
==== Front
Heliyon
Heliyon
Heliyon
2405-8440
Elsevier

S2405-8440(24)12700-4
10.1016/j.heliyon.2024.e36669
e36669
Research Article
Optimizing energy management strategies for microgrids through chaotic local search and particle swarm optimization techniques
Li Ping 2000010@zzuli.edu.cn
a⁎
Liu Qian b
a Zhengzhou University of Light Industry, Zhengzhou, 450000, China
b Henan Agricultural University, Zhengzhou, 450046, China
⁎ Corresponding author. 2000010@zzuli.edu.cn
22 8 2024
15 9 2024
22 8 2024
10 17 e3666931 8 2023
16 8 2024
20 8 2024
© 2024 The Authors
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
The advent of multi-Microgrid (MG) energy systems necessitates the optimization of management strategies to curtail operational costs. This paper introduces an innovative MG energy management strategy that integrates Chaotic Local Search (CLS) with Particle Swarm Optimization (PSO) to fulfill this requirement. Our approach leverages PSO for extensive global exploration and subsequently employs CLS to refine local searches, thereby ensuring the attainment of optimal global outcomes. To further enhance performance, we have crafted a PSO algorithmic framework underpinned by chaotic local search principles, aimed at circumventing regions of local optima. The study presents a comprehensive MG energy system model that encompasses a photovoltaic generation unit, battery energy storage, and a micro gas turbine. The experimental data corroborates that our proposed algorithm secures optimal solutions within a range of 48.2–51.7, outperforming others in achieving these optimal resolutions. When juxtaposed with Scenario 1, there is a significant reduction in both operational and primary energy conversion costs by 24.22 % and 31.39 %, respectively. In comparison to Scenario 2, these figures are reduced by an additional 3.08 % and 6.05 %, respectively. The research findings underscore the strategy's exceptional performance in optimization tasks, as illustrated by the simulation outcomes. The methodology's application to a micro-energy network substantiates its practical relevance. Collectively, this research offers a holistic solution for the optimization of MG energy systems, effectively merging theoretical progress with tangible practical applications.

Keywords

Chaotic local search
Particle swarm algorithm
Microgrid energy management
Economic operating costs
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pmc1 Introduction

In alignment with the nation's “dual carbon” objectives, there is an escalating focus on the development of efficient, sustainable, and low-carbon renewable energy sources, particularly solar and wind power. These renewable sources are projected to take a central role in shaping the future energy landscape [1]. Microgrids (MGs) act as pivotal orchestrators of distributed power sources, capitalizing on interconnected power systems to achieve large-scale optimization of energy allocation. This positions MGs as an essential mechanism for both local renewable energy consumption and its transmission within the grid-connected framework [2]. Nonetheless, conventional single MGs have a restricted ability to integrate a variety of renewable energy generation systems. To transcend this limitation, MGs can be interconnected within a region through the implementation of suitable protocols, creating a multi-microgrid system. Such a system fosters the synergistic use of multiple energy sources across different regions, significantly amplifying the uptake of renewable energy [3]. MGs are versatile in their operational modes, capable of functioning either grid-connected or in an independent manner, harnessing local generation and energy storage for autonomous power management [4]. As the electricity market undergoes progressive liberalization and reform, MGs are increasingly permitted to engage in electricity trading as entities with distinct interests [5].

In the conventional framework of the electricity market, each participant possesses an individual optimization objective, tailored to fulfill their specific interests and attain a Nash equilibrium [6]. However, the intricate architecture and mathematical intricacies inherent in interconnected Microgrids (MGs), compounded by the absence of comprehensive global information and other contributing factors, render the discovery of a Nash equilibrium a challenging endeavor. This complexity results in diminished convergence efficiency and an overall instability within the model. Within the multi-MG marketplace, the interests of each stakeholder are heterogeneous, giving rise to a myriad of intricate and varied trading dynamics [7]. Consequently, pricing mechanisms are instituted as a strategic game-theoretic approach to dissect these interactions. Nonetheless, when scrutinizing the competitive interactions among diverse entities within a multi-MG ecosystem, it is imperative to incorporate constraints pertaining to distributed power generation—a factor that is frequently overlooked in such analyses. Distributed control frameworks and multi-agent consensus algorithms have gained widespread adoption in the management of AC-DC hybrid MG collectives, streamlining operations scheduling, power distribution coordination, and the collective management of multiple MGs and photovoltaic systems [8]. Decentralized control strategies, predicated on consensus algorithms, offer a distinct advantage over centralized control paradigms by minimizing the requisite communication infrastructure, thereby expanding their utility and application across various sectors.

At present, a robust body of research on microgrid energy management is being advanced by scholars worldwide. In the realm of hybrid energy storage systems for photovoltaic power generation, Literature [9] implements a Particle Swarm Optimization (PSO) algorithm to address strategic planning. Moving forward, Literature [10] constructs and addresses an economic operation model utilizing CPLEX, which integrates carbon emission costs in the pursuit of minimizing the daily operational expenditures of a micro-energy network. The work in Literature [11] is dedicated to enhancing the operational parameters within a single-machine hybrid microgrid context. Literature [12] employs linear programming, model predictive control, and optimization techniques. In literature [13], a mixed-integer linear programming model based on comprehensive demand response is solved with the assistance of CPLEX. Literature [14] thoroughly examines the operational characteristics and mathematical model of the distributed micro-source in the system, emphasizing energy storage architecture for mathematical modeling. Literature [15] explores the technical and economic model, as well as sensitivity analysis of available resources in rural communities in India. Literature [16] proposes an enhanced PSO algorithm that considers adaptive variation and employs time-varying acceleration coefficients and compression factors as weight reference factors. Literature [17] introduces a hybrid AC-DC microgrid power management method based on a sagging control strategy, consisting of two stages: family update operator and separation operator. Presently, in pursuit of minimizing the proportion of energy storage batteries and elevating the penetration of renewable energy sources, scholars are leveraging Homer simulation to optimize microgrid configurations. Some notable efforts encompass microgrid energy management strategies, energy optimization models, and real-time microgrid operation control tactics within grid-connected modes. However, many of these initiatives are typically focused on singular objectives, such as predefined energy storage targets, accompanied by relatively simplistic economic analyses of the microgrid. PVsyst software, on the other hand, finds its primary application in modeling and simulating photovoltaic power generation systems. This model delves into a comprehensive analysis of myriad factors influencing power generation. Nonetheless, it's worth noting that PVsyst software entails a higher number of control variables. The trajectory of simulation software continues to be one of ongoing enhancement and refinement.

This research constructs a comprehensive Microgrid (MG) energy system model at the Point of Common Coupling (PCC), integrating photovoltaic (PV) devices, battery energy storage systems, and micro-gas turbine units. Our approach strategically harnesses the Particle Swarm Optimization (PSO) algorithm for extensive global exploration to navigate the solution space effectively. To complement this, we introduce a chaotic algorithm specialized for local exploration, which significantly refines the search for optimal solutions. We have crafted an algorithmic flowchart for PSO that integrates chaotic local search principles, thereby amplifying the local search capabilities of the particle swarm and circumventing the entrapment in local optima. The results substantiate that our hybrid algorithm markedly enhances both the convergence velocity and the overall convergence proficiency, as evidenced by a comparative analysis. This novel methodology has been validated for its efficacy in curtailing operational costs, ensuring the steadfast performance of the micro-energy network, optimizing the harnessing of renewable energy resources, and streamlining the economic dispatch of the network.

