Multiobjective Load Dispatch for Coal-Fired Power Plants under Renewable-Energy Accommodation Based on a Nondominated-Sorting Grey Wolf Optimizer Algorithm

Abstract

Coal-fired power plants are widely used to achieve a power balance in grids with renewable energy, which leads to new requirements for speediness in load dispatch. This paper presents a nondominated-sorting grey wolf optimizer algorithm (NSGWO) for the multiobjective load dispatch of coal-fired power plants that employed efficient nondominated sorting, a reference-point selection strategy, and a simulated binary crossover operator. The optimization results of the benchmark functions indicated that the NSGWO algorithm had a better accuracy and a better distribution than the traditional multiobjective grey wolf optimizer algorithm. Regarding the load dispatch of economy, environmental protection, and speediness strategies, the NSGWO had the best performance of all the simulated algorithms. The optimal-compromise solutions of the economy and speediness strategies of the NSGWO algorithm had a good distribution, which elucidated that this novel algorithm was favorable to allowing coal-fired power plants to accommodate renewable energy.

1. Introduction

Modern coal-fired power plants are clean and highly efficient, and their emissions of NOx and coal consumption are controlled at remarkably low levels [1]. Regarding the national energy distribution in China, coal-fired power plants still need to output power for basic loads in the grid. A coal-fired power plant may have different units (such as water-cooling and air-cooling units [2]) and units of various scales. Differences in unit characteristics may result in variable performances; therefore, it is necessary to optimize the operation strategy of coal-fired power plants with multiple units on the basis of both the economy and environmental-protection objectives. Furthermore, regarding the long-term objective of carbon neutrality [3], coal-fired power plants will play a new role in the power grid. Considering the development of renewable energy including solar power and wind power, coal-fired power plants, which have a strong capability to accommodate renewable energy, are widely used to achieve a power balance in power grids. In terms of the uncertainty of renewable energy [4], it is necessary to operate coal-fired power plants under conditions of rapid load change. Therefore, the optimal operation of coal-fired power plants should be further investigated while considering the factors of the economy, environmental protection, and speediness.

1.1. Load-Dispatch Strategy

Units of various types and scales in a coal-fired power plant usually receive the load command from the grid, which limits the optimal operation of the coal-fired power plant. Regarding the load-dispatch strategy, the coal-fired power plant receives the total load command from the grid, and then dispatches it to different units [5]. Based on previously mentioned factors, namely the economy, environmental protection, and speediness, the load dispatch of a coal-fired power plant is a multiobjective optimization of load command between different units.

Several researchers have employed optimal load-dispatch strategies for power generation systems. Dai et al. [6] proposed a multiobjective economic load-dispatch method for large coal-fired power plants using a hierarchical clustering and retrieval strategy based on the fuzzy c-means clustering algorithm. They concluded that the data-mining-based method was able to achieve plant-level optimal load dispatching while meeting the actual requirements of the grid. Ishraque et al. [7] introduced various optimization strategies of load dispatch for microgrid power systems with renewable energy sources. They considered that the proposed load-following method was the best dispatch strategy to achieve lower operating costs and lower pollutant emissions. Cui et al. [8] considered that the heat load and power load could be optimized simultaneously in microgrids. Regarding the differences in real-time response of heat load and power load, they performed short-term and long-term simulations and elucidated that the total cost could be reduced, as could the peak load of the power grid. Li et al. [9] also focused on investigating multiobjective load dispatch between the heat load and power load, and concluded that the proposed method could balance the production cost and pollutant-emission objectives under the fluctuation of the thermoelectric load demand. Xu et al. [10] introduced the load-dispatch strategy to save coal for coal-fired power plants using a support-vector machine. Their results indicated that the proposed load-dispatch strategy performed better than the normal separate load demand on a single unit in a coal-fired power plant. In addition, novel load-dispatch strategies have been studied for similar power-generation systems. Liao et al. [11] investigated the short-term load-dispatching method for a hydropower plant with multiple turbines, and simulation results indicated that it could achieve optimal operation while simultaneously meeting the practical operation requirements of the hydraulic and electric constraints. Jamal et al. [12] introduced a bioinspired computational heuristic algorithm for dispatch loads for a power-generation system that included coal-fired thermal-generating units and wind-power units. They considered that the proposed algorithm had advantages in accuracy, convergence, and robustness in solving load-dispatch problems. Furthermore, Bie et al. [13] conducted a load-dispatch investigation based on the uncertainty of renewable energy, and the results showed that the real-time load-dispatch model could calculate the load command while considering the primary and secondary regulation of the grid.

In general, much research has been published on load-dispatch optimization, but few studies considered the present application of a coal-fired power plant with renewable-energy accommodation, which is significant in terms of the multiobjective load dispatch of modern coal-fired power plants.

1.2. Multiobjective Optimization Algorithms

Among the load-dispatch strategies, researchers consider the optimization algorithm to be the key technology needed to achieve the objectives of energy saving and reducing pollutant emissions. In addition to the traditional optimization methods (including dynamic programming, Lagrange relaxation, and the exhaustive method), intelligent optimization algorithms have already been introduced in recent investigations.

Intelligent optimization algorithms usually employ an evolutionary mechanism or heuristic algorithm to enhance the convergence speed and robustness ability of the optimal solution procedure. Ghasemi et al. [14] proposed a differential evolution algorithm (DE) based on the mathematical model of sociopolitical evolution. They performed comparative simulations to verify the effectiveness of the algorithm, and concluded that this algorithm was a reliable tool for solving the load-dispatch problem. Neto et al. [15] proposed a method that combined a differential-evolution algorithm and the greedy randomized adaptive search procedure algorithm. The simulation results showed that the algorithm could solve the nonsmooth load-dispatch problem by considering the effect of valve-point loading. Secui et al. [16] proposed a new heuristic algorithm called modified symbiosis (MSOS) that could solve the large-scale-unit load-optimization problem with a valve-point effect. They concluded that the proposed algorithm had a good performance based on the simulation results of five power-generation systems of different constraints and dimensions.

Furthermore, other heuristic algorithms, including the bat algorithm and the population extremum algorithm, have been introduced to solve the multiobjective optimization of load dispatch. Kavousi-Fard et al. [17] proposed an improved bat algorithm to solve the nonconvex load-optimization problem, which was then examined on IEEE test systems to show its high abilities and satisfying performance. Chen et al. [18] used the feasible-solution method to deal with constraints, and proposed a constrained multiobjective population extremal optimization algorithm called CMOPEO-EED. They considered the proposed algorithm to be a viable option to solve the load-dispatch problem with renewable power generation.

