A New Migration and Reproduction Intelligence Algorithm: Case Study in Cloud-Based Microgrid

Abstract

Inspired by the migration and reproduction of species in nature to explore suitable habitats, this paper proposed a new swarm intelligence algorithm called the Migration and Reproduction Algorithm (MARA). This new algorithm discusses how to transform the behavior of an organism looking for a suitable habitat into a mathematical model, which can solve optimization problems. MARA has some common features with other optimization methods such as particle swarm optimization (PSO) and the fireworks algorithm (FWA), which means MARA can also solve the optimization problems that PSO and FWA are used to, namely, high-dimensional optimization problems. MARA also has some unique features among biology-based optimization methods. In this paper, we articulated the structure of MARA by correlating it with natural biogeography; then, we demonstrated the performance of MARA on sets of 12 benchmark functions. In the end, we applied it to optimize a practical problem of power dispatching in a multi-microgrid system that proved it has certain value in practical applications.

1. Introduction

Intelligent optimization has received extensive attention and has been applied in many areas including control [1,2,3,4,5], forecasting [6,7,8], and simulation, which solves practical problems and attains better solutions reasonably and efficiently. For example, in the intelligent control of the robot field, we can use intelligent optimization to plan a robot’s moving path and also improve the efficiency of the search path, plan the best path, and then optimize it [9,10,11]; in the transportation field, we can use intelligent optimization to model a multi-path road network, establish an intelligent transportation system, and manage the transportation in real time, improving the safety and mobility of the entire transportation system [12,13]; in power system field, intelligent optimization can solve the problems in power and automation systems to make the systems operate optimally [14,15,16]. To solve the field road segmentation problem in an agricultural machinery trajectory semantic segmentation model, Hai et al. [17] proposed an improved multivariate oscillating slime mold algorithm (MOSMA) based on the slime mold algorithm (SMA) to improve the segmentation performance of the model. Bao et al. [18] combined a neural network and deep learning and other artificial intelligence learning algorithms to improve the efficiency of spacecraft complex impact damage detection and damage feature classification. Aiming at solving the magnetotelluric inversion problem, Wang et al. [19] proposed a hybrid intelligent algorithm combining the particle swarm algorithm and the differential evolution algorithm and integrated the Nelder–Mead optimization algorithm to further develop the optimization results locally. However, the problem of the linear algorithm and single nonlinear intelligent algorithms is that they are prone to fall into local minima and low optimization efficiency.

At present, intelligent swarm optimization is divided into several main algorithms based on its technical roots: the evolution algorithm (EA), the artificial immune system (AIS), swarm intelligence (SI), and the artificial neural network (ANN). The algorithm we proposed in this paper belongs to SI, as a new bionic optimization algorithm with robustness to local minima and universality to the optimization problem.

Biogeography Mathematical models described how species migrate from one island to another, how new populations arise, and how populations become extinct [20]. (To be clear, a species is defined as a group of animals that can reproduce with each other; a population is all individuals of the same species living in a natural area at the same time.) If an island is a suitable habitat, we considered that this area has a high Habitat Suitability Index (HSI) [21]. HSI features include rainfall, temperature, food diversity, land area, etc. In this paper, the HSI characteristics for evaluating habitat suitability are described as a characteristic index that we consider to be the independent variable, while HSI is considered to be the dependent variable. As long as we have a quantifiable measure for the given problem, the HSI can be similar to an index for evaluating solutions. Inspired by natural mechanisms and combining them with problem solving, a new swarm intelligence algorithm, the Migration and Reproduction Algorithm (MARA), is proposed.

MARA considers an optimization problem feasible region, Ω, as a natural habitat where populations live. We solve an optimization problem and find a better solution by simulating the behaviors of natural species that are constantly migrating and reproducing in search of a more suitable habitat to survive.

The rest of the paper is organized into three parts: First, we expounded on the structure of MARA by correlating it with natural bio-geography; second, we demonstrated the performance of MARA on sets of six benchmark functions; third, we applied MARA to the practical optimization of multi-microgrid power dispatch in a cloud environment to test its ability to solve a currently practical problem; finally, we wrapped up the paper by discussing its drawbacks and future work.

2. Migration and Reproduction Algorithm

MARA is divided into three stages including migration, reproduction, and natural selection. In the following, we will illustrate these stages in detail.

