Economic optimization scheduling of microgrid group based on chaotic mapping optimization BOA algorithm

Abstract

Due to the intermittency and volatility of distributed power sources, the microgrid system has poor stability and high operation cost. Therefore, the study proposes an economic optimization scheduling strategy based on the chaotic mapping butterfly optimization algorithm and the mathematical model of microgrid group system. The study creates simulation trials of function poles and microgrid group operation to confirm the strategy’s efficacy. According to the experimental findings, the multimodal function of the enhanced butterfly optimization method had a variance of 0.0000E + 00, and the function’s optimal value was less than 10–30, and the calculation time is 4.5s. The variance on the fixed dimensional function was 0.0000E + 00 and the optimal value of the function was 10 − 3.5,and the calculation time is 4.7s. The algorithmic curve all digging depth was maximum and convergence speed was fastest. The microgrid group system had the lowest economic cost of 4029.32 yuan in the grid-connected mode and 3343.39 yuan in the off-grid mode. The study proves that the energy coordination and economic management of this strategy are greatly optimized, which can effectively protect the energy storage equipment and guarantee the smooth power consumption of the system. This provides an innovative theoretical basis for optimization scheduling of microgrid group.

Introduction

At present, China’s energy industry is in a critical stage of green transformation, and the development of renewable energy is becoming more and more important. The microgrid group system (MGS) composed of many microgrids shows good potential for renewable energy utilization. A microgrid is a single, autonomous power generation and distribution system that consists of loads, energy storage (ES) devices, control devices, and distributed power generation. It can address the issue of connecting multiple distributed power sources with different types and large base loads to the grid [1]. For a lithium iron phosphate battery ES system, Yang et al. presented a multi-objective planning optimization model to encourage the cost-effective and efficient functioning of the microgrid. Optimal control of the ES system was achieved by combining China’s certified emission reduction with power supply planning. The model enhanced the energy supply reliability by 0.15% and decreased the economic operation cost by 18.81%, according to the testing data [2]. By balancing the grid transmission within wireless sensor networks, Karthik et al. presented a deep learning model based on fog computing for power transmission in microgrids. The outcomes indicated that this approach effectively reduced the energy utilization and improved the microgrid throughput and data processing rate [3]. Hasan et al. proposed a grid-connected hybrid energy system based on residential areas in order to solve the environmental problems caused by energy use. The approach maximized the use of renewable energy by hybridizing and optimizing multiple energy systems. Experimental data indicated that the system reduced energy costs by 53%, operating costs by 54%, and total emissions by 31% [4]. Garcia-Torres et al. proposed a stochastic framework for microgrid optimization to address the energy prediction uncertainty of microgrids by constructing complex microgrids to serve system operators flexibly. It was demonstrated that the framework reduced the risk of penalty bias in the regulatory services market and improved the regulation of microgrid services [5]. To mitigate the negative impacts of load variations in microgrids, Santra et al. presented an inertial injection controller based on fractional order proportional-integral differentiation. By giving the microgrid inertial power, the technique enhanced the system’s transient performance. Experimental results demonstrated that the scheme effectively scaled down the stabilization time and steady state error [6]. At present, scientists’ research on microgrid optimization scheduling mostly stays in single microgrid system, which cannot better solve the impact of the volatility and randomness of distributed power. Therefore, the research focus is shifted to the optimization scheduling of microgrid group and group intelligent optimization algorithms with the ability to solve large-scale complex problems quickly.

