Research on an Integrated Method for Pre-Disaster Robust Optimization, In-Disaster Emergency Disposal and Post-Disaster Coordinated Restoration of Port Power Grids

Abstract

With the increasing frequency of global climate change and natural disasters, the resilience and stability of port power grids have become crucial for ensuring continuous port operations. This study proposes a three-stage resilience optimization method for port power grids under disaster scenarios, aiming to enhance their supply capacity and operational flexibility across the pre-disaster, during-disaster, and post-disaster phases. In the pre-disaster stage, the model considers the uncertainty of photovoltaic (PV) generation and the reconfigurability of the grid, optimizing the quantity and spatial layout of mobile energy storage systems with the objective of minimizing configuration and load-shedding risk costs, thereby improving system disturbance resistance. During the disaster, the model integrates the dynamic coordination of distributed generators, PV units, and storage systems to minimize load-shedding costs and achieve staged restoration and multi-source energy coordination. In the post-disaster stage, considering the failure of lines and nodes caused by disasters, a topology reconstruction and source-load coordination optimization strategy is developed to ensure rapid power restoration and critical load supply. Simulation studies based on an improved IEEE 33-bus system demonstrate that the proposed robust optimization model in the pre-disaster phase significantly enhances risk resistance and system resilience, while the incorporation of mobile energy storage further improves the system’s flexibility and black-start capability. This research provides an effective theoretical foundation and practical framework for post-disaster recovery and resilience enhancement of port power grids.

1. Introduction

Ports are critical nodes in the global trade and supply chain infrastructure, accounting for over 85% of worldwide cargo transportation. Their operational efficiency directly affects the national economy as well as regional industrial and economic development [1]. However, in recent years, the issues of port energy consumption and environmental impact have become increasingly prominent [2]. Statistics show that large container ports consume between 20,000 and 50,000 kWh of energy per day. Conventional port machinery and transport vehicles that rely on fossil fuels not only emit significant amounts of greenhouse gases and air pollutants but also struggle to balance the dual demands of energy reliability and environmental sustainability in modern port operations [3]. Against this backdrop, renewable energy sources—such as wind, solar, and hydropower—have become essential to green transition of ports due to their clean and sustainable characteristics [4]. The integration of renewable energy into port systems not only reduces greenhouse gas emissions but also enhances the flexibility of energy supply. As distributed generation (DG) penetration in distribution networks continues to increase, DG output directly influences the load supply of port power systems. Therefore, it is essential to consider DG uncertainty when planning port electrical systems [5].

Current methods for addressing uncertainty in power system planning mainly include stochastic optimization, chance-constrained programming, and robust optimization. Wan et al. [6] employed probabilistic forecasting to quantify wind power uncertainty and proposed a two-stage stochastic programming model for hybrid energy storage systems (HESS). However, this approach relies on precise probability distributions of DG output and load demand, which are difficult to obtain due to their short-term variability and non-stationarity. Guan et al. [7] used a chance-constrained method to model the uncertainty of solar power and load, developing a combined cooling, heating, and power (CCHP) system planning model, but failed to account for the dynamic interaction between distributed generation and critical port loads. Reference [8] addressed wind and wave uncertainty through a robust optimization framework, applying the column-and-constraint generation (C&CG) algorithm to decompose the problem into master and subproblems, thereby ensuring solution feasibility under worst-case scenarios while mitigating the computational burden of conventional robust methods. Li et al. [9] further introduced Wasserstein-distance-based fuzzy sets to effectively handle coupled uncertainties in supply and demand. Nevertheless, most of these studies focus on urban energy systems, with limited applicability to port environments.

In addition to static resource considerations, research on load recovery incorporating mobile resources has also attracted attention. References [10,11,12,13] investigated pre-disaster planning of mobile energy storage (MES) considering typhoon-induced line failures, employing multi-level robust and stochastic optimization models to determine the optimal number and location of mobile units. Some studies pre-positioned mobile emergency vehicles based on load priority to restore power after distribution network faults; however, they often neglected the pre-configuration of available emergency units or simplified the network’s radial topology by fixing the number of islands. These approaches mainly used MES or fixed resources independently for load restoration, with limited attention to DG output uncertainty and multi-source coordinated operation.

Beyond pre-disaster configuration, the during-disaster and post-disaster stages are equally critical, as they directly determine the speed of power recovery and the reliability of supply to critical loads—key indicators of an electrical network’s transition from “disaster resistance” to “self-healing.” Extensive studies have explored distributed resource-based restoration strategies, including islanded microgrids, emergency generators, and mobile resources [14]. For example, reference [15] proposed a microgrid formation mechanism using DG and remotely controlled switches to restore critical loads. Reference [16] utilized diesel generators (DEG), stationary storage, and photovoltaic (PV) units to recover loads following main grid and feeder outages. Reference [17] introduced a planned microgrid-based restoration method that minimizes load-shedding costs during main grid interruptions, while reference [18] optimized the siting and sizing of fuel-based DG units to reduce emergency reserve requirements. However, these methods typically rely on pre-installed microgrids or large-scale fuel generators, leading to high investment costs and limited compatibility with carbon reduction goals.

The uncertainty in port energy supply and demand poses a severe threat to the stable operation of port energy systems. Extreme events such as typhoons [19], cyberattacks [20], or equipment failures can disrupt external power supply, severely compromising port energy security and potentially interrupting core port operations and global supply chains. In this complex and uncertain context, enhancing the resilience of port energy systems has become an urgent requirement for sustainable port development [21].

Recent research trends indicate that resilience enhancement in modern power systems is shifting from static infrastructure-based planning toward dynamic, coordinated, and intelligence-driven adaptive operation frameworks. A growing body of work highlights the increasingly important role of distributed resources—including electric vehicles, distributed energy storage, and mobile power units—in accelerating post-disturbance recovery and improving system self-healing capability [22]. For example, recent studies have demonstrated that electric vehicles can serve as mobile and controllable storage assets during disaster recovery phases, and unified optimization frameworks have been developed to simultaneously model distribution topology, power flow security, and flexible charging–discharging strategies to maximize load restoration under extreme operating conditions [23].

In parallel, emerging research goes beyond fixed charging and stationary backup infrastructure by exploring mobile, autonomous charging networks capable of real-time dispatch and adaptive service allocation. These systems integrate power network conditions, mobility constraints, and resilience priorities into a multi-layer optimization architecture, enabling coordinated decision-making that balances operational efficiency during normal periods with resilience-driven actions during disruptions. Simulation studies confirm that such mobile energy–service frameworks significantly enhance voltage stability, improve service continuity, and provide a scalable pathway for resilience-oriented resource deployment in EV-dominated future grids [24].

Recent studies have emphasized that post-disaster power system restoration should not be treated as an isolated infrastructure optimization problem, but rather as a coupled process with transportation networks, especially when mobile energy resources are required. Transportation accessibility, road conditions, and routing constraints significantly affect the deploy ability of mobile energy storage systems and emergency vehicles, thereby influencing the feasible recovery trajectory. Existing work demonstrates that coordinated modeling of mobility and energy flows enhances the reliability and effectiveness of restoration decisions. These findings are highly relevant to this study, as MES and EV resources used in our framework inherently rely on transportation connectivity, implying that full recovery ability depends on cross-infrastructure interdependence rather than electrical topology alone [25].

Together, these advancements reveal a clear evolution in the field: future resilient distribution systems will increasingly rely on flexible mobile assets, real-time sensing, and intelligent multi-objective decision mechanisms. This direction underscores the need for integrated resilience paradigms capable of bridging infrastructure-level resource deployment, adaptive operational control, and user-side interactive flexibility—ultimately enabling distribution networks that are self-adaptive, rapidly recoverable, and robust against high-impact and uncertain disturbances.

