# Strategic Energy Management in Multi-Source Microgrids: A Bertrand Duopoly Game Algorithm Model

Abderrahmane Redouane, Rachid Saou, Sofiane Touati, Youcef Belkhier, Amrane Oukaour, Samira Chouikhi, Lyes Khoukhi

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Abderrahmane Redouane, Rachid Saou, Sofiane Touati, Youcef Belkhier, Amrane Oukaour, et al. Strategic Energy Management in Multi-Source Microgrids: A Bertrand Duopoly Game Algorithm Model. Results in engineering, 2025, pp.107765. 10.1016/j.rineng.2025.107765. hal-05323188

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# Strategic energy management in multi-source microgrids: A Bertrand duopoly game algorithm model

Abderrahmane Redouane<sup>a</sup>, Rachid Saou<sup>a</sup>, Sofiane Touati<sup>b</sup>, Youcef Belkhier<sup>c,∗</sup> <sup>●</sup>,
Amrane Oukaour<sup>d</sup>, Samira Chouikhi<sup>e</sup>, Lyes Khoukhi<sup>f</sup>

## ARTICLE INFO

### Keywords:

- Energy management

- Grid-connected

- Multi-source

- Renewable energy

- Game theory

- Bertrand duopoly

## ABSTRACT

This paper presents an innovative energy management strategy for a grid-connected multi-source system, utilizing a Bertrand duopoly game model. This approach integrates a pricing mechanism and energy availability to govern the interactions between suppliers and the consumer. We propose a negotiation-based strategy that simulates these interactions, with all suppliers participating on equal terms by simultaneously setting their selling prices. In response, to the varying prices, the consumer selects the most favorable energy source. This strategy balances energy distribution, minimizing costs for the consumer while maximizing energy utilization for the suppliers. The performance and effectiveness of the proposed algorithm were validated through extensive MATLAB/Simulink simulations. The results indicate that the model enhances operational efficiency and promotes sustainable energy management practices within hybrid energy systems. The integration of game theory allows for the optimization of interactions among stakeholders and addresses the complexities of decision-making in multi-source systems.

## 1. Introduction

Multi-sources systems have grown in popularity over the last few decades due to their capacity and flexibility, making them ideal for integrating renewable energy. A multi-source system can be considered as a medium-to low voltage distribution grid that can be connected to the main grid or operated in an isolated mode; it includes a small-scale generation system incorporating renewable energy resources with energy storage devices (ESDs) that provide electrical energy to consumers [1]. By integrating storage units, microgrids improve efficiency and reliability; these units help mitigate the intermittent nature of the renewable power generation and also maintain equilibrium between power supply and demand [2]. However, this evolution introduces new challenges that complicate decision-making processes [3]. As a result, major challenges have emerged regarding the reliability, operation, control, and management of these systems. To address these issues, advanced methods have gained increasing interest, focusing particularly

on optimizing decision-making processes as well as control and management strategies [4].

In the literature, several studies have dealt with energy management using various techniques to achieve optimal microgrid performance. Paper [5] proposes a real-time energy management system (RT-EMS) applied to a grid-connected PV system with battery storage, where pricing signals indirectly manage the system's generated power based on demand variations. The aim of the RT-EMS is to maximize profit, and Lagrange multipliers (LP) are utilized to address the optimization problem. However, the LP approach faces computational challenges when the number of variables and constraints increases. Reference [6] proposes an EMS that is used to control the batteries in a grid-connected PV/Battery system, aiming to reduce consumption costs by taking into account main grid price fluctuations, renewable generation, and demand variation. The optimization problem is addressed via mixed-integer linear programming (MILP); nevertheless, MILP struggles with nonlinear effects, which can lead to excessive complexity.

<footnote><sup>a</sup> Université de Bejaia, Faculté de Technologie, Laboratoire de Génie Électrique, Bejaia 06000, Algeria</footnote>

<footnote><sup>b</sup> Research Unit LaMOS, Exact Sciences Faculty, University of Bejaia, Bejaia, Algeria</footnote>

<footnote><sup>c</sup> University of Brest, UMR CNRS 6027 IRDL, 29238 Brest, France</footnote>

<footnote><sup>d</sup> Laboratoire de Recherche en Sciences du Numérique GREYC, Université de Caen Normandie, 6 Boulevard du Maréchal Juin, 14032 Caen, France</footnote>

<footnote><sup>e</sup> Computer Science and Digital Society Laboratory, University of Technology of Troyes, Troyes, France</footnote>

<footnote><sup>f</sup> GREYC Laboratory, ENSICAEN, University of Caen Normandie, 14000 Caen, France</footnote>

<footnote>* Corresponding author.
E-mail address: youcef.belkhier@univ-brest.fr (Y. Belkhier).</footnote>
An advanced real-time EMS for a grid-connected microgrid is proposed in [7], which aims to optimize the use of each distributed energy resource, minimizing costs and carbon emissions while maintaining load demand. A genetic algorithm (GA) is employed to solve the optimization problem. In [8], an EMS based on Fuzzy-Logic is presented for a grid-connected PV/battery system to optimize revenues through energy exchanges with the main grid using a time-of-use pricing strategy. The EMS employs a hierarchical Genetic Algorithm (HGA) to fine-tune Fuzzy-Logic membership functions. However, GA may have limitations requiring careful parameter settings and possibly longer convergence times. Researchers in [9] propose an EMS for dispatching battery storage systems in grid-connected microgrids, accounting for battery state-of-charge (SOC), renewable energy production, power demand and energy costs. The objective of this paper is to reduce energy costs from the main grid, using parallel particle swarm optimization (PPSO), which, however, has constraints in parameter setting and computational complexity.

Further studies explore novel EMS designs: Reference [10] introduces a novel EMS for a remote hybrid microgrid, focusing on operational cost optimization using mixed integer non-linear programming (MINLP) which despite its computational demands, as it requires a high number of iterations, yields effective results in complex scenarios. An EMS based on a predictive control (PC) and dynamic programming (DP) method for a grid connected electric-hydrogen microgrid is proposed in [11]. This approach efficiently distributes and maximizes the microgrid output power, while maintaining energy balance and optimizing system costs. This method's efficiency is demonstrated in comparison with conventional DP and MILP-MPC. Paper [12] also proposes a DP-based EMS to control the energy generation systems of a microgrid. The aim of this technique is to optimize revenues from selling surplus renewable power while reducing costs to meet consumption demand. To solve the optimization problem, a dynamic programming approach is used. However, the DP approach encounters complex implementation challenges due to the high number of functions. A Fuzzy Logic-based Energy Management System (EMS-FL) for a grid-connected microgrid is proposed in [13]. The EMS-FL aims to reduce operating costs and average peak consumption by controlling the charge and discharge operations of the battery storage system. The optimization process of the Fuzzy-Logic membership functions is performed using non-dominated sorting genetic algorithm (NSGA-II). The downside of this approach is its sensitivity to the parameter settings such as population size, crossover, and mutation, which require careful tuning to achieve optimal results. A comparison of two fuzzy logic tuning techniques for the EMS of a grid-connected hybrid PV and storage system is proposed in [14]. The suggested rule-based fuzzy method offers fast execution and real-time applicability, enabling cost savings by efficiently managing energy generation, storage, and consumption. Reference [15] proposes an EMS for grid-connected microgrids with the goals of optimizing energy consumption, reducing the cost of purchased power from the main grid, and improving power quality. The three objective functions are addressed via a new multi-agent system. The control scheme of this method consists of several levels with coordinated control, making this approach complex to implement.

Among the approaches mentioned above that have proven effective are metaheuristic algorithms, multi-agent systems, and fuzzy logic. One particularly interesting approach adopted in the literature is game theory, which examines conflicts arising from decisions made by multiple interacting participants and extends the principle of decision theory to contexts involving multiple players or decision-makers [16]. This approach serves as a powerful analytical tool, especially well-suited for addressing interactions among different decision-makers involved in multi-objective optimization processes. A notable advantage of this method is that it allows solving complex problems in less time and with fewer computations compared to the previously mentioned approaches [17,18].

In [20], a pricing model based on game theory is presented for the

electricity market. Using a Stackelberg game and an iterative pricing algorithm, the study accounts for interactions between sellers and buyers. Results show a 12.61 % increase in seller profits and a 4.36 % reduction buyers costs. The work in [21] proposes an energy management system based on a multi-leader-multi-follower Stackelberg model. Microgrids optimize production and pricing, while customers decide on consumption and storage. This leads to 55 % increases in microgrid profits and a 30.79 % rise in customer consumption.

In article [22], an intelligent energy management model integrating renewable energy is proposed. It aims to simultaneously minimize bills and peak demand. A cooperative game theory approach is used to find a Pareto-optimal solution, treating the problem as a mixed-integer nonlinear program. The results demonstrate the model's effectiveness through various case studies. The paper [23] addresses the coordination between diverse consumer types and a power company for managing electricity consumption using an adverse selection model from contract theory. It proposes a load curve optimization method amid competition among power companies based on the Bertrand model. By identifying consumer types and constructing relevant utility functions, the study facilitates incentive systems for load shifting, ensuring a separating equilibrium in pricing. To provide a clearer view of these contributions, Table 1 summarizes the main approaches, studied systems, and objectives reported in the literature.