Butterfly optimization algorithm (BOA) is a population-intelligent optimization algorithm inspired by butterfly food searching and mating behavior, which has the advantages of simple operation and few tuning parameters, as well as the problems of poor convergence, low solution accuracy, and insufficient global exploration capability [7]. Yadav et al. proposed a hybrid flash BOA for clinical data extraction of kidney diseases. The method introduced and tuned kernel soft plus extreme learning machine to accurately predict chronic kidney diseases. The experimental results indicated that the algorithm performed well [8]. Sun et al. proposed a binary monarch BOA in order to solve the problems of small BOA search range and easy to be trapped in local optimization. The method performed meta-heuristic feature selection by transforming the search space, updating the step parameter, and subset ordering. It was revealed that the algorithm improved the classification efficiency and global search capability [9]. Shams et al. proposed an improved BOA that combined search space jumps and constant impedance methods for maximum power point tracking for load conditions with varying solar shadows. Experimental results demonstrated that the algorithm improved the steady state efficiency by 99.85%, the load variation by 86.15% accordingly, and the average tracking time was less than 1 s [10]. To overcome the challenging optimization challenge for AI data, Alweshah et al. suggested an enhanced monarch BOA. By distributive integrating Lévy flights and supplementing the crossover operator, the approach selected features for the dataset. The system was shown to have an average high classification accuracy of 93%, significantly narrowing the selection range [11]. Sharma et al. presented a hybrid sine-cosine BOA for the problem of insufficient BOA search capability. The method realized exploratory and exploitative search by combining BOA and positive cosine algorithms. The experimental results indicated that the superior results of this algorithm were 75% on average and the similar results were 20% on average [12]. After the experiments of many scholars and scientists, the performance of the BOA algorithm for searching for superiority has been improved to a certain extent by the improvement strategies such as binary search and Lévy flight. However, the global exploration efficiency of the existing improved algorithms is still low, the convergence is weak, and the accuracy needs to be improved.

In this context, the study attempts to formulate an economic optimization scheduling strategy for microgrid group based on chaotic mapping improved BOA algorithm. The study breaks through the limitations of individual microgrid system, overcomes the shortcomings of traditional BOA algorithm, and innovatively establishes the MGS architecture. The new BOA algorithm is used to seek the global optimal solution, which is expected to realize low-cost and high-efficiency microgrid group operation.

Methods and materials

Improved BOA based on chaotic mapping

To attempt to mimic the natural behaviors of butterflies, such as foraging and courtship, Arora et al. (2018) introduced the BOA algorithm, a meta-heuristic optimization method that has the advantages of simple principles, low parameter needs, and straightforward implementation. Nevertheless, the fundamental BOA algorithm is prone to local optimality and has poor convergence accuracy [13]. In this regard, a new BOA algorithm (Tent simplex method butterfly optimization algorithm, TSMBOA) is proposed by combining chaos theory and simplex form method. On the one hand, the chaotic idea is introduced into the algorithm. Tent chaotic mapping is added to the population at the startup phase to play with its traversal, randomness, and sensitivity, increasing the population’s diversity and the algorithm’s efficiency when looking for the best solution [14]. However, for the purpose to increase the population’s capacity to solve global optimization problems, the simplex form approach technique is employed to continually optimize the replacement of underperforming people [15].In the BOA algorithm, butterflies exchange information by sensing the intensity of the surrounding fragrance to determine the direction of the target, and make a global search by constantly updating the position. When the fragrance cannot be perceived, the butterflies will randomly move and search in the local area. The execution process of BOA algorithm includes three main steps: first, randomly initialize the butterfly group and its location information; second, conduct iteration to perform global or local search; and finally, stop iteration and output the best solution. The global search calculation formula is shown in Eq. (1).

$$x_{k}^{{d+1}}=x_{k}^{d}+({r^2} \times {m^ * } - x_{k}^{d}) \times {f_k}$$

(1)

In Eq. (1), $x_{k}^{d}$ represents the kth butterfly position vector of the current iteration, ${m^ * }$ is the optimal solution, ${f_k}$ is the fragrance content of the k butterfly, r is the constant, and its value interval is [0,1]. The local search calculation formula is shown in Eq. (2).

$$x_{k}^{{d+1}}=x_{k}^{d}+({r^2} \times (x_{q}^{d} - x_{l}^{d}) - x_{k}^{d}) \times {f_k}$$

(2)

