Optimal Scheduling of Microgrids Based on an Improved Dung Beetle Optimization Algorithm

Abstract

More distributed energy resources are being integrated into microgrid systems, making scheduling more complex and challenging. In order to achieve the utilization of renewable energy and peak load shifting on a microgrid system, an optimal scheduling model is established. Firstly, a microgrid operation model including a photovoltaic array, wind turbine, micro gas turbine, diesel generator, energy storage, and grid connection is constructed, considering the demand response and the uncertainty of wind and solar power. The modeling demand response is determined via a price–demand elasticity matrix, whereas the uncertainty of wind and solar power is established using Monte Carlo sampling and a K-means clustering algorithm. Secondly, a multi-objective function that includes operational and environmental treatment costs is constructed. To optimize the objective function, an Improved Dung Beetle Optimization algorithm (IDBO) is proposed. A tent mapping, non-dominated sorting, and reverse elite learning strategy is proposed to improve the Dung Beetle Optimization algorithm (DBO); therefore, the IDBO is developed. Finally, the proposed model and algorithm are validated through some simulation experiments. A benchmark function test proves that IDBO has a fast convergence speed and high accuracy. The microgrid system scheduled by IDBO has the lowest total cost, and its ability to achieve peak load shifting and improve the utilization of renewable energy is proved through tests involving different scenarios. The results show that compared with traditional optimal scheduling models and algorithms, this approach is more reliable and cost-effective.

1. Introduction

The energy demand is rising, and fossil resources like coal, diesel, and natural gas are still important for both production and daily life [1,2]. However, excessive use of fossil resources is causing an increasing environmental problem, which threatens natural ecosystems [3]. As a result, renewable energy (RE) sources like wind and solar power are highly anticipated. The microgrid (MG) is seen as a system that allows fossil and REs to work in a coordinated way.

MG is a small power-generation and distribution system composed of distributed generations (DGs), Battery Energy Storage Systems (BESSs), and loads. Fossil-energy-generation units, such as diesel generators (DEs) and micro gas turbines (MTs), together with RE-generation units, such as wind turbines (WTs) and photovoltaic panels (PVs), collectively form the DGs. They operate collaboratively to provide a reliable power supply [4]. MG was proposed to achieve flexible and efficient use of distributed energy sources and to address the challenge of integrating DGs into the grid. The development and promotion of MG can greatly advance the integration of fossil and RE, ensuring a highly reliable energy supply to loads in multiple forms. This is an effective way to transition from traditional grids to smart grids. The scale of distributed RE generation is expanding, and its role in MG is becoming prominent.

However, RE generation depends on real-time weather conditions, making it unpredictable. Directly integrating RE will lead to a negative impact on the safety and stability of MG. Fossil energy generation also faces uncertainties, such as fluctuating operational costs and environmental treatment costs, which may impact grid interaction [5,6]. As a result, the multiple power sources involved mean that managing and optimizing the scheduling of microgrids has become a complex problem involving many constraints. In recent years, the demand response (DR) and the uncertainty of wind and solar power have been proposed to describe issues resulting from unpredictability in MG scheduling. But this will make the MG operation more complex to schedule. Finding efficient ways to resolve these problems is a challenge [7]. Bio-inspired optimization algorithms have gained attention for their ability to tackle microgrid optimization issues. Common algorithms, such as particle swarm optimization (PSO) [8,9], genetic algorithm (GA) [10,11], and grey wolf optimizer (GWO) [12], have been shown to be feasible in MG optimal scheduling.

Researchers have explored various bio-inspired optimization algorithms for MG optimal scheduling. Peng et al. [13] introduced a model for analyzing the cost-effectiveness of MG operation using a Distributed Economic Model Predictive Controller (DEMPC). This method is effective; however, the solving process requires each subsystem to obtain the optimal solutions of other subsystems, making the process complex. Nguyen et al. [14] proposed an optimization model for an MG with PV, WT, MT, DE, and BESS, aiming to reduce operating costs using Improved Archimedes Optimization (IAO). While it performs well in global searches, it can sometimes encounter local optima and may struggle with multi-objective problems. Yang et al. [15] used the Analytical Target Cascading (ATC) method but offered limited discussion on RE sources like PVs and WTs; as such, their model is overly restrictive for solving a complex MG system that contains diverse DG units. Ghodusinejad et al. [16] presented an MG model using the Pareto Envelope-based Selection Algorithm II (PESA-II) for optimal scheduling. This algorithm only checks undefeated members and its parents; the process is simple. But fossil energy sources are not sufficiently addressed. Other researchers, like Hu et al. [17], also focused on optimal scheduling, considering fossil energy and using the Amended Multiverse Optimizer Algorithm (AMVOA), considering the uncertainty of wind and solar power. However, the optimization process of AMVOA includes two stages. The running speed is slow, which has a negative impact on the overall optimization efficiency. Bolurian et al. [18] introduced a model that optimizes costs, emissions, and power demand using a column and constraint generation algorithm (C and CG). But this model only involved WT and BESS. Lou et al. [19] created a hybrid MG model with solar thermal and photovoltaic systems, using Non-dominated Sorting Genetic Algorithm II (NSGA-II), but this model applies only to specific cases like scenarios involving thermal energy. Bo et al. [20] focused on a very innovative scenario involving RE and hydrogen, solving their model with distributionally robust optimization (DRO). He et al. [21] used Multi-Objective War Strategy Optimization (MOWSO) for MG optimal scheduling, focusing on operational and environmental costs. A very new algorithm was used, which has a strong parallelism ability and a simple process. But there is a lack of discussion on fossil fuels. Other studies such as that by Wang et al. [22], who used an Improved Grey Wolf Optimization (I-GWO) for an MG with PV, WT, MT, DE, and BESS. Although I-GWO worked well for generating the initial population, it was prone to becoming stuck in local optima. Meng et al. [23] applied the Multi-Objective Sparrow Search Algorithm (MOSSA) to optimize operational and environmental costs, considering the DR of MG. MOSSA uses fused sinusoidal search strategy to improve searching ability. Although the performance has been improved, trigonometry will make the calculation process more complex. Qiu et al. [24] used data-driven robust optimization (RO) to develop a comprehensive system, but they did not account for environmental costs. Lin et al. [25] used Lagrange Multipliers (LMs), though their comparison lacked sufficient experiments in different scenarios. Zhang et al. [26] proposed a model using the Improved Bacterial Foraging Optimization Algorithm (I-BFO) but faced challenges with local optima, considering the uncertainty of wind and solar power. Finally, Zhang et al. [27] applied the Improved Butterfly Optimization Algorithm (I-BOA). The I-BOA showed faster convergence speed, though improvements could be made in generating the initial population. Chen et al. [28] introduced a probability distribution and employed Monte Carlo sampling to generate a large number of scenarios. The number of scenarios was then reduced through scenario reduction techniques, comprehensively modeling the uncertainty of RE. Nammouchi et al. [29] accomplished energy management in MG by progressively establishing deterministic optimization models and robust optimization models, addressing the uncertainties in RE units and user consumption. They innovatively proposed the concepts of “good deviation” and “bad deviation” to enhance the flexibility of the model. Rui et al. [30] managed the energy scheduling of MG by establishing a clear two-stage optimization model. Utilizing a rolling optimization approach, they updated the optimization decisions at each time step based on the latest forecast information, thereby effectively addressing the uncertainties in PV generation and load. By applying Stackelberg game theory, they simulated the interactive relationship between the Energy Management Operator (EMO) and the MG, ensuring that both parties could identify optimal strategies within the game. Vidan et al. [31] effectively addressed the uncertainties in the bidding strategies of thermal generators and battery storage by establishing two robust models. They thoroughly investigated the impact of robustness parameters on the models and rigorously introduced mathematical formulations. A summary of the MG literature review is presented in

Table 1. Summary of MG literature review.