In this context, game theory emerges as a promising approach for modeling the interactions among actors in a hybrid energy system. The comparative study presented in Appendix A shows a comprehensive overview of the main methods used in energy management systems. It covers a range of techniques, including meta-heuristic algorithms, mixed integer linear programming, dynamic programming, fuzzy logic, game theory. We highlighted main advantages and limitations for each technique. This comparison easily showed that game theory is the most appropriate for studying strategic behaviour and interactions between customers when they have to make a decision regarding their profits.

This paper focuses on the application of game theory to the energy management of a hybrid energy system connected to the distribution grid. The system consists of four actors (three energy providers and one consumer). Among the classical approaches commonly used in the literature to model market behaviors in non-cooperative games, we identify the Bertrand model, Cournot model, and the Stackelberg model. The Bertrand model, in particular, focuses on price competition among several energy suppliers. A static non-cooperative game with incomplete information, based on the Bertrand model, is applied to the system, relying on the pricing mechanism and the available energy quantities. A negotiation technique is proposed to model the interactions between the providers and the consumer. Each provider equally participates in the system's operation, simultaneously controlling its selling price, while the consumer chooses its source of supply in response to price variations.

The proposed approach balances energy distribution between suppliers and consumer, thereby reducing costs for the latter while maximizing energy utilization for the providers. This model analyzes the strategic interactions between suppliers and consumers, highlighting how pricing decisions influence energy allocation in a multi-source system. By applying the Bertrand model to energy management, our study can optimize the allocation of energy resources while considering price dynamics, which is essential in a context where renewable energy sources are increasingly integrated. Unlike most published works, which merely use data as inputs for the algorithm without integrating them into a simulation system, our method stands out due to its practical application. Indeed, the algorithm is implemented in a MATLAB simulation environment, with online inputs for enhanced realism and outputs that actively control the system. The generated results are not just theoretical; they are actively used to control and adjust the system in real-time. This integrated approach enhances dynamic energy management, allowing operations to adapt according to fluctuations in demand and supply, thus achieving optimal efficiency in energy distribution. The main contributions of this study are as follows:
<table><caption>**Table 1**
Summary of related work on energy management system.</caption><thead><tr><th>Ref</th><th>Approach</th><th>Key Description</th><th>Input Parameters</th><th>Decision Making</th></tr></thead><tbody><tr><td>[5]</td><td>Lagrange multipliers (LP)</td><td>Proposes a RT-EMS for grid-connected PV with battery storage, where pricing signals adjust power output based on demand.</td><td>PV energy, Battery state of charge, electricity price</td><td>Maximize the total revenue,</td></tr><tr><td>[6]</td><td>Mixed Integer Linear Programming (MILP)</td><td>Proposes an EMS for grid-connected PV-battery systems to reduce consumption costs by considering the grid price fluctuations, renewable generation, and demand variations.</td><td>PV energy, Battery energy Electricity tariff</td><td>Reduced energy consumption cost by 3.3 %</td></tr><tr><td>[7]</td><td>Genetic Algorithm (GA)</td><td>Proposes a RT-EMS for grid-connected microgrids to optimize distributed energy resource usage, minimizing costs and carbon emissions while meeting demand.</td><td>Load power balance, Grid tariff, wind power profile, Emission constraint</td><td>Increased economic and environmental performance</td></tr><tr><td>[8]</td><td>Fuzzy-Logic with Hierarchical Genetic Algorithm (FL-HGA)</td><td>Proposes a FL-EMS for grid-connected PV-battery systems to maximize revenues via time-of-use pricing and energy exchanges with the grid</td><td>PV system, Battery State of charge, energy price</td><td>The profit of optimized EMS is close to 90 % of the optimal performance</td></tr><tr><td>[9]</td><td>Parallel Particle Swarm Optimization (PPSO)</td><td>Proposes an EMS for dispatching battery storage in grid-connected microgrids, considering battery SOC, renewable output power, demand, and energy costs, aiming to reduce grid energy costs.</td><td>State-of-charge, wind and photovoltaic output power, energy costs</td><td>Energy purchasing cost was reduced by 1.63 %, while processing time decreased by 55.89 %</td></tr></tbody></table>

<table><caption>**Table 1 (continued)**</caption><thead><tr><th>Ref</th><th>Approach</th><th>Key Description</th><th>Input Parameters</th><th>Decision Making</th></tr></thead><tbody><tr><td>[10]</td><td>Mixed Integer Non-Linear Programming (MILP)</td><td>Proposes a novel EMS for remote hybrid microgrids, optimizing operational costs for various distributed generation (DG) units in islanded mode by sending droop characteristic parameters to ensure optimal power sharing.</td><td>Water desalination units, Power balance, PV panels, Dispatched DG, droop characteristic</td><td>Reduced operating cost by 52.4 %</td></tr><tr><td>[11]</td><td>Predictive Control with Dynamic Programming (PC-DP)</td><td>Proposes an EMS for grid-connected electric-hydrogen microgrids using PC and DP to maximize output power, maintain energy balance, and optimize costs.</td><td>Battery state of charge, Hydrogen state of charge, Load demand, PV power</td><td>Reduced operating Cost by 90.05 %</td></tr><tr><td>[12]</td><td>Dynamic Programming (DP)</td><td>Proposes an EMS to control microgrid energy production, aiming to maximize revenue from surplus renewable sales while reducing consumption costs.</td><td>Battery state of charge, wind energy, electricity price</td><td>Battery charging and discharging depend on the electricity price</td></tr><tr><td>[13]</td><td>Fuzzy Logic with Nondominated Sorting Genetic Algorithm (FL-NSGA II)</td><td>Proposes a FL-EMS for grid-connected microgrids to reduce operating costs and average peak consumption by managing battery charge/discharge.</td><td>Renewable energy sources, consumers demand, Electricity price, battery state of charge</td><td>Cost and Average Peak decreased by 7.5 % and 16.1 % during offline tuning, and by 2.1 % and 18.4 % during online validation, respectively.</td></tr><tr><td>[14]</td><td>Fuzzy Logic</td><td>Proposes an FL-based EMS for grid-connected hybrid solar photovoltaic and battery storage system, with the objective of maximizing cost savings</td><td>Grid energy cost, Battery state of charge</td><td>Compared to the 16-rule EMS, the 12-rule fuzzy EMS reduced daily operational costs by 17.5 %, and achieved a total cost reduction of</td></tr></tbody></table>

(continued on next page)
<table><caption>**Table 1 (continued)**</caption><thead><tr><th>Ref</th><th>Approach</th><th>Key Description</th><th>Input Parameters</th><th>Decision Making</th></tr></thead><tbody><tr><td></td><td></td><td>through dynamic energy balancing, storage, and consumption</td><td></td><td>23.5 % over 20 years</td></tr><tr><td>[15]</td><td>Multi-Agent System (MAS)</td><td>Proposes an EMS for grid-connected microgrids to optimize energy consumption, reduce grid power costs, and improve power quality.</td><td>Renewable energy sources, consumers demand, Electricity price, battery state of charge</td><td>Electricity bills can be reduced by 2-6.5 %, Minimized power purchased from the main grid, Improved power quality</td></tr><tr><td>[19]</td><td>Game Theory (GT)</td><td>Proposes two EMS mechanisms for smart buildings using negotiation techniques and a multi-leader-follower game to optimize costs and reduce peak consumption.</td><td>Energy suppliers, consumers demand, Electricity price</td><td>Reduced total daily cost, minimized individual demand Costs</td></tr><tr><td>[20]</td><td>Game Theory (GT)</td><td>Proposes EMS based pricing model for electricity markets using a multi-leader-follower game and iterative pricing to reach equilibrium.</td><td>Prosumer, consumers demand, energy purchase price, energy sale price</td><td>Seller profits was increased by 12.61 %, buyer utility sacrifice was reduced by 4.36 %</td></tr><tr><td>[21]</td><td>Game Theory (GT)</td><td>Proposes an EMS based multi-leader-follower game where microgrids optimize production and pricing, and customers decide consumption and storage.</td><td>Microgrids generated power, consumers demand, electricity tariff</td><td>profit grid profit was improved by 55 %, Customers consume 30.79 % more energy, increasing microgrid energy utilization</td></tr><tr><td>[22]</td><td>Game Theory (GT)</td><td>Proposes an intelligent EMS integrating renewable energy to minimize bills and peak demand, using cooperative game theory to find a Pareto-optimal solution</td><td>Renewable energy generated power, battery state of charge, load demand, electricity price</td><td>Consumers monthly energy bills were reduced by over 50 %, while the company utility has curtailed its peak demand value</td></tr></tbody></table>

<table><caption>**Table 1 (continued)**</caption><thead><tr><th>Ref</th><th>Approach</th><th>Key Description</th><th>Input Parameters</th><th>Decision Making</th></tr></thead><tbody><tr><td></td><td></td><td>Proposes a coordination model between consumers and power companies using adverse selection and the Bertrand model to optimize load curves, enable incentive-based load shifting, and ensure a separating equilibrium in pricing</td><td></td><td>Consumers shift their consumption from peak to off-peak hours depending on the tariffs</td></tr><tr><td></td><td></td><td>Proposed an EMS for a grid-connected multi-source system using a Bertrand duopoly model, allowing consumers to minimize their bills according to supplier pricing and maximizing energy utilization for the suppliers</td><td></td><td>Average bill of the consumer was reduced by 21.93 % during winter day and by 20.82 % during summer day</td></tr><tr><td></td><td></td><td>Consumer demand, Suppliers generated power, production costs, electricity tariff</td><td></td><td></td></tr></tbody></table>

* A Betrand duopoly EMS model is proposed for grid-connected multi-sources based on negotiation technique.

