In the context of a two-part electricity pricing mechanism, this paper establishes a SES planning model for multiple PV stations with the objective of minimizing overall costs. The approach involves three main steps: initially constructing an optimization configuration model for SES, developing a model aimed at maximizing the benefits of prosumers, and finally, formulating a collaborative SES planning model for PV stations under cooperative conditions.

The configuration of SES entails rational planning to maximize its efficiency and benefits. The objective function F _es for optimizing the configuration include transaction cost with the grid C ₁, initial installation cost C ₂, and operational cost C ₃. The specific calculation formula is as follows: F es = ⁡ min C 1 + C 2 + C 3 (1)

Regarding transaction costs with the grid, a two-part electricity pricing mechanism is adopted. This mechanism includes both an energy charge based on actual electricity consumption and a demand power charge based on the user’s maximum monthly demand power. SES engages in arbitrage by charging during periods of low electricity prices and discharging during periods of high prices. By shifting load demands, the user’s demand is reduced, consequently lowering costs for the consumer and enhancing the benefits of SES. Therefore, the transaction cost C ₁ between SES and the grid can be expressed as follows: C 1 = ∑ h ∈ H λ h b q h b Δ H + Q ⁡ max q h b (2) where λ h b , Q , H , Δ H are respectively the time-of-use price, demand power price, dispatching period, and dispatching interval. q h b denotes the power bought from superior grid at time h.

The initial installation cost of SES, denoted as C ₂, primarily comprises capacity and power costs (Wang YX. et al., 2024; Wang D. et al., 2024). These two components constitute the main expenses for the initial installation of the SES system and can be expressed as follows: C 2 = λ e E rate + λ p P rate (3) where λ e , λ p are respectively the unit capacity and power costs, and E rate , P rate represent maximum rated capacity and power, respectively.

As for operation and maintenance cost of SES, it is closely associated with the charging and discharging power of the system. Variations in SES charging and discharging power directly impact the frequency and complexity of maintenance. Therefore, the operational cost of SES C ₃ can be expressed as follows: C 3 = ∑ h ∈ Η C om q h ch + q h dis Δ H (4) where C om denotes unit power operation and maintenance cost coefficient of SES, and q h ch , q h dis are respectively the charge and discharge power of SES at time h.

In the process of planning SES, to ensure the stable operation of the PV station, certain inherent constraints must be satisfied (Ma et al., 2022; Jiao et al., 2021). These constraints include the following: 0 ≤ q h ch ≤ q max ch 0 ≤ q h dis ≤ q max dis q h b ≥ 0 q h ch q h dis = 0 S o C h = 1 − ρ S o C h − 1 + η ch q h ch E rate − η dis q h dis E rate Δ H E rate = γ P rate 0.2 ≤ S o C h ≤ 1 S o C 0 = S o C H q h ch + q h b = q h dis (5)

Equation 5 represent the charging and discharging power constraints of the SES, where q max ch and q max dis denote the maximum charging and discharging power of SES, respectively. Equation 5 signifies the non-negativity constraint on the power absorbed by SES from the upper grid. Equation 5 represents the state of charge constraint, indicating that SES cannot simultaneously charge and discharge. Equation 5 describes the state of charge for SES, determined by the charging and discharging power at the current time and the state of charge at the previous time. Here, S o C h and S o C h − 1 represent the state of charge for SES in time periods h and h-1, respectively. ρ is the self-discharge coefficient, indicating the proportion of energy loss per hour, while η ch and η dis represent the charging and discharging efficiencies of SES, respectively. Equation 5 represents the energy rate constraint for SES, where γ is the energy rate coefficient. Equation 5 indicates that the remaining energy of SES should be between 0.2 and 1 to extend the lifespan of SES. Equation 5 states that the charging and discharging quantities of SES should remain consistent within an optimization cycle H. Equation 5 represents the power balance constraint for SES. For this problem, without loss of generality, let q h ch *   q h dis *   q h b *   denote the optimal solution of F es .

For prosumer n, its operational cost, denoted as F _pv, encompass the transaction costs of buying and selling electricity within time period h. Utilizing the two-part pricing mechanism, the operational cost can be calculated as follows: F pv = ⁡ min ∑ h ∈ H λ h b p n , h b − λ h s p n , h s Δ H + Q ⁡ max p n , h b (6) where p n , h b and p n , h s respectively represent the power bought and sold by prosumer n to the grid at time h.

