Mathematical Problems in Engineering
Volume 2019 (2019), Article ID 5831362, 17 pages
https://doi.org/10.1155/2019/5831362

Research Article

Modified Antipredatory Particle Swarm Optimization for Dynamic Economic Dispatch with Wind Power

Kai Chen, Li Han, Shuhuan Wang, Junjie Lu, and Liping Shi

School of Electrical and Power Engineering, China University of Mining and Technology, Xuzhou 221008, China

Correspondence should be addressed to Li Han; dannyli717@163.com

Received 22 May 2019; Revised 11 August 2019; Accepted 29 August 2019; Published 23 October 2019

Academic Editor: Gaetano Zizzo

Copyright © 2019 Kai Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

A modified antipredatory particle swarm optimization (MAPSO) algorithm with evasive adjustment behavior is proposed to solve the dynamic economic dispatch problem of wind power. The algorithm adds the social avoidance inertia weight to the conventional antipredatory particle swarm optimization (APSO) speed update formula. The size of inertia weight is determined by the distance between the global worst particle and other particles. After normalizing the distance, the inertia weight is controlled within the ideal range by using the characteristics of sigmoid function and linear decreasing method, which improves the ability of particles to avoid the worst solution. Then, according to the characteristics of the acceleration coefficient which can adjust the local and global searching ability of particles, acceleration coefficients of nonlinear change strategy is proposed to improve the searching ability of the algorithm. Finally, the proposed algorithm is applied to several benchmark functions and power grid system models, and the results are compared with those reported using other algorithms, which prove the effectiveness and superiority of the proposed algorithm.

1. Introduction

As environmental issues become more prominent, wind energy, as an important renewable energy source, has been paid more and more attention. However, the volatility and randomness of wind power make the dynamic economic dispatching (DED) model more and more complicated, which also brings many difficulties to the solution of the model. Therefore, the method for solving the dynamic economic dispatching model has always been a research hotspot [1, 2].

In recent years, artificial intelligence algorithms have been widely used in dynamic economic dispatching models. For example, genetic algorithm (GA) [3–9], simulated annealing algorithm (SA) [10], tabu search algorithm (TS) [11, 12], differential evolution algorithm (DE) [13–19], particle swarm optimization (PSO) [20–34], artificial bee colony algorithm (ABC) [35], artificial immune system algorithm (AIS) [36, 37], evolutionary programming algorithm (EP) [38–40], complementary quadratic programming algorithm (cQP) [41], biogeography-based optimization algorithm (BBO) [42, 43], teaching learning-based optimization algorithm (TLBO) [44, 45], charged system search algorithm (CSSA) [46], flower pollination algorithm (FPA) [47], rooted tree optimization (RTO) [48], synergic predator-prey optimization (SPPO) [49], kinetic gas molecule optimization (KGMO) [50], grey wolf optimization (GWO) [51], modified crow search algorithm (MCSA) [52], Maclaurin series-based Lagrangian algorithm (MSL) [53], pattern search algorithm (PS) [54], harmony search algorithm (HS) [55], covariance matrix adapted evolution strategy algorithm (CMAES) [56], improved bacterial foraging algorithm (IBFA) [57], and mine blast algorithm (MBA) [58].

In the intelligent optimization algorithm, particle swarm optimization is favored by scholars because of its advantages of low parameter setting, fast convergence, robustness, and easy implementation. Some scholars have studied the acceleration coefficients c1g and c2g of particle swarm optimization. For example, Snganthan [59] indicated that a better solution can be obtained when the two acceleration coefficients are constant. Trelea [60] further used the convergence factor to improve the convergence ability of the basic PSO and proves the correlation between the two acceleration coefficients. On this basis, Venter and Sobieszczanski-Sobieski [61] proved through experiments that a small cognitive coefficient and a large social coefficient can improve the performance of the PSO algorithm. However, the acceleration coefficients of the abovementioned literatures for particle swarm optimization are all the fixed value, and the influence of the acceleration coefficient of the variation strategy on the performance of the algorithm is not considered. In response to this, Ratnaweera et al. [62] proposed time-varying acceleration coefficients (TVACs), which is to increase the global search ability of the algorithm by using larger individual cognitive coefficients and smaller social coefficients in the early iteration (c1g is larger, and c2g is smaller), and as the number of iterations increases, the individual cognitive coefficient decreases linearly and the social coefficient increases linearly so that the algorithm can obtain smaller individual cognitive coefficients and larger social coefficients to speed up the convergence rate of PSO (c1g is smaller, and c2g is larger). This change process allows the particles to be distributed in every corner of the search space at an early stage and can quickly converge to the global best at a later stage. However, the linear change strategy makes the value of the acceleration coefficient too uniform in the whole change interval, and the speed of change does not change according to the number of iterations, which cause the particles to fall into prematureness in advance and difficult to jump out in the later stage. Therefore, this linear change strategy still cannot effectively raise the performance of the algorithm. Another part of the literature studies the movement characteristics, displacement, and velocity update formula of particles. Kheshti et al. [63] proposed a double-inertia weighted particle swarm optimization algorithm which uses two different inertia weights at the beginning and the end of the algorithm. Both of inertia weights enhance the local and global search ability of the particles and improve the accuracy of the algorithm. Zou et al. [64] used the new displacement update formula to guide the particle’s search activity, expanding the particle’s search space and reducing the possibility of particles falling into local optimum. Elsayed et al. [65] added a crossover operation followed by a greedy selection process while replacing the mean best position of the particles with the personal best position of each particle in the velocity updating equation of random drift particle swarm, which increases the diversity of population and improves the performance of the algorithm. Yao et al. [66] introduced quantum computing theory and used quantum bit and angle to depict the state of particles rather than using particle position and velocity of the classical PSO, which shows a stronger search ability and quicker convergence speed. However, these studies all make improvements from the aspects of the optimal solution of particles and do not consider the influence of the worst solution on particle optimization. Thus, Selvakumar Thanushkodi [67] added the worst solution idea to the particle swarm optimization and proposed the antipredatory particle swarm optimization. The algorithm likens the local worst solution and the global worst solution found by the particles in the iteration to the natural predator. The particles will dodge the predator and avoid the worst solution during the optimization process. However, in each iteration, the particles cannot judge whether to make evasive behavior or the degree of evasion according to the distance between the particles and the worst solution at each iteration; that is to say, no matter how far the particles is from the worst solution, it will make the same degree of evasive behavior, which inevitably increases the complexity of the algorithm and the limitation of search space.

