With the transformation of the economic growth model of modern society, the structure of electric power production is also changing, along with the upgradation of industrial and energy structures. In recent years, the global emphasis on new energy has increased, which is in line with the development of the times to eliminate the dependence on a single source of energy but also to contribute to the improvement of the ecological environment. In the face of changes in the power supply and demand, microgrids have become the choice for power systems to cope with energy security and low-carbon development. Microgrids involve the comprehensive utilization of wind, solar, natural gas, electricity, and other energy sources, and their scheduling problem is complex. Microgrid optimal scheduling refers to the reduction of operating costs, improvement of new energy utilization, enhancement of the stability of the power grid, reduction of environmental pollution, energy loss, etc., by rationally arranging the power of each power source and load of the microgrid under the premise of ensuring the safe and stable operation of the system.

The microgrid scheduling problem is a classical unit-combination (UC) problem. So far, there are many scholars have conducted research on microgrid (MG) scheduling problems, such as Lagrangian relaxation , mixed-integer linear programming , metaheuristic optimization algorithms , et al. In , an improved quantum particle swarm optimization (QPSO) using a differential evolutionary algorithm was proposed, the experiments show that the improved QPSO has better performance and is more suitable than the QPSO and CLQPSO to solve the demand problem of microgrids. In , an improved real-coded genetic algorithm (GA) and enhanced mixed-integer linear programming (MILP)-based, developed GA-based, and MILP-based optimizers were proposed for application to a test microgrid model under different operating strategies. The experimental results show that voltage and load violations can be corrected very accurately. In , a hybrid improved GSA-PSO was proposed by combining the gravitational search algorithm (GSA) and particle swarm optimization (PSO) algorithms (MGSA-PSO), which implements and analyzes load scheduling optimization. Experimental results demonstrate the effectiveness of the proposed scheme by analyzing the effects of different numbers of electric vehicles and different charging modes. In , a chaotic cuckoo search SVR (SVRCCS) model based on the tent chaotic mapping function was proposed. The numerical results from the dataset tests show that the proposed SVRCCS model outperforms other alternative models. In , an improved mayfly algorithm incorporating Levy flights is proposed, the experimental results demonstrate that the proposed IMA algorithm can solve the CEED problem in grid-connected microgrids. In , a cuckoo search algorithm (CSA) was used to optimize the economic dispatch of microgrids, the experimental results show that CSA has good global convergence and provides a better optimal solution and emission costs. In , an enhanced population-aware particle swarm optimization (QS-PSO) was used to determine the full-cycle optimal scheduling solution for microgrids, the experimental results demonstrate the effectiveness of the proposed scheduling model and QS-PSO application. In , a metaheuristic adaptive elephant swarm optimization (SA-EHO) was proposed for the implementation of an ideal scheduling model for microgrids (MGs) with EVs and RESs. The experimental results demonstrate that the proposed algorithm performed well. In , a variant of the Non-dominated Sorting Genetic Algorithm (NSGA)-II algorithm was used along with the introduction of an external penalty function to deal with the constraints and facilitate the solution of the multi-objective optimization model. The proposed algorithm was applied to the dynamic economic dispatch model of a microgrid. The economy of the proposed scheme was verified by comparing its economic dispatch effects under different operating conditions. In , a BSA based on an adaptive taxing flight strategy (LF-BSA) was proposed, the experimental results verify the feasibility of LF-BSA and the effectiveness of multi-objective optimization. In , the Whale Optimization Algorithm (WOA) was used to perform all ELDs, emission scheduling, and CEEDs on islanded and renewable energy-integrated microgrids, the experimental results demonstrate that the proposed method outperforms other optimization techniques. In , a symbiotic organism search (SOS)-based staging algorithm and an improved multi-intelligent actor (MA) consensus algorithm (IMACA) were proposed, the experimental results verify that the proposed algorithm has superior performance and can obtain the most economically efficient solution. In , an improved PSO algorithm with adaptive inertia weights and shrinkage factors was proposed, the results show that the improved PSO algorithm effectively reduces the comprehensive objective cost and achieves better optimization results. In , an improved butterfly optimization algorithm (IBOA) based on a partial tent chaotic map, Cauchy variation, and simplex method was proposed, the experimental results show that the IBOA can effectively reduce the system cost of electricity, promote the effective utilization of renewable energy, and improve the operational stability of the microgrid cluster system. In , Giza pyramid construction (GPC) is proposed to realize the optimal design of isolated microgrids. Net current cost (NPC), levelized cost of energy (LCOE), loss of power probability and availability index are used as objective functions. In , an improved mayfly optimization algorithm was proposed for solving microgrid problems. These results were compared with those of recent state-of-the-art algorithms using the same microgrid model. Experimental results show that the proposed algorithm is competitive and robust.