2.1. Migration

There are two attributes of information carried by each population: coordinate position and velocity. The velocity attribute is a geometric vector reflecting the direction and distance of migration. In every iteration, the population’s velocity vector will be changed by the current situation. The velocity equation is constrained by three aspects: first, the learning efficiency of a specific population; second, the relative coordinate position between the specific population habitat and the recorded local best habitat of the same species; finally, the relative coordinate position between the specific population habitat and the recorded global optimal habitat. Therefore, every population owns its variable velocity vector.

Population migration can be described as follows:

$$X{(t+1)}_{T,m,i}=X{(t)}_{T,m,i}+\dot{V}{(t+1)}_{T,m,i}$$

(1)

where $V{(t)}_{T,m,i}$ and $X{(t)}_{T,m,i}$ are the velocity vector and the coordinate position of population i which belongs to m species of generation T in literation t.

The evaluation of the population depends on its present habitat and species.

• The population belongs to the global optimal if this population resides in the best-known HSI habitat compared with all species.
• The population belongs to the local optimal if this population resides in the best-known HSI habitat compared with the same species m.
• The population belongs to the ordinary if the habitat where this population resides is not the best-known area compared with the same species and other species.

A specific population has its velocity attribute which updates in every iteration. The velocity vector-updating equations, summarized into three types of population, are given as follows:

If the population belongs to the global optimal

$$V{(t+1)}_{T,m,i}=rand\ast V{(t)}_{T,m,i},$$

(2)

If the population belongs to the local optimal

$$V{(t+1)}_{T,m,i}=L_m\ast V{(t)}_{T,m,i}+W(X_{glo}-X{(t)}_{T,m,i}),$$

(3)

If the population belongs to the ordinary

$$V{(t+1)}_{T,m,i}=L_m\ast V{(t)}_{T,m,i}+W(X_{glo}-X{(t)}_{T,m,i})+W(X_{sel,m}-X{(t)}_{T,m,i}),$$

(4)

L_m is described in the following equation:

$$L_m=\sqrt{{\displaystyle \sum _{n=1}^D{(X_{gl{o}_n}-X_{sel{,\mathrm{m}}_{\mathrm{n}}})}^2}}/{\displaystyle \sum _{m=1}^N\sqrt{{\displaystyle \sum _{n=1}^D{(X_{gl{o}_n}-X_{sel{,\mathrm{m}}_{\mathrm{n}}})}^2}}}$$

(5)

where $X_{glo}$ refers to the recorded global best HSI habitat in all species; $X_{sel,\mathrm{m}}$ refers to the known local best HSI habitat belonging to m species; and $L_m$ is the learning efficiency index of m species. $rand$ refers to a set of random vectors whose elements with a boundary of −1 to 1 accord with uniform distribution. $D$ is the dimension of habitat space which is a positive correlation with the complexity of practical problems. $N$ is the quantity of different species. $W$ is a positive inertia index, which represents the ability to search in the region Ω.

From Equations (3) and (4), we can realize that MARA has the character of information sharing called mutual benefit. Each population can record and share the known local best habitat, as a direct experience, among the same species. We comprehensively selected the best one as the recorded global optimal habitat, as an indirect experience shared with all species, from the known local optimal. Through the sharing mechanism, the population can dynamically adjust the migration strategy to get close to the recorded global optimal habitat or to explore a better one on the way to it.

The pseudo-code of the migration process is presented as shown in Algorithms 1:

Algorithms 1 The pseudo-code of the migration process

Create and initialize an nx-dimensional species position; repeat for each population i belongs to species m ∈ 1, ……, N do   //set the local best-known HSI habitat belongs to m species               If f (Xm,i) < f (Xsel,m)                       Xsel,m = Xm,i               end               //set the global best-known HSI habitat               If f (Xsel,m) < f (Xsel,m)                       Xglo = Xsel,m               end           end until stopping condition is true; for each population i belongs to species m ∈ 1, ……, N do         create population i velocity vector using Equations (2)–(4) end

2.2. Reproduction

After migration, each population resides in a new habitat, and the HSI may be better or worse. The recorded global optimal HSI changing for the better or being unchanged means the population belonging to the global optimal has not found a better habitat. It is easy for populations to judge HSI in new habitats, but they often ignore changes in the extent and trend of HSI after migrating a certain distance. In order to make full use of this information, we defined the maximum variable rate of HSI, including the maximum rate of ascent and maximum rate of decline, that is, when a population migrates from one habitat to another, its HSI has the maximum rate of change, compared with all species.