Synthesizing above all researches, this paper proposes a modified antipredatory particle swarm optimization (MAPSO) based on the evasive adjustment behavior of the particles. Firstly, a nonlinear change strategy is introduced to solve the problem that the linear change strategy of acceleration coefficient makes the particles prone to precocity. Then, before the global worst position part of the antipredation particle swarm velocity formula, the avoiding inertia weight is added to adjust the degree of dodge of the particles. The equation of avoiding inertia weight considers the monotonicity of the sigmoid function and introduces the linear decreasing coefficients to control the avoiding inertia weight within the ideal range of variation. Finally, the effectiveness of the proposed algorithm is verified by test functions and standard power system cases.

2. Mathematical Model for Economic Dispatch considering Wind Power

2.1. Cost Function

As a renewable and clean energy source, wind power does not generate fuel costs. Therefore, on the premise of not considering wind power fuel cost, the goal of dynamic economic dispatching of power system with wind farms is to reasonably distribute the output of each generator set in the power grid to minimize the total cost of power generation during dispatching under the condition of meeting the constraints of load and unit constraints. Usually, the objective of economic dispatch of power systems with wind power is usually to minimize the sum of fuel cost of thermal power units, valve-point effect cost, overestimation cost, and underestimation cost of wind power uncertainty. The total cost expression is as follows:

where is the total cost of dispatching in single period, F is the fuel cost, R is the cost of wind power uncertainty, n is the number of thermal power units, N is the total number of thermal power units, m is the number of wind farms, M is the total number of wind farms, Pn is the active power output of the nth thermal power unit in a single period, and is the actual output of the mth wind farm.

The objective function of dynamic economic dispatch needs to consider the sum of costs of each period. The expression is as follows:

where h is the dispatching periods, H is the total number of dispatching periods, and C is the total cost of dynamic economic dispatch in H period.

2.1.1. Fuel Cost

The total fuel cost includes fuel cost of thermal power units and energy consumption cost of steam turbine valve-point effect, and the expression is set as

where an, bn, and cn are the fuel cost coefficient, dn and en are the valve-point effect cost coefficients, and is the lower limit of the active power output of the nth thermal power unit.

2.1.2. Wind Power Cost

The expressions of overestimation and underestimation cost caused by the randomness of wind power are as follows:

where and are the penalty coefficients for overestimation and underestimation of wind power cost, respectively, and is the available output of the mth wind farm.

2.2. Constraints

2.2.1. System Active Power Balance Constraints

The power balance constraint equation considering the network loss is expressed as

The power balance constraint equation considering the network loss and wind power is expressed as

where PD is the predicted value of active power in single load period and PL is the network loss in single load period. The network loss calculation formula is

where , , and are network loss parameters.

2.2.2. Unit Active Output Constraints

where and are, respectively, the upper and lower limits of the output of the nth unit and is the maximum installed capacity of the mth wind farm.

2.2.3. Unit Ramp Rate Constraints

where URn and DRn are the up and down ramp rates of the nth unit, respectively, and ΔT is the dispatch time interval and takes 1 h.

2.2.4. System Spinning Reserve Constraints

(1)Positive spinning reserve capacity constraint with wind power system:

(2)Negative spinning reserve capacity constraint with wind power system:

where and are, respectively, the demand coefficients of wind power prediction errors for positive and negative spinning reserve; and are, respectively, the demand coefficients of the system load prediction errors for positive and negative spinning reserve; Un and Dn are, respectively, the positive and negative spinning reserve capacity provided by the nth thermal power unit; and T₁₀ is the response time of spinning reserve, taking 10 minutes.

3. Modified Antipredation Particle Swarm Optimization Algorithm

3.1. Antipredation Particle Swarm Optimization (APSO)

Antipredation particle swarm optimization (APSO) algorithm regards the particle optimization process as the foraging process of birds. Birds will try their best to avoid the attack of natural enemies while searching for food in nature. APSO adds the position of the natural enemies as the worst position to the PSO model and divides the cognitive experience in the PSO algorithm into two parts: good cognitive experience and bad cognitive experience. The former is the best location memory of the particle’s own experience, and the latter is the memory of the worst position (predator’s position) experienced by the particle itself. Similarly, social experience is divided into two parts: good social experience and bad social experience. The former is the best position memory experienced by the group, and the latter is the worst position memory experienced by the group. Thus, the speed and displacement update formula of APSO algorithm is obtained as follows:

where is the velocity of the ith particle at the tth iteration; ω is the inertia weight, indicating the degree of succession of the particle to the tth iteration speed; c1g is the acceleration coefficient of the particle flying to its own best position; c1b is the acceleration coefficient of the particle flying away its own worst position; c2g is the acceleration coefficient of the best position of the particle flying to the group; c2b is the acceleration coefficient of the worst position of the particle flying away from the group; r₁, r₂, r₃, and r₄ are random numbers uniformly distributed on [0, 1]; is the individual optimal value of the ith particle at the tth iteration; is the individual worst value of the ith particle at the tth iteration; is the global optimal value of the particle swarm at the tth iteration; is the global worst value of the particle swarm at the tth iteration; and is the displacement of the ith particle at the tth iteration.

In equation (13), the velocity update formula is composed of five parts: the first part reflects the influence of the current velocity on the particle, which is related to the current state of the particle; the second part reflects that the particle is affected by its own best position, namely, the particle’s good cognitive experience; the third part reflects the influence of its own worst position, i.e., the bad cognitive experience of particles; the fourth part reflects that particles are affected by the best position of the group, that is, the good social experience of the particle; and the fifth part means the influence of particles on the worst position of the group, i.e., the bad social experience of particles.

3.2. Modified Antipredation Particle Swarm Optimization (MAPSO)

3.2.1. Nonlinear Time-Varying Acceleration Coefficient

In order to improve the performance of the algorithm, Ratnaweeera et al. proposed a linear strategy to adjust the acceleration coefficients c1g and c2g and experimentally obtained that c1g is linearly decreasing from 2.5 to 0.5 and c2g is linearly increasing from 0.5 to 2.5 (the black dotted part in Figure 1), and the algorithm has the best optimization effect. The reason for adopting of linear strategy is that, at the beginning of iteration, particles need to occupy every corner of the search space by relying on their own cognitive experience and maintain the diversity of particles, so a larger c1g and a smaller c2g are set. At the later stage of iteration, the global optimal value approaches the convergence value, and each particle needs to approach the optimal solution and search around it. At this time, c1g is small, while c2g is large. However, the linear time-varying acceleration coefficient changes too fast in the early iteration so that the particles have not yet undergone too many local searches and have to move closer to the global optimal solution on a large scale. In general, the global optimal value of the population in the early search is often the local optimal value, especially for the multipeak problems, which lead to the premature convergence of the particles to the local optimum; while the speed of change of the acceleration coefficient slows down in the late iteration, which makes the particles learn too much from themselves, and is rarely affected by other particles, further aggravating the premature algorithm. Based on this, this paper proposes a nonlinear time-varying acceleration coefficient method as a new adjustment strategy for c1g and c2g (the solid line part of Figure 1). The improved expression is as follows:

where the nonlinear parameters η and δ are used to limit the range of variation of c1g and c2g. Since the nonlinear strategy has the same trend as the linear strategy, only the rate of change is different, so this paper also uses the same range of variation as the linear variation strategy, and in order to satisfy it, the η and δ are supposed to be 2 and 0.5, respectively; θ is the current number of iterations; and θmax is the total number of iterations.