This paper a multi-strategy enhanced dwarf mongoose optimization algorithm (EDMO) for microgrid scheduling problem is proposed. The main contributions are as follows:

(1) The dwarf mongoose optimization algorithm was enhanced with the golden sine strategy, adaptive t-distribution variational strategy, and Lévy flight strategy.

(2) A multi-strategy enhanced dwarf mongoose optimization algorithm (EDMOA) was proposed.

(3) Comparison with well-known algorithms, and it is experimentally verified that the proposed EDMOA performs better than the other algorithms.

(4) The EDMOA was successfully applied to solve the microgrid optimal scheduling problem.

In this section, the mathematical model of the microgrid optimal dispatch problem is proposed which includes the wind turbine, photovoltaic, diesel generator, micro gas turbine, battery, and the objective functions. The main objective of optimal microgrid scheduling is to minimize the operating cost of the microgrid in grid-connected mode.

A wind turbine (WT) , which is mathematically modeled as shown in Eq. (1):

In Eq. (1), PWT is the output power of the wind turbine, P_r is the rated power of the wind turbine, v_ci is the cut-in wind speed of the wind turbine, v_r denotes the rated wind speed of the wind turbine, and v_co denotes the cut-out wind speed of the wind turbine, a, b, c, and d denote the wind speed parameters.

The mathematical model of the PV output power is given by Eq. (2).

where P_pv denotes the PV power output, R_pv denotes the PV power output under the standard test, q_pv denotes the derating coefficient of the PV, I_T denotes the actual solar irradiation intensity, I_STC denotes the solar irradiation intensity under the standard test, αp denotes the temperature coefficient of the PV panels, Tc denotes the temperature of the PV panels in the current time interval, and Tstc denotes the temperature of the PV cells under the standard test. PV cell temperature under test.

The diesel generator is a type of power machine that uses diesel fuel and a diesel engine as the prime mover to drive the generator to generate electricity. The entire set is generally composed of a diesel engine, generator, control box, fuel tank, starting and control batteries, protection device, emergency cabinet, and other components. Diesel generators are economical and efficient power generation equipment with a wide range of applications in modern society. The model is given in Eq. (3).

where C_DM(t), C_DF(t), and C_DN(t) denote the operation and maintenance, fuel, and pollution treatment costs of the diesel generator at moment t, respectively. P_DE(t) denotes the power generation of the diesel generator at time t, K_DM denotes the coefficient of operation and maintenance cost of the diesel generator, denotes the amount of release of the k pollutants generated by the operation of the diesel generator, Ck denotes the treatment of the k pollutant cost coefficient, and denotes the coefficient of the diesel generator.

A microturbine (MT) is a gas turbine with a power range of 25–300kW. It is characterized by its small size, light weight, high power density, low noise, low emission, high reliability, and low maintenance cost compared to conventional gas turbines and has broad application prospects in the fields of distributed power generation, standby power generation, cogeneration, and mobile power. The working principle of a micro gas turbine is the same as that of a traditional gas turbine, which converts the thermal energy generated by fuel combustion into mechanical energy and then converts it into electric energy through a generator. It is mainly composed of four core components: compressor, combustion chamber, turbine, and generator. The output power of the micro gas turbine was adjustable and exhibited a high response speed. The model is given in Eq. (4).

where η_MT (t) represents the operating efficiency of the micro-gas turbine, and is the active output power of the micro-gas turbine. The operation and maintenance costs, pollutant treatment costs and fuel costs will be generated during the operation of the micro-gas turbine, and the model is shown in Eq. (5).

where C_MM (t), C_MF (t), and C_MN (t) denote the operation and maintenance, fuel, and pollution treatment costs of the diesel generator at moment t, respectively. P_MT (t) denotes the power generation of the diesel generator at moment t, K_M denotes the operation and maintenance cost coefficient of the diesel generator, K_M denotes the amount of k pollutant releases generated by the operation of the diesel generator, C_k denotes the cost factor for treating k pollutant releases, and C_k denotes the cost factor for treating k pollutants. Cost coefficient of pollutant treatment.

