The purpose of energy storage system planning is to store the surplus electricity generated during the process of new energy generation, thereby reducing the costs associated with curtailed wind and solar power, enhancing the economic efficiency of power system operation, and ultimately lowering the overall cost of distribution networks. Additionally, energy storage systems can perform peak shaving and valley filling, significantly decreasing voltage fluctuations in distribution networks during peak electricity usage periods, ensuring that the voltage at each node remains within a preset range, thus enhancing the stability of the power system.

Meanwhile, it is essential to consider the coordinated optimization between energy storage devices and the distribution network to achieve the minimization of total costs. Without such coordination, the operational costs of energy storage systems could exceed the costs associated with renewable energy curtailment, failing to improve the economic efficiency of the power system and potentially resulting in a counterproductive outcome. The objective function for this optimization can be formulated as follows: min ⁡ J = C 1 + ∑ k = 1 n C 2 E k (1)

In the formula, C 1 represents the operational cost of the distribution network, C 2 denotes the operation and maintenance cost of the introduced distributed energy storage devices, and E k is the capacity of the energy storage device installed at the k th node in the distribution network. (If no energy storage device is installed at that node, then the corresponding E k = 0 ).

The expression for the operational cost of the distribution network is as follows: C 1 = 365 ∑ i 4 p i f sub + f loss + f DSR + f RDG + f DG (2)

In the formula, p i represents the probability of occurrence for the i th typical scenario, where the calculation methods for each cost are as follows:

(1) cost of purchasing electricity from the higher-level grid f sub

Where C sub is the cost per unit of electricity purchased from the higher-level grid, P sen is the active power input from the higher-level grid.

(2) electricity network loss cost f loss

Where C loss represents the cost per unit of electricity network loss, I i j is the current flowing between nodes i and j , Z i j is the impedance between nodes i and j .

(3) interruptible load compensation cost f DSR

Where C DSR represents the compensation price for interruptible load, P DSR i is the size of the i th interruptible load participating in demand-side response.

(4) curtailment penalties of wind and solar power f RDG

Where C RDG represents the penalty cost per unit of curtailed wind and solar power, P RDG max i is the total active power generated by the i th RDG connected to the distribution network, P RDG i is the active power consumed by the distribution network from this RDG.

(5) renewable energy operation and maintenance cost f DG

Where C DG represents the operation and maintenance cost per unit of energy produced by renewable sources.

The operational and maintenance cost expression for distributed energy storage systems is given by: C 2 = f ESS + f op (8) the calculation methods for each cost are as follows:

(6) annual fixed investment cost of the energy storage system f ESS

Where b represents the annual discount rate of the energy storage system, y ESS is the service life cycle of the energy storage system, x ESS represents number of energy storage units installed, c ESS represents installation cost per energy storage unit.

(7) annual fixed investment cost of the energy storage system f op

Where f i represents the operating cost of the distributed energy storage system in the i th scenario, p i is the probability of occurrence of the i th typical scenario.

For the established objective function, the corresponding constraints are formulated as follows:

Constraint1: After configuring energy storage system in the renewable energy integrated distribution network, the system must satisfy the power flow balance constraint under steady-state conditions. P i = V i ∑ j = 1 n V j G i j ⁡ cos ⁡ θ i j + B i j ⁡ sin ⁡ θ i j Q i = V i ∑ j = 1 n V j G i j ⁡ sin ⁡ θ i j − B i j ⁡ cos ⁡ θ i j i = 1 , 2 , ⋯ , n (11)

Where V i and V j are the voltages at nodes i and j , G i j is the real part of the admittance matrix at position i , j , B i j is the imaginary part of the admittance matrix at position i , j , θ i j is the phase angle difference between nodes i and j , n is the total number of nodes in the network, P i and Q i represent the active power and reactive power injected at the node, respectively.

Constraint2: The distribution network must satisfy the output constraints. Specifically, the active and reactive power input from the main grid to the distribution network must remain within their respective upper and lower bounds. Additionally, the active power delivered by distributed renewable energy sources to the grid, as well as the power of interruptible loads on the demand side, must also adhere to their corresponding upper and lower limits. 0 ≤ P gen ≤ P genmax − Q genmax ≤ Q gen ≤ Q genmax P RDGmin ≤ P RDG ≤ P RDGmax P DSRmin ≤ P DSR ≤ P DSRmax (12)

In the formula, P gen and Q gen represent the active power and reactive power input from the main grid to the distribution network, respectively. P RDG denotes the active power delivered by distributed renewable energy sources to the grid, with P DSR represents the power of interruptible loads on the demand side.

