Improving Sparrow Search Algorithm for Optimal Operation Planning of Hydrogen–Electric Hybrid Microgrids Considering Demand Response

Abstract

Microgrid operation planning is crucial for ensuring the safe and efficient output of distributed energy resources (DERs) and stable operation of the microgrid power system. The integration of hydrogen fuel cells into microgrids can increase the absorption rate of renewable energy, while the incorporation of lithium batteries facilitates the adjustment of microgrid power supply voltage and frequency, ensuring the three-phase symmetry of the system. This paper proposes an economic scheduling method for a grid-connected microgrid that considers demand response and combines hydrogen and electricity. Based on the operating costs of renewable energy, maintenance and operation costs of nonrenewable energy, interaction costs between the microgrid and main grid, and pollution control costs, an optimization model for dispatching a hydrogen–electric hybrid microgrid under grid-connected mode is established. The primary objective is to minimize the operating cost, while the secondary objective is to minimize the impact on the user’s power consumption comfort. Therefore, an improved demand response strategy is introduced, and an enhanced sparrow search algorithm (ISSA) is proposed, which incorporates a nonlinear weighting factor and improves the global search capability based on the sparrow search algorithm (SSA). The ISSA is used to solve the optimal operation problem of the demand-response-integrated microgrid. After comparison with different algorithms, such as particle swarm optimization (PSO), whale optimization algorithm (WOA), sooty tern optimization algorithm (STOA), and dingo optimization algorithm (DOA), the results show that the proposed method using demand response and ISSA achieves the lowest comprehensive operating cost for the microgrid, making the microgrid’s operation safer and with minimum impact on user satisfaction. Therefore, the feasibility of the demand response strategy is demonstrated, and ISSA is proved to have better performance in solving optimal operation planning problems for hydrogen–electric hybrid microgrids.

1. Introduction

With the continuous development of society, the demand for electricity is increasing for human beings. In the entire power generation system of society, the power grid, as the main power generation system, bears a large part of the power generation responsibility. However, due to the increasing attention paid to sustainable development, many drawbacks of the power grid have gradually emerged [1,2,3,4]. On the one hand, the operation of the power grid is accompanied by significant environmental pollution [5,6,7,8], and on the other hand, the high operating cost of the power grid [9] makes it difficult to achieve real-time operation scheduling, which contradicts the increasing demand for electricity reliability from users. Therefore, in this situation, where environmental and economic pressures are exerting a double layer of pressure, distributed generation technology is rapidly developing [10,11,12]. Furthermore, the various advantages of distributed energy sources increasingly meet environmental and economic needs. Firstly, distributed generation has less pollution, which can effectively reduce environmental pollution. Secondly, it effectively uses clean energy sources, such as wind, solar, and geothermal energy in the local area to increase energy utilization efficiency. Thirdly, the cost of investment and construction is lower.

However, with the rapid development of distributed generation technology, many problems have gradually emerged. In the operation of the power grid, many distributed energy sources are connected to it, which produces a series of disadvantages related to system security protection, power quality, and reliable operation [13]. In order to solve the phenomenon of the impact of integrating DER into grid operation and make full use of the economic and environmental benefits brought by distributed generation technology, microgrid systems composed of renewable energy have been widely applied and developed [14,15,16,17,18]. A microgrid is a system unit that includes energy storage systems, loads, and DER. It can be connected to the power grid for grid-connected operation, and it can also operate independently from the power grid. It plays an important role in promoting the effective absorption of renewable energy and ensuring power supply reliability [19,20,21,22,23].

In the power grid system, because the generated energy cannot be stored in a large capacity, in order to solve the coordination and control problem, almost all power generation, transformation, transmission, and distribution must be completed at the same time. In a microgrid system, it includes a series of uncontrollable renewable energy sources, such as wind power and photovoltaic power. The complexity of the system scheduling and control mechanism is relatively high. However, the microgrid system has high efficiency, high reliability, flexible operation management, and can be connected to the distribution network for grid-connected operation. Therefore, optimizing the scheduling of microgrids involves many functional objectives and is relatively complex.

Therefore, an increasing number of researchers are using different scheduling strategies and metaheuristic algorithms to solve the optimal operation problem of microgrids. In [24], Alireza et al. proposed a robust metaheuristic optimization algorithm, evolutionary particle swarm optimization (E-PSO), to solve the optimization configuration and technoeconomic analysis of a standalone multicarrier microgrid (SMCMG). The paper [25] used PSO to solve the minimum-cost problem of complex microgrid models, including microgas turbines (MT), fuel cells (FC), waste-to-energy plants, wind turbine systems, boilers, anaerobic digestion-reformer systems, inverters, rectifiers, and some energy storage units. In [26], Chenyu et al. proposed to improve the economics of microgrids using twin-delayed deep deterministic policy gradient algorithm (TD3). Meanwhile, in [27], Hongsheng et al. proposed an improved particle swarm optimization algorithm considering demand response and a Gaussian disturbance mechanism to improve the economics and environmental friendliness of microgrids. Arul et al. [28] proposed a new gray wolf optimization algorithm based on reverse gradient to clarify the optimized operation of sustainable and unsustainable energy in microgrids by calculating the operation of microgrids in grid-connected mode. Prakash et al. [29] proposed an algorithm that combines gradient boosting decision tree (GBDT) and seagull optimization algorithm (SOA) to achieve the optimal operation of a microgrid system under grid-connected mode. This microgrid includes MT, WT, and photovoltaic (PV) systems. The goal is to reduce the cost incurred during the operation and maintenance of the microgrid system connected to the grid and to reduce the hourly power generation fluctuations of the microgrid. Yehia et al. [30] proposed an energy management system based on grasshopper optimization algorithm (GOA) to find the optimal power generation of distributed generators in microgrids, which involves reducing the cost of total power generation. In this study, the proposed unit was applied to a microgrid with five power generation units: industrial, commercial, feed, and residential loads. In addition, this study also used muddy soil fish optimization algorithm (MSFOA) based on fish foraging patterns. The main purpose of the algorithm is to reduce the cost incurred during the production process and to reduce the cost of importing energy into the grid. Karthik et al. [31] evaluated the optimal planning setting of independent microgrids under system fuzziness and studied many operating strategies to investigate system performance. The proposed optimal planning system is mainly aimed at improving energy utilization efficiency and reducing system fuel costs and gas emissions. Trong-The et al. [32] proposed an improved sparrow search algorithm (ESSA) based on elite reverse learning strategy and the firefly algorithm (FA) mutation strategy for microgrid optimization operation planning. Based on variables such as environmental costs, power interactions, investment depreciations, and maintenance systems, the optimal planning and total operation cost scheduling cycle of the microgrid with distributed power generation were modeled, and a multiobjective economic optimization model of the power grid was established to propose suggestions for the feasibility of microgrid operation. Yixing et al. [33] proposed an improved whale optimization algorithm (IWOA) with an adaptive weight strategy and a Levy flight trajectory based on the original whale optimization algorithm (WOA). The algorithm significantly improved the optimization results and opened up new directions for algorithmic improvements. In addition, to minimize emissions and operating costs, Amin et al. [34] introduced the concept of a temporary microgrid and proposed a new optimal model for microgrids, demonstrating the advantages of this new type of microgrid. This new microgrid will be used for planning the next generation of intelligent sustainable integrated grids [35].

