Among various natural disasters, lightning strikes occur frequently and pose a significant threat to overhead distribution lines, particularly in regions with complex terrain and volatile weather conditions. Given their high frequency and severe impact on the operational reliability of distribution networks, this study focuses specifically on lightning hazards as the primary type of extreme weather under consideration.

To effectively assess the risk of lightning hazards, this study considers two critical environmental factors: climate conditions and terrain types.

Based on long-term meteorological observations, the climate is classified into three representative states: rain/snow, cloudy, and sunny. The occurrence of lightning hazards is closely related to climate conditions. For example, rain and snow are typically accompanied by high humidity and atmospheric instability, which increase the likelihood of thunderstorms. Cloudy weather often indicates the onset of convective activity and is also associated with a higher risk of lightning. In contrast, sunny conditions are generally more stable, resulting in a lower probability of lightning strikes.

Likewise, the terrain along distribution lines is divided into four types: flatlands, hills, mountains, and highlands. Terrain characteristics not only shape local atmospheric dynamics—such as humidity levels and wind patterns—but also affect the exposure of power lines to lightning. For example, areas with higher elevation or rugged topography are more prone to lightning activity.

Given the memoryless property of distribution lines in operation, the number of faults k occurring within a unit period of time can be modeled using a Poisson distribution, expressed as: (1)p(X=k)=λkk!e−λwhere X is a discrete random variable representing the number of failures of a distribution line within a unit time, X=k denotes the event that exactly k failures occur during that period, and λ is the average line failure rate per unit time.

Considering the vulnerability of distribution lines to extreme weather conditions, the line fault model is reformulated as: (2)p(X=k)=(ωλ)kk!e−ωλwhere ω is the extreme weather impact factor and ωλ is the line failure rate per unit of time.

Then, the failure probability of a distribution line within a unit time can be defined as the probability that at least one failure occurs, which is given by: (3)p(X≠0)=1−p(X=0)=1−e−ωλwhere p(X=0) represents the probability that no failure occurs during the time period.

The overall framework of the optimal configuration model for DG and ESS is shown in Fig. 1. First, a comprehensive line vulnerability analysis model is employed to identify lines with lower failure probability, which are used to determine candidate access locations for DG and ESS. Then, an optimization model is formulated to minimize the annual total cost and network losses, subject to constraints including power flow, node voltage, branch current, and the operational limits of ESS and SOP. The CPLEX solver is used to obtain the optimal installation locations and capacities of DG and ESS.

The DG and ESS planning model proposed in this paper is formulated as a multi-objective mixed-integer optimization problem. While heuristic algorithms are commonly used to solve such problems and obtain a Pareto-optimal solution set [20], they often suffer from slow convergence and sensitivity to initial conditions. To improve computational efficiency, this study adopts a linear weighting approach to transform the multi-objective problem into a single-objective optimization problem, which can be effectively solved using a commercial solver.

The objective function combines two key indices: the annual comprehensive cost and the annual network losses. Specifically, the annual comprehensive cost includes the investment and operation costs of DG and ESS, the electricity purchase cost from the main grid, and the cost associated with carbon emissions. This modeling approach ensures both economic efficiency and network performance in the planning of DG and ESS installations.

The objective function can be represented as: (5)f=minω1×(CDG+CESS+CBuy+CCO2)+ω2×Flosswhere CDG is the annual investment and operating costs of DG; CESS is the annual investment and operating costs of ESS; CBuy is the cost of purchasing power from the main grid; CCO2 is the cost of carbon emissions; Floss is the annual network losses. ω1 and ω2 are weighting factors that balance the relative importance between the economic objective and the technical objective of network loss minimization. In this study, both investment and operating costs and annual network losses are considered equally important, and their magnitudes are on a similar scale, so the weights are set to ω1=0.5 and ω2=0.5.

The expression of CDG is as follows: (6)CDG=∑j∈NcanγjSDG,j[r(r+1)n(r+1)n−1×Cinv,j+Cope,j]where Ncan is the set of nodes that are candidates for installation of distributed generation; γj denotes whether distributed generation is installed at node j; SDG,j is the capacity of distributed generation installed at node j; r is the fixed annual interest rate; n is the planning horizon; Cinv,j is the investment cost per unit of capacity of distributed generation installed at node j; and Cope,j is the operating cost per unit of capacity of distributed generation installed at node j.