* The negotiation process for energy supply between providers and consumer incorporates consideration of both consumer comfort and providers' profitability, ensuring a balanced approach that optimizes overall satisfaction and economic efficiency."

The structure of this paper is as follows: Section II provides a detailed description of the system, Section III outlines the problem formulation, the game model is introduced in Section IV, simulation results are presented and discussed in Section V, and finally, the conclusions are drawn in Section VI

## 2. System description

The system consists of two energy sources and a variable load connected to the main grid at the point of common coupling (PCC), as illustrated in Fig. 1. The first source includes a wind energy conversion system with a battery storage system, while the second source incorporates a photovoltaic energy conversion system with its own battery storage system. Both sources are connected to the DC bus via a DC/DC converter. The renewable energy resources are controllable, while the energy storage systems are controlled. Finally, the system is connected to the variable load and the main grid via a controlled 5L-APNC DC/AC converter [24]. The operation of the entire system is as follows: when wind or solar energy production is insufficient, the storage system powers the load, and if the energy is still not sufficient, the load purchases energy from the main grid. In the case where the energy produced
<figure><img src="image_3.png" /><figcaption>Fig. 1. The system architecture.</figcaption></figure>

by the wind turbine or the solar panels exceeds the load demand, the excess energy is stored in the storage system or sold to the main grid. The references used to control both storage systems are provided by the energy management algorithm.

## 2.1. Wind energy modeling and control

The captured wind turbine energy can be defined as [25-27]:

$$P_w = 0.5 \rho \pi R^2 C_p(\lambda) V_w^3 \quad (1)$$

where: $\rho$ is the air density, $R$ is the turbine blades radius, $C_p$ is the wind turbine power coefficient $\lambda$ is the tip speed ratio, $V_w$ is the wind velocity. Fig. 2 depicts Wind energy conversion system local control.

## 2.2. Solar energy modeling and control

The photovoltaic cell's output power $p_{pv}$ is mathematically formulated as follows [28,29]:

$$P_{pv} = V_{pv} \left[ \frac{G}{1000} [I_{sc} + K_i(T - 298)] - I_o \left[ e^{\frac{V_{pv} + R_S I_{pv}}{\lambda K_i T}} - 1 \right] - \frac{V_{pv} + R_S I_{pv}}{R_p} \right] \quad (2)$$

where: $G$ represents solar irradiation, $K_i$ denotes cell's short-circuits current coefficient, $I_{sc}$ indicates short-circuit current, $I_o$ symbolizes diode's reverse saturation current, $q$ is the electron charge, $A$ is the diode ideality factor, $K_z$ is Boltzmann's constant, $R_p$ and $R_s$ denote the parallel and series resistor respectively. Finally, $T$ represents PV cell temperature.

The solar energy conversion system local control is shown in Fig. 3.

## 2.3. Battery energy modeling and control

The charge and discharge behavior of the lithium-ion battery can be described as follows [30,31]:

$$E_{bat-dis} = E_0 - K_b \left( \frac{Q}{It + 0.1 \cdot Q} \right) I_{bat} - K_b \left( \frac{Q}{Q - It} \right) It + A_b e^{(-B \cdot It)} \quad (3)$$

$$E_{bat-dis} = E_0 - K_b \left( \frac{Q}{Q - It} \right) I_{bat}^* - K_b \left( \frac{Q}{Q - It} \right) It + A_b e^{(-B \cdot It)} \quad (4)$$

where: $E_0$ represents the constant voltage (V), $K_b$ denotes the polarization coefficient of the resistance (Ω), $Q$ is the maximum battery capacity (Ah), $It$ is the extracted capacity (Ah), $I_{bat}^*$ is the low frequency current (A), $A_b$ is the exponential voltage (V) and $B$ is the exponential capacity (Ah$^{-1}$).

The state of charge (SOC) can be expressed as follows:

<figure><img src="image_4.png" /><figcaption>Fig. 2. Wind turbine conversion system local control.</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Fig. 3. Solar energy conversion system local control.</figcaption></figure>

$$SOC(t) = SOC_{init} - \frac{1}{Q_{max}} \int_{0}^{t} I_{bat} dt \quad (5)$$

Fig. 4 illustrates the battery energy conversion system local control.

## 2.4. Grid-side converter modeling and control

The DC-Link voltage model can be expressed as follows:

$$I_c = C_{DC} \frac{dV_{DC}}{dt} = I_{in} - I_{DC} \quad (6)$$

where: $I_{in}$ is the DC-Link input current, $I_c$ is the DC-Link capacitor current, and $I_{DC}$ is the DC-Link output current.

The direct and quadratic output voltages of the DC/AC converter are defined as follow [29,32]:

$$\left\{ \begin{array}{l} V_{id} = R_f I_{gd} + L_f \frac{dI_{id}}{dt} - \omega L_f I_{gq} + V_{gd} \\ V_{iq} = R_f I_{gq} + L_f \frac{dI_{iq}}{dt} + \omega L_f I_{gd} + V_{gq} \end{array} \right. \quad (7)$$

where: $V_{idq}$ are the DC/AC converter direct and quadratic output voltages, $V_{gdq}$ represent the grid direct and quadratic voltages, $I_{gdq}$ denote the grid direct and quadratic currents, and $R_f$, $L_f$ are the filter inner resistance and inductance respectively. Fig. 5 shows the grid-side DC/AC converter local control.

## 3. Problem formulation

In this market, each actor (supplier) seeks to achieve their individual objectives by adjusting their pricing strategy, without cooperation with other suppliers. The objective function (utility) of each supplier is defined based on parameters such as production cost, selling price, and the purchasing price of the consumer. These parameters allow each supplier to optimize their strategy locally, adjusting prices to maximize their utility based on the consumer's response. The strength of this algorithm lies in its ability to achieve local optimization for each supplier, independently of the others, without requiring cooperation or communication between market actors. Instead of a central agent coordinating the system, each supplier autonomously adjusts their price and observes the consumer's demand in response, and selecting the strategy that maximizes their utility.

This decentralized approach reflects the reality of local energy networks, where market actors have limited knowledge of the system and make decisions based on their own interests. The consumer plays a key role by reacting to the proposed prices, which allows suppliers to discover optimal strategies through repeated interactions. The absence of a central coordinator ensures scalability and flexibility, as suppliers optimize their behavior based solely on local interactions and responses.

➤ The primary objective of the system is to maximize the consumption of each energy source produced locally.

• Thus, for the first source (Wind energy/battery storage) the optimization problem consists of minimizing the following objective function F:

$$\min F_1 = \left[ (P_{S_1} - \xi_{c,S_1})^2 \right] \quad (8)$$

Where $P_{S_1}$ and $\xi_{c,S_1}$ represent, respectively, the estimated power generated by the first supplier (source) and the power demanded by consumer c from the first supplier (source).

• The second source (Solar energy/battery storage) optimization problem consists of minimizing the following objective function F:

$$\min F_2 = \left[ (P_{S_2} - \xi_{c,S_2})^2 \right] \quad (9)$$

Where $P_{S_2}$ and $\xi_{c,S_2}$ represent, respectively, the estimated power generated by the second supplier (source) and the power demanded by consumer c from the second supplier (source).

• The optimization problem for the consumer (Load) aims to minimize the following objective function C:

$$\min C = \sum_{s \in \{S_1, S_2, g\}} (\rho_s \xi_{c,s}) \quad (10)$$

Where $\rho_s$ is the price proposed by suppliers s, and $\xi_{c,s}$ is the amount of power requested from supplier s.

➤ The optimization problems are subject to various constraints.

• The first constraint ensures that the power demanded from a supplier does not exceed the maximum generated power, expressed as:

$$\xi_{c,s} \le P_s \quad \forall s \in \{S_1, S_2, g\} \quad (11)$$

• In addition to the constraints related to suppliers, it is necessary to define additional constraints for consumer c, where $\xi_{c,max}$ represents their total maximum consumption. Consumption is divided into two parts: the minimum required consumption $\xi_{c,min}$ and the

<figure><img src="image_6.png" /><figcaption>Fig. 4. Battery energy conversion system local control.</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>Fig. 5. Grid-side DC/AC converter local control.</figcaption></figure>

variable consumption $\xi_c$. The energy demanded $\xi_c$ is expressed as the sum of the energies demanded from all suppliers:

$$ \xi_c = \sum_{s \in \{S_1, S_2, \dots, S_g\}} \xi_{c,s} \qquad (12) $$

Thus, the following constraints are defined:

$$ \xi_{c,min} \le \xi_c \le \xi_{c,max} \qquad (13) $$

* For each battery storage system, certain constraints must be considered. Based on the current conditions of the case study, the charging and discharging power are subject to specific limits:

$$ 0 \le P_b \le P_b^{charge,max} \qquad (14) $$

$$ 0 \ge P_b \ge P_b^{discharge,max} \qquad (15) $$

* In addition, the state of charge (SOC) is constrained by predefined upper and lower boundaries:

$$ SOC_{min} \le SOC \le SOC_{max} \qquad (16) $$

## 4. Game model

The situation deals with a short-term electricity market (24 h). The market is supplied by three sources: wind energy/batteries, photovoltaic energy/batteries, and a conventional grid. Energy suppliers offer a unit selling price for each 4-h slot (6 time slots over 24 h). A slot is denoted by the interval $[T_i, T_{i+1}]$. A consumer (i.e., a load) purchases energy from this market. During each 4-h time slot, the selling prices are determined at the beginning of the slot and then fixed for the entire duration. The consumer then determines, at each moment $t \in [T_i, T_{i+1}]$, the amount of energy to purchase from each supplier based on:

1. Proposed prices;

2. The quantity available from each provider: energy quantities depend on natural factors such as wind speed for wind energy and sunlight for photovoltaic energy;

3. The load demand: variable over time.

The load demand varies over time and is unknown in advance by the market players. The energy quantities produced by the suppliers also fluctuate over time and are unknown in advance, as they depend on natural factors. At the beginning of each time slot, unit prices are determined. To achieve this, at each instant $T_i$, the producers propose a price and provide information about their available power at that moment. This information is then transmitted to the load, which allocates its demand based on this data and its purchasing cost.