Currently, most research on the optimal operation of microgrids focuses on improving the efficiency of renewable energy sources and controlling pollution costs, without considering the reduction in load demand through operating strategies. In addition, different metaheuristic algorithms have different effects on different problems, and some metaheuristic algorithms are prone to low accuracy and slow response when solving the optimal operation of microgrids. Furthermore, the uncertainty and volatility of wind power and photovoltaic power have an impact on microgrids. The introduction of lithium batteries can facilitate the adjustment of the microgrid’s power supply voltage and frequency, ensuring that the system does not experience three-phase asymmetry faults. Therefore, to improve the efficiency of clean energy, reduce pollution, and ensure the safe and stable operation of the entire microgrid, this paper proposes a demand response strategy and an improved sparrow search algorithm (ISSA) that incorporates nonlinear weighting factors and enhances global search capabilities to solve the optimal operation problem of a hydrogen–electric hybrid grid-connected microgrid. Economic, environmental, and user comfort factors are considered to establish the mathematical model and objective function of the microgrid, including the differences in electricity prices, load demand, and pollution emissions.

2. Demand Response Model

Demand response refers to the behavior of users who can flexibly adjust their own electricity usage time and duration to participate in microgrid optimization. Through this approach, the electricity usage of the microgrid can be adjusted to address issues of high or low electricity usage during certain periods. In this section, by optimizing the load curve from the demand side, the controllable load in each period is controlled by time shifting, and the load curve is smoothed to make electricity usage more stable for users. At the same time, by adopting the ISSA method under the constraint of controllable load capacity, the optimal transfer quantity is obtained to improve the safe operation of the microgrid while reducing its operating costs. The demand response model studied in this paper is a demand response model for a grid-connected microgrid that includes wind turbine (WT) and photovoltaic (PV) power. In this model, the demand response of the microgrid is analyzed.

2.1. Demand Response Objective Function

We fully utilize the low-cost clean energy generated by photovoltaics (PV) and wind turbines (WT). To minimize the system net load, which is defined as the total electricity consumption throughout the day minus the total available renewable energy generation throughout the day, the system net load is minimized as the objective function, as shown in Equation (1):

$$min{F}_1=\sum _t^T\left|P_{Load}\left(t\right)-P_{PV}\left(t\right)-P_{WT}\left(t\right)\right|$$

(1)

where $P_{Load}\left(t\right)$ represents the optimized load demand at time t, $P_{PV}\left(t\right)$ represents the PV generation power at time t, $P_{WT}\left(t\right)$ represents the WT generation power at time t, T is the total time period, and $F_1$ is the net load value.

During a microgrid scheduling period, renewable energy generation often falls short of the load demand and requires purchasing power from the main grid. The time-of-use tariff mechanism is considered to minimize the user’s electricity cost within the load dispatchable space, as shown in Equation (2):

$$min{F}_2=\sum _t^T[C_b\left(t\right)P_b\left(t\right)-C_S\left(t\right)P_S\left(t\right)]$$

(2)

where $C_b\left(t\right)$ and $C_S\left(t\right)$ represent the purchase and selling electricity prices at time t, $P_b\left(t\right)$ and $P_S\left(t\right)$ represent the purchased and sold power at time t, and $F_2$ represents the interaction cost between the microgrid and the main grid.

To convert the multiobjective problem into a single-objective problem, the values of the different dimensional objective functions are normalized and a trade-off function is selected, as shown in Equation (3) [36]:

$$minD=\sqrt{\sum _{l=1}^2(F_l-F_{l.best})}$$

(3)

where $F_{1.best}$ and $F_{2.best}$ represent the optimal net load value and optimal electricity cost of the single-objective optimization equation, respectively, and $F_1$ and $F_2$ represent the calculated net load value and electricity cost in practice.

2.2. User Satisfaction Function

The demand response strategy may affect the user’s electricity satisfaction to some extent. The electricity satisfaction function defined in this paper is shown in Equation (4) [36], where the electricity satisfaction of the unscheduled load curve is defined as 100%.

$$STL=1-\frac{{\displaystyle \sum _t^T}\left|P_{Load}\left(t\right)-P_{load}\left(t\right)\right|}{2\times {\displaystyle \sum _t^T}{P}_{load}\left(t\right)}$$

(4)

where $STL$ represents the electricity satisfaction, and $P_{load}\left(t\right)$ represents the load demand at time t before optimization.

2.3. Demand Response Constraints

When loads participate in demand response, there are also constraints on load scheduling. The maximum value of the transferable load in each time period is determined by the current demand response scheme, and the minimum transferable load is set to 0. The transfer capacity constraint is shown in Equation (5). The total load before and after the transfer remains unchanged, as shown in Equation (6):

$$\left\{\begin{array}{c}0\le V_{in}\left(t\right)\le V_{in.max}\left(t\right)\hfill \\ 0\le V_{out}\left(t\right)\le V_{out.max}\left(t\right)\hfill \end{array}\right.$$

(5)

$$\sum _t^T{P}_{Load}\left(t\right)=\sum _t^T{P}_{load}\left(t\right)$$

(6)

where $V_{in}\left(t\right)$ and $V_{out}\left(t\right)$ represent the transferred load in and out of the system during time period t, and $V_{in.max}\left(t\right)$ and $V_{out.max}\left(t\right)$ represent the maximum allowable load that can be transferred in and out during time period t.

