A novel economic load dispatch method of microgrid based on hybrid slime mould and genetic algorithm

Abstract

The economic load dispatch problem of microgrid strives to optimize the allocation of total power demand among generating units under specific constraints. Many optimization techniques have been used to solve this problem in power systems; however, achieving the optimal solution is considered difficult due to the involvement of a nonlinear objective function and large search domain. In order to achieve economic load dispatch more quickly and accurately, a novel economic load dispatch method of microgrid based on hybrid slime mould and genetic algorithm (GSMA) is proposed in this paper. Objective function models and their constraints based on wind, photovoltaic, energy storage and fuel power generation are presented. For the early iterations of the method, crossover and mutation of the genetic algorithm are used to increase the diversity of the population. When the number of iterations reaches the threshold, the slime mould algorithm is used to improve the adaptability to complex objective functions. The velocity matrix is introduced to adjust the direction and speed of the individual movement to enhance the searching ability in GSMA. For performance evaluation, GSMA is compared with slime mould algorithm (SMA), grey wolf optimizer (GWO), sparrow search algorithm (SSA), Harris Hawks optimization (HHO), whale optimization algorithm (WOA) and particle swarm optimization (PSO) using standard optimization functions. The experimental results show that GSMA converges to the optimal solution faster than other algorithms. The algorithms are used for economic load dispatch on the simulation test system. The GSMA spends minimum dispatch cost and achieves the best dispatch results compared to other algorithms. It further demonstrates the effectiveness of the new method in solving the economic load dispatch problem of microgrid.

Introduction

In recent years, wind and solar energy have been more and more widely used in power systems. The application of these renewable energy sources not only makes the construction of power grids more flexible, but also greatly reduces the operating costs of power grids and environmental pollution [1]. However, the microgrid contains uncontrollable renewable energy, which has profound uncertainty. In power systems, economic load dispatch is a large, complex and nonlinear problem. For microgrid, economic load dispatch is also one of the challenges to be solved [2]. The goal of microgrid economic load dispatch is to rationally and efficiently allocate various energy resources in microgrid to reduce costs, improve energy efficiency and reduce carbon emissions [3]. The optimal economic load dispatch algorithms play a key supporting role in promoting the application and development of microgrid [4].

The microgrid consists of a variety of distributed power sources, including wind power modules, photovoltaic (PV) power modules, energy storage modules and fuel power modules. Its simplified structure diagram is shown in Fig. 1 [5]. The different characteristics of these distributed power sources increase the complexity of dispatch. And the inclusion of renewable energy sources requires coordinating the matching between different energy sources and load demand within the microgrid, which increases the difficulty of dispatch [6]. The need for fast and accurate economic load dispatch algorithms is becoming more and more urgent with the increasing application of microgrid [7].

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Fig. 1

Physical structure of microgrid

In the past thirty years, the initial grid economic load dispatch algorithms were mainly based on traditional static optimization methods, such as linear programming [8] and integer programming [9]. These algorithms enabled economic dispatch by balancing various constraints such as power supply and demand balance, transmission line losses and minimum generation cost. With the increasing complexity of power systems, traditional static optimization algorithms were unable to effectively respond to the dynamic changes in system operation [10]. Therefore, dynamic optimization algorithms were applied. These algorithms were based on various optimization techniques, such as dynamic programming [11] and model predictive control [12]. They considered power system uncertainty, fast response to demand and real-time scheduling problems to improve the economy and stability of the system. With the rapid development of artificial intelligence and machine learning, intelligent algorithms have been applied in the economic dispatch of power grids. These algorithms include neural networks [13] and metaheuristic algorithms [14]. Through learning and optimization, the intelligent algorithms can automatically adjust the output of generating units and grid operation based on historical data and real-time information, making grid operation more economical and efficient. Li et al. presented an improved mayfly algorithm for environmental economic dispatch (EED) of power grids [15]. Prakash et al. gave a multi-objective optimal economic dispatch for integrated grid-connected microgrids with fuel cells and cogeneration renewable energy based on whale optimization algorithm (WOA) [16]. Deng et al. introduced an improved krill herd algorithm for the economic dispatch of integrated energy systems [17]. Dassa and Recioui presented a genetic algorithm (GA)-based method for demand side management and dynamic economic dispatch [18]. Xu and Yu gave a multi-objective universe optimization algorithm to solve the EED problem [19]. The above algorithms can solve the corresponding dispatch problem well. However, these algorithms are more complex in structure and have more parameter settings. A sinusoidal mapping optimization load-shedding technique based on chaotic sticky mushroom algorithm (CSMA) was presented by Abid [20]. When the network size is large and complex, the performance of CSMA performs better. Nevertheless, the algorithm runs for a long time. A bird swarm algorithm (BSA) based on adaptive levy flight strategy for dealing with multi-objective optimal dispatch problem for microgrids was presented by Ma [21]. The method can deal with multi-objective optimal dispatch problem which has a faster rate of convergence, but the complexity of the algorithm is slightly higher.