Figure 1: Change trend chart of c1g and c2g.

Figure 1 intuitively shows the change process of linear acceleration coefficient and nonlinear acceleration coefficient.

As can be seen from Figure 1, at the initial iteration stage of the algorithm, c1g and c2g change slowly so that the particles can have more time to learn from themselves at the beginning of the iteration and are less affected by other particles. In this way, the particles can avoid blindly moving closer to other particles, which can effectively ameliorate the precocity phenomenon; in the later stage of iteration, c1g and c2g change faster, and the social experience of particles accumulates rapidly, which accelerate the convergence speed of particles to the global optimal solution. In summary, the setting of the nonlinear acceleration coefficient can improve the premature phenomenon of the particles in the early stage and speed up the convergence of the particles to the optimal solution in the later stage.

3.2.2. Adjustment of Avoid Inertia Weight

c2b in APSO is used to determine the extent to which particles avoid the global worst solution. It is defined as a constant in this algorithm, which may cause particles to avoid the global worst solution even when they are far away.

Figure 2(a) shows the optimization process of two particles i and j in APSO. The superscript t represents the optimization result of the tth iteration, and the superscript represents the optimization result of the iteration; black circle represents the global worst solution. It can be seen from the optimization result of the tth iteration that the particle i is closer to gworst and the particle j is farther away from gworst. In APSO, in the process of avoiding the global worst solution gworst, the particle does not adjust the avoidance degree according to the distance between itself and gworst, so the avoidance extent of i and j is the same. In fact, the particle j is far from gworst and does not need to make the same avoidance extent as particle i, or it may cause the problem that the particle j in Figure 2(a) oversteps the global optimal solution due to its excessive avoidance extent.

(a)

(b)

Figure 2: The evasive principle of (a) APSO and (b) MAPSO.

In order to solve this problem, the inertia weight is defined based on the distance between the particle and the global worst solution gworst to adjust the evading degree of the particle. As shown in Figure 2(b), when the particle j is far away from gworst, the particle’s evasive inertia weight is small, and the degree of evasive behavior of the particle is small, avoiding the problem that the particle j is over the optimal solution in Figure 2(a); when the particle i is closer to gworst, the particle's evasive inertia weight ωgw is large, and the degree of evasive behavior of the particle is large, thereby increasing the search range of the particles i.

The expression of is shown in the following equation:

where

where α and β are linearly decreasing coefficients; μ is the inclination coefficient; is the distance between the particle i and the global worst solution gworst; and are the minimum and maximum distances of the global worst solution to the particle swarm, respectively; is the normalized distance of ; and I is the number of particles.

When the particles are moving, and should exhibit an inverse relationship, i.e., the distance between the particles and gworst is too close ( is small), then is larger, which makes the particles are completely away from gworst; on the contrary, is smaller, making the particles slightly away from gworst. However, due to the monotonous increment of the sigmoid function itself, the inverse relationship cannot be met. Therefore, the linear decreasing strategy is added to the sigmoid function in equation (17) and the values of α and β are defined, which not only makes show an inverse ratio with the change of the distance but also is controlled within an ideal range related to α and β.

The fractional part in equation (17) is the sigmoid function, and the variation of depends on , which is the global worst value and the distance between the particles. However, if is directly expressed in , then in the actual process, the difference in dimension and order of magnitude caused by different optimization models will cause to be too large and the fractional part can only be taken to 1. In order to avoid this phenomenon, this paper normalizes and controls it between [−1, 1], The normalized value is expressed by , as shown in equation (18).

Since the value of is between [−1, 1], the value range of sigmoid function is the red bold curve () of Figure 3. However, this red part is only a small part of the whole value range of sigmoid function (). As can be seen from equation (17), this phenomenon will lead to a very small range of values of , so it is necessary to multiply a coefficient μ before to expand the range of the domain of the sigmoid function so that the range of the sigmoid function can be taken as wide as possible.

Figure 3: The value area of sigmoid function.

The red bold curve () in Figure 3 is the range of the sigmoid function before the coefficient μ is defined, and the region () represented by in the figure almost covers the entire value range of the sigmoid function, so it is reasonable to define the coefficient μ.

Based on the analysis in this section, the velocity and displacement expressions of MAPSO are given by the following equations:

Equation (20) changes the trend of c1g and c2g into a nonlinear strategy based on the speed update formula of APSO and adds before the global worst position, which not only improves the premature of the particles but also expands the search scope, allowing the algorithm to more effectively find the optimal solution to the problem.

4. Constraint Processing and Algorithm Solving Steps

4.1. Constraints Processing

When dealing with DED model of power system, it is necessary to comprehensively consider the output distribution of units and the dynamic adjustment of units in different periods. Therefore, the upper and lower power bounds of thermal power units are dynamically adjusted before the dispatching calculation in each period to satisfy the unit ramp and power limitation constraints. Combined with equations (8) and (10), the following treatment is carried out:

4.2. Algorithm Steps and Flow Chart

According to the dynamic economic dispatch model of wind farms in this paper, the steps of MAPSO are as follows:

 Step 1: initialize the particle population and randomly generate the initial velocity and displacement of each particle, input system, unit parameters, and algorithm control parameters. This paper uses linear decreasing inertia weight (LDIW) [68]:

 where ωmax is set to 0.9 and ωmin is set to 0.4. Step 2: limit the initial velocity and displacement of each particle according to the constraints processing of Section 4.1 so that the particle swarm can search in the feasible region. Step 3: calculate the fitness value of the initial position of the particle and retain the individual optimal and worst positions pbest and pworst, global optimal, and worst positions gbest and gworst. Step 4: calculate according to equation (17) and update the displacement and velocity of each particle by equations (20) to (21). Step 5: calculate the fitness value, update pbest, pworst, gbest, and gworst, and compare with the initial fitness value of step 3, and if the updated fitness value is dominant, it is retained. Step 6: repeat steps 4 to 5 until the iteration terminates. Step 7: output the optimal result.