Source	Renewable Energy	Fossil Energy	Energy Storage	DR and Uncertainty	Objective Function	Strategy
[13]	PV, WT	DE	Yes	No	Operating	DEMPC
[14]	PV, WT	MT, DE	Yes	No	Operating	IAO
[15]	No	MT, DE	Yes	No	Operating	ATC
[16]	PV, WT	No	Yes	No	Net present	PESA-II
[17]	PV, WT	DE	Yes	Uncertainty	Net present	AMVOA
[18]	WT	No	Yes	Uncertainty	Operating, environmental	C and CG
[19]	PV	Thermal	No	No	Operating, environmental	NSGA-II
[20]	Hydrogen, PV	No	Yes	No	Operating	DRO
[21]	PV, WT	No	Yes	No	Operating, environmental	MOWSO
[22]	PV, WT	MT, DE	Yes	No	Operating, environmental	IGWO
[23]	PV, WT	DE	Yes	DR	Operating, environmental	SSA
[24]	PV, WT	DE	Yes	No	Operating	RO
[25]	PV, WT	MT, DE	Yes	No	Operating	LM
[26]	PV, WT	MT, DE	Yes	Uncertainty	Operating, environmental	IBFO
[27]	PV, WT	MT, DE	Yes	No	Operating	IBOA
[28]	PV, WT	No	Yes	DR, Uncertainty	Operating, user satisfaction	CPLEX
[29]	PV	No	Yes	Uncertainty	Energy management	Gurobi
[30]	PV	No	Yes	No	Energy trading, profits	MIP
[31]	No	Thermal	Yes	Uncertainty	Profits	RO

.

The Dung Beetle Optimizer (DBO) is a new swarm bio-inspired algorithm introduced in 2022 [32]. It mimics the behavior of dung beetles, selecting random candidate solutions and simulating the best path for delivering food. The algorithm divides the beetles into four groups, each using different search strategies. By adding complexity to the search process, DBO reduces the impact of environmental changes on finding the best solution. Although DBO is relatively new, it has already been applied in many studies. Chen et al. [33] used DBO for 3D path planning of drones, initializing the population with a chaotic strategy. Wei et al. [34] utilized DBO to optimize the dimensions of ship hulls, achieving certain results by combining the algorithm with a random forest surrogate model. Alnafisah et al. [35] established a financial futures prediction model, integrating deep learning with DBO. It is evident that DBO algorithm exhibits high versatility, showing good performance in path planning, industrial production and finance, etc. However, there is currently a significant gap in applying DBO for optimal scheduling in MG.

In this study, we aim to minimize total costs, including operational costs and environmental costs, by modeling an MG system that includes PV, WT, DE, MT, BESS, and grid interactions. To flexibly adjust the load and improve system efficiency, DR, as well as the uncertainty of wind and solar power, is factored into MG. To enhance the DBO algorithm’s robustness in addressing initial population randomness and optimize its global search performance, three innovative strategies are introduced: Tent chaotic mapping initialization, non-dominated sorting, and elite reverse learning.

The main contributions and novelties of this study are summarized as follows:

• A more comprehensive microgrid scheduling model. In response to the shortcomings of existing research in terms of system efficiency, power supply reliability, and economic optimization, an integrated optimal scheduling model is constructed, considering DR and the uncertainties of wind and solar power. Under the time-of-use pricing mechanism, DR is divided into two categories for modeling: reducible load and transferable load. Additionally, the Monte Carlo scenario sampling combined with the K-means clustering algorithm is proposed to simulate the uncertainties of wind and solar power output. This approach effectively addresses the issue of insufficient consideration of uncertainty factors in traditional optimization models.
• Based on Improved Dung Beetle Optimization algorithm (IDBO). An improved DBO is developed to address the limitations of traditional optimization algorithms in terms of convergence speed, solution accuracy, and the ability to escape local optima. Tent chaotic mapping initialization, non-dominated sorting, and elite reverse learning strategy are introduced to improve the performance of DBO. These enhancements significantly improve the algorithm’s global search capability and convergence performance, providing a more effective tool for solving complex microgrid optimal scheduling issues.
• The proposed model and algorithm are verified from multiple scenarios. Compared to existing methods, simulation experiments for various operational scenarios are designed. Through comparative analysis, the feasibility and superiority of the proposed model and the IDBO are validated. By comparing some indicators, the feasibility of the proposed model and algorithm is verified, providing a new idea for improving the reliability and economy of microgrid system.

2. Microgrid Operation Structure

The objective of this study is to optimize the operation of the microgrid to enhance the utilization of RE, reduce total costs, and ensure the stability and reliability of the MG. To achieve this, it is essential to model the various components of the MG and propose an optimization scheduling strategy. The MG consists of several key components, and the behavior of these components has a direct impact on the overall optimization of the microgrid. The MG system includes DGs such as PV, WT, DE, MT, BESS, grid interactions, and the loads, as shown in Figure 1. The power output of each DG source can be expressed as P_PV, P_WT, P_MT, P_DE, P_ES, and P_grid, where P_PV is the output power of the PV, P_WT is the output power of the WT, P_MT is the output power of the MT, P_ES is the output power of the DE, P_ES is the output power of the BESS, and P_grid is the output power of the grid connection.

2.1. Model of Distributed Generations

2.1.1. Photovoltaic Array

As one of the main sources of RE, solar power can harness sunlight in real time and convert it into electricity for load use. The model for the output power of a PV is shown as follows:

$$P_{\mathrm{P}\mathrm{V}}=P_m{\displaystyle \frac{I}{I_{STC}}}\left[1+\alpha \left(T-T_M\right)\right]$$

(1)

where P_PV is the current output power of the PV, P_m is the output power of the PV under standard test conditions, I is the current solar irradiance, I_STC is the solar irradiance under standard test conditions, α is the temperature coefficient of the PV, T is the current temperature of the PV, and T_M is the temperature of the PV under standard test conditions.

2.1.2. Wind Turbine

Wind energy is another main source of RE. The mathematical model for the output power of a WT unit is shown as follows:

$$P_{\mathrm{W}\mathrm{T}}=\left\{\begin{array}{cc}\hfill 0,& v<v_{ci}\hfill \\ \hfill a^{\prime }{v}^3+b^{\prime }{v}^2+c^{\prime }v+d^{\prime },& v_{ci}\le v<v_m\hfill \\ \hfill P_r,& v_m<v<v_{co}\hfill \\ \hfill 0,& v\ge v_{co}\hfill \end{array}\right.$$

(2)

where P_WT is the current output power of the WT, P_r is the rated power of the WT, v is the current wind speed, v_ci is the cut-in wind speed, v_m is the rated wind speed, v_CO is the cut-out wind speed, and a′, b′, c′, and d′ are wind speed parameters.

2.1.3. Micro Gas Turbine

MTs are small, efficient distributed generation units, typically with a single-unit power output ranging from 25 kW to 500 kW. The operational efficiency of an MT is shown as follows:

$${\eta }_{\mathrm{M}\mathrm{T}}={0.0753\left[{\displaystyle \frac{P_{\mathrm{M}\mathrm{T}}\left(t\right)}{65}}\right]}^3-{0.3095\left[{\displaystyle \frac{P_{\mathrm{M}\mathrm{T}}\left(t\right)}{65}}\right]}^2+0.4174\left[{\displaystyle \frac{P_{\mathrm{M}\mathrm{T}}\left(t\right)}{65}}\right]+0.1068$$

(3)

where ${\eta }_{\mathrm{M}\mathrm{T}}$ is the operational efficiency of the MT, and P_MT(t) is the active output power of the MT.

Additionally, three components are included in the cost model of an MT: the environmental treatment cost model, the fuel cost model, and the maintenance cost model.

$$C_{\mathrm{M}\mathrm{T}}=C_{\mathrm{M}\mathrm{T}o}+C_{\mathrm{M}\mathrm{T}m}+C_{\mathrm{M}\mathrm{T}P}$$

(4)

where C_MT is the total operational cost of the MT, C_MTO is the fuel cost, C_MTm is the maintenance cost, and C_MTP is the environmental treatment cost.