To allow species to have a higher probability to explore better habitats, populations living in marked habitats with globally optimal HSI and maximum HSI mutation rate will breed near these habitats.

Variation Rate of HSI and Reproduction

The current migration situation is decided on the initial characters of the coordinate position and the velocity vector of a new generation.

The variation rate of HSI can be described as follows:

$$F=(HSI{(t+1)}_{T,m,i}-HSI{(t)}_{T,m,i})/\sqrt{{\displaystyle \sum _{\mathrm{n}=1}^D{(X{(t+1)}_{T,m,i_n}-X{(t)}_{T,m,i_n})}^2}},$$

(6)

where $HSI{(t)}_{T,m,i}$ reveals the suitability of the habitat where population i resides, which belongs to m species of generation T in literation t.

The $F$ of population i accords with the maximum rise rate:

$$X_{\max ,i}=X_{T,\max }+[-K_{\max },K_{\max }] i\in (1,2,3\dots \dots Q),$$

(7)

$$K_{\max }=\sqrt{{\displaystyle \sum _{n=1}^D{(X_{gl{o}_n}-X_{{\mathrm{T},\max }_{\mathrm{n}}})}^2}}/{\displaystyle \sum _{m=1}^N\sqrt{{\displaystyle \sum _{n=1}^D{(X_{gl{o}_n}-X_{sel{,\mathrm{m}}_{\mathrm{n}}})}^2}}},$$

(8)

$$V_{\max ,i}=I_{sub\_\max ,i}{V}_{T,\max } i\in (1,2,3\dots \dots Q),$$

(9)

$$I_{sub\_\max ,i}=1-(HS{I}_{T,\max }-HS{I}_{sub-\max ,i})/{\displaystyle \sum _{n=1}^Q(HS{I}_{T,\max }-HS{I}_{sub-\max ,n})},$$

(10)

where $X_{T,\max }$ and $V_{T,\max }$ are the coordinate position and the velocity vector of a population whose habitat accords with maximum rise rate. $K_{\max }$ is a deviation which is superposed over the coordinate position. $X_{\max ,i}$ and $V_{\max ,i}$ are the coordinate position and the velocity vector of offspring population i. $I_{sub\_\max ,i}$ is the velocity learning rate of offspring population i, which means the degree of offspring population inherited from parents’ velocity. $Q$ is the quantity of the offspring population.

The $F$ of population i accords with the maximum decline rate:

$$X_{\min ,i}=X_{T,\min }+[-K_{\min },K_{\min }] i\in (1,2,3\dots \dots P),$$

(11)

$$K_{\min }=\sqrt{{\displaystyle \sum _{n=1}^D{(X_{gl{o}_n}-X_{{\mathrm{T},\min }_n})}^2}}/{\displaystyle \sum _{m=1}^N\sqrt{{\displaystyle \sum _{n=1}^D{(X_{gl{o}_n}-X_{sel{,\mathrm{m}}_{\mathrm{n}}})}^2}}},$$

(12)

$$V_{\min ,i}=I_{sub\_\min ,i}{V}_{T,\min } i\in (1,2,3\dots \dots P),$$

(13)

$$I_{sub\_\min ,i}=1-(HS{I}_{T,\min }-HS{I}_{sub-\min ,i})/{\displaystyle \sum _{n=1}^P(HS{I}_{T,\min }-HS{I}_{sub-\min ,n})},$$

(14)

where $X_{T,\min }$ and $V_{T,\min }$ are the coordinate position and the velocity vector of a population whose habitat accords with maximum decline rate. $K_{\min }$ is a deviation. $X_{\min ,i}$ and $V_{\min ,i}$ are the coordinate position and the velocity vector of the offspring population i. $I_{sub\_\min ,i}$ is the velocity learning rate of the offspring population i. $P$ is the quantity of the offspring population.