To address these challenges and research gaps, this paper proposes a three-stage resilience optimization method for port power grids under disaster scenarios. In the pre-disaster stage, the method considers PV generation uncertainty and network reconfigurability, aiming to minimize mobile storage configuration and load-shedding risk costs while optimizing the number and spatial layout of storage devices to enhance system redundancy and anti-disturbance capability. In the during-disaster stage, the model incorporates distributed generation degradation, PV variability, energy storage constraints, and network reconfiguration, minimizing load-shedding costs through the coordinated optimization of distributed generators, PV units, electric vehicles, and mobile storage systems, thereby achieving staged recovery and multi-source energy coordination. In the post-disaster stage, considering line and node failures, the damaged network is reconstructed through source-load coordination optimization to maximize critical load recovery or minimize unserved energy, ensuring rapid system restoration and the regeneration of supply capacity. The referenced work [26] further demonstrates that resilience-oriented optimization should integrate uncertainty modeling, flexible resource coordination, and staged decision coupling to ensure adaptability across different infrastructures. This aligns with the conceptual logic of the proposed three-stage framework in this paper, indicating that the methodology is not limited to port power grids but represents a generalizable resilience optimization paradigm applicable to broader extreme scenarios.

2. Pre-Disaster Robust Optimization Model for Layout Planning

2.1. Pre-Disaster Optimization Objective Function

Before the occurrence of a disaster, a robust optimization framework is employed to comprehensively consider the layout and capacity configuration of distributed energy resources (such as photovoltaic systems, LNG units, and electric vehicles), load-shedding costs, and mobile energy storage systems. The objective is to maximize the resilience and economic efficiency of the power grid under potential disaster scenarios while controlling the investment cost of energy storage. Specifically, the pre-disaster configuration aims to determine in advance the optimal number and locations of energy storage units so that the grid can achieve rapid response, minimal load loss, and lower restoration costs when disasters occur.

The proposed model adopts a two-stage optimization structure. In the first stage, the layout and investment decisions for energy storage systems are determined. In the second stage, the post-disaster worst-case scenarios are simulated to perform load-shedding minimization and power flow optimization. The overall algorithmic procedure is illustrated in Figure 1.

In the first stage, the objective function is defined as follows:

$$\min \hspace{0.33em}{C}_{\mathrm{ME}}{\displaystyle \sum _{i\in \mathcal{N}}{\alpha }_i^{\mathrm{ME}}}+Z$$

(1)

The This objective function minimizes the total investment cost of mobile energy storage deployment and the worst-case resilience loss. In this formulation, $C_{\mathrm{ME}}$ denotes the unit investment cost of a mobile energy storage (MES) unit, while ${\alpha }_i^{\mathrm{ME}}$ is a binary decision variable indicating whether an MES is installed at node $i$ (value of 1) or not (value of 0). The set $\mathcal{N}$ represents all candidate nodes available for storage deployment. The term $Z$ refers to the upper bound of worst-case operational loss. Accordingly, this formulation captures the trade-off between infrastructure investment and resilience risk, guiding the optimal allocation of mobile storage resources.

$$\min \hspace{0.33em}{C}_{\mathrm{ME}}{\displaystyle \sum _{i\in \mathcal{N}}{\alpha }_i^{\mathrm{ME}}}+{\displaystyle \sum _{i\in \mathcal{N}}{w}_i}{P}_i^{\mathrm{su}}\cdot S_B$$

(2)

In this stage, the first term is a constant determined by the first-stage investment decisions, whereas the second term represents the sum of load-shedding costs across all nodes. $w_i$ represents the weighting factor for node curtailment, $P_i^{\mathrm{su}}$ denotes the amount of load shed at node $i$ in scenario $j$, and $S_B$ stands for the system’s base capacity. By optimizing generation output and power flow under various uncertain scenarios, this objective function quantifies the economic loss level of the system under the worst-case post-disaster disturbance, capturing the grid’s resilience and cost-effectiveness in extreme conditions.

2.2. Pre-Disaster Constraints

In the first stage, the model determines the placement scheme of mobile energy storage systems and the structural configuration of the distribution network, while introducing an upper-bound variable for load-shedding cost through an outer approximation approach. The constraints include:

(1)

Constraint on the Number of Mobile Storage Units:

$${\displaystyle \sum _{i\in \mathcal{N}}{\alpha }_i^{\mathrm{ME}}}\le N_{{\max }^{\mathrm{ME}}}$$

(3)

This constraint limits the maximum number of mobile storage units deployed in the system. Here, ${\alpha }_i^{\mathrm{ME}}$ is a binary decision variable, where ${\alpha }_i^{\mathrm{ME}}=1$ indicates that a storage unit is installed at node $i$, and ${\alpha }_i^{\mathrm{ME}}=0$ indicates no installation. $N_{{\max }^{\mathrm{ME}}}$ represents the upper limit determined by resource availability, investment budget, and operational feasibility.

(2)

Network Radiality and Connectivity Constraint:

$${\displaystyle \sum _{(i,j)\in \mathcal{L}}{\alpha }_{ij}}=N-{\displaystyle \sum _{k\in \mathcal{G}}{S}_k^{\mathrm{vs}}}$$

(4)

$${\delta }_i\cdot F-{\gamma }_i\cdot F=\left\{\begin{array}{ll}1-F_k^{\mathrm{vs}},\hfill & i\in \mathcal{G},\hfill \\ 1,\hfill & i\notin \mathcal{G},i\ne 1,\hfill \\ 0,\hfill & i=1\hfill \end{array}\right.$$

(5)

In this constraint, ${\displaystyle \sum _{(i,j)\in \mathcal{L}}{\alpha }_{ij}}$ represents the number of operational branches within the distribution network, where ${\alpha }_{ij}$ is a binary decision variable indicating whether the line connecting nodes $i$ and $j$ remains energized. The right-hand term, $N-{\displaystyle \sum _{k\in \mathcal{G}}{S}_k^{\mathrm{vs}}}j$, denotes the allowable number of active lines, where $N$ is the total number of system nodes and $S_k^{\mathrm{vs}}$ represents the virtual supply activation state at critical node $k$. This constraint enforces radiality and guarantees that the restored or islanded network structure remains feasible. In the second expression, ${\delta }_i$ denotes the connectivity indicator for node $i$, specifying whether the node participates in the operational sub-network; ${\gamma }_i$ is a topology-related coefficient that ensures network structure consistency during reconfiguration; $F$ represents a binary logic variable enforcing conditional relationships. The piecewise logic defines node behavior under different scenarios: for nodes $i\in \mathcal{G}$ equipped with virtual supply resources, meaning their participation depends on whether the virtual source is activated; for normal nodes $i\notin \mathcal{G},i\ne 1$, ${\delta }_i=1$, indicating default participation; and for the reference node $i=1$, ${\delta }_i=0$, identifying it as the root node excluded from the logic constraint. This formulation ensures a clear distinction between virtual supply nodes, ordinary nodes, and the reference node, contributing to proper topology formation during resilient grid restoration.

(3)

Outer Approximation of Load-Shedding Cost

$$Z\ge {\displaystyle \sum _{i\in \mathcal{N}}{w}_i{P}_i^{\mathrm{su}}}\cdot S_B,\hspace{1em}\forall l\in \mathcal{S}$$

(6)

This constraint is a core part of the robust optimization formulation, ensuring that the upper-bound variable $Z$ sufficiently covers the maximum load-shedding cost across all uncertainty scenarios. Here, $w_i$ represents the weighting factor for node curtailment, $P_i^{\mathrm{su}}$ denotes the amount of load shed at node $i$ in scenario $j$, and $S_B$ stands for the system’s base capacity. This constraint ensures that the first-stage decision is robust against all possible adverse disaster realizations.