Storage batteries work by using electrochemical reactions to convert chemical energy into electrical energy . When the battery is discharged, the electrochemical reaction causes the active substances inside the battery to change, thereby releasing electrical energy. When the battery was recharged, the electrical energy provided by the external power supply restored the active material inside the battery to its original state. The role of the battery as a reserve energy, in the case of renewable energy, cannot meet the load of the microgrid system for the system energy supply, thereby stabilizing the system power supply and ensuring stable and safe operation of the system. The model is given in Eq. (6).

where SOC (t) indicates the remaining capacity of the battery at moment t and P_bess (t) indicates the charging and discharging power of the battery at time t. When the result is positive, it indicates charging, and when the result is negative, it indicates discharging, which indicates the charging efficiency, which indicates the discharging efficiency.

(1) Power-balance constraint

Power constraint is one of the most important constraints in power system dispatching, which can ensure the safe and stable operation of the power grid and improve its economic operation of the power grid. This requires the sum of the generation power and load power of each node in the grid to be equal. The model is given in Eq. (7).

(2) Diesel generator output constraints:

The diesel generator output constraints refer to the diesel generator set in the operation process, and its output does not exceed a certain limit. The role of diesel generator output constraints is to ensure the safe operation of diesel generator sets, extend the service life of diesel generator sets, and improve their economic efficiency of diesel generator sets. The diesel generator output constraints are an important constraint in the operation of diesel generator sets, and the safe, reliable, and economic operation of diesel generator sets plays a vital role. The model is given in Eq. (8).

(3) The micro gas turbine Output constraints

The output force constraint of a micro gas turbine is an important constraint in the operation process of a micro gas turbine and plays a vital role in the safe, reliable, and economical operation of micro gas turbines. The model is given in Eq. (9).

(4) Transmission power constraints for the contact lines

The transmission power constraint of the contact line refers to the fact that the transmission power of the contact line does not exceed a certain limit under normal operation conditions. The model is given in Eq. (10).

(5) Energy storage device constraints

The energy storage device constraint refers to the fact that during the operation of the energy storage device, parameters such as output power, charging and discharging power, voltage, and current must not exceed certain limit values to ensure the safe operation of the energy storage device. The model is given in Eq. (11).

In Eqs. (7)-(11), , represent the upper and lower limits of the output power of the diesel engine, and , represent the upper and lower limits of the output power of the micro-gas turbine; , represent the upper limit of the climbing power of the diesel generator and the upper limit of the climbing power of the micro-gas turbine; and , represent the upper and lower limits of the power transmitted by the contact line, respectively; , denote the upper and lower limits of the output power of the energy storage device, and the positive value indicates the power input, while the negative value indicates the power output; SOC_max(t) and SOC_min(t) denote the upper and lower limits of the energy storage capacity at the time of t, respectively.

(1) The microgrid operating cost

The mathematical model of the microgrid operating cost is expressed by Eq. (12).

where C_grid(t), C_bess(t), C_MT(t), and C_DE(t) denote the total cost of interaction between the microgrid and the main grid at moment t, the cost of maintenance of the energy storage, the total operating cost of the microgas turbine, and the total operating cost of the diesel generator, respectively; P_bess(t) denotes the power of the energy storage at moment t. P_sell(t) and _Pbuy(t) denote the power sold and purchased at moment t for the microgrid and the large grid, respectively; and c_buy (t) and c_sell (t) denote the price of electricity purchased and the price of electricity sold at moment t for the microgrid and the large grid, respectively.

(2) The microgrid Environmental protection cost

The model of the environmental protection cost of microgrid is given by Eq. (13).

where C_GN(t) denotes the pollutant treatment cost of the large power grid, denotes the release amount of pollutants of type k generated by the operation of the large power grid, and C_k denotes the cost coefficient of treating pollutants of type k.

(3) Objective function of the microgrid optimal scheduling.

The objective function of the microgrid optimal scheduling problem model is to minimize the total cost, including the operating and environmental costs. The model is given in Eq. (14).

The Dwarf Mongoose Optimization (DMO) Algorithm is proposed by Jeffrey O. in 2022 . It is inspired by the group foraging behavior of dwarf mongooses and mainly simulates the foraging, scouting, and babysitting behaviors of dwarf mongooses.

The initialization of the DMO starts with the selection of candidate meerkat individuals that are randomly generated between the upper and lower bounds of the given problem. The population is initialized as shown in Eq. (15):

where X denotes the candidate solution, X_i,j denotes the position of the ith mongoose in the jth dimension, and the mathematical model is given by Eq. (16). N and d denote the population and dimension sizes of the problem, respectively.

where unifrnd is used to generate uniformly distributed random numbers and ub and lb denote the upper and lower bounds of the given problem, respectively. d denotes the dimensional size of the problem.