The global best population:

$$X_{sub\_glo,i}=X_{glo}+[-K_{glo},K_{glo}] i\in (1,2,3\dots \dots U),$$

(15)

$$K_{glo}=\min (\sqrt{{\displaystyle \sum _{n=1}^D{(X_{gl{o}_{\mathrm{n}}}-X_{sel{,\mathrm{m}}_{\mathrm{n}}})}^2}})>0 m\in (1,2\dots \dots N),$$

(16)

$$V_{sub\_glo,i}=I_{sub-glo,i}{V}_{glo} i\in (1,2,3\dots \dots U),$$

(17)

$$I_{sub-glo,i}=1-(HS{I}_{glo}-HS{I}_{sub-glo,i})/{\displaystyle \sum _{n=1}^U(HS{I}_{glo}-HS{I}_{sub-glo,n})},$$

(18)

where $X_{glo}$ and $V_{glo}$ are the coordinate position and the velocity vector of a population whose habitat accords with the global best HSI. $K_{glo}$ is a deviation. $X_{sub\_glo,i}$ and $V_{sub\_glo,i}$ are the coordinate position and the velocity vector of the offspring population i. $I_{sub-glo,i}$ is the velocity learning rate factor of the offspring population i. $U$ is the quantity of the offspring population.

The pseudo-code of the reproduction process is presented in Algorithms 2:

Algorithms 2 The pseudo-code of the reproduction process

Offspring population reproduction;               //population i accords whit maximum rise rate:                           create offspring populations using Equations (7)–(9)                   //population i accords whit maximum decline rate:                           create offspring populations using Equations (11)–(13)                   //the global best population:                           create offspring populations using Equations (15)–(17)

2.3. Nature Selection

The scale of species will expand after the progress of migration and reproduction. If generations reach the threshold of reproduction time, $G$, a vast majority of the population will be eliminated by natural selection for its HSI is inferior, except for those residing in the habitat that meets requirements. The surviving become the initial population of their species in the next iteration t + 1.

The pseudo-code of the Nature Selection process as shown in Algorithms 3:

Algorithms 3 The pseudo-code of the Nature Selection process

if reaching the threshold G is true         for each population i belongs to species do             Xm,i = Xsel,m         end end

2.4. Differences between MARA and Other Swarm-Based Optimization Algorithms

In this section, we will point out some of the distinctive features of MARA. First, MARA has migration, reproduction, and natural selection mechanisms and introduces a search method in biocoenosis. Second, MARA differs from PSO, in that an inferior population will be eliminated by the natural selection mechanism and sharing experience information among different species. Third, MARA has a similar method to FWA to increase the number of feasible solutions. However, the MARA reproduction mechanism is not only based on static population HSI but also has a close correlation with displacing equations. Moreover, we note that although MARA is a swarm-based optimization algorithm, it clearly distinguishes from those that simply involve displacement or diversification strategies. The deviation in Equations (8), (12), and (16) and the velocity learning rate factor in Equations (10), (14), and (18) reveal that the overall situation of exploration to direct position (solutions) changes extent flexibly, enabling MARA to have a great exploration capacity in surroundings between the global best position and the local best one. For these regulating indexes, whole species can achieve a balance between the global search ability and the local search ability.

3. Experiments on General Benchmarks

To show the advantages of MARA, we compared its performance on six benchmark functions with two other optimization algorithms. We did some rough tuning on each of the optimization algorithms to obtain reasonable performance, did not tune parameters deliberately, and referred to those in the reference, as shown in

Table 1. Parameter settings.

Optimization Method	Parameter
PSO	$C_1=C_2=1.5,\omega =0.5$
FWA	$\widehat{A}=40,p=5,m=50,a=0.04,b=0.8$
MARA	$Q=5,P=5,U=5,G=3,W=0.8$

. Each algorithm had an initial population quantity = 100 and ran for 200 iterations.

The benchmark functions we optimized are representative of those used in the literature for the comparison of optimization abilities. The details of the 12 test functions are shown in

Table 2. Benchmark functions with ranges and dimensions.

