Research on Enhancing Disaster-Resilient Power Supply Capabilities in Distribution Networks Through Coordinated Clustering of Distributed PV Systems and Mobile Energy Storage System

Abstract

To enhance the power supply resilience of distribution networks with high-penetration distributed photovoltaic (PV) integration during extreme disasters, deploying Mobile Energy Storage Systems (MESSs) proves to be an effective countermeasure. This paper proposes an optimized operational strategy for distribution networks, integrating coordinated clustering of distributed PV systems and MESS operation to ensure power supply during both pre-disaster prevention and post-disaster restoration phases. In the pre-disaster prevention phase, an improved Louvain algorithm is first applied for PV clustering to improve source-load matching efficiency within each cluster, thereby enhancing intra-cluster power supply security. Subsequently, under the worst-case scenarios of PV output fluctuations, a robust optimization algorithm is utilized to optimize the pre-deployment scheme of MESS. In the post-disaster restoration phase, cluster re-partitioning is performed with the goal of minimizing load shedding to ensure power supply, followed by reoptimizing the scheduling of MESS deployment and its charging/discharging power to maximize the improvement of load power supply security. Simulations on a modified IEEE 123-bus distribution network, which includes two MESS units and twenty-four PV systems, demonstrate that the proposed strategy improved the overall restoration rate from 68.98% to 86.89% and increased the PV utilization rate from 47.05% to 86.25% over the baseline case, confirming its significant effectiveness.

1. Introduction

Under global pledges to cut emissions and the accelerating shift to low-carbon electricity, renewable energy sources such as wind, solar, and hydropower have expanded rapidly, and renewables accounted for about 30% of global electricity generation in 2023, indicating that power systems are increasingly being operated with a high share of renewable generation [1]. Because renewable generation is highly sensitive to weather and other external factors, uncertainty has been greatly increased, and control and optimization have become essential for safe and reliable operation, as reflected by advanced control of wind power interfacing converters [2] and optimized hydropower generation forecasting/prediction modeling [3]. In distribution networks, the fixed number and locations with a high penetration of distributed PV units have made local hosting-capacity limits and voltage-related constraints more evident in low-voltage distribution systems [4]. In addition, their random and fluctuating output creates challenges for PV integration [5,6]. Recent outage statistics and studies have shown that improving the resilience of distribution networks under extreme weather is especially important, compared with transmission networks [7,8,9,10]. Therefore, improving disaster response and supply security across the full disaster cycle has become a key issue that must be addressed in distribution networks with a high penetration of distributed PV.

To meet power supply needs under disaster conditions, MESS can be flexibly deployed and quickly dispatched and can both charge and discharge. During fault isolation and restoration in distribution networks, it can provide power and energy support for critical loads and has been recognized as an important means of improving distribution system resilience. In existing studies, the disaster process is typically divided into stages such as prevention, resistance, adaptation, and recovery [11,12]. Meanwhile, MESS is planned through pre-disaster positioning and post-disaster dynamic scheduling, including routing–power co-optimization [13]. In the pre-disaster stage, the number of mobile storage units and their connection nodes are optimized based on disaster scenario forecasts and factors such as power–transportation network coupling, to improve their availability during disasters [14,15]. In the post-disaster stage, the travel routes and charging/discharging of mobile storage are dynamically optimized by considering the repair process and the time-coupled characteristics of multiple resources, so that load loss is reduced and recovery is accelerated [16]. However, the supply-support performance of MESS strongly depends on its connection location. If pre-disaster positioning does not consider post-disaster boundary changes and shifts in supported loads, storage resources may not match actual needs in space and time, and their benefits may not be fully realized [17].

Given that PV systems are widely dispersed and are difficult to monitor and control in a centralized manner, the idea of cluster/community partitioning can be adopted to enable aggregated management of distributed PV [18]. A well-designed clustering scheme not only offers dimension-reduced modeling and scalable scheduling, but also provides a structural basis for defining the boundaries of supply-support units and enabling localized autonomous operation after disaster-induced topological islanding, thereby improving within-cluster source–load matching and coordinated control [19]. Existing studies have mainly focused on normal operation and have proposed cluster-based methods for output assessment and PV integration/loss analysis [20], as well as clustering-based voltage control strategies for systems with a high PV penetration [21]. Flexible resources such as electric vehicles have also been considered to improve PV integration and operational coordination [22]. Overall, these studies have provided an important foundation for aggregated management of distributed PV and restoration operation of distribution networks. However, as highlighted by recent reviews on mobile energy storage for resilience enhancement, cluster partitioning criteria that target extreme disaster scenarios with supply security as the core objective, as well as coordination mechanisms with cross-area support by MESS, still need systematic investigation [23].

To leverage the dimension-reduction and within-cluster coordination benefits of distributed PV clustering, as well as the spatiotemporal flexibility of MESS, a method for improving disaster resilience and supply security in distribution networks is proposed. A two-stage optimization framework is then developed, covering pre-disaster prevention and post-disaster restoration. The main innovations and contributions are summarized as follows:

(1) Distributed PV Cluster Partitioning Mechanism for Power Supply Assurance: A comprehensive index system that integrates the electrical topology, voltage security, and power support capabilities of distribution networks is developed. A weighting method is used for flexible regulation, which reduces computational complexity while achieving the integrated utilization of distribution facilities. The clusters under this index better align with power supply needs during disasters, providing structured “supply units” for subsequent localized autonomous supply and cross-regional support.

(2) Robust pre-positioning of MESS under worst-case PV output: Under PV output uncertainty, a pre-disaster robust pre-positioning model is formulated. The total cost, including load curtailment cost and mobile storage investment cost, is minimized. Connection locations and capacity sizes are determined to ensure supply security even under the worst-case PV condition, which improves resource availability and the minimum supply level during disasters.

(3) Post-disaster re-clustering and coordinated routing–power scheduling: After fault isolation, topology and source–load states change, and re-clustering is triggered with the objective of minimizing supply-related load curtailment. The MESS travel routes and charging/discharging power are co-optimized to enhance within-cluster supply capability and improve cross-cluster support efficiency. The spatiotemporal mismatch between pre-disaster placement and post-disaster support needs is mitigated.

Finally, simulations under extreme disasters are conducted on a modified IEEE123-bus distribution system with a high penetration of distributed PV. The proposed method is compared with baseline strategies, including no clustering, no MESS, and non-robust pre-positioning. Improvements are validated in terms of critical-load supply capability and load curtailment level.

The remainder of this paper is organized as follows: Section 2 elaborates on the Distributed PV Cluster Partitioning method based on the improved Louvain algorithm and the coordinated strategy with MESS; Section 3 establishes the mathematical models for the pre-disaster prevention and post-disaster restoration phases, and describes the solution methods; Section 4 validates the effectiveness of the proposed strategy through simulations on a modified IEEE123-node distribution network; Section 5 summarizes the research conclusions; Section 6 proposes limitations and future work.