The foraging routes of the dwarf mongoose were determined by the alpha females produced in the alpha group. The probability that each female individual in the alpha group will become a leader is determined using Eq. (17):

The number of mongooses in the alpha group corresponds to the n-bs. Where bs is the number of babysitters; fit(i) denotes the fitness value of the ith individual, the alpha female’s vocalization that keeps the family within a path is denoted by peep. Each individual in the alpha group forages for food, and the formula for updating the location of the food source is shown in Eq. (18):

where X_{i, j} denote the location of the new food source, X_i denotes the location of the ith individual, and phi is a random number between [-1, 1]; the female alpha keeps the whole group within a path by making a sound, denoted by "peep,” and the value of peep is set to 2 in this paper. x_k denotes the alpha group of X_k denotes the attendant individuals in the alpha group.

Sleeping mounds are the resting places of the dwarf mongoose population and are calculated using the following formula:

The formula for calculating the mean value of a sleep mound is as follows:

Members of the scouting group were responsible for finding the next sleeping mound, as the dwarf mongoose would not return to a place where it had previously slept, which ensured that territories were explored and that scouting and foraging took place simultaneously. This movement was modeled as an overall success or failure assessment for finding a new sleeping mound. In other words, this movement depends on the overall performance of the dwarf mongoose. The rationale is that if the dwarf mongoose colony forages far enough away, it will find a new sleep mound. The formula for updating the position of an individual in a scouting group is as follows.

where C denotes the parameter controlling the mobility of the dwarf meerkat population, which decreases linearly with the number of iterations, as shown in Eq. (22), phi is a random number between [-1, 1], r represents a random number between , 1], M is the direction vector determining the movement of the dwarf meerkat towards the new sleep mound, as shown in Eq. (23) and φ is given by Eq. (20).

where t denotes the number of iterations, and Max_t denotes the maximum number of iterations.

where n denotes the number of members in the scout group, X_i denotes the position of the ith individual, and sm_i denotes the value of the sleep mound.

Babysitters are usually subordinate group members who stay with the pups and rotate periodically so that the alpha female can lead the rest of the group in daily foraging. The alpha female usually returns to the nurse the pups at midday and evening. The number of nannies depends on the size of the population, and they affect the algorithm by reducing the overall population size by a set percentage. This population was modeled by a percentage reduction in the number of nanny groups. The nanny exchange parameter was used to reset the scouting and food source information held by family members. The babysitter fitness weight was set to zero, which ensured that the alpha group's average weight was reduced in the next iteration, implying that the group's movement was impeded, thus emphasizing exploitation.

The Gold-SA (Gold-SA) is proposed by Tanyildizi et al. in 2017, which is inspired by the sine function in mathematics. The advantages of gold-SA are fast convergence, good robustness, and ease of implementation. Gold-SA utilizes the sine function and the unit circle relationship to the unit circle to search for the unit circle. The position update formula is given in Eq. (24).

where r₁ is the determining parameter of the moving distance of the next-generation individual, r₁ is a random number between [0, 2π], r₂ is the parameter that determines the moving direction of the next-generation individual, and the range of the value is [0, π]; and x₁, x₂ are the golden section coefficients, and their mathematical expressions are shown in Eq. (25).

where a = − π, b = π, and t is the golden fraction ratio.

Adaptive t-distribution mutation (ATS) is a mutation operator for optimization problems. It is a mutation operator based on t-distribution, in the iterative process of the algorithm, the parameter of the degrees of freedom of the t-distribution will be adaptively adjusted according to the progress of the algorithm, in the early stage of the algorithm, n is smaller, the t-distribution of the t-distribution of the t-distribution of the tail is longer, with a larger jump ability, which is conducive to the algorithm to jump out of the local optimal solution. In the later stages of the algorithm, n is larger, and the t-distribution has a shorter tail, which has a strong local search ability and is conducive to the algorithm converging to the global optimal solution. The mathematical model is given by Eq. (26).

where t denotes the current iteration number, T denotes the maximum iteration number, and trnd() denotes the t-distribution.

The Lévy flight is a heuristic strategy for solving optimization problems . This method adopts a Lévy distribution to realize stochastic search, integrates short-range search and long-range exploration, effectively enhances the global search capability of the algorithm, and improves the efficiency of jumping out of the local optimum. The model is given in Eq. (27).

where d denotes the dimensions of the problem.

After the introduction of the adaptive t-distribution variation strategy and Levy flight strategy, the position update method for the alpha group is given by Eq. (28).

After the introduction of the adaptive t-distribution variation strategy and Levy flight strategy, the position update method of the reconnaissance group is given by Eq. (29) and Eq. (30).