3. Hydrogen–Electric Hybrid Microgrid Model

Due to the fact that traditional microgrids only consider the generation of wind turbines, photovoltaics, and some nonrenewable distributed power sources, their operation is not stable enough, and there is also severe emission of pollutants. In order to ensure the safety of microgrid operation and reduce its pollutant emissions, this paper considers using proton exchange membrane fuel cells as hydrogen fuel cells (HFCs), lithium batteries as representative energy storage systems (ESSs), controllable distributed energy sources represented by microgas turbines (MT), and renewable energy sources (RESs) represented by photovoltaic generators (PV) and wind turbines (WT). Moreover, this paper’s research is based on the grid-connected mode, in which the microgrid interacts with the main grid. Figure 1 shows the typical structure of a hydrogen–electric hybrid microgrid under grid-connected mode.

3.1. Objective Function of Microgrid

Microgrid operational planning refers to developing specific plans and strategies for the operation of a microgrid system based on its characteristics and requirements. The purpose is to achieve the highest economic efficiency in microgrid operation. Therefore, the objective of the hydrogen–electric hybrid microgrid system studied in this paper is to minimize the sum of operational costs, pollutant emission costs, and compensation costs for demand response, as expressed in Equation (7):

$$min{C}_{all}=c_1+c_2+c_2$$

(7)

where $C_{all}$ represents the overall cost of the microgrid, $c_1$ is the operating cost function of the microgrid, $c_2$ is the pollution emission cost function of the microgrid, and $c_3$ is the cost function for demand response compensation. The operating cost function $c_1$ is defined by Equation (8):

$$c_1=F_2+\sum _t^T\left(C_{ESS}\left(t\right)+C_{PV}\left(t\right)+C_{WT}\left(t\right)+C_{HFC}\left(t\right)+C_{MT}\left(t\right)\right)$$

(8)

where $C_{ESS}\left(t\right)$ is the maintenance cost of the lithium battery in time period t, $C_{PV}\left(t\right)$ is the maintenance cost of the photovoltaic system in time period t, $C_{W}T\left(t\right)$ is the maintenance cost of the wind turbine in time period t, $C_{HFC}\left(t\right)$ is the fuel and maintenance cost of the hydrogen fuel cell in time period t, and $C_{MT}\left(t\right)$ is the fuel and maintenance cost of the microgas turbine in time period t.

The calculation of pollution emission cost is based on the cost of pollution control caused by the pollutants emitted to the environment. The distributed power source microgas turbine in the microgrid generates corresponding pollutant gases, including CO${}_2$, SO${}_2$, CO, and NO${}_{\mathrm{x}}$. $c_2$ is the pollution emission cost function, and Equation (9) defines the mathematical model of the pollutant emission cost function. Equation (10) gives the calculation cost of the MT emission of various pollutants.

$$c_2=\sum _t^T{C}_{MT.con}\left(t\right)$$

(9)

$$C_{MT.con}\left(t\right)=(C_{E.C{O}_2}+C_{E.S{O}_2}+C_{E.CO}+C_{E.N{O}_X})·P_{MT}\left(t\right)$$

(10)

where $C_{MT.con}\left(t\right)$ is the pollutant emission cost of the microgas turbine in time period t, and $P_{MT}\left(t\right)$ is the output power of the microgas turbine in time period t.

The calculation of demand response cost is based on providing certain compensation for users’ participation in demand response and compensating users by transferring the load. The compensation cost function $c_3$ is given by Equation (11):

$$c_3=z·P_{trans}\left(t\right)$$

(11)

where z is the compensation amount per 1 kW/hour of load transfer, and $P_{trans}\left(t\right)$ is the transferred load.

3.2. Distributed Energy Resources Model

Photovoltaic (PV) power generation is a technology that directly converts solar energy into electrical energy through the photovoltaic effect of the semiconductor interface. The key component of this technology is the solar cell. After being connected in series and encapsulated for protection, the solar cells can form large-area solar cell modules, which can be combined with power controllers and other components to form a photovoltaic power generation device. The output power of PV panels is mainly affected by factors such as light intensity and ambient temperature, and its mathematical expression is shown in Equation (12):

$$P_{PV}=P_{STC}\frac{S_{PV}[1+M(T_e-T_{STC})]}{G_{STC}}A·E_{PV}$$

(12)

where $P_{PV}$ is the actual output power of the photovoltaic panel, $P_{STC}$ is the output power of the photovoltaic panel under standard test conditions, $S_{PV}$ is the light intensity, $G_{STC}$ is the light intensity under standard test conditions, M is the temperature power coefficient, $T_e$ is the ambient temperature, $T_{STC}$ is the temperature under standard test conditions, A is the area of the photovoltaic panel, and $E_{PV}$ is the energy conversion efficiency.

The principle of wind turbine (WT) power generation is to use wind power to drive the rotation of the blades and then increase the speed through the gearbox to promote the generator to generate electricity. Its mathematical expression is shown in Equation (13):

$$P_{WT}=\left\{\begin{array}{cc}0,\hfill & 0\le v\le v_s,v_e\le v\hfill \\ \frac{P_{rate}\left(v-v_s\right)}{v_r-v_e},\hfill & v_s\le v\le v_r\hfill \\ P_{rate},\hfill & v_r\le v\le v_e\hfill \end{array}\right.$$

(13)

where $P_{WT}$ is the output power of the wind turbine, $P_{rate}$ is the rated power of the wind turbine, v is the actual wind speed, $v_r$ is the rated wind speed, $v_s$ is the cut-in wind speed, and $v_e$ is the cut-out wind speed.

A hydrogen fuel cell (HFC) is a power generation device that directly converts the chemical energy of hydrogen and oxygen into electrical energy. Its basic principle is the reverse reaction of water electrolysis. HFC has no pollution in the environment, and its combustion only releases water. In addition, HFC can adjust its power output to any desired reference value within the power limitation range. Its output power is shown in Equation (14), and the cost is shown in Equation (15):

$$P_{HFC}=e·v_{H_{2}FC}$$

(14)

$$C_{HFC}=a_k·P_{HFC}^2\left(t\right)+b_k·P_{HFC}\left(t\right)+c_k$$

(15)

where $P_{HFC}$ is the output power of the hydrogen fuel cell, $v_{H_{2}FC}$ and $C_{HFC}$ are the power generation cost of the fuel cell, and $a_k$, $b_k$, and $c_k$ are the coefficients of the cost function.