The unique aspects and contributions of this paper are as follows.1) Integration of Optimization Techniques: The paper introduces a novel approach by combining chaotic optimization with PSO, enhancing local search capabilities and mitigating issues related to local minima entrapment.

2) Harmonic Distortion Analysis: The study delves into harmonic distortion arising from partial linearity in the system, providing insights into total harmonic distortion influenced by frequency and power supply magnitude.

3) MG Energy System Model: The research establishes a comprehensive MG energy system model at the Point of Common Coupling (PCC), incorporating key elements such as photovoltaic devices, battery energy storage, and micro-gas turbines.

4) Enhanced Global Optimization: Our algorithm combines chaotic local search with PSO to escape local optima effectively in micro-energy network optimization.

5) Application-specific Adaptations: Tailored for micro-energy networks with optical storage, our algorithm is optimized to meet their unique needs, enhancing its effectiveness in real-world scenarios.

This paper mainly consists of five parts, the first part is the introduction, the second part is the advanced energy systems, the third part is the methodology, the fourth part is the result analysis and discussion, and the fifth part is the conclusion.

2 Advanced energy systems

2.1 Micro-energy system structure composition

Energy consumption culminates in diverse forms, including thermal energy for heating and cooling, electrical power, and gas. These compact, integrated energy systems facilitate the efficient utilization, strategic storage, and seamless interconversion of an array of energy sources [18]. Moreover, micro-energy systems are capable of participating in energy trading with external networks, providing energy services to customers, and generating revenue through mechanisms of energy conversion, storage, and distribution [19]. A schematic representation of a quintessential micro-energy system is depicted in Fig. 1.Fig. 1 Micro-energy system structure.

Fig. 1

2.2 Model for micro-grid energy management

Within a micro-energy system, the efficient and sustainable transformation, transmission, storage, and distribution of multiple energy sources are paramount. The integration of an energy storage system is pivotal, as it enables the strategic planning of energy consumption and generation across temporal and spatial dimensions, culminating in the enablement of energy sharing [20]. This underscores the system's critical role in promoting the agile conversion and exhaustive exploitation of diverse energy sources.

As energy harvesting and power electronics continue to advance, the progression of distributed energy generation systems that harness wind and solar energy is gaining momentum. This manuscript introduces an integrated photovoltaic storage energy system adept at transitioning seamlessly between grid-connected and off-grid operational modes. This system is conceptualized as a streamlined DC microgrid, as portrayed in Fig. 2. It comprises a photovoltaic (PV) module array, an ensemble of lithium-ion battery module systems, a micro gas turbine unit, and a versatile inverter system that facilitates dual-mode functionality for both grid-connected and off-grid applications.Fig. 2 Simplified DC MG system.

Fig. 2

2.2.1 Operational mode of the photovoltaic array

The photovoltaic (PV) array serves as the core local power generation unit, supplying energy to both the local load and the battery energy storage system. The PV modules are interfaced with the DC bus through a DC-DC boost converter to ensure efficient energy transfer. Standard practice involves equipping the PV system with a Maximum Power Point Tracking (MPPT) mechanism for optimal energy extraction. In instances where the local load's power demand surpasses the PV system's output, the system defaults to MPPT mode, prompting the battery energy storage system to discharge and bridge the power deficit. Conversely, when the PV system's power output surpasses the local load's consumption, the system remains in MPPT mode, with the surplus electrical energy directed towards charging the battery energy storage system, thereby utilizing the excess power effectively. Should the battery energy storage system reach full charge capacity and the PV system continues to generate excess power beyond the local load's needs, the PV system transitions to a constant voltage mode. This mode is instrumental in preserving the stability of the local DC bus voltage, aligning the power output of the PV system with the load's consumption.

2.2.2 Key parameters of the battery energy storage system

In this paper, a lithium battery energy storage system is used. Lithium battery energy storage systems have small volumes, mature technologies, high utilization rates, are suitable for high-capacity installations, and can smooth out fluctuations in photovoltaic power generation. The battery energy storage module is connected to the DC bus via a bidirectional DC converter. To ensure safety, the voltage of the battery energy storage module should be less than or equal to 48VDC, and an isolated transformer is used. State of Charge (SoC) and output power are critical parameters of the battery. SoC represents the ratio of the remaining battery capacity to the total capacity. If Pbat is the output power of the battery energy storage system, the power constraints of charge and discharge are shown in Equation (1).(1) −Pbat_max≤Pbat≤Pbat_max

where Pbat_max is the maximum charge and discharge power of the lithium battery.

To prevent damage to the lithium battery due to over-discharge or over-charge, it is necessary to ensure that the lithium battery is in a safe State of Charge (SoC) during operation. The SoC of the lithium battery is shown in Equation (2).(2) soc(t)=soc(t−1)−Pes.c/dΔtC

where soc(t) represents the state of charge of the lithium battery at time t. soc(t−1) represents the state of charge of the lithium battery at time (t-1). Pes.c/d represents the power of the lithium battery energy storage system at the previous moment. C represents the capacity of the energy storage device, and the conditions satisfied by C are shown in equation (3).(3) Csoc−min≤Csoc≤Csoc−max

where Csoc−min is the minimum allowable value for discharging a Li-ion battery. Csoc−max is the maximum allowable value for charging a Li-ion battery.

2.2.3 Micro gas turbine

The mathematical model of the micro gas turbine (MT) is shown in Equation (4) and Equation (5).(4) Qmt,h(n)=Pmt,e(n)⋅(1−ηmt,e(n)−ηL)ηmt,e(n)

(5) Vmt(n)=Pmt,e(n)⋅Δnηmt,e(n)⋅RLHVT(n)

where Pmt,e(n) is the electrical power output of MT at time n. ηmt,e(n) is the power generation efficiency of MT. Qmt,h(n) is the residual heat of MT. ηL is the heat loss efficiency. Vmt(n) is the volume of natural gas consumed by MT. and RLHVT(n) is the low level heat value constant of natural gas.