Particle-swarm optimization (PSO) is another commonly used algorithm in multiobjective optimization. Hosseinnezhad et al. [19] used an improved PSO algorithm (θ-PSO) to solve the economic load-optimization problem that could easily deal with various constraints. The findings of this study elucidated that the advantage of computational efficiency of the θ-PSO algorithm could be a promising alternative approach to solving the load-dispatch problem in practical power systems. Regarding the combined advantages of the PSO algorithm and other intelligent algorithms, researchers have investigated hybrid algorithms based on PSO. Narimani et al. [20] proposed a hybrid algorithm based on the PSO algorithm and leapfrog algorithm for multiregional economic and environmental-protection load distribution. They verified the effectiveness of the hybrid algorithm using different test systems to balance the multiobjectives of generation costs and environmental issues. Zou et al. [21] proposed a new global particle-swarm-optimization algorithm, which adopted an updated method that relied on the global optimal particle to guide all particle-search activities and a slightly perturbed randomization method for particle flight trajectories with uniform distribution. Mandal et al. [22] introduced variable acceleration coefficients into the PSO algorithm and then proposed a self-organized hierarchical particle-swarm-optimization algorithm. They found that the results of the proposed method were superior in terms of fuel costs, emission output, and losses.

In general, intelligent optimization algorithms have advantages in solving the multiobjective optimization problems of load dispatch at power plants, though their application under the conditions of renewable energy accommodation should be further investigated.

1.3. Grey Wolf Optimizer

Former studies show that various intelligent algorithms have been employed to solve the multioptimization issues in power-generation systems and similar applications. However, considering the rapid load-change conditions present under renewable-energy accommodation, the load dispatch of current coal-fired power plants requires rapid and accurate convergence.

The grey wolf optimizer (GWO) is a swarm-intelligence optimization algorithm proposed by Mirjalili et al. [23]. Compared with other natural heuristic optimization algorithms such as particle-swarm optimization and differential evolution, GWO has attracted attention due to its fast convergence speed, strong robustness, and high solution accuracy with a simple structure. Zhang et al. [24] used a GWO to plan the path of an unmanned combat aerial vehicle, and they concluded that it was more competent than other state-of-the-art evolutionary algorithms in terms of quality, speed, and stability of solutions. Although the grey wolf algorithm has superior performance on most data sets, it may not have the best performance on all. In view of the defects of a GWO, many scholars have made relevant improvements to the performance of the GWO algorithm. Madhiarasan et al. [25] optimized the initial population of a GWO to improve the diversity of the population. Saremi et al. [26] improved the search mechanism of a GWO. The results indicated that the proposed GWO algorithm was able to significantly improve the performance of multiobjective optimization. Long et al. [27] proposed an updating method for the key parameter of a GWO in the application of constrained optimization problems. They considered that the simulations for both benchmark functions and engineering applications could demonstrate the good performance of the proposed GWO algorithm. Moreover, hybrid optimization algorithms combining GWO with other intelligent optimization algorithms have already been introduced by researchers [28,29,30]. However, few scholars have conducted in-depth research on the evolution mechanism of the GWO for the present application of load-dispatch optimization. Song et al. [31] introduced a GWO to solve the economic emission problem of two different power systems, and used a price penalty factor to combine the goals of economic dispatch and emission dispatch into a single function. Wong et al. [32] solved the economic scheduling problem under practical constraints with a GWO and tested them on two test systems with practical constraints. Moradi et al. [33] used a GWO to solve nonlinear and nonconvex economic scheduling problems while considering the valve-point effect and transfer loss, and compared the obtained results with some existing heuristic methods. Jangir et al. [34] investigated the economic constrained-emission dispatch problem based on a nondominated-sorting GWO, and they considered that it could be used in a power grid with wind power. However, they did not consider the speediness requirement of units in the coal-fired power plant, which indicated that the solving procedure of nondominated-sorting GWO should be further optimized.

The above shows that limited work has been conducted on the multiobjective optimization algorithm while considering the rapid and accurate convergence requirements under the unit rapid load-change condition. The more constraints that are considered, the more complex the load-dispatch optimization will be. Therefore, further research on the GWO needs to be carried out according to the specific problems of the multiobjective optimization of the load dispatch of coal-fired power plants.

1.4. Present Investigation

On the basis of the previous literature review, although much research has been published on load-dispatch optimization, few studies considered the present application of a coal-fired power plant with a renewable-energy accommodation. Moreover, limited work has been conducted on the multiobjective optimization algorithm that considered the rapid and accurate convergence requirements under the unit’s rapid load-change condition. Therefore, an investigation into the multiobjective optimization of the load dispatch of a coal-fired power plant was conducted using a novel grey wolf algorithm.

In the present paper, the proposed nondominated-sorting grey wolf algorithm showed that it may help solve various optimization problems in similar applications. The organization of this study was arranged as follows. Section 2 shows the methods for the multiobjective optimization problem of load dispatch. Section 3 details the solving procedure of the novel grey wolf algorithm. Section 4 presents the simulation results and further discussion. Section 5 presents the conclusions of this study.

2. Methods

A theoretical model for the multiobjective optimization of the load dispatch of a coal-fired power plant, as well as optimization constraints and performance evaluation indices, were proposed in order to achieve the requirements related to the economy, environmental protection, and speediness.

2.1. Multiobjective Optimization Problem of a Coal-Fired Power Plant

2.1.1. Problem Description

The load-dispatch problem of a coal-fired power plant can generally be described as per Equation (1):

$$\{\begin{array}{l}\min \mathrm{F}(x)=\min {({\mathrm{f}}_1(x),{\mathrm{f}}_2(x),\dots ,{\mathrm{f}}_m(x))}^T\\ st.\{\begin{array}{l}{\mathrm{g}}_i(x)\le 0,i=1,2,\dots p\\ {\mathrm{h}}_j(x)=0,j=1,2,\dots ,q\end{array}\end{array}$$

(1)

where F(x) is the objective function of the economy performance, and the sub-functions represent the economy characteristics of units in the coal-fired power plant. Further, ${\mathrm{g}}_i(x)$ and ${\mathrm{h}}_j(x)$ represent inequality constraints and equality constraints, respectively. Specifically, pollutant emission is an inequality constraint when it is not selected as an optimization objective.