In order to quickly and accurately achieve economic load dispatch of microgrid, a novel economic load dispatch method of microgrid based on hybrid slime mould and genetic algorithm (GSMA) is proposed in this paper. Genetic algorithm (GA) is used to increase the diversity of the population for the early iterations of GSMA. When the number of iterations reaches the threshold, the slime mould algorithm (SMA) is used to improve the adaptability to complex objective functions. Meanwhile, to enhance the searching ability, the velocity matrix is introduced to adjust the direction and speed of the individual movement. The main contributions of this paper are as follows:

1. A novel economic load dispatch method of microgrid based on hybrid slime mould and genetic algorithm (GSMA) is proposed. Combining GA with SMA for optimal scheduling, enrich the population with GA to avoid the algorithm from falling into local optimum. Use SMA to improve the dynamic adaptation performance of the algorithm.

2. The velocity matrix is proposed to control the individual movement in GSMA. The relationship between the local optimal position of the particle and the current position is utilized to regulate the particle position update, thus enhancing the exploration capability of the algorithm.

Using 23 standard optimization functions in the standard optimization functions, GSMA is compared with other algorithms. It is found that GSMA ranks first in optimization searching performance in the test. It performs well in multimodal functions; at the same time, it has the fastest rate of convergence in unimodal test functions. The comparison reveals that GSMA spends minimum dispatch cost and achieves the best dispatch results compared to other algorithms. It further demonstrates the effectiveness of the new method in solving the economic load dispatch problem.

The rest of the paper is organized as follows. In Sect. "Problem formulation for microgrid economic load dispatch ", a mathematical model for microgrid dispatch is developed. Sect. "Economic load dispatch method based on hybrid slime mould and genetic algorithm" proposes a novel GSMA method and gives the detailed steps of the method. The comparisons of the proposed method with other algorithms are shown in Sect. "Applications of GSMA". Finally, conclusions are drawn in Sect. "Conclusions".

Problem formulation for microgrid economic load dispatch

Objective function

The sum of the microgrid’s generation cost and environmental pollution control cost are considered in this paper. $f_1(x)$ is the generation cost of microgrid. It is calculated as follows:

1

$$f_1(x)=C_1+C_2+C_3$$where ${\text{C}}_1$ is the fuel cost for each distributed power source, ${\text{C}}_2$ is the cost of inter-purchase of electricity between the microgrid and the main grid and ${\text{C}}_3$ is the cost of daily operation and maintenance of the distributed power source.

1. The fuel cost for each distributed power source is calculated as follows:

   2

   $$C_1=\sum _{t=1}^n\left(a_i{P}_i^2+b_i{P}_i+c_i\right)$$where $a_i$, $b_i$ and $c_i$ are the fuel cost coefficient of the i^th unit, $P_{\text{i}}$ is the output power of the i^th unit and $n$ is the number of generation units.

1. The cost of inter-purchase of electricity is shown as follows:

   3

   $$C_2=\sum _{t=1}^{n}H[C_u(t)P_g(t)-C_s(t)P_l(t)]$$where H is the elapsed time for electricity consumption, $C_u(t)$ is the purchased electricity price of the microgrid in time period $t$, $P_g(t)$ is the active power output from the main grid to the microgrid in time period $t$, $C_s(t)$ is the purchased electricity price of the main grid in time period $t$, $P_l(t)$ is the active power output from the microgrid to the main grid in time period $t$.

1. The cost of daily operation and maintenance of distributed power sources is calculated as follows:

   4

   $$C_3=\sum _{t=1}^N{K}_{\mathit{OM}}(i)M_i(t)$$where $K_{\mathit{OM}}(i)$ is the maintenance factor of the i^th distributed power supply, $M_i(t)$ is the maintenance cost per unit of power consumed by that distributed power supply in time period $t$.