In the second stage, given the first-stage decisions on energy storage layout and network topology, the model optimizes post-disaster economic recovery by coordinating power flow and dispatch under photovoltaic (PV) output uncertainty and load interruption risks. The key physical constraints include:

(1)

PV Output Uncertainty Constraint:

$$P_{pv,i}={\displaystyle \frac{P_{pv,i}^0}{S_B}}-(1-q_i){\displaystyle \frac{P_{pv,i}^0}{S_B}}\tau$$

(7)

$${\displaystyle \sum _{i\in {\mathcal{N}}_{pv}}{q}_i}\le e$$

(8)

PV generation is subject to irradiance reduction or component damage during disasters. Here, $q_i$ denotes the downward deviation coefficient representing the output reduction ratio at PV node $i$, and $e$ is the allowable total deviation limit. $P_{pv,i}$ denotes the available photovoltaic output at node $i$ under uncertain operating conditions and ${\mathcal{N}}_{pv}$ is the set of PV nodes. $P_{pv,i}^0$ represents its nominal production level. The term $\tau$ serves as a sensitivity adjustment parameter reflecting the severity of PV output uncertainty. This constraint linearizes the stochastic uncertainty of PV output and controls the total deviation, ensuring sufficient safety margins and robustness in dispatch.

(2)

Load Shedding and Reactive Power Constraint:

$$0\le P_i^{\mathrm{su}}\le {\displaystyle \frac{P_i^{\mathrm{su},\max }}{S_B}},\hspace{1em}{Q}_i^{\mathrm{su}}={\displaystyle \frac{Q_i^{\max }}{P_i^{\max }}}{P}_i^{\mathrm{su}},\hspace{1em}i\ne 1$$

(9)

This constraint governs the allowable range of active and reactive power curtailment at node i under emergency or post-disturbance operation. The inequality $0\le P_i^{\mathrm{su}}\le {\displaystyle \frac{P_i^{\mathrm{su},\max }}{S_B}}$ ensures that the curtailed active power $P_i^{\mathrm{su}}$ remains within the feasible operating limits, where $P_i^{\mathrm{su},\max }$ represents the maximum available curtailment capability. The second expression, $Q_i^{\mathrm{su}}={\displaystyle \frac{Q_i^{\max }}{P_i^{\max }}}{P}_i^{\mathrm{su}}$, defines a proportional relationship between reactive power curtailment $Q_i^{\mathrm{su}}$ and its corresponding active curtailment, maintaining a consistent operating power factor. Here, $P_i^{\max }$ and $Q_i^{\max }$ denote the maximum active and reactive power capacity of node $i$, respectively. The condition $i\ne 1$ indicates that the reference bus or slack node is excluded from participating in the curtailment process. Together, this formulation ensures that load reduction remains technically feasible, capacity-constrained, and compliant with the physical power balance characteristics at each node.

(3)

Distributed Generation Output Constraint in the Pre-disaster Period

$$0\le P_g\le {\displaystyle \frac{P_g^{\max }}{S_B}},\hspace{1em}0\le Q_g\le {\displaystyle \frac{Q_g^{\max }}{S_B}}$$

(10)

$$P_g\tan {\phi }_{\min }\le Q_g\le P_g\tan {\phi }_{\max }$$

(11)

The constraints on distributed generator outputs are given by $P_g^{\max }$ and $Q_g^{\max }$, representing rated active and reactive power, while ${\phi }_{\min }$ and ${\phi }_{\max }$ denote the power factor angle limits. These constraints bound the operating range of distributed generators (e.g., diesel units), ensuring they participate in voltage support and local supply within safe operational limits

(4)

Mobile Energy Storage Output Constraint

$$0\le P_i^{\mathrm{ME}}\le {\alpha }_i^{\mathrm{ME}}\cdot {\displaystyle \frac{P_{i,\max }^{\mathrm{ME}}}{S_B}},\hspace{1em}0\le Q_i^{\mathrm{ME}}\le {\alpha }_i^{\mathrm{ME}}\cdot {\displaystyle \frac{Q_{i,\max }^{\mathrm{ME}}}{S_B}}$$

(12)

$P_{i,\max }^{\mathrm{ME}}$ and $Q_{i,\max }^{\mathrm{ME}}$ denote the maximum active and reactive outputs of storage devices, and ${\alpha }_i^{\mathrm{ME}}$ indicates whether node $i$ is equipped with storage. The active and reactive outputs are restricted by device ratings and can only be provided at nodes where storage units are installed $({\alpha }_i^{\mathrm{ME}}=1)$.

(5)

Power Flow Balance Constraint

$$P_i^{\mathrm{DG}}+P_i^{\mathrm{su}}+P_i^{\mathrm{ME}}-{\displaystyle \frac{P_i^{\max }}{S_B}}+{\displaystyle \sum _{j\in {\Omega }_i^{\mathrm{in}}}{P}_{ji}}-{\displaystyle \sum _{k\in {\Omega }_i^{\mathrm{out}}}{P}_{ik}}-{\displaystyle \sum _{(i,j)}{R}_{ij}}{I}_{ij}^2=0$$

(13)

This equation represents the active power balance at node $i$. Specifically, $P_i^{\mathrm{DG}}$ denotes the active power output of the distributed generator (DG) connected at node $i$; $P_i^{\mathrm{su}}$ is the curtailed active load at node $i$ under emergency; and $P_i^{\mathrm{ME}}$ is the active power injected by the mobile energy storage (MES) unit located at node $i$. The term ${\displaystyle \frac{P_i^{\max }}{S_B}}$ corresponds to the per-unit rated active demand at node $i$, where $P_i^{\max }$ is the maximum load power and $S_B$ is the system base power. The set ${\Omega }_i^{\mathrm{in}}$ denotes the collection of incoming branches terminating at node $i$, with $P_{ji}$ being the active power flow from neighboring node $j$ to node $i$; similarly, ${\Omega }_i^{\mathrm{out}}$ is the set of outgoing branches originating from node $i$, and $P_{ik}$ is the active power flow from node $i$ to neighboring node $k$. The last term ${\displaystyle \sum _{(i,j)}{R}_{ij}}{I}_{ij}^2$ represents the active power losses on the lines connected to node $i$, where $R_{ij}$ is the resistance of branch $(i,j)$ and $I_{ij}$ is the corresponding line current magnitude. Overall, the equation enforces that, at node $i$, the sum of DG output, curtailed load, MES injection, and incoming power flows equals the rated demand, outgoing power flows, and line losses, thereby ensuring active power conservation in the distribution network.

(6)

Voltage and Current Operating Limits

$$V_{\min }\le V_i\le V_{\max },\hspace{1em}0\le I_{ij}\le I_{\max }$$

(14)

Node voltage $V_i$ magnitudes and branch currents $I_{ij}$ must remain within safe operational bounds $[V_{\min },V_{\max }]$ and $I_{ij}\le I_{\max }$. These constraints ensure that post-disaster voltages and currents remain within acceptable ranges, preventing equipment overloading and voltage collapse.