2. Disaster-Resilient Power Supply Strategy Based on Distributed PV Cluster Partitioning and MESS Coordination

2.1. Framework for Resilience Enhancement in Power Supply Assurance Through Coordinated Distributed PV Cluster Partitioning and MESS

Existing research indicates that flexible resources, including MESS, can play a pivotal role in both prevention and restoration phases [24]. Given that large-scale distributed PV integration into distribution networks presents challenges such as numerous positions, extensive coverage, and difficult regulation, this paper proposes a fault restoration framework that considers the division of PV clusters and the coordination of MESS. Through a two-stage coordinated optimization of pre-disaster prevention and post-disaster restoration, this framework leverages the dimensionality reduction advantage of cluster partitioning and the flexibility characteristics of MESS to collectively enhance the distribution networks’ resilience against extreme disasters. The overall architecture is illustrated in Figure 1:

2.1.1. Pre-Disaster Prevention Phase

The pre-disaster prevention phase centers on the core logic of “coordinated interaction between distributed PV cluster partitioning and MESS pre-deployment,” aiming to enhance source-load matching efficiency and mitigate the impact of PV output uncertainty on power supply assurance. Its ultimate goal is to maximize the power supply assurance capability of the distribution network before an extreme disaster occurs, under the premise of optimal MESS pre-deployment. The specific steps are as follows:

(a) To address the high degree of source-load mismatch caused by the volatility of distributed PV output, an improved Louvain algorithm is employed to conduct distributed PV cluster partitioning (detailed in Section 2.2). The partitioning process balances the electrical structure of clusters with their power supply balancing capability, aiming for a high degree of matching between PV output and load demand within each cluster. This also lays the foundation for the subsequent precise deployment of MESS.

(b) To tackle the uncertainty of PV output, based on the cluster partitioning results, a robust optimization algorithm is applied to optimize the pre-deployment of MESS. Under the worst-case PV output scenario, a robust optimization model is established with the objectives of minimizing load curtailment and MESS deployment costs (detailed in Section 3.1.1). The model incorporates constraints such as intra-cluster MESS pre-deployment and distribution network power flow. It solves for the optimal deployment nodes for MESS within each cluster, ensuring efficient support for cluster loads under extreme PV output conditions and achieving the coordinated power supply assurance objective of integrating cluster partitioning with MESS deployment.

2.1.2. Post-Disaster Restoration Phase

The post-disaster restoration phase aims to minimize load curtailment through the “coordinated re-partitioning of PV clusters and cross-cluster scheduling of MESS.” The specific steps are as follows:

(a) Based on fault line isolation results and the post-disaster actual status of sources and loads, the PV clusters are re-partitioned to cover critical loads, providing defined power supply units for MESS dispatch.

(b) Building on the cluster re-partitioning results, the spatio-temporal output allocation of MESS across clusters is optimized. Considering constraints such as MESS charge/discharge limits and distribution network power flow constraints, a mixed-integer programming model is established with the objective of minimizing load curtailment (model formulation detailed in Section 3.2). This model solves for the access locations, charge/discharge power, and dispatch paths of MESS for each time period, ultimately constructing a coordinated power supply assurance mechanism that “maximizes intra-cluster supply capability and optimizes inter-cluster power support.”

2.2. Distributed PV Cluster Partitioning Based on an Improved Louvain Algorithm

Distribution networks incorporating large-scale distributed PV exhibit the complex characteristic of transitioning from dual-source power supply to multi-source supply. The integration of MESS further enhances the system’s flexibility. To reduce computational complexity, this paper employs an improved Louvain algorithm for Distributed PV Cluster Partitioning [25]. In the traditional Louvain algorithm, the modularity metric is defined as follows:

$$M={\displaystyle \frac{1}{2m}}{\displaystyle \sum _{i,j}\left(A_{ij}-\frac{k_i{k}_j}{2m}\right)}{\delta }_{ij}$$

(1)

where $A_{ij}$ is the connection weight between node $i$ and node $j$, $k_i={\displaystyle \sum _j{A}_{ij}}$ is the sum of the weights of all edges connected to node $i$, $m={\displaystyle \frac{1}{2}}{\displaystyle \sum _{i,j}{A}_{ij}}$ is the overall grid weight, ${\delta }_{ij}$ is the identity function, if node $i$ belongs to the same cluster as node $j$, it is 1; otherwise, it is 0.

The high-penetration integration of large-scale distributed PV into distribution networks is prone to triggering two core operational issues: first, the risk of voltage limit violations, where rapid power fluctuations lead to sudden changes in nodal power injections, causing voltages to deviate significantly from the rated range; second, the problem of power imbalance, where the mismatch between PV output and load demand reduces operational efficiency and weakens the system’s power regulation flexibility. Traditional Louvain algorithm-based clustering only considers network topology and does not adequately address operational constraints introduced by PV integration.

From a practical perspective, the updated index system follows the logic from “structural foundation” to “risk avoidance.” The electrical distance parameter serves as the physical foundation for cluster partitioning, ensuring tight electrical connections within clusters. This provides structural support for subsequent autonomous operation and power coordination. Voltage sensitivity parameters are used to capture how changes in node power affect voltage, ensuring that cluster boundaries avoid voltage violation-prone regions, thus reducing voltage security risks from PV output fluctuations. Additionally, the reactive/active power support indicators quantify nodes’ potential for power regulation, ensuring sufficient power flexibility within each cluster, enhancing adaptability to the randomness and intermittency of PV output. Based on these multi-dimensional indicators, the improved modularity metric is constructed as follows:

$$M={\omega }_1{D}_{ij}+{\omega }_2{S}_i^{\mathrm{V}}+{\omega }_3{Q}_i^{\mathrm{su}}+{\omega }_4{E}_i^{\mathrm{su}}$$

(2)

where $D_{ij}$ is the electrical distance parameter between node $i$ and node $j$, $S_i^{\mathrm{V}}$ is the voltage sensitivity parameter that characterizes the magnitude of voltage amplitude change at one node resulting from a change in power injection at another node, $Q_i^{{\mathrm{su}}^{\prime }}$ and $E_i^{{\mathrm{su}}^{\prime }}$ represent reactive power support and active power support respectively, where both serve as indicators of flexibility support, and ${\omega }_1, {\omega }_2, {\omega }_3, {\omega }_4$ represent the weight coefficients for each indicator, The specific calculations for each indicator can be found in Reference [26].

3. Mathematical Model and Solution for the Two-Stage Power Supply Assurance Strategy

3.1. Model for Distributed PV Cluster Partitioning and MESS Pre-Deployment in the Pre-Disaster Prevention Phase

In the pre-disaster prevention phase, the improved Louvain algorithm proposed in Section 2.2 is employed for cluster partitioning of distributed PV resources to enhance the source-load matching degree within clusters; subsequently, a two-stage robust optimization model is established with the objectives of minimizing the deployment cost of MESS and minimizing the load curtailment cost, to determine the optimal cluster partitioning scheme and the collaborative pre-deployment scheme for MESS, thereby achieving maximization of the pre-disaster power supply assurance capability in the distribution networks.