* The selling price of the grid, $\rho_g$, is known by all.

* The purchasing price, $\alpha$, from the grid is known by all.

* The production cost, $c_1$, of $S_1$ is known only by $S_1$.

* Similarly, $c_2$ is known only by $S_2$.

Let $s \in \{S_1, S_2\}$ denote the index of the sources, and $g$ the index for the grid. $S_1$ represents the wind energy source, and $S_2$ the photovoltaic source. We model the electricity market as a Bertrand duopoly, where the renewable energy producers are $S_1$ and $S_2$. The strategy of a source $s$ consists of determining the selling price per kWh, denoted $\rho_s$. The production cost of energy for source $s$ is noted as $c_s$. The grid also offers electricity at a fixed price, the highest of the three, denoted $\rho_g$, and purchases the electricity produced by the sources at a fixed price $\alpha$, the lowest of the three. Each source produces a quantity of energy $P_s$, which is either stored in batteries or sold at a low price to the grid if the batteries are fully charged.

The market parameters follow these constraints: $c_s < \alpha < \rho_s < \rho_g$

* The production cost is lower than the electricity purchase price from the conventional grid. This assumption ensures that it is always beneficial to generate electricity from renewable sources.

* The selling price must be at least equal to the electricity purchase price from the conventional grid. This assumption ensures that selling renewable energy to the load is more advantageous than selling it to the conventional grid. Otherwise, producers would only sell their energy to the grid.

The selling price is always lower than the grid's selling price. Otherwise, customers would source their electricity directly from the grid.

The load (consumer) purchases a fixed amount of energy, denoted as $\xi_c$. It can be seen as a decision-maker aiming to minimize its purchasing costs. To simplify the operation, we model its behavior using a demand function, which is integrated into the game model. Thus, the load starts by purchasing the largest possible amount to meet its needs from the source offering the lowest price. If the selected source does not provide enough energy, it will complement the purchase with the next cheapest offer, and so on. Let $\xi_s$ denote the amount of energy purchased by the consumer $c$ (load) from source $s$, and $\xi_g$ the amount of energy purchased from the grid. The vector $\xi = (\xi_1, \xi_2, \dots, \xi_g)$ allows us to represent the quantity purchased by the load based on the market's technical parameters. These parameters are positive. Additionally, we have $\xi_c = \xi_1 + \xi_2 + \dots + \xi_g$, $\xi_1 \le P_1$, and $\xi_2 \le P_2$.

Now, we distinguish between two scenarios:

1. $P_1 + P_2 \le \xi_c$: In this case, the supply of renewable energy is less than the load's demand. We will necessarily have $\xi_g = \xi_c - (\xi_1 + \xi_2)$. The remaining $\xi_1 + \xi_2$ must be ensured, and as long as the price is lower than the conventional grid price, both sources have an incentive to set the highest possible price. Therefore, we will have: $\xi_1 = P_1$ and $\xi_2 = P_2$.

2. $P_1 + P_2 > \xi_c$: In this case, the supply of renewable energies exceeds the load's demand. We distinguish the following cases:

$$ (a) \rho_1 > \rho_2: \\ \xi_c \le P_2: \xi_2 = \xi_c \qquad (17) $$
$$ \xi_c > P_2 : P_2 = \xi_2, \xi_1 = \xi_c - \xi_2; \qquad (18) $$

$$ (b) \rho_1 < \rho_2: \qquad \xi_c \le P_1 : \xi_1 = \xi_c; \qquad (19) $$

$$ \xi_c > P_1 : \xi_1 = P_1, \xi_2 = \xi_c - \xi_1; \qquad (20) $$

(c) $\rho_1 = \rho_2$: The load does not differentiate between the offers but favors the supplier with the largest quantity, taking the maximum from that supplier.

$$ \bullet P_1 < P_2: \qquad \xi_2 = \min(P_2, \xi_c)_{\max} \xi_1 = \min(P_1, \xi_c - P_2) \qquad (21) $$

$$ \bullet P_1 > P_2: \qquad \xi_1 = \min(P_1, \xi_c), \xi_2 = \min(P_2, \xi_c - P_1) \qquad (22) $$

$$ \bullet P_1 = P_2 \qquad \text{On average, we will have: } \xi_2 = \xi_1 = \xi_c / 2 \qquad (23) $$

In the first case, we note that competition occurs only on the quantity given by the minimum between $\xi_c$ and $(\xi_1 + \xi_2)$.

We will denote $\xi = \min(\xi_c, \xi_1 + \xi_2)$ as the quantity at stake, or the quantity subject to competition, with the remainder being supplied by the conventional grid. In the second case, we distinguish three situations: either the load purchases its entire demand from the source offering the lowest price (case I), as that source has sufficient supply; or the load completes its demand by choosing another source (case II). The grid is not considered, as its price is always higher than that of the sources.

The demand distribution function of the load is given by the function $D: D(\xi_c, P_1, P_2)(\rho_1, \rho_2) = (\xi_1, \xi_2, \xi_g)$, indicating the quantity purchased based on the proposed prices and the parameters of the market. Unlike a classic Bertrand model, where a demand function gives the quantity demanded by the market based on prices, in this situation, we have a function that allocates a given demand among the producers based on prices. This function is clearly defined in the section above.

We can now formulate the profit of the sources as equal to their revenue minus their production cost, plus what they sell to the grid:

The supplier profit is formulated as follows:

$$ \begin{aligned} u_s^c(\xi_c, P_1, P_2)(\rho_1, \rho_2) &= \rho_s \xi_s + \alpha(P_s - \xi_s) - c_s P_s \\ &= (\rho_s - \alpha) \xi_s + (\alpha - c_s) P_s \end{aligned} \qquad (24) $$

We can also formulate the consumer's bill as follows:

$$ F_c^c(\xi_c, P_1, P_2)(\rho_1, \rho_2) = \rho_1 \xi_1 + \rho_2 \xi_2 + \rho_g \xi_g \qquad (25) $$

## 4.1. Nash equilibrium

* **Supplier:** Each supplier determines its price $\rho_s$ independently, without prior knowledge of the prices set by competing suppliers. Additionally, they have no visibility into or access to the pricing decisions of each other.

* **Strategy:**
  * Each supplier chooses its price $\rho_s$ based on consumer behavior, without knowing the other supplier's decision.

* The consumer determines the quantity of energy $\xi_c$ to demand based on the prices proposed by the suppliers.

* **Utility:**
  * Each supplier aims to maximize its utility $u_s^c(\xi_c, P_1, P_2)(\rho_1, \rho_2)$, which is a function of the price it offers and the corresponding quantity of energy requested by the consumer.

* **Nash Equilibrium Condition:**
  * Suppliers iteratively adjust their prices, increasing or decreasing $\rho_s$, until they reach a state where no further price modification results in an increase in utility.

* The iterative process continues until $\Delta\rho_s \le 0$, indicating that further price adjustments no longer yield improvements in the respective utilities. This condition signifies that a Nash equilibrium has been reached.

At the Nash equilibrium, the above condition must be satisfied simultaneously for all suppliers. In other words, $(\rho_1^*, \rho_2^*)$ is a Nash equilibrium, if:

$$ \begin{cases} u_1(\rho_1^*, \rho_2^*) \ge u_1(\rho_1', \rho_2^*), \quad \rho_{min1} \le \rho_1' \le \rho_{max1} \\ u_2(\rho_1^*, \rho_2^*) \ge u_2(\rho_1^*, \rho_2'), \quad \rho_{min2} \le \rho_2' \le \rho_{max2} \end{cases} \qquad (26) $$

Where: $\rho_s'$ denotes $\rho_s^{up}$ or $\rho_s^{down}$, depending on whether the supplier chooses to increase or decrease its price.

This condition means that the utility of supplier $s$ when selecting price $\rho_s'$ must be at least equal to the utility obtained by choosing another price $\rho_s'$. Consequently, the selection of price $\rho_s'$ is optimal for the supplier.

## 4.2. Energy demand management algorithm

In the previously defined market, each supplier seeks to maximize its satisfaction according to its objective, which is in this case (utility function). To simulate the market, this section proposes an iterative algorithm as shown in Fig. 6, where the suppliers adjust their prices iteratively by exploiting two paths: increasing the price, and decreasing it.

The iterative algorithm performs as described in the algorithm below. After initializing all the parameters, each supplier $s$ selects a random price smaller than $\rho_{max}$ and sends it to consumer $c$. This latter determines the different energy amounts to be requested from different suppliers.