Microgas turbines (MT) have advantages, such as high efficiency, clean and reliable operation, and flexible system configuration. They can serve as backup power generation units for microgrids in case of insufficient electricity supply, ensuring the reliability of microgrid power supply. The working principle of MT involves using a centrifugal compressor to continuously intake air from the outside and compress it. The compressed air is heated in a heat exchanger and then enters the combustion chamber. After mixing with fuel injected into the combustion chamber, the high-temperature gas produced by combustion flows into the turbine, driving it to rotate and generating electricity with the help of a generator. The cost is expressed in Equation (16):

$$C_{MT}=k·P_{MT}\left(t\right)$$

(16)

where $C_{MT}$ is the fuel cost of MT, $P_{MT}$ is the output power of the diesel generator, and k is the fuel cost coefficient.

Lithium batteries are used as ESS for charging and discharging in microgrids to maintain the power balance and energy buffering of the entire microgrid. They can also be used to conveniently adjust the voltage and frequency of the microgrid and stabilize the system’s three-phase symmetry. The state of charge (SOC) is used to represent the remaining amount of lithium battery charge as a ratio of the total amount of electricity, as shown in Equation (17), during the operation of lithium batteries.

$$SO{C}_t=\frac{C_e}{c}=1-\frac{\int Idt}{C}$$

(17)

where $SO{C}_t$ represents the remaining capacity of the lithium battery at time t, $C_e$ represents the remaining energy, and C represents the total energy.

The relationship between SOC and charging and discharging is shown in Equation (18):

$$\left\{\begin{array}{c}SOC(t+1)=(1-Esor)SOC\left(t\right)+P_{ch}{\eta }_{ch}\hfill \\ SOC(t+1)=(1-Esor)SOC\left(t\right)-\frac{P_{dis}}{{\eta }_{ch}}\hfill \end{array}\right.$$

(18)

where $SOC(t+1)$ represents the SOC of the lithium battery at time $t+1$, Esor represents the self-discharge rate of the lithium battery, $P_{c}h$ represents the charging power of the lithium battery, ${\eta }_{ch}$ represents the charging efficiency of the lithium battery, $P_{dis}$ represents the discharging power of the lithium battery, and ${\eta }_{ch}$ represents the discharging efficiency of the lithium battery.

3.3. Constraints of DER

In the model of a hydrogen–electricity hybrid microgrid, to ensure stable and safe operation of the entire microgrid system, the power generation unit needs to satisfy corresponding limiting conditions when outputting electric energy under the influence of its own devices. During operation, the microgrid must satisfy the constraint of power balance, expressed in Equation (19):

$$P_{all}\left(t\right)-P_{ESS}\left(t\right)+P_{LG}\left(t\right)=P_{load}\left(t\right)$$

(19)

where $P_{all}\left(t\right)$ is the power generation of the DRE microgrid at time t, $P_{LG}\left(t\right)$ is the interaction power between the microgrid and the main grid at time t, $P_{ESS}\left(t\right)$ is the output or input power of the lithium battery at time t, and $P_{load}\left(t\right)$ represents the load on the microgrid at time t.

HFC and MT also need to satisfy their respective upper and lower power limits, as well as ramp-up power limits, which are represented by Equations (20) and (21):

$$P_i^{min}\le P_i\left(t\right)\le P_i^{max}$$

(20)

$$P_i\left(t\right)-P_i(t-1)\le p_i\mathsf{\Delta }t$$

(21)

where $P_i\left(t\right)$ is the output power of the i-th distributed power source during time t, and $P_i^{min}$ and $P_i^{max}$ represent the upper and lower limits of the i-th distributed power source’s output power. $p_i$ represents the maximum climb rate of the i-th controllable generator unit, and $\mathsf{\Delta }t$ represents the operating time increment.

In addition, the power constraint for the interaction between the microgrid and the main grid is represented by Equation (22):

$$P_{LG.min}\le \left|P_{grid}\left(t\right)\right|\le P_{LG.max}$$

(22)

where $P_{LG.max}$ and $P_{LG.min}$ are the upper and lower limits of the interaction power between the microgrid and the main grid.

Furthermore, there exist power and capacity limitations on the charging and discharging of the lithium battery during its operation, as represented by Equations (23) and (24):

$$\left\{\begin{array}{c}0\le P_{ch}\left(t\right)\le P_{ch.max}\hfill \\ 0\le P_{dis}\left(t\right)\le P_{dis.max}\hfill \end{array}\right.$$

(23)

$$SO{C}_{min}\left(t\right)\le SOC\left(t\right)\le SO{C}_{max}\left(t\right)$$

(24)

where $P_{ch}\left(t\right)$ and $P_{dis}\left(t\right)$ represent the charging and discharging power of the lithium battery, and $P_{ch.max}$ and $P_{dis.max}$ represent the upper limits of the charging and discharging power of the lithium battery. $SO{C}_{max}\left(t\right)$ and $SO{C}_{min}\left(t\right)$ represent the upper and lower limits of the battery capacity during time t.

4. Algorithm for Solving the Optimization Model

The optimal operation of microgrids, which involves achieving the best output from distributed power sources while minimizing costs, is a complex multiobjective problem. This paper introduces demand response and uses metaheuristic algorithms to solve the optimal operation problem of microgrids. Therefore, this section proposes an improved sparrow search algorithm (ISSA) that incorporates nonlinear weight factors and enhances global search capabilities.

4.1. Improved Sparrow Search Algorithm

The sparrow search algorithm (SSA) [37] simulates the foraging behavior of sparrows, which are typically divided into producers and followers to accomplish foraging.

