO algorithm has an advantage of simple easy operation. However, like other algorithms, it also has the inevitable disadvantage, such as easily trapping in local optimum. The SOS algorithm has slow convergence speed in the later stage. In order to improve the performance when the PSO algorithm and SOS algorithm solve complex problems, an ISPSO algorithm is proposed in this paper. The ISPSO algorithm is based on the strategy of mutualistic symbiosis in nature, which makes the particles support each other during the whole iterative process. What is more, the strategy is advantageous to creeping towards the optimal individual. It can make the whole swarm prevent the phenomenon of evolutionary stagnation. In this paper, inspired by the proverb “two heads are better than one,” the ISPSO algorithm consists of three swarms. The concrete strategies are as shown in Figure1.Figure 1
Concrete strategies.From Figure1, it can be seen that swarm-1 and swarm-2 execute the iterative optimization separately. Swarm-3 updates the velocity by the fitness value and velocity obtaining from the above two swarms. At last, the global best value is chosen from the three fitness values.The descending inertia weight linearly is used by the particles in swarm-1 and swarm-2 [24]. The expression of inertia weight is as shown in formula (7). The updating of velocity and position is as formula (6).(7)w=wmax−wmax−wminMt,where wmax represents the maximum value of inertia weight, wmin is the minimum value of inertia weight, and M expresses the maximum number of iterations.The velocity of particles in swarm-3 is updated by the fitness value and velocity obtained from swarm-1 and swarm-2. The specific expression is as shown in the following formula:(8)vi,jt+1=wλλ1v1i,jt+λλ2v2i,jt+vi,jt+c1r1pi,j−xi,jt+c2r2pg,j−xi,jtxi,jt+1=xi,jt+vi,jt+1,where λ1 is the fitness value of swarm-1. Similarly, λ2 is the fitness value of swarm-2. λ=λ1+λ2 represents the sum of two fitness values. v1i,j expresses the velocity of swarm-1. v2i,j expresses the velocity of swarm-2.


## 3.3. Analysis of Improved Symbiosis Particle Swarm Optimization Algorithm

In order to further explain the proposed ISPSO algorithm, the six standard test functions are used as an example [25, 26]. Through the platform MATLAB, the convergence, robustness, and time complexity are discussed. The minimum values of six standard test functions are all zero. The expression of standard test functions is as shown in the following formula:(9)fSphere=∑i=1Dxi2,xi∈−100,100D,fRastrigin=∑i=1Dxi2−10cos2πxi+10,xi∈−10,10D,fGirewank=14000∑i=1Dxi2−∏i=1Dcosxii+1,xi∈−50,50D,fSchwefel=∑i=1Dxi+∏i=1Dxi,xi∈−10,10D,fElliptic=∑i=1D106i−1/D−1xi2,xi∈−100,100D,fRosenbrock=∑i=1D−1100xi2−xi+12+xi−12,xi∈−100,100D,where D indicates the number of independent variables.The parameters of ISPSO algorithm are as shown in the following formula:(10)N=30,D=10,c1=c2=2,wmax=0.9,wmin=0.4,M=1000.The parameters of PSO algorithm are as shown in the following formula:(11)N=30,D=10,c1=c2=2,w=0.5,M=1000.The parameters of ICPSO algorithm are as shown in the following formula:(12)N=30,D=10,c1=c2=2,wmax=0.9,wmin=0.4,MaxC=10,M=1000.The parameters of SOS algorithm are as shown in the following formula:(13)N=30,D=10,M=1000.The convergent curves of six standard test functions can be described as in Figure2.Figure 2
Convergent curves.From Figure2, it can be seen that the proposed ISPSO algorithm has global convergence properties for the six standard test functions. What is more, the convergence is superior to PSO algorithm, ICPSO algorithm, and SOS algorithm. In its searching later stages, the proposed ISPSO algorithm performs the fastest convergence, because of the strategy of mutualistic symbiosis in nature. It improves the diversity of population. Besides, every particle performs global search well, avoiding getting into a local optimum.In this paper, the robustness refers to the proportion reaching the threshold specified when the algorithm has the fifty operating independently. The threshold settings of six standard test functions are shown in Table1. The calculation of success rate is as the following formula:(14)Sr=50−n50×100%,where Sr is the probability of success and n represents the number greater than the threshold value.Table 1
Threshold settings of six standard test functions.
Standard functionsfSpherefRastriginfGirewankfSchwefelfEllipticfRosenbrockThresholds1.0e−0031.0e+0011.0e−0031.0e−0011.0e+0011.0e+001The analysis of robustness is shown in Table2.Table 2
Analysis of robustness.
AlgorithmsfSphere (%)fRastrigin (%)fGirewank (%)fSchwefel (%)fElliptic (%)fRosenbrock (%)ISPSO100921006882100PSO506276443694ICPSO10090100607696SOS10010092668098From Table2, it can be obviously seen that, for the standard test functions Sphere, Girewank, and Rosenbrock, the proposed ISPSO algorithm can be at the threshold of a hundred percent. Regarding it as a whole, compared with the PSO algorithm, the ICPSO algorithm, and the SOS algorithm, the success rate of reaching the threshold is higher by obtaining the proposed ISPSO algorithm.In theory, the computational complexity of the proposed ISPSO algorithm, PSO algorithm, ICPSO algorithm, and SOS algorithm is as shown in the following formula:(15)MISPSO=3ON×D×M≈ON×D×M,MPSO=ON×D×M,MICPSO=ON×D×M,MSOS=ON×D×M.From formula (15), it can be seen that the computational complexity of the proposed ISPSO algorithm and PSO algorithm is in the same order of magnitude theoretically.Table3 lists the run times of six standard test functions obtained by the proposed ISPSO algorithm, PSO algorithm, ICPSO algorithm, and SOS algorithm. The calculation method uses the command statement (tic, toc) of MATLAB.Table 3
Run times of two algorithms.
AlgorithmsfSpherefRastriginfGirewankfSchwefelfEllipticfRosenbrockISPSO4S5S6S6S5S4SPSO1S1S1S1S1S1SICPSO1S1S1S1S1S1SSOS2S2S3S2S3S2SAs seen in Table3, it can be known that the run times obtained by the proposed ISPSO algorithm are in the same order, compared with the PSO algorithm, the ICPSO algorithm, and the SOS algorithm. The strategy of introduction has not increased the computational complexity of the algorithm, which is consistent with the theoretical analysis.