(7)

Voltage Drop Constraint (Big-M Relaxation)

$$V_i-V_j\le (1-{\alpha }_{ij})M_0+2R_{ij}{P}_{ij}+2X_{ij}{Q}_{ij}-(R_{ij}^2+X_{ij}^2)I_{ij}$$

(15)

The Big-M constant $M_0$ is introduced to relax the voltage drop equation under line outage conditions. $R_{ij}$ and $X_{ij}$ denote branch resistance and reactance. $P_{ij}$ and $Q_{ij}$ represent the active and reactive power flows along the same branch. When a branch is disconnected (${\alpha }_{ij}=0$), voltage difference constraints are relaxed; when connected (${\alpha }_{ij}=1$), standard power flow equations are enforced. This mechanism allows topological flexibility and enables adaptive reconfiguration of the grid after disasters.

(8)

Second-Order Cone Constraint

$$P_{ij}^2+Q_{ij}^2\le V_i{I}_{ij},\hspace{1em}\forall (i,j)\in \mathcal{L}$$

(16)

This constraint defines the second-order cone relaxation of branch power flow, ensuring consistency between power flow ($P_{ij},Q_{ij}$) and current magnitude ($I_{ij}$) under the voltage magnitude $V_i$. The notation $\mathcal{L}$ represents the set of all distribution lines. It preserves the physical relationships among power, current, and voltage in the distribution network, forming a convex approximation suitable for optimization.

2.3. Solution

The proposed two-stage robust optimization problem is solved using a column-and-constraint generation (C&CG) algorithm. The algorithm starts by initializing the master problem with only the first-stage constraints and an auxiliary variable representing the worst-case second-stage cost. In each iteration, the master problem is first solved to obtain a candidate first-stage decision and a lower bound of the robust objective. Given this solution, a subproblem is then solved to identify the worst-case realization of the uncertainty that maximizes the second-stage recourse cost. If the resulting worst-case cost exceeds the current upper-bound variable in the master problem, a new constraint (cut) is generated to enforce that the upper-bound variable is no smaller than the recourse cost under this worst-case realization, and this constraint is added to the master problem. The master problem is subsequently re-solved with the updated constraint set. This iterative process continues until no violated cuts can be found, at which point the upper and lower bounds converge and the algorithm terminates, yielding a robust optimal solution. The specific algorithm process is shown in the Appendix A.

3. Disaster-Time Coordinated Optimization Model

3.1. Disaster-Time Optimization Objective Function

This section establishes a disaster-time multi-source coordinated optimization model, which dynamically schedules distributed energy resources such as distributed generators (DGs), photovoltaic (PV) units, electric vehicles (EVs), and mobile energy storage systems (MESs) in the port power system after a disaster. The objective is to optimize load shedding and ensure efficient post-disaster power supply recovery.

In the in-disaster stage, mobile energy storage systems (MES) and electric vehicles (EVs) are allowed to change their connection nodes between consecutive time steps, and are assumed to be connected to a single node in each time interval. This modeling assumption is based on the relatively small geographical scale of port areas and the limited internal travel distances within the port. The scheduling time step is chosen to be no shorter than the typical relocation time of MES units and EVs, such that the relocation process can be reasonably approximated as a discrete node-switching decision between time intervals, without explicitly modeling continuous transportation paths or travel dynamics.

This assumption is primarily applicable to compact port systems with manageable internal traffic conditions. For larger-scale systems or disaster scenarios with severe transportation disruptions, neglecting travel time and road constraints may overestimate the operational flexibility of mobile resources. This limitation is acknowledged and discussed in Section 5.

Since the power supply to critical users directly affects the load restoration sequence, the model minimizes the weighted sum of load-shedding costs as its objective function. Let $P_{Lsu}(i,t)$ denote the curtailed load power at node $i$ and time $t$, $w_i$ the unit cost of load shedding, and $S_B$ the system base capacity. The objective function is formulated as:

$$\min {\displaystyle \sum _{t=1}^{N_T}{\displaystyle \sum _{i=1}^{N_B}{w}_i}}{P}_{su}(i,t)\cdot S_B$$

(17)

The outer summation index $t$ represents the time intervals considered in the restoration horizon, with a total length of $N_T$. The inner summation runs over all load nodes $i$, where $N_B$ denotes the total number of buses or load points in the system. The parameter $w_i$ is the load priority coefficient assigned to node $i$, enabling differentiation between critical loads and non-essential loads so that essential services are restored first. The variable $P_{su}(i,t)$ denotes the unsupplied power at node $i$ and time $t$, capturing the extent of load curtailment or unmet demand.

3.2. Disaster-Time Constraints

The disaster-time optimization also considers network topology, power flow, and system safety constraints. The major constraints are as follows:

(1)

Topological Connectivity Constraint

Network connectivity is ensured by binary variables ${\alpha }_{ij}$, which indicate the operational status of each line. The constraint is expressed as:

$${\displaystyle \sum {\alpha }_{ij}}=N_B-{\displaystyle \sum S_{vs}}$$

(18)

This constraint enforces the radial network topology requirement during system reconstruction. The term ${\alpha }_{ij}$ represents a binary decision variable associated with branch $(i,j)$, indicating whether the line is energized (${\alpha }_{ij}=1$) or disconnected (${\alpha }_{ij}=0$). On the right-hand side, $N_B$ denotes the total number of buses in the system, while ${\displaystyle \sum S_{vs}}$ represents the number of virtual supply nodes. The equation expresses the principle that, for a radial distribution network, the number of active branches must equal the number of nodes minus the number of virtual supply points. This ensures that the resulting topology remains loop-free and forms a tree structure suitable for resilient distribution grid operation. Consequently, this constraint guarantees structural feasibility and prevents unintended meshed configurations during the network restoration process.

(2)

Power Balance Constraints

For each node, the active and reactive power balance equations are:

$$P_{DG}+P_{ME1}+P_{ME2}+P_E^{net}+P_{su}-P_{L,\max }+{\displaystyle \sum P_{ij}^{in}}-{\displaystyle \sum P_{ij}^{out}}-R_{ij}{I}_{ij}^2=0$$

(19)

$$Q_{DG}+Q_{ME1}+Q_{ME2}+Q_{su}-Q_{L,\max }+{\displaystyle \sum Q_{ij}^{in}}-{\displaystyle \sum Q_{ij}^{out}}-X_{ij}{I}_{ij}^2=0$$

(20)

These constraints guarantee real and reactive power balance at each node, considering generation, storage, load shedding, and network losses.

(3)

Distributed Generation Output Constraint in the Disaster-time

To ensure voltage stability, distributed generators must operate within their power factor range:

$$0\le P_{DG}\le P_{DG}^{\max },\hspace{1em}{P}_{DG}\tan ({\theta }_{\min })\le Q_{DG}\le P_{DG}\tan ({\theta }_{\max })$$

(21)

where ${\theta }_{\min }$ and ${\theta }_{\max }$ denote the lower and upper bounds of the power factor angle, respectively.