In SSA, the best individual in the population is given priority in obtaining food during the search process. As producers, they have a larger foraging search range than followers. The update method for the position of the explorer in each iteration is shown in Equation (25):

$$X_{i,j}^{t+1}=\left\{\begin{array}{c}{X}_{i,j}^t·exp\left(\frac{-i}{\alpha ·ite{r}_{max}}\right),if{R}_2<ST\hfill \\ X_{i,j}^t+Q·L,if{R}_2\ge ST\hfill \end{array}\right.$$

(25)

when $R_2<ST$, it means that there are no predators around, and the producer can perform a global search. If $R_2\ge ST$, it means that some sparrows have detected predators, and all sparrows must take appropriate actions. During foraging, some followers monitor the producers at all times. Once the producers find better food, they will immediately leave their current location to compete for food. If they win the competition, they can immediately obtain the food; otherwise, they need to continue to execute Equation (26):

$$X_{i,j}^{t+1}=\left\{\begin{array}{c}Q·exp\left(\frac{X_{worst}^t-X_{i,j}^t}{i^2}\right),ifi>\frac{n}{2}\hfill \\ {X_P}^{t+i}+|X_i,j^t-X_P^{t+i}|·A^{+}·L,otherwise\hfill \end{array}\right.$$

(26)

where $X_P^{t+i}$ is the optimal producer position, $X_{worst}^t$ is the current global worst position, and n is the population size. A is a 1 × d matrix, and the random amplitude of each element is 1 or $-1$. The definition of $A^{+}$ is shown in Equation (27):

$$A^{+}=A^T{\left(A{A}^T\right)}^{-1}$$

(27)

when $i>\frac{n}{2}$, it indicates that the i-th follower with a lower fitness value is in a poor state and needs to fly to other places for food. Assuming that 10–20% of the individuals in the population are aware of the danger, the initial positions of these individuals are randomly generated in the population, as shown in Equation (28):

$$X_{i,j}^{t+1}=\left\{\begin{array}{c}{X}_{best}^t+\beta ·|X_{i,j}^t-X_{best}^t|,if{f}_i>f_g\hfill \\ X_{i,j}^t+K·(\frac{X_{i,j}^t-X_{worst}^t}{f_i-f_W+\omega }),if{f}_i=f_g\hfill \end{array}\right.$$

(28)

where $X_{best}^t$ is the current global optimal position, $\beta$ is the step size control parameter (its value is a random number that obeys the normal distribution with a mean of 0 and a variance of 1), K is a random number in [$-1$,1], f is the fitness value, and $f_i$, and $f_g$ are the current optimal and worst fitness values, respectively. $\omega$ is a constant to avoid denominator 0. Therefore, $f_i$ > $f_g$ indicates that the sparrow is at the edge of the population. When $f_i$ = $f_g$, it indicates that the sparrow in the middle of the population is aware of the danger and needs to be close to other sparrows to avoid predation. Here, K represents the direction of the sparrow’s movement and is also the step control parameter.

In SSA, the way sparrows converge to the current best solution is by directly jumping to the vicinity of the current best solution, rather than moving towards it. This leads to the weakness of SSA in a global search and makes it prone to getting stuck in local optima. As can be seen from Equation (25), the global search capability of SSA is relatively weak.

The first improvement made to the SSA is the modification of Equation (25) by removing the step of converging to the origin and changing the movement towards the optimal position to movement towards the vicinity of the optimal position. In addition, a nonlinear weight factor is introduced in this paper, as shown in Equation (29):

$$\mathsf{\Delta }=2-{(\frac{t_{iter}}{T_{max}})}^2$$

(29)

The fully improved equation is presented as Equation (30):

$$X_{i,j}^{t+1}=\left\{\begin{array}{c}{X}_{i,j}^t·(1+\mathsf{\Delta }·Q),if{R}_2<ST\hfill \\ X_{i,j}^t+Q,if{R}_2\ge ST\hfill \end{array}\right.$$

(30)

The introduction of a nonlinear weighting factor can effectively enhance the optimization ability of SSA. By applying a nonlinear weighting factor that changes dynamically with the increase in iteration times, the global search ability and local search ability of SSA can be adjusted dynamically, which can accelerate the convergence speed of the algorithm and improve the optimization accuracy. Next, Equation (26) is retained, and Equation (28) is improved by randomly approaching the discoverer in all dimensions for each sparrow. The specific formula is shown in Equation (31):

$$X_{i,j}^{t+1}=\left\{\begin{array}{c}{X}_{i,j}^t+\beta ·(X_{i,j}^t-X_{best}^t),if{f}_i>f_g\hfill \\ X_{best}^t+\beta ·(X_{worst}^t-X_{best}^t),if{f}_i=f_g\hfill \end{array}\right.$$

(31)

The improvement removes the cumbersome parts of the original algorithm. When $f_i$ > $f_g$, the sparrow will flee to a random position between itself and the optimal position. When $f_i$ = $f_g$, the sparrow at the optimal position will flee to a random position between the optimal and worst positions.

Such improvements actually accelerate the update speed and accuracy of the entire algorithm, greatly enhancing the global search ability, which is precisely the weakest point of the original algorithm. Therefore, the improvement of the entire algorithm not only enhances the global search ability but also greatly strengthens the accuracy and convergence speed of the algorithm. Within the set number of iterations, the individual positions are continuously updated according to the formula mentioned above, in order to ultimately find the feasible optimal solution and complete the optimal planning of the objective. The ISSA process is illustrated in Figure 2.

Algorithm 1 gives the pseudocode of ISSA.

4.2. Test Function

This section verifies the performance of the proposed ISSA. Ten benchmark functions from CEC2017 [38] are used to test the performance of ISSA, as shown in

Table 1. The equation of the selected test function.