According to the proposed prices. Then, $c$ sends back his decision to the suppliers. At this stage, each supplier will try to discover two different strategies while increasing and decreasing the prices by $\Delta\rho_s > 0$. The two prices are forwarded to $c$ which tries to find optimal strategies for each new price. When suppliers receive the new requested energy amounts for their two proposed prices, they calculate the utility for each case, and then compare them. The price that generates the higher utility will be maintained. Each supplier decreases $\Delta\rho_s$ by $\epsilon > 0$ and the process repeats until $\Delta\rho_s \le 0$. The selection of $\epsilon$ value impacts the number of iterations as well as the dimension of explored space. Hence, with a large value, the algorithm converges more quickly, while with a small value, the optimal strategies (i.e., prices and requested energy amounts) are more accurate and very close to the optimal. Unlike $\epsilon$, with a larger $\Delta\rho_s$, the algorithm takes more iterations to converge with better optimal strategy, and vice versa. The convergence of the algorithm is guaranteed since by $\Delta\rho_s > 0$ and $\epsilon > 0$.

# 5. Results and discussion

## 5.1. Case study results and discussion

For our study, we have selected the site of Béjaïa as a representative case due to its geographical position and renewable energy potential. Béjaïa, a coastal city located in northern Algeria (36°44'15" N, 4°58'17"
<figure><img src="image_8.png" /><figcaption>Fig. 6. Energy request flow chart.</figcaption></figure>

E), see Fig. 7, benefits from a high level of solar irradiance and is currently experiencing a growing domestic energy demand. In this context, the development of an electricity market represents an interesting opportunity to improve energy efficiency and encourage the use of renewable energy sources. Such an initiative would reduce dependence on the national power grid and promote the optimal integration of

renewable resources.

The annual sunshine duration in Béjaïa exceeds 2000 h [34]. The annual solar energy received on a horizontal surface of one square meter is approximately 1.7 MWh/m². In addition, wind speeds in the areas located at altitudes above 100 m range between 6 m s⁻¹ and 9.5 m s⁻¹, indicating a moderate to high wind energy potential. This information is

<figure><img src="image_9.png" /><figcaption>Fig. 7. The selected study area [33].</figcaption></figure>
illustrated in Fig. 8.

In this study, simulations were conducted using real environmental data corresponding to the city of Béjaïa, Algeria (36°44'15"N, 4°58'17"E). The input data include solar irradiance, ambient temperature, and wind speed, which are critical for evaluating the performance of the hybrid energy system.

The environmental data were obtained from [36,37], which provides reliable and location-specific historical weather data. For this case study, hourly data for the year 2024 were used to ensure realistic simulation results.

The simulations were conducted to verify the behavior and performance of the proposed system. The simulation parameters and variables are provided in Appendix B. To test the robustness of the proposed method, two different day profiles (winter/summer) were selected.

### Scenario 1: (Winter Day)

Regarding the timescale, it is assumed that each second during the operational test corresponds to one hour in real time. The solar irradiation profile and temperature are illustrated in Fig. 9, while the wind speed is represented in Fig. 10.

The pricing mechanism is depicted in Fig. 11, while the forecasted demand profile for the load is illustrated in Fig. 12.

The power produced by the wind generator is shown in Fig. 13.a, while that produced by the photovoltaic system is presented in Fig. 13.b. Fig. 13.c shows the power supplied/charged by the battery connected to the wind energy system. Similarly, Fig. 13.d illustrates the power supplied/charged by the battery associated with the photovoltaic system. Fig. 13.e presents the power supplied and absorbed by the grid, while Fig. 13.f shows the alignment between the predicted energy consumption ($\xi_{c,ref}$) and the measured consumption ($\xi_c$), illustrating the accuracy and reliability of the energy management system in meeting the energy needs of the load. Fig. 13.g illustrates the power profiles of the energy system, providing a comprehensive view of the interactions between the various sources and the load. It also enables the observation of the evolution of produced, consumed, stored powers, as well as those exchanged with the grid.

Based on the simulation results presented in Fig. 13.g and summarized in Table 2, the analysis shows that throughout the day, the renewable energy sources were utilized according to their selling price, availability, and energy demand. Each period is analyzed as follows:

* From 0 h to 4 h, the selling price of $S_1$ is lower compared to $S_2$ ($\rho_{S1} < \rho_{S2}$), and ($P_w + P_{b,w} = \xi_c$), so the energy demand is met by $S_1$.

* Between 4 h and 8 h, the selling price of $S_2$ becomes lower than that of $S_1$ ($\rho_{S1} > \rho_{S2}$). During the period from 4 h to 5 h, the storage system $P_{b,pv}$ of $S_2$ intervenes since $P_{pv}$ is equal to zero ($P_{b,pv} = P_l$). Simultaneously, $S_1$ charges its storage system via wind turbine ($P_{b,w} = P_w$). From 5 h to 5h30 the load purchases energy from $S_1$ as $S_2$ is not sufficient ($P_{b,pv} + P_w = \xi_c$). From 5h30 to 7 h the combined power $P_{b,pv}$, $P_w$, and $P_{b,w}$ meets the load demand ($P_{b,pv} + P_w + P_{b,w} = \xi_c$), and from 7 h to 8 h, the addition of $P_{pv}$ ensures the demand is met ($P_{b,pv} + P_w + P_{b,w} + P_{pv} = \xi_c$).

* Between 8 h and 12 h, the selling price of $S_1$ is lower ($\rho_{S1} < \rho_{S2}$). During the period from 8 h to 9 h, the load purchases energy from $S_1$ and fills the energy deficit from $S_2$ ($P_{b,pv} + P_w + P_{b,w} + P_{pv} = \xi_c$). From 9 h to 9h30, the load purchases energy from $S_1$ and fills the energy deficit from $S_2$ ($P_w + P_{b,w} + P_{pv} = \xi_c$). Simultaneously, the excess energy from $S_2$ is used to charge its storage system. From 9h30 to 12 h, the load purchases energy from $S_1$ ($P_w + P_{b,w} = \xi_c$). While, $S_2$ charges its storage system and the surplus is sold to the main grid.

* Between 12 h and 20 h, the selling price of $S_2$ is lower ($\rho_{S1} < \rho_{S2}$). During the period from 12 h to 14 h, $S_2$ supplies the load ($P_{pv} = \xi_c$), and the surplus is used to charge its storage system. While, $S_1$ uses all its energy to charge its storage system. From 14 h to 16h30, $S_2$ supplies the load demand ($P_{pv} + P_{b,pv} = \xi_c$). While, $S_1$ continues to charge its storage system. From 16h30 to 17h30, the load purchases energy from $S_2$ and $S_1$ ($P_{pv} + P_{b,pv} + P_w = \xi_c$). From 17h30 to 20 h, the load purchases energy from $S_2$ and $S_1$ ($P_{pv} + P_{b,pv} + P_w + P_{b,w} = \xi_c$).

* From 20 h to midnight, the selling price of $S_1$ is lower ($\rho_{S1} < \rho_{S2}$). The load purchases energy from $S_1$ and deficit of energy is filled by $S_2$ and main grid ($P_w + P_{b,w} + P_{b,pv} + P_g = \xi_c$).

Fig. 14 illustrates how the various energy sources are utilized to meet the load demand over a 24-h period. This variation directly responds to changes in demand, the availability of sources, and their selling prices. The grid is utilized only when both sources are insufficient, indicating a preference for renewable sources whenever possible.

Fig. 15 provides an overview of the performance and impact of the number of energy suppliers on the variation of the consumer's energy bill. Increasing the number of available energy sources creates price competition and results in a noticeable reduction in energy costs throughout the day. During periods when the renewable sources (PV and wind) are available and offer competitive prices, the consumer switches to these sources, which explains the lower values of $F_{c3}$ and $F_{c2}$ compared to $F_{c1}$. The differences between the curves prove that competition between suppliers forces the system to use the cheapest

<figure><img src="image_10.png" /><figcaption>Fig. 8. Solar (a) and wind (b) potential of Béjaïa [35,36].</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>Fig. 9. Profile of a winter day: (a) Solar irradiation, (b) Temperature.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 10. Wind speed profile.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 11. Pricing mechanism.</figcaption></figure>

source at each time step, which results in an overall decrease in the bill.

Fig. 16 confirms the same tendency and highlights more clearly the average bill for each 4-h period. The average bill calculated using only the grid (blue) is always the least economical. When a second source is added (yellow color), the average bill decreases, indicating that the system shifts part of the demand to the cheaper source whenever possible. With three suppliers (Grid + PV + Wind, green color), the average bill is further reduced. Table 3 highlights the reduction rate of the average bill. During every time interval, the average bill with two suppliers ($\bar{F}_{c2}$) is lower than the average bill with only the grid one ($\bar{F}_{c1}$), and the average bill is further reduced when three suppliers ($\bar{F}_{c3}$) are available. The reduction rate column confirms this trend: the use of two

suppliers provides a constant cost reduction of approximately 9.83 %, while the introduction of a third supplier significantly increases the reduction rate, reaching 21.93 %. This demonstrates that the diversification of energy sources considerably improves cost efficiency and enables the consumer to benefit from cheaper energy offers throughout the day.

The utility of $S_1$ (Wind/battery) and $S_2$ (PV/battery) fluctuates throughout the day as shown in Fig. 17, reflecting their role in energy supply based on price competitiveness and the quantity of energy involved.

Fig. 18 demonstrates the stability of the DC bus voltage. Despite fluctuations in the load profile and the profiles of the sources, the DC bus
<figure><img src="image_14.png" /><figcaption>Fig. 12. Forecasted load demand profile.</figcaption></figure>

voltage remains consistently balanced at its desired operational value of 500 V. Fig. 19 illustrates the stable phase voltage, along with the current that adjusts to load variations and remains in phase with the voltage. Fig. 20 shows the state of charge of the storage system from both sources.