The golden sine strategy can be used to solve various continuous, discrete, and multi-objective optimization problems. It can effectively find the optimal or near-optimal solution to a problem. It exhibits the characteristics of fast convergence and good robustness. The dwarf Mongoose optimization algorithm was combined with the golden sine strategy, and the performance of the algorithm was improved by combining adaptive T-distribution variation and Levy flight. Table 1 and Figure 1 show the flow chart of the EDMO algorithm.

The experimental parameters set are listed in the following Table 2. In addition, this experiment uses MATLAB2021b, and a Windows 10 64-bit operating system PC with an Intel Core i7-9700 processor @3.00GHz and 16.0 GB RAM served as the experimental platform. The proposed DMOWOA parameter settings: n=50, bs=3, Max_t=1500, t=0, N=20; L=8. Cited references related to all algorithm parameters compared.

To verify the performance of EDMO in solving the microgrid optimal scheduling problem, EDMO was compared with the ABC , AOA , ChOA , GWO , MPA , PSO , SA , SWO , and WOA . For fairness of the comparison, the maximum number of iterations for each algorithm was 15000 and 20 independent runs. Four indicators were selected as statistical criteria: minimum (Min), maximum (Max), mean (Mean), and standard deviation (SD). Wilkerson's signed rank test was used for statistical analysis, with the symbol "+" indicating better than the comparison algorithm, the symbol "-" indicating worse than the comparison algorithm, and the symbol "≈" indicates the comparison algorithm.

Table 6 shows the statistics of the results of 20 independent runs of each algorithm, from Table 5.6, it shows that EDMO is ranked first in the minimum value, followed by SA, SWO, PSO, DMO, MPA, ABC, GWO, WOA, ChOA, and the worst one is AOA. EDMO is ranked first in terms of the maximum value, mean value, and standard deviation. This verifies that EDMO outperformed the compared algorithms.

The results of Wilkerson's signed-rank test are shown in Table 7, from which it can be seen that EDMO is statistically superior to DMO, ABC, AOA, ChOA, GWO, MPA, PSO, SA, SWO, and WOA. Therefore, it can be statistically verified that EDMO is an effective method for solving the optimal dispatch problem of microgrid.

Figure 2 shows the box-line plots of each algorithm, from which it can be seen that the data fluctuation of EDMO for 20 independent runs is very small, indicating that the algorithm is stable. This is followed by SA, DMO, and PSO. The data fluctuations of AOA, ChOA, and WOA were very large, indicating weak stability. Thus, the stability and robustness of EDMO were verified.

Figure 5 shows the convergence graphs of each algorithm, from which it can be seen that EDMO obtains the minimum cost and converges faster. Figure 3 shows the scheduling results of each unit for EDMO with the minimum total cost, and Figure 4 shows the predicted values of PV, wind, and load power for EDMO with the minimum total cost. Figure 6 shows the unit power scheduling for each algorithm in the case of total cost minimization. Figure 7 shows the fitted curve plot of the predicted values of each algorithm, from which it can be seen that EDMO and SA have the best fit between the predicted and actual curves, and the predictive ability of EDMO basically matches the actual power generation, which verifies that EDMO has good predictive ability.

For the microgrid optimal scheduling problem, this paper an EDMO is proposed. In the EDMO, the golden sinusoidal strategy is introduced, which utilizes the relationship between the sinusoidal function and the unit circle, searches the unit circle, and narrows the search space through the golden ratio to approximate the optimal solution of the algorithm. In addition, an adaptive t-distribution variation strategy was introduced to improve the population diversity in the late stage of the algorithm, and the Lévy flight strategy was introduced to enhance the ability of the algorithm to jump out of the local optimum. The EDMO was applied to the microgrid optimal scheduling problem. The experimental results show that the EDMO yields the least costly economic scheduling scheme compared to other well-known algorithms, it can accurately predict the generation power of each unit. According to the Wilcoxon signed-rank test, EDMO exhibits good robustness and excellent optimization-finding ability. Therefore, the validity and superior performance of EDMO for solving the microgrid optimal scheduling problem is verified, and it is an effective alternative method for solving this problem. In future work, other strategies will be to improve the performance of the proposed algorithm to solve problems in other energy fields.

This study was supported by the National Natural Science Foundation of China under Grant No. U20464.

Weiping Meng wrote the original manuscript. Shijian Chen Algorithm Design. Yongquan Zhou Supervision, revised. All authors have read and approved the final manuscript.

The authors were unable to specify which data were used.

The authors declare that they have no known competing financial interests or personal relationships that could have influenced the work reported in this study.