Algorithm	Equation
$F_1\left(x\right)$	$x^2+{10}^6{\displaystyle \sum _{i=2}^D}{x}_i^2$
$F_2\left(x\right)$	${\displaystyle \sum _{i=1}^D}{\left|x_i\right|}^{i+1}$
$F_3\left(x\right)$	${\displaystyle \sum _{i=1}^D}{x}_i^2+{({\displaystyle \sum _{i=1}^D}0.5x_i)}^2+{({\displaystyle \sum _{i=1}^D}0.5x_i)}^4$
$F_4\left(x\right)$	${\displaystyle \sum _{i=1}^{D-1}}(100{(x_i^2-x_{i+1})}^2+(x_{i-1}^2))$
$F_5\left(x\right)$	${\displaystyle \sum _{i=1}^D}(x_i^2-10cos\left(2\pi x_i\right)+10)$
$F_7\left(x\right)$	$min({\displaystyle \sum _{i=1}^D}{(\widehat{x_i}-{\mu }_0)}^2),dD+s{\displaystyle \sum _{i=1}^D}{(\widehat{x_i}-{\mu }_1)}^2)+10(D-{\displaystyle \sum _{i=1}^D}cos(2\pi \widehat{z_i}))$
$F_9\left(x\right)$	${sin}^2\left(\pi w_1\right)+{\displaystyle \sum _{i=1}^{D-1}}{(w_i-1)}^2(1+10{sin}^2(\pi w_i+1))+{(w_D-1)}^2(1+{sin}^2\left(2\pi w_D\right))$
$F_{11}\left(x\right)$	${\displaystyle \sum _{i=1}^D}{\left({10}^6\right)}^{\frac{i-1}{D-1}}{x}_i^2$
$F_{13}\left(x\right)$	$-20exp(-0.2\sqrt{\frac{1}{D}{\displaystyle \sum _{i=1}^D}{x}_i^2})-exp(\frac{1}{D}{\displaystyle \sum _{i=1}^D}cos(2\pi x_i))+20+e$
$F_{15}\left(x\right)$	${\displaystyle \sum _{i=1}^D}\frac{x_i^2}{4000}-{\displaystyle \prod _{i=1}^D}cos\left(\frac{x_i}{\sqrt{i}}\right)+1$

. The experimental results of ISSA are compared with those of SSA, PSO, WOA, DOA, STOA, WSSA, and ESSA.

Table 2. The average results of ISSA on 10 different benchmark functions are compared with SSA, WSSA, and ESSA, respectively.

Algorithms	SSA	WSSA	ESSA	ISSA
$F_1\left(x\right)$	$2.7\times {10}^{-72}$	$2.9\times {10}^{-47}$	$3.0\times {10}^{-67}$	$2.4\times {10}^{-203}$
$F_2\left(x\right)$	$6.7\times {10}^{-51}$	$2.3\times {10}^{-49}$	$2.4\times {10}^{-61}$	$3.5\times {10}^{-69}$
$F_3\left(x\right)$	$3.2\times {10}^0$	$4.9\times {10}^{-1}$	$2.8\times {10}^1$	$2.8\times {10}^{-1}$
$F_4\left(x\right)$	$2.5\times {10}^{-4}$	$3.7\times {10}^{-5}$	$6.7\times {10}^{-5}$	$4.2\times {10}^{-9}$
$F_5\left(x\right)$	$2.6\times {10}^{-11}$	$7.1\times {10}^{-9}$	$8.9\times {10}^{-12}$	$4.2\times {10}^{-11}$
$F_7\left(x\right)$	$4.2\times {10}^{-2}$	$7.1\times {10}^{-2}$	$3.3\times {10}^{-2}$	$2.9\times {10}^{-4}$
$F_9\left(x\right)$	$3.8\times {10}^{-9}$	$2.7\times {10}^{-9}$	$2.9\times {10}^{-9}$	$7.6\times {10}^{-9}$
$F_{11}\left(x\right)$	$3.1\times {10}^{-4}$	$4.9\times {10}^{-4}$	$2.6\times {10}^{-3}$	$5.1\times {10}^{-5}$
$F_{13}\left(x\right)$	$4.1\times {10}^0$	$5.4\times {10}^0$	$3.6\times {10}^0$	$2.6\times {10}^0$
$F_{15}\left(x\right)$	$1.1\times {10}^{-1}$	$4.3\times {10}^{-2}$	$4.9\times {10}^{-2}$	$3.3\times {10}^{-2}$

shows the average results of ISSA on the ten different benchmark functions, compared with SSA, weight sparrow search algorithm (WSSA), and enhanced sparrow search algorithm (ESSA) [32]. Figure 3, Figure 4 and Figure 5 show the convergence curves of ISSA and the left-side SSA, WSSA, and ESSA for each benchmark function. The function landscape views of $F_1\left(x\right)$, $F_4\left(x\right)$, and $F_7\left(x\right)$ from CEC2017 are shown on the right side of the figures.

Algorithm 1 Pseudocode of ISSA.

Require:   T: the maximum iterations   $PD$: the number of producers   $SD$: the number of sparrows who perceive danger   $R_2$: the alarm value   n: the number of sparrows   Initialize a population of n sparrows and define its relevant   Parameters  Ensure: $X_{best}$,$f_g$    1: while$(t<T)$do    2:    Rank the fitness values and find the current best individual and the current worst individual.    3:    $R_2$ = $rand\left(1\right)$    4:    for do i = 1:PD    5:        Using Equation (30) update the sparrow’s location;    6:    end for    7:    for do i = (PD + 1):n    8:        Using Equation (26) update the sparrow’s location;    9:    end for     10:    for $\mathbf{do}l$ = 1:SD     11:        Using Equation (31) update the sparrow’s location;     12:    end for     13:    Get the current new location;     14:    If the new location is better than before, update it     15:    t = $t+1$     16: end while     17: return $X_{best}$,$f_g$

Table 2 indicates that ISSA performs better than most of the test functions, with smaller function values being found. Figure 3, Figure 4 and Figure 5 show that ISSA significantly improves the convergence speed, accuracy, and global search ability compared with SSA, WSSA, and ESSA, indicating that ISSA has superior algorithmic-solving capabilities.

Table 3. The average results of ISSA on 10 different benchmark functions are compared with POS, WOA, DOA, and STOA, respectively.