In order to ensure that the prosumers can stably participate in the operation of the optical storage park, the following constraints need to be met (Zhang et al., 2022; Liu J. et al., 2023). p n , h b ≥ 0 p n , h s ≥ 0 p n , h b p n , h s = 0 p n , h load + p n , h s − p n , h b = p n , h pv (7) where Equation 7 represent the non-negativity constraints on the power bought and sold by prosumer n within time period h. Equation 7 ensures that prosumer n cannot purchase and sell electricity at the same time. Equation 7 represents the power balance constraint, where p n , h load is the load demand of prosumer n at time h, and p n , h pv is the PV output power of prosumer n at time h. Without loss of generality, let   p n b *   p n s * denote the optimal solution of F pv .

In order to reduce the operational cost of PV prosumers and optimize SES configuration, collaboration between prosumers and SES is essential. A planning model for minimizing the costs of PV prosumers needs to be established. Here, p n , h e represents the energy transferred to SES by prosumer n at time h, and q n , h e represents the energy received by SES from prosumer n at time h. τ n denotes the payment made by prosumer n to SES, while π n represents the fee accepted by SES from prosumer n. The energy-sharing configuration mechanism determines the amount of energy transferred between prosumers and SES, while the payment mechanism ensures a fair and mutually agreed upon cost-sharing for the exchanged energy. In the process of solving the SES planning for multiple PV prosumers sharing model, the energy sharing constraint shown in Equation 8 and the energy payment mechanism shown in Equation 9 need to be satisfied, as mentioned in (Wu et al., 2024). p n , h e = q n , h e (8) τ n = π n (9) where Equation 8 ensures that the energy sharing plan between prosumers and SES is consistent, that is, the energy transmitted to SES by prosumers at the same time h is the same as the energy obtained by SES from prosumers. Similarly, Equation 9 ensures that the prosumer’s payment to SES is consistent with the payment obtained by SES from the prosumer at time h.

Firstly, according to Equations 1–5 and Equations 6–9, we can know the respective cost functions of SES and prosumers when they do not cooperate. Then, according to the payment mechanism of prosumers and SES shown in Equation 9, the overall objective function of SES and prosumers when they cooperate is as follows: min ⁡ C = C 1 + C 2 + C 3 − ∑ n ∈ N π n + ∑ h ∈ H λ h b p n , h b − λ h s p n , h s Δ H + Q ⁡ max p n , h b + ∑ n ∈ N τ n (10)

SES needs to meet the following constraints when cooperating with prosumers: q h ch + q h b = ∑ n ∈ N q n , h e + q h dis p n , h load + p n , h s = p n , h pv + p n , h b + p n , h e (11)

Furthermore, Equation 9 ensures that ∑ n ∈ N τ n − ∑ n ∈ N π n = 0 , meaning the sum of payments between PV prosumer stations and SES does not impact the overall energy costs of the entire PV stations. This implies that, in the proposed energy cooperation model, individual interests are not contradictory to societal interests. By optimizing the objective function while satisfying these constraints, the model presented in this paper aims to achieve the most effective energy cooperation strategy that minimizes costs for both SES and PV prosumers simultaneously.

This paper introduces data from five prosumers within a time range of 7:00–19:00, covering a period of 12 h. The load demand curve for prosumers is depicted in Figure 3A, while Figure 3B illustrates the PV output power (Liu et al., 2017). The electricity purchase price from the grid follows a two-part pricing system, where valley periods are 0:00 - 8:00 with electricity price 0.37CNY/kWh, flat periods are 12:00 - 17:00/21:00 - 24:00 with electricity 0.82CNY/kWh, and peak periods are 8:00 12:00/17:00 - 21:00 with electricity price 1.36CNY/kWh. The demand power charge is 38 CNY per month, and the on-grid power is 0.3CNY/kWh. The unit capacity cost of SES is 0.650 CNY/kWh, the unit power cost of SES is 0.245 CNY/kW, SES unit power operational cost is 0.1 CNY/kW, with charging and discharging efficiencies of 0.94 and 1.05, and an energy rate coefficient of 0.3. When using the AADMM algorithm for solution, the algorithm’s iteration precision is set to 0.0001, the penalty parameter is α 1 = 0.1 , and the initial judgments are x n 1 = p n b * , p n s * , 0 , y 1 = q h ch * , q h dis * , q h b * , 0 , and δ = 0.

When prosumers do not engage in energy cooperation with SES, the net load curve of prosumer is depicted in Figure 4A. When the PV output power of prosumers satisfies the load demand, the excess PV output is sold to the grid. However, when the PV output power is insufficient to meet the load demand, electricity needs to be purchased from the superior grid to fulfill the demand. In this scenario, the utilization efficiency of PV is low, and it fails to achieve maximum energy efficiency and local consumption of PV power, which is easy to cause the power grid reverse heavy overload.