Flow chart is shown in Figure 4.

Figure 4: Flow chart of algorithm.

5. Case Analysis

Benchmark functions and economic dispatch cases were used in this section to test and verify the MAPSO. The algorithm runs on the personal computer (Windows 7, 64 bits, Intel Core i5 2.6 GHz, 4G RAM) version of MATLAB 2014a.

5.1. Benchmark Functions

In order to verify the effectiveness of the MAPSO algorithm, five test functions are used to prove the search performance of the MAPSO.

(1)Sphere function:

(2)Schwefel function:

(3)Rastrigin function:

(4)Griewank function:

(5)Styblinski function:

Among these test functions, the minimum value of f₁∼f₄ is 0 and the minimum value of f₅ is −39.16599n, where n is the dimension of the function (for example, when n = 30, the minimum value is about −1170). The algorithm parameters are set as follows: the total number of particles is 40, the maximum number of iterations is 1000, and the function dimension is 30. The time-varying acceleration coefficients particle swarm optimization (TVAC-PSO), the nonlinear time-varying acceleration coefficients particle swarm optimization (NTVAC-PSO), the time-varying acceleration coefficients antipredator particle swarm optimization (TVAC-APSO), the time-varying acceleration coefficients antipredator particle swarm optimization (NTVAC-APSO), and the proposed MAPSO are used to calculate the benchmark functions 50 times independently and compared the optimal value, average value, and standard deviation of the results, as shown in Table 1.

Table 1: Comparison of five algorithms for benchmark functions.

Number
Function
Method
Average value
Optimal value
Standard deviation

1
Sphere
TVAC-PSO
3.24E − 01
1.25E – 01
1.38E − 01

NTVAC-PSO
1.38E − 01
4.79E – 02
7.37E − 02

TVAC-APSO
0.00E + 00
0.00E + 00
0.00E + 00

NTVAC-APSO
0.00E + 00
0.00E + 00
0.00E + 00

MAPSO
0.00E + 00
0.00E + 00
0.00E + 00

2
Schwefel
TVAC-PSO
4.32E + 00
2.04E + 00
1.11E + 00

NTVAC-PSO
3.59E + 00
2.02E + 00
1.07E + 00

TVAC-APSO
0.00E + 00
0.00E + 00
0.00E + 00

NTVAC-APSO
0.00E + 00
0.00E + 00
0.00E + 00

MAPSO
0.00E + 00
0.00E + 00
0.00E + 00

3
Rastrigin
TVAC-PSO
5.57E + 01
2.33E + 01
1.39E + 01

NTVAC-PSO
4.98E + 01
3.24E + 01
1.12E + 01

TVAC-APSO
4.18E + 01
1.53E + 01
1.13E + 01

NTVAC-APSO
4.10E + 01
1.37E + 01
1.08E + 01

MAPSO
3.70E + 01
1.06E + 01
1.01E + 01

4
Griewank
TVAC-PSO
3.85E − 02
1.76E – 02
1.08E − 02

NTVAC-PSO
1.31E − 02
7.70E – 03
5.45E − 03

TVAC-APSO
1.23E − 02
3.10E – 03
1.42E − 02

NTVAC-APSO
8.63E − 03
3.10E – 03
6.46E − 03

MAPSO
8.34E − 03
2.40E – 03
3.49E − 03

5
Styblinski
TVAC-PSO
−9.52E + 02
−1.07E + 03
4.15E + 01

NTVAC-PSO
−9.71E + 02
−1.04E + 03
3.85E + 01

TVAC-APSO
−1.00E + 03
−1.08E + 03
3.50E + 01

NTVAC-APSO
−1.00E + 03
−1.08E + 03
3.72E + 01

MAPSO
−1.02E + 03
−1.11E + 03
3.47E + 01

It can be seen from Table 1 that, in terms of the average value and the optimal solution, the results of the MAPSO are superior to the other four algorithms, even multimodal functions such as f₃ and f₄, which have multiple local minimum values, perform equally well. In terms of standard deviation, the standard deviation of f₁ and f₂ is 0, and in the function f₃–f₅, the results of MAPSO are also smaller than the other four algorithms. Taking f₅ as an example, the standard deviation of MAPSO is 34.7. Compared with TVAC-PSO, NTVAC-PSO, TVAC-APSO, and NTVAC-APSO, the standard deviation of MAPSO decreases by 6.8, 3.8, 0.3, and 2.5, respectively. This shows the stability of the MAPSO algorithm and proves the superiority of the MAPSO algorithm in the global optimal solution search ability.

In order to visually demonstrate the convergence characteristics of MAPSO and PSO, Figure 5 shows the convergence curves of two PSO variants and MAPSO.

(a)

(b)

(c)

(d)

(e)

Figure 5: Convergence curve of optimal value of MAPSO and PSO: convergence characteristics of (a) f₁, (b) f₂, (c) f₃, (d) f₄, and (e) f₅.

As can be seen from Figure 5 that the iteration times of the MAPSO algorithm are the least when it converges to the optimal solution, which means that the proposed algorithm can not only improve the convergence speed but also increase the accuracy. In addition, during the iteration process, the fluctuation of the MAPSO convergence curve is relatively more frequent than other PSO variants, which shows that the MAPSO algorithm can expand the search space of particles and improve the activity of the particles so that the particles are not easy to fall into local optimum.

5.2. Economic Dispatch

In order to further prove the superiority of the MAPSO algorithm, the performance of the algorithm is validated by analyzing and simulating the standard 5 units and standard 10 units of IEEE 39-bus in this section. The parameters of the algorithm are set as follows: the number of particles is 30; the total number of iterations is 100; c1g and c2g are set according to formula (15)∼(16); c1b and c2b are set to 0.4 and 0.2, respectively; ω is updated according to formula (23); α and β are taken to 0.4 and 0.2, respectively; and μ is taken to 9.

5.2.1. MAPSO Test of 5 Generators Unit System without Wind Power

Taking the five-unit power system without wind power as an example, the dispatching model considers fuel cost (3) and constraints (5), (7), (8), and (10), dispatching period H = 24 h, and time interval is 1 h. The parameters of units, loads, and network losses in the system are obtained from [10]. The dynamic economic dispatching model is optimized 30 times by using the MAPSO algorithm, and the optimal total cost is 43439 USD. The output, network loss, and cost distribution of each unit within 24 hours are shown in Table 2.