Constraint3: the power flow constraints of the distribution network must be satisfied. U min ≤ U i ≤ U max I min ≤ I i ≤ I max U i 2 + I i 2 ≥ P i 2 + Q i 2 (13)

Where U i and I i are the voltage and current at node i , P i and Q i are the total active power and reactive power injected at node i respectively.

Constraint4: the constraints related to the energy storage devices must be satisfied, meaning that the charging and discharging power of each energy storage device is limited by its maximum power capacity and its own energy capacity. Additionally, the stored energy at any given time must remain within a reasonable range. 0 ≤ P ESS − d ≤ P ESSmax 0 ≤ P ESS − c ≤ P ESSmax W ESS = P ESS − c t ESS − c − P ESS − d t ESS − d 0.2 S i C ≤ W ESS ≤ 0.9 S i C (14)

Where P ESSmax represents the maximum power of the energy storage device, P ESS − d is the discharging power, P ESS − c is the charging power, t ESS − d is the discharging time, t ESS − c is the charging time, W ESS is the remaining energy in the energy storage device, S i C is the energy storage capacity installed for the i th synchronous condenser.

For the objective function constructed in Section 3.1, the model is a complex dynamic differential-algebraic equation, representing a nonlinear parameter optimization problem considering the power grid. It involves the optimization of multiple node-mounted synchronous condenser capacities, which is difficult to solve using classical mathematical programming methods. Considering computational speed and accuracy, and to ensure global search capability, the Beetle Swarm Optimization (BSO) algorithm is employed in this paper to solve the objective function.

The Beetle Swarm Optimization (BSO) algorithm is a novel optimization method that combines the advantages of both the Beetle Antennae Search (BAS) algorithm and the Particle Swarm Optimization (PSO) algorithm. It leverages their complementary strengths, offering high optimization efficiency and global search capability (Wu et al., 2024; Xuan and Li, 2024). In this algorithm, the initial beetle particles are generated through Monte Carlo random sampling. By incorporating an adaptive inertia weight adjustment strategy into the optimization iteration process, the population-based optimization process with a population size of N _p is improved. In this population, the capacity of the synchronous condenser corresponds to the position of the beetles. In the k + 1 th iteration, the velocity V i , j k + 1 and position X i , j k + 1 of the j th element of the i th beetle are given by: V i , j k + 1 = ω k V i , j k + c 1 k r 1 p i , j k − X i , j k + c 2 k r 2 g j k − X i , j k (15) X i , j k + 1 = X i , j k + λ V i , j k + 1 + 1 − λ ξ i , j k (16) ξ i , j k + 1 = δ k V i , j k sign f X r , i , j k − f X l , i , j k (17)

Where p i , j k represents the j th element of the best position in the history of the i th beetle at the k th iteration, g j k represents the j th element of the global best position at the k th iteration, r 1 and r 2 are random numbers within the range 0 , 1 , ω k is the inertia weight, λ is a positive constant, ξ i , j k is the position increment factor. ω k + 1 = ω max − k + 1 k max ω max − ω min (18) c 1 k + 1 = 2 ⁡ sin 2 π k max − k + 1 2 k max (19) c 2 k + 1 = 2 ⁡ sin 2 π k + 1 2 k max (20)

Where ω max is the maximum inertia weight, ω min is the minimum inertia weight, c 1 k + 1 is the self-learning factor, c 2 k + 1 is the swarm learning factor, k max is the maximum allowed number of iterations. δ k = η δ δ k − 1 (21) X r , i , j k = X i , j k − 1 + d 2 × V i , j k − 1 (22) X l , i , j k = X i , j k − 1 − d 2 × V i , j k − 1 (23)

Where δ k is the search step size at the k th iteration, sign * is the sign function, X l , i , j k is the position of the f X r , i , j k left antenna of the beetle, X r , i , j k is the position of the right antenna of the beetle, f X l , i , j k is the fitness value of the left antenna of the beetle,is the fitness value of the right antenna of the beetle, η δ is the variable step size factor, d is the distance between the left and right antennae of the beetle. The flowchart of the Beetle Swarm Optimization (BSO) algorithm is shown in Figure 2.