Algorithms	POS	WOA	DOA	STOA	ISSA
$F_1\left(x\right)$	$5.8\times {10}^{-12}$	$2.8\times {10}^{-57}$	$2.9\times {10}^{-67}$	$4.2\times {10}^{-86}$	$2.4\times {10}^{-203}$
$F_2\left(x\right)$	$8.3\times {10}^{-22}$	$1.9\times {10}^{-51}$	$2.3\times {10}^{-59}$	$3.9\times {10}^{-54}$	$3.5\times {10}^{-69}$
$F_3\left(x\right)$	$3.9\times {10}^1$	$1.1\times {10}^1$	$5.3\times {10}^{-1}$	$6.8\times {10}^{-1}$	$2.8\times {10}^{-1}$
$F_4\left(x\right)$	$2.5\times {10}^1$	$2.3\times {10}^2$	$7.3\times {10}^{-2}$	$4.6\times {10}^{-6}$	$4.2\times {10}^{-9}$
$F_5\left(x\right)$	$2.5\times {10}^{-2}$	$3.9\times {10}^{-9}$	$1.6\times {10}^{-10}$	$7.3\times {10}^{-12}$	$4.2\times {10}^{-11}$
$F_7\left(x\right)$	$4.2\times {10}^{-1}$	$4.6\times {10}^{-1}$	$8.1\times {10}^{-2}$	$7.3\times {10}^{-4}$	$2.9\times {10}^{-4}$
$F_9\left(x\right)$	$6.5\times {10}^{-2}$	$5.9\times {10}^{-2}$	$7.6\times {10}^{-8}$	$2.9\times {10}^{-8}$	$7.6\times {10}^{-9}$
$F_{11}\left(x\right)$	$9.1\times {10}^{-1}$	$3.6\times {10}^{-1}$	$2.9\times {10}^{-4}$	$2.6\times {10}^{-5}$	$5.1\times {10}^{-5}$
$F_{13}\left(x\right)$	$3.9\times {10}^0$	$9.0\times {10}^0$	$7.3\times {10}^0$	$1.7\times {10}^0$	$2.6\times {10}^0$
$F_{15}\left(x\right)$	$4.8\times {10}^{-1}$	$4.3\times {10}^{-1}$	$2.9\times {10}^{-1}$	$7.3\times {10}^{-2}$	$3.3\times {10}^{-2}$

presents the test results of ISSA and other metaheuristic algorithms.

Table 3 shows that ISSA performs better than other algorithms in optimizing functions $F_1\left(x\right)$, $F_2\left(x\right)$, $F_3\left(x\right)$, $F_4\left(x\right)$, $F_7\left(x\right)$, $F_9\left(x\right)$, and $F_{15}\left(x\right)$. It greatly improves the accuracy of functions $F_1\left(x\right)$ and $F_2\left(x\right)$ and finds smaller function values.

5. Calculation and Analysis

The subject of this study is a hydrogen–electric hybrid microgrid system. An optimization model with bounded rationality in user electricity decision making and ISSA were used to solve and analyze the optimal operation problem. The interconnected microgrid system is shown in Figure 6, connected to the large power grid, with renewable clean energy sources consisting of PV and WT and distributed controllable energy sources consisting of HFC and MT. The energy storage system is ESS, and there are 31 households using the system. All calculations in this paper are based on MATLAB.

5.1. Related Calculation and Analysis Data

Based on the dataset provided in [39,40], and with reference to the mathematical model described earlier, the 24-h electricity load data for typical weather conditions in summer, as well as the output power data of WT and PV without dispatch, are selected as shown in Figure 7.

The purchasing and selling electricity prices for each moment in the microgrid are shown in

Table 4. Market price of electricity [41].

Types	Price/[USD·(kWh)${}^{-1}$]
	Peak Period	Through Period	Normal Period
Buy	0.84	0.19	0.51
Sell	0.42	0.10	0.26

, and the parameters for each distributed power source are presented in

Table 5. Generation parameters of each DER in the microgrid [42].

Types	Minimum Power/(kW)	Maximum Power/(kW)	Maintenance Costs/(USD/kW)	Climb Rates/(kW/min)
Large Grid	−240	240	0.001	/
PV	0	180	0.012	/
WT	0	170	0.036	/
MT	15	300	0.107	2
HFC	5	250	0.205	3
ESS	−150	150	0.005	/

. The cost calculation of polluting gases is illustrated in

Table 6. Pollutant emission factors for each DER in the microgrid [32].

Types of Pollutant	Pollution Costs (USD/kg)	Emission Factors of MT/(kg/kWh)
CO${}_2$	0.0041	0.184
SO${}_2$	0.875	9.3 × 10${}^{-7}$
NO${}_{\mathrm{x}}$	1.25	6.19 × 10${}^{-4}$
CO	0.145	1.7 × 10${}^{-4}$

. Based on time-of-use pricing [40] and Figure 7, we know that the peak hours are from 11:00 to 16:00 and 18:00 to 21:00; the off-peak hours are from 07:00 to 10:00, 16:00 to 18:00, and 21:00 to 24:00; and the valley hours are from 00:00 to 07:00.

5.2. Analysis of Demand Response Results

This paper analyzes four different response schemes for transferable load capacity, which are set at 0% (Scheme 1), 10% (Scheme 2), 20% (Scheme 3), and 30% (Scheme 4) of the original load curve. In addition, three different microgrid operation strategies are implemented to balance the generation of MT and HFC (Strategy 1), reduce the generation of MT while increasing that of HFC (Strategy 2), and reduce the generation of HFC while increasing that of MT (Strategy 3). The compensation for demand response is set at 0.3 (USD/kWh). The optimal solution is selected by comparing the operational cost of the microgrid and user satisfaction. Figure 8 shows the changes in residential load demand for the four schemes, and

Table 7. The optimization results under different operation schemes and strategies.

Schemes	Electricity Satisfaction	Compensation for Demand Response/(USD)	Comprehensive Operation Cost/(USD)
			Strategy 1	Strategy 2	Strategy 3
1	100%	0	3420.91	3692.82	3301.26
2	96.53%	175.39	3349.65	3619.77	3212.14
3	94.14%	298.63	3246.34	3510.24	3121.83
4	89.39%	425.13	3224.39	3473.64	3098.49

shows the operating costs of the microgrid under four different schemes and three different strategies. The cost calculation is performed using Equation (7).

From Figure 8 and Table 7, it can be observed that user participation in demand response reduces electricity consumption during peak hours and increases it during off-peak hours. This leads to a more balanced load for the microgrid and lower operational costs, but it may also result in decreased user satisfaction.

When optimizing solely for user satisfaction, Scheme 1 has the highest user satisfaction rate of 100%, but it is also the most expensive option. Scheme 2 decreases user satisfaction by 3.47%, while having low demand response compensation costs, but its overall operational cost reduction compared with Scheme 1 is relatively small. Scheme 3 decreases user satisfaction by 5.86% and has a slightly lower overall operational cost than Scheme 2, but it comes with increased demand response compensation costs. Scheme 4 decreases user satisfaction by 10.61% and has the lowest operational cost, but it greatly reduces user satisfaction.