Table 2: Unit output, network loss, and cost during each period of no wind five-unit system.

Period of time
Output of thermal generating units in different periods (MW)
Network loss (MW)
Cost (USD)

G1
G2
G3
G4
G5

1
22.66
98.54
112.67
40.00
139.76
3.64
1265.98

2
10.00
98.54
112.67
78.06
139.76
4.03
1444.85

3
10.00
92.43
112.67
124.91
139.76
4.76
1389.84

4
10.14
98.54
112.67
174.91
139.76
4.49
1659.36

5
10.00
92.52
112.67
209.81
139.76
6.77
1587.79

6
15.09
98.54
152.67
209.82
139.76
7.88
1857.62

7
23.63
98.54
112.67
209.82
189.76
8.42
1965.28

8
12.71
98.54
112.67
209.82
229.52
9.27
1797.29

9
35.55
98.55
126.70
209.82
229.52
10.14
2043.16

10
64.02
98.54
112.67
209.82
229.52
10.57
1996.61

11
75.00
104.04
112.68
209.82
229.52
11.06
2038.04

12
75.00
98.54
112.67
236.01
229.52
11.74
2206.63

13
64.03
98.54
112.67
209.81
229.52
10.57
1996.61

14
49.63
98.54
112.67
209.82
229.52
10.18
1977.71

15
19.63
91.59
112.67
209.81
229.52
9.23
1862.85

16
10.00
75.20
112.67
159.81
229.52
7.21
1892.85

17
10.00
87.59
112.67
124.91
229.52
6.69
1615.27

18
33.64
98.54
129.22
124.91
229.52
7.83
1854.72

19
47.46
98.54
112.67
174.91
229.52
9.10
2061.87

20
64.02
98.54
112.67
209.82
229.52
10.57
1996.62

21
39.36
98.54
112.67
209.82
229.52
9.91
1944.65

22
10.00
98.54
112.67
209.82
181.89
7.92
1863.93

23
20.18
98.54
112.67
161.74
139.76
5.89
1695.06

24
10.00
82.13
110.71
124.91
139.76
4.50
1424.73

Total
192.39
741.76
2300.68
2772.78
4252.66
4703.01
43439.34

In order to highlight the performance of the MAPSO algorithm, Table 3 lists the optimal results of the proposed algorithm compared with those of other algorithms.

Table 3: Comparison of simulation results.

Algorithm
Minimum cost (USD)
Average cost (USD)
Maximum cost (USD)

MSL [53]
49216.81
—
—

SA [10]
47356
—
—

IRCGA [9]
47185
—
—

PS [54]
46530
—
—

DE [9]
45800
—
—

GA [35]
44862.42
44921.76
45893.95

AIS [36]
44385.43
44758.84
45553.77

HS [55]
44367.23
—
—

PSO [35]
44253.24
45657.06
46402.52

ABC [35]
44045.83
44064.73
44218.64

EAPSO [34]
43820
44082
44982

CMAES [56]
43536
43915
44191

MAPSO
43439.34
43751.13
44034

It can be clearly seen that the proposed algorithm is superior to other algorithms in terms of average, maximum, and minimum, which further illustrates the effectiveness of the algorithm for model solving.

5.2.2. MAPSO Test of 10 Generators Unit System without Wind Power

The dispatching model is the same as the ten-unit model. The parameters of units, loads, and network losses in the system are obtained from [69]. The dynamic economic dispatching model is run 30 times by using the MAPSO algorithm, and the optimal total cost is 2474831 USD. The output, network loss, and cost distribution of each unit within 24 hours are shown in Table 4.

Table 4: Unit output, network loss, and cost during each period of no wind ten-unit system.

Period of time
Output of thermal generating units in different periods (MW)
Network loss (MW)
Cost (USD)

G1
G2
G3
G4
G5
G6
G7
G8
G9
G10

1
150.00
135.00
73.00
148.42
164.72
140.39
90.25
120.00
23.82
10.00
19.54
61859

2
150.00
135.00
73.00
98.42
214.72
160.00
117.31
120.00
53.82
10.00
22.48
65257

3
150.00
135.00
131.70
147.78
243.00
140.85
130.00
120.00
48.44
40.00
28.61
72718

4
150.00
135.00
197.70
178.95
243.00
160.00
130.00
120.00
72.05
55.00
35.56
80414

5
150.00
135.00
277.70
228.95
223.73
143.62
128.03
115.31
64.08
53.20
39.59
84957

6
150.00
135.00
340.00
278.95
243.00
153.16
130.00
120.00
71.43
55.00
48.22
93839

7
150.00
210.38
340.00
280.62
243.00
160.00
130.00
120.00
74.16
47.03
53.23
100183

8
150.00
257.45
340.00
300.00
243.00
160.00
130.00
120.00
80.00
55.00
58.61
106791

9
230.00
337.45
340.00
300.00
243.00
160.00
130.00
120.00
80.00
54.41
70.70
123188

10
306.22
379.12
340.00
298.26
243.00
159.62
130.00
120.00
80.00
44.76
79.72
136391

11
311.06
459.12
340.00
300.00
243.00
160.00
130.00
120.00
75.42
54.54
88.07
148757

12
391.06
435.70
340.00
300.00
243.00
160.00
130.00
120.00
67.94
55.00
92.73
156969

13
311.06
425.77
340.00
292.12
243.00
160.00
130.00
120.00
80.00
55.00
84.72
143974

14
231.06
345.77
340.00
300.00
243.00
160.00
130.00
120.00
80.00
44.38
70.77
123555

15
151.06
265.77
340.00
300.00
243.00
160.00
130.00
117.59
80.00
47.15
58.65
107111

16
150.00
185.77
268.46
250.00
225.38
158.69
130.00
120.00
54.77
55.00
43.90
90665

17
150.00
135.73
309.78
244.61
218.43
148.98
127.98
120.00
24.77
39.62
39.69
84755

18
150.88
135.00
340.00
294.61
243.00
155.13
130.00
117.58
54.77
55.00
48.21
93681

19
192.12
215.00
340.00
300.00
243.00
160.00
130.00
120.00
80.00
55.00
58.48
106556

20
272.12
295.00
340.00
300.00
243.00
160.00
130.00
120.00
80.00
55.00
70.64
123440

21
260.65
319.08
340.00
300.00
243.00
159.19
130.00
120.00
71.19
52.74
70.85
124113

22
180.65
239.08
260.00
300.00
238.33
123.65
130.00
120.00
41.19
43.98
49.30
98371

23
150.00
159.39
180.00
252.32
243.00
125.10
130.00
90.00
20.00
13.98
32.25
77864

24
150.00
135.00
122.09
231.39
223.82
101.22
130.00
83.04
22.69
10.00
25.58
69405

Total
4788
5746
6653
6225
5640
3630
3064
2804
1481
1061
1290
2474813

The MAPSO algorithm is compared with the algorithms used in the same model and data in recent years [19, 37, 57, 69–71]. The specific results are shown in Table 5.