The expression of CESS is as follows: (7)CESS=∑i∈NcanνiPESS,i[r(r+1)n(r+1)n−1×CP,i+CE,i]where Ncan is the set of candidate nodes for ESS installation; νi denotes whether ESS is installed at node i; PESS,i is the installed capacity of ESS at node i; r is the fixed annual interest rate; n is the planning horizon; CP,i is the unit investment cost of ESS at node i; and CE,i is the unit operating cost of ESS at node i.

The expression of CBuy is as follows: (8)CBuy=365×∑t=124Cb,t(Pload,t−PDG,t−PESS,t)Δtwhere Cb,t is the real-time electricity purchase price from the grid at time t; Pload,t is the total active power demand of all load points at time t; PDG,t is the total active power output of all DG units at time t; and PESS,t is the total active power output of all ESS units at time t.

The expression of CCO2 is as follows: (9)CCO2=365×m×e1000×∑t=124(Pload,t+Ploss,t−PDG,t−PESS,t)Δtwhere m the environmental penalty cost for carbon emissions, representing the penalty incurred per ton of carbon dioxide emitted; e is the carbon emission factor, indicating the amount of carbon dioxide (in kilograms) emitted per kilowatt-hour of electricity generated; and Ploss,t is the total active power loss of the network at time t.

The expression of Floss is as follows: (10)Floss=365×∑t=124∑i,j∈ΩIij,t2rijΔtwhere Ω is the set of all branches of the distribution network; Iij,t is the current magnitude of branch ij at time t; rij is the resistance of branch ij.

(1)

Powe flow constraints: (11)ui,t=Ui,t2,uj,t=Uj,t2 (12)lij,t=Iij,t2 (13)uj,t=ui,t−2(rijPij,t+xijQij,t)+(rij2+xij2)lij,t (14)∑i:i→j(Qij,t−xijlij,t)−qj,t=∑k:j→kQjk,t (15)∑i:i→j(Pij,t−rijlij,t)−pj,t=∑k:j→kPjk,t (16)ui,tlij,t=Pij,t2+Qij,t2 (17)pj,t=Pload,j−Pj,tDG−Pj,tESS−Pj,tSOP (18)qj,t=Qload,j−Qj,tDG−Qj,tSOP where j→k indicates that node j is the upstream node of node k; Ui,t and Uj,t are the voltages of node i and node j at time t, respectively; Pjk,t and Qjk,t represent the active and reactive power flowing from node j to node k at time t, respectively; pj,t and qj,t denote the net active and reactive loads at node j at time t, respectively; rij and xij are the resistance and reactance of the branch between nodes i and j; Pload,j and Qload,j are the active and reactive load at node j, respectively; Pj,tDG and Qj,tDG are the active and reactive power generated by the DG at node j at time t, respectively; Pj,tESS is the charging/discharging power of the ESS installed at node j at time t, where positive values indicate discharging and negative values indicate charging; Pj,tSOP and Qj,tSOP are the active and reactive power injected by the SOP to node j at time t.

Eq. (16) is nonlinear and nonconvex, but it can be reformulated into a standard second-order cone constraint through second-order cone relaxation: (19)‖2Pij,t2Qij,tlij,t−ui,t‖2⩽lij,t+ui,t

The IEEE 33-bus distribution system is adopted for analysis and validation, as illustrated in Fig. 2. The system operates at a voltage level of 12.66 kV, with an allowable voltage range of [0.95, 1.05] p.u. Detailed information on branch impedance and load power is available in [21]. Two dual-port SOPs are configured between nodes 12–22 and 18–33, respectively. Each SOP has a loss coefficient of 0.02. The rated output power of the ports at nodes 12 and 22 is 0.002 MVA, while that of the ports at nodes 18 and 33 is 0.01 MVA [22].

The specific probabilities associated with different terrain-climate combinations, as referenced in Table 1, are detailed in Table 3.

The service life of DG is assumed to be 20 years, with a fixed annual interest rate of 0.1. The investment cost is $500/kVA, and the annual operating cost is $70/kVA. For ESS, the service life is set at 15 years, with the same fixed annual interest rate of 0.1. The investment cost is $226.38/kW, and the annual operating cost is $175/kW [23].

Time-of-use electricity prices are listed in Table 4 [24], and the output coefficients of DG at each time interval are illustrated in Fig. 3.

The vulnerability indices of all 32 distribution lines are obtained from the Monte Carlo simulation results, as shown in Fig. 4. According to the law of large numbers, a larger number of sampling trials M yields probability estimates that more closely approximate their true values. Therefore, in this study, M = 100,000 is selected to ensure sufficient statistical stability of the Monte Carlo simulation results.