## 3.4. An Economic Load Dispatch Problem Based on Improved Symbiosis Particle Swarm Optimization Algorithm

The ELD problem can be solved by the proposed ISPSO algorithm, the detailed description of which is as follows:Figure 3
Basic flowchart.(1)
The velocity and position of all particles in the three swarms can be initialized by formula (16), which is shown as follows specifically:(16)vi,j=randN,1∗Pi_max−Pi_min20,xi,j=randN,1∗Pi_max−Pi_min+Pi_min,whererandN,1 represents a random number between 0 and 1.(2)
With the above subjected functions (1) and (2), every particle calculates the objective function values by its own different iteration strategy, which is according to formula (5).(3)
The velocity and position of all particles in the three swarms can be updated through the proposed ISPSO algorithm in this paper. The particles of swarm-1 and swarm-2 are updated according to formulas (6) and (7), while the particles of swarm-3 according to formulas (7) and (8).(4)
The local optimal values outputted from every swarm are compared with each other. Then, the global optimal value is got.The basic flowchart of ELD problem solved by the proposed ISPSO algorithm is as shown in Figure3.


## 4. Experiment and Simulation

In order to verify the validity and efficiency of the proposed ISPSO algorithm for solving the ELD problem, the two different kinds of practical examples are used to do experiments and simulations. The practical example 3-machine-6-bus system in the literature is experimented and simulated on MATLAB, which is compared with the PSO algorithm, the ICPSO algorithm, and the SOS algorithm.The total load power of whole system isP=850MW. The constraint conditions and characteristics parameters of consumption are as shown in Table 4, belonging to each generator. In the first case, the valve-point effect is not considered. On the contrary, the other case is that the valve-point effect is considered.Table 4
Constraint conditions and characteristics parameters of consumption.
GeneratorPi_minMWPi_maxMWaibicieifi11006000.001567.925613000.031521004000.001947.853102000.0423502000.004827.97781500.063