2.1.4. Diesel Generator

Compared to other DGs, DEs have a faster startup speed, making them suitable as auxiliary power sources to enhance the stability of MG. Three components are included in the cost model of a DE: the environmental treatment cost model, the fuel cost model, and the maintenance cost model.

$$C_{\mathrm{D}\mathrm{E}}=C_{\mathrm{D}\mathrm{E}o}+C_{\mathrm{D}\mathrm{E}m}+C_{\mathrm{D}\mathrm{E}P}$$

(5)

where C_DE represents the total operational cost of the DE, C_DEO is the fuel cost, C_DEm is the maintenance cost, and C_DEP is the environmental treatment cost.

2.1.5. Energy Storage

Energy storage maintains the balance between supply and demand.

$$SOC\left(t\right)=\left\{\begin{array}{c}SOC\left(t-1\right)+{\displaystyle \frac{1}{{\eta }^{-}}}{P}_{\mathrm{E}\mathrm{S}}\left(t\right), P_{\mathrm{E}\mathrm{S}}\left(t\right)\le 0\\ SOC\left(t-1\right)+{\eta }^{+}{P}_{\mathrm{E}\mathrm{S}}\left(t\right), P_{\mathrm{E}\mathrm{S}}\left(t\right)>0\end{array}\right.$$

(6)

where SOC(t) is the state of charge of the battery at time t, and P_ES(t) is the charging or discharging power of the battery at time t.

2.2. Microgrid Demand Response Model

DR refers to the regulation and optimization of the electricity consumption behavior of users through some measures, thereby optimizing the load curve, enhancing the MG’s ability to absorb wind and solar power, and achieving the peak load shifting. DR is based on the price–demand elasticity matrix E.

$$E=\left[\begin{array}{cccc}{e}_{11}& e_{12}& \cdots & e_{1n}\\ e_{21}& e_{22}& \cdots & e_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ e_{n1}& e_{n2}& \cdots & e_{nn}\end{array}\right]$$

(7)

$$e_{ij}={\displaystyle \frac{\raisebox{1ex}{$∆P_L^i$}\!\left/ \!\raisebox{-1ex}{$P_L^i$}\right.}{\raisebox{1ex}{$∆p_j$}\!\left/ \!\raisebox{-1ex}{$p_{j0}$}\right.}}$$

(8)

where e_ij is the elasticity coefficient of the i-th row and the j-th column, indicating the elasticity coefficient of the load at time i corresponding to the price at time j. $∆P_L^i$ is the load variation at time i after DR, $P_L^i$ is the initial load at time i, Δp_j is the price variation at time j after DR, and p_j0 is the initial price at time j.

Since different types of loads show varying sensitivities to the same price changes, DR is further divided into reducible loads and transferable loads. Reducible loads are adjusted by comparing price changes before and after DR to decide whether to reduce the load. Transferable loads are freely adjusted by users based on the price after DR.

$$∆P_L^i=P_L^i\left[{\sum }_{j=1}^{24}E\left(i,j\right){\displaystyle \frac{p_j-p_{j0}}{p_{j0}}}\right]$$

(9)

where $∆P_L^i$ is the initial shiftable or curtailable load at time i, and p_j represents the electricity price at time j.

3. Microgrid Multi-Objective Optimization Model

3.1. Objective Function

The scheduling model’s objective function is formulated to minimize the total system cost, encompassing both operational expenditures and environmental mitigation expenses. The objective function is defined as follows:

$$F=C_1+C_2$$

(10)

where F represents the total cost of the MG, C₁ is the operational cost, and C₂ is the environmental treatment cost.

3.1.1. Operational Cost

The expression for operational cost is as follows:

$$C_1=C_{\mathrm{M}\mathrm{T}om}+C_{\mathrm{D}\mathrm{E}om}+a\times C_{\mathrm{g}\mathrm{r}\mathrm{i}\mathrm{d}}+C_{\mathrm{P}\mathrm{V}om}+C_{\mathrm{W}\mathrm{T}om}+C_{\mathrm{E}\mathrm{S}}$$

(11)

where C_MTom is the operation and maintenance cost of the MT; C_DEom is the operation and maintenance cost of the DE; C_grid is the cost of interacting with the grid; a is the grid connection coefficient, which can be either 0 or 1 (1 for grid-connected, 0 for isolation); C_PVom is the operation and maintenance cost of PVs; C_WTom is the operation and maintenance cost of WTs; and C_ES is the cost of BESS.

The operation and maintenance cost of the MT C_MTom is divided into fuel cost C_MTo and maintenance cost C_MTm.

$$C_{\mathrm{M}\mathrm{T}om}=C_{\mathrm{M}\mathrm{T}o}+C_{\mathrm{M}\mathrm{T}m}$$

(12)

$$\left\{\begin{array}{c}{C}_{\mathrm{M}\mathrm{T}o}=C{\displaystyle \frac{∆t}{LHV}}{\displaystyle \sum _{t=1}^T}{\displaystyle \frac{P_{\mathrm{M}\mathrm{T}}\left(t\right)}{{\eta }_{\mathrm{M}\mathrm{T}}}}\hfill \\ C_{\mathrm{M}\mathrm{T}m}={\sum }_{t=1}^T{K}_{\mathrm{M}\mathrm{T}}{P}_{\mathrm{M}\mathrm{T}}\left(t\right)\hfill \end{array}\right.$$

(13)

where P_MT(t) is the output power of the MT, ${\eta }_{\mathrm{M}\mathrm{T}}$ is the efficiency of the DE, LHV is the lower heating value, C is the fixed natural gas price (2.5 CNY/m³), and K_MT is the maintenance coefficient of the MT.

The operation and maintenance cost of the DE is as follows:

$$C_{\mathrm{D}\mathrm{E}om}=C_{\mathrm{D}\mathrm{E}o}+C_{\mathrm{D}\mathrm{E}m}$$

(14)

$$\left\{\begin{array}{c}{C}_{\mathrm{D}\mathrm{E}o}=c_1{P}_{\mathrm{D}\mathrm{E}}^2+c_2{P}_{\mathrm{D}\mathrm{E}}+c_3\\ C_{\mathrm{D}\mathrm{E}m}={\sum }_{t=1}^T{K}_{\mathrm{D}\mathrm{E}}{P}_{\mathrm{D}\mathrm{E}}\left(t\right)\end{array}\right.$$

(15)

where P_DE is the output power of the DE, and c₁, c₂, c₃ are the combustion coefficients, whereas K_DE is the maintenance coefficient of the DE.

When the grid connection coefficient a = 1, the cost function for energy interaction with the grid is as follows:

$$C_{\mathrm{g}\mathrm{r}\mathrm{i}\mathrm{d}}={\sum }_{t=1}^T{{(P}_{\mathrm{g}\mathrm{r}\mathrm{i}\mathrm{d}}\left(t\right)G}_{\mathrm{g}\mathrm{r}\mathrm{i}\mathrm{d}}\left(t\right)\times \mathsf{\Delta }t)$$

(16)

where P_grid(t) is the power exchanged with the grid, and G_grid(t) represents the current purchase and sale price of electricity.

Since solar and wind are clean energy sources, PVs and WTs only include maintenance costs C_PVom and C_WTom.

$$\left\{\begin{array}{c}{C}_{\mathrm{P}\mathrm{V}om}={\sum }_{t=1}^T{K}_{\mathrm{P}\mathrm{V}m}{P}_{\mathrm{P}\mathrm{V}}\left(t\right)\\ C_{\mathrm{W}\mathrm{T}om}={\sum }_{t=1}^T{K}_{\mathrm{W}\mathrm{T}m}{P}_{\mathrm{W}\mathrm{T}}\left(t\right)\end{array}\right.$$

(17)

where K_PVm is the maintenance coefficient for PVs, and K_WTm is the maintenance coefficient for WTs.