In the energy cooperation mode, SES stores the surplus energy from prosumers’ high PV output, sells excess PV output to the grid after reaching maximum SES capacity, and shares energy with prosumers during periods of low PV output. The on-grid power of PV under energy cooperation between prosumers and SES is illustrated in Figure 4B. Through collaborative energy sharing, the PV grid power for prosumers significantly decreases. For instance, at 13:00-15:00, prosumer 2 has high PV output and low load demand, leading to the surplus PV output being sold to the grid. Overall, energy cooperation redistributes excess PV output to SES, achieving on-site consumption of PV power, and ensuring the resilience operation of the distribution network.

Table 1 presents a comparison of the economic cost for SES and prosumers under non-cooperative and cooperative modes. It is observed that SES, through peak-valley price arbitrage and on-site consumption of PV power, reduces the costs of the PV prosumers and increases the self-sufficiency. The self-sufficiency rate of prosumers increases from 71.05% to 92.91%, and the comprehensive electricity cost decreases from 1090.78 CNY to 1042.82 CNY. This implies that more PV power is locally consumed by prosumers, reducing reliance on traditional power plants, achieving “self-production and self-consumption,” improving energy utilization efficiency, and reducing the overall operational cost, ultimately maximizing energy efficiency. Figure 5 shows the power balance curve of prosumer 1 with or without cooperation. As shown in Figure 5, when there is no cooperation, user 1 can only trade with the power grid, which increases the dependence on the power grid and cannot carry out peak-valley arbitrage. When user 1 cooperates with SES, the user can transfer the excess power to SES.

In the energy cooperation mode, the optimal energy-sharing curve among prosumers is illustrated in Figure 6. Prosumer 4, with low PV output and high load demand, absorbs energy to meet the demand. Other prosumers store surplus energy in SES during periods of high PV production for subsequent self-load demands. The SES planning model for the PV station optimizes SES operational scheduling strategies, taking SES state of charge as an optimization parameter to maximize economic benefits. Based on the provided data, the optimal SES capacity configuration is determined to be 333.50 kWh. Taking prosumer 2 as an example, the power optimization results are shown in Figure 7A. When its own PV output cannot meet the load demand, prosumer 2 needs to purchase electricity from the grid or SES. On the contrary, when its own PV output remains after meeting the load demand, prosumer 2 sells the remaining electricity to the grid or transmits it to SES for use by other prosumers. The SES charging and discharging power, along with the state of charge curve, are depicted in Figure 7B. The SES state of charge pattern corresponds to the time-of-use pricing periods, with discharging occurring during peak pricing hours (8:00 -9:00 and 18:00 -19:00) and coinciding with peak pricing periods. Charging occurs during standard pricing hours (13:00 -17:00) and aligns with non-peak pricing periods. The state of charge curve follows the “low charge, high discharge” characteristic typical of SES.

This paper presents AADMM to iteratively solve the SES planning for multiple PV prosumers sharing. A comparison between ADMM and AADMM algorithms is presented in Table 2. The following conclusions can be drawn from Table 2. Firstly, different penalty parameter selections require varying iteration counts and time to achieve the same algorithmic accuracy. Hence, the initial choice of the penalty parameter can impact the algorithm’s iteration count and time. Secondly, for the proposed model in this paper, smaller initial penalty factors result in better convergence performance. The adaptive penalty parameter proposed in AADMM significantly reduces both the iteration count and iteration time compared to the fixed penalty parameter. Furthermore, the algorithm’s improvement varies with different parameter initializations. For both ADMM and AADMM, a value of 0.3 is a preferable parameter choice. The adaptive penalty parameter in AADMM accelerates the algorithm’s convergence process during the iterative stage. For instance, with a value of 0.3, the algorithm converges after 28 iterations. The convergence curves for residual errors at different time steps are illustrated in Figure 8A. At the same time, as shown in Figure 8B, taking prosumer 2 at 15: 00 as an example, as the number of iterations increases, the power transmitted by the prosumer 2 to the SES and the power received by the SES from prosumer 2 gradually tend to be consistent, reaching a global optimum. In addition, in order to show the influence of different penalty factor initial values on AADMM, this paper shows the iterative effect at α = 0.3, 0.5. As shown in Figures 9, 10, regardless of the initial value of the penalty factor, AADMM can self-adjust and ultimately achieve global optimization. This corresponds to Figure 8 and Table 2.