### 4.1. Case 1

Without considering the valve-point effect, it is can be described asei=fi=0. The remaining setting of parameters is as shown in Table 4. Because of the randomness of initialization, in order to obtain the objective and fair conclusion, each algorithm runs 50 times independently. The outputted results are analysis and statistics, which are as shown in Table 5 in detail. By experiments, the power allocation result of every generator is as displayed in Table 6.Table 5
Outputted results without considering the valve-point effect.
AlgorithmsBest ($)Worst ($)Mean ($)ISPSO6.85e+037.03e+036.94e+03PSO8.19e+038.19e+038.19e+03ICPSO7.53e+038.19e+037.98e+03SOS8.19e+038.19e+038.19e+03Table 6
Power allocation results without considering the valve-point effect.
Power allocationISPSOPSOICPSOSOS1(MW)3993943873932(MW)3503343303343(MW)101122133123Total load power (MW)850850850850Total cost ($)6.93e+038.19e+037.69e+038.19e+03The better results are in bold in Table5. From Table 5, it can be seen clearly that the best result, worst result, and average result obtained by the proposed ISPSO algorithm are all the best compared with obtaining by PSO algorithm, ICPSO algorithm, and SOS algorithm. And then, it can be concluded that the proposed ISPSO algorithm is the most effective in solving the ELD problem, which is compared with the PSO algorithm, the ICPSO algorithm, and the SOS algorithm.From Table6, it can be seen that, compared with the PSO algorithm and the SOS algorithm, the total cost reduces 1.26e+03$, which adopts the proposed ISPSO algorithm. Compared with the ICPSO algorithm, the total cost reduces 0.76e+03$ adopting the proposed ISPSO algorithm. It is further clarified that, for solving the ELD problem without considering the valve-point effect, the proposed ISPSO algorithm is more useful.The curve of evolutionary process using the proposed ISPSO algorithm is as shown in Figure4.Figure 4
Curve of evolutionary process using the ISPSO algorithm.The curve of evolutionary process with the PSO algorithm is as shown in Figure5.Figure 5
Curve of evolutionary process with the PSO algorithm.The curve of evolutionary process with the ICPSO algorithm is as shown in Figure6.Figure 6
Curve of evolutionary process with the ICPSO algorithm.The curve of evolutionary process with the SOS algorithm is as shown in Figure7.Figure 7
Curve of evolutionary process with the SOS algorithm.The comparisons among the proposed ISPSO algorithm, PSO algorithm, ICPSO algorithm, and SOS algorithm are as shown in Figure8.Figure 8
Comparisons among the four algorithms.Figure4 shows that the evolutionary process of searching the optimal value using the proposed ISPSO algorithm. Similarly, the evolutionary process of searching the optimal value with the PSO algorithm, the ICPSO algorithm, and the SOS algorithm is as shown in Figures 5–7. Figure 8 shows the comparisons among the four algorithms. From Figures 4–8, it can be found that the proposed ISPSO algorithm can quickly search the optimal value. What is more, it can break away from the local optimum and obtain the relatively high alignment accuracy. And so to illuminate, the proposed ISPSO algorithm in this paper is effective in solving the ELD problem without considering the valve-point effect.


### 4.2. Case 2

Considering the valve-point effect, the parameters set are as shown in Table4. Due to the randomness of initialization, in order to get the objective and fair conclusion, each algorithm separately runs 50 times. The outputted results analyzed and counted are as shown in Table 7 in great detail. Through experiments, the power allocation result of every generator is as displayed in Table 8.Table 7
Outputting results considering the valve-point effect.
AlgorithmsBest($)Worst($)Mean($)ISPSO6.88e+037.29e+037.08e+03PSO7.92e+03a8.34e+038.24e+03ICPSO7.62e+038.45e+038.21e+03SOS8.12e+038.32e+038.23e+03Table 8
Power allocation results considering the valve-point effect.
Power allocationISPSOPSOICPSOSOS1(MW)4984995004902(MW)2512512442593(MW)101100106101Total load power (MW)850850850850Total cost ($)6.99e+038.24e+037.95e+038.23e+03In Table7, in order to directly observe, the better results are in bold. From Table 7, it can be seen that clearly the best results, worst results, and mean results got through the proposed ISPSO algorithm are superior to those got through the PSO algorithm, the ICPSO algorithm, and the SOS algorithm. It can be explained effectively that the proposed ISPSO algorithm is more suitable for solving the ELD problem considering the valve-point effect, compared with the PSO algorithm, the ICPSO algorithm, and the SOS algorithm.From Table8, it can be clearly seen that the total cost reduces 1.25e+03$ using the proposed ISPSO algorithm, compared with the PSO algorithm. Compared with the ICPSO algorithm, it reduces 0.96e+03$, while, compared with the SOS algorithm, it reduces 1.24e+03$. It is further explained that, for solving the ELD problem considering the valve-point effect, the proposed ISPSO algorithm is the most effective.The curve of evolutionary process obtained through the proposed ISPSO algorithm is as shown in Figure9.Figure 9
Curve of evolutionary process through the ISPSO algorithm.The curve of evolutionary process got by the PSO algorithm is as shown in Figure10.Figure 10
Curve of evolutionary process with the PSO algorithm.The curve of evolutionary process got by the ICPSO algorithm is as shown in Figure11.Figure 11
Curve of evolutionary process with the ICPSO algorithm.The curve of evolutionary process got by the SOS algorithm is as shown in Figure12.Figure 12
Curve of evolutionary process with the SOS algorithm.The comparisons among the four algorithms are as shown in Figure13.Figure 13
Comparisons among the four algorithms.The evolutionary process of searching the optimal value through the proposed ISPSO is as shown in Figure9. Also, Figures 10–12, respectively, show the evolutionary process of searching the optimal value with the PSO algorithm, the ICPSO algorithm, and the SOS algorithm. Figure 13 shows the comparison among the four algorithms. From Figure 13, it can be observed that, for solving the ELD problem considering the valve-point effect, the proposed ISPSO algorithm in the convergence rate and the ability to jump out to local optimum algorithm is the best compared with the PSO algorithm, the ICPSO algorithm, and the SOS algorithm. It can be absolutely ensured that the proposed ISPSO algorithm is the best in solving the ELD problem considering the valve-point effect.