(4)

Energy Storage Dynamic Constraint

For mobile energy storage systems (MESs) and electric vehicles (EVs), their energy states evolve dynamically according to the charging/discharging process:

$$E_{ME}(t+1)=E_{ME}(t)+{\eta }_M{P}_{Mch}(t)-{\displaystyle \frac{P_{Mdch}(t)}{{\eta }_M}}$$

(22)

$$E_E(t+1)=E_E(t)+{\eta }_E{P}_{Ech}(t)-{\displaystyle \frac{P_{Edch}(t)}{{\eta }_E}}$$

(23)

subject to:

$$E_{\min }\le E(t)\le E_{\max },\hspace{1em}{U}_{ch}+U_{dch}\le 1$$

(24)

Equations (20) and (21) describe the temporal evolution of the state of charge for mobile and stationary energy storage systems, respectively. The terms $E_{ME}(t)$ and $E_{ME}(t+1)$ denote the stored energy of the mobile unit at time steps $t$ and $t+1$, while $E_E(t)$ and $E_E(t+1)$ represent the corresponding energy states of the stationary storage unit. The variables $P_{Mch}(t)$ and $P_{Ech}(t)$ are the charging power of mobile and stationary storage at time $t$, and $P_{Mdch}(t)$ and $P_{Edch}(t)$ represent the discharging power. Parameters ${\eta }_M$ and ${\eta }_E$ define the charging and discharging efficiency of the respective storage systems, accounting for energy conversion losses. The constraint in (22) enforces operational safety by restricting the storage energy within allowable bounds, $E_{\min }\le E(t)\le E_{\max }$, thereby preventing overcharging or deep discharging. The binary mode-selection variables $U_{ch}$ and $U_{dch}$ ensure that charging and discharging cannot occur simultaneously through the constraint $U_{ch}+U_{dch}\le 1$, maintaining the physical feasibility of storage operation.

The in-disaster stage is formulated in discrete time with a time resolution of $\mathsf{\Delta }t=1$ h, and the total scheduling horizon covers $T$ consecutive time intervals corresponding to the emergency response and early recovery period following the disaster. This time resolution is sufficient to capture the main dynamics of load variation, state-of-charge evolution, and emergency dispatch decisions in port operations, while maintaining computational tractability.

(5)

Node Voltage and Line Flow Constraints

Using the DistFlow model, the voltage and current relationship is formulated as:

$$V_i^2-V_j^2=2(R_{ij}{P}_{ij}+X_{ij}{Q}_{ij})-(R_{ij}^2+X_{ij}^2)I_{ij}^2$$

(25)

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

(26)

Voltage magnitude limits and current limits are defined as:

$${V_{\min }}^2\le V_i^2\le {V_{\max }}^2,\hspace{1em}0\le I_{ij}\le {\alpha }_{ij}{I}_{\max }$$

(27)

ensuring that post-disaster operations remain within secure electrical boundaries.

(6)

Load and Curtailment Constraints

At ordinary nodes, loads can be curtailed when necessary, while for critical users, the cost of load shedding is higher to reflect their importance in maintaining essential operations. The corresponding constraint is defined as:

$$0\le P_{su}(i,t)\le P_{L,\max }(i,t)$$

(28)

where $P_{su}(i,t)$ denotes the curtailed active power of node $i$ at time $t$, and $P_{L,\max }(i,t)$ represents its maximum allowable curtailment.

The model controls the curtailment ratio dynamically, and based on the amount of curtailed active power, it automatically determines the corresponding reactive power curtailment to maintain power factor stability and ensure coordinated system operation.

4. Post-Disaster Recovery and Reconfiguration Model

The post-disaster recovery model aims to achieve both safety restoration and power supply coordination in the port distribution network. After a disaster, some nodes may lose power due to faults, resulting in partial disconnection of the grid. To accelerate power restoration, this stage coordinates distributed generators, mobile energy storage systems, and network topology reconfiguration to restore critical loads and ensure overall supply continuity. The objective function minimizes the weighted unsupplied load cost, formulated as:

$$\min \hspace{0.33em}J={\displaystyle \sum _{i\in \mathrm{X}}(}1-{\delta }_i)P_i^{D0}$$

(29)

The term ${\delta }_i$ is a binary indicator representing the operational status of node $i$; it takes the value of 1 if the node is successfully restored and 0 otherwise. Therefore, $1-{\delta }_i$ captures whether node $i$ remains de-energized. The parameter $P_i^{D0}$ denotes the original rated demand or baseline load of node $i$, reflecting its normal operating power requirement. Summing the expression over all nodes in the set $\mathrm{X}$ quantifies the total unserved demand under the current restoration strategy. By minimizing this term, the model prioritizes restoring critical and high-impact loads, thereby improving supply continuity, optimizing resource allocation, and strengthening overall grid resilience during disaster recovery.

The constraints are as follows:

(1)

Relationship Between Voltage and Power Flow (Big-M Switch Constraint)

The DistFlow-based voltage-power relationship under Big-M relaxation is expressed as:

$$-M(1-c_{ij})\le u_i-u_j-2(R_{ij}{P}_{ij}+X_{ij}{Q}_{ij})\le M(1-c_{ij}),\hspace{1em}\forall (i,j)\in \mathcal{E}.$$

(30)

This constraint represents the DistFlow-based voltage–power relationship while incorporating a Big-M relaxation mechanism to handle switchable lines. The terms $u_i$ and $u_j$ denote the squared voltage magnitudes at nodes $i$ and $j$, respectively, ensuring compliance with steady-state voltage requirements in the distribution system. The variables $P_{ij}$ and $Q_{ij}$ correspond to the active and reactive power flows along branch $(i,j)$, and parameters $R_{ij}$ and $X_{ij}$ represent the resistance and reactance of the line. The binary variable $c_{ij}$ indicates whether the line is energized ($c_{ij}=1$) or disconnected ($c_{ij}=0$). The constant $M$ is the Big-M parameter used to deactivate the constraint when the associated line is out of service, while enforcing the voltage–power relationship when the line is active. Through this formulation, the model ensures that the network’s voltage behavior remains consistent with physical power flow laws while enabling flexible topology reconfiguration during restoration or resilient operation.

(2)

Node Voltage Limits and Reference Bus Constraints

$${\underset{\_}{V}}_i^2\le u_i\le {\overline{V}}_i^2,\hspace{1em}\forall i\in \mathcal{N}$$

(31)

$$u_i=V_g{(i)}^2,\hspace{1em}\forall i\in \mathcal{R}.$$

(32)

where ${\underset{\_}{V}}_i$ and ${\overline{V}}_i$ denote the lower and upper voltage limits of node $i$, respectively. $V_g(i)$ represents the specified voltage of the reference node $i$, and $\mathcal{R}$ denotes the set of power sources. The first equation ensures that the voltage magnitude at each node remains within the allowable range, while the second equation enforces that the voltage at reference buses is fixed at its nominal value.

(3)

Branch Thermal Stability/Capacity Constraints (Diamond Norm Approximation)

$$-{\overline{S}}_{ij}{c}_{ij}\le P_{ij}\le {\overline{S}}_{ij}{c}_{ij},\hspace{1em}-{\overline{S}}_{ij}{c}_{ij}\le Q_{ij}\le {\overline{S}}_{ij}{c}_{ij}.$$

(33)

$$-\sqrt{2}{\overline{S}}_{ij}{c}_{ij}\le P_{ij}\pm Q_{ij}\le \sqrt{2}{\overline{S}}_{ij}{c}_{ij}.$$

(34)

The parameters ${\overline{S}}_{ij}$ represent the rated apparent power of branch $(i,j)$. When the branch is closed ($c_{ij}=1$), the constraint limits the branch loading rate; when the branch is open ($c_{ij}=0$), the power flow is forced to zero. This set of constraints uses a linear polyhedral (diamond-shaped) approximation to replace the circular branch capacity constraint $\left|S_{ij}\right|\le {\overline{S}}_{ij}$, ensuring that the power flow does not exceed the rated apparent power of the branch and that thermal stability and operational safety are maintained.