A lower vulnerability index represents a lower risk of lightning-induced failures. To ensure reliability in DG and ESS deployment, lines with vulnerability index values not exceeding 0.05 are selected as feasible installation candidates. A total of 24 lines meet this criterion, corresponding to Nodes 4, 6–9, 11–17, 19–21, 23, 24, and 26–32, which form the candidate set for the subsequent optimization.

The optimal configuration model is solved using the CPLEX solver, resulting in the DG and ESS allocation scheme presented in Table 5.

The economic performance of the proposed configuration scheme is summarized in Table 6, with a total cost of $2,669,400.

Additionally, the annual network losses amount to 455.76 MWh. The hourly distribution of daily network losses is illustrated in Fig. 5.

With the integration of DG and ESS, the voltage levels at each node are shown in Fig. 6. All node voltages remain within the specified limits.

The 24-h charging and discharging profile of the ESS is illustrated in Fig. 7. During Periods 1–6, the ESS charges, taking advantage of low-load intervals to gradually increase the SOC. Subsequently, in Periods 7–9, the ESS remains idle, maintaining a stable SOC while avoiding unnecessary cycling. From Periods 10–14, the system enters a high-demand stage, and the ESS discharges to support network load and reduce grid power purchases. After this stage, the ESS briefly returns to charging in Periods 15–18 to restore part of its stored energy. During the later peak demand from Periods 20–23, the ESS discharges again to compensate for the shortage of local supply and mitigate peak load stress on the distribution network. Finally, the ESS returns to the idle state in Period 24. Overall, the schedule effectively achieves peak shaving and valley filling while maintaining feasible SOC operation throughout the day.

To further demonstrate the necessity of incorporating line vulnerability into planning decisions, a benchmark DG-ESS planning model [25] is adopted for comparison. The benchmark approach optimizes DG and ESS solely based on economic operation and voltage constraints, without considering the increased failure risk of distribution lines under extreme weather conditions.

There are four planning strategies designed to quantitatively reflect how line vulnerability influences DG-ESS allocation decisions and overall distribution network performance.

Case 1.1: DG and ESS are planned using the proposed vulnerability-aware methodology, where climate-induced line vulnerability is explicitly incorporated into the decision process.

All four cases are applied to the IEEE 33-bus test system. The key metrics include annual total cost, annual network losses, average outage duration considering climate-adjusted line failure rate, and maximum node voltage deviation.

The comparison results are presented in Table 7. Firstly, the effectiveness of ESS can be evaluated by comparing Case 1.1 with Case 1.2 and Case 2.1 with Case 2.2. In both planning methods, including ESS significantly enhances system performance. For the proposed method, ESS deployment reduces the average outage duration by 21.66%, decreases annual network losses by 9.91%, and improves voltage regulation by 65.71%, at the expense of a modest cost increase of 4.87%. A similar pattern appears in the benchmark method, where ESS lowers the outage duration by 19.23%, cuts network losses by 10.56%, and reduces voltage deviation by 65.36%, with only a 4.92% increase in annual cost. These results indicate that ESS provides fast-response power support during contingencies and performs peak shaving and valley filling during normal operation. Therefore, the investment in ESS, despite slightly higher costs, is economically justified because the gains in operational flexibility, voltage quality, and supply continuity substantially outweigh the marginal increase in annual cost.

Then, the impact of incorporating line vulnerability can be evaluated by comparing Case 1.1 with Case 2.1 and Case 1.2 with Case 2.2. When DG and ESS are jointly planned, the vulnerability-aware strategy reduces the average outage duration by 45.37% compared with the benchmark method, while maintaining slightly lower annual cost and voltage deviation and keeping network losses nearly unchanged. Even for DG-only planning, considering line vulnerability still shortens the outage duration by 43.67%. These results demonstrate that proactively avoiding vulnerable lines enables more resilient resource allocation, prevents substantial performance degradation under weather-induced contingencies, and significantly enhances the distribution network’s ability to withstand climate-driven uncertainties.

Besides, to examine whether the preceding Monte Carlo-based vulnerability evaluation increases computational burden, the computation time of each case is also analyzed. The results show that the total computation time of Case 1.1 and Case 1.2, which includes the Monte Carlo simulation performed in the preprocessing stage, remains comparable to that of the benchmark cases. This indicates that the additional vulnerability assessment introduces negligible computational overhead while improving the robustness of planning decisions.

In summary, ESS contributes to operational flexibility and enhanced reliability, while vulnerability-aware planning ensures robust resource placement. The combination of the two features in Case 1.1 achieves the best performance in terms of cost-resilience balance, clearly demonstrating the superiority and practical necessity of the proposed planning approach.