Table 5: Algorithm optimization results and comparison in 24 h.

Algorithm
Computing time (s)
Cost (USD)

MAPSO
12.26
2474813

IMOEA/D-CH [70]
—
2480200

MAMODE [19]
505
2492451

IBFA [57]
5.2
2481733

HMO-DE-PSO [71]
—
2484000

RCGA/NSGA-II [69]
1080
2516800

AIS [37]
53.56
2519700

PSO [37]
68.47
2572200

EP [37]
72.68
2585400

From Table 5, it can be seen that, compared with the IMOEA/D-CH algorithm, MAMODE algorithm, HMO-DE-PSO algorithm, and RCGA/NSGA-II algorithm, the cost of MAPSO is reduced by 5387 USD, 17638 USD, 9187 USD, and 4187 USD. Similarly, compared with traditional algorithms such as AIS, PSO, and EP, the cost savings are also considerable. In terms of computing time, MAPSO only takes 12.26 seconds, which is much lower than MAMODE and RCGA/NSGA-II. Compared with traditional algorithms, the computing time of MAPSO is also significantly reduced. Compared with IBFA, the computing time of both algorithms is close, but the cost is reduced by 6920 USD. Synthesizing the above data analysis., it can be seen that the proposed MAPSO algorithm is more effective and superior for solving high-dimensional and multiconstrained dynamic economic dispatching models. It can not only save time but also find more economical and reasonable dispatching schemes for decision makers in the process of dispatching without losing accuracy while guaranteeing speed.

5.2.3. MAPSO Test of 10 Generators Unit System with Wind Power

In this test system, a wind farm connected to the grid is considered. The total installed capacity of the wind farm connected to the grid is 100 MW, with a total of 50 wind turbines. The dispatching model considers fuel cost (3) and constraints (6)–(10), dispatching period H = 24 h, and time interval is 1 h. The parameters of units and network losses in the system are obtained from [70]. The predicted values of wind power and system load in each period are shown in Table 6 and are 0.2 and 0.3, respectively, while and are 0.05, respectively. The dynamic economic dispatching model is optimized 30 times by using the MAPSO algorithm, and the optimal total cost is 2355671 USD. The output, network loss, and cost distribution of each unit within 24 hours are shown in Table 7. The power distribution of each unit is shown in Figure 6, which verifies that it satisfies the power balance constraints.

Table 6: Wind power and system load forecast at different periods.

Period of time

PD

1
55
1036

2
50
1110

3
65
1258

4
48
1406

5
38
1480

6
48
1628

7
55
1702

8
48
1776

9
32
1924

10
20
2022

11
40
2106

12
50
2150

13
65
2072

14
72
1924

15
90
1776

16
100
1554

17
85
1480

18
68
1628

19
60
1776

20
70
1972

21
75
1924

22
90
1628

23
80
1332

24
75
1184

Table 7: Unit output, network loss, and cost during each period of wind ten-unit system.

Period of time
Output of thermal generating units in different periods (MW)
Network loss (MW)
Cost (USD)

G1
G2
G3
G4
G5
G6
G7
G8
G9
G10

1
150.00
135.00
73.00
60.00
159.31
160.00
111.17
120.00
20.00
10.11
17.75
59133

2
150.00
135.00
115.96
60.00
209.31
160.00
130.00
90.00
20.00
10.00
20.58
62973

3
150.99
135.00
176.52
110.00
171.37
147.17
130.00
120.00
50.00
27.42
25.69
69439

4
150.00
135.00
256.52
160.00
221.37
160.00
128.69
109.93
20.27
49.28
33.27
78288

5
150.00
135.00
277.17
210.00
243.00
160.00
130.00
111.27
28.33
34.64
37.48
83064

6
151.34
138.96
340.00
260.00
243.00
160.00
106.44
120.00
58.33
46.98
45.41
91564

7
150.00
137.52
340.00
287.57
241.86
160.00
130.00
120.00
80.00
49.25
49.30
94644

8
170.31
201.55
340.00
300.00
233.82
160.00
130.00
120.00
80.00
46.99
54.97
102375

9
250.13
281.55
340.00
300.00
243.00
160.00
130.00
120.00
80.00
55.00
67.81
119564

10
299.50
361.55
340.00
300.00
243.00
160.00
130.00
120.00
80.00
45.96
77.83
133788

11
301.93
440.45
340.00
300.00
243.00
160.00
130.00
120.00
60.51
55.00
84.42
144372

12
374.93
387.89
340.00
300.00
243.00
160.00
130.00
120.00
80.00
51.28
87.35
147870

13
313.60
345.89
340.00
298.78
243.00
160.00
130.00
119.01
80.00
55.00
78.21
134511

14
237.77
265.89
340.00
297.55
243.00
160.00
130.00
120.00
79.98
41.94
64.59
115564

15
157.77
185.89
340.00
295.06
225.21
160.00
125.30
120.00
80.00
49.05
52.14
99136

16
150.00
135.00
260.00
245.60
224.16
145.60
130.00
102.97
55.58
43.54
38.15
83394

17
150.21
135.00
201.58
195.60
243.00
160.00
129.99
120.00
51.71
43.23
34.94
79544

18
150.00
156.04
281.58
232.25
243.00
160.00
125.77
120.00
80.00
55.00
44.05
90295

19
150.00
236.04
340.00
282.25
243.00
160.00
130.00
118.98
63.64
46.41
54.38
102082

20
230.00
316.04
340.00
300.00
243.00
160.00
130.00
120.00
80.00
52.06
68.78
120282

21
211.14
287.77
340.00
300.00
235.31
160.00
124.40
120.00
80.00
55.00
64.46
115275

22
150.00
207.77
260.00
250.00
221.80
155.23
130.00
120.00
50.45
35.82
43.15
90214

23
150.00
135.00
181.34
200.00
179.14
136.87
128.09
116.27
20.45
33.11
28.17
72575

24
150.00
135.00
141.08
158.79
134.22
131.00
130.00
87.42
50.45
12.97
22.15
65728

Total
4650
5166
6645
5704
5373
3756
3060
2776
1430
1005
1195
2355671

Figure 6: Power balance constraint verification.