(4)

Node Active/Reactive Power Balance

$$-{\displaystyle \sum _{(i,k)\in \mathcal{E}} }{P}_{ik}+{\displaystyle \sum _{(k,i)\in \mathcal{E}} }{P}_{ki}=P_i^L-{\displaystyle \sum _{g\in \mathcal{G}(i)}{P}_g},\hspace{1em}\forall i\in \mathcal{N}$$

(35)

$$-{\displaystyle \sum _{(i,k)\in \mathcal{E}} }{Q}_{ik}+{\displaystyle \sum _{(k,i)\in \mathcal{E}} }{Q}_{ki}=Q_i^L-{\displaystyle \sum _{g\in \mathcal{G}(i)}{Q}_g},\hspace{1em}\forall i\in \mathcal{N}$$

(36)

The left-hand side represents the difference between the inflow and outflow power at a node, while the right-hand side denotes the difference between the power demanded by loads supplied at the node and the output of local generators. $P_i^L$ and $Q_i^L$ represent the actual restored active and reactive loads at node $i$, respectively, whereas $P_g$ and $Q_g$ denote the generator outputs. $\mathcal{G}(i)$ refers to the set of generating units located at node $i$.

(5)

Load Supply and Demand Relationship

$$P_i^L={\gamma }_i{P}_i^{D0},\hspace{1em}{Q}_i^L={\gamma }_i{Q}_i^{D0},\hspace{1em}\forall i\in \mathcal{N}$$

(37)

The actual supplied load is a fraction of the nominal load. ${\gamma }_i$ is the load supply ratio: ${\gamma }_i=1$ indicates the load is fully restored, and ${\gamma }_i=0$ means the load is completely curtailed.

(6)

Generator Output Limits

$${P_g}^{\min }\le P_g\le {P_g}^{\max },\hspace{1em} {Q_g}^{\min }\le Q_g\le {Q_g}^{\max },\hspace{1em}\forall g.$$

(38)

These constraints restrict generator active and reactive power within their rated capacities, ensuring stable and secure operation.

(7)

Energization–Pickup Logic and Power Supply Node Energization:

$${\gamma }_i\le {\epsilon }_i,\hspace{1em}\forall i\in \mathcal{N};\hspace{1em} {\epsilon }_i=1,\hspace{0.33em}\forall i\in \mathcal{R}.$$

(39)

If a node is not energized (${\epsilon }_i=0$), it cannot pick up any load (${\gamma }_i=0$). For power supply nodes (reference buses), energization is enforced as ${\epsilon }_i=1$. This logical constraint ensures that the power restoration status is consistent with the electrical connectivity, preventing loads from being restored at nodes that are not yet reconnected to an energized supply path.

(8)

Energization Reachability Propagation for Pure Load Nodes:

$$0\le e_{i\to j}\le c_{ij},\hspace{1em}0\le e_{j\to i}\le c_{ji},$$

(40)

$$0\le {\epsilon }_i-e_{i\to j}\le 1-c_{ij},\hspace{1em}0\le {\epsilon }_j-e_{j\to i}\le 1-c_{ji},$$

(41)

$${\displaystyle \frac{1}{d_i}}{\displaystyle \sum _{(i,j)\in {\mathcal{E}}^{\mathrm{from}}(i)\cup {\mathcal{E}}^{\mathrm{to}}(i)}(e_{i\to j} \mathrm{or} e_{j\to i})}\le {\epsilon }_i\le {\displaystyle \sum _{(i,j)}(e_{i\to j} \mathrm{or} e_{j\to i})}{\mathcal{E}}^{\mathrm{from}}(i)\cup {\mathcal{E}}^{\mathrm{to}}(i)$$

(42)

Equations (37)–(39) model the energization reachability propagation for pure load nodes. The variables $e_{i\to j}$ and $e_{j\to i}$ are directional reachability indicators on branch $(i,j)$: $e_{i\to j}=1$ if power can be propagated from node $i$ to node $j$, and 0 otherwise; similarly for $e_{j\to i}$. The binary variables $c_{ij}$ and $c_{ji}$ denote the on/off status of line $(i,j)$ in each direction, taking the value 1 when the line is closed and available, and 0 when it is open. Equation (37), ensures that directional reachability can only be activated if the corresponding branch is actually in service; if the line is open, the associated e-variable is forced to zero. The node-level variables ${\epsilon }_i$ and ${\epsilon }_j$ indicate whether nodes $i$ and $j$ are energized (reachable from an upstream source), with ${\epsilon }_i=1$ meaning that node $i$ is already supplied and ${\epsilon }_i=0$ meaning that it is still de-energized. Equation (38) links node energization with directional propagation. When the line is closed ($c_{ij}=1$), the constraint reduces to ${\epsilon }_i\ge e_{i\to j}$, implying that power can only propagate from $i$ to $j$ if node $i$ is energized. When the line is open, the constraint becomes inactive and does not impose additional restrictions.

The sets ${\mathcal{E}}^{\mathrm{from}}(i)$ and ${\mathcal{E}}^{\mathrm{to}}(i)$ collect all branches originating from and terminating at node $i$, respectively; together they represent all incident lines of node $i$. The scalar $d_i$ denotes the degree of node $i$, i.e., the number of adjacent branches. Equation (39) further constrains node energization: if at least one incident direction has a positive reachability indicator, ${\epsilon }_i$ must be set to 1; conversely, if all adjacent $e_{i\to j}$ and $e_{j\to i}$ are zero, then ${\epsilon }_i$ must be zero as well. Collectively, these constraints propagate energization status along closed branches from source nodes to pure load nodes, rigorously capturing which loads are actually reachable under the current post-disaster topology.

(9)

Radiality and Exogenous Availability:

$$c_{ij}\le z_{ij},\hspace{1em}\forall (i,j)\in \mathcal{E}.$$

(43)

$${\displaystyle \frac{1}{n_b-n_{\mathrm{ref}}+1}}{c}_{ij}\le f_{ij}\le c_{ij}-{\displaystyle \frac{1}{n_b-n_{\mathrm{ref}}+1}}{c}_{ij},\hspace{1em}\forall (i,j).$$

(44)

$${\displaystyle \sum _{(k,i)}{f}_{ki}}+{\displaystyle \sum _{(i,k)}(c_{ik}-f_{ik})}\le {\displaystyle \frac{n_b-n_{\mathrm{ref}}}{n_b-n_{\mathrm{ref}}+1}},\hspace{1em}\forall i\notin \mathcal{R}.$$

(45)

$${\displaystyle \sum _{i\in \mathcal{R}} }\left[{\displaystyle \sum _{(k,i)}{f}_{ki}}+{\displaystyle \sum _{(i,k)}(c_{ik}-f_{ik})}\right]\le {\displaystyle \frac{n_b-n_{\mathrm{ref}}}{n_b-n_{\mathrm{ref}}+1}}.ik$$

(46)

$$f_{ij}\ge 0,\hspace{1em}\forall (i,j)\in \mathcal{E}.$$

(47)

This set of constraints ensures that the reconfigured distribution network maintains a radial (loop-free) structure while also adhering to exogenous availability limitations. Specifically, $z_{ij}$ represents the exogenous availability of a branch, where a value of 1 indicates that the branch is operational and 0 indicates that it is damaged or out of service, while $f_{ij}$ is an auxiliary flow variable representing network connectivity. $c_{ik}$ denotes the switch status of the line between nodes $i$ and $k$, where 1 means the line is closed and 0 means it is open. $f_{ik}$ is an auxiliary virtual flow variable used to propagate connectivity and enforce radiality, rather than representing actual power flow. The parameters $n_b$ and $n_{\mathrm{ref}}$ denote the total number of buses and the number of energized nodes, respectively, and their ratio ($n_b-n_{\mathrm{ref}}+1$) defines the maximum allowable number of operational branches in a radial structure. The constraints guarantee that only branches marked as available can be closed, that the total number of closed branches does not exceed to prevent loop formation, and that the conservation of flow through $f_{ij}$ enforces a directed spanning-tree structure extending from the power sources to the load nodes. Together, these conditions ensure that the post-reconfiguration network remains both connected and operational, enabling reliable power delivery from source nodes to all energized loads.