3.1.1. Pre-Disaster Phase Objective Function

To account for the worst-case scenario of distributed PV output, the specific expression of the objective function in this stage is as follows:

$$\left\{\begin{array}{l}{\min }_{{\alpha }^{\mathrm{ME}},{\alpha }_j}{\max }_{P^{\mathrm{PV}}\in \mathrm{U}}{\min }_{\mathrm{Y}}\left({\displaystyle \sum _{i\in {\Omega }_N}{C}^{\mathrm{ME}}}{\alpha }_i^{\mathrm{ME}}+{\displaystyle \sum _{i\in {\Omega }_N}{w}_i}{P}_i^{\mathrm{cut}}\cdot S_B\right)\hfill \\ \mathrm{Y}=\left(P_i^{\mathrm{cut}},Q_i^{\mathrm{cut}},P_i^{\mathrm{DG}},Q_i^{\mathrm{DG}},P_{ij},Q_{ij},U_i,I_{ij}\right)\hfill \end{array}\right.$$

(3)

where the outer layer ${\min }_{{\alpha }^{\mathrm{ME}},{\alpha }_j}$ optimizes the MESS affiliation variables ${\alpha }^{\mathrm{ME}}$ and the branch switching decision variables ${\alpha }_j$, minimizing the system’s decision-level cost; the middle layer ${\max }_{P^{\mathrm{PV}}\in \mathrm{U}}$, for the uncertainty set of distributed PV output $P^{\mathrm{PV}}$, selects the PV output scenario $\mathrm{U}$ that is most unfavorable to the system, ensuring the robustness of the strategy under extreme operating conditions. The inner layer ${\min }_Y$ minimizes the system operating cost under the given worst-case distributed PV output scenario and outer-layer decision variables. The detailed decision flow of this three-layer optimization structure is illustrated in Figure 2. $C^{\mathrm{ME}}$ is the unit deployment cost of MESS, ${\alpha }_i^{\mathrm{ME}}$ is the MESS pre-deployment decision variable, $w_i$ is the load curtailment cost weight for node $i$, and $S_H$ is the base power. $\mathrm{Y}=\left(P_i^{\mathrm{cut}},Q_i^{\mathrm{cut}},P_i^{\mathrm{DG}},Q_i^{\mathrm{DG}},P_{ij},Q_{ij},U_i,I_{ij}\right)$ are the active power curtailment, reactive power curtailment, active power generation, reactive power generation at node $i$, active power flow, reactive power flow on branch $ij$, voltage at node $i$, and current $ij$ on branch $ij$.

3.1.2. Constraints

(1)

Constraints on Pre-deployment of MESS Within Clusters

$${\displaystyle \sum _{(i,j)\in {\mathcal{E}}_c}{\alpha }_{ij}}\ge (1-{\lambda }_C)\cdot |{\mathcal{E}}_c|\hspace{1em}(\forall c\in {\Omega }_C)$$

(4)

$${\displaystyle \sum _{i\in N_c}{\alpha }_{i,0}^{\mathrm{ME}}}\le 1\hspace{1em}(\forall c\in {\Omega }_C)$$

(5)

$${\displaystyle \sum _{c\in {\Omega }_C}{N}_c^{\mathrm{ME}}}=N_{\max }^{\mathrm{ME}}$$

(6)

$$U_{m,t}^{\mathrm{Mch}}+U_{m,t}^{\mathrm{Mdch}}\le 1$$

(7)

$$0\le P_{m,t}^{\mathrm{Mch}}\le U_{m,t}^{\mathrm{Mch}}\cdot P_{\max }^{\mathrm{Mch}}/S_B$$

(8)

$$0\le P_{m,t}^{\mathrm{Mdch}}\le U_{m,t}^{\mathrm{Mdch}}\cdot P_{\max }^{\mathrm{Mdch}}/S_B$$

(9)

$$P_{c,t}^{\mathrm{pv}}+{\displaystyle \sum _{m\in M}{\alpha }_{m,c,t}^{\mathrm{ME}}}(P_{m,t}^{\mathrm{Mdch}}-P_{m,t}^{\mathrm{Mch}})=P_{c,t}^{\mathrm{load}}$$

(10)

the symbols and meanings of these equations are shown in the

Table 1. Formula Explanation.

Equation	Symbols	Constraint Meaning
(4)	${\mathcal{E}}_c$: Set of branches within the $c$ cluster. ${\alpha }_{ij}$: State variable (0/1) of branch $(i,j)$. ${\lambda }_C$: Permissible disconnection ratio of branches within a cluster (0 to 1). ${\Omega }_C$: Set of all clusters.	Defines the topological connectivity constraint within clusters. It prevents excessive fragmentation during optimization, ensuring clusters remain internally connected to avoid isolated sub-grids and enhance supply reliability.
(5) and (6)	${\alpha }_{i,0}^{\mathrm{ME}}$: The 0–1 decision variable for deploying MESS at node $i$ in cluster $C$. ${\displaystyle \sum _{c\in {\Omega }_C}{N}_c^{\mathrm{ME}}}$: Number of MESS instances deployed within the cluster. $N_{\max }^{\mathrm{ME}}$: Maximum total MESS deployment limit.	Govern the pre-deployment of MESS. Each cluster may deploy at most one MESS unit, and the total number of MESS units deployed in the system is fixed.
(7)–(9)	$U_{m,t}^{\mathrm{Mch}},U_{m,t}^{\mathrm{Mdch}}$: Charging/discharging status. $P_{m,t}^{\mathrm{Mch}},P_{m,t}^{\mathrm{Mdch}}$: Actual charge/discharge power. $P_{\max }^{\mathrm{dch}},P_{\max }^{\mathrm{Mdch}}$: Upper and lower power limits	Define MESS charge/discharge operational limits. The status indicators are mutually exclusive. The charge/discharge power must remain within specified maximum limits.
(10)	$P_{c,t}^{\mathrm{pv}}$: Total active power output from PV systems in cluster $C$. $P_{c,t}^{\mathrm{load}}$: Total active load in cluster $C$. ${\displaystyle \sum _{m\in M}{\alpha }_{m,c,t}^{\mathrm{ME}}}(P_{m,t}^{\mathrm{Mdch}}-P_{m,t}^{\mathrm{Mch}})$: Net power output from MESS within the cluster.	Enforces active power balance within a cluster. This constraint guides the optimal clustering of resources during pre-disaster pre-deployment to achieve a self-balancing power supply within each cluster.

below.

(2)

PV Uncertainty Constraints

PV output is significantly influenced by natural factors such as sunlight intensity and ambient temperature, introducing substantial uncertainty. Its fluctuations directly impact the robustness of pre-disaster pre-deployment plans. The range of these fluctuations is quantified using boxed uncertainty sets, with the specific expression as follows:

$$\mathrm{U}=\left\{P^{\mathrm{P}\mathrm{V}}\mid (1-\tau ){\widehat{P}}^{\mathrm{PV}}\le P^{\mathrm{P}\mathrm{V}}\le {\widehat{P}}^{\mathrm{P}\mathrm{V}}\right\}$$

(11)

$$P^{\mathrm{P}\mathrm{V}}\tan ({\cos }^{-1}({\delta }_{\max }^{\mathrm{PV}}))\le Q^{\mathrm{PV}}\le P^{\mathrm{P}\mathrm{V}}\tan ({\cos }^{-1}({\delta }_{\min }^{\mathrm{PV}}))$$

(12)

where $P^{\mathrm{PV}}$ denotes the actual active power output of the PV system, $Q^{\mathrm{PV}}$ denotes the actual reactive power output of the PV system, ${\widehat{P}}^{\mathrm{P}\mathrm{V}}$ denotes the predicted PV power output, $\tau$ denotes the uncertainty used to characterize the maximum deviation of the PV output from the predicted value, $\mathrm{U}$ denotes the box-shaped uncertainty set of the PV output, covering the most severe power fluctuation scenarios. ${\delta }_{\max }^{\mathrm{PV}}$ and ${\delta }_{\min }^{\mathrm{PV}}$ denote the maximum and minimum values of the power factor of the PV inverter.