Similarly, in order to verify the search performance of the MAPSO algorithm, the optimal results are compared with those of IMOEA/D-CH and NSGA-II-CH algorithms in [70], as shown in Table 8.

Table 8: Comparison of optimization results of each algorithm.

Algorithm
Cost (USD)

MAPSO
2355671

IMOEA/D-CH [70]
2359800

NSGA-II-CH [70]
2377700

It can be seen from Table 8 that the MAPSO algorithm is still superior to the other two algorithms in solving the dynamic economic model with wind power. Compared with the IMOEA/D-CH algorithm and NSGA-II-CH algorithm, the cost is reduced by 4129 USD and 20229 USD, respectively. Therefore, the advantages of the MAPSO algorithm are reflected in the results.

In order to embody the rationality of wind power integration into the power system, Table 9 gives the comparison of the optimization results of the dispatching model before and after the application of the MAPSO algorithm to wind power integration.

Table 9: Data comparison of two models.

Models
Total network loss (MW)
Total cost (USD)

Model without wind power
1290.1
2474813

Model with wind power
1195.0
2355671

From Table 9, it is clear that the dispatching model with wind farms is lower than that without wind farms, regardless of network loss or cost. In this paper, Tables 4 and 7 are used to extract three periods from 24 periods for comparative analysis: for example, in the first moment, the network loss of the model without wind farms is 19.54 MW, and that of the model with wind farms is 17.75 MW, which is reduced by 1.79 MW and 9.2%. In the 16th period, the network loss of the model without wind farms is 43.9 MW, and that of the model with wind farms is 38.15 MW, which is reduced by 5.75 MW and 13.1%. In the 24th period, the network loss of the model without wind farms was 25.58 MW, and that of the model with wind farms was 22.15 MW, reduced by 3.43 MW and 13.4%, and the total network loss of the 24 periods was reduced by 95.1 MW. Similarly, in these three periods, the cost savings of wind power grid-connected models are 2726 USD, 7271 USD, and 3633 USD, respectively. The total cost of 24 periods decreased by 119142 USD. It can be seen that the addition of wind power is very beneficial to the efficiency and cost of the system operation.

5.2.4. MAPSO Test of 10 Generators Unit System considering Wind Power Uncertainty

Considering the randomness of wind power, the model adds overestimation and underestimation of penalty costs on the basis of Section 5.2.3; therefore, the model considers (3) and (4) and constraints (6)–(10), dispatching period H = 24 h, and time interval is 1 h. The parameters of units and network losses in the system are obtained from [70]. The total installed capacity of the wind farm connected to the grid is 100 MW, with a total of 50 wind turbines. The predicted values of wind power and system load in each period are shown in Table 6. and are 0.2 and 0.3, respectively, while and are 0.05, respectively. and are 30 and 5, respectively. The dynamic economic dispatching model is run 30 times by using the MAPSO algorithm, and the optimal total cost is 2368764 USD. The thermal power and wind power output, network loss, and cost distribution of each unit within 24 hours are shown in Table 10.

Table 10: Unit output, network loss, and cost of a ten-unit system considering wind power uncertainty.

Period of time
Output of the thermal power and wind power in different periods (MW)
Network loss (MW)
Cost (USD)

G1
G2
G3
G4
G5
G6
G7
G8
G9
G10
W

1
150.0
135.0
73.0
60.0
189.4
57.0
130.0
79.1
71.3
55.0
54.6
18.2
60057

2
160.6
135.0
135.5
60.0
159.9
107.0
112.3
109.1
41.3
55.0
54.8
20.6
64758

3
150.0
135.0
215.5
106.4
182.9
122.5
82.3
120.0
53.4
51.1
65.3
26.0
70225

4
150.0
135.0
241.0
156.4
223.9
143.0
112.3
115.6
64.6
38.8
58.3
32.8
78433

5
150.0
135.0
321.0
204.3
238.6
127.5
95.3
113.0
80.0
10.0
43.3
37.7
83397

6
221.0
135.0
333.9
243.1
188.6
160.0
125.3
120.0
80.0
10.0
56.4
45.5
93751

7
150.0
177.9
316.2
293.1
238.6
160.0
128.0
109.3
76.1
40.0
61.2
49.1
96149

8
150.0
257.9
340.0
300.0
232.0
160.0
130.0
101.5
59.4
51.8
49.0
55.5
104448

9
230.0
334.9
340.0
300.0
243.0
160.0
130.0
101.9
80.0
35.3
37.0
68.0
121455

10
281.1
370.3
340.0
295.8
243.0
160.0
126.1
120.0
71.4
55.0
35.9
76.5
133219

11
361.1
372.2
337.0
300.0
239.7
155.5
122.2
117.3
67.8
55.0
60.8
82.6
143785

12
319.6
401.9
340.0
300.0
240.6
160.0
121.4
120.0
74.6
55.0
100.0
82.7
142804

13
300.0
354.8
340.0
300.0
242.1
160.0
130.0
119.5
80.0
25.0
96.6
75.7
132993

14
241.0
274.8
340.0
300.0
232.6
160.0
108.9
107.7
80.0
55.0
87.8
63.9
116751

15
161.0
208.4
340.0
300.0
243.0
129.3
130.0
120.0
50.0
55.0
91.2
52.5
99796

16
150.0
135.0
310.4
250.0
229.7
147.4
120.7
100.8
20.0
28.6
100.0
38.4
84286

17
150.0
135.0
340.0
200.0
179.7
138.7
130.0
99.2
20.0
34.3
88.2
35.4
81265

18
150.0
183.8
330.1
250.0
229.7
160.0
100.0
101.6
42.2
55.0
69.9
44.5
92439

19
184.9
225.5
340.0
300.0
206.1
160.0
130.0
120.0
52.1
50.2
61.2
54.5
103093

20
227.9
301.0
340.0
294.0
243.0
160.0
130.0
120.0
80.0
42.0
100.0
66.3
118000

21
213.0
307.2
340.0
300.0
220.9
129.6
130.0
120.0
76.3
49.6
100.0
63.3
114992

22
150.0
227.2
275.6
256.3
216.0
126.4
100.0
120.0
80.0
19.6
100.0
43.1
91493

23
150.0
147.2
195.6
247.8
166.0
76.4
86.4
120.0
77.8
10.0
83.0
28.6
74200

24
150.0
140.9
115.6
197.8
175.0
82.3
86.4
106.5
65.8
10.0
76.1
22.5
66973

Total
4701
5366
6940
5815
5204
3303
2798
2682
1544
946
1730
1184
2368764

It can be seen from Table 10 that the optimal output, network loss, and cost of each unit in each dispatching period considering wind power uncertainty. The total optimal output of ten thermal generators (G1–G10) and one wind farm (W) under each load is equal to the system load, which satisfies the power balance constraints and the upper and lower power limits of each unit, so the results obtained by MAPSO are reasonable.