The environmental pollution control cost is denoted as $f_2(x)$ and shown as:

5

$$f_2(x)=\sum _{t=1}^T\sum _{k=1}^n{C}_k[Y_{grid,k}{P}_{\mathit{buy}}(t)+Y_{mt,k}{P}_{\mathit{MT}}(t)]$$where $C_k$ is the cost coefficient for treating pollutants of type $k$, $Y_{grid,k}$ is the emission of pollutants of type $k$ generated by the operation of the main grid. $P_{\mathit{buy}}(t)$ is the purchased power between the microgrid and the main grid at moment $t$.$Y_{mt,k}$ is the emission of pollutants of type $k$ generated by the operation of the diesel engine and $P_{\mathit{MT}}(t)$ is the power generation of the diesel engine at moment $t$.

The objective function constructed in this paper is shown as:

6

$$minf(x)=f_1(x)+f_2(x)$$where $f(x)$ is the total cost of microgrid dispatch, which consists of the sum of the microgrid’s generation cost and environmental pollution control cost.

Constraints

To ensure the safe and stable operation of the system, the work process of each distributed power source must be constrained so as to meet the requirements of the optimization index [22]. The equality constraints and inequality constraints are as follows:

1. Microgrid system power balance constraints are described as follows [23]:

   7

   $$P_d+P_{\text{loss}}=\sum _{i=1}^n{P}_i+P_{\text{grid}}$$

where $P_d^{}$ is the power used by the system load, $P_{\mathit{loss}}$ is the transmission line losses power, $P_{\mathit{grid}}^{}$ is the active power exchanged between the microgrid and main power grid.

$P_{\mathit{loss}}$ can be evaluated using Korn’s formula of B-coefficients as given in Eq. (8)[24].

8

$$P_{\mathit{loss}}=\sum _{i=1}^n\sum _{j=1}^n{P}_i{B}_{\mathit{ij}}{P}_j+\sum _{i=1}^n{B}_{0i}{P}_i+B_{00}$$where $B_{\mathit{ij}}$, $B_{0i}$, $B_{00}$ are the B-matrix coefficients for $P_{\mathit{loss}}$, which can be generally assumed to be constants under a normal operating condition.

The equation for $P_{\mathit{grid}}$ is shown as:

9

$$P_{\mathit{grid}}=P_g(t)+P_l(t)$$

1. Exchange power constraints between the microgrid and the main grid are given as follows [25]:

   10

   $$P_{\text{grid}}^{min}\le P_{\text{grid}}\le P_{\text{grid}}^{max}$$

where $P_{\mathit{grid}}^{\mathit{min}}$ is the minimum value of the power exchanged between the microgrid and the main power grid, $P_{\mathit{grid}}^{\mathit{max}}$ is the maximum value of the power exchanged between the microgrid and the main power grid.

1. Output power constraints for each distributed power supply are shown as follows [26]:

   11

   $$P_i^{min}\le P_i\le P_i^{max}$$where $P_i^{min}$ is the minimum value of the active power output of the ith distributed power source, $P_i^{max}$ is the maximum value of the active power output of the ith distributed power source.

Economic load dispatch method based on hybrid slime mould and genetic algorithm

Slime mould algorithm

SMA mainly simulates the self-organizing behaviour and information transfer mechanism of slime mould organisms. It has the characteristics of few parameters and high optimization seeking ability. By imitating this behaviour, SMA can be used in optimization problems and path planning problems. SMA adjusts different search modes according to the fitness value, and its multi-exploration mechanism gives the algorithm a strong global optimization capability. SMA mainly consists of approaching food, wrapping food and oscillation [27]. It is described in detail as follows:

When approaching food, the slime mould can rely on the smell of the food in the air. The mathematical expression is defined as:

12

$$\overrightarrow{X\left(q,+,1\right)}=\left\{\begin{array}{c}\overrightarrow{X_b\left(q\right)}+\overrightarrow{\mathit{vb}}\bullet \left(\overrightarrow{W},\bullet ,\overrightarrow{X_A\left(q\right)},-,\overrightarrow{X_B\left(q\right)}\right),r<p\\ \overrightarrow{\mathit{vc}}\bullet \overrightarrow{X\left(q\right)},r\ge p\end{array}\right)$$where $\overrightarrow{\mathit{vc}}$ represents the parameter reduced from 1 to 0, $\overrightarrow{\mathit{vb}}$ represents the oscillation factor, $q$ denotes the current iteration, $\overrightarrow{X_b}$ represents the optimal individual with the strongest global smell found so far, $\overrightarrow{X}$ represents the location of the slime mould individual, $\overrightarrow{X_A}$ and $\overrightarrow{X_B}$ represent the two randomly selected individuals from the slime mould population, $\overrightarrow{W}$ represents the weight of the slime mould.