2.2.4 Integrated grid-connected inverter and DC bus

The inverter connects the DC bus, the local load, and the distribution network. Let the inverter output power be Uinv, the local load power be U1, and the grid-connected power be Ugrid. All three satisfy Equation (6).(6) Uinv=U1+Ugrid

2.3 The impact of models on cost reduction

The proposed model is designed to optimize the scheduling of micro-energy networks integrated with optical storage, prioritizing the enhancement of renewable energy utilization as a core objective. By maximizing the capture and integration of renewable energy sources, notably photovoltaic power generation, the model aims to diminish reliance on traditional energy sources and their associated costs. Specifically, three key aspects will be delved into.1 Mitigating peak load demands

Enhanced absorption of renewable energy plays a crucial role in mitigating peak load demands within the micro-energy network. By optimizing the utilization of renewable energy sources, such as photovoltaic power generation, the system can capitalize on abundant energy availability during peak sunlight hours. This surplus renewable energy can be directly utilized to meet the peak energy demands of the network, thereby reducing the reliance on conventional energy sources, such as micro-gas turbines or grid-purchased electricity, during peak load periods. Consequently, effective management of peak load demands through increased renewable energy absorption contributes to significant cost savings by minimizing the need for expensive energy procurement from external sources.2 Reducing reliance on grid-purchased electricity during high-priced periods

The optimization of renewable energy absorption also leads to a reduction in the reliance on grid-purchased electricity, particularly during high-priced periods. By maximizing the utilization of locally generated renewable energy, the micro-energy network can strategically allocate resources to align with time-of-use pricing mechanisms. During periods of elevated electricity prices, the system can prioritize the utilization of renewable energy sources, thereby reducing the need to purchase electricity from the grid at premium rates. This proactive approach to energy management not only minimizes operational costs but also enhances the overall economic efficiency of the micro-energy network by capitalizing on cost-effective renewable energy resources.3 Facilitating efficient energy storage and distribution

Furthermore, the optimization of renewable energy absorption facilitates efficient energy storage and distribution within the micro-energy network. By integrating energy storage systems, such as batteries, with the renewable energy infrastructure, surplus energy generated during periods of peak production can be stored for later use during periods of high demand or low renewable energy availability. This dynamic balancing of energy supply and demand enhances the overall resilience and reliability of the micro-energy network while reducing the need for grid-purchased electricity as a backup power source. Moreover, efficient energy storage and distribution mechanisms enable the system to capitalize on favorable market conditions, such as off-peak pricing, by storing excess energy for later use, thereby further optimizing operational costs and enhancing economic viability.

3 Methodology

This study presents a groundbreaking energy management strategy for Microgrids (MGs) that integrates Chaotic Local Search (CLS) with Particle Swarm Optimization (PSO). This novel approach adeptly mitigates the limitations of local search within particle swarm algorithms, circumventing the entrapment in local minima. The fusion of CLS and PSO is instrumental in attaining global optimization across micro-energy networks equipped with optical storage. The PSO algorithm lays down the groundwork for global exploration, leveraging the collective intelligence of particles to efficiently traverse the search space. In tandem, the chaotic local search introduces an element of stochasticity and exploratory power, enhancing the PSO's capacity to evade local optima and uncover more advantageous solutions. This strategic integration harnesses the combined strengths of both methodologies. PSO offers a resilient framework for extensive global exploration, while chaotic local search introduces diversity and randomness, averting premature convergence, and propelling the discovery of optimal solutions. This cohesive integration significantly bolsters the algorithm's convergence velocity and global optimization prowess, leading to reduced operational expenditures and elevated system performance. The simulation outcomes corroborate the exceptional efficacy of this methodology in optimization endeavors. Ultimately, the proposed strategy is deployed within a micro-energy network context, ensuring a reduction in operational costs while accounting for constraints related to micro gas turbine power, photovoltaic generation, and battery storage.

Drawing from the model delineated above, this paper's MG energy system encompasses a photovoltaic power generation segment, a battery storage segment, a micro gas turbine power generation segment, a Hondo load, and an integrated distribution network. We introduce an MG energy management strategy that amalgamates chaotic local search with the particle swarm algorithm, merging the strengths of both. This strategy not only remedies the local search inadequacies and the propensity for local minima encountered in particle swarm algorithms but also addresses the precision and temporal consumption concerns associated with chaotic local search outcomes. The algorithmic fusion's flowchart is illustrated in Fig. 3.Fig. 3 The algorithm flow chart after fusion.

Fig. 3

3.1 Chaotic local search

Chaos is an intrinsically regular and seemingly random nonlinear phenomenon with extreme sensitivity to initial conditions, ergodicity, and random-like properties [21]. Chaos can traverse all states in the phase space without repeating, and can traverse every point in the problem state space in a sufficiently long time. Therefore, chaotic variables are mapped to the problem state space by using chaotic ergodic property, and the search process of optimal solution on the problem state space is transformed into the ergodic process of chaotic trajectory. Chaotic orbits exhibit extreme sensitivity to initial conditions. Arbitrarily close points in the orbit will gradually separate and become unrelated, and the form of the orbit is determined by controlling parameters for the growth rate. Logistic is usually used to generate mapping variables. When the growth rate control parameter μ = 4, the chaotic orbit forms a full mapping of the interval [0,1]. Chaotic motion can traverse each point in the phase space without repetition for a sufficiently long period of time, so this property can be exploited for optimal solution search. The PSO algorithm is widely used in optimization solution problems because of its low requirement for the objective function and its good global search capability. However, it also has shortcomings such as early convergence and weak local search capability [18]. In order to overcome the lack of local search ability of PSO algorithm and avoid falling into local minima, chaos optimization is incorporated into the PSO algorithm. Chaos optimization is used to perform local search of particle swarm to jump out of the local optimal search region, avoid falling into local minima and achieve the optimal value on the global scale [22]. The process of chaotic local search algorithm mainly includes the steps of chaotic variable generation, chaotic variable modulation to the problem space, solving the objective function using chaotic sequence values, optimal solution determination, and quadratic optimization. The specific process steps are shown below.(1) Mapping the decision variable interval to the chaotic variable interval. Let the decision variable ixn(x=1,2,⋯,t),ixn∈[imin,x,imax,x] at moment n, then the corresponding value of its mapping to the chaotic variable interval is shown in Equation (7).

(7) jxn=ixn−imin,xxit−xmax,i,x=1,2,⋯,t

imin,x is the minimum value of ix and imax,x is the maximum value of ix. jxn is the corresponding value of ix in the interval of chaos variables. xit represents the decision variable ix at a specific time t within the chaotic local search algorithm.(2) Substitute yxn into the logistic chaotic mapping of Equation (8), take μ=4, and iterate to generate chaotic variables.

(8) jxn+1=4jxn(1−jxn),x=1,2,⋯,t

jxn+1 is the chaotic mapping value of ix at time n+1.(3) Transform jxn+1 into a decision variable ixn+1 according to Equation (9).

(9) ixn+1=imin,x+jxn+1(imax,x−imin,x),x=1,2,⋯,t

ixn+1 is the decision variable value of ix at time n+1.(4) The adaptation value is calculated by substituting ixn+1 into the objective function and compared. If the new solution is better than the current optimal solution or reaches the maximum number of chaotic iteration steps, it will be used as the local search result. Otherwise, let n=n+1 and continue the search in step (2).

In this paper, the concept of the constraint determination function is introduced as a crucial step in the optimization algorithm. This function is employed to ensure that the initial vector satisfies the constraints of the optimization problem. Fig. 4 clearly illustrates the overall functional process of the constraint determination function.Fig. 4 Flowchart of constraint determination function.