In order to reflect the rationality of considering the randomness of wind power, Table 11 lists the data comparison results considering and not considering the randomness of wind power dispatching models.

Table 11: Data comparison of two models.

Models
Total cost (USD)

Not considering wind power uncertainty
2355671

Considering wind power uncertainty
2368764

From Table 11, it is obviously that the cost of model considering wind power uncertainty is 13093 USD higher than that of model not considering wind power uncertainty. This is because the former model adds overestimation and underestimation costs to the original model. Although the cost increases, the impact of wind power stochastic characteristics on the grid cannot be neglected in the actual wind power grid operation. Therefore, the slight increase in costs is reasonable. On the contrary, it also shows that the MAPSO algorithm has good search performance in solving the problem of the wind power stochastic model.

6. Conclusions

Dynamic economic dispatching models are built. Fuel cost of thermal power units, valve-point effect cost, and overestimation and underestimation costs of wind power uncertainty are considered in the objective function of the model. Network loss and positive and negative spinning reserve demand added by uncertainty of wind power grid connection are considered in the constraints. For solving the model, a modified antipredator particle swarm optimization algorithm is proposed, and the following work is done:

(1)The algorithm considers the distance between the global worst particle and other particles, introduces the inertia weight to control the degree of particle movement, and constructs the formula by using the characteristics of sigmoid function, normalization, and linear decreasing. In view of the shortcomings of time-varying acceleration coefficient, a nonlinear time-varying acceleration coefficient is proposed, which improves the local search and global search ability of the particle.(2)In order to verify the effectiveness of the improved algorithm, several benchmark functions and power grid system models are used to analyze the proposed algorithm. The simulation results show that the MAPSO algorithm is practical and superior in model solving.

Abbreviations

MAPSO: | Modified antipredatory particle swarm optimization
APSO: | Antipredatory particle swarm optimization
DED: | Dynamic economic dispatching
PDF: | Probability density function
GA: | Genetic algorithm
SA: | Simulated annealing
DE: | Differential evolution
ABC: | Artificial bee colony
AIS: | Artificial immune system
EP: | Evolutionary programming
MSL: | Maclaurin series-based Lagrangian
PS: | Pattern search
HS: | Harmony search
CMAES: | Covariance matrix adapted evolution strategy
IBFA: | Improved bacterial foraging algorithm
TVAC: | Time-varying acceleration coefficients
NTVAC: | Nonlinear time-varying acceleration coefficient
LDIW: | Linear decreasing inertia weight
IRCGA: | Improved real-coded genetic algorithm
EAPSO: | Enhanced adaptive particle swarm optimization
IMOEA/D-CH: | Improved multiobjective evolutionary algorithm based on decomposition with constraints handling
MAMODE: | Multiobjective differential evolution with expanded double selection and adaptive random restart
RCGA/NSGA-II: | Real-coded genetic algorithm and nondominated sorting genetic algorithm-II
HMO-DE-PSO: | Hybrid multiobjective DE and PSO
NSGA-II-CH: | Nondominated sorting genetic algorithm-II with constraints handling.

Nomenclature

: | The total cost of dispatching in single period
: | The fuel cost of thermal unit n
an, bn, and cn: | The fuel cost coefficients of thermal unit n
: | The energy consumption cost caused by the valve-point effect of thermal unit n
dn and en: | The coefficients of thermal unit n due to valve-point effect
Pn: | The active power output of thermal unit n
and : | The maximum and minimum limits of the output of thermal unit n
N: | The total number of thermal units
: | The available output of the mth wind farm
: | The actual output of the mth wind farm
: | The penalty cost of overestimation of wind power output
: | The penalty cost of underestimation of wind power output
H: | The total number of dispatching periods
C: | The total cost of dynamic economic dispatch in H period
M: | The total number of wind farms
PD: | The predicted value of active power in single load period
PL: | The network loss in single load period
, , and : | The loss coefficients of network loss
: | The maximum installed capacity of the mth wind farm
URn and DRn: | The up and down ramp rates of the nth unit, respectively
ΔT: | The dispatch time interval
and : | The demand coefficients of the system load prediction errors for positive and negative spinning reserve
and : | The demand coefficients of wind power prediction errors for positive and negative spinning reserve, respectively
Un and Dn: | The positive and negative spinning reserve capacity provided by the nth thermal unit, respectively
T₁₀: | The response time of spinning reserve
k and c: | Shape factor and scale factor of Weibull distribution, respectively
: | The rated capacity of the wind farm
V, , , and : | Actual, cut-in, rated, and cut-out wind speeds, respectively
: | The probability of occurrence of events
kr and kp: | The penalty coefficients for overestimation and underestimation
: | The velocity of the ith particle at the tth iteration
ω: | Inertia weight of the particle
ωmax and ωmin: | The maximum and minimum inertia weight of the particle
c1g and c1b: | The acceleration coefficients of the particle flying to its own best position and flying away its own worst position, respectively
c2g and c2b: | The acceleration coefficients of the best position of the particle flying to the group and the worst position of the particle flying away from the group, respectively
r₁, r₂, r₃, and r₄: | The random number uniformly distributed on [0, 1], respectively
and : | The individual optimal value and worst value of the ith particle at the tth iteration, respectively
and : | The global optimal value and worst value of the particle swarm at the tth iteration
: | The displacement of the ith particle at the tth iteration
η and δ: | The nonlinear parameters
θ and θmax: | The current number and total number of iterations, respectively
: | The particle’s evasive inertia weight
Α and β: | The linear decreasing coefficients of
μ: | The inclination coefficient of
: | The distance between the particle i and the global worst solution gworst
lmin and lmax: | The minimum and maximum distances of the global worst solution to the particle swarm, respectively
: | The normalized value of li
I: | The number of particles.

Data Availability

The data used to support the findings of this study will be provided by the author upon request.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Acknowledgments

This study was funded by National Natural Science Foundation of China (61703404) and China Postdoctoral Science Foundation (2016M591956).

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