(3)

Load curtailment constraints

$$0\le P_i^{\mathrm{cut}}\le P_i^{\mathrm{l}\mathrm{oad}},\hspace{1em}{Q}_i^{\mathrm{cut}}={\displaystyle \frac{Q_i^{\mathrm{load}}}{P_i^{\mathrm{load}}}}{P}_i^{\mathrm{cut}}$$

(13)

It indicates that the active load curtailment $P_i^{\mathrm{cut}}$ at node $i$ should not exceed the original active load $P_i^{\mathrm{load}}$ of that node, and the reactive load curtailment $Q_i^{\mathrm{cut}}$ at node $i$ is proportional to the active load curtailment.

(4)

Constraints of radial topology in distribution networks

The Virtual Power Flow [27] (VPF) method is employed to ensure the reconfigured grid meets radial operation requirements, allowing the formation of multiple islands. Nonlinear constraints are relaxed using the Big-M method:

$${\displaystyle \sum _{ij\in {\Omega }_L}{\alpha }_{ij}}=N_{\mathrm{B}}-N_{\mathrm{island}}$$

(14)

$${\displaystyle \sum _{j\in \delta (i)}{F}_{ij}}-{\displaystyle \sum _{j\in \gamma (i)}{F}_{ji}}=\left\{\begin{array}{ll}1-F_i^{\mathrm{VS}},\hfill & \mathrm{if} i\in {\Omega }_{\mathrm{DG}}\hfill \\ 1,\hfill & \mathrm{if} i\notin {\Omega }_{DG} \mathrm{and} i\ne i_{\mathrm{r}}\hfill \\ 0,\hfill & \mathrm{if} i=i_{\mathrm{r}}\hfill \end{array}\right.$$

(15)

$$-M\cdot S_i^{\mathrm{V}\mathrm{S}}\le F_i^{\mathrm{VS}}\le M\cdot S_i^{\mathrm{V}\mathrm{S}},\hspace{1em}-M\cdot {\alpha }_{ij}\le F_{ij}\le M\cdot {\alpha }_{ij}$$

(16)

where ${\Omega }_{\mathrm{L}}$ denotes the set of distribution networks branches, $N_{\mathrm{B}}$ represents the total number of nodes, ${\alpha }_{ij}$ is the branch switching variable (0–1), $N_{\mathrm{island}}$ indicates the number of isolated islands, and this constraint ensures the total number of operational branches equals “total number of nodes minus number of isolated islands”; In Equations (15) and (16): $F_i^{\mathrm{VS}}$ denotes the virtual injection power at node $i$, $\delta (i)$ denotes the set of branch endpoints flowing out from node $i$, $\gamma (i)$ denotes the set of branch starting points flowing into node $i$, ${\Omega }_{\mathrm{DG}}$ denotes the set of nodes containing distributed power sources, $F_{ij}$ denotes branch virtual power flow, $i$ is node, $i_r$ is root node, $S_i^{\mathrm{VS}}$ is virtual power source 0–1 variable indicator, $M$ is a sufficiently large constant. This constraint defines node virtual power flow and virtual power source injection power, enabling flow from the root node or distributed power sources to loads. This ensures the power flow operation logic of the radial topology after distribution networks reconfiguration.

(5)

Current Safety Operation Constraints

Considering the dynamic characteristics of the grid topology resulting from changes in feeder switch status during grid reconfiguration, nonlinear voltage constraints are linearized by introducing the large-M relaxation technique. The constraints are as follows:

$${\displaystyle \sum _j{P}_{ij}}-{\displaystyle \sum _j(P_{ji}-I_{ji}{R}_{ji})}=P_i^{\mathrm{DG}}+P_i^{\mathrm{MESS}}-P_i^{\mathrm{load}}+P_i^{\mathrm{cut}}$$

(17)

$${\displaystyle \sum _j{Q}_{ij}}-{\displaystyle \sum _j(Q_{ji}-I_{ji}{X}_{ji})}=Q_i^{\mathrm{DG}}+Q_i^{\mathrm{MESS}}-Q_i^{\mathrm{l}\mathrm{oad}}+Q_i^{\mathrm{cut}}$$

(18)

$$V_i^{\mathrm{sqr}}-V_j^{\mathrm{sqr}}-2(R_{ij}{P}_{ij}+X_{ij}{Q}_{ij})+(R_{ij}^2+X_{ij}^2){\displaystyle \frac{P_{ij}^2+Q_{ij}^2}{V_i^{\mathrm{sqr}}}} \le M_1(1-{\alpha }_{ij})$$

(19)

$$V_i^{\mathrm{sqr}}-V_j^{\mathrm{sqr}}-2(R_{ij}{P}_{ij}+X_{ij}{Q}_{ij})+(R_{ij}^2+X_{ij}^2){\displaystyle \frac{P_{ij}^2+Q_{ij}^2}{V_i^{\mathrm{sqr}}}} \ge M_1({\alpha }_{ij}-1)$$

(20)

$$V_{i,\min }^{\mathrm{sqr}} \le V_i^{\mathrm{sqr}}\le V_{i,\max }^{\mathrm{sqr}}$$

(21)

$$0 \le I_{ij}^{\mathrm{sqr}}\le {\alpha }_{ij}{I}_{ij,\max }^{\mathrm{sqr}}$$

(22)

$$P_{ij}^2+Q_{ij}^2 \le V_i^{\mathrm{sqr}}{I}_{ij}^{\mathrm{sqr}}$$

(23)

where $R_{ij}$ and $X_{ij}$ represent the equivalent resistance and reactance of branch $ij$, respectively; $V_i^{\mathrm{sqr}}$ denotes the square term of the voltage amplitude at node $i$, whose permissible fluctuation range is jointly defined by $V_{i,\min }^{\mathrm{sqr}}$ and $V_{i,\max }^{\mathrm{sqr}}$; $I_{ij}^{\mathrm{sqr}}$ denotes the square term of the steady-state current effective value for branch $ij$, whose maximum allowable value is controlled by ${\alpha }_{ij}{I}_{ij,\max }^{\mathrm{sqr}}$, ${\alpha }_{ij}$ is the binary switch state variable; Equations (19) and (20) achieve linearized modeling of nonlinear voltage drop constraints by introducing the relaxation coefficient $M_1$. Theoretically, this parameter must satisfy:

$$M_1>\max \left\{{\displaystyle \frac{V_i^{\mathrm{sqr}}-V_j^{\mathrm{sqr}}}{1-{\alpha }_{ij}}},{\displaystyle \frac{V_j^{\mathrm{sqr}}-V_i^{\mathrm{sqr}}}{{\alpha }_{ij}-1}}\right\}$$

(24)

At the same time, due to the nonlinear relationship between voltage, current, and power in Equation (23), it is typically transformed into a second-order cone model using the second-order cone method to facilitate solution:

$${‖\begin{array}{c}2P_{ij}\\ 2Q_{ij}\\ I_j^{\mathrm{sqr}}-V_i^{\mathrm{sqr}}\end{array}‖}_2\le I_{ij}^{\mathrm{sqr}}+V_i^{\mathrm{sqr}}$$

(25)

3.2. Power Supply Model for Distributed PV Cluster Re-Partitioning and MESS Re-Scheduling in the Post-Disaster Restoration Phase

As shown in Figure 1 for the post-disaster restoration phase, considering the impact of distribution network topology changes, the distributed PV clusters are re-partitioned, and MESS is re-scheduling to achieve maximization of critical load restoration. Therefore, a mixed-integer programming model is established with the objective of minimizing load curtailment, solving for the access positions of MESS to clusters, charge/discharge power, and scheduling paths, to realize rapid load restoration.