Fig. 4

This flowchart outlines the steps involved in determining constraints for the optimization process. It starts with initializing parameters, followed by evaluating the constraints. If the constraints are satisfied, the process ends; otherwise, it continues until the constraints are met.

In general, optimization problems in engineering technology are often mathematical models with constraints. The following process can be used to implement the algorithm for solving such maximum (minimum) value problems.① Constraint determination. The initial vector I is a random sequence of uniformly distributed 0–1. It is modulated into the vector Temp_I on the problem space using the modulation method. The Temp_I vector is then substituted into the constraint determination function to determine whether the current initial value satisfies the constraint. If it does, the subsequent steps of the algorithm are executed, otherwise the execution ends and the initial value is reselected until an initial value satisfying the constraint is found.

② If the initial vector satisfies the constraint, the initial vector is used to calculate the result of the objective function Max_F. It is used as the initial value of the optimal result.

③ Generate the logistic chaotic sequence using the iterative method. The corresponding result Temp_F is calculated by mapping it to the problem space definition domain and substituting it into the objective function and comparing it with Max_F. If Temp_F > Max_F, the current chaotic sequence is assigned to the vector Max_I and Temp_F is assigned to Max_F.

④ Determine whether the current optimal solution satisfies the requirements. The search space is further narrowed to find the optimal solution based on the search space obtained from the first chaos optimization. Then the second chaos optimization is performed until the optimal solution satisfies the requirements.

The constraint determination function plays a critical role in guiding the optimization algorithm by ensuring that the initial vectors adhere to the defined constraints. The combination of constraint determination with the subsequent chaotic local search contributes to the algorithm's ability to escape local minima and find optimal solutions in a global search space. This step is crucial for maintaining the validity of the solutions throughout the optimization process.

3.2 Improvement of particle swarm algorithm

3.2.1 Basic principle of particle swarm algorithm

Assume that the target search space is D-dimensional, and each particle swarm consists of T particles. The x-th particle can be expressed as a D-dimensional vector, i.e., Ix=(ix1,ix2,⋯，ixD). Its particle “flight” speed can also be expressed as a D-dimensional vector, i.e., Vx=(vx1,vx2,⋯,vxD). Then the individual extreme value of the particle can be expressed as pbest=(px1,px2,⋯,pxD). The global extreme value of the whole particle swarm can be expressed as gbest=(g1,g2,⋯, gD). When the individual extreme value and the global extreme value are found, the particles will be updated by Equation (10) and Equation (11) to update the velocity and position.(10) Vxdn+1=ωVxdn+c1r1(Pxdn−Ixdn)+c2r2(Pgdn−Ixdn)

(11) Ixdn+1=Ixdn+Vxdn+1

where ω,c1andc2 denote the population cognitive coefficient. ω is the inertia weight, which enables the particle to maintain the motion inertia. The learning factor c1,andc2 are the parameters that regulate the relative importance of pbest, gbest. Generally, c1andc2 are taken as 2. pbest denotes the position where the particle itself finds the optimal solution. gbest denotes the position where the population finds the optimal solution. r1andr2 are random variable subjects to a uniform distribution in the interval [0,1].

3.2.2 Improvement of learning factor

The learning factors c1,c2 are important parameters that control the motion of particles toward pbest and gbest. They have a great influence on particle self-awareness and population awareness, respectively. When c1=0, the self-awareness of the particle is missing and the algorithm has the ability to expand the search space. However, due to the lack of self-awareness, i.e., local search capability, the algorithm tends to be premature and results in a local optimum situation. When c2=0, the particles will lose the group cognition, i.e., global search. In this case, the particles in the algorithm will blindly conduct a random search, and the chance of finding the optimal value of the population will be reduced. In order to obtain a better equilibrium of population search, a learning factor strategy based on simultaneous randomization is proposed in this paper.

Let c1=c2=z, and z take the 9 sets of data 0.5, 0.7, 0.9, 1.1, 1.3, 1.5, 1.7, 1.9 and 2.0. The three distinct test functions were sequentially explored, and the random interval for composing the learning factor, showing relatively effective search outcomes, was determined through comprehensive study and analysis of experimental results. In this study, Quadric-Noise, Zakharov, and Rosenbrock are selected as the test functions, and different learning factors of z are iteratively searched for. The specific mathematical descriptions of the three test functions are shown in Table 1.Table 1 Test function formula.

Table 1Test Functions	Test function expressions	Particle search range	Optimal solution	
Quadric_ Noise [22]	f1(i)=∑x=1Dxix4+rand[0,1)	[-100,100]D	0	
Zakharov [23]	f2(i)=∑x=1Dix2+(∑x=1D0.5xix)2+(∑x=1D0.5xix)4	[-5,10]D	0	
Rosenbrock [24]	f3(i)=∑x=1D[100(ix+1−ix2)2+(ix−1)2]	[-30,30]D	0	

For z taking different simultaneous learning factors, the optimization-seeking simulations are performed by each of the three different test functions in Table 1 in turn. The number of particle populations is 30. The number of iterations of the test function is 1000. The number of simulations for each group is 30. The optimal average values obtained from the simulation experiments of the three different test functions into line graphs are shown in Fig. 5, Fig. 6, Fig. 7, respectively.Fig. 5 Quadric_Noise optimization results.

Fig. 5

Fig. 6 Zakharov optimization results.

Fig. 6

Fig. 7 Rosenbrock optimization results.

Fig. 7

From the analysis of simulation results in Fig. 5, it can be seen that for Quadric_noise function, when the interval of learning factor z is [1.3, 2.0], the accuracy of PSO algorithm's iterative optimization results is better than the other two test functions. As for Zakhatov function, according to the simulation result analysis in Fig. 6, when the interval distribution of learning factor z is [1.3, 2.0], the accuracy of the result of iterative optimization by PSO algorithm is better than that of the other two test functions.

The optimization test results of the above three different test functions are studied and analyzed, and the learning factor is improved with balanced consideration. Compared with the traditional PSO, this paper introduces a learning factor strategy based on synchronous randomization to obtain the optimal equilibrium state of population search. This ensures a better balance between individual and group experiences, leading to a more accurate optimal solution. The learning factor z is randomly and evenly distributed in the interval [1.3, 2.0]. This can make the individual experience and the group experience can achieve a better balance state, so that the optimal solution is more accurate. The expression of the improved learning factor is shown in Equation (12).(12) c1′=c2′=1.3+(2.0−1.3)rand()

Rand(.) is a random function. The random numbers returned are uniformly distributed real numbers that are greater than or equal to 0 and less than 1.

Substituting Equation (12) into Equation (10), the “flight” speed of the particle population can be expressed as shown in Equation (13).(13) Vidt+1=Vidt+c1′r1(Pidt−Xidt)+c2′r2(Pgdt−Xidt)

3.2.3 Parameter settings

To ensure the transparency and reproducibility of our proposed algorithm, we detail the key parameter settings in this section. The selection of these parameters is aimed to ensure the efficiency and effectiveness of the algorithm.