In Eq. (2), $x_{q}^{d}$ is the q butterfly position vector of the current iteration, and $x_{l}^{d}$ is the l butterfly position vector of the current iteration. Since the BOA algorithm explores the optimal solution through continuous iteration, its accuracy depends on the number and distribution of the original population, the study introduces chaotic mechanism in the initialization stage, plays its advantages of ergodicity, convergence and sensitivity to the initial value, comprehensively explore the search space and improve the quality of the initial population [16]. The study uses Tent chaotic mapping to generate the original chaotic sequence. The applied equation is shown in Eq. (3).

$$x_{{n+1}}^{k}=\left\{ \begin{gathered} \alpha x_{n}^{k},x_{n}^{k}{\text{<0}}{\text{0.5}} \hfill \\ \alpha (1 - x_{n}^{k}),x_{n}^{k} \geqslant 0.5 \hfill \\ \end{gathered} \right.$$

(3)

In Eq. (3), $\alpha$ is a tunable parameter whose value interval ranges from $(0,2]$. n is the chaos variable number. k is the population size. When the value is 2, the degree of chaos is the highest and $x_{{n+1}}^{k}$ will traverse the interval with equal probability, unordered variants and will not repeat. Tent chaotic mapping produces n initial chaotic sequences after inverse mapping to the corresponding individual search space. The individual search space variables are shown in Eq. (4).

$$y_{n}^{k}=1{b_n}+(u{b_n} - 1{b_n})x_{n}^{k}$$

(4)

In Eq. (4), $y_{n}^{k}$ denotes the position of the kth butterfly in the nth dimensional space (DS). $u{b_k}$ is the upper limit of the nth DS. $1{b_n}$ is the lower limit of the nth DS. $x_{n}^{k}$ is the optimal chaos variable. So that the individual butterflies are evenly distributed on the search space, the initial position of the butterfly population with more diversity is computationally generated. In the iteration stage, the paper updates and adjusts the poor individual positions after the global search and the local search, so as to improve the quality of the optimal solution of the algorithm and the convergence efficiency of the population solution. The simplex form method is a search technique proposed by Nelder et al. for solving linear programming problems through geometric forms, which has the advantages of simple operation, small computational volume, and strong local search capability [17]. The operation of simplex form method is shown in Fig. 1.

Fig. 1

The operation process of simplex method

In Fig. 1, in a multiDS, the simplex algorithm constructs a polyhedral simplex by constructing polyhedral simplexes with multiple vertices. The fitness values of each vertex are evaluated to identify the optimal solution ${x_g}$, the suboptimal solution ${x_b}$ and the worst solution ${x_s}$ of the butterfly population. The worst solution of the simplex performs operations such as reflection, expansion and contraction to form a new polyhedral simplex by replacing the worst vertices with more optimal ones. This repeated iteration progressively approaches the optimal point [18]. The midpoint solution formulas for ${x_g}$ and ${x_b}$ are given in Eq. (5).

$${x_c}=\frac{{{x_g}+{x_b}}}{2}$$

(5)

In Eq. (5), ${x_g}$ is the optimal vertex. ${x_b}$ is the suboptimal vertex. ${x_c}$ denotes the midpoint of the two. The reflection operation is performed on the optimal point, and the reflection point solution is shown in Eq. (6).

$${x_r}={x_c}+a({x_c} - {x_s})$$

(6)

In Eq. (6), ${x_r}$ denotes the reflection point. ${x_c}$ is the midpoint between the optimal point and the secondary point. ${x_s}$ is the optimal point. a is the reflection coefficient, which takes the value of 0.5. The expansion operation is carried out and the reflection direction is accurate when the optimal point value exceeds the reflection point value. The expansion point solution is shown in Eq. (7).

$${x_e}={x_c}+\gamma ({x_r} - {x_c})$$

(7)

In Eq. (7), ${x_e}$ is the expansion point. ${x_r}$ is the reflection point. ${x_c}$ is the midpoint between the optimal point and the secondary point. $\gamma$ is the expansion coefficient, which takes the value of 1.5. If the expansion point value is less than the optimal point value, ${x_e}$ is used instead of ${x_s}$. Instead, ${x_r}$ is used instead of ${x_s}$. The compression procedure is carried out when the reflection point value exceeds the ideal point value, indicating that the reflection is occurring in the incorrect direction. The compression point solution is shown in Eq. (8).