3.2.1. Post-Disaster Phase Objective Function

The objective function for the post-disaster restoration phase aims to minimize the curtailment in critical load during the fault duration while maximizing the restoration ratio of critical loads. The specific expression is as follows:

$$\min {\displaystyle \sum _{t\in T}{\displaystyle \sum _{c\in C}\left(w_c^{\mathrm{ess}}{P}_{c,t}^{\mathrm{cut},\mathrm{ess}}+w_c^{\mathrm{ord}}{P}_{c,t}^{\mathrm{cut},\mathrm{ord}}\right)}}$$

(26)

where $T$ denotes the set of fault restoration periods, $C$ represents the total number of clusters after dynamic partitioning, $w_c^{\mathrm{ess}}$ and $w_c^{ord}$ respectively indicate the cost curtailment for critical/non-critical load units within cluster $C$, while $P_{c,t}^{\mathrm{cut},\mathrm{ess}}$ and $P_{c,t}^{\mathrm{cut},\mathrm{ord}}$ denote the power curtailment for critical/non-critical loads in cluster $C$ during period $t$.

3.2.2. Constraints

(1)

Spatio-Temporal Dispatch Constraints of MESS

Considering that each MESS unit can belong to only one cluster during each time period, the constraints are as follows:

$${\displaystyle \sum _{c=1}^{N_C}{\alpha }_{m,c,t}^{\mathrm{ME}}}=1\hspace{1em}\forall m\in {\Omega }_{\mathrm{ME}},\forall t\in {\Omega }_T$$

(27)

$${\alpha }_{m,c,t}^{\mathrm{ME}}+{\alpha }_{m,c^{\prime },t+\Delta t}^{\mathrm{ME}}\le 1\hspace{1em}\forall c\in I_i,c^{\prime }\notin I_i,\Delta t<T_{\mathrm{trans}}(c,c^{\prime })$$

(28)

$$T_{\mathrm{trans}}(c,c^{\prime })=\max \left(t_{\mathrm{base}},{\displaystyle \frac{L(i_c,i_{c^{\prime }})}{v}}\right)$$

(29)

where ${\alpha }_{m.c,t}^{\mathrm{ME}}$ denotes a variable ranging from 0 to 1, indicating that MESS $m$ joins cluster $C$ during time period $t$; $I_i$ represents the set of clusters contained within island $i$; $T_{\mathrm{trans}}(c,c^{\prime })$ denote the cross-island mobility time thresholds, $L(i_c,i_{c^{\prime }})$ represents the shortest path between the two representative nodes, and $t_{\mathrm{base}}$ is the minimum movement time. This constraint ensures that MESS cannot simultaneously appear in two different clusters before completing a transfer cycle.

Additionally, the state of charge (SoC) of MESS should fluctuate within safe limits:

$$E_{m,t}^{\mathrm{ME}}=E_{m,t-1}^{\mathrm{ME}}+P_{m,t}^{\mathrm{Mch}}\cdot {\eta }^{\mathrm{ME}}-P_{m,t}^{\mathrm{Mdch}}/{\eta }^{\mathrm{ME}}$$

(30)

$$E_{\min }^{\mathrm{ME}}/S_B\le E_{m,t}^{\mathrm{ME}}\le E_{\max }^{\mathrm{ME}}/S_B$$

(31)

where $E_{m,t}^{\mathrm{ME}}$ denotes the actual energy storage capacity of the $m$-th MESS unit at time $t$, ${\eta }^{\mathrm{ME}}$ represents the charge–discharge efficiency of the MESS, $E_{\min }^{\mathrm{ME}}$ and $E_{\max }^{\mathrm{ME}}$ denote the minimum and maximum allowable energy storage capacities of the MESS, respectively. Equation (30) indicates that the energy storage capacity of the MESS at time $t$ is jointly determined by the capacity at the previous time, the energy gain during the charging process, and the energy loss during the discharging process. Equation (31) imposes constraints through normalization, limiting the actual energy storage capacity of the MESS unit within a safe range. Finally, the charging and discharging power constraints for the MESS unit are detailed in the pre-disaster constraint section.

(2)

PV output constraints

$$0\le P_i^{\mathrm{PV}}\le P_i^{\mathrm{PV},\max },\hspace{1em}0\le Q_i^{\mathrm{PV}}\le Q_i^{\mathrm{PV},\max }$$

(32)

where $P_i^{\mathrm{PV}}$ and $Q_i^{\mathrm{PV}}$ represent the active power output and reactive power output of distributed PV at node $i$, while $P_i^{\mathrm{PV},\max }$ and $Q_i^{\mathrm{PV},\max }$ denote the maximum active power capacity and reactive power capacity of distributed PV at node $i$.

(3)

Power balance constraint

During fault conditions, the distribution networks is segmented into multiple isolated islands, each requiring compliance with power balance constraints. Additionally, to ensure source-load balance within each cluster of an isolated island and reduce reliance on inter-cluster power support, the active power constraints for clusters and isolated islands are mathematically expressed as follows:

$$P_{c,t}^{\mathrm{pv}}+{\displaystyle \sum _{m\in M}{\alpha }_{m,c,t}^{\mathrm{ME}}}(P_{m,t}^{\mathrm{Mdch}}-P_{m,t}^{\mathrm{Mch}})+P_{c,t}^{\mathrm{cut}}=P_{c,t}^{\mathrm{load}}$$

(33)

$${\displaystyle \sum _{c\in {\mathrm{L}}_k}}\left(P_{c,t}^{\mathrm{pv}}+{\displaystyle \sum _{m\in M}{\alpha }_{m,c,t}^{\mathrm{ME}}}(P_{m,t}^{\mathrm{Mdch}}-P_{m,t}^{\mathrm{Mch}})+P_{c,t}^{\mathrm{cut}}\right)={\displaystyle \sum _{c\in {\mathrm{L}}_k}}{P}_{c,t}^{\mathrm{load}}$$

(34)

where $P_{c,t}^{\mathrm{pv}}$ denotes the active power output from distributed PV in cluster $C$ at time $t$; $P_{m,t}^{\mathrm{Mdch}}$ denotes the active power discharge from MESS $m$ at time $t$; $P_{m,t}^{\mathrm{Mch}}$ denotes the active power charging of MESS $m$ at time $t$; ${\displaystyle \sum _{m\in M}{\alpha }_{m,c,t}^{\mathrm{ME}}}(P_{m,t}^{\mathrm{Mdch}}-P_{m,t}^{\mathrm{Mch}})$ denotes the net active power output from all MESS across all clusters; $P_{c,t}^{\mathrm{load}}$ denotes the active load of cluster $C$ at time $t$; $P_{c,t}^{\mathrm{cut}}$ represents the active power curtailment of cluster $C$ at time $t$; ${\mathrm{L}}_k$ indicates the $k$-th island formed post-disaster. The reactive power balance constraint follows the same principle.

Tidal constraints and load curtailment constraints are evident in pre-disaster constraints (13) and (17)–(23).