Inertia Weight (ω): The inertia weight is a pivotal parameter that governs the movement of particles within the search space, dictating the ability of particles to maintain their current search direction. We set its initial value to 0.9 and linearly decrease it to 0.4. This strategy facilitates extensive exploration in the early stages of the algorithm and gradually shifts focus to local search as iterations progress, thus avoiding entrapment in local optima.

Individual Learning Factor (c1): This parameter represents the ability of particles to utilize their own best position information. It is set to 2.0, a common value that helps balance the particle's memory capability with the exploration of new solutions.

Social Learning Factor (c2): The social learning factor determines the extent to which the best solution from the particle swarm is utilized. Also set to 2.0, it facilitates information exchange among particles, thereby accelerating convergence.

Search Interval [imin,imax]: The search interval defines the range within which particles search across each dimension. We set it to [−100, 100], a widely used range suitable for a variety of optimization problems.

Maximum Number of Iterations (W): The maximum number of iterations determines the run time of the algorithm. Set at 1000, it provides the algorithm with ample time to explore the solution space and identify high-quality solutions.

Maximum Number of Chaotic Search Steps (Max C): Chaotic search is an effective local search technique that enhances the global search capability of the algorithm. We set it to 10 % of the maximum number of iterations to ensure the algorithm maintains diversity while conducting effective local searches.

Number of Particles (w): The number of particles affects the search capability and computational complexity of the algorithm. Set at 30, it provides a balance between computational efficiency and adequate search capability.

Logistic Map Parameter (for Chaotic Local Search): This parameter governs the generation of chaotic sequences and is crucial for the effectiveness of chaotic search. We set it to 3.9, a commonly used value that produces rich chaotic characteristics.

Number of Chaotic Iterations (for Chaotic Local Search): The number of chaotic iterations determines the depth of chaotic search. Set at 50, it allows the algorithm to conduct an in-depth search within local areas without incurring excessive computational costs.

These parameter settings are designed to provide a balanced search capability, ensuring that the algorithm can effectively explore the solution space and avoid becoming trapped in local optima.

3.3 Optimization algorithm model

PSO algorithm regards each search problem as a particle with only speed and position in an N-dimensional space, which iteratively searches for the optimal solution. The fitness of the current position of each particle is evaluated by the objective function. In each iteration process, particles determine their movement in the next step by tracking their current best position (pbest) and the global best position (gbest) of the whole particle swarm.

Let I be a swarm of w particles in T-dimensional space, denoted as I=[i1,i2,⋯,iw]. The position of the x-th particle at time n is denoted as ixn=(ix1n,ix2n,⋯,ixtn), and its velocity is denoted as qxn=(qx1n,qx2n,⋯,qxtn). The individual best position is denoted as uxn=(ux1n,ux2n,⋯,uxtn). The global best position of the particle swarm I is denoted as uan=(ua1n,ua2n,⋯,uatn). After determining the individual best position uxn and the global best position uan, each particle updates its position according to Equation (14) and updates its velocity according to Equation (15).(14) vxdn+1,z+1=ωvxdn,z+c1r1(uxdn,z−qxdn,z)+c2r2(uadn,z−qxdn,z)

(15) ixdn+1,z+1=ixdn,z+qxdn+1,z+1

where d∈[1,T].x∈[1,w]. z is the number of iterations. r1 and r2 are random numbers uniformly distributed on the (0,1) interval. c1 and c2 are the learning factors, and ω is the inertia factor.

The steps of the algorithm implementation process are shown below.(1) Select model scenarios and the objective function. Different model scenarios have different system parameters. Build a data module according to the model scenario system parameters, and use the data to initialize the parameters and select the objective function.

(2) Initialize the swarm of particles. The position and velocity of each particle are initialized with random values uniformly distributed from 0 to 1. And set parameters such as inertia factor ω, learning factors c1 and c2, search interval [imin,imax], the maximum number of iterations W, the maximum number of chaotic search steps Max C, and the number of particles w.

(3) The adaptation value of each particle is calculated by the objective function, and the adaptation degree of each particle is evaluated. The current position and adaptation value of each particle is stored in its corresponding pbest. The position and adaptation value of the whole optimum is stored in gbest.

(4) Update the position and velocity of each particle based on Equation (13) and Equation (14), respectively.

(5) Calculate the corresponding objective function value by substituting each particle in the objective function, incorporating the introduced learning factor strategy for optimal search. A certain percentage of p% of the particles in the particle population with the best value of the fitness function is retained, and a chaotic local search is performed on them. The pbest and gbest of each particle are updated accordingly. The present study introduces a learning factor strategy based on synchronous randomization to attain the optimal equilibrium state of population search, thereby facilitating a more balanced integration of individual and group experiences. As a result, the proposed method outperforms the traditional PSO, leading to more accurate optimal solutions.

(6) If the chaotic local search meets the preset accuracy requirement or reaches the maximum number of iterations, the search is terminated, and the results are output. Otherwise, the search proceeds to the next step. In this paper, the termination condition is set as the maximum number of iterations. In the second step, the maximum number of iterations is defined as M. When the number of iterations exceeds M, the search is stopped, and the optimal solution is output. Otherwise, the process proceeds to step 7.

(7) Shrink the search area according to Equation (16) and Equation (17). Where r∈(0,1), ia,x means the x-th dimensional vector value of the current pbest.

(16) imin,x=max{imin,x,ia,x−r*(imax,x−imin,x)}

(17) imax,x=max{imax,x,ia,x−r*(imax,x−imin,x)}

imin,x is the minimum value of the current pbest x dimension. imax,x is the maximum value of the current pbest x dimension.(8) Generate a random 1-p% proportion of particles in the shrunken new region to replace the poorly performing particles in the original particle population. Then turn to step (2) to continue the execution. The implementation flow of the particle swarm algorithm based on chaotic local search is shown in Fig. 8.Fig. 8 Particle swarm algorithm flow based on chaotic local searched.

Fig. 8

3.4 Daily operation optimization of micro-energy networks

3.4.1 Objective function

The objective function of this paper is the lowest operating cost minF(n) of micro-energy network. This cost includes routine running cost F1(n) and primary energy conversion cost F2(n). Its expression is shown in Equation (18).(18) minF(n)=F1(n)+F2(n)

minF(n) is the minimum operation cost of micro-energy network.(1)Operating costs F1(n).

The operating cost includes the purchased electricity cost Ce(n) and purchased gas cost Cgas(n) of the micro-energy network. The specific expressions are shown in Equation (19), Equation (20) and Equation (21).(19) F1(n)=Ce(n)+Cgas(n)

(20) Ce(n)=∑n=124(Cgrid⋅Pgrid)

(21) Cgas(n)=CNG⋅VMT(n)=CNG⋅Pmt,e(n)⋅Δnηmt,e(n)⋅RLHVT(n)

where Cgrid and Pgrid are the purchase price and purchase power respectively. CNG is the price of natural gas.(2)Primary energy conversion cost F2(n), which is expressed as shown in Equation (22).