## 4.1. Case 1

Without considering the valve-point effect, it is can be described asei=fi=0. The remaining setting of parameters is as shown in Table 4. Because of the randomness of initialization, in order to obtain the objective and fair conclusion, each algorithm runs 50 times independently. The outputted results are analysis and statistics, which are as shown in Table 5 in detail. By experiments, the power allocation result of every generator is as displayed in Table 6.Table 5
Outputted results without considering the valve-point effect.
AlgorithmsBest ($)Worst ($)Mean ($)ISPSO6.85e+037.03e+036.94e+03PSO8.19e+038.19e+038.19e+03ICPSO7.53e+038.19e+037.98e+03SOS8.19e+038.19e+038.19e+03Table 6
Power allocation results without considering the valve-point effect.
Power allocationISPSOPSOICPSOSOS1(MW)3993943873932(MW)3503343303343(MW)101122133123Total load power (MW)850850850850Total cost ($)6.93e+038.19e+037.69e+038.19e+03The better results are in bold in Table5. From Table 5, it can be seen clearly that the best result, worst result, and average result obtained by the proposed ISPSO algorithm are all the best compared with obtaining by PSO algorithm, ICPSO algorithm, and SOS algorithm. And then, it can be concluded that the proposed ISPSO algorithm is the most effective in solving the ELD problem, which is compared with the PSO algorithm, the ICPSO algorithm, and the SOS algorithm.From Table6, it can be seen that, compared with the PSO algorithm and the SOS algorithm, the total cost reduces 1.26e+03$, which adopts the proposed ISPSO algorithm. Compared with the ICPSO algorithm, the total cost reduces 0.76e+03$ adopting the proposed ISPSO algorithm. It is further clarified that, for solving the ELD problem without considering the valve-point effect, the proposed ISPSO algorithm is more useful.The curve of evolutionary process using the proposed ISPSO algorithm is as shown in Figure4.Figure 4
Curve of evolutionary process using the ISPSO algorithm.The curve of evolutionary process with the PSO algorithm is as shown in Figure5.Figure 5
Curve of evolutionary process with the PSO algorithm.The curve of evolutionary process with the ICPSO algorithm is as shown in Figure6.Figure 6
Curve of evolutionary process with the ICPSO algorithm.The curve of evolutionary process with the SOS algorithm is as shown in Figure7.Figure 7
Curve of evolutionary process with the SOS algorithm.The comparisons among the proposed ISPSO algorithm, PSO algorithm, ICPSO algorithm, and SOS algorithm are as shown in Figure8.Figure 8
Comparisons among the four algorithms.Figure4 shows that the evolutionary process of searching the optimal value using the proposed ISPSO algorithm. Similarly, the evolutionary process of searching the optimal value with the PSO algorithm, the ICPSO algorithm, and the SOS algorithm is as shown in Figures 5–7. Figure 8 shows the comparisons among the four algorithms. From Figures 4–8, it can be found that the proposed ISPSO algorithm can quickly search the optimal value. What is more, it can break away from the local optimum and obtain the relatively high alignment accuracy. And so to illuminate, the proposed ISPSO algorithm in this paper is effective in solving the ELD problem without considering the valve-point effect.