$${x_t}={x_c}+\beta ({x_s} - {x_c})$$

(8)

In Eq. (8), ${x_t}$ denotes the compression point. ${x_s}$ is the optimal point. ${x_c}$ is the midpoint between the optimal point and the secondary point. $\beta$ is the compression coefficient, which takes the value of 0.5. If the value of compression point is less than the value of the minimum point, replace ${x_s}$ with ${x_t}$. The contraction operation is carried out when the values of the minimum point and reflection point are larger than each other and the optimum point is less than the reflection point. Equation (9) presents the contraction point solution.

$${x_w}={x_c}+\beta ({x_s} - {x_c})$$

(9)

In Eq. (9), ${x_w}$ denotes the contraction point, ${x_s}$ is the minimum point and ${x_c}$ is the midpoint between the optimal point and the secondary point. $\beta$ is the shrinkage coefficient, which takes the value of 0.5. If the value of the shrinkage point is less than the value of the minimum point, ${x_w}$ is used instead of SSS. Instead, ${x_r}$ is used instead of ${x_s}$. Figure 2 depicts the TSMBOA algorithm’s general flow.

Fig. 2

TSMBOA algorithm operation flow chart

In Fig. 2, the TSMBOA algorithm first initializes all the parameters and initializes the position of the butterfly population in the search space by Tent chaotic mapping. Secondly, according to the spatial location of the butterfly, calculate the butterfly individual fitness value as well as the scent concentration. Entering the iteration stage, set the switch probability p and perform conditional determination. If the frequency is greater than p, search for optimization in the local range. On the contrary, perform global range exploration. Next, the simplex algorithm is used to adjust the butterfly individual with insufficient optimization performance to update the butterfly individual and the global optimal solution. If the maximum iterations is reached, the optimal solution is output. Otherwise, return the individual butterfly adaptation value, repeat the global search, local search and simplex operation steps until the optimal solution is obtained.

Optimization scheduling strategy for microgrid group economy

In view of the uncertainty and volatility of wind power generation, a micro-grid group architecture with wind power generation as the core and equipped with gas turbine and diesel engine as the supplementary power supply was established, whose structure is shown in Fig. 3. The microgrid system, which incorporates multiple independent microgrid systems and is connected to the main grid via the Internet, is expanded and deepened by the MGS [19]. Based on solar photovoltaic, diesel engines, wind turbines, small gas turbines, ES devices and electrical loads, the microgrid is able to realize self-supply of energy. At the same time, they communicate and work together with each other and with the main grid through an information management center [20].

Fig. 3

Microgrid group structure diagram

The research design objective function aims to reduce microgrid group operating expenses and environmental impact costs, and maximize the economic benefits of scheduling the MGS. Equation (10) illustrates the cost of running and maintaining the microgrid’s generating apparatus.

$${\text{C}}_{{\text{i}}}^{{{\text{EM}}}}(t)=k_{i}^{{WT}}(t)P_{i}^{{WT}}(t)+k_{i}^{{PV}}(t)P_{i}^{{PV}}(t)+k_{i}^{{DE}}(t)P_{i}^{{DE}}(t)+k_{i}^{{MT}}(t)P_{i}^{{MT}}(t)$$

(10)

In Eq. (10), ${k_i}(t)$ is the O&M cost factor per unit of power of the equipment. ${P_i}(t)$ is the equipment power. The microgrid power exchange cost is shown in Eq. (11).

$$C_{i}^{{ET}}(t)=P_{i}^{{mg}}(t)\delta _{i}^{{mg}}(t)+P_{i}^{{grid}}(t)\delta _{i}^{{grid}}(t)$$

(11)

In Eq. (11), $P_{i}^{{mg}}(t)$ is the energy flow power (EFP) between microgrids. $\delta _{i}^{{mg}}(t)$ is its power cost at a certain moment. $P_{i}^{{grid}}(t)$ is the EFP between the microgrid and the distribution grid. $\delta _{i}^{{grid}}(t)$ is the power cost of the latter at a certain moment. There are limitations with the MGS. The system power balance constraints are shown in Eq. (12).