(22) F2(n)=μe⋅Pgrid+μr⋅VMT

where μe is the unit of electrical energy and μr is the cost factor generated by the primary energy consumed by natural gas. The primary energy conversion cost (F2(n)) in Equation (22) captures the energy conversion dynamics between different system nodes. Specifically, F2(n) is composed of two components: μe⋅Pgrid and μr⋅VMT. This equation inherently incorporates the energy conversion constraints between the purchased electricity Pgrid and the micro gas turbine VMT.

3.4.2 Constraint conditions

3.4.2.1 The micro-energy network power and capacity constraints are shown in Equation (23)

(23) δmtγminmtPmtN,e⩽Pmt,e⩽δmtPmtN,e

where, δmt is the start/stop status of the micro gas turbine. The value of 0 indicates the stop status and the value of 1 indicates the operation status. γminmt is the minimum load rate of the micro gas turbine. PmtN,e is the rated output of the micro gas turbine.

3.4.2.2 The PV output force constraint is shown in Equation (24)

(24) 0⩽Pgt(n)⩽Pgt,max(n)

where, Pgt(n) is the output of PV. Pgt,max(n) is the maximum output of PV. This constraint ensures that the generated power (Pgt(n)) remains within the permissible range, contributing to the overall energy conversion dynamics of the micro-energy network.

3.4.2.3 Battery constraint

1) The battery charge state constraint is shown in Equation (25).

(25) socmin⩽soc⩽socmax

soc indicates the charged state of the battery. socmin indicates the minimum charge status of the battery. socmax indicates the maximum charged state of the battery.2) Battery charging and discharging power constraints as shown in Equation (26).

(26) 0⩽Pes.c/d⩽Pes.c/dmax

where, Pes.c/d is the charge/discharge power. Pes.c/dmax is the maximum charge/discharge power.3) The energy storage capacity is kept constant throughout the scheduling cycle, as shown in Equation (27).

(27) EesH=Ees0

where Ees0andEesH are the stored energies at the beginning and end of the dispatch, respectively.

The pseudocode for the integrated Chaos Local Search and Particle Swarm Optimization (CLS-PSO) algorithm which proposed in this paper is as follows.Inputs:- Objective Function F

- Constraint Function G

- Dimension D

- Number of Particles w

- Maximum Number of Iterations MaxIter

- Maximum Number of Chaotic Search Steps MaxC

- Learning Factor Range zRange

	
Initialization:1 Initialize the swarm of particles I with w particles in D-dimensional space.

2 Initialize the position and velocity of each particle randomly within the search space [i_min, i_max].

3 Set parameters such as inertia factor ω, learning factors c1 and c2, search interval [i_min, i_max], and the number of particles w.

	
Algorithm:1 Evaluate the adaptation value of each particle using the objective function F and store the current position as p_best.

2 Determine the global best position g_best of the entire particle swarm.

3 Update the position and velocity of each particle based on Equations (10), (11), incorporating the introduced learning factor strategy for optimal search.

4 Perform a chaotic local search on a certain percentage p% of the particles in the population with the best fitness values.

5 Update p_best and g_best based on the results of the chaotic local search.

6 Repeat steps 3–5 until the termination condition is met (e.g., maximum number of iterations reached).

7 Output the best solution found.

	
Chaotic Local Search:1 Generate chaotic variables according to Equations (8), (9).

2 Transform the chaotic variables into decision variables.

3 Calculate the adaptation value using the objective function and compare with the current optimal solution.

4 If the new solution is better or the maximum number of chaotic search steps is reached, use it as the local search result; otherwise, continue the search.

	
Improvement of Learning Factor:1 Randomly select z from the range zRange.

2 Incorporate z into Equations (13), (14) to update the particle flight speed.

Constraint Determination Function:1 Initialize the initial vector I as a random sequence of uniformly distributed 0–1.

2 Modulate I into Temp_I using the modulation method.

3 Determine if Temp_I satisfies the constraints. If not, reselect until a satisfying initial value is found.

4 Use the satisfying initial value to calculate the objective function result Max_F.

5 Generate the logistic chaotic sequence and calculate Temp_F.

6 Update Max_I and Max_F if Temp_F is better.

7 Narrow the search space based on the first chaos optimization and perform the second chaos optimization until the optimal solution satisfies the requirements.

	

4 Result analysis and discussion

To substantiate the effectiveness of the proposed algorithm, this study scrutinizes its efficacy through two lenses: a comparison of algorithm performance and an evaluation of the update curves of adaptive value iteration. Subsequently, by applying the algorithm model in diverse scenarios, we further validate its efficiency, aiming to minimize the operating cost of the micro-energy network. Operation cost includes daily operation cost F1(n) and primary energy conversion cost F2(n).

4.1 Comparison of algorithm performance

In order to facilitate a more intuitive comparative analysis of the improved algorithms proposed in this study, the algorithms presented in this paper, as well as those documented in the literature [25,26], were chosen for testing the MG energy management strategy problem. Literature [25] introduces an enhanced PSO algorithm based on Bat Swarm Algorithm (BSA), addressing the modification of BSA's mutation and crossover operators within the original PSO framework. This adaptation enhances convergence speed through localized interactions. Literature [26] presents an adaptive control strategy grounded in the PSO algorithm. The study employs steady-state power of the filter power supply as the benchmark output power for the battery, specifically concerning steady-state current output. The approach constructs a control structure centered on power difference, complete with an optimized PID controller. The experimental assessment entails a comparison and analysis of the PSO algorithm against the designated test using MATLAB/Advisor. Fig. 9 illustrates a comparative representation of the algorithm's progression as discussed in this paper, along with references [25,26]. The population size employed was 200, and the number of iterations amounted to 1000.Fig. 9 Iterative convergence of the algorithm.

Fig. 9

Table 2 provides a detailed analysis of the optimal solutions for the trio of algorithms under discussion. It presents a comparative examination of the optimal solutions yielded by each algorithm. The results from this study reveal that the optimal solution generated by our proposed algorithm oscillates within a narrow band of 48.2–51.7, showcasing exceptional performance in consistently attaining the optimal solution. As illustrated in Fig. 9, while the objective function values for all algorithms exhibit a gradual decline and eventual stabilization with escalating iterations, the algorithm referenced in literature [25] is inclined to converge prematurely at a local optimum. In stark contrast, our proposed algorithm incorporates chaotic optimization techniques throughout its iterative process. This integration significantly amplifies the local search capability, engendering a more rapid convergence rate and an enhanced capacity to identify superior solution values. The algorithm introduced in this paper has proven its proficiency in securing the global optimal solution when addressing the optimal scheduling challenges within micro-energy networks.Table 2 Comparison of optimal solutions of algorithms.

Table 2Algorithm name	Number of iterations	Optimum value	Worst value	Average value	
Proposed	600	48.2	51.7	49.5	
PSOBSA [25]	500	49.3	53.5	51.2	
PSO-PID [26]	300	63.4	71.1	65.3	

Table 3 shows the results of the T-test comparing the proposed algorithm with each of the comparison algorithms in terms of convergence speed and optimization accuracy. The level of significance of the difference can be assessed from the P-value results.Table 3 Analysis of statistical test results.