The equation for $\text{p}$ can be described as:

13

$$p=tanh|f(l)-DF|l\in 1,2,\cdots ,n$$where $f(l)$ represents the fitness of $\overrightarrow{X}$, $\mathit{DF}$ represents the best fitness in the iteration.

The equation for $\overrightarrow{\mathit{vb}}$ is given as:

14

$$\overrightarrow{\mathit{vb}}=[-a,a]$$

15

$$a=\text{arctanh}\left(-\left(\frac{t}{max\_t}\right)+1\right)$$

The equation for $\overrightarrow{W}$ is shown as:

16

$$\overrightarrow{W\left(\text{SmellIndex}\right)}=\left\{\begin{array}{c}1+r·log\left(\frac{\text{BF}-f(i)}{\text{BF}-\text{WF}+\epsilon }+1\right),\text{condition}\\ 1-r·log\left(\frac{\text{BF}-f(i)}{\text{BF}-\text{WF}+\epsilon }+1\right),\text{others}\end{array}\right)$$

17

$$\text{SmellIndex}=\text{sort}(S)$$where condition represents the first half of $f(i)$ ranked in the overall, $r$ represents a random value in the interval [0,1], $\mathit{BF}$ and $\mathit{WF}$ represent the best fitness value and the worst fitness value in the current iteration, $\epsilon$ represents a very small value, $max\_t$ represents the maximum number of iterations. Finally, $\mathit{SmellIndex}$ represents the vector of sorted fitness values.

For wrapping food, the algorithm mainly simulates the adjustment of foraging patterns of slime mould to different food concentrations. When a region has a high food concentration, slime mould will tend to search in this region. On the contrary, if the food concentration is low, they will leave this area and move to search in areas with higher food concentration [28]. The equation for updating the position of slime mould is described as follows:

18

$$\overrightarrow{X_{}^{\ast }}=\left\{\begin{array}{c}rand\bullet \left(u,b,-,l,b\right)+lb,rand<z\\ \overrightarrow{X_b\left(t\right)}+\overrightarrow{\mathit{vb}}\bullet \left(W,\bullet ,\overrightarrow{X_A\left(t\right)},-,\overrightarrow{X_B\left(t\right)}\right),r<p\\ \overrightarrow{\mathit{vc}}\bullet \overrightarrow{X\left(t\right)},r\ge p\end{array}\right)$$where z represents the proportion of randomly distributed slime mould individuals to the total, $p=0.03$[29], $\mathit{lb}$ and $\mathit{ub}$ represent the lower and upper boundaries of the search range, $\mathit{rand}$ and $r$ represent random values in the range of [0,1].

The pseudocode of the SMA is shown as follows:

Genetic algorithm

Genetic algorithm (GA) is an algorithm that simulates the mechanism of biological evolution in nature [30]. The general idea of the algorithm is based on the law of “survival of the fittest” in Darwin’s theory of biological evolution. In the genetic algorithm, after the initial population is formed through coding, the task of genetic manipulation is to apply certain operations to the individuals of the population according to their adaptability to the environment, so as to realize the evolutionary process of survival of the fittest. Genetic operations include three basic genetic operators: selection, crossover and mutation [31]. The mutation mechanism is used to avoid the algorithm from falling into a local optimum and to enhance the search capability. The operations of individual genetic operators are performed under random perturbations. Consequently, the rules of migration of individuals in the population to the optimal solution are randomized [32]. But this randomized operation is different from the traditional random search method. Genetic operation performs efficient directed operation instead of directionless search as in general random search methods. Moreover, the genetic algorithm is highly extensible and easy to be used in combination with other algorithms.

The hybrid model of slime mould and genetic algorithm

SMA, as a heuristic algorithm, the main feature is that it will adjust different search patterns based on the fitness values. By simulating the network formation and oscillation behaviour of slime moulds, SMA can effectively explore the complex search space. Meanwhile, it is highly robust to environmental changes and disturbances. The adaptive dynamic updating mechanism of SMA enables it to approach the global optimal solution quickly with the increase of iterations.

GA is also a heuristic algorithm with better global search capability. When the number of iterations is less, GA is able to find the current optimal solution faster. In addition, GA can evaluate all individuals in the population simultaneously, which makes it suitable for parallelization and distributed processing. In order to achieve microgrid economic load dispatch more quickly and accurately, the hybrid model of slime mould and genetic algorithm is presented in this paper.