$$P_{i}^{{load}}(t)=P_{i}^{{WT}}(t)+P_{i}^{{PV}}(t)+P_{i}^{{DE}}(t)+P_{i}^{{MT}}(t)+P_{i}^{{BT}}(t)+P_{i}^{{mg}}(t)+P_{i}^{{grid}}(t)$$

(12)

In Eq. (12), $P_{i}^{{load}}(t)$ denotes the microgrid power load. The diesel engine power regulation limits are shown in Eq. (13).

$$R_{{i,down}}^{{DE}}\Delta t \leqslant P_{i}^{{DE}}(t) - P_{i}^{{DE}}(t - 1) \leqslant R_{{i,up}}^{{DE}}\Delta t$$

(13)

In Eq. (13), $R_{{i,down}}^{{DE}}$ is the descending power of the diesel engine. $R_{{i,up}}^{{DE}}$ is the rising power of the diesel engine. Small gas turbine power regulation limits are shown in Eq. (14).

$$R_{{i,down}}^{{MT}}\Delta t \leqslant P_{i}^{{MT}}(t) - P_{i}^{{MT}}(t - 1) \leqslant R_{{i,up}}^{{MT}}\Delta t$$

(14)

In Eq. (14), $R_{{i,down}}^{{MT}}$ is the descending power of the small gas turbine. $R_{{i,up}}^{{MT}}$ is the rising power of the small gas turbine. In the ES device constraint, the ES unit charge state is shown in Eq. (15).

$$SOC_{i}^{{\hbox{min} }} \leqslant SOC(t) \leqslant SOC_{i}^{{\hbox{max} }}$$

(15)

In Eq. (15), $SOC_{i}^{{\hbox{min} }}$ is the ES unit charge minimum. $SOC_{i}^{{\hbox{max} }}$ is the ES unit charge power maximum value. In the ES device constraints, the ES unit charging and discharging power constraints are shown in Eq. (16).

$$\left\{ \begin{gathered} 0 \leqslant P_{i}^{{ch}}(t) \leqslant P_{i}^{{\hbox{max} }} \hfill \\ 0 \leqslant P_{i}^{{dis}}(t) \leqslant P_{i}^{{\hbox{max} }} \hfill \\ \end{gathered} \right.$$

(16)

In Eq. (16), $P_{i}^{{\hbox{max} }}$ is the upper limit of the charge power of the ES unit. The limit of power exchange between microgrid groups is shown in Eq. (17).

$$P_{{\hbox{min} }}^{{mg}} \leqslant P_{i}^{{mg}}(t) \leqslant P_{{\hbox{max} }}^{{mg}}$$

(17)

In Eq. (17), $P_{{\hbox{min} }}^{{mg}}$ is the lowest electrical energy interaction of the microgrid group. $P_{{\hbox{max} }}^{{mg}}$ is the highest electrical energy interaction of microgrid group. The interaction power constraints between microgrids are shown in Eq. (18).

$$P_{{\hbox{min} }}^{{grid}} \leqslant P_{i}^{{grid}}(t) \leqslant P_{{\hbox{max} }}^{{grid}}$$

(18)

In Eq. (18), $P_{{\hbox{min} }}^{{grid}}$ is the minimum value of the interaction power between microgrids. $P_{{\hbox{max} }}^{{grid}}$ is the maximum value of the interaction power between microgrids. There are two main modes of microgrid operation: grid-connected operation (GCO) and off-grid operation. This study develops corresponding optimization scheduling strategy for different operation modes [21]. The microgrid group economicoptimization scheduling strategy for GCO mode is shown in Fig. 4.