Benchmark Function		Range	${\mathit{f}}_{\mathbf{min}}$	Dimensions
Griewank $f_1$	$f(x)={\displaystyle \sum _{i=1}^d\frac{x_i^2}{4000}-}{\displaystyle \prod _{i=1}^d\cos (\frac{x_i}{\sqrt{i}})}+1$	[±5]	0	30, 50
Levy $f_2$	$\begin{array}{ll}f(x)=& {\sin }^2(\pi w_1)+{\displaystyle \sum _{i=1}^{d-1}{(w_1-1)}^2[1+10{\sin }^2(\pi w_1+1)]+}\\ & {(w_d-1)}^2[1+{\sin }^2(2\pi w_d)] where w_i=1+\frac{x_i-1}{4}\end{array}$	[±5]	0	30, 50
Ackley $f_3$	$\begin{array}{ll}f(x)=& -a\exp (-b\sqrt{\frac{1}{d}{\displaystyle \sum _{i=1}^d{x}_i^2}})-\exp (-b\sqrt{\frac{1}{d}{\displaystyle \sum _{i=1}^d\cos (c{x}_i)}})\\ & \begin{array}{ccc}& & \end{array}+a+\exp (1) where a=20,b=0.2,c=2\pi \hfill \end{array}$	[±5]	0	30, 50
Rosenbrock $f_4$	$f(x)={\displaystyle \sum _{i=1}^{d-1}[100{(x_{i+1}-x_i^2)}^2+{(x_i-1)}^2]}$	[±2.048]	0	30, 50
Zakharov $f_5$	$f(x)={\displaystyle \sum _{i=1}^d{x}_i^2+{({\displaystyle \sum _{i=1}^d}0.5i{x}_i)}^2}+{({\displaystyle \sum _{i=1}^{d}0.5i{x}_i})}^4$	[±2.048]	0	30, 50
Sphere $f_6$	$f(x)={\displaystyle \sum _{i=1}^d{x}_i^2}$	[±2.048]	0	30, 50
Shubert $f_7$	$f(x)=({\displaystyle \sum _{i=1}^{5}i\cos ((i+1)x_1+i)})({\displaystyle \sum _{i=1}^{5}i\cos ((i+1)x_2+i)})$	[±5]	−186.7309088	2
Beale $f_8$	$\begin{array}{ll}f(x)=& {(1.5-x_1+x_1{x}_2)}^2+{(2.25-x_1+x_1{x}_{{}_2}^2)}^2\\ & +{(2.625-x_1+x_1{x}_{{}_2}^3)}^2\end{array}$	[±5]	0	2
Drop-wave $f_9$	$f(x)=-\frac{1+\cos (12\sqrt{x_1^2+x_2^2})}{0.5(x_1^2+x_2^2)+2}$	[±5]	−1	2
Booth $f_{10}$	$f(x)={(x_1+2x_2-7)}^2+{(2x_1+x_2-5)}^2$	[±5]	0	2
LevyN13 $f_{11}$	$\begin{array}{ll}f(x)=& {\sin }^2(3\pi x_1)+{(x_1-1)}^2[1+{\sin }^2(3\pi x_2)]\\ & +{(x_2-1)}^2[1+{\sin }^2(2\pi x_2)]\end{array}$	[±2.048]	1.34978 × 10−31	2
Coalville $f_{12}$	$\begin{array}{ll}f(x)=& 100{(x_1^2-x_2)}^2+{(x_1-x_2)}^2+{(x_3-1)}^2+90{(x_3^2-x_4)}^2\\ & +10.1({(x_2^2-1)}^2+{(x_4^2-1)}^2)+19.8((x_2-1)(x_4-1))\end{array}$	[±2.048]	0	4

and

Table 3. Results of algorithms for different dimensions of scalable f₁–f₆ problems.

Function Number	Algorithm	Calculation 20 Times
		Dimension = 30		Dimension = 50
		Avg.	Std.	Avg.	Std.
$f_1$	PSO	0.011785804	0.008488843	0.061667621	0.022917967
	FWA	0.101092822	0.049650421	0.140462274	0.063959911
	MARA	6.7341 × 10−14	8.06892 × 10−14	1.8137 × 10−10	1.48813 × 10−10
$f_2$	PSO	0.904922375	0.561842052	2.518104766	0.942174474
	FWA	3.397330857	0.359140704	6.138812196	0.732744864
	MARA	0.507326752	0.204809122	1.141485276	0.297463896
$f_3$	PSO	1.729108764	0.52665594	2.299285672	0.317300079
	FWA	1.706420577	0.766897915	2.032193936	0.481059678
	MARA	1.602559448	0.332637407	1.369326755	0.559561803
$f_4$	PSO	39.90653705	14.4747126	131.5349857	18.41397261
	FWA	205.730317	112.2328165	710.0739199	710.0739199
	MARA	24.45018011	2.601112533	46.09457226	1.725310705
$f_5$	PSO	0.683691858	0.326381088	4.460063563	1.39632072
	FWA	9.759598735	2.730051265	20.03706161	2.769946403
	MARA	6.65506 × 10−11	7.37389 × 10−11	0.001518519	0.000929729
$f_6$	PSO	0.038681795	0.019648355	0.565557727	0.123206603
	FWA	1.878704395	0.84128266	4.677972552	1.313075453
	MARA	3.84309 × 10−26	1.30309 × 10−25	1.5962 × 10−11	1.0514 × 10−11