Table 3Algorithm comparison	Convergence speed	Optimization accuracy	
Proposed vs. PSOBSA [25]	Mean difference: 55	Mean difference: 2.3	
Standard deviation: 10	Standard deviation: 1.2	
t-value: 4.72	t-value: 3.81	
Degrees of freedom: 78	Degrees of freedom: 78	
P-value: <0.001	P-value: <0.001	
Proposed vs. PSO-PID [26]	Mean difference: 55	Mean difference: 16.1	
Standard deviation: 10	Standard deviation: 3.9	
t-value: 5.27	t-value: 4.12	
Degrees of freedom: 78	Degrees of freedom: 78	
P-value: <0.001	P-value: <0.001	

4.2 Comparison of adaptive value iteration update curve

The algorithm proposed in this article was compared with literature [27,28], and [29]. Literature [27] is a multi-objective quantum behavior particle swarm optimization with improved objective weight coefficients. Literature [28] is a quantum particle swarm optimization algorithm. Literature [29] is an optimization model based on particle swarm optimization algorithm and genetic algorithm. The maximum number of iterations is 2000. Search for spatial boundaries: [−100,100] for each dimension. Termination criteria: Reaching convergence or maximum number of iterations. Fig. 10 shows the iterative update curve of the four algorithms, and the summary is shown in Table 4.Fig. 10 Comparison of algorithm fitness change curve.

Fig. 10

Table 4 Comparison of algorithm fitness change data.

Table 4Algorithm type	Fitness	The number of iterations that converge	
PSO	5.57 × 108	1136	
MOQPSO [27]	4.08 × 108	1063	
QPSO [28]	4.03 × 108	813	
GAPSO [29]	3.36 × 108	711	
Proposed	2.15 × 108	545	

The analysis of Table 4 and Fig. 10 is detailed below.(1) In QPSO, compared with MOQPSO, the convergence rate increases and the optimization results are better. It shows that the time-varying compression factor improves the efficiency and jumps out of the local search.

(2) Compared with MOQPSO, GAPSO optimizes the results and increased the number of iterations. The results show that adaptive variation jumps out of local search, but reduces the convergence speed of the algorithm.

(3) Compared with QPSO, GAPSO shows that the optimization results are better, indicating that the global convergence of adaptive variation is better.

(4) Overall, compared with the other three algorithms, the proposed algorithm has the best optimization results, and the convergence rate is only slightly lower than the algorithm in QPSO. The proposed algorithm demonstrates the integration of PSO algorithm and chaotic local search, resulting in a significant improvement in convergence speed and global convergence ability. This achievement aligns with the intended design purpose.

4.3 Application of algorithmic models in different scenarios

This study introduces a sophisticated scheduling model crafted to optimize micro-energy networks, leveraging optical storage and gas turbines. Considering the variance in dimensions among scheduling models across different scenarios, the criterion for comparison is established based on the minimal operational costs. These encompass both the routine daily operational expenditures and the costs associated with primary energy conversion. The pursuit of minimized operational costs is a testament to a refined micro-energy management strategy. To counteract the propensity for particle swarms to converge on local optima, the study employs a chaotic local search methodology to refine the Particle Swarm Optimization (PSO) algorithm. The proposed algorithm is then engaged to resolve the micro-energy model, assisting in identifying the most efficacious amalgamation of photovoltaic power generation, gas turbine power generation, and battery storage to satisfy user energy demands at designated operational intervals.(1) Scenario Description and Parameter Configuration

For the purpose of examining variations in system operational costs, the study simulates a micro-energy network model predicated on a quintessential summer day. The model's efficacy is substantiated through its application across three distinct scenarios, thereby evaluating its practical utility. The delineation of these scenarios is presented in Table 5, with the peak-to-valley time-sharing tariff and natural gas pricing detailed in Table 6. The parameters governing the micro-energy network and energy storage apparatus are enumerated in Table 7.Table 5 Scenario settings.

Table 5Scenario	Micro gas turbine	Photovoltaic	Energy storage	
Scenario 1	Yes	No	No	
Scenario 2	Yes	Yes	No	
Scenario 3	Yes	Yes	Yes	

Table 6 Time-of-use price and natural gas price.

Table 6Category	Unit price/(Yuan·(kWh)-1)	Time period	
Peak tariff	0.896	09:00–11:00, 19:00–20:00	
Parity tariff	0.572	12:00–18:00, 21:00–23:00	
Valley tariff	0.236	00:00–09:00	
Natural gas prices	3.53	/	

Table 7 Parameters of each device.(2) Modeling of battery energy storage system

Table 7Equipment parameter	Symbol	Unit	Value	
Efficiency (Mechanical to Electrical)	ηmt,e	%	33 %	
Loss Coefficient	ηL	%	9 %	
Minimum Generation Power Coefficient	γminmt		0.4	
Maximum Unit Output	PmtN,θ	kW	220	
Maximum Charging Power of Energy Storage System	Pes,cmax	kW	35	
Maximum Discharging Power of Energy Storage System	Pes,dmax	kW	35	
Minimum SOC of Energy Storage System	socmin		0.1	
Maximum SOC of Energy Storage System	socmax		0.9	
Initial Energy of Energy Storage System	Ees0	kJ	25	

Energy storage systems are integral to the functionality of microgrid systems, providing essential support to stabilize and smooth the transition between different operational modes. They act as a buffer, absorbing surplus energy during periods of peak production and releasing it during times of low demand, thereby ensuring the microgrid's stable and consistent operation.

Furthermore, these storage systems facilitate a robust power interchange with the main grid. Under typical operating conditions, the battery is governed by P/f and Q/V droop control strategies, enabling it to adjust its active and reactive power outputs dynamically in response to fluctuations in busbar voltage and frequency. This real-time compensation is pivotal for preserving the microgrid's stability and for ensuring a harmonious connection with the broader electrical grid.

The battery serves as a crucial component in balancing the intermittent nature of renewable energy sources, such as photovoltaic power generation, with the demand variability within the micro-energy network. During periods of surplus renewable energy generation, the battery efficiently stores excess energy for later use, thereby mitigating the impact of intermittency and ensuring a reliable power supply. Conversely, when renewable energy generation is insufficient to meet demand, the battery can discharge stored energy to supplement the shortfall, reducing the reliance on non-renewable energy sources and grid-purchased electricity.

Moreover, the optimization of the battery's charging and discharging cycles takes into account factors such as aging and maintenance costs. By implementing intelligent charging and discharging strategies, the battery's lifespan can be prolonged, minimizing the need for frequent replacements and reducing overall operational costs. Fig. 11, Fig. 12 provide simulation schematic diagrams of the P/f and Q/V sag control of the battery in Simulink, respectively. These visual representations offer a clear understanding of how the battery's output is regulated to maintain a stable microgrid operation and optimize power exchange with the main grid.Fig. 11 The simulation schematic diagram of P/f sagging control of battery.