Fig. 4

Optimization scheduling strategy for grid connected operation mode

In Fig. 4, the GCO mode optimization scheduling strategy is divided into three stages. The preferred power supply units (PSU) are photovoltaic arrays and wind power in the microgrid system when it is in the low-peak power consumption period. When the electricity provided by renewable energy sources surpasses the necessary power consumption in the current phase, the ES device receives the remaining power first. When the ES reaches its capacity limit, the extra power is traded externally, with priority given to selling to the microgrids with the highest power demand. If all microgrids have sufficient power supply, they are then exchanged with the distribution grid for economic returns. During normal load periods, when customer electricity demand is greater than PV and wind power generation, the ES device is first selected as the PSU. If the power demand still exceeds the load, small gas turbines, diesel engines and the distribution network are evaluated and the PSUs are optimized. On the other hand, when the renewable power exceeds the demand, power is delivered to the storage unit before selling the extra power to the outside world to realize the economic benefits. Being in a situation where solar and wind power cannot meet the surging demand for electricity during peak hours, ES devices are first used to release the energy. When the capacity limit of the ES discharge is reached, it is necessary to choose between small gas turbines, diesel engines and the distribution grid, and select the best way to supply power.The distribution grid will be used to supply electricity if the tiny gas turbine and diesel engine are unable to meet demand because of restrictions on power control. The optimization scheduling strategy for microgrid group economy in off grid operation mode is shown in Fig. 5.

Fig. 5

Optimization scheduling strategy for off grid operation mode

In Fig. 5, the off grid operation mode optimization scheduling strategy is similarly divided into three phases. During the low peak period, solar PV and wind turbines are prioritized for power supply. When the power generated by both is greater than the required load of the consumer, the excess power is prioritized to be stored in the ES device. Once the storage reaches its capacity limit, the extra power is used to exchange economic benefits between microgrid groups. During the normal load period, as the demand for electricity grows and the new energy generation cannot meet the user’s demand criteria, the ES equipment is prioritized as the PSU. If the ES reaches the discharge limit, the small gas turbine is activated to generate power, or power is purchased from the power-rich microgrid part. During peak periods, the electricity consumption of the consumers increases dramatically. Each microgrid may be in a state of oversupply, and the ES system is first selected to supply power. When the maximum ES power supply is reached, a small gas turbine is used to supply power. If it is still unable to meet the user’s electricity load, the diesel engine climbing constraint is lifted, and power is supplied by the diesel engine to maintain the balance of power supply.

Results

Experimental setup and data set

To validate the comprehensive performance of the TSMBOA algorithm and the feasibility of the optimization scheduling strategy, two simulation experiments are designed. The unified computing environment of the experiments is Windows 10 operating system and the simulation software is MATLAB2022b. Three benchmark test functions are selected for the experiments. The extreme value optimization of particle swarm optimization (PSO), whale optimization algorithm (WOA), BOA and TSMBOA algorithms are compared. The experimental test object is three interconnected microgrid systems. Each system consists of solar photovoltaic panels, wind turbines, diesel engines, small gas turbines, ES devices, and electrical loads. The microgrid group economic optimization scheduling is compared under different operation modes. The functions stand devation (STD), convergence curve (CC), average value (AV), and best value (BV) are chosen as the algorithm performance evaluation indices. The microgrid zone group operation cost and equipment output power are chosen as the optimization scheduling strategy’s assessment indices. The experiment fixes all the parameters and sets the function to run 50 times, the learning factor of PSO algorithm is 2, the inertia weight is 0.6, the convergence factor of WOA algorithm is [2, 0], the log constant is 1, the BOA algorithm is 0.01, the power index is 0.1, p is 0.6, the TSMBOA algorithm is 0.01, $\alpha$ is 0.1, and the simplex search 10 times. The lower and upper limit interval of the power generation unit is [0, 15] and [50, 240]. The rated capacity of the ES unit is set to be 150 kWh, the maximum allowable charging and discharging power is set to be 50 kWh, the charge capacity interval is set to be [30, 135], and the charge installation cost is set to be 120, 000 yuan. Table 1 displays the tariffs for various time periods.