. f₁, f₂, and f₃ are multimodal functions to test the algorithm’s ability to escape local optima; f₄, f₅, and f₆ are unimodal functions, mainly used to test the convergence and optimization performance of the improved algorithm; f₇–f₁₂ are fixed-dimension multimodal benchmark functions.

To decrease the influence of random parameters, different algorithms ran 20 independent times in each benchmark function. From the results, as shown in Table 3 and

Table 4. Results of benchmark functions (f₇–f₁₂).

Function Number	Calculation 20 Times
	Metric	PSO	FWA	MARA
$f_7$	Avg.	−186.7309088	−181.768768	−186.7309088
	Std.	0	4.159307203	0
$f_8$	Avg.	2.57304 × 10−11	0.023584372	0
	Std.	4.55116 × 10−32	0.016860899	0
$f_9$	Avg.	−0.968122664	−0.992645658	−1
	Std.	0.032705457	0.019784592	0
$f_{10}$	Avg.	0	0.024346211	0
	Std.	0	0.019335457	0
$f_{11}$	Avg.	1.34978 × 10−31	0.023085346	1.34978 × 10−31
	Std.	0	0.027431161	0
$f_{12}$	Avg.	1.12524 × 10−23	3.979559541	3.14997 × 10−10
	Std.	5.02115 × 10−23	1.881842918	6.88597 × 10−10

, it can be seen that the MARA algorithm demonstrated its excellent solving ability and strong stability, according to the average or standard deviation of final fitness [12,13,14,15,16,17,18,19,20,21,22,23,24]. Even when the dimension was 50, MARA still performed the best on each test function. Table 4 shows that MARA achieved better optimization results than PSO and FWA, especially for functions $f_7$–$f_{11}$, achieving a theoretical optimal value $f_{\min }$. Except for the $f_{12}$ problem, MARA obtained better optimal and average values than the other two algorithms. For the f6 problem, although the optimal value obtained by MARA was not better than PSO, it was superior to the FWA. Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12 show the best fitness curves of each algorithm, revealing that MARA is capable of maintaining good balance between the exploratory and exploitative tendencies on problems with many variables, with good convergence performance and search accuracy.

We take the benchmark results with a neutral stance. First, every algorithm has a process to set parameters. A small tuning parameter value might influence algorithm performance significantly. Second, we always apply algorithms to solve practical problems, which usually consist of continuous and discrete variables, and the performance might have opposite conclusions from the benchmark tests. Third, different assessment criteria surely result in different conclusions. Therefore, the simulation results of the benchmarks can reflect some abilities incomprehensively, but not fully.

4. MARA for Practical Engineering Problems

In this section, we reviewed the coordinating economic dispatching problem for the multi-microgrid system, which we later used as a test problem for the MARA theory.

With the popularization and application of multi-microgrid technology, considering power interaction, the optimal operation of multi-microgrid systems is the basis to ensure the economic and environmental benefits of microgrid operation. In this paper, we constructed a multi-system with three microgrids that realize the process of energy interaction and information transmission among microgrids. The structure of this multi-system is shown in the Figure 13. According to the coordinate dispatching criterion, we formed unified management. Due to the complexity of scheduling objects and control strategies, intelligence optimization can provide more possibilities for collaborative dispatching in multi-microgrid systems.

Each microgrid contains a controllable distribution generator (DG), load, and energy storage system (ESS). Through the microgrid management center, hierarchical management reduces the difficulty of flexible dispatching, ensures the interest of the microgrid itself, and has the characteristics of autonomy.

4.1. Power Dispatching Strategy and Model

4.1.1. Power Dispatching Strategy

The power dispatching strategies in this paper are considered in three situation

(1)

Each microgrid meets the power load demand and has surplus power

If the distribution network is working in the peak period of the electricity selling price, the microgrid sells the surplus power to the Power Distribution Center (PDC); if the distribution network is working in the flat and valley period of the electricity selling price, the surplus power will be preferentially charged to the respective microgrid ESS until reaching the upper limit, and then, the remaining surplus power will be sold to the PDC. This dispatching strategy can not only reduce the operation cost of each microgrid but also alleviate the pressure on the distribution network during peak power consumption.