Fig. 11

Fig. 12 The simulation schematic diagram of Q/V sagging control of battery.(3) Optimization calculation results

Fig. 12

To provide a comprehensive evaluation of the proposed scheduling model's effectiveness, we employed rigorous metrics and methodology to assess the observed cost reductions in various scenarios compared to the baseline scenario. The total cost reduction was calculated by considering the cumulative reduction in electricity purchase cost, gas purchase cost, operation cost, and energy conversion cost across different scenarios. By analyzing the combined effect of these cost components, we obtain a holistic view of the overall economic benefits achieved through our scheduling optimization model. Additionally, the operating cost reduction specifically focused on the reduction in daily operational expenses and primary energy conversion costs. It provides insights into the efficiency gains and cost savings resulting from the implementation of our model. Our methodology involved detailed calculations based on the specific parameters and tariffs outlined in Table 4, Table 5, Table 6, ensuring accuracy and consistency in the assessment process. The results of the economic operation cost of the micro-energy network using the algorithm of this paper in the above three different scenarios are shown in Table 8. And the output power is shown in Table 9.Table 8 Economical operation cost of the micro energy grid (cost/million yuan).

Table 8Scenario	Electricity purchase	Gas purchase	Operation	Energy conversion	Total cost	
1	2086.20	3085.24	5175.89	1689.66	6865.55	
2	1102.18	2798.75	3922.14	1159.24	5081.38	
3	1078.33	2785.41	3801.27	1089.08	4890.35	

Table 9 Microsource output power in different cases.

Table 9Scenario	Microsource output power	
MT	Grid	PV	Battery	
1	2783.6	4792.4	0	0	
2	2524.5	3209.7	2243.1	0	
3	2520.7	3208.8	2243.1	156.4	

As evident from Table 8, Table 9, Scenario 1, which does not include photovoltaic and energy storage, exhibits increased purchased power from the grid and higher output power from micro-gas turbines, resulting in a higher energy conversion cost compared to other scenarios. Therefore, Scenario 1 has the highest cost among all the scenarios. In scenario 1, PV module is added, and because of the lower cost and pollution of PV, its operation cost and primary energy conversion cost are reduced compared with scenario 1, by 24.22 % and 31.39 % respectively, thus reducing the total operation cost. Scenario 3 adds an energy storage module to Scenario 2, which further reduces the operating cost and energy conversion by 3.08 % and 6.05 %, respectively, due to the battery discharging at peak and charging at trough, while coordinating the grid and micro-combustion engine. Scenario 3 shows a 28.77 % reduction in total cost over scenario 1.

In order to further verify the robustness of the algorithm, this paper compares the standard deviation with the uniformity index. The standard deviation change curves of each scenario at different starting iteration numbers are shown in Fig. 13.Fig. 13 Curve of the robustness judgment.

Fig. 13

From Fig. 13, it can be seen that the robustness of Scenario 3 is slightly better than that of Scenario 2 when the number of starting iterations is less than 20. As the number of starting iterations moves backward, it can be found that the robustness of Scenario 3 is much better than that of Scenario 2 and Scenario 1. This is due to the fact that the learning factor strategy based on simultaneous randomization introduced in this paper significantly improves the global performance and robustness of the algorithm.

Fig. 14 presents the optimal power dispatch results for different scenarios. The findings reveal that micro-combustion turbines are less efficient than grid-purchased power during low-priced hours. Thus, grid-purchased electricity is the primary energy source, and the shortage is complemented by micro-combustion turbines. Conversely, during the high-load summer season, photovoltaic (PV) power generation may be insufficient to meet the full power demands. Therefore, the micro-combustion turbine operates at the lowest power and charges the excess power to the battery, which aids in meeting the peak load demand and reducing the operating cost. During normal electricity price periods, both micro-combustion turbines and grid-purchased electricity have comparable benefits, and their output is dependent on the energy supply cost to the load. Additionally, during periods of excess generation capacity, the battery is charged as much as possible to enable energy storage during peak periods and provide power support during peak demand periods. Finally, during peak hours, the benefits of micro-combustion turbines outweigh those of grid-purchased electricity, and the batteries are discharged to reduce the cost. To sum up, the results provide valuable insights into the optimal power dispatch strategy for micro-energy networks with optical storage. The strategy involves using grid-purchased electricity during low-priced hours, utilizing micro-combustion turbines and battery storage during high-load summer seasons, and discharging batteries during peak hours to reduce costs.Fig. 14 Optimal dispatch result of power flow for different cases.

Fig. 14

Fig. 15(a) and (b) show the power generated by the simulated battery in Simulink based on the literature [29] and the algorithm proposed in this paper, respectively. Among them, Fig. 15 (a) is the power of literature [29], and Fig. 15 (b) is the power of the proposed algorithm [29]. By comparing the Figures, it can be seen that the proposed algorithm is used to generate a more stable battery power; after a slight fluctuation within 6 s, the stability is quickly restored.Fig. 15 The power results generated by the simulation battery in Simulink.

Fig. 15

5 Conclusion

This research endeavors to introduce a scheduling optimization model tailored for micro-energy networks integrated with optical storage. The core objective of the model is to curtail daily operational expenses and the costs associated with primary energy conversion, concurrently boosting the assimilation rate of renewable energy sources. To attain an optimal solution on a global scale for the objective function, we have strategically merged the chaotic local search method with Particle Swarm Optimization (PSO) algorithms. The resultant algorithm exhibits notable enhancements in terms of convergence velocity and global optimization capacity, yielding superior solution values through its search processes. Upon applying the model across various scenarios, the proposed scheduling strategy has demonstrated a substantial reduction in overall costs, achieving a 28.77 % decrease in Scenario 3 when contrasted with Scenario 1. Moreover, in comparison to Scenario 2, the model efficaciously diminishes operational costs by 3.08 % and energy conversion expenses by 6.05 %. This model introduces an innovative algorithm for the optimal scheduling of micro-energy networks, ensuring reliable network operation while simultaneously minimizing financial outlays. It is important to acknowledge a potential limitation inherent in the proposed methodology: its resilience under extreme or unforeseen circumstances, such as abrupt shifts in energy demand or supply. Although the model is designed to reduce operational costs and augment the uptake of renewable energy, it may encounter difficulties in swiftly adjusting to sudden environmental variations or alterations in system constraints. The efficacy of the algorithm could also be contingent upon the precision of the input parameters, the accessibility of real-time data, and the dependability of energy storage systems. Future research endeavors should be directed towards scrutinizing the robustness of the algorithm within a multitude of problem domains and toward refining parameter optimization strategies. This will not only amplify the algorithm's overall effectiveness but also broaden its applicability in the realm of micro-energy network management.

Data availability statement

Data wil be made available on request.

CRediT authorship contribution statement

Ping Li: Writing – review & editing, Writing – original draft, Validation, Data curation, Conceptualization. Qian Liu: Writing – review & editing, Visualization, Software, Resources.

Declaration of competing interest

The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:Ping Li reports a relationship with 10.13039/501100004262 Zhengzhou University of Light Industry that includes: employment, non-financial support, and travel reimbursement. If there are other authors, they declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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