Table 1 Time of use electricity price table

Algorithm performance optimization analysis

The results of the benchmark function test of the algorithm are shown in Figure Fig. 6. In Fig. 6, on the function F1, the AV and STD values of TSMBOA algorithm rank first among all algorithms, with 6.1461E-08 and 4.2729E-08, respectively, with a high degree of data dispersion and large difference, and the poor stability and robustness of unimodal function problems. On the function F2, the TSMBOA algorithm AV and STD values were significantly lower than the other algorithms, with 1.3498E-32 and 0.0000E + 00, respectively. On the function F3, the TSMBOA algorithm AV and STD values are also almost smaller than the other algorithms, with 3.0749E-04 and 0.0000E + 00, respectively. This shows that the proposed algorithm can find the multimodal function and the fixed dimension function with the optimal solution stably and maintain the consistency of the output results, showing good stability and robustness.

Fig. 6

Bar graph of the benchmark function test results of the algorithm

A comparison of the CCs of the algorithms on the test functions is shown in Fig. 7. On the function F1, the TSMBOA algorithm curve is a deep digging trend, compared with the rest of the algorithms to find the highest optimization accuracy (OA) and the fastest convergence speed (CS). On the function F2, the butterfly class algorithms have better convergence performance, and the CCs show an obvious decreasing trend. The PSO and WOA algorithms have a gentle curve. The curve of TSMBOA algorithm is the steepest, and its function BV is less than 10–30. Its OA is superior, its convergence efficiency is faster, and its robustness is higher. On the function F3, all algorithms are able to find the BV quickly. The TSMBOA curve has the largest digging depth and the fastest CS, and the function BV is less than the other algorithms is 10 − 3.5. It is known that compared with PSO, WOA and BOA comparison algorithms, the TSMBOA algorithm has a higher OA and CS, and it is able to avoid falling into the local extreme value.

Fig. 7

Convergence curve for solving test functions

The 50 independent calculations on the test function are shown in Table 2. In Table 2, the PSO algorithm has the shortest operation time because it only involves the particle speed update and position calculation. However, the proposed algorithm can easily fall into the local optimal value and no convergence. Among WOA, BOA and TSMBOA algorithms, WOA algorithm has the longest operation time and the lowest efficiency. However, the operation time of TSMBOA algorithm is relatively short, and the time consuming is 4.3s, 4.5s and 4.7s on functions F1, F2 and F3, respectively, and the operation efficiency is better than that of BOA. In general, the proposed algorithm has significantly improved in the calculation and convergence efficiency, solution accuracy and stability, and the improvement strategy adopted in the research is effective.

Table 2 The algorithm independently performs 50 calculations

Optimization scheduling strategy results

The predicted changes in the electrical energy of the microgrid system are shown in Fig. 8. The curve trends of photovoltaic and wind turbine show negative correlation changes, alternating intermittently to supply electricity to the microgrid output. According to the load curve fluctuation, the electricity load grows to the highest peak in the morning and afternoon hours, and then drops slightly in the midday hours until it reaches the lowest valley at night. The peak and valley values of the three microgrid systems are slightly different. Microgrid 1 reaches a peak of 340 kW at 18:00 and a trough of 130 kW at 02:00. Microgrid 2 reaches a peak of 380 kW at 14:00 and a trough of 240 kW at 06:00. Microgrid 3 reaches a peak of 390 kW at 22:00 and a trough of 190 kW at 03:00.

Fig. 8

Prediction of energy changes in microgrid systems

Combined with the power consumption load curve, the study conducts a comparative analysis for the microgrid group equipment discharges, and the results of the microgrid group equipment discharges under the off grid operation mode are shown in Fig. 9. Wind turbines and PV arrays, as the main force, bear the vast majority of the discharge. The diesel engine takes over the power supply during the overload phase of the electrical load. Microgrid 3 system power is more adequate than other microgrids. During the hours of 0:00 to 1:00, 5:00 to 8:00, and 20:00 to 24:00, the microgrid power supply exceeds the demand, and the ES devices carry out the output power supply. Among them, during the hours of 6:00 to 7:00 and 8:00 to 24:00, the ES equipment reaches the upper limit of output power, and the diesel engine takes over the discharge. In the 2:00 to 4:00 and 9:00 to 19:00 h, the microgrid system is full of power, and power is delivered to the ES equipment. Once the storage capacity limit is reached, additional power is resold to the outside world. It is sold to the Microgrid 2 system during the 3:00 to 4:00 h and to the Microgrid 1 system during the 12:00 to 19:00 h.