The cost model for the energy storage system C_ES is represented as follows:

$$C_{\mathrm{E}\mathrm{S}}={\sum }_{t=1}^T{C}_p\left|P_{\mathrm{E}\mathrm{S}}\right|∆t$$

(18)

$$\left\{\begin{array}{c}{C}_p={\displaystyle \frac{G_{\mathrm{E}\mathrm{S}}{K}_{\mathrm{E}\mathrm{S}}}{t_{\mathrm{E}\mathrm{S}}}}\\ G_{\mathrm{E}\mathrm{S}}=\delta {\displaystyle \frac{{\left(1+\delta \right)}^n}{{\left(1+\delta \right)}^n-1}}\end{array}\right.$$

(19)

where P_ES is the output power of the BESS (positive during charging, negative during discharging), C_p is the present value of total investment costs, G_ES is the total investment cost of the energy storage system, t_ES is the annual operating time of the energy storage system, K_ES is the investment recovery coefficient, $\delta$ is the depreciation rate of the storage unit, and n is the designed life span.

3.1.2. Environmental Treatment Cost

The expression for environmental treatment cost is as follows:

$$C_2=C_{\mathrm{M}\mathrm{T}P}+C_{\mathrm{D}\mathrm{E}P}$$

(20)

$$\left\{\begin{array}{c}{C}_{\mathrm{M}\mathrm{T}P}={\sum }_{y=1}^M{D}_y{H}_y{P}_{\mathrm{M}\mathrm{T}}\left(t\right)∆t\\ C_{\mathrm{D}\mathrm{E}P}={\sum }_{y=1}^M{D}_y{H}_y{P}_{\mathrm{D}\mathrm{E}}\left(t\right)∆t\end{array}\right.$$

(21)

where C_MTP is the environmental treatment cost for the MT, C_DEP is the environmental treatment cost for the DE, P_MT(t) is the output power of the MT, P_DE(t) is the output power of the DE, and D_yH_y is the cost coefficient for treating major pollutants.

3.2. Constraints

3.2.1. Power Balance Constraints

$$P_L+P_{\mathrm{E}\mathrm{S}}=P_{\mathrm{M}\mathrm{T}}+P_{\mathrm{D}\mathrm{E}}+P_{\mathrm{g}\mathrm{r}\mathrm{i}\mathrm{d}}+P_{\mathrm{P}\mathrm{V}}+P_{\mathrm{W}\mathrm{T}}$$

(22)

where P_L is the daily load of the MG.

3.2.2. Output Constraints

$$\left\{\begin{array}{c}{P}_{\mathrm{M}\mathrm{T}}^{min}\left(t\right)\le P_{\mathrm{M}\mathrm{T}}\left(t\right)\le P_{\mathrm{M}\mathrm{T}}^{max}\\ P_{\mathrm{D}\mathrm{E}}^{min}\left(t\right)\le P_{\mathrm{D}\mathrm{E}}\left(t\right)\le P_{\mathrm{D}\mathrm{E}}^{max}\left(t\right)\\ P_{\mathrm{W}\mathrm{T}}^{min}\left(t\right)\le P_{\mathrm{W}\mathrm{T}}\left(t\right)\le P_{\mathrm{W}\mathrm{T}}^{max}\left(t\right)\\ P_{\mathrm{P}\mathrm{V}}^{min}\left(t\right)\le P_{\mathrm{P}\mathrm{V}}\left(t\right)\le P_{\mathrm{P}\mathrm{V}}^{max}\left(t\right)\\ P_{\mathrm{E}\mathrm{S}}^{min}\left(t\right)\le P_{\mathrm{E}\mathrm{S}}\left(t\right)\le P_{\mathrm{E}\mathrm{S}}^{max}\left(t\right)\end{array}\right.$$

(23)

where $P_i^{min}$ and $P_i^{max}$ are the upper and lower limits of power for each DG.

3.2.3. Climbing Constraints

$$\left\{\begin{array}{c}\left|P_{\mathrm{M}\mathrm{T}}\left(t\right)-P_{\mathrm{M}\mathrm{T}}\left(t-1\right)\right|\le r_{\mathrm{M}\mathrm{T}}\\ \left|P_{\mathrm{D}\mathrm{E}}\left(t\right)-P_{\mathrm{D}\mathrm{E}}\left(t-1\right)\right|\le r_{\mathrm{D}\mathrm{E}}\end{array}\right.$$

(24)

where r_MT is the upper limit of the ramp rate for the MT, and r_DE is the upper limit for DEs.

3.2.4. Grid Constraints

$$P_{\mathrm{g}\mathrm{r}\mathrm{i}\mathrm{d}}^{min}\left(t\right)\le P_{Grid}\left(t\right)\le P_{\mathrm{g}\mathrm{r}\mathrm{i}\mathrm{d}}^{max}\left(t\right)$$

(25)

The constraints apply only when the grid connection parameter a = 1.

3.2.5. Battery Energy Storage System Constraints

$${SOC}_{\mathrm{E}\mathrm{S}}^{min}\left(t\right)\le SOC\left(t\right)\le {SOC}_{\mathrm{E}\mathrm{S}}^{max}\left(t\right)$$

(26)

where ${SOC}_{\mathrm{E}\mathrm{S}}^{min}$ and ${SOC}_{\mathrm{E}\mathrm{S}}^{max}$ are the lower and upper limits of the storage capacity.

3.3. Expression of Optimization Model

Therefore, the optimization problem of the MG can be formulated as follows: decision variable X = {P_DG,i} subject to constraint $Y=\{P_L+P_{\mathrm{E}\mathrm{S}}=P_{\mathrm{M}\mathrm{T}}+P_{\mathrm{D}\mathrm{E}}+P_{\mathrm{g}\mathrm{r}\mathrm{i}\mathrm{d}}+P_{\mathrm{P}\mathrm{V}}+P_{\mathrm{W}\mathrm{T}},P_{DG,i}^{min}\left(t\right)\le P_{DG,i}\le P_{DG,i}^{max}\left(t\right), \left|P_{\mathrm{M}\mathrm{T},\mathrm{D}\mathrm{E}}\left(t\right)-P_{\mathrm{M}\mathrm{T},\mathrm{D}\mathrm{E}}\left(t-1\right)\right|\le r_{\mathrm{M}\mathrm{T},\mathrm{D}\mathrm{E}}, {SOC}_{\mathrm{E}\mathrm{S}}^{min}\left(t\right)\le SOC\left(t\right)\le {SOC}_{\mathrm{E}\mathrm{S}}^{max}\left(t\right)\}$, with the objective of minimizing the total operational cost Z = {F_min}.

3.4. Uncertainty of Wind and Solar Power

Wind power and solar power are affected by various factors such as weather conditions and equipment maintenance, making wind speed and solar irradiance inherently random and uncertain. This results in the highly random nature of the output power from both WT and PV systems, thereby compromising the reliability of MG optimal scheduling.

3.4.1. Monte Carlo Sampling

Based on the wind speed and solar irradiance data, daily distribution curves with equal probabilities can be obtained. These curves accurately characterize the uncertainty of wind and solar power. Using the wind speed and solar irradiance data from a specific day at the School of Electrical and Information Engineering, Beijing University of Civil Engineering and Architecture, Monte Carlo sampling was performed. Figure 2 shows 1000 wind and solar output scenarios generated using Monte Carlo sampling, where different colors show the different scenarios.

3.4.2. Scenario Reduction

Monte Carlo sampling can generate a large number of WT and PV time-series scenarios with equal probabilities. Whereas this effectively shows the uncertainty of wind and solar power, the large number of sample scenarios significantly increases the computational complexity of algorithms and reduces their efficiency. Therefore, scenario reduction methods are required to select a small number of representative scenarios and fit their occurrence probabilities. The proposed approach effectively addresses the inherent uncertainties associated with wind and solar power generation while simultaneously improving the computational efficiency of the algorithm.

Common reduction techniques include PCA and DBSCAN, but they are not capable of addressing the uncertainty of wind and solar power in MGs. Firstly, PCA is a dimensionality reduction technique primarily used for data compression and feature extraction, rather than directly performing cluster analysis. While PCA can help simplify data, it cannot directly handle clustering problems in wind and solar power. Secondly, although DBSCAN excels at handling noise and clusters of complex shapes, in the analysis of MGs, the data typically have less noise, and the clusters are relatively regular, making DBSCAN’s advantages less pronounced. In contrast, the backward scenario reduction and the K-means clustering algorithm are frequently used to address wind and solar scenario reduction in MGs. The backward scenario reduction method eliminates scenarios with minimal impact on the overall characteristics step by step until the desired number of scenarios is reached. This method is intuitive and simple to implement. The K-means clustering algorithm employs iterative optimization to automatically classify and aggregate similar scenarios, efficiently reducing a vast set of scenarios into a compact collection of representative ones. It is computationally efficient and more suitable for handling large-scale data.