(2)

DG output in each microgrid is insufficient

If the electricity purchase price belongs to the peak period, the ESS will preferentially output until reaching the upper limit of discharge, and then, the remaining insufficient power shall be purchased from the PDC; if the distribution network is working in the flat and valley period of the electricity purchase price, the insufficient power will be directly purchased from the PDC to reduce the discharging times of the ESS. This dispatching strategy can help the distribution network to improve the utilization rate of electric energy in power flat and valley periods.

(3)

Some microgrids can suffice their load demand and some microgrids cannot

The surplus power of the former gives priority to the latter and then dispatches redundant or insufficient power according to the first strategy and the second strategy above.

4.1.2. Power Dispatching Model

Referring to the multi-microgrid structure [25,26], the three microgrids are close to each other and belong to the same distribution grid area. Each microgrid and the distribution network can interact with each other through a power line, as shown in Figure 14. DG models, in this paper, include micro gas turbines and fuel cells.

4.2. Construct Objective Function

The optimized dispatching strategy aims to minimize the comprehensive operation cost of N microgrids in an operation cycle T [27]. The objective function can be described as follows:

$$\min \mathrm{Cos}t=\min {\displaystyle \sum _{\mathrm{t}=1}^T{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{N}}(\alpha C_{i,t}^{\mathrm{grid}}+\beta C_{i,t}^{\mathrm{gen}}+\delta C_{i,t}^{\mathrm{emission}})}},$$

(19)

where $C_{i,t}^{\mathrm{gen}}$ is the microgrid i cost of equipment’s operation and maintenance at time t, and $C_{i,t}^{\mathrm{emission}}$ is the cost of environmental governance for pollution. $C_{i,t}^{\mathrm{grid}}$ is the cost of purchase or the income from power interaction among microgrids and PDC. T and N represent the total scheduling period and the number of microgrids. PV and WT are clean energy; there is no cost for environmental governance. Therefore, the objective function mainly depends on the controllable power generator’s operation and power interaction. $\alpha$, $\beta$, and $\delta$ are weighted indexes, and in this paper, we set them as 1/3.

4.3. Result and Discussion

To verify that MARA is more feasible and effective, we use it to optimize the same multi-microgrid system model compared with PSO and FWA.

The initial parameters of MARA are as follows: quantity of offspring population $Q=5$, $P=5$, and $U=5$; threshold of reproduction time $G=3$; inertia factor $W=0.8$. PSO: $C_1=C_1=1.5,\omega =0.5$. For the FWA, the parameters were set to $\widehat{A}=40$, $p=5$, $m=50$, $a=0.04$, and $b=0.8$. Each algorithm had an initial population quantity = 100 and ran for 300 literation 20 times independently, and the minimum cost was chosen as the best result of the respective algorithm, as shown in Figure 15.

The convergence speed of PSO, in the same practical model, is close to that of MARA, but the global search ability is worse. The optimized result of FWA is slightly higher than that of MARA, but the convergence speed is poor. When the literation reached 83, MARA has the best result than other algorithms and finally converges to 4019.

5. Conclusions

We proposed a new algorithm called the Migration and Reproduction Algorithm (MARA), which was inspired by species migration and reproduction. We applied MARA to benchmark functions and practical dispatching problems in multi-microgrid systems in the cloud environment and showed the results compared with other swarm-based algorithms. The results show that MARA has better performance in several benchmark functions and practical problems, which shows that MARA is more effective in solving optimization problems, but we do not claim that MARA is the best optimization solution over others. MARA has some common features with PSO, which mean MARA has a clearly changed solution method, excellent robustness, and is easily programmed. However, the PSO may fall into a local optimal solution, because the strategy of PSO is that all particles search in feasible regions in every iteration, which may significantly reduce the efficiency. MARA makes up for this shortage by purposefully and sensitively increasing population diversity according to real-time search situations (republication strategy). Nevertheless, similar to many swarm intelligence algorithms, MARA is sensitive to the initial population location, which influences optimal results to a certain extent. MARA is still in its infancy, and there are still some issues that need to be solved. Therefore, in the next step, we will study how the initial parameters affect the optimal results and find general parameters that can avoid tedious settings.