First, the optimization problem is defined in YALMIP. Then, the beetle swarm is initialized, and the iteration count along with parameters is set. During each iteration, the position and velocity of each individual beetle are updated. The corresponding objective function value for each beetle is calculated using YALMIP, and the individual and global best solutions are updated accordingly. After multiple iterations, an approximate optimal solution is found, effectively solving the original problem. This approach combines the global search capability of the beetle swarm optimization (BSO) algorithm with the precise modeling and solving capabilities of YALMIP.

Based on the optimal planning scheme of distributed energy storage systems obtained in the previous section, the economic costs of each component of the rural distribution network before and after energy storage planning are presented in Table 2. By comparing the economic costs under both scenarios, the improvement in the economic performance of the rural distribution network with renewable energy integration after energy storage planning is analyzed.

From Table 2, it can be observed that after implementing the distributed energy storage planning, the rural distribution network integrated with renewable energy demonstrates significant advantages in terms of economic performance, renewable energy accommodation capability, and operational efficiency. Most notably, the total operating cost of the distribution network is reduced by 681,400 RMB, approximately 11.4%, thereby enhancing overall economic benefits and highlighting the financial return capacity of energy storage-based compensators.

The energy storage system effectively reduces electricity procurement costs from the upper grid by storing surplus power during periods of excessive renewable generation and discharging during peak load times. This results in a cost reduction of 608,800 RMB, or about 12.6%. Furthermore, the system mitigates the variability of wind and solar power outputs, significantly reducing the waste of renewable energy and the associated economic losses caused by curtailed wind and solar generation. Specifically, after the implementation of energy storage planning, the penalty cost due to renewable curtailment decreases by as much as 85.4%, indicating that the energy storage system substantially improves the utilization rate of renewable energy generation and reduces resource wastage. At the same time, the integration of energy storage improves the power flow distribution within the grid, minimizes transmission losses, and reduces network loss costs to 26.5% of their original level, further enhancing the operational efficiency of the distribution network.

Therefore, implementing distributed energy storage planning for rural distribution networks incorporating renewable energy sources represents a crucial approach to achieving a balance between technical and economic considerations, as well as promoting environmental and social sustainability. In the short term, the integration of energy storage leads to significant reductions in electricity procurement costs, network losses, and renewable curtailment costs, thereby improving economic returns while markedly enhancing renewable energy utilization and grid operational efficiency. In the long run, energy storage planning not only enhances the grid’s capacity to integrate renewable energy but also reduces dependence on traditional fossil fuels, providing critical support for building a secure, economical, green, and reliable new power system. It also serves as an essential foundation for achieving national energy transition objectives.

By combining the node voltage data of the distribution network across different time periods before and after the implementation of distributed energy storage planning, this paper analyzes the improvement in voltage stability of the rural distribution network integrated with renewable energy. Under four typical scenarios, the voltage stability at each node is effectively enhanced, with generally similar results. Therefore, this study only presents the voltage profile comparisons before and after planning for two representative scenarios, while the others are omitted for brevity. The voltage curves of the distribution network before and after distributed energy storage planning under Scenario one and Scenario 2 are illustrated in Figures 4, 5, respectively.

In the three-dimensional voltage profile graphs, deeper colors indicate a greater deviation from the baseline voltage values. Prior to the implementation of distributed energy storage planning, node voltages across the distribution network exhibited significant fluctuations during various time periods, especially during midday and evening peak hours when power demand was high, leading to noticeable voltage drops or rises at some nodes. During off-peak hours, voltage levels notably decreased, with some nodes even falling below the lower voltage limit of 0.95, thus increasing voltage variability within the grid and posing challenges to the stable operation of the distribution network.

After the introduction of distributed energy storage, the overall voltage profiles became smoother, with significantly reduced voltage differences between nodes and substantially diminished voltage fluctuations. By discharging during peak load times and absorbing excess power during off-peak times, the energy storage systems effectively balanced system power, mitigating voltage imbalances caused by load fluctuations. Moreover, the amplitude of voltage fluctuations over time was markedly reduced, bringing node voltage values closer to the nominal voltage (1.0) without any significant voltage dips, thereby enhancing the voltage stability of the system. The introduction of such composite devices not only avoids the issue of unstable voltage fluctuations that can occur with energy storage systems alone but also precludes the need for additional reactive power compensation equipment to address this deficiency. This demonstrates that these devices optimize both the economy and operational efficiency of the distribution network.