Fig. 9

Off grid mode microgrid group equipment output

The results of microgrid group equipment discharge in GCO mode are shown in Fig. 10. Wind turbines and PV arrays act as the main actors, taking the vast majority of the discharge. During the overload phase of the electrical load, the power is exchanged by the distribution grid. During the hours of 0:00 to 1:00, 5:00 to 8:00, and 20:00 to 24:00, the Microgrid 3 system power consumption is in short supply, and the ES equipment compensates for the power supply. Among them, in the hours from 6 to 9 and 8 to 24, the ES devices reach the maximum limit of power supply and the system needs to buy power from outside. During the hours of 2 to 4 and 9 to 19 o’clock, the microgrid system has enough power to charge the ES devices and sell the excess power after reaching the charging limit. In this case, it is sold to the Microgrid 2 system during the 3:00 to 4:00 h and to the Microgrid 1 system during the 12:00 to 19:00 h.

Fig. 10

Grid connected mode microgrid group equipment output

The study applies the algorithm to solve and analyze the microgrid group economic cost in different modes and the microgrid group economic cost is shown in Fig. 11. The operating cost decreases with the iterations of the algorithm, and the algorithm solves for the final economic cost when the curve is flat. In off-grid mode, the costs obtained by the PSO, WOA, BOA and TSMBOA algorithms are 4399.85, 4161.65, 4156.88 and 4029.32 yuan, respectively. In grid-connected mode, the costs obtained from the PSO, WOA, BOA and TSMBOA algorithms are 4338.14, 4161.94, 4105.38 and 3343.39 yuan, respectively. The proposed algorithm shows the highest economic benefit in both modes, and the calculated economic cost is lower than that of the comparison algorithm. It can be seen that by optimizing the operation and scheduling of the microgrid group, the overall operation cost of the system can be effectively reduced, and the economy and sustainability of the microgrid group can be improved.

Fig. 11

Operating costs of microgrid clusters

Discussion and conclusion

Aiming at the problem of poor power supply stability of the existing single microgrid system, the study proposes an economic optimization scheduling strategy for microgrid group based on the improved BOA algorithm. The performance of the algorithm is evaluated through simulation experiments, and the optimization scheduling of microgrid group is analyzed. The experimental data shows that the variance of TSMBOA algorithm on unimodal functions is 4.2729E-08, with an operation time of 4.3 s. On multimodal functions, the variance is 0.0000E + 00, with an optimal value less than 10–30 and an operation time of 4.5 s. On fixed dimensional functions, the variance is 0.0000E + 00, with an operation time of 4.7 s and an optimal value of 10 –3.5 s. This was due to the introduction of Tent chaotic mapping initialization and simplex form method search, the algorithm’s OA and CS were greatly improved. Compared to the hybrid flash BOA proposed by Yadav et al. the TSMBOA algorithm achieved 98.5% accuracy and better prediction performance. The experimental results indicated that microgrid group had lowest economic cost of 4029.32 yuan in grid-connected mode and 3343.39 yuan in off-grid mode. This was because the microgrid system could interact with electricity to the distribution grid and the rest of the microgrids in the grid-connected mode. While in off-grid mode, the microgrid system could not be connected and integrated into the distribution grid and had to rely on the internal equipment of the system such as diesel engine to generate electricity. It is proved that this strategy greatly improves the algorithm’s ability to find the optimal solution and effectively reduces the economic cost of the MGS. The disadvantage is that the improvement algorithm takes a long time to optimize, and the robustness needs to be improved, and the proposed system does not fully consider the real-time data processing requirements. In the future, the research will focus on strengthening the computing efficiency of the algorithm, designing the multi-layer optimal dispatching strategy, and realizing the real-time data collection and processing combined with the edge computing technology, so as to comprehensively improve the overall operation of the micro-grid group.