Although the restoration model in this work accounts for resource mobility and reachability constraints, the transportation system is currently simplified and does not reflect dynamic road accessibility or routing constraints during disasters. As highlighted in recent studies, full collaboration between transportation and power systems enables more realistic deployment strategies for mobile resources. Therefore, future extensions of this work will incorporate joint transportation–power network optimization, road failure modeling, and interdependent recovery coordination mechanisms.

5. Case Study Analysis

The modified IEEE 33-bus system was adopted for simulation analysis, incorporating photovoltaic (PV) and LNG distributed resource nodes as well as critical users such as quay cranes and yard cranes. In the pre-disaster model, three control groups were established to compare the effects of mobile energy storage (MES) deployment and the use of robust optimization. When MES was not considered, the storage units were by default assigned to node 1. For PV output uncertainty, the downward deviation coefficient $q_i$ is bounded by $0\le q_i\le 0.3$. The uncertainty budget is set to $e=0.6$, which implies that at most two out of the five PV units can simultaneously experience their maximum output reduction. This setting captures the spatially correlated but non-universal degradation of PV generation under disaster conditions. For the IEEE 33-bus test system, the robust subproblem contains approximately 680 linear and quadratic constraints, together with 32 second-order cone (SOC) constraints arising from the branch flow model. The resulting problem is formulated as a mixed-integer second-order cone program (MISOCP) and solved using IBM ILOG CPLEX. On a standard workstation, the average wall-clock time for one C&CG run is approximately 20–60 s, depending on the convergence tolerance and the number of C&CG iterations. The results of the four experimental cases are shown in Figure 2, Figure 3, Figure 4 and Figure 5.

Table 1. Critical Port Loads in the Port Distribution Network.

Bus Index	Load Description	Rated Active Power (MW)	Load Type	Load-Shedding Penalty
3	Container terminal control system	0.42	Critical	10
4	Refrigerated container area	0.35	Critical	10
6	Quay crane system	0.58	Critical	10
10	Port communication center	0.21	Critical	10
11	Emergency lighting and safety systems	0.18	Critical	10
15	Port traffic management system	0.26	Critical	10
17	Customs and inspection facilities	0.31	Critical	10
19	Fuel pumping and auxiliary services	0.29	Critical	10
24	Cold-chain logistics warehouse	0.47	Critical	10
26	Data center and control room	0.22	Critical	10
28	Security and monitoring system	0.19	Critical	10
33	Emergency response facilities	0.25	Critical	10

presents the Critical Port Loads in the Port Distribution Network.

The results indicate that, under deterministic optimization with MES deployment, the system primarily focuses on economic performance and energy balance under average scenarios. In this case, the two MES units are concentrated near the main supply nodes, effectively reducing power flow fluctuations and PV curtailment. However, when feeder faults or generation deviations occur, the power supply to critical loads can be compromised. Without MES deployment, the system relies entirely on fixed generation and network connectivity, making it prone to power interruptions at key loads under extreme events. In contrast, the two-stage robust optimization model determines MES locations in the first stage and performs corrections for the worst-case scenarios in the second stage, resulting in a more defensive and decentralized deployment pattern. The two MES units are placed near load-dense regions and potential network cut points, forming multiple local microgrid units with islanding capability. Even when part of the network is damaged, these microgrids can continue supplying power to critical loads. When MES is not deployed, the robust model still maintains feasibility under worst-case disturbances through more conservative power flow and reserve constraints, though at the expense of reduced economic efficiency. Overall, the two-stage robust optimization demonstrates significant advantages in pre-disaster planning for risk mitigation and resilience enhancement, while the integration of mobile energy storage further strengthens system flexibility and black-start capability, providing reliable support for continuous power supply to critical loads during disaster scenarios.

Table 2. Comparative Experiment.

Scenario	Mobile Energy Storage Pre-Configuration Nodes	Pre-Layout Cost/Yuan
Two-stage Robust Optimization with Mobile Energy Storage Pre-configuration Considered	4.23	5085.8454
Two-stage Robust Optimization without Mobile Energy Storage Pre-configuration Considered	1	8085.7593
Deterministic Optimization with Mobile Energy Storage Pre-configuration Considered	5.25	4666.23939
Deterministic Optimization without Mobile Energy Storage Pre-configuration Considered	1	7665.8638

presents the pre-deployment costs of mobile energy storage for the four experimental cases. Without MES pre-deployment, although the system avoids the investment cost of MES location planning, it incurs higher load-shedding costs, resulting in higher overall costs. In comparison, the two-stage robust optimization model, which accounts for the worst-case scenarios, produces a more conservative yet resilient strategy with higher pre-deployment costs than the deterministic optimization model.

Table 3. Cost Breakdown Comparison.

Optimization Strategy	CAPEX (Yuan)	OPEX (Yuan)	Risk Cost (Yuan)	Total Cost (Yuan)
Two-stage robust optimization with MES pre-configuration	1000	4085.85	0	5085.85
Two-stage robust optimization without MES pre-configuration	0	3085.76	5000.0	8085.76
Deterministic optimization with MES pre-configuration	1000	2666.24	1000.0	4666.24
Deterministic optimization without MES pre-configuration	0	2665.86	5000.0	7665.86

presents a breakdown of the total system cost into capital expenditure (CAPEX), operating expenditure (OPEX), and risk-related cost (load shedding penalty) for different optimization strategies.

Table 4. Comparison Between the Proposed Method and the No-MES Baseline.

Metric	Proposed Method (with MES)	No-MES Baseline
Total load shedding (MWh)	3.8	5.4
Critical load supply ratio (%)	96.2	82.7
Maximum single-node load shedding (MW)	0.62	1.15
Number of switching operations	14	9

compares the proposed method with a baseline strategy in which mobile energy storage systems are not deployed. Under the same disaster scenario, the proposed approach significantly reduces total load shedding and improves the supply ratio of critical port loads. Although a slightly higher number of switching operations is observed, the results demonstrate that the enhanced operational flexibility provided by MES is effective in improving system resilience during disasters.

During the disaster, all distributed resources are jointly optimized and dispatched to minimize load losses while ensuring priority power supply for critical users. In this stage, the state of charge (SOC) and node connection locations of the mobile energy storage systems (MESS) evolve dynamically, as shown in Figure 6. In the proposed framework, EVs are treated as controllable energy resources that can provide charging and discharging services during disaster conditions. Transportation demand, driver preferences, and individual travel behaviors of EV users are not explicitly modeled. This simplification allows the study to focus on the power-system resilience benefits enabled by EV flexibility, but it also limits the direct applicability of the results to real-world scenarios with strong coupling between transportation and energy systems.

From the results, the following observations can be made:

• Time 1–2: Both MESS units remain at high charge levels (around 100%) and are connected to different nodes, indicating that immediately after the disaster, they were rapidly deployed to critical load points to provide emergency power support.
• Time 3–6: The SOC levels decrease significantly, especially for MESS2 (red line), whose energy drops rapidly from about 90% to 50%, suggesting continuous discharging to sustain local load supply. MESS1 also experiences a decline, though to a lesser extent.
• Time 7–8: Both MESS units show further SOC reduction along with changes in connection nodes (abrupt node index shifts), indicating that they have relocated to new affected areas during the mid-disaster period to support newly impacted zones.
• Time 9–10: A rise in SOC is observed, marking the charging phase, which reflects that after partial grid restoration, the mobile storage units begin recharging to recover their energy capacity.
• Time 11: Both MESS units stabilize at medium charge levels and are connected to different nodes, showing that the system has largely recovered and the storage units have transitioned into standby or local stabilization modes.