For a coal-fired power plant, the relationship between the pollutant emission g_i and the unit load $P_i$ can be expressed as the quadratic function relationship shown below

$${\mathrm{g}}_i(P_i)={\alpha }_i{P}_i^2+{\beta }_i{P}_i+{\gamma }_i$$

(2)

where α_i, β_i, and γ_i are the emission-characteristic coefficients of the unit.

Regarding the requirements of the relevant national policies of environmental protection, it is necessary to employ the environmental protection index as an optimization object instead of an inequality constraint. Thus, the objective function of load dispatch when considering environmental protection is defined as shown in Equation (3):

$$\min \mathrm{G}=\min {\displaystyle {\sum }_{i=1}^N{\mathrm{g}}_i(P_i)}$$

(3)

where G represents the pollutant discharge of the whole plant.

The power grid has specific requirements for the load response rate of coal-fired power plants, specially under the condition of renewable-energy accommodation. Therefore, the speediness of load variation should be considered in the load dispatch of coal-fired power plants. The minimum time consumption of load dispatch under ideal conditions T_ideal is shown in Equation (4):

$$T_{ideal}=\left|P_D-{\displaystyle {\sum }_{i=1}^N{P}_{i,now}}\right|/{\displaystyle {\sum }_{i=1}^N{v}_i}$$

(4)

where P_D is the total load command of all units, MW; and v_i represents the load changing speed of unit i, MW/min.

The sum of squares of the differences between the load-dispatch times of each unit and the minimum time under ideal conditions T(P) can be calculated as follows:

$$\mathrm{T}(P)={\displaystyle {\sum }_{i=1}^N{((P_i-P_{i,now})/v_i-T_{ideal})}^2}$$

(5)

Therefore, combining former indicators with the economic indicators, the multiobjective optimization of load dispatch for a coal-fired power plant can be described as shown in Equation (6):

$$\begin{array}{l}\min (F,G,T),\min (F,G),\min (F,T)\\ st.\{\begin{array}{l}{\displaystyle {\sum }_{i=1}^N{P}_i-P_D}\\ P_i^{\min }\le P_i\le P_i^{\max }\\ \left|P_i-P_i^{now}\right|/t_i\le v_i\end{array}\end{array}$$

(6)

2.1.2. Constraint-Violation Degree

The constraint-violation degree is introduced to show whether an individual x violates the i-th inequality constraint or the j-th equality constraint, which is expressed in Equations (7) and (8):

$$Co{n}_i(x)=\max \{g_i(x),0\},1\le i\le p$$

(7)

$$Co{n}_j(x)=\max \{h_j(x),0\},1\le j\le q$$

(8)

Thus, the total constraint-violation degree of individual x is shown in Equation (9):

$$\nu (x)={\displaystyle {\sum }_{i=1}^{p}Co{n}_i(x)+{\displaystyle {\sum }_{j=1}^{q}Co{n}_j(x)}}$$

(9)

In the multiobjective optimization of the load dispatch for a coal-fired power plant, the equation constraint describes that the sum of the loads of each unit must be equal to the total load command, while the inequality constraint shows the load-regulation range.

2.1.3. Feasible and Infeasible Solutions

When an individual x satisfies both the inequality constraints and the equality constraints in the load-dispatch model, this individual is called a feasible solution; otherwise, it is an infeasible solution. The region composed of all feasible solutions is called the feasible region Ω, and the set composed of infeasible solutions is called the infeasible region.

2.1.4. Dominance

For the two specific solutions x₁ and x₂, the economy, speediness, and environmental-protection indicators corresponding to solution x₁ are better than those corresponding to solution x₂. Thus, solution x₁ is called a nondominated solution, while solution x₂ is called a dominant solution, which can be written as $x_1\succ x_2$.

All individuals in the solution set that cannot be dominated by other individuals are called nondominated solutions, and this set of nondominated solutions is called the Pareto front (PF) of optimization solutions.

2.2. Constraints for Multiobjective Optimization

Constraint processing should be conducted to balance the relationship between the objective function and the constraints. In this paper, the penalty-function method was used to deal with constraints. By adding the penalty function $F(x,M)$ into the objective function, the constrained optimization problem was transformed into an unconstrained optimization problem. Parameter M in the penalty function represents a large positive number, called the penalty factor, which acts as a penalty for the solution that violates the constraints. Using the external penalty function, the multiobjective function with the penalty function is expressed as follows:

$$\min F(x,M)=\min (f_1(x)+M_{1}v(x),f_2(x)+M_{2}v(x),\dots ,f_m(x)+M_{m}v(x))$$

(10)

2.3. Evaluation Index of Multiobjective Optimization

For the multiobjective optimization of a load dispatch, evaluation indices were employed to compare the simulation results, which may have shown the convergence performance and comprehensive performance of the optimization algorithm.

2.3.1. Convergence Evaluation Index

The generational distance (GD) was introduced to evaluate the convergence performance, which represented the degree of the solution set P to the real Pareto frontier P*, as shown in Equation (11):

$$GD(P,P^{*})=\frac{1}{\left|P\right|}\sqrt{{\displaystyle {\sum }_{i=1}^{\left|P\right|}{d}_i^2}}$$

(11)

where $\left|P\right|$ is the number of solutions in P, and d_i represents the Euclidean distance from the i-th solution in P to the nearest solution in P*. Therefore, a small GD value meant the convergence performance was better.

2.3.2. Comprehensive Evaluation Index

The inverse generational distance (IGD) was used to represent the degree of the real Pareto frontier P* to the solution set P [35]. It evaluated the convergence and diversity of the algorithm simultaneously, which was defined as:

$$IGD(P,P^{*})=\frac{1}{\left|P^{*}\right|}\sqrt{{\displaystyle {\sum }_{i=1}^{\left|P^{*}\right|}{d}_i^{\prime 2}}}$$

(12)

where $d_i^{\prime }{}^2$ represents the nearest Euclidean distance from the i-th individual in P* to the solution in P. Here, a small IGD showed that the optimization algorithm was better in both the convergence performance and the diversity of solutions.

2.3.3. Optimal-Compromise Solution

For the multiobjective optimization of a load dispatch, the fuzzy set theory is suggested to find an optimal-compromise solution from the Pareto fronts. The fuzzy membership degree for the load-dispatch optimization can be expressed as:

$${\phi }_{pi}=\{\begin{array}{ll}1& f_{pi}\le f_i^{\min }\\ (f_i^{\max }-f_{pi})/(f_i^{\max }-f_i^{\min })& f_i^{\min }<f_{pi}<f_i^{\max }\\ 0& f_{pi}\ge f_i^{\max }\end{array}$$

(13)

where ${\phi }_{pi}$ represents the satisfaction of the i-th objective of the p-th optimal solution, $i\in \left\{1,2,\dots ,m\right\}$; $f_{pi}$ represents the value of the i-th objective of the p-th optimal solution; and $f_i^{\max }$ and $f_i^{\min }$ represent the maximum and minimum values of the i-th objective in all optimal solutions, respectively. When ${\phi }_{pi}$ was equal to 1, it meant that the individual was completely satisfied with the i-th target value.

The standardized satisfaction calculation method of each solution p in the nondominated solution set is shown in Equation (14), where the solution corresponding to the largest ${\phi }_{pi}$ was the optimal-compromise solution:

$${\phi }_p={\displaystyle {\sum }_{k=1}^m{\phi }_{pk}}/{\displaystyle {\sum }_{p=1}^N{\displaystyle {\sum }_{k=1}^m{\phi }_{pk}}}$$

(14)

Regarding the previous objectives and constraints of the multiobjective optimization of the load dispatch for a coal-fired power plant, the sequence diagram is shown in Figure 1.

3. Solving Procedure of the Optimization Algorithm

The multiobjective grey wolf algorithm was employed to find the load-dispatch-optimization solution for a coal-fired power plant. However, the issues of local optimum and insufficient stability may limit the performance of this optimization algorithm. In addition, the influence of the alpha wolf may result in a poor distribution of the obtained solutions., and the traditional nondominated sorting may consume more time in the iteration process. Considering the multiobjective optimization of a load dispatch with multiple complex constrains, a nondominated-sorting grey wolf optimizer (NSGWO) algorithm was proposed that employed efficient nondominated sorting (ENS), a reference-point selection strategy, and a simulated binary crossover operator.

3.1. Fast Nondominated Sorting Based on ENS

Fast nondominated sorting is a cyclic stratification process. If this process is conducted in sequence, it will require too much time to converge. In fact, the frontier number of individuals is determined by the dominated individual, which could infer the following equation:

$$front(p)=1+\underset{q\in Q}{\max }front(q)$$

(15)

where $front(p)$ represents the number of the frontier where individual p is located, and Q is the set composed of all individuals that dominate individual p.

The first step of the ENS was to arrange all individuals in the population according to their first-dimension target values from small to large. If the first-dimension target values were the same, we compared their second-dimension index values, and so on. In the sorted population, it was impossible for the individual in the latter order to dominate the individual in front of it. This meant that the individual that dominated p must have been assigned to the frontier when examining p. The number of the frontier where p is located was calculated using Equation (15).

On this basis, starting from the first frontier F₁, it was judged in turn whether each frontier contained an individual that dominated p. The first frontier that does not contain the dominant p was the frontier where p resides. If there were individuals dominating p up to the last frontier F_k, then p was assigned to a new frontier F_k+1. If all individuals were mutually nondominant, then a nondominant comparison was required between them. In this case, since there was only one frontier, the complexity was $T_{worst}=m\cdot N(N-1)/2=O(m{N}^2)$. The algorithm could achieve the best performance if the population satisfied the following conditions: there were $\sqrt{N}$ frontiers in total, and each frontier contained $\sqrt{N}$ individuals. Moreover, an individual was dominated by all individuals whose frontier number was smaller than its under this condition. For a specific individual p, if it belonged to frontier F_i, it was necessary to find an individual dominating it in each frontier $F_i(j<i)$. This indicated that at least i − 1 nondominated comparisons were required. For all $\sqrt{N}$ individuals in F_i, a total of $\sqrt{N}(i+1)$ comparisons were required. In addition, all individuals of F_i needed to be compared pairwise, which meant that a total of $\sqrt{N}(\sqrt{N}-1)/2$ comparisons were required. Considering all frontiers, the best period of complexity is shown in Equation (16):

$$T_{best}=m\cdot {\displaystyle {\sum }_{i=1}^{\sqrt{N}}\left[\sqrt{N}(i-1)+\sqrt{N}(\sqrt{N}-1)/2\right]}=m\cdot \left[N(\sqrt{N}-1)/2+N(\sqrt{N}-1)/2\right]=O(M{N}^{1.5})$$

(16)

3.2. Selection Strategy Based on Reference Point

To select N individuals from the set of offspring and parents ${\mathrm{R}}_t=P_t{\displaystyle \cup Q_t}$, it should firstly divide R_t into multiple nondominated layers $(F_1,F_2,\dots ,F_k)$ by nondominated sorting. Then, a new population S_t is established from the start of F₁ until its size is equal to or exceeds N for the first time. In order to maintain the diversity of solutions, a selection strategy based on reference point was used to select $n_L=N-{\displaystyle {\sum }_{i=1}^{L-1}{N}_i}$ individuals, where N_i is the number of individuals in the i-th frontier, and $n_L$ is the number of individuals that needed to be selected from the last frontier satisfying the former requirements. Moreover, the systematic approach was introduced to generate a set of uniform weight vectors $\left\{{\lambda }_1,{\lambda }_2,\dots ,{\lambda }_R\right\}$ so that the reference points could be determined. In order to compare the target values of each dimension, it was necessary to adaptively normalize the target value of each dimension. The ideal point $\stackrel{-}{\mathrm{z}}=(z_1^{\min },z_2^{\min },\dots ,z_m^{\min })$ of defining population S_t was composed of the minimum value $z_1^{\min },i=1,2,\dots ,m$ of each dimension of S_t. Based on the ideal point, the objective function could be transformed to $f_i^{\prime }(x)=f_i(x)-z_i^{\min }$. Then, a hyperplane could be built by extra points obtained from the objective function; this process can be expressed as:

$$ASF(x,w)={\max }_{i=1}^m{f}_i^{\prime }(x)/w_i,x\in S_t$$

(17)

$$Z^{i,\max }=s:\arg {\min }_{s\in S_t}ASF(s,w),w=(\tau ,\dots ,\tau ),\tau ={10}^{-6}$$

(18)

For the i-th target, an extra target vector $z^{i,\max }$ was generated. Thus, m extra vectors were generated by m targets which further constituted an m-dimensional linear hyperplane, and then the intercept $a_i,i=1,2,\dots ,m$ could be calculated. The objective function can be normalized as shown in Equation (19):

$$f_i^n(x)=(f_i^{\prime }(x)-z_i^{\min })/(a_i-z_i^{\min })$$

(19)

3.3. Simulated Binary Crossover Operator

Under the influence of the top three levels of wolves, a GWO easily falls into a local optimum. Therefore, a simulated binary crossover (SBX) operator was introduced in this paper. Assuming that the two parent individuals are $x^1(x_1^1,x_2^1,\dots x_n^1)$ and $x^2(x_1^2,x_2^2,\dots x_n^2)$, the SBX operator could generate the offspring individuals $c^1(c_1^1,c_2^1,\dots c_n^1)$ and $c^2(c_1^2,c_2^2,\dots c_n^2)$ using the following equations:

$$\{\begin{array}{l}{c}_i^1=0.5\times [(1+\beta )·x_i^1+(1-\beta )·x_i^2]\\ c_i^2=0.5\times [(1-\beta )·x_i^1+(1+\beta )·x_i^2]\end{array}$$

(20)

where β is randomly determined by the distribution factor η according to Equation (21):

$$\beta =\{\begin{array}{l}{(rand\times 2)}^{1/(1+\eta )},rand\le 0.5\\ {(1/(2-rand\times 2))}^{1/(1+\eta )},rand>0.5\end{array}$$

(21)

With the increase in iterations, the probability of generating individuals according to the grey wolf hunting mechanism increased, thereby enhancing the development ability and convergence of the algorithm. In this paper, the variable a was controlled by a nonlinear decreasing strategy to balance the exploration and development capabilities of the algorithm, as shown in Equation (22):

$$a=2(1-{(t/iter)}^2)$$

(22)

where iter is the maximum number of iterations.

3.4. Solving Procedure of NSGWO

The solving procedure of the proposed nondominated-sorting grey wolf optimizer algorithm is presented in Figure 2. There were five specific steps of this algorithm, which are detailed as follows.

Step 1: Set the number N of individuals in the grey wolf population and the maximum number of iterations iter. Initialize parameters a, A, and C.

Step 2: Initialize the grey wolf population P, and generate a reference point Z.

Step 3: Calculate the target value of each dimension of individuals in the population.

Step 4: Perform the nondominated sorting based on ENS, and select the first three levels of wolves α, β, δ.

Step 5: Update the population P_t and generate the offspring Q_t. When the random number is greater than the set probability threshold, binary crossover mutation is conducted. Otherwise, update the population based on the evolution mechanism of GWO.

Step 6: Select the first N individuals based on the reference point in the merged population ${\mathrm{R}}_t=P_t{\displaystyle \cup Q_t}$, and then generate a new population S_t.

Step 7: If the maximum number of iterations is not reached, repeat steps (2–4). Otherwise, output the optimization results.

4. Results and Discussion

4.1. NSGWO Algorithm Performance Test

In order to test and evaluate the performance of the NSGWO algorithm, three benchmark functions [36] were employed in this paper, the detailed structures of which are shown in

Table 1. Benchmark functions for multiobjective optimization test of NSGWO algorithm.

Function Name	Objective Function	Variable Range
ZDT1	$\{\begin{array}{l}{f}_1(x)=x_1\\ f_2(x)=g(x)h(f_1(x),g(x))\\ g(x)=1+\frac{9}{29}{\displaystyle {\sum }_{i=2}^{30}{x}_i}\\ h(f_1(x),g(x))=1-\sqrt{f_1(x)/g(x)}\end{array}$	$0\le x_i\le 1,1\le i\le 30$
UF2	$\{\begin{array}{l}{f}_1=x_1+\frac{2}{\left|J_1\right|}{\displaystyle \sum _{j\in J_1}{y}_j^2}\\ f_2=1-\sqrt{x}+\frac{2}{\left|J_2\right|}{\displaystyle \sum _{j\in J_2}{y}_j^2}\\ J_1=\left\{j\right|jisevennumber,2\le j\le n\},J_2=\{j|jisoddnumber,2\le j\le n\}\\ y_j=\{\begin{array}{l}{x}_j-[0.3x_1^2\cos (24\pi x_1+4j\pi /n)+0.6x_1]\cos (6\pi x_1+j\pi /n),j\in J_1\\ x_j-[0.3x_1^2\cos (24\pi x_1+4j\pi /n)+0.6x_1]\cos (6\pi x_1+j\pi /n),j\in J_2\end{array}\end{array}$	$0\le x_i\le 1,1\le i\le 30$
UF4	$\{\begin{array}{l}{f}_1=x_1+\frac{2}{\left|J_1\right|}{\displaystyle \sum _{j\in J_1}\mathrm{h}(y_j})\\ f_2=1-x_2+\frac{2}{\left|J_2\right|}{\displaystyle \sum _{j\in J_2}\mathrm{h}(y_j})\\ y_j=x_j-\sin (6\pi x_1+j\pi /n),j=2,3,\dots ,n\\ h(t)=\left|t\right|/(1+e^{2\left|t\right|})\end{array}$	$0\le x_i\le 1,1\le i\le 30$

.

Then, test simulations were conducted for the multiobjective optimization algorithm; the results for the GD and IGD values of the former benchmark functions are presented in

Table 2. GD values of optimization results for benchmark functions.

Function	Algorithm	Mean Value	Worst Value	Optimal Value
ZDT1	NSGWO	0.00490	0.01250	0.00090
	MOGWO	0.00754	0.06144	0.00119
	MOPSO	0.06150	0.36040	0.00370
UF2	NSGWO	0.00500	0.00600	0.00420
	MOGWO	0.05949	0.11078	0.02178
	MOPSO	0.10420	0.15110	0.06570
UF4	NSGWO	0.04160	0.04260	0.04000
	MOGWO	0.05756	0.06230	0.05198
	MOPSO	0.07910	0.08800	0.07030

and

Table 3. IGD values of optimization results for benchmark functions.

Function	Algorithm	Mean Value	Worst Value	Optimal Value
ZDT1	NSGWO	0.00690	0.01340	0.00390
	MOGWO	0.20824	0.69615	0.01391
	MOPSO	0.06310	0.34060	0.00820
UF2	NSGWO	0.01600	0.01980	0.01250
	MOGWO	0.06961	0.08307	0.05542
	MOPSO	0.09580	0.11190	0.08190
UF4	NSGWO	0.03430	0.03530	0.03220
	MOGWO	0.06200	0.06890	0.05861
	MOPSO	0.07733	0.08944	0.06682

, respectively. Here, the results of the multiobjective grey wolf optimizer (MOGWO) and the multiobjective particle-swarm optimizer (MOPSO) are also shown for performance comparisons. For the unimodal function ZDT1, the IGD value of the NSGWO optimization result was 0.00690, which was an order of magnitude smaller than the results of other optimization algorithms, indicating that the improved algorithm had a strong ability to jump out of the local optimum. The UF series functions were all multimodal functions, which provided many local optimal solutions, and therefore the requirements for the algorithm were relatively high. Regardless of the IGD value or the GD value, the results of the improved algorithm were better than those of the other algorithms in the UF4 function. However, the IGD of the improved algorithm was smaller, as was the difference between the maximum and minimum values of the GD value, indicating that the improved algorithm had a better stability. The above test results showed that the improved algorithm NSGWO had good stability and the ability to jump out of the local optimum.

Figure 3 shows the Pareto fronts of optimization results of the benchmark function, in which the horizontal and vertical coordinates represent a certain target value of the benchmark function. For the ZDT1 function, the solutions of MOGWO and NSGWO were close to the real Pareto fronts. For the UF2 function, most of the MOGWO solutions were far from the real Pareto fronts, while most of the NSGWO solutions were concentrated on the real Pareto fronts. Moreover, the results indicated that the UF4 function was relatively complex. Although the optimization results of the NSGWO algorithm in Figure 3c were not close to the real Pareto fronts, they were still better than those of MOGWO. In general, the NSGWO had better accuracy and a better distribution of the obtained solutions.

4.2. Multiobjective Optimization of Load Dispatch Based on NSGWO

The NSGWO algorithm was employed in the multiobjective optimization of the load dispatch for coal-fired power plants while considering the economy, environmental protection, and speediness. To better show the performance of the NSGWO algorithm in load dispatch, four intelligent algorithms, including the real-number coding genetic algorithm (RCGA), particle-swarm optimizer (PSO), modified nondominated-sorting genetic algorithm (NSGA-II), and MOGWO, were employed in the following simulations. Their configurations in terms of the population size and maximum number of iterations are summarized in

Table 4. Configurations of NSGWO and other intelligent algorithms used in simulations of multiobjective optimization.

	Simulation I 1			Simulation II 1			Simulation III 1
	NSGWO	RCGA	PSO	NSGWO	NSGA-II	MOGWO	NSOGWO	MOGWO
Population size	200	200	200	100	100	100	100	100
Number of iterations	500	500	500	300	300	300	1000	1000

¹ Simulations I–III represent the optimization of economy and environmental protection, the optimization of economy and speediness for various power-scale units, and the optimization of economy and speediness for similar-scale units, respectively.

, and were selected through analyses of presimulations.

4.2.1. Optimization of Economy and Environmental Protection

The NSGWO algorithm was employed for the multiobjective optimization of the load dispatch for coal-fired power plants while considering economy and environmental protection. The penalty function was used in the following simulations to deal with the equality constraints in the load-dispatch model. Firstly, a coal-fired power plant with six units was investigated; its coal consumption and pollutant-emission characteristics are listed in

Table 5. Characteristics of dispatched units based on economy and environmental protection.

Unit	Coal Consumption Characteristics	Pollutant Emission Characteristics	$P_i^{\min }$ /MW 1	$P_i^{\max }$ /MW 1
1	$f_1(P_1)=0.010P_1^2+2.0P_1+10+\left|1.5\sin [6.28(P_1^{\min }-P_1)]\right|$	$g_1(P_1)=6.49\times {10}^{-4}{P}_1-0.056P_1+4.091$	5	50
2	$f_2(P_2)=0.012P_2^2+1.5P_2+10+\left|1.0\sin [8.98(P_2^{\min }-P_2)]\right|$	$g_2(P_2)=5.64\times {10}^{-4}{P}_2-0.061P_2+2.543$	5	60
3	$f_3(P_3)=0.004P_3^2+1.8P_3+20+\left|1.0\sin [14.78(P_3^{\min }-P_3)]\right|$	$g_3(P_3)=4.59\times {10}^{-4}{P}_3-0.051P_3+4.258$	5	100
4	$f_4(P_4)=0.006P_4^2+1.0P_4+10+\left|0.5\sin [20.94(P_4^{\min }-P_4)]\right|$	$g_4(P_4)=6.38\times {10}^{-4}{P}_4-0.030P_1+5.326$	5	120
5	$f_5(P_5)=0.004P_5^2+1.8P_5+20+\left|0.5\sin [25.13(P_5^{\min }-P_5)]\right|$	$g_5(P_5)=4.59\times {10}^{-4}{P}_5-0.051P_5+4.258$	5	100
6	$f_6(P_6)=0.010P_6^2+1.5P_6+10+\left|0.5\sin [18.48(P_6^{\min }-P_6)]\right|$	$g_6(P_6)=5.15\times {10}^{-4}{P}_1-0.056P_1+6.131$	5	60

¹ Variables $P_i^{\min }$ and $P_i^{\max }$ represent lower limit and upper limit of output power.

[37]. The total command of these six units was 283.4 MW, and their load-changing speed was limited to 5% of the current output power. To better show the performance of the NSGWO algorithm in the load dispatch, the RCGA and PSO were introduced for a comparison with the optimization results. Then, the simulations were conducted; the results are shown in

Table 6. Optimization results of economic load dispatch for coal-fired power plant without valve-point effect of steam turbines.

Algorithm	P1 /MW	P2 /MW	P3 /MW	P4 /MW	P5 /MW	P6 /MW	Coal Consumption Cost/(USD h −1)	Pollution Emission /(th −1)
NSGWO	15.7	33.0	58.4	94.4	46.7	35.2	94.5382	0.2321
RCGA	11.5	30.6	59.9	98.2	51.3	35.5	95.6516	0.2199
PSO	12.8	27.0	55.5	100.5	45.4	44.5	95.6517	0.2207

and

Table 7. Optimization results of load dispatch under economy and environmental-protection strategies for coal-fired power plant considering valve-point effect of steam turbines.

Item	Economy Strategy			Environmental-Protection Strategy
	NSGWO	RCGA	PSO	NSGWO	RCGA	PSO
Cost/(USD h −1)	95.2184	96.1129	99.0861	100.7577	101.5641	101.9237
Emission/(th −1)	0.229	0.214	0.214	0.193	0.194	0.195
P1/MW	21.4	11.5	9.9	40.5	41.1	37.1
P2/MW	30.9	30.6	36.3	40.7	46.3	46.7
P3/MW	50.5	59.9	48.4	50.2	54.4	56.4
P4/MW	93.0	98.2	87.4	37.7	39	36.5
P5/MW	46.7	51.3	66.4	58.8	54.4	52.2
P6/MW	41.0	35.5	39.0	55.5	51.5	57.8

.

The results showed that the performance of the NSGWO algorithm was better than that of the other algorithms in reducing power-plant coal consumption and power-plant emissions, which indicated the advantages of the NSGWO algorithm in the optimal dispatch of multiobjective economic and environmental-protection loads. Table 6 and Table 7 also show that the valve-point effect of steam turbines had a great influence on the optimal dispatch results. The optimization results of Unit 1 and Unit 6 in the two models were quite different. When the valve-point effect was considered, the optimal dispatch results of these two units were 21.4 MW and 41.0 MW, respectively. However, the optimal dispatch results changed to 15.7 MW and 35.2 MW without considering the valve-point effect of the steam turbines.

In order to further illustrate the advantages of NSGWO, the Pareto fronts were determined for the NSGWO and MOGWO algorithms without considering the valve-point effect, as shown in Figure 4. The solutions of NSGWO were more widely distributed, and the solutions of MOGWO were basically dominated by the solutions of NSGWO. The comparison results show that the NSGWO algorithm had a certain application value in the multiobjective optimization of the load dispatch for coal-fired power plants.

4.2.2. Optimization of Economy and Speediness for Various Power-Scale Units

Considering the requirements of rapid load change of coal-fired power plants under a renewable-energy accommodation, simulations were conducted to show the performance of the NSGWO algorithm in the multiobjective optimization of economy and speediness. Here, 10 units of various power scales are selected to be dispatched in the following simulations, the characteristics of which are shown in

Table 8. Characteristics of 10 dispatched units with various power scales based on economy and speediness.

Unit	Coal-Consumption Characteristics	${\mathit{P}}_{\mathit{n}\mathit{o}\mathit{w}}$/MW	${\mathit{v}}_{\mathit{i}}$/(MW min −1)	${\mathit{P}}_{\mathit{i}}^{\mathbf{min}}$/MW	${\mathit{P}}_{\mathit{i}}^{\mathbf{max}}$/MW
1	$f_1(P_1)=0.00510P_1^2+2.2034P_1+15$	15	7	15	60
2	$f_2(P_2)=0.00396P_2^2+1.9101P_2+25$	20	8	20	80
3	$f_3(P_3)=0.00393P_3^2+1.8518P_3+40$	30	10	30	100
4	$f_4(P_4)=0.00382P_4^2+1.6966P_4+32$	35.8	15	25	120
5	$f_5(P_5)=0.00212P_5^2+1.8015P_5+29$	50.3	8	50	150
6	$f_6(P_6)=0.00261P_6^2+1.5354P_6+72$	75.4	6	75	280
7	$f_7(P_7)=0.00289P_7^2+1.2643P_7+49$	124	11	120	320
8	$f_8(P_8)=0.00148P_8^2+1.2130P_8+82$	251.85	10	125	445
9	$f_9(P_9)=0.00127P_9^2+1.1954P_9+105$	298.75	12	250	520
10	$f_{10}(P_{10})=0.00135P_{10}^2+1.1285P_{10}+100$	298.9	9	250	550

[38]. The total dispatch command of the 10-unit power plant was 1500 MW, and the penalty-factor values of the two-dimensional objective of economy and speediness in the NSGWO algorithm were 0.5 and 2.1, respectively.

Table 9. Optimization results of load dispatch under economy and speediness strategies for 10 units of various power scales.

Item	Optimal-Compromise Solution			Economy Strategy			Speediness Strategy
	NSGWO	NSGA-II	MOGWO	NSGWO	NSGA-II	MOGWO	NSGWO	NSGA-II	MOGWO
fh/(th −1) 1	3118.20	3120.77	3124.15	3115.02	3115.22	3123.25	3130.11	3131.29	3126.27
T/min 1	4.91	4.98	4.49	7.48	7.12	5.13	3.52	3.59	3.77
P1/MW	15.64	20.96	24.48	15.14	15.00	23.50	34.21	35.94	25.66
P2/MW	37.58	37.88	45.04	26.00	27.33	45.24	44.566	45.50	46.30
P3/MW	45.16	48.03	48.21	36.20	34.21	45.48	59.17	61.00	52.29
P4/MW	65.82	69.80	76.64	58.34	55.86	76.33	80.19	70.07	77.91
P5/MW	79.48	76.81	76.43	76.74	82.85	75.24	77.36	78.82	78.27
P6/MW	101.45	94.49	97.10	109.45	109.63	95.54	94.91	96.26	97.04
P7/MW	159.34	156.95	152.18	146.84	149.06	152.97	157.04	155.41	156.39
P8/MW	298.22	294.80	296.78	302.62	306.39	301.44	284.90	286.40	289.48
P9/MW	354.25	358.46	345.97	364.46	356.67	339.19	339.36	341.79	344.03
P10/MW	343.06	341.81	337.18	364.15	363.01	345.06	328.23	328.81	332.64

¹ Variables f_h and T represent coal consumption per hour and adjustment time, respectively.

presents the optimization results of the load dispatch under the economy and speediness strategies. To better explain the advantages of nondominated sorting, the optimization of the NSGA-II was also performed, as shown in Table 9.

The results indicated that the optimal solutions of the NSGWO algorithm had the minimum adjustment times. For the minimum coal consumption of the economy strategy, the optimal solutions of the NSGWO algorithm were better than those of the other algorithms. Moreover, we found that the optimal-compromise solution of the NSGWO algorithm was able to dominate the solutions of the NSGA-II algorithm.

Figure 5 shows the Pareto fronts of the NSGWO algorithm and the MOGWO algorithm under the combined dispatch of economy and speediness. Here, a speediness index was employed to better present the optimal solutions that was defined as the square of the adjustment time. It can be seen that the optimization results of the NSGWO algorithm were more widely distributed than those of the MOGWO algorithm. The solution that dominated the optimization results of the MOGWO algorithm was found in the Pareto fronts of the NSGWO algorithm. In general, this indicated that the performance of the NSGWO algorithm in the load-dispatch optimization of economy and speediness was the best among the simulated algorithms.