3.3. Model Solution

3.3.1. Pre-Disaster Model Solution

The pre-disaster prevention collaborative scheduling model (Equations (3)–(25)) constitutes a two-stage robust optimization model featuring a three-tiered “min-max-min” structure. Its decision variables contain both integer and continuous components, making direct computation challenging. Furthermore, the results of Distributed PV Cluster Partitioning directly influence the pre-deployment of MESS devices. Grid reconfiguration during robust optimization may alter cluster power boundaries, necessitating dynamic cluster structure adjustments. To address these challenges, the Constraint and Constraint Generation (C&CG) algorithm decomposes the original problem into a Master Problem and Subproblem. Iterative solutions approximate optimal outcomes, while a dynamic cluster update mechanism re-partitions PV clusters after each iteration based on the current grid topology.

The original problem is summarized as follows:

$$\begin{array}{l}{\min }_{x\in \mathrm{X}}\hspace{0.33em}{C}^{T}x+{\max }_{\mathrm{u}\in \mathrm{U}}\left\{{\min }_{y\in \mathrm{Y}}\hspace{0.33em}{f}^{T}y\right\}\\ \mathrm{s}.\mathrm{t}.\left\{\begin{array}{l}\mathrm{A}x\le \mathrm{b},\hfill \\ \mathrm{D}x+\mathrm{E}y\le \mathrm{d},\hfill \\ \Vert F_{i}x+{\mathrm{G}}_{i}y-{\mathrm{g}}_i{\Vert }_2\le h_i\hspace{0.33em}(i=1,\dots ,n)\hfill \end{array}\right.\end{array}$$

(35)

where Variable $x$ includes the MESS pre-deployment variable ${\alpha }^{\mathrm{ME}}$ and the branch switching decision variable ${\alpha }_j$. Variable $\mathrm{u}$ includes the PV output variable $P^{\mathrm{PV}}$. Variable $y$ includes inner-layer control variables such as $P_i^{\mathrm{cut}},Q_i^{\mathrm{cut}},P_{ij},Q_{ij},U_i,I_{ij}$. $\mathrm{U}$ represents the PV output uncertainty set, and $\mathrm{X},\mathrm{Y}$ denotes the feasible region of variable.

The Master Problem involves optimizing the pre-deployment of MESS within a given cluster partitioning, as shown in Equation (36). This problem is fundamentally a Mixed-Integer Second-Order Cone Programming (MISOCP) problem, which can be directly solved using commercial solvers. Its core objective is to minimize the deployment cost of MESS and the load curtailment cost under PV output prediction scenarios while maintaining the current cluster structure and satisfying cluster, topology, and power flow constraints.

$$\left\{\begin{array}{l}{\min }_{x,\mathsf{\eta },y^k}\hspace{1em}{C}^{T}x+\mathsf{\eta }\hfill \\ \mathrm{s}.\mathrm{t}. Ax\le b\hfill \\ \hspace{1em}\hspace{1em}Dx+E{y}^k\le d,\hspace{1em}k=1,2,\dots ,n_{iter}\hfill \\ \hspace{1em}\hspace{1em}\Vert F_{i}x+G_i{y}^k-g_i{\Vert }_2\le h_i,\hspace{1em}i=1,2,\dots ,n_{cone};\hspace{1em}k=1,2,\dots ,n_{iter}\hfill \\ \hspace{1em}\hspace{1em} \mathsf{\eta }\ge f^T{y}^k,\hspace{1em}k=1,2,\dots ,n_{iter}\hfill \\ \hspace{1em}\hspace{1em}x\in X,\hspace{1em}{y}^k\in Y\hfill \end{array}\right.$$

(36)

where η is the slack variable representing the upper bound of the objective function, $n_{iter}$ denotes the current iteration round, and $y^k$ represents the inner-layer control variable candidate solution generated in the $k$-th iteration. After solving the Master Problem, the optimal MESS deployment $x^{*}=({\alpha }^{\mathrm{ME}*},{\alpha }_j^{*})$ and the lower bound of the objective function $C^T{x}^{*}$ for the current iteration are obtained.

The Subproblem involves searching for the worst-case PV scenario under a given MESS deployment $x^{*}$, as shown in Equation (37). Its core objective is to validate the robustness of the main problem’s solution by identifying the extreme scenario $\mathrm{U}$ that maximizes the system’s supply assurance cost within the PV output uncertainty set $u^{*}=P^{\mathrm{PV}*}$.

$${\max }_{P^{\mathrm{PV}}\in \mathrm{U}}{\min }_Y\left({\displaystyle \sum _{i\in B_N}{w}_i}{P}_i^{\mathrm{cut}}\right)$$

(37)

Apply Lagrange duality theory to the inner-layer ${\min }_Y$ problem of the Subproblem, transforming the three-layer nested problem into a two-layer optimization and eliminating the inner-layer minimization. Define Lagrange multipliers λ, μ, ${\rho }_i$, then the dual form of the Subproblem is:

$$\left\{\begin{array}{cc}{\max }_{\lambda ,\mu ,\nu ,\rho }\hfill & {\lambda }^T\left(h-T{\alpha }^{\mathrm{ME},*}-H{P}^{\mathrm{PV}}\right)+{\mu }^T{b}_s+{\displaystyle \sum _{i=1}^{n_{\mathrm{c}}}{\nu }_i}{h}_{si}\hfill \\ \mathrm{s}.\mathrm{t}.\hfill & W^T\lambda +D_s^T\mu +{\displaystyle \sum _{i=1}^{n_{\mathrm{c}}}{G}_i^T}{\rho }_i=f\hfill \\ \hfill & \Vert {\rho }_i{\Vert }_2\le {\nu }_i,\hspace{1em}i=1,\dots ,n_{\mathrm{c}}\hfill \\ \hfill & \lambda ,\mu ,\nu \ge 0,\hspace{1em}{P}^{\mathrm{PV}}\in \mathrm{U}\hfill \end{array}\right.$$

(38)

where $\mathrm{U}$ denotes the set of PV output uncertainties satisfying Equations (11) and (12). The Subproblem objective function contains a non-convex bilinear term ${\lambda }^{T}H{P}^{\mathrm{PV}}$ involving PV output $P^{\mathrm{PV}}$ and Lagrange multiplier λ, which is linearized using binary variables and the large-M relaxation. After introducing binary variable $z_j\in \left\{0,1\right\}$, the PV output can be expressed as:

$$P_j^{\mathrm{PV}}=(1-z_j){P_j^{\mathrm{PV}}}_{\min }+z_j{P_j^{\mathrm{PV}}}_{\max }$$

(39)

By introducing auxiliary variables $w_j=z_j{\lambda }_j$, the bilinear terms are transformed into linear constraints via the Big-M relaxation:

$${\displaystyle \sum _j{\lambda }_j}{H}_j{P}_j^{\mathrm{PV}}={\displaystyle \sum _j\left[(1-z_j){\lambda }_j{H}_j{P_j^{\mathrm{PV}}}_{\min }+z_j{\lambda }_j{H}_j{P_j^{\mathrm{PV}}}_{\max }\right]}$$

(40)

After solving the Subproblem, the worst-case PV scenario $u^{*}=P^{\mathrm{PV}*}$ and the optimal value of the Subproblem $ob{j}_{\mathrm{SP}}$ are obtained, and the upper bound of the objective function $UB=ob{j}_{\mathrm{SP}}$ is updated. If $\mathrm{UB}-\mathrm{L}\mathrm{B} < {\epsilon }_{gap}$, the iteration converges; otherwise, a cutting-plane constraint must be added to the main problem, and the next iteration round begins.