The number of clusters is important in scenario reduction algorithms. Too many clusters will reduce computational efficiency, whereas too few will fail to represent the uncertainty of wind and solar power. Based on an analysis of different cluster numbers, 1000 wind and solar output scenarios generated through Monte Carlo sampling are reduced to five scenarios using both methods, allowing for a comprehensive comparison of their advantages and disadvantages, as shown in Figure 3 and Figure 4. The probabilities of the five scenarios after reduction are shown in

Table 2. The probabilities of the five scenarios after reduction.

Scenario	Backward Reduction		K-means Clustering Reduction
	WT	PV	WT	PV
Scenario 1	0.025	0.162	0.229	0.225
Scenario 2	0.041	0.077	0.179	0.192
Scenario 3	0.125	0.1	0.161	0.202
Scenario 4	0.134	0.041	0.209	0.175
Scenario 5	0.675	0.62	0.204	0.206

.

From Figure 3 and Figure 4, it can be observed that the backward scenario reduction method performs poorly when dealing with a large sample size of 1000 scenarios. Its output exhibits excessive fluctuations and bad probability distribution, making it insufficient to describe the uncertainty of wind and solar power output. In contrast, the K-means clustering algorithm better captures the variability and uncertainty of wind and solar power output, providing more reliable and practical guidance for the optimization scheduling and economic operation of MGs incorporating WT and PV.

The K-means clustering algorithm was chosen as the scenario reduction technique primarily because it can efficiently process large-scale sample data and significantly reduce computational complexity. By clustering the output scenarios of wind and solar power into representative scenarios, K-means preserves the core characteristics of the original data while reducing the number of samples. This enhances the computational efficiency and convergence speed of the optimization algorithm, making it particularly suitable for addressing the randomness and uncertainty issues in microgrid optimization.

4. Improved Dung Beetle Optimization Algorithm

4.1. Dung Beetle Optimization Algorithm

DBO is a novel swarm bio-inspired algorithm that mimics the behavior of dung beetles [32]. The beetles are divided into four subpopulations, each performing different strategies.

4.1.1. Ball-Rolling Beetles

The behavior of ball-rolling beetles can be divided into two modes: obstacle mode and non-obstacle mode. When a beetle is moving and encounters an unobstructed path, its direction is determined by the sun. The position update expression for a beetle in motion is as follows:

$$x_i^{t+1}=x_i^t+\alpha k{x}_t^{t-1}+b\left|x_i^t-x_{worst}^t\right|$$

(27)

where t is the current iteration count, and $x_i^t$ denotes the position of the i-th beetle in the population at the t-th iteration. $x_{worst}^t$ is the worst position in the current iteration. k is the deflection coefficient, where $k\mathsf{ϵ}\left(\mathrm{0,0.2}\right]$. b is any constant within (0, 1). The value of α can be ±1. When α equals 1, the beetle moves without deviation; when α equals −1, the beetle deviates from its original direction.

When a beetle encounters an obstacle that prevents further movement, a unique step size is required to regain direction. DBO uses the tangent function to update position.

$$x_i^{t+1}=x_i^t+\tan \left(\theta \right)\left|x_i^t-x_{worst}^t\right|$$

(28)

where $\theta$ is the deflection angle of the beetle’s special step size. When $\theta$ is $0,\pi /2,\pi$, the beetle’s position will not be updated.

4.1.2. Egg-Laying Beetles

When egg-laying beetles move to a safe and suitable location, a boundary selection can be established, which is defined by

$$\left\{\begin{array}{c}{Lb}^{\ast }=\max \left\{x_{gbest}^t\left(1-R\right),Lb\right\}\\ {Ub}^{\ast }=\max \left\{x_{gbest}^t\left(1+R\right),Ub\right\}\\ R=1-{\displaystyle \frac{t}{T_{max}}}\end{array}\right.$$

(29)

where Lb and Ub are the lower and upper boundary vectors of the area, whereas $x_{gbest}^t$ indicates the global best position vector of the current population. The variable R is defined with T_max being the maximum iteration count. Lb* and Ub* are the lower and upper boundary vectors of the egg-laying area, which can be adjusted dynamically through iterations based on R. Therefore, the dynamic iterative process for egg-laying beetles is expressed as follows:

$$B_i^{t+1}=x_{gbest}^t+b_1\times \left(B_i^t-L{b}^{\ast }\right)+b_2\times \left(B_i^t-U{b}^{\ast }\right)$$

(30)

$B_i^{t+1}$ is the position information of the ith egg-laying beetle at the t-th iteration, whereas b₁ and b₂ are two independent random vectors of size 1 × D, and D is the dimension of the optimization.

4.1.3. Small Beetles

Small beetles will search for food. During foraging, the area is dynamically refreshed with each iteration. The boundary of foraging area is defined as follows:

$$\left\{\begin{array}{c}{Lb}^{\prime }=\max \left\{x_{lbest}^t\left(1-R\right),Lb\right\}\\ {Ub}^{\prime }=\max \left\{x_{lbest}^t\left(1+R\right),Ub\right\}\\ R=1-{\displaystyle \frac{t}{T}}\end{array}\right.$$

(31)

where $x_{lbest}^t$ is the local best position, whereas Lb′ and Ub′ are the lower and upper boundary vectors of the foraging area. The individual position update expression for foraging beetles is as follows:

$$x_i^{t+1}=x_i^t+C_1\left(x_i^t-{Lb}^{\prime }\right)+C_2\left(x_i^t-{Ub}^{\prime }\right)$$

(32)

where C₁ is a random number following a normal distribution, and C₂ is a random vector in (0, 1).

4.1.4. Thief Beetles

Thief beetles steal food from other beetles by following the global best position. The position information of the thief beetles can be described as follows:

$$x_i^{t+1}=x_{lbest}^t+Sg\left(\left|x_i^t-x_{gbest}^t\right|+{|x}_i^t-x_{lbest}^t\right)$$

(33)

where g is a random vector of size 1 × D that follows a normal distribution, whereas S is an arbitrary constant. The position of the i-th beetle in the population at the t-th iteration is indicated by $x_i^t$, whereas $x_i^{t+1}$ is the current position vector, and $x_{lbest}^t$, $x_{gbest}^t$ are the position vectors of the local best point and global best point.

Thus far, limited research has applied DBO to MGs. Improving DBO to meet the needs of MG optimization can help to fully harness its potential.

4.2. Improved Dung Beetle Optimization

4.2.1. Tent Chaotic Mapping

In DBO, the initial population is usually generated randomly. This will result in an uneven distribution and reduce diversity. Tent chaotic mapping is used for improving initialization. Tent chaotic mapping has a more uniform distribution function; the expression is as follows:

$$Z_{k+1}=\left\{\begin{array}{c}{\displaystyle \frac{Z_k}{\beta }},Z_k\in \left(0,\beta \right)\\ {\displaystyle \frac{\left(1-Z_k\right)}{1-\beta }},Z_k\in \left[\beta ,1\right]\end{array}\right.$$

(34)

where k is the current population count, Z_k is the k-th population, with a certain initial value, and β is the chaos parameter with a certain initial value.

4.2.2. Non-Dominated Sorting Method

The selection process of DBO lacks clear evaluation and ranking of the superiority among individuals in the population, making it difficult to distinguish the advantages of different individuals in multi-objective optimization problems. This may make the overall performance and convergence speed of the algorithm worse. To address this, a non-dominated sorting method is introduced to improve the selection process of the DBO. The non-dominated sorting method is implemented as follows:

• Non-dominated sorting;
• Crowding distance calculation.