This temporal–spatial evolution reflects the coordinated strategy of the mobile energy storage system during the disaster—“discharge first for support, then relocate, and finally recharge”. It demonstrates the flexible dispatch and strong support capability of MESS in maintaining reliable power supply under extreme conditions.

In addition to mobile energy storage, electric vehicles (EVs) and distributed generation units also participate in collaborative power supply during the disaster stage, further enhancing the reliability and resilience of the overall system.

From Figure 7 and Figure 8, it can be observed that the three electric vehicles (EVs) initially discharge at maximum power immediately after the disaster to support the grid, then gradually reduce their output and either switch to standby mode or relocate to other nodes for additional support, eventually exiting the scheduling process entirely in the later stage. This reflects the dynamic participation characteristics of EVs during disaster-time optimal scheduling: in the early stage, they concentrate on maximum-power discharge to sustain loads; in the middle stage, they flexibly adjust their power output according to system conditions; and in the later stage, they are dispatched adaptively based on the progress of system recovery. This behavior highlights the role of EVs as flexible energy resources in multi-source coordinated dispatch during disasters. Meanwhile, distributed generation (DG) units serve as stable sources of fundamental power supply. Located at load-dense or critical nodes, they ensure basic voltage stability and reliable power delivery to key users. Unlike EVs, distributed generators cannot be relocated or dispatched flexibly, and therefore maintain a constant power output. Their steady operation provides a foundational power backbone that supports flexible resources such as EVs and mobile energy storage systems, enabling coordinated, resilient operation of the disaster-affected grid.

After the disaster caused partial damage to the distribution network, service restoration and network reconfiguration optimization were carried out. The core objective of this stage is to maximize the recoverable load and minimize unserved energy under conditions where some lines, nodes, or distributed generation sources are out of service. By coordinating switching operations and power supply-load balancing, the network seeks to restore as much service as possible. Figure 9 illustrates the post-disaster reconfiguration results.

In Figure 9, different colors, symbols, numbers, lines, and dashed lines are used to represent various states and connections in the power grid. Red represents lines that are faulty or disconnected. Blue indicates nodes or lines that are operating normally but are not critical loads. These are secondary loads or parts that are not prioritized for recovery. Green is typically used to represent activated devices or restored connections, such as grid sections that have been restored. Black represents regular connection lines in the grid that are functioning normally and are not experiencing any faults. Dashed lines indicate backup connections, playing a supportive role in the grid’s topology adjustments or recovery process.

It can be seen that after several lines were disconnected due to damage, the system reselected and closed appropriate tie-lines through optimization, forming new radial power-supply paths. This process enabled rapid network reconfiguration and load restoration. The optimized topology significantly enhanced system connectivity, allowing more critical nodes to be re-energized. Consequently, several local self-supply zones and main supply areas were established, operating in a coordinated manner. These results verify that the proposed post-disaster restoration model effectively improves both the power restoration capability and the resilience of the distribution network under disaster scenarios.

To intuitively evaluate the power supply recovery effect of the post-disaster reconfiguration optimization model, this paper plots a comparison diagram of node active load recovery, which is shown Figure 10.

Figure 10 illustrates the comparison of active loads across all bus nodes during the post-disaster service restoration stage. The blue curve represents the actual supplied load of nodes after optimization, while the red curve denotes the original pre-disaster load demand. It can be observed that the load of some nodes has achieved full recovery, whereas certain nodes still suffer from a certain degree of load curtailment or even a complete lack of power supply. On the whole, following the post-disaster reconfiguration and optimization, the power system has realized power recovery for most critical nodes under damaged conditions. Particularly, the power supply capacity has been significantly enhanced around nodes numbered 20–25. This indicates that under the constraints of limited resources and damaged topology in the post-disaster scenario, the optimization model can effectively reconfigure power supply paths and prioritize the satisfaction of critical load demands, thereby improving the power supply recovery rate and operational resilience of the distribution network.

It is equally crucial to assess the rationality of power flow distribution and operational safety of the network under the post-disaster reconfiguration scheme. This paper conducts a statistical analysis on the load rates of closed branches after reconfiguration, as shown in Figure 9.

As can be seen from Figure 11, the load rates of most branches are lower than the rated value, indicating that the overall operation of the system lies within a safe range after post-disaster reconfiguration. Only a few individual branches have load rates close to or slightly exceeding the rated level, which demonstrates that these lines assume a heavy responsibility in power transmission during the power supply restoration process. On the whole, the post-disaster reconfiguration and optimization have achieved effective power flow distribution and load balancing. While ensuring the power supply recovery rate, the system maintains a favorable safety margin, which verifies that the proposed model has achieved a sound balance between power supply reliability and operational safety.

This paper also conducts a statistical analysis on the node power supply and load curtailment structure, as shown in Figure 12 and Figure 13. Figure 12 is a stacked bar chart of power supply and load curtailment for each bus node, while Figure 13 is a pie chart illustrating the power supply ratio at the system level. It can be seen from the results that most nodes have achieved partial or full load recovery, yet some nodes still undergo load curtailment due to branch damage or voltage constraints. The overall power supply ratio of the system reaches 54.6%, which indicates that despite network damage caused by the disaster, the optimization model can still realize the recovery of more than half of the total load. This result verifies that the proposed post-disaster service restoration optimization model can effectively coordinate the network structure and power source output under the conditions of limited resources and damaged topology, thereby significantly improving the system’s power supply recovery capability and resilience performance.

6. Conclusions and Outlook

This study proposes an integrated three-stage resilience optimization framework for port distribution systems under extreme weather events, covering pre-disaster planning, in-disaster coordinated operation, and post-disaster network restoration. Comprehensive case studies based on a modified IEEE 33-bus port system demonstrate that the proposed method can effectively reduce load shedding, enhance the supply guarantee for critical loads, and provide efficient decision support under various uncertainties. The main contributions are summarized as follows:

(1)

A unified three-stage resilience optimization framework for port power systems is proposed.

The framework integrates pre-disaster robust planning, in-disaster multi-source coordination, and post-disaster network reconfiguration, enabling joint optimization of resource allocation, energy flow, and network topology adjustments—addressing the limitations of existing studies that treat these stages separately.

(2)

A robust pre-disaster mobile energy storage allocation model is developed.

By considering uncertainties in PV generation, loads, and potential fault conditions, a two-stage robust optimization model determines the spatial deployment of mobile storage units that remain effective under worst-case scenarios, improving the system’s disturbance resistance.

(3)

A unified power-flow and topology-constrained model for in-disaster operation and post-disaster restoration is formulated.

The model coordinates mobile energy storage relocation, electric vehicles, PV systems, and distributed generators to maximize system supply and ensure safe operation, while a reachability-based radial reconfiguration method improves critical-load restoration rates. This work extends existing resilience-oriented planning studies by explicitly addressing port-specific operational characteristics

Although the proposed integrated framework has demonstrated its effectiveness in representative port scenarios, several limitations remain, including restricted disaster types, simplified device models, and the gap between theoretical optimization and real-world implementation. Future work may focus on enriched uncertainty modeling, multi-energy integration, and intelligent deployment to enhance robustness and practical applicability. Possible directions for future research include:

(1)

Enhanced modeling of multi-hazard uncertainties and component failures.

Future studies may incorporate compound disaster scenarios—such as storm surges coupled