To achieve synergy between cluster partitioning and robust optimization, after each iteration of the C&CG algorithm, the improved Louvain algorithm is re-executed based on the current grid topology to update the cluster structure. The specific steps are as follows:

(1) Extract branch switching states ${\alpha }_{ij}^{*}$ from the optimal solution $x^{*}$ of the main problem to construct the distribution networks topology $G^k$ for the current iteration.

(2) Recalculate the electrical distance metric $D_{ij}$, voltage sensitivity metric $S^{\mathrm{V}}$, reactive power support metric $Q_i^{\mathrm{su}}$, and active power support metric $E_i^{\mathrm{su}}$ based on the topology $G^k$.

(3) Perform cluster partitioning using the improved modularity metric $M$. If ${\displaystyle \frac{|M^{k+1}-M^k|}{M^k}}\le {\epsilon }_M$ and $\mathrm{UB}-\mathrm{L}\mathrm{B}< {\epsilon }_{gap}$, the algorithm converges, outputting the optimal cluster partition $C^{*}=C^{k+1}$ and MESS deployment $x^{*}$. Otherwise, update constraints (4)–(6) according to $C^{k+1}$.

The specific process is shown in Figure 3:

3.3.2. Post-Disaster Model Solution

The post-disaster restoration phase model employs a multi-period mixed-integer linear programming (MILP) approach, aiming to minimize total load curtailment across all time periods. Constraints encompass MESS scheduling, power balance between clusters and isolated islands, and power flow safety. During solution, the model first performs Distributed PV Cluster Partitioning based on post-disaster fault topology and source/load states. It is then converted into standard MILP format and solved directly using the CPLEX solver.

4. Case Study

4.1. Case Study Scenario

This paper employs an enhanced IEEE123-bus distribution system for simulation, categorizing all nodes into critical and non-critical loads. The system topology and equipment locations are shown in Figure 4. The rated parameters of each power source are listed in Appendix A,

Table A1. Power Source Parameters.

Power Source	Maximum Active Power/kW	Maximum Reactive Power/kvar	Charge/Discharge Efficiency	Energy Storage Capacity/(kW·h)
MESS1, MESS2	200	170	0.98	600

. System parameters are detailed in

Table A2. System Parameters.

Parameter	Value
Rated Voltage	12.66 kV
Voltage Upper Limit	13.92 kV
Voltage Lower Limit	11.39 kV
System Rated Capacity	10 MVA
Unit Deployment Cost of MESS	500/yuan
Unit Curtailment Cost for Critical Loads	5/yuan
Unit Curtailment Cost for Non-Critical Loads	1/yuan

. The predicted output of PV units before the disaster is shown in

Table A3. Pre-Disaster Predicted Output of PV Units.

PV Unit	Predicted Power/kW
PV1, PV7, PV10, PV11,	320
PV2, PV8	230
PV3, PV4, PV12, PV19	380
PV5, PV6, PV14, PV17	180
PV13, PV20	450
PV15	510
PV16, PV18	150
PV21, PV22, PV23, PV24	280

, while the predicted output after the disaster is illustrated in Figure A1. The predicted change rate of active power for loads is depicted in Figure A2. Branch parameters and rated load parameters are referenced from [28]. Assume both MESSs units are fully charged at the initial time and located at Node 1. The traffic grid topology of the MESSs corresponds to the distribution networks system.

4.2. Pre-Disaster Results Analysis

To validate the superiority of the proposed pre-disaster Distributed PV Cluster Partitioning and MESS coordination strategy, simulation experiments are conducted under conditions simulating a PV output uncertainty $\tau$ of 0.2 (with the worst-case scenario allowing at most 14 out of the 24 PV units to experience simultaneous output reduction), cluster partitioning weights ${\mathsf{\omega }}_1=0.4,{\mathsf{\omega }}_2=0.3,{\mathsf{\omega }}_3=0.2,{\mathsf{\omega }}_4=0.1$, and loss of main grid supply to the distribution networks. A total of three comparative plans are established: Plan 1: without cluster partitioning and without MESS pre-deployment; Plan 2: with MESS pre-deployment only; Plan 3: with cluster partitioning and MESS pre-deployment. The solved MESSs deployment positions, pre-deployment costs, and branch disconnection results from cluster partitioning for each plan are shown in Figure 5:

4.2.1. Cost Analysis

To validate the effectiveness of the proposed method, this paper establishes three comparative plans for economic comparison in the pre-disaster prevention phase: The simulation results demonstrate that incorporating MESSs pre-deployment can significantly reduce costs, with Plan 2 achieving a 26.41% cost reduction compared to Plan 1. Furthermore, integrating cluster partitioning further optimizes resource allocation, resulting in an additional 7.88% cost reduction in Plan 3 relative to Plan 2. This verifies the superiority of the pre-disaster plan combining cluster partitioning and MESSs pre-deployment.

4.2.2. Analysis of Branch Disconnection Rationality in Cluster Partitioning

In the pre-disaster prevention phase, all three plans simulate the disconnection of certain branches to prioritize the power supply assurance of critical loads. The advantages and disadvantages of the three plans are analyzed below in terms of the number and locations of disconnected branches. As shown in

Table 2. Comparison of Pre-Disaster cluster partitioning and MESS Pre-Deployment Plans.

Plan	Pre-Deployment Location	Pre-Deployment Cost (CNY)	Number of Disconnected Branches
1	Node 1, Node 1	18,273.3986	19
2	Node 28, Node 71	13,447.6374	17
3	Node 25 in Cluster 3, Node 72 in Cluster 9	12,387.7475	11

, without cluster partitioning, Plan 1 disconnects 19 branches, while Plan 2 disconnects 17 branches. The difference between the two is mainly due to the fact that with the addition of MESS in Plan 2, the connections between the island formed by nodes 34–37 and the island formed by nodes 24–33 remain intact. Therefore, it can be concluded that MESS effectively enhances the power supply assurance capability of the distribution network during the pre-disaster phase. However, both plans suffer from excessive disconnections leading to topological fragmentation, which is detrimental to the overall power supply assurance of the system.

In contrast, Plan 3 disconnects significantly fewer branches than the previous two plans. This clearly demonstrates that through cluster partitioning, the degree of source-load matching and the electrical boundaries are optimized. Critical loads are integrated into clusters with stronger power supply assurance capabilities. Consequently, when prioritizing the power supply to critical loads, only non-critical inter-cluster connecting branches need to be disconnected, thereby avoiding the fragmentation issue caused by excessive disruption to the network topology.

4.2.3. Analysis of MESSs Deployment Logic

As indicated in Table 2, the pre-deployment positions of MESS in Plan 2 and Plan 3 exhibit a certain degree of convergence. Both plans choose to deploy one MESS unit within the island formed by nodes 21–37. This location is situated in an area with multiple loads but limited PV resources. Deploying here allows upward coverage of loads from nodes 33 to 36 and downward coverage of loads at nodes 24 to 27. However, for the other MESS unit, Plan 2 opts for deployment at node 71 to cover the critical loads at nodes 70 and 71, whereas Plan 3 deploys it at node 72 within Cluster 9, providing power supply assurance for critical loads at nodes 72, 75, and 77.