4.2.3. Elite Reverse Learning Strategy

Becoming trapped in local optima is another disadvantage of DBO. The reverse learning strategy operates by evaluating a solution, calculating its opposite, and then selecting the better one. However, normal reverse learning has its limitations, as it relies only on the upper and lower bounds of the population. It will increase the computational complexity in later stages of the algorithm. To address this, the boundaries are adjusted to use the best solution from the current generation and the worst solution from the previous generation. This approach makes better use of the information from top-performing individuals, and is called the elite reverse learning strategy.

$$\left\{\begin{array}{c}{K}_{sd}=\log \left[1.2+{\displaystyle \frac{1.5\cdot T}{T_{max}}}\right]\\ x_{lbest}^t=K_{sd}\cdot \left(x_{lbest}^t+x_{lworst}^t\right)-x_{lbest}^t\end{array}\right.$$

(35)

where K_sd is the current disturbance factor, $x_{lbest}^t$ is the current optimal position vector, and $x_{lworst}^t$ is the previous generation’s worst position vector.

4.3. Algorithm Solution Process

The pseudocode of IDBO is shown in Algorithm 1. The flowchart of the MG optimization scheduling model based on IDBO is shown in Figure 5.

Algorithm 1. Improved Dung Beetle algorithm.

Require: The maximum iterations Tmax, the size of the particle’s population N. Ensure: Optimal position and its fitness value. 1: Initializing algorithm parameters: Population size Np, Max generation Nm, Dimension D 2: Improved Chaotic Initialization of Population 3:   function Non-dominated Sorting of the Population 4:   Initialization 5:   Calculation of dominance relationship                   $\mathrm{I}\mathrm{f} \forall \mathrm{i}\in \left\{1,\dots ,\mathrm{M}\right\}, {\mathrm{f}}_{\mathrm{i}}\left(\mathrm{p}\right)\le {\mathrm{f}}_{\mathrm{i}}\left(\mathrm{q}\right) \mathrm{a}\mathrm{n}\mathrm{d} \exists \mathrm{j}\in \left\{1,\dots ,\mathrm{M}\right\},{\mathrm{f}}_{\mathrm{j}}\left(\mathrm{p}\right)<{\mathrm{f}}_{\mathrm{j}}\left(\mathrm{q}\right),\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n} \mathrm{p} \mathrm{d}\mathrm{o}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{s} \mathrm{q}$ 6:   function Frontier Construction 7:      Determine the first frontier 8:      Building other frontiers 9:      repeat 10:   end function 11:   function Crowding Degree Calculation 12:      Boundary individual processing 13:      Distance Calculation ${\mathrm{d}}_{\mathrm{k}}={\mathrm{d}}_{\mathrm{k}}+{\displaystyle \frac{{\mathrm{f}}_{\mathrm{k}+1}\left(\mathrm{m}\right)-{\mathrm{f}}_{\mathrm{k}-1}\left(\mathrm{m}\right)}{{\mathrm{f}}_{\mathrm{m}\mathrm{a}\mathrm{x}}\left(\mathrm{m}\right)-{\mathrm{f}}_{\mathrm{m}\mathrm{i}\mathrm{n}}\left(\mathrm{m}\right)}}$ 14:   end function 15: while ($t\le T_{max}$) do 16: for $i\leftarrow 1$ to N do 17:   if i == Ball-rolling Beetle then 18:      $\delta =rand\left(1\right)$ 19:      if $\delta <0.9$ then 20:      Select α value by −1 or 1 21:      Update the ball-rolling beetle by $x_i^{t+1}=x_i^t+\alpha k{x}_t^{t-1}+b\left|x_i^t-x_{worst}^t\right|$ 22:      else 23:      Update the ball-rolling beetle by $x_i^{t+1}=x_i^t+\tan \left(\theta \right)\left|x_i^t-x_{worst}^t\right|$ 24:      end if 25:   end if 26:   if i == Egg-laying Beetles then 27:      Update the egg-laying beetles by $B_i^{t+1}=x_{gbest}^t+b_1\times \left(B_i^t-L{b}^{\ast }\right)+b_2\times \left(B_i^t-U{b}^{\ast }\right)$ 28:   end if 29:   if i == Small Beetles then 30:      Update the small beetles by $x_i^{t+1}=x_i^t+C_1\left(x_i^t-{Lb}^{\prime }\right)+C_2\left(x_i^t-{Ub}^{\prime }\right)$ 31:   end if 32:   if i == Thief Beetles then 33:      Update the thief beetles by $x_i^{t+1}=x_{lbest}^t+Sg\left(\left|x_i^t-x_{gbest}^t\right|+{|x}_i^t-x_{lbest}^t\right)$ 34:   end if 35: end for 36: if the newly generated position is better than before then 37:      Update it 38: end if 39: Non-dominated Sorting of the Current Population 40: function Reverse Learning for Elite Individuals 41:   Select elite individuals 42:   Calculate the boundaries of elite individuals 43:   Generate reverse solution X′ = rand() × (TLb + TUb) − X 44: end function 45: t = t + 1 46: end while 47: return

In this study, the optimization scheduling of the MG is achieved through the IDBO. Each beetle’s position represents the power output of various DG units and energy storage systems within the MG. The movement of the beetles simulates the exploration behavior of the algorithm in the process of searching for the optimal solution. The beetles’ “movement” is limited by several constraints, such as power balance, output power range, and climb rate constraints. In this metaphor, the beetles’ behavior can be seen as traversing the solution space by following specific rules, avoiding infeasible regions, and utilizing the reverse learning strategy to escape local optima, ultimately searching for the global optimal solution. Therefore, the IDBO provides an efficient solution for optimization scheduling of MGs.

It is worth mentioning that the main reason for choosing the DBO is its unique exploration and exploitation mechanisms, which can effectively solve complex optimization problems, rather than simply due to its novelty or natural metaphor. Many nature-inspired algorithms rely solely on superficial metaphors, lacking substantive scientific contributions, and even obscuring underlying mathematical principles through obscure terminology [36]. In contrast, the core mechanisms of DBO (such as rolling, dancing, and foraging behaviors) have been carefully designed and transformed into mathematically rigorous optimization strategies, making a clear scientific contribution. Supported by both empirical and theoretical evidence, DBO performs excellently in search efficiency and in handling multimodal problems, avoiding the “just another nature-inspired algorithm” trap. The selection of DBO is justified by its dual merits: as a novel algorithm, it not only contributes significant scientific value to the optimization domain but also exhibits remarkable potential in addressing complex problem-solving challenges.

5. Case Simulation

5.1. Algorithm Testing

To test the performance of IDBO, six benchmark functions were selected, including two unimodal benchmark functions, two multimodal benchmark functions, two fixed-dimension multimodal benchmark functions, and a multi-objective function to comprehensively assess its optimization capabilities [28]. Through testing, the basic performance of IDBO can be validated. Benchmark functions are shown in

Table 3. Benchmark functions.

Function	Dim	Range	Optimal Value
$F_1={\sum }_{i=1}^n{\left({\sum }_{j-1}^i{x}_j\right)}^2$	30	[−100, 100]	0
$F_2={\sum }_{i=1}^n{\left[\left(x_i+0.5\right)\right]}^2$	30	[−100, 100]	0
$F_3={\displaystyle \sum _{i=1}^n}[x_i^2-10\cos \left(2\pi x_i\right)+10]$	30	[−5.12, 5.12]	0
$F_4={\displaystyle \frac{1}{4000}}{\displaystyle \sum _{i=1}^n}{x}_i^2-{\displaystyle \prod _{i=1}^n}\cos \left({\displaystyle \frac{x_i}{\sqrt{i}}}\right)+1$	30	[−600, 600]	0
$F_5={\displaystyle \sum _{i=1}^{11}} {\left[a_i-{\displaystyle \frac{x_1(b_i^2+b_i{x}_2)}{b_i^2+b_i{x}_3+x_4}}\right]}^2$	4	[−5, 5]	0.0003
$F_6=-{\displaystyle \sum _{i=1}^5} \left[\right(X-a_i\left)\right(X-a_i{)}^T+c_i{]}^{-1}$	4	[0, 10]	−10.1532
ZDT 1	2	[−4, 4]	(0, 1) to (1, 0)

.