### Scenario 2: (Summer Day)

Regarding the time scale, it is assumed that one second during the simulation corresponds to one hour in real time. The solar irradiation profile and temperature are presented in Fig. 21, while the wind speed is illustrated in Fig. 22.

Fig. 23 and Fig. 24 illustrate the pricing mechanism and the forecasted load demand profile, respectively.

Fig. 25.a and 25.b respectively illustrate the power generated by the wind energy system and the photovoltaic system. The power supplied/ charged by the battery connected to the wind energy system is illustrated in Fig. 25.c, while the power supplied/charged by the battery connected to the photovoltaic system is presented in Fig. 25.d. Finally, the power supplied and absorbed by the grid is shown in Fig. 25.e. Fig. 25.f shows the alignment between the predicted energy consumption ($\xi_{c,ref}$) and the measured consumption ($\xi_{c}$), demonstrating the accuracy and reliability of the energy management system in meeting the load's energy needs. Fig. 25.g illustrates the power variations within the energy system, providing a comprehensive view of the interactions between different sources and the load. It also enables the observation of the evolution of produced, consumed, and stored powers, as well as those exchanged with the grid.

Based on the simulation results illustrated in Fig. 25.g and summarized in Table 4, it appears that throughout the day, the utilization of renewable energy sources has been optimized according to their selling cost, availability, and energy needs. A detailed analysis for each period is presented below:

* From 0 h to 4 h, the selling price of $S_1$ is lower ($\rho_{S1} < \rho_{S2}$). The load purchases energy from $S_1$ and deficit of energy is filled by $S_2$ ($P_w + P_{b,w} + P_{b,pv} = \xi_c$).

* Between 4 h and 8 h, the selling price of $S_2$ becomes lower than that of $S_1$ ($\rho_{S1} > \rho_{S2}$). From 4 h to 4h30 and 5 h to 6 h the load is supplied by $S_2$ and $S_1$ ($P_{b,pv} + P_w + P_{b,w} = \xi_c$), except between 4h30 to 5 h the energy surplus of $S_1$ is utilized to charge its storage system. From 6 h to 8 h, the load purchases energy from $S_2$ and the energy deficit is filled by $S_1$ ($P_{pv} + P_{b,pv} + P_w + P_{b,w} = \xi_c$).

* Between 8 h and 12 h, the selling price of $S_1$ is lower ($\rho_{S1} < \rho_{S2}$). During the period from 8 h to 10 h, the load purchases energy from $S_1$ and the deficit of energy is filled by $S_2$ ($P_w + P_{b,w} + P_{pv} = \xi_c$). While the energy surplus of $S_2$ is used to charge its storage system. From 10 h to 12 h, the load is supplied by $S_1$ and $S_2$ ($P_w + P_{b,w} + P_{pv} = \xi_c$), and

the energy excess from $S_2$ is utilized to feed the storage system, and the rest of the excess is sold to the main grid.

* Between 12 h and 20 h, the selling price of $S_2$ is lower ($\rho_{S1} > \rho_{S2}$). During the period from 12 h to 15h30, The load is supplied by $S_2$ ($P_{pv} = \xi_c$), the energy surplus is utilized to supply its storage system. Whereas, $S_1$ used its energy to charge its storage system. From 15h30 to 18h30, The load purchases energy from $S_2$ ($P_{pv} + P_{b,pv} = \xi_c$), and $S_1$ charges its storage system. From 18h30 to 19h30h, the load is supplied by $S_2$ and $S_1$ ($P_{pv} + P_{b,pv} + P_w + P_{b,w} = \xi_c$), and from 19h30 to 20 h the main grid intervenes to fill the energy deficit ($P_{pv} + P_{b,pv} + P_w + P_{b,w} + P_g = \xi_c$).

* From 20 h to midnight, the selling price of $S_1$ is lower ($\rho_{S1} < \rho_{S2}$). The load purchases energy from $S_1$ and deficit of energy is filled by $S_2$ and main grid ($P_w + P_{b,w} + P_{b,pv} + P_g = \xi_c$).

Fig. 26 illustrates the use of different energy sources to meet the load demand over a 24-h period. This distribution varies based on demand fluctuations, source availability, and their unit cost. The use of the electrical grid occurs only when renewable sources are insufficient, highlighting the priority given to renewable energy whenever possible.

Fig. 27 illustrates the relationship between the number of energy suppliers and the variability of the consumer's energy bill. As the number of available energy sources increases, price competition is introduced, leading to a reduction in energy costs throughout the day. When renewable sources (PV/battery and wind/battery) are available and offer more competitive prices, the system prioritizes these sources, as evidenced by the lower values of $F_{c3}$ and $F_{c2}$ compared to $F_{c1}$. The differences between the cost curves clearly demonstrate that supplier competition compels the system to select the most cost-effective energy source at each time step, thereby reducing the overall energy bill.

Fig. 28 reinforces this trend by presenting the average energy bill over successive 4-h intervals. The average bill with only the grid (blue color) consistently represents the highest cost. With the integration of a second source (yellow color), the average bill is reduced, indicating that the system reallocates part of the demand to the more economical option when available. The addition of a third source (grid + PV + wind, in greencolor) further decreases the average bill, confirming the advantages of supplier diversification.

Table 5 quantifies these reductions by showing the average bill and its corresponding reduction rate for each configuration. In all time intervals, the average bill with two suppliers ($\bar{F}_{c2}$) is lower than with the grid alone ($\bar{F}_{c}$), and further decreases when three suppliers ($\bar{F}_{c3}$) are available. The reduction rate column confirms this pattern: the integration of two suppliers yields a consistent reduction of approximately 9.55 %, while the addition of a third supplier enhances the cost savings, achieving a reduction rate of 20.82 %. These findings demonstrate that increasing the diversity of energy sources significantly improves cost efficiency and allows consumers to benefit from more competitive
<figure><img src="image_15.png" /><figcaption>**Fig. 13.** (a) Wind energy system generated power, (b) Solar energy system generated power, (c) Power supplied/charged by the battery connected to the wind energy system, (d) Power supplied/charged by the battery connected to the solar energy system, (e) Power supplied/absorbed by the grid, (f) Load demand profile (predicted/measured), (g) Power profiles of the overall system.</figcaption></figure>

energy prices throughout the day.

Finally, Fig. 29 presents the utility of sources $S_1$ (Wind/battery) and $S_2$ (PV/battery) over time, reflecting their dynamic contribution to the energy supply based on both price competitiveness and available energy quantities.

Fig. 30 highlights the stability of the DC bus voltage, which remains

consistently regulated at its setpoint of 500 V despite fluctuations in the load profile and energy sources.

Fig. 31 illustrates the stability of the phase voltage, with the current adjusting to load variations while remaining synchronized with the voltage. Finally, Fig. 32 presents the state of charge of the storage system for both energy sources.
<table><caption>**Table 2**
Simulation results of a winter day.</caption><thead><tr><th>Modes</th><th>Time</th><th>Source used</th></tr></thead><tbody><tr><td rowspan="10">Mode (1)</td><td rowspan="10">ρ<sub>1</sub> &gt; ρ<sub>2</sub></td><td>[4 h - 5 h]</td><td>P<sub>b,pv</sub></td></tr><tr><td>[5 h - 5h30]</td><td>P<sub>b,pv</sub> + P<sub>w</sub></td></tr><tr><td>[5h30 - 7 h]</td><td>P<sub>b,pv</sub> + P<sub>w</sub> + P<sub>b,w</sub></td></tr><tr><td>[7 h - 8 h]</td><td>P<sub>b,pv</sub> + P<sub>w</sub> + P<sub>b,w</sub> + P<sub>pv</sub></td></tr><tr><td>[12 h - 14 h]</td><td>P<sub>pv</sub></td></tr><tr><td>[14 h - 16h30]</td><td>P<sub>pv</sub> + P<sub>b,pv</sub></td></tr><tr><td>[16h30 - 17h30]</td><td>P<sub>pv</sub> + P<sub>b,pv</sub> + P<sub>w</sub></td></tr><tr><td>[16h30 - 20 h]</td><td>P<sub>pv</sub> + P<sub>b,pv</sub> + P<sub>w</sub> + P<sub>b,w</sub></td></tr><tr><td>[0 - 4 h]</td><td>P<sub>w</sub> + P<sub>b,w</sub></td></tr><tr><td rowspan="4">Mode (2)</td><td rowspan="4">ρ<sub>1</sub> &lt; ρ<sub>2</sub></td><td>[8 h - 9 h]</td><td>P<sub>b,pv</sub> + P<sub>w</sub> + P<sub>b,w</sub> + P<sub>pv</sub></td></tr><tr><td>[9 h - 9h30]</td><td>P<sub>w</sub> + P<sub>b,w</sub> + P<sub>pv</sub></td></tr><tr><td>[9h30 - 12 h]</td><td>P<sub>w</sub> + P<sub>b,w</sub></td></tr><tr><td>[20 h - 24 h]</td><td>P<sub>w</sub> + P<sub>b,w</sub> + P<sub>b,pv</sub> + P<sub>g</sub></td></tr></tbody></table>

<figure><img src="image_16.png" /><figcaption>**Fig. 14.** Profile of powers purchased from sources.</figcaption></figure>

## 5.2. Comparative analysis with related works

To better highlight the originality and effectiveness of our contribution, a comparative study was conducted with several existing works in the literature that also rely on game theory for energy management ([19]–[23]). This comparison addresses both the pursued objectives and the characteristics of the studied markets see (Table 6).