In summary, the pre-disaster plan integrating cluster partitioning with MESS coordination demonstrates clear advantages in terms of overall pre deployment costs, coverage scope, and the rationality of MESS placement.

4.3. Post-Disaster Results Analysis

To validate the effectiveness of the proposed post-disaster restoration strategy, simulations compared two approaches: a node-level dispatch strategy disregarding cluster division (hereafter “Node-Level Strategy”) and a cluster-level dispatch strategy incorporating distributed PV cluster division (hereafter “Cluster-Level Strategy”). First, assuming the disaster occurs at 9:00, six typical time periods from 9:00 to 14:00 during the post-disaster restoration phase are selected as the scheduling window. Due to the disaster impact, lines (1–2), (58–59), and Line (78–82) are disconnected, forming three isolated islands: Island 1 (Nodes 2–58), Island 2 (Nodes 59–81, 103–123), and Island 3 (Nodes 82–102). Node-level strategy topology diagram: Cluster-level strategy topology diagram: The cluster partitioning results are shown in Figure 6:

4.3.1. Overall Performance Comparison

The overall results of the two strategies are shown in

Table 3. Overall Performance Comparison of Two Strategies.

Metric	Node-Level Strategy	Cluster-Level Strategy
Total Load Curtailment (MW)	10.8403	4.3946
Overall Restoration Rate (%)	61.20	86.27
Total Cost (CNY)	53,010.98	31,351.97
Total PV Output (MW)	19.745	27.681
PV Utilization Rate (%)	47.05	86.25
MESS Utilization Rate (%)	47.77	83.38

: The Node-Level Strategy achieves a load curtailment of 10.8403 MW at a cost of 53,010.98 yuan; the Cluster-Level Strategy achieves a curtailment of 4.3946 MW at a cost of 31,351.97 yuan. In terms of performance, the Node-Level Strategy suffers from low restoration rates and underutilized resources due to fine-grained physical constraints and inter-node scheduling; the Cluster-Level Strategy, targeting “intra-cluster supply-demand balance” and “inter-cluster power complementarity,” leverages dimensionality reduction advantages to enhance restoration rates and utilization, such as elevating PV utilization from 47.05% to 86.25%, thereby improving coordinated absorption and avoiding PV wastage.

4.3.2. MESS Dispatch Analysis

As shown in Figure 7a, under the Node-Level Strategy, mobile energy storage demonstrates characteristics of “single-point emergency response.” While it addresses power supply at local nodes, the lack of global coordination results in SoC imbalance and low utilization (overall only 47.77%), reflecting the limitations of dispersed and insufficient scheduling. In contrast, the Cluster-Level Strategy shown in Figure 7b employs repartitioned clusters as dispatch units, enabling cross-cluster coordinated support. The MESS dispatch path closely aligns with load and PV fluctuations, significantly increasing the total discharge volume. Through coordinated PV-storage charging and discharging, the utilization rate is raised to 86.25%, avoiding ineffective movement and capacity waste. This demonstrates the clear advantages of cluster partitioning in dimensionality reduction optimization and efficient resource utilization.

4.3.3. Load Restoration Analysis

The load restoration under the Node-Level Strategy exhibits the characteristic of “local priority at the expense of global imbalance.” Combined with the data mentioned earlier, the total load curtailment reaches 10.8403 MW, and the restoration rate is only 68.98%. As observed in Figure 8a, this strategy, which treats individual nodes as optimization units, results in some critical loads still facing curtailment due to barriers in power allocation between nodes. In contrast, the load restoration under the Cluster-Level Strategy, shown in Figure 8b, first maximizes the absorption of internal PV resources within each cluster, and then utilizes MESS to provide composite support, achieving “self-balancing power supply within clusters and inter-cluster power support.” This reduces the total load curtailment to 4.3946 MW and increases the restoration rate to 86.89%. In terms of load restoration, the Cluster-Level Strategy demonstrates clear advantages in enhancing post-disaster power supply efficiency and balancing resource allocation.

5. Conclusions

In this study, we tackle the challenge of power supply security in distribution networks with high-penetration distributed PV integration under extreme disasters, proposing a two-stage optimization strategy comprising “pre-disaster prevention and post-disaster restoration”. During the pre-disaster prevention phase, an improved Louvain algorithm is utilized for PV clustering, and combined with robust optimization to complete the pre-deployment of MESS. In the post-disaster fault restoration phase, issues related to cluster re-partitioning and MESS re-scheduling are addressed, and a MIP approach is employed to solve for the charging and discharging power of MESS. Finally, the effectiveness of the strategy is validated via case studies on a modified IEEE 123-node distribution network with twenty-four PV systems and two MESS units. The core findings are as follows:

(1) The proposed distributed PV cluster partitioning method combines a comprehensive multi-dimensional metric with the Louvain algorithm, balancing both safety and economic efficiency. It enables the integrated utilization of distribution facilities and effectively avoids issues such as insufficient regulation capacity in certain clusters caused by partitioning based solely on a single metric. Pre-disaster simulation results indicate that compared to a non-clustering strategy, the proposed approach reduces pre-disaster costs by 26.41% and decreases the number of disconnected branches between clusters from 17 to 11.

(2) The two-stage robust optimization model in the pre-disaster phase coordinates the electrical structure and power supply balance capability of PV clusters. Under the worst-case PV output scenarios, the model achieves precise coordination between cluster partitioning and MESS deployment. This not only effectively prevents excessive fragmentation of the distribution network but also reduces costs by 7.88% compared to a strategy with “MESS pre-deployment only,” laying a more sufficient and economical foundation for post-disaster rapid power restoration.

(3) The post-disaster cluster re-partitioning and cross-cluster MESS dispatch mechanism achieves spatio-temporal optimal allocation of electrical energy. This raises the PV utilization rate from 47.05% to 86.25% and the MESS utilization rate from 47.77% to 83.38%. It effectively overcomes the drawback of “local priority at the expense of global imbalance” inherent in node-level scheduling, greatly enhancing the accommodation level of distributed energy resources and the utilization efficiency of emergency resources.

6. Limitations and Future Work

While this study offers obvious advancements, it has certain limitations that can be addressed in future research. These limitations are related to the assumptions made in the model and the lack of adaptability in the clustering approach.

6.1. Limitation: Simplified Constraints on Transportation Network and MESS Deployment

The study assumes that MESS can be dispatched across clusters without obstruction, overlooking real-world challenges such as road damage, traffic congestion, and transportation time.

Future Work: Future research should incorporate real-time traffic data to better simulate transportation constraints and optimize MESS deployment schedules, particularly in disaster scenarios where infrastructure may be compromised. Dynamic routing algorithms can be implemented to adapt to changing conditions, such as road blockages or increased traffic, thus improving the robustness of MESS deployment during post-disaster recovery.

6.2. Limitation: Fixed Weights for Cluster Partitioning Metrics

The clustering approach in this study uses fixed weights for parameters such as electrical distance and voltage sensitivity. These weights are set without considering the variations in grid topology or renewable energy penetration, potentially limiting the adaptability of the clustering algorithm.

Future Work: Future research could explore the development of adaptive weighting methods that adjust based on the specific characteristics of the distribution grid and renewable energy penetration levels. This could allow for a more flexible and context-aware clustering approach that is better suited to varying grid conditions. Additionally, integrating machine learning techniques to dynamically adjust weights based on real-time data and predictive analytics could further improve the accuracy of cluster partitioning, especially during extreme weather events.