A parameter named $\text{GA}\_\text{maxIter}$ is given in this section. $\text{GA}\_\text{maxIter}$ is a threshold for the number of iterations that will divide the algorithm into two parts. GA is used to carry out iterative operations initially. When the number of iterations is equal to $\text{GA}\_\text{maxIter}$, GA stops iterating, returns the currently found optimal solution as the result. When the number of iterations is bigger than $\text{GA}\_\text{maxIter}$, the optimization is continued using SMA. For each iteration, individuals are first ranked for fitness according to Eq. (17) and then the current optimal position is updated. The algorithm stops when the maximum number of iterations is reached. The value of $\text{GA}\_\text{maxIter}$ needs to be adjusted according to the complexity, time constraints and solution requirements of the specific problem.

In order to improve the search capability of the method and to speed up the convergence, the velocity matrix $V$ is introduced to control the movement of the individual. V is defined by

19

$$V_{}^{}=R\times \frac{B{P}_{\text{n}}-X_{\mathit{mn}}}{1-W}$$where $m$ is the index of the particle, $n$ represents the dimension of the particle, $W$ represents the weight of the particle, $R$ represents a random number ranging from 0 to 1, ${\text{BP}}_{\text{n}}$ represents the current position of the optimal individual in dimension $n$ and $X_{\text{mn}}$ represents the current position of the particle. When updating the position of an individual, the new position of the individual is obtained by adding the velocity matrix V with the current position of the individual. This new position will be used in the next iteration for fitness calculation and further search. By updating the velocity matrix V, the movement of the individual can be controlled according to the guidance of the historical optimal position in the searching process combined with the individual’s own velocity information, which helps the individual to better explore and utilize the information in the searching space, and improves the algorithm’s ability to perform both local searching and global searching.

The novel economic load dispatch method of microgrid based on hybrid slime mould and genetic algorithm (GSMA) runs in steps as follows:

Step 1: Set parameters and define $\text{GA}\_\text{maxIter}$.

Step 2: Calculate the fitness function.

Step 3: Rank the individual fitness, update the weights by Eqs. (16) and (17).

Step 4: Update the velocity matrix V by Eq. (19), and get the new velocity of the individual by adding V to the individual velocity.

Step 5: Calculate the current iteration number. If the current iteration number is less than or equal to $\text{GA}\_\text{maxIter}$, use GA for cross mutation operation to get the next generation population. Otherwise, use SMA to adjust the optimal position of the current individual.

Step 6: Termination condition judgement. If the maximum number of iterations is reached, output the optimal solution. If not, return to Step 2.

The flowchart of GSMA is shown in Fig. 2.

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Fig. 2

The flowchart of GSMA

Applications of GSMA

GSMA performance evaluation

GSMA is compared with SMA, grey wolf optimizer (GWO)[33], sparrow search algorithm (SSA)[34], Harris Hawks optimization (HHO)[35], whale optimization algorithm (WOA)[36] and particle swarm optimization (PSO)[37] using 23 standard optimization functions. These standard optimization functions include unimodal, multimodal and fixed-dimension multimodal functions.

The unimodal functions are shown in Table 1 [38].

Table 1. Unimodal testbench functions

Function	Dim	Interval	$f_{\mathit{min}}$
$F_1(x)={\sum }_{i=1}^n{x}_i^2$	30	[− 100, 100]	0
$F_2(x)=\sum _{i=1}\left|x_i\right|+\prod _{i=1}^n\left|x_i\right|$	30	[− 10, 10]	0
$F_3(x)=\sum _{i=1}^i{\left(\sum _{j-1}^i{x}_j\right)}^2$	30	[− 100, 100]	0
$F_4(x)=\underset{i}{max}\left\{\left|x_i\right|,1\le i\le n\right\}$	30	[− 100, 100]	0
$F_5(x)=\sum _{i=1}^{n-1}\left[100{\left(x_{i+1}-x_i^2\right)}^2+{(x_i-1)}^2\right]$	30	[− 30, 30]	0
$F_6(x)=\sum _{i=1}^n{\left[(x_i+0.5)\right]}^2$	30	[− 100, 100]	0
$F_7(x)=\sum _{i=1}^{n}i{x}_i^4+\text{random}[0,1)$	30	[− 1.28, 1.28]	0

The multimodal functions are shown in Table 2 [38].