<table><caption>**Table 3**
Average bill and reduction rates over 24 h for different supply scenarios (winter day).</caption><thead><tr><th>Time</th><th>Average bill F<sub>c1</sub> (pu/kW)</th><th>Average bill F<sub>c2</sub> (pu/kW)</th><th>Average bill F<sub>c3</sub> (pu/kW)</th><th>Reduction rate (%) (F<sub>c1</sub>/F<sub>c2</sub>)</th><th>Reduction rate (%) (F<sub>c1</sub>/F<sub>c3</sub>)</th></tr></thead><tbody><tr><td>[0 h - 4 h]</td><td>-3,6955</td><td>-3326</td><td>-3,0808</td><td>10</td><td>16,634</td></tr><tr><td>[4 h - 8 h]</td><td>-6994</td><td>-6,2946</td><td>-5,8682</td><td>10</td><td>16,096</td></tr><tr><td>[8 h - 12 h]</td><td>-7532</td><td>-6,7788</td><td>-5,6698</td><td>10</td><td>24,724</td></tr><tr><td>[12 h - 16 h]</td><td>-4,9662</td><td>-4,4696</td><td>-3,3891</td><td>10</td><td>31,757</td></tr><tr><td>[16 h - 20 h]</td><td>-6888</td><td>-6,1992</td><td>-5,6006</td><td>10</td><td>18,69</td></tr><tr><td>[20 h - 24 h]</td><td>-10,408</td><td>-9,4701</td><td>-7,9415</td><td>9,0116</td><td>23,698</td></tr></tbody></table>

<figure><img src="image_17.png" /><figcaption>**Fig. 15.** Consumer bill.</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>**Fig. 16.** Average bill.</figcaption></figure>
<figure><img src="image_19.png" /><figcaption>Fig. 17. Utility of the sources.</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>Fig. 18. DC-bus voltage.</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>Fig. 19. Voltage and current of the load.</figcaption></figure>

### 5.2.1. Comparison criteria

The comparison is based on three groups of criteria:

#### 1. Article Objectives:

a) Reduction of energy cost: aiming to minimize the consumer's bill.

b) Maximization of producer profit: ensuring the economic viability of producers.

c) Promotion of renewable energy integration: encouraging the penetration of renewable sources.

d) Demand smoothing: mitigating demand peaks to reduce grid stress.

#### 2. Market Characteristics:

a) Price competition: determining whether electricity suppliers compete on price.

b) Complete information: verifying if all technical and economic data are shared among actors.

c) Sequential decisions: determining if the market equilibrium is reached through an iterative/negotiated process.

#### 3. Obtained Results: Quantitative or qualitative assessment of energy savings, producer profit, renewable energy integration, and demand smoothing, when available.

### 5.2.3. Discussion

This comparison highlights several important points:

The works [19,20], and [22] share with our approach the same objectives of reducing energy costs and integrating renewable energies.

Our approach achieves an average cost reduction of ≈ 21.93 % (winter) and 20.82 % (summer), which is significantly higher than the results reported in [20] (≈ 4 %) and comparable to [21] (≈ 20 %).

<figure><img src="image_22.png" /><figcaption>Fig. 20. Batteries state of charge.</figcaption></figure>
<figure><img src="image_23.png" /><figcaption>Fig. 21. Profile of a summer day: (a) Solar irradiation, (b) Temperature.</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>Fig. 22. Wind speed profile.</figcaption></figure>

<figure><img src="image_25.png" /><figcaption>Fig. 23. Pricing mechanism.</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>Fig. 24. Forecasted load demand profile.</figcaption></figure>
<figure><img src="image_27.png" /><figcaption>**Fig. 25.** (a) Wind energy system generated power, (b) Solar energy system generated power, (c) Power supplied/charged by the battery connected to the wind energy system, (d) Power supplied/charged by the battery connected to the solar energy system, (e) Power supplied/absorbed by the grid, (f) Load demand profile (predicted/measured), (g) Power profiles of the overall System.</figcaption></figure>
<table><caption>**Table 4**
Simulation results of a summer day.</caption><thead><tr><th>Modes</th><th>Time</th><th>Source used</th></tr></thead><tbody><tr><td rowspan="7">Mode (1)</td><td rowspan="7">ρ₁ > ρ₂</td><td>[4 h - 4h30]</td><td>P<sub>b,pv</sub> + P<sub>w</sub> + P<sub>b,w</sub></td></tr><tr><td>[4h30 - 5 h]</td><td>P<sub>b,pv</sub> + P<sub>w</sub></td></tr><tr><td>[5 h - 6 h]</td><td>P<sub>b,pv</sub> + P<sub>w</sub> + P<sub>b,w</sub></td></tr><tr><td>[6 h - 8 h]</td><td>P<sub>pv</sub> + P<sub>b,pv</sub> + P<sub>w</sub> + P<sub>b,w</sub></td></tr><tr><td>[12 h - 15h30]</td><td>P<sub>pv</sub></td></tr><tr><td>[15h30 - 18h30]</td><td>P<sub>pv</sub> + P<sub>b,pv</sub></td></tr><tr><td>[18h30 - 19h30]</td><td>P<sub>pv</sub> + P<sub>b,pv</sub> + P<sub>w</sub> + P<sub>b,w</sub></td></tr><tr><td rowspan="4">Mode (2)</td><td rowspan="4">ρ₁ < ρ₂</td><td>[19h30 - 20 h]</td><td>P<sub>pv</sub> + P<sub>b,pv</sub> + P<sub>w</sub> + P<sub>b,w</sub> + P<sub>g</sub></td></tr><tr><td>[0 - 4 h]</td><td>P<sub>w</sub> + P<sub>b,w</sub> + P<sub>b,pv</sub></td></tr><tr><td>[8 h - 10 h]</td><td>P<sub>w</sub> + P<sub>b,w</sub> + P<sub>pv</sub></td></tr><tr><td>[10 h - 12 h]</td><td>P<sub>w</sub> + P<sub>b,w</sub> + P<sub>pv</sub></td></tr><tr><td></td><td>[20 h - 24 h]</td><td>P<sub>w</sub> + P<sub>b,w</sub> + P<sub>b,pv</sub> + P<sub>g</sub></td></tr></tbody></table>

The cooperative approach of [22] shows higher savings (≈ 50 %); however, it relies on a centralized and penalizing framework where consumers bear an additional cost during demand peaks. This constraint does not exist in our model, which is based on price competition among suppliers.

Unlike [22], our approach explicitly integrates Bertrand-type competition, a realistic mechanism absent from cooperative frameworks.

Finally, the work [19] focuses on demand smoothing, an objective that is outside the scope of our study but could complement our contribution in future extensions.

Overall, this comparison contextualizes our contribution relative to existing works and demonstrates its originality. Our model distinguishes itself by achieving significant savings within a realistic, non-cooperative competitive market framework, while ensuring the integration of renewable energies and system stability.

## 6. Conclusion

The continuous growth in global domestic energy demand has highlighted the necessity for more efficient and sustainable energy systems. The development of competitive electricity markets appears to be an effective approach for increasing energy efficiency and promoting the integration of renewable energy sources. By encouraging decentralized energy production and consumption, these markets reduce dependence on the national power grid and promote the use of locally available renewable resources. This approach not only enhances grid reliability and resilience but also reduces energy costs and supports environmental sustainability.

The integration of renewable energy sources has significantly affected the energy sector, offering opportunities to increase efficiency, reduce energy costs, and lower carbon emissions. However, these developments also present challenges, including fluctuations in energy generation, grid stability issues, and complexities in managing interactions between customers and energy suppliers.

<figure><img src="image_28.png" /><figcaption>**Fig. 27.** Utility of the sources.</figcaption></figure>

<figure><img src="image_29.png" /><figcaption>**Fig. 28.** Average bill.</figcaption></figure>

<table><caption>**Table 5**
Average bill and reduction rates over 24 h for different supply scenarios (Summer day).</caption><thead><tr><th>Time</th><th>Average bill F<sub>c1</sub> (pu/kW)</th><th>Average bill F<sub>c2</sub> (pu/kW)</th><th>Average bill F<sub>c3</sub> (pu/kW)</th><th>Reduction rate (%) (F<sub>c1</sub>/F<sub>c2</sub>)</th><th>Reduction rate (%) (F<sub>c1</sub>/F<sub>c3</sub>)</th></tr></thead><tbody><tr><td>[0 h - 4 h]</td><td>-6,1095</td><td>-5,4985</td><td>-5,1844</td><td>10</td><td>15,141</td></tr><tr><td>[4 h - 8 h]</td><td>-8,176</td><td>-7,3584</td><td>-6,8833</td><td>10</td><td>15,81</td></tr><tr><td>[8 h - 12 h]</td><td>-8,3444</td><td>-7,51</td><td>-6,4056</td><td>10</td><td>23,235</td></tr><tr><td>[12 h - 16 h]</td><td>-7,3639</td><td>-6,6275</td><td>-5,0087</td><td>10</td><td>31,983</td></tr><tr><td>[16 h - 20 h]</td><td>-8,2242</td><td>-7,4045</td><td>-6,5854</td><td>9,9667</td><td>19,926</td></tr><tr><td>[20 h - 24 h]</td><td>-11,428</td><td>-10,586</td><td>-9,2714</td><td>7,3604</td><td>18,869</td></tr></tbody></table>

<figure><img src="image_30.png" /><figcaption>**Fig. 26.** Profile of powers purchased from sources.</figcaption></figure>
<figure><img src="image_31.png" /><figcaption>Fig. 29. Utility of the sources.</figcaption></figure>

<figure><img src="image_32.png" /><figcaption>Fig. 30. DC-Bus voltage.</figcaption></figure>

In response to these challenges, this paper presents an energy management strategy for a grid-connected multi-source system. The proposed management algorithm is designed to simultaneously minimize consumer costs and maximize energy utilization by suppliers. A Bertrand duopoly game model is employed to implement a dynamic pricing mechanism, where energy suppliers set their prices based on availability, and consumers respond by selecting the most economical option. This negotiation-based approach promotes equitable supplier participation and ensures balanced energy distribution.