From the benchmark function tests, IDBO showed clear advantages in both speed and accuracy. In tests on the unimodal benchmark functions F₁ and F₂, IDBO quickly converged to a stable state in the early stages of iteration and was much faster than other algorithms, including DBO. The results showed that IDBO had excellent speed and accuracy. For the multimodal functions F₃ and F₄, which have multiple local optima where other algorithms often become stuck, IDBO maintained good performance. It effectively avoided these traps and found better solutions, with better fitness function values than other algorithms. In tests with fixed-dimension multimodal functions F₅ and F₆, IDBO performed especially well, particularly with F₆, where other algorithms became stuck in local optima. IDBO not only escaped the local optima but also quickly approached the global optimal solution, maintaining its advantage in complex search spaces. From the results of the multi-objective test function ZDT1, it can be observed that the approximate Pareto front is relatively close to the true Pareto front in most regions, which indicates that IDBO exhibits good convergence and diversity in the multi-objective optimization process, effectively approximating the true optimal solution set. A comparison of the results of different test benchmark functions is shown in Figure 6.

5.2. Case Simulation Without DR and Uncertainty of Wind and Solar Power

To verify the proposed method, a basic MG system was selected as a simulation case study. The MG system includes MT, DE, PV, WT, BESS, and grid connection, without considering the uncertainty of wind and solar power output and the impact of DR on load. The parameters of DGs are listed in

Table 4. Parameters of DGs.

Parameters	MT	DE	PV	WT
Power upper limit/KW	60	60	30	10
Power lower limit/KW	3	6	0	0
Climb rate/KW	1.5	1.5	0	0
Maintenance factor/(CNY/KW·h)	0.0293	0.128	0.0096	0.45

. The pollutant treatment coefficients for the MT and DE are shown in

Table 5. Pollutant treatment coefficients.

Pollutant Type	Pollutant Discharge Factor/(g/kWh)		Governance Cost/(CNY/kg)
	MT	DE
CO2	724	689	0.21
CO	0.047	0.03	10.29
NO2	0.0036	0.0018	14.842

. The parameters of BESS are shown in

Table 6. Parameters of BESS.

Parameters	Value	Parameters	Value
Maximum capacity/KW·h	150	Minimum capacity/KW·h	5
Basic capacity/KW·h	50	Maximum exchange power/KW	30
Maintenance factor/(CNY/KW·h)	0.026	Charge/discharge efficiency	0.9

, and the time-of-use pricing is shown in

Table 7. Time-of-use pricing.

Price Type	Valley Time	Normal Time	Peak Time
	0:00–6:00 22:00–24:00	6:00–9:00 14:00–17:00	9:00–14:00 17:00–22:00
Purchase electricity/[CNY/(KW·h)]	0.38	0.82	1.35
Sell electricity/[CNY/(KW·h)]	0.15	0.36

.

Figure 7 shows the Pareto front, illustrating a relationship between environmental costs and operational costs. As environmental costs decrease, operational costs rise; conversely, higher environmental costs correlate with lower operational costs. This relationship arises because lower environmental costs typically indicate greater reliance on RE. Whereas these energy sources are environmentally friendly, their operation costs are usually higher. In contrast, when an MG primarily relies on fossil fuels (such as gas and diesel), the fuel costs may be lower, but environmental costs significantly increase due to the large amount of pollutant emissions from the combustion of fossil fuel.

Figure 8 shows the output of the MG at the lowest total cost without considering DR and the uncertainty of wind and solar power, illustrating the output strategies of different DGs.

The BESS charges actively during the valley time. By charging the battery at low electricity prices, the system prepares for the high demand in peak time. During the normal time, the BESS output fluctuates slightly, indicating that the system discharges based on electricity prices and load demand to meet the load. During peak time, the output power of the battery significantly increases, indicating that the system utilizes stored energy to supply power during high-demand periods. This aligns with the basic strategy of energy storage systems: charging when electricity prices are low or there is excess generation, and discharging when electricity prices are high or during peak load.

The PV output is nearly zero during the valley time due to the lack of sunlight, but it gradually increases between 6:00 a.m. and 8:00 a.m., reaching higher generation levels between 2:00 p.m. and 4:00 p.m. This is related to the intensity and duration of sunlight, and as the sun sets, the output of the PV system drops quickly, which fits the typical characteristics of solar systems.

The WT output exhibits significant fluctuations, as wind power generation largely depends on wind speed. In the morning, when the wind speed is low, WT output is minimal, whereas during the evening and midnight, as wind speed increases, WT output increases.

The MT and DE have lower output during the valley time, as electricity prices are lower and the system tends to reduce fuel consumption, maintaining a low output. During the normal time, the output of the MT and DE gradually increases, meeting the load and maintaining a stable supply. During peak time, in response to high demand, the MT and DE are activated to provide backup or peak-load power, leading to a significant increase in their output.

The grid interaction power fluctuates with changes in electricity prices. This indicates that the system is charging or purchasing cheaper electricity during the valley time to release the stored energy during peak time. During peak time, the power shows bidirectional fluctuations, indicating that the system has both the time to purchase electricity from the grid and the time to feed surplus electricity back into the grid. This dynamically balances the relationship between load and generation, which helps maintain the stability of the MG system.

To further validate the performance of IDBO, GWO and DBO were selected for comparison with IDBO. The simulation experiments used an optimization step of 1 h, with a 24 h period as the optimization cycle. The minimum total cost was set as the objective function. The comparison of converge curves is shown in Figure 9.

It can be seen that the IDBO significantly outperforms the GWO and DBO. Specifically, the initial value of IDBO is about 18.6% better than GWO and 5.5% better than DBO, enabling the algorithm to converge to the optimal value more quickly in fewer iterations. Non-dominated sorting allows for clearer evaluation and ranking of individuals within the population, which can improve convergence speed. As a result, IDBO reached the optimal range in fewer than 100 iterations. Additionally, the elite reverse learning strategy enhances convergence accuracy of IDBO in later iterations and improves the quality of the final value, which is about 56.6% better than GWO and 37.9% better than DBO. IDBO not only demonstrated a faster convergence speed, but also achieved higher solution accuracy and stronger local optima escape capability. These results fully demonstrate the significant advantages of the IDBO algorithm in terms of convergence, accuracy, and robustness.

A comprehensive evaluation framework was established to assess the performance of IDBO, DBO, and GWO, incorporating convergence half-life, relative convergence speed, and stability as key metrics, in addition to initial and final values. Convergence half-life is the number of iterations required to reach half of the final value. Relative convergence speed represents the relative rate of change of function values after each iteration.

Table 8. Indicator results comparison of different algorithms.

Algorithm	Initial Value/CNY	Final Value/CNY	Convergence Half-Life	Relative Convergence Speed	Running Time/s
IDBO	3891.18	1002.12	14	0.19%	23.54
DBO	4107.02	1382.01	47	0.25%	20.25
GWO	4616.79	1569.39	21	0.62%	31.34

compares the indicators of the three algorithms, including initial values, final values, convergence half-life, and relative convergence speed.

Table 9. Data comparison of different algorithms after 100 runs.

Parameters	IDBO	DBO	GWO
Avg initial value/CNY	4096.84	4237.22	4511.71
Avg final value/CNY	1111.81	1602.26	1686.31
Standard deviation	96.9	116.03	25.5

compares the data after running the algorithms 100 times.

The IDBO demonstrates excellent performance in both initial and final values, achieving lower initial and final values, indicating its outstanding optimization capability. Additionally, IDBO also performs better in terms of convergence speed and stability compared to DBO and GWO. The GWO performs worse in this case, showing good stability but obtaining high initial and final values, along with slower convergence speed. Additionally, the running time of IDBO is slightly worse than DBO, which is a negative impact resulting from the increased algorithmic complexity. However, considering the overall performance, IDBO still maintains a significant competitive edge.

The performance indicators after 100 runs also show that IDBO has lower initial and final values. Although the GWO is slightly advantageous in terms of overall stability, IDBO still demonstrates better performance in this case.