Table 2. Multimodal testbench functions

Function	Dim	Interval	$f_{\mathit{min}}$
$F_8(x)=\sum _{i=1}^n-x_{i}sin(\sqrt{|x_i|})$	30	[− 500,500]	− 2094.9145
$F_9(x)=\sum _{i=1}^n\left[x_i^2-10cos(2\pi x_i+10)\right]$	30	[− 5.12,5.12]	0
$\begin{array}{c}{F}_{10}(x)=-20exp\left(-0.2\sqrt{\frac{1}{n}\sum _{i=1}^n{x}_i^2}\right)-exp\left(\frac{1}{n}\sum _{i=1}^{n}cos\left(2\pi x_i\right)\right)+20+e\end{array}$	30	[− 32,32]	0
$F_{11}(x)=\frac{1}{4000}\sum _{i=1}^n{x}_i^2-{\prod }_{i=1}^{n}cos\left(\frac{x_i}{\sqrt{i}}\right)+1$	30	[− 600,600]	0
$\begin{array}{c}{F}_{12}(x)=\frac{\pi }{n}\left\{10sin(\pi y_1)+\sum _{i=1}^{n-1}{(y_i-1)}^2[1+10{sin}^2(\pi y_{i+1})]+\right)\\ \left({(y_n-1)}^2\right\}+\sum _{i=1}^{n}u(x_i,10,100,4)\end{array}$$y_i=1+\frac{x_i+1}{4}u(x_i,a,k,m)=\left\{\begin{array}{c}k{(x_i-a)}^m{x}_i>a\\ 0-a<x_i<a\\ k{(-x_i-a)}^m{x}_i<-a\end{array}\right)$	30	[− 50,50]	0
$\begin{array}{c}{F}_{13}(x)=0.1\{{sin}^2(3\pi x_i)+\sum _{i=1}^i{(x_i-1)}^2[1+{sin}^2(3\pi x_i+\\ 1)]+{(x_n-1)}^2\left[1+{sin}^2(2\pi x_n)\right]\}+\sum _{i=1}^{n}u(x_i,5,100,4)\end{array}$	30	[− 50,50]	0

The fixed-dimension multimodal functions are shown in Table 3 [38].

Table 3. Fixed-dimension multimodal testbench functions

Function	Dim	Interval	$f_{\mathit{min}}$
$F_{14}(x)={\left(\frac{1}{500}+\sum _{j=1}^{25}\frac{1}{j+\sum _{i=1}^2{(x_i-a_{\mathit{ij}})}^6}\right)}^{-1}$	2	[− 65,65]	1
$F_{15}(x)=\sum _{i=1}^{11}{\left[a_i-\frac{x_1(b_i^2+b_i{x}_2)}{b_i^2+b_i{x}_3+x_4}\right]}^2$	4	[− 5,5]	0.00030
$F_{16}(x)=4x_1^2-2.1x_1^4+\frac{1}{3}{x}_1^6+x_1{x}_2-4x_2^2+4x_2^4$	2	[− 5,5]	− 1.0316
$\begin{array}{c}{F}_{17}(x)={\left(x_2-\frac{5.1}{4{\pi }^2}{x}_1^2+\frac{5}{\pi }{x}_1-6\right)}^2+10\left(1-\frac{1}{8\pi }\right)cos{x}_1+10\end{array}$	2	[− 5,5]	0.398
$\begin{array}{c}{F}_{18}(x)=[1+{(x_1+x_2+1)}^2(19-14x_1+3x_1^2-14x_2+6x_1{x}_2\\ +3x_2^2)]\times [30+{(2x_1-3x_2)}^2(18-32x_1+12x_1^2\\ +48x_2-36x_1{x}_2+27x_2^2)]\end{array}$	2	[− 2,2]	3
$F_{19}(x)=-\sum _{i=1}^4{c}_{i}exp\left(-\sum _{j=1}^3{a}_{\mathit{ij}}{(x_j-p_{\mathit{ij}})}^2\right)$	3	[1, 3]	− 3.86
$F_{20}(x)=-\sum _{i=1}^4{c}_{i}exp\left(-\sum _{j=1}^6{a}_{\mathit{ij}}{(x_j-p_{\mathit{ij}})}^2\right)$	6	[0,1]	− 3.32
$F_{21}(x)=-\sum _{i=1}^5{\left[(x-a_i){(x-a_i)}^T+C_i\right]}^{-1}$	4	[0,10]	− 10.1532
$F_{22}(x)=-\sum _{i=1}^7{\left[(x-a_i){(x-a_i)}^T+C_i\right]}^{-1}$	4	[0,10]	− 10.4028
$F_{23}(x)=-\sum _{i=1}^{10}{\left[(x-a_i){(x-a_i)}^T+C_i\right]}^{-1}$	4	[0,10]	− 10.5363

In Tables 1, 2 and 3, dim represents the dimension of the function, interval represents the definition domain of the function and $f_{\text{min}}$ represents the optimal value of the function.