Simulation results obtained using MATLAB/Simulink demonstrate that the proposed model reduces the consumer's average electricity bill by approximately 21.93 % during winter days and 20.82 % during summer days. Furthermore, the results show that increasing the number of participating suppliers leads to further reductions in consumer costs, indicating that the efficiency of the electricity market improves as competition increases. Additionally, the model allows consumers to rely

on the national grid only when local energy sources are insufficient, thereby enhancing grid reliability and overall system resilience. These findings confirm the effectiveness and practicality of the proposed approach in achieving cost efficiency and operational stability in a multi-source energy system.

Energy markets are now facing new challenges of various natures: reduction of load peaks, integration of renewable energies, smart consumption/production, the coexistence of multiple producers and distributors, as well as the possibilities of cooperation and competition. All these specificities must be further studied and considered in the design of future energy management systems.

Future work will focus on extending the proposed energy management strategy to a decentralized framework involving multiple interconnected microgrids and multiple consumers. This will enable the investigation of cooperative energy exchange and competition among microgrids operating under dynamic pricing conditions.

## Funding

This research received no external funding.

## CRediT authorship contribution statement

**Abderrahmane Redouane:** Writing – original draft, Visualization, Validation, Software, Resources, Methodology, Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization. **Rachid Saou:** Writing – original draft, Validation, Supervision, Software, Resources, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization. **Sofiane Touati:** Writing – review & editing, Visualization, Validation, Supervision, Resources, Project administration, Methodology, Funding acquisition, Data curation, Conceptualization. **Youcef Belkhier:** Writing – review & editing, Validation, Supervision, Software, Resources, Project

<figure><img src="image_33.png" /><figcaption>Fig. 31. Voltage and current of the load.</figcaption></figure>
<figure><img src="image_34.png" /><figcaption>Fig. 32. Batteries state of charge.</figcaption></figure>

<table><caption>**Table 6**

Comparative analysis of works using game theory and the proposed article.</caption><thead><tr><th rowspan="2">Article</th><th colspan="4">Article Objectives</th><th colspan="3">Market Characteristics</th><th colspan="4">Results Obtained</th></tr><tr><th>Reduce Energy Cost</th><th>Increase Producer Profit</th><th>Promote Renewables Energy</th><th>Smooth Demand</th><th>Price Competition</th><th>Complete Information</th><th>Game Characteristic</th><th>Cost Reduction</th><th>Profit Increase</th><th>Renewable Usage</th><th>Demand Smoothing</th></tr></thead><tbody><tr><td>Proposed Model [19]</td><td>✓</td><td>—</td><td>✓</td><td>—</td><td>✓</td><td>—</td><td>Static Non-cooperative</td><td>≈ 20 %</td><td>n.a.</td><td>✓</td><td>—</td></tr><tr><td>[20]</td><td>✓</td><td>✓</td><td>✓</td><td>—</td><td>✓</td><td>✓</td><td>Sequential Non-cooperative</td><td>n.d.</td><td>n.a.</td><td>✓</td><td>✓</td></tr><tr><td>[21]</td><td>—</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>Sequential Non-cooperative</td><td>≈ 4 %</td><td>≈ 12 %</td><td>✓</td><td>—</td></tr><tr><td>[22]</td><td>✓</td><td>—</td><td>✓</td><td>✓</td><td>—</td><td>✓</td><td>Sequential Non-cooperative</td><td>≈ 50 %</td><td>n.a.</td><td>✓</td><td>✓</td></tr><tr><td>[23]</td><td>✓</td><td>—</td><td>—</td><td>✓</td><td>✓</td><td>—</td><td>Static Cooperative</td><td>≈ 10 %</td><td>n.a.</td><td>—</td><td>✓</td></tr></tbody></table>

✓: Feature verified by the study.

—: Feature not targeted or not verified by the study.

n.a.: Numerical value or information not available in the study.

administration, Methodology, Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization. **Amrane Oukaour:** Writing – original draft, Validation, Supervision, Resources, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization. **Samira Chouikhi:** Writing – review & editing, Visualization, Validation, Supervision, Resources, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis. **Lyes Khoukhi:** Writing – review & editing, Visualization, Validation, Supervision, Software, Methodology,

Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix A

Appendix A.1

Table A.1
<table><caption>**Table A.1**
Comparison of different approach employed in EMS.</caption><thead><tr><th>Approach</th><th>Advantages</th><th>Drawbacks</th></tr></thead><tbody><tr><td>Genetic Algorithm (GA)</td><td>An appropriate convergence speed and easy adaptability to different types of optimization problems<br>Provides near-optimal solutions within a reasonable time</td><td>Parameters must be defined for mutation, selection, and crossover operations during the solving process</td></tr><tr><td>Particle Swarm Optimization (PSO)</td><td>Efficiency in Solving Optimization Problems<br>Ease of adaptation to different types of optimization problems, with near-optimal solutions provided within a reasonable computational time.</td><td>Reaching the optimal solution is not always guaranteed<br>Complex computations involved in solving an optimization problem<br>PSO demonstrates a slow convergence rate during the iterative process.</td></tr><tr><td>Mixed Integer Linear Programming (MILP)<br>Dynamic Programming (DP)</td><td>Enables the efficient and rapid resolution of complex problems by ensuring a globally optimal solution through linear constraints defined within a feasible convex region.<br>The approach is both simple and effective in achieving the global optimum<br>Divide the problem into subproblems, then optimize each to find the overall optimal solution</td><td>Its applicability is limited in cases where the objective functions are neither continuous nor differentiable<br>Computationally and memory intensive, often used in offline for benchmarking<br>Due to the presence of numerous recursive functions, the process is computationally time-consuming</td></tr><tr><td>Fuzzy Logic (FL)</td><td>Ease of implementation<br>Provides the optimal solution</td><td>It can sometimes be difficult to maintain accuracy<br>The performance depends heavily on the quality of the defined rules</td></tr><tr><td>Game Theory (GT)</td><td>It is straightforward and easy to use<br>Mathematically, it determines the most optimal strategy for a given set of conditions, maximizing the outcomes</td><td>Complexity increases with a large number of actors<br>Results depend on the model used</td></tr></tbody></table>

# Appendix B

## Appendix B.1

### Table B.1

<table><caption>**Table B1**
System variables.</caption><thead><tr><th>Variables</th></tr></thead><tbody><tr><td>P<sub>s</sub>: estimated power provided by supplier (kW)</td></tr><tr><td>ξ<sub>c</sub>: power requested by consumer c from s (kW)</td></tr><tr><td>ρ<sub>s</sub>: supplier selling price (pu)</td></tr><tr><td>α: grid purchase price (pu)</td></tr><tr><td>ρ<sub>g</sub>: grid selling price (pu)</td></tr><tr><td>c<sub>s</sub>: supplier production cost (pu)</td></tr><tr><td>u<sub>s</sub>: supplier utility</td></tr><tr><td>F<sub>c</sub>: consumer bill</td></tr><tr><td>SOC: battery state of charge (%)</td></tr><tr><td>V<sub>DC</sub>: DC-Link voltage (V)</td></tr><tr><td>V<sub>l,a</sub>: consumer phase voltage (V)</td></tr><tr><td>I<sub>l,a</sub>: consumer phase current (A)</td></tr></tbody></table>

## Appendix B2

### Table B.2

<table><caption>**Table B2**
System parameters.</caption><thead><tr><th>Parameter</th><th>Value</th></tr></thead><tbody><tr><td>P<sub>w,max</sub>: Maximum power of wind energy system</td><td>20 kW</td></tr><tr><td>P<sub>pv,max</sub>: Maximum power of solar energy system</td><td>80 kW</td></tr><tr><td>P<sub>b,max</sub>: Maximum power of battery storage system</td><td>20 kW</td></tr><tr><td>ρ<sub>1,max</sub>: Maximum energy selling price of the first source</td><td>0.17 pu/kW</td></tr><tr><td>ρ<sub>1,min</sub>: Minimum energy selling price of the first source</td><td>0.13 pu/kW</td></tr><tr><td>ρ<sub>2,max</sub>: Maximum energy selling price of the second source</td><td>0.18 pu/kW</td></tr><tr><td>ρ<sub>2,min</sub>: Minimum energy selling price of the second source</td><td>0.11 pu/kW</td></tr><tr><td>ρ<sub>g</sub>: Grid energy selling price</td><td>0.2 pu/kW</td></tr><tr><td>α: Grid energy purchase price</td><td>0.09 pu/kW</td></tr><tr><td>c<sub>1</sub>: Energy production cost of the first source</td><td>0.03 pu/kW</td></tr><tr><td>c<sub>2</sub>: Energy production cost of the second sources</td><td>0.05 pu/kW</td></tr></tbody></table>
# Data availability

Data will be made available on request.

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