## 4.2. Case 2

Considering the valve-point effect, the parameters set are as shown in Table4. Due to the randomness of initialization, in order to get the objective and fair conclusion, each algorithm separately runs 50 times. The outputted results analyzed and counted are as shown in Table 7 in great detail. Through experiments, the power allocation result of every generator is as displayed in Table 8.Table 7
Outputting results considering the valve-point effect.
AlgorithmsBest($)Worst($)Mean($)ISPSO6.88e+037.29e+037.08e+03PSO7.92e+03a8.34e+038.24e+03ICPSO7.62e+038.45e+038.21e+03SOS8.12e+038.32e+038.23e+03Table 8
Power allocation results considering the valve-point effect.
Power allocationISPSOPSOICPSOSOS1(MW)4984995004902(MW)2512512442593(MW)101100106101Total load power (MW)850850850850Total cost ($)6.99e+038.24e+037.95e+038.23e+03In Table7, in order to directly observe, the better results are in bold. From Table 7, it can be seen that clearly the best results, worst results, and mean results got through the proposed ISPSO algorithm are superior to those got through the PSO algorithm, the ICPSO algorithm, and the SOS algorithm. It can be explained effectively that the proposed ISPSO algorithm is more suitable for solving the ELD problem considering the valve-point effect, compared with the PSO algorithm, the ICPSO algorithm, and the SOS algorithm.From Table8, it can be clearly seen that the total cost reduces 1.25e+03$ using the proposed ISPSO algorithm, compared with the PSO algorithm. Compared with the ICPSO algorithm, it reduces 0.96e+03$, while, compared with the SOS algorithm, it reduces 1.24e+03$. It is further explained that, for solving the ELD problem considering the valve-point effect, the proposed ISPSO algorithm is the most effective.The curve of evolutionary process obtained through the proposed ISPSO algorithm is as shown in Figure9.Figure 9
Curve of evolutionary process through the ISPSO algorithm.The curve of evolutionary process got by the PSO algorithm is as shown in Figure10.Figure 10
Curve of evolutionary process with the PSO algorithm.The curve of evolutionary process got by the ICPSO algorithm is as shown in Figure11.Figure 11
Curve of evolutionary process with the ICPSO algorithm.The curve of evolutionary process got by the SOS algorithm is as shown in Figure12.Figure 12
Curve of evolutionary process with the SOS algorithm.The comparisons among the four algorithms are as shown in Figure13.Figure 13
Comparisons among the four algorithms.The evolutionary process of searching the optimal value through the proposed ISPSO is as shown in Figure9. Also, Figures 10–12, respectively, show the evolutionary process of searching the optimal value with the PSO algorithm, the ICPSO algorithm, and the SOS algorithm. Figure 13 shows the comparison among the four algorithms. From Figure 13, it can be observed that, for solving the ELD problem considering the valve-point effect, the proposed ISPSO algorithm in the convergence rate and the ability to jump out to local optimum algorithm is the best compared with the PSO algorithm, the ICPSO algorithm, and the SOS algorithm. It can be absolutely ensured that the proposed ISPSO algorithm is the best in solving the ELD problem considering the valve-point effect.


## 5. Conclusions

In this paper, an ISPSO algorithm was proposed for solving the ELD problem. The ISPSO algorithm successfully introduced the strategy of mutualistic symbiosis in nature, which was that biological populations would benefit and influence each other. Besides, the ISPSO algorithm adopted three swarms inspired by the proverb “two heads are better than one,” where swarm-1 and swarm-2 carried on iteration optimization, and swarm-3 utilized the position information of the above two swarms to self-update. Moreover, the descending inertia weight linearly is utilized by the particles to improve the defects of precocious convergence. To solve the ELD problem well, it was converted into a mathematical model using certain conditions in accordance with the reality. Then, the specific analysis of ISPSO was through the standard test functions, and two different kinds of practical examples were employed for the evaluation experiments. The compared simulation results with PSO algorithm, ICPSO algorithm, and SOS algorithm illustrated the effectiveness of the proposed ISPSO algorithm in solving the ELD problem.



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*Source: 8869477-2021-01-05.xml*