The qualitative analysis results of GSMA in handling unimodal functions and multimodal functions are presented in Fig. 3. The graph consists of average fitness and convergence curve. The average fitness indicates the trend of the average fitness of the slime mould population with the iteration process. The convergence curve shows the process of convergence to the optimal slime mould fitness value. It analyses the changes in the position and fitness of the slime mould during the foraging process intuitively. Two examples of each type of function are selected in Fig. 3, and the effects of the remaining similar functions are similar to those in the figure. Among them, functions F1 and F7 belong to the unimodal functions, functions F8 and F10 belong to the multimodal functions and functions F15 and F23 belong to the fixed-dimensional multimodal functions. The first column of Fig. 3 represents the parameter space of the standard optimization functions, the second column is the evaluation fitness, and the third column is the convergence curve of GSMA.

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Fig. 3

Qualitative analysis results for functions F1, F7, F8, F10, F15, F23

By observing the average fitness curves, the trend of the fitness of the slime moulds during the iterative process can be visualized. In the unimodal testbench functions, multimodal testbench functions and fixed-dimension multimodal testbench functions, the average fitness curve of the slime mould has almost no oscillatory state in the iterative process. And the change of the average fitness value shows a decreasing trend, thus ensuring the rapid convergence of the slime mould in the prophase and the precise search in the anaphase.

The convergence curve reveals that the average fitness of the optimal fitness values searched for by the slime mould varies with iteration. By observing the downward trend of the curve, we can observe the rate of convergence of the slime mould and the point at which it switches between exploration and exploitation. In Fig. 3, we can see that GSMA has a very fast convergence rate in unimodal testbench functions, multimodal testbench functions and fixed-dimension multimodal testbench functions. The convergence curve is smooth without large fluctuations or oscillations, which means that GSMA can find or nearly find the optimal solution in fewer iterations, proving that the algorithm maintains good stability during the iteration process and can approach the optimal solution smoothly.

All the algorithms were performed with the same parameters to ensure the fairness of the experiment. The number of populations is 30; the dimension is set to 30. In order to maintain the consistency of function evaluation, the maximum number of iterations is 1000. The detailed parameters of each algorithm are shown in Table 4 [39].

Table 4. Parameter settings for all comparison algorithms

Algorithm	Year	Parameter
PSO	1995	$c1=c2=2,w=1$
GWO	2014	$a=[0,2]$
WOA	2016	$a_1=\left[2,,,0\right],a_2=\left[-,2,,,-,1\right];b=1$
HHO	2019	$N=30,T=1000$
SMA	2020	$z=0.03,a=\left[-,\text{1,1}\right],b=[\text{0,1}]$
SSA	2020	$SD=0.2,PD=0.7,SD=0.2$

The comparison of the convergence curves for the unimodal functions is shown in Fig. 4, the comparison of the convergence curves for the multimodal functions is shown in Fig. 5 and the comparison of the convergence curves for the fixed-dimension multimodal functions is shown in Fig. 6. Figure 4 shows that GSMA has the fastest convergence to the optimal solution among all the compared algorithms. For F7, GSMA can search for the fitness value with the highest accuracy, while some algorithms still fail to obtain a better solution after a certain number of iterations. This is caused by the local optimal stagnation, which indicates that GSMA can still show better exploration ability under poorer exploration conditions. In addition, Figs. 5 and 6 show that GSMA performs well in multimodal functions and is able to find the optimal solution with a fast convergence trend. Therefore, it can be concluded that GSMA has the best overall performance in this group of experiments, which reflects its strong stability.

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Fig. 4

Comparison of the convergence curves for the unimodal functions

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Fig. 5

Comparison of the convergence curves for multimodal functions

[See PDF for image]

Fig. 6

Comparison of the convergence curves for the fixed-dimension multimodal functions

GSMA applied to the grid dispatch problem

In order to verify the feasibility of GSMA, it is applied to a microgrid scheduling system as shown in Fig. 1. This test system includes a variety of distributed power sources including PV, WT, MT and battery energy storage system (BESS). The parameters of the distributed generation sources are shown in Table 5 [40]. The discharge coefficient and cost of pollutants are shown in Table 6 [40].