Taking both user satisfaction and operational strategies into account, Strategy 1 appears to be the most constructive, while Strategy 2, which reduces MT’s generation while increasing HFC’s, raises microgrid environmental friendliness but significantly increases operational costs. On the other hand, Strategy 3, which increases MT’s generation while decreasing HFC’s, lowers operational costs but also reduces microgrid environmental friendliness and increases pollutant emissions. Therefore, Scheme 3 and Strategy 1 strike a balance between improving microgrid economics, environmental friendliness, and user satisfaction, making them the optimal choices for microgrid operations.

5.3. Analysis of Demand Response Results

Figure 9 presents the output power of distributed renewable energy (DRE) sources, excluding PVs and WTs, as well as the buying and selling power with the main grid, under the operation of the microgrid optimized using Scheme 3 and Strategy 1 in ISSA.

From Figure 9, it can be seen that during the time period of 00:00–07:00, the load demand of the entire microgrid system decreases. At this time, the price of purchasing electricity from the main grid is lower than the price of generating electricity from nonrenewable distributed power sources. Therefore, the ESS is charged while meeting the load demand. During the period of 07:00–10:00, the electricity usage is at a normal level, and the prices of purchasing and selling electricity from the main grid are not much different from the price of generating electricity from nonrenewable distributed power sources. From 10:00–16:00, the load demand continues to increase, and it is the peak period of electricity usage from the main grid. The price of selling electricity to the main grid is higher than the price of generating electricity from nonrenewable distributed power sources. At this time, the PV generation reaches its peak, but due to the high demand for electricity, the ESS begins to generate electricity. From 16:00–18:00, the main grid is in a normal period, and the PV output begins to decrease. The amount of electricity sold to the main grid decreases, and the ESS is charged. From 18:00–21:00, it is the peak period of electricity usage for the whole day, and the PV generation basically stops. Therefore, the MT, HFC, and ESS are all at a high level of electricity generation. If the DER cannot meet the load demand at this time, the microgrid can only balance the load by purchasing electricity from the main grid. From 21:00–24:00, the load demand begins to decrease, and therefore, the output power of the ESS, MT, and HFC begins to decrease.

To verify the feasibility of ISSA, Figure 10 shows the power generation proportion of various DRE sources in the microgrid under different algorithms.

From Figure 10, it can be observed that in the microgrid controlled by ISSA, the power generation proportion of MT and HFC is relatively high, nearly 48%. The power generation proportion of ESS is 11%, while the proportion of buying power from the main grid is the lowest compared with other algorithms, reduced by 3%, 3%, 2%, 2%, and 3% compared with SSA, POS, WOA, DOA, and STOA, respectively. This indicates that the microgrid controlled by ISSA can reduce dependence on the main grid and minimize interaction with the main grid.

To further verify the improvement of ISSA, we conducted 20 iterations using ISSA, SSA, POS, WOA, DOA, and STOA, as well as E-PSO [24], ESSA [32], and IWOA [33], and compared and analyzed the average values. Using the objective function given in Equation (7), the best fitness curves obtained from 20 runs of each algorithm using ISSA are shown in Figure 11 and Figure 12, and the worst fitness curves are shown in Figure 13 and Figure 14. The fitness values for the worst, best, and average objective functions obtained after running each algorithm 20 times are shown in

Table 8. The best and worst of each algorithm and the average cost.

Types of Algorithms	Cost/(USD)
	Worst	Best	Average
SSA	3482.29	3225.77	3341.36
ISSA	3327.41	2647.34	2879.53
POS	3456.25	3194.27	3320.51
WOA	3490.26	3165.27	3297.19
DOA	3493.65	3050.44	3145.73
STOA	3401.53	2851.26	3014.59
ESSA	3469.36	3259.71	3376.29
E-PSO	3503.64	3186.82	3324.76
IWOA	3486.62	3094.36	3256.73

.

From Table 8, it can be observed that the worst and best cost of ISSA are smaller than those of other algorithms. The average cost computed by ISSA is reduced by 16.1%, 15.3%, 14.5%, 9.2%, 4.7%, 16.2%, 15.5%, and 13.1% compared with SSA, POS, WOA, DOA, STOA, ESSA, E-PSO, and IWOA, respectively. These results indicate that the proposed ISSA can lower the operating cost of the microgrid and can be used to solve the optimal operational planning problem of the hydrogen-electricity hybrid microgrid.

6. Conclusions

This study establishes a grid-connected hydrogen–electric hybrid microgrid model that includes WT, PV, MT, HFC, and ESS. The optimal operating objective is to minimize economic and environmental costs while maximizing user satisfaction. To address this problem, demand response is introduced, and an improved weight sparrow search algorithm is proposed, which uses a nonlinear weight factor and enhances global search capability. The optimization results show that introducing demand response can reduce microgrid operating costs by sacrificing a small portion of user satisfaction, and selecting the correct operating strategy can significantly reduce microgrid operating and environmental costs. In addition, using an appropriate metaheuristic algorithm for the microgrid can greatly reduce operating costs. Furthermore, using lithium batteries as energy storage systems in the coordinated control effectively regulates microgrid power quality and ensures the symmetry of renewable energy and microgrid systems. This study also introduces an improved sparrow search algorithm (ISSA), which outperforms other optimization algorithms, including SSA, PSO, WOA, DOA, STOA, ESSA, E-PSO, and IWOA, in terms of microgrid operating costs, reducing them by 16.1%, 15.3%, 14.5%, 9.2%, 4.7%, 16.2%, 15.5%, and 13.1%, respectively. Moreover, the DER outputs under different algorithms are analyzed, and the results show that using ISSA can increase the output ratio of RES and reduce the electricity purchased from the main grid. Thus, achieving optimal operation in microgrids is not only important for economic dispatch but also for balancing various distributed power sources, including WT, PV, ESS, MT, and HFC.

By introducing demand response, selecting the correct operating strategy, and using ISSA to solve the optimization problem of the hydrogen–electric hybrid microgrid, this study provides a reference for improving the optimal operation of the microgrid.