5.3. Case Simulation with DR and Uncertainty of Wind and Solar Power

The output of wind and solar energy is highly random, necessitating appropriate modeling to enhance the efficiency of scheduling optimization. The K-means clustering algorithm effectively reduces computational complexity by categorizing different generation scenarios, thereby shortening running time. The clustered scenarios enable MG operators to predict and respond to various generation fluctuations based on historical data, leading to more accurate scheduling decisions. This approach not only increases scheduling efficiency but also enhances the MG’s adaptability to sudden changes. Based on the reduction scenarios as proposed in Section 3.3, a simulation analysis was conducted, considering DR.

It can be observed from Figure 10 that after the DR, the distribution of load exhibits a pronounced optimization effect. Specifically, the load during the valley and normal time increases, whereas the load during the peak time significantly decreases. This change indicates that the DR effectively guides users to reduce electricity demand during peak time while encouraging increased consumption during valley time, thereby smoothing the load curve. This adjustment in load distribution not only helps alleviate the pressure on the grid during peak time, reducing over-reliance on DG units, but also enhances the overall operational efficiency and stability of the grid. Furthermore, through its peak load shifting, the DR promotes the rational allocation of electricity resources, reduces operational costs of the power system, and provides greater flexibility for the integration of renewable energy sources.

Five scenarios obtained after K-means clustering reduction in Section 3.3 and the load curve considering DR were selected for scheduling simulation, and the data are shown in Figure 11. Figure 11a–e represent the solar and wind power of five scenarios. Additionally, the most typical scenario with the highest probability (Scenario 1) obtained through K-means clustering was selected for comparison simulation with a scenario that does not consider DR and the uncertainty of wind and solar power, using IDBO for optimization.

Table 10. Results comparison of minimal total cost in different scenarios.

Scenario	IDBO/CNY	DBO/CNY	GWO/CNY
1	954.24	1302.04	1420.69
2	997.48	1402.99	1426.54
3	996.01	1325.97	1411.65
4	963.95	1370.78	1425.76
5	986.61	1438.19	1459.79

shows the results comparison of minimal total cost in different scenarios.

Table 10 shows that IDBO still demonstrates its outstanding global optimization capability and stability. The final value is significantly lower than the other two comparison algorithms. This result not only confirms the efficacy of IDBO in solving complex optimization problems but also validates the results in Section 5.1 and Section 5.2, further proving the unique advantages of IDBO in microgrid optimization scheduling. IDBO is better at handling nonlinear, multi-constraint optimization problems, especially in complex scenarios like microgrids that involve coordination of various energy sources and uncertainties, where its performance is particularly remarkable.

Table 11. Average final value comparison of different algorithms after 100 runs in different scenarios.

Scenario	IDBO		DBO		GWO
	Avg Final Value/CNY	Standard Deviation	Avg Final Value/CNY	Standard Deviation	Avg Final Value/CNY	Standard Deviation
1	953.68	89.21	1312.24	114.3	1426.87	23.94
2	988.25	92.3	1415.20	120.22	1433.31	27.85
3	994.32	93.02	1361.79	124.91	1427.26	31.56
4	962.88	91.25	1344.55	116.78	1428.45	26.51
5	984.57	90.33	1422.31	119.85	1462.65	21.2

compares the average final value after running the algorithms 100 times in five different scenarios. IDBO performs well across all five scenarios, with lower average total costs compared to other algorithms. Additionally, while achieving lower final values, IDBO also maintains lower standard deviations, indicating more stable results. This shows that IDBO is more reliable when handling different datasets, exhibiting higher stability and robustness. It effectively reduces operational uncertainty, validating its superior performance in MG optimal scheduling.

Figure 12 shows the comparison results of convergence curves considering DR and uncertainty, using IDBO for optimization. It can be seen that compared to the traditional scheduling model, which does not consider DR and uncertainty, this approach results in a significant reduction in the MG’s daily total cost by CNY 58.28. This substantial cost reduction is mainly attributed to the flexible adjustment of the load curve by the DR and the uncertainty factors. The DR incentivizes users to reduce electricity consumption during peak time and increase consumption during valley and normal time, achieving a peak load shifting, thereby reducing the operational pressure on the system. Meanwhile, the IDBO improves the system’s economy and reliability through precise prediction and optimization scheduling of wind–solar output fluctuations.

Figure 13 shows the output of the MG at the lowest total cost when considering DR and uncertainty of wind and solar power. Through the figure, it is proved that after the DR and uncertainty, the system’s scheduling strategy becomes more flexible and intelligent. It can dynamically adjust the operating mode based on the characteristics of different periods, thereby improving the system’s reliability and independence while ensuring economic efficiency. It can be seen that while the scheduling strategy of the MG remains consistent with Section 5.2, significant differences exist at the execution part. The most noticeable change is that the MG reduces its dependence on the grid, indicating that the system places greater use of DGs in the optimization scheduling process. By leveraging the DR and uncertainty, the MG is better able to coordinate its internal resources, reduce the need for electricity purchase from the grid, and, thereby, increase the system’s independence and self-sufficiency. Furthermore, the activity level of the BESS is significantly enhanced in the optimized scheduling, particularly notable at 8:00 a.m. At this time, the electricity price remains in the normal time, and the PV output is relatively abundant, so the BESS actively utilizes the PV output to charge. This strategy not only reduces charging costs but also improves the consumption of RE, avoiding the waste of solar energy. Simultaneously, the discharging behavior of the BESS during peak time further mitigates the power supply pressure on the system, achieving a peak load shifting.

6. Conclusions

It is crucial to fully consider economic issues and RE when scheduling MGs. In this paper, we constructed an MG system model including PV, WT, MT, DE, BESS, and grid connection. Based on the price–demand elasticity matrix, Monte Carlo sampling, and K-means clustering algorithm, DR and uncertainty of wind and solar power were fully modeled. The objective function was the total cost, including operational costs and environmental treatment costs. A novel algorithm IDBO was proposed to solve the issue. Compared to other algorithms, the designed IDBO managed to obtain lower total costs, achieve peak load shifting, and improve the utilization of RE. The optimization scheduling method proposed in this study holds significant practical value, particularly in scenarios such as smart MGs and virtual power plants (VPPs). By integrating the strategies, the economic efficiency and stability of the system can be significantly enhanced. The method presented in this paper can assist MG operators in reducing energy procurement costs and optimizing the utilization of renewable energy, thereby improving the system’s self-sufficiency. Furthermore, with the opening of the electricity market and the advancement of smart grid technologies, the proposed method will enable MGs to operate more efficiently in dynamic market environments. Especially when faced with uncertainties such as complex weather conditions and load fluctuations, this method will enhance the system’s robustness and flexibility, ensuring the reliability and economic efficiency of power supply.

Although our study achieved certain results in MG optimization scheduling, there are still some limitations that provide directions for future research improvements. Firstly, our study primarily focuses on optimizing the MG system in specific scenarios; it does not fully consider the impact of different geographical environments, climate conditions, and load characteristics on MG operations. Therefore, the generalizability of the research findings in more complex or diverse microgrid scenarios still requires further validation. Additionally, Monte Carlo sampling and K-means clustering may introduce certain biases. Specifically, potential uncertainty factors such as extreme weather events will cause information loss during the clustering process, lead to insufficient description of scenarios.

In the future, we aim to extend our research to more complex and diversified MG systems, further exploring the applicability and scalability of the method proposed in this paper. Specifically, we will investigate the feasibility of our approach in scenarios such as urban, rural, and industrial MG systems, each with distinct load profiles, energy sources, and operational constraints. By doing so, we can evaluate the robustness and adaptability of our method under different conditions, ensuring its practical utility in real-world applications. Additionally, we plan to explore the integration of our proposed method with emerging technologies and concepts in the energy domain. One particularly promising direction is its application to VPPs, which have garnered significant attention as a cutting-edge topic in modern power systems. By applying our method to VPPs, we aim to optimize their coordination and control, enhancing their ability to participate in energy markets, provide grid services, and improve overall system efficiency.