4.2.3. Optimization of Economy and Speediness for Similar-Scale Units

Moreover, a real coal-fired power plant with eight 600 MW units and two 660 MW units was investigated under a renewable-energy accommodation, which may further explain the performance of the NSGWO algorithm in the load dispatch. These 10 units had similar power scales, so a higher optimization accuracy was needed in the algorithm. The coal-consumption characteristics, as well as other initial conditions and constraints, are shown in

Table 10. Characteristics of 10 dispatched units based on economy and speediness with 10 similar-scale units.

Unit	Coal-Consumption Characteristics	${\mathit{P}}_{\mathit{n}\mathit{o}\mathit{w}}$/MW	${\mathit{v}}_{\mathit{i}}$/(MW min −1)	${\mathit{P}}_{\mathit{i}}^{\mathbf{min}}$/MW	${\mathit{P}}_{\mathit{i}}^{\mathbf{max}}$/MW
1	$f_1(P_1)=0.000128P_1^2-0.10413P_1+361.508$	409.164	10	240	600
2	$f_2(P_2)=0.000118P_2^2-0.10897P_2+360.777$	343.598	10	240	600
3	$f_3(P_3)=0.000104P_3^2-0.11813P_3+348.179$	321.472	12	240	600
4	$f_4(P_4)=0.000115P_4^2-0.11699P_4+350.690$	368.000	10	240	600
5	$f_5(P_5)=0.000190P_5^2-0.12183P_5+352.533$	359.787	10	240	600
6	$f_6(P_6)=0.000154P_6^2-0.11906P_6+350.421$	420.531	8	240	600
7	$f_7(P_7)=0.000150P_7^2-0.11031P_7+352.498$	362.807	8	240	600
8	$f_8(P_8)=0.000144P_8^2-0.10936P_8+346.564$	333.882	10	240	600
9	$f_9(P_9)=0.000272P_9^2-0.12038P_9+342.560$	400.494	5	264	660
10	$f_{10}(P_{10})=0.000292P_{10}^2-0.12938P_{10}+340.501$	427.870	5	264	660

, in which the unit for coal consumption is grams of coal per kWh. In addition, the penalty-factor values of the two-dimensional targets of economy and speediness in the algorithms were 0.5 and 1.1, respectively, and the total power command of this 10-unit power plant was 4000 MW.

Table 11. Optimization results of load dispatch under economy and speediness strategies for a real coal-fired power plant with 10 similar-scale units.

Item	Optimal-Compromise Solution		Economy Strategy		Speediness Strategy
	NSGWO	MOGWO	NSGWO	MOGWO	NSGWO	MOGWO
fw/(gkWh −1) 1	329.70	330.34	328.02	328.19	331.30	331.71
T/min 1	11.16	8.17	22.85	23.21	5.32	6.68
P1/MW	378.15	405.97	302.75	323.48	434.55	385.14
P2/MW	394.44	420.79	353.30	415.80	379.64	376.09
P3/MW	455.35	419.57	595.65	600.00	385.35	401.64
P4/MW	462.83	414.01	527.16	543.27	413.77	412.89
P5/MW	353.74	351.06	382.36	309.99	359.38	372.00
P6/MW	450.43	438.11	447.16	427.13	455.26	465.65
P7/MW	368.90	371.26	376.00	350.80	383.84	379.13
P8/MW	385.39	408.06	383.56	347.93	371.17	384.94
P9/MW	366.62	376.87	310.88	336.77	392.15	393.99
P10/MW	383.84	394.51	321.08	344.28	425.31	421.73

¹ Variables f_w and T represent coal consumption per kWh and adjustment time, respectively.

shows the optimization results for the real coal-fired power plant with 10 similar-scale units. It indicates that the minimum coal consumption of NSGWO was 328 g/(kWh), which was better than that of MOGWO. The minimum adjustment time of NSGWO was 1.36 min, which was also shorter than that of MOGWO. Furthermore, the solutions of the MOGWO algorithm for these two minima were dominated by the NSGWO algorithm, which meant that the distribution of the solutions of the NSGWO algorithm was better than that of the MOGWO algorithm.

Figure 6 illustrates the Pareto fronts of coal consumption and adjustment time of the former simulations for NSGWO and MOGWO. It shows that the optimization results of NSGWO were basically at the lower left of MOGWO, indicating that the optimization effect of NSGWO was better. Most of the solutions of the MOGWO algorithm are concentrated in the upper left, while the solutions of the NSGWO algorithm are more uniformly distributed. This indicates that the solutions of the NSGWO algorithm had a better distribution in the combined load dispatch while considering the requirements of both economy and speediness for coal-fired power plants.

The results of this study showed that the proposed nondominated-sorting GWO algorithm achieved high accuracy and had a good distribution of solutions, which may extend its applications in multiobjective optimization. We caution that the presented analyses for the NSGWO and other intelligent algorithms were somewhat simplified, and further investigations are needed. The optimization procedure proposed in this study may help researchers conduct other investigations into the optimization of multiobjective problems and the operation of power-generation systems.

5. Conclusions

Multiobjective optimizations of load dispatch under a renewable energy accommodation were conducted based on the NSGWO algorithm. On the basis of the performed analyses, the conclusions of this paper can be summarized as follows:

(1)

The proposed NSGWO algorithm employed efficient nondominated sorting, a reference-point selection strategy, and a simulated binary crossover operator in order to avoid falling into the local optimum, enhance the solution diversity, and shorten the convergence time. The optimization results of benchmark functions indicated that the NSGWO algorithm had a better accuracy and a better distribution than the MOGWO algorithm.

(2)

Regarding the load dispatch of economy and environmental-protection strategies, the performance of the NSGWO algorithm was better than those of the other simulated algorithms in reducing power-plant coal consumption and emissions. In addition, the widely distributed solutions of the NSGWO algorithm could dominate those of the MOGWO algorithm, which elucidated the application value of the NSGWO algorithm in the multiobjective load dispatch of coal-fired power plants.

(3)

The NSGWO algorithm could achieve lower coal consumption and a shorter adjustment time than the MOGWO algorithm could for coal-fired power plants with various power-scale units and similar-scale units. The optimal-compromise solutions of the NSGWO algorithm had a better distribution in the combined load dispatch of economy and speediness strategies; thus, it is favorable for coal-fired power plants to accommodate renewable energy.