Table 5. Distributed generation sources parameters

ID	Type	Min power(kW)	Max power(kW)	O&M costs($)
1	MT	6	30	0.128
2	PV	0	25	0.0015
3	WT	0	15	0.0309
4	Grid	− 30	30	0
5	BESS	− 30	30	0.0296

Table 6. Discharge coefficient and cost of pollutants

ID	Type	Cost of pollutants ($/kg)	Pollutant emission factor(g/kW•h)
			MT	PV	WT	Grid	BESS
1	CO2	0.023	724	0	0	889	10
2	SO2	6	0.0036	0	0	1.8	0.0002
3	NOX	8	0.2	0	0	1.6	0.001

The dispatching cycle is set to 1 day, with 1 h as the dispatching unit. The whole day can be divided into 24 units. The electric load and generation curve of WT, PV, MT and BESS in 24 h are shown in Fig. 7. The WT is set up in open areas with high wind speeds, and the PV is set up in areas with no obstructions and plenty of light. The microturbines are placed close to a heat or cold source, making them suitable for integration with the thermal system. They can provide electricity and heat according to the load demand and work in the “cold/heat-ordered power” mode. The BESS is located at the centre of the microgrid for connection to other power sources and loads. It is mainly used to balance the gap between power generation and demand due to load variations, to store excess power, to cope with load fluctuations and instability of renewable energy sources and to ensure the smooth operation of the grid.

[See PDF for image]

Fig. 7

Electric load and generation curve of WT, PV, MT and BESS in 24 h

GSMA, SMA, GWO, SSA, HHO, WOA and PSO are used to solve this problem. For this test system, the maximum number of iterations selected for all the algorithms is 500. For GSMA, the mutation rate is taken as 0.02, and the $\text{GA}\_\text{maxIter}$ is set to be 30. The simulation results are described in Fig. 8, which shows the convergence curves of the algorithms when the population size is 30. The horizontal axis is the number of iterations, and the vertical axis is the dispatch cost.

[See PDF for image]

Fig. 8

Cost convergence characteristic of microgrid economic load dispatch

The convergence process of the algorithms can be seen in Fig. 8. In the iterations, the initial position of the convergence curve of GSMA is smaller than the algorithms such as SMA, HHO and WOA, which has a strong searching ability, and it converges very quickly to the neighbourhood of the optimal solution. After 500 iterations, all algorithms reach steady state, but GSMA converges more accurate than others.

The microgrid economic load dispatch cost of all mentioned algorithms is presented in Table 7. 30 randomized tests were conducted for all the algorithms. It can be seen that GSMA outperforms the other algorithms in terms of the best solution and average value. GWO has the highest average cost compared to GSMA, while PSO has the worst solution among these algorithms. The final average cost ranking of the algorithms is GWO > SMA > HHO > PSO > SSA > WOA > GSMA. It is a good proof of the effectiveness of GSMA in the field of economic load dispatch for microgrids.

Table 7. Comparison of the microgrid economic load dispatch cost

	Optimization techniques	Best solution	Worst solution	Average value
Cost($)	PSO	13,962.31521	14,076.63738	14,002.95
	SSA	13,962.51603	14,053.20946	13,999.83
	HHO	13,969.92928	14,050.39278	14,003.84
	SMA	13,965.76531	14,058.00774	14,011.92
	GWO	13,977.75169	14,070.64585	14,027.14
	WOA	13,965.97962	14,041.16809	13,991.87
	GSMA	13,960.34449	13,982.12794	13,971.15

Conclusions

A novel economic load dispatch method of microgrid based on hybrid slime mould and genetic algorithm has been proposed in this paper. By introducing the $\text{GA}\_\text{maxIter}$, the GA has been fully utilized to increase the algorithm population diversity. At the same time, the powerful dynamic adaptability of SMA has been used to improve the optimization ability. Velocity matrix has been introduced into the algorithm to control individual motion, which has improved the convergence rate of the method. In order to verify the effectiveness of the proposed method, GSMA has been simulated using 23 standard optimization functions and used to solve the microgrid economic load dispatch problem. The simulation results have been shown that GSMA has the fastest rate of convergence; the lowest cost outlay and good stability have been compared to other algorithms. The GSMA can fulfil the microgrid economic load dispatch task well.

In the future, GSMA can be extended to other areas including image processing, data clustering, etc. It can also be combined with other optimization algorithms to further improve its rate of convergence on other optimization problems, such as feature selection using GSMA and ORPD problems in power systems.

Acknowledgements

Not applicable.

Author contributions

Dr. Wei Ba and Mr. Wei Sun carried out basic design, simulation work and prepared draft paper. Dr. Qi Li and Ms. Chunjiang Zhao participated in checking results and discussions, sequence of paper and helped to prepare the manuscript. All authors read and approved the final manuscript.

Funding

This work was supported by the Fundamental Research Funds for the Central Universities (DUT24LAB120).

Availability of data and materials

In this article, the necessary general information has been cited from the reference in the manuscript body. Also, if more information and questions are requested in the review process, we are willing to provide data and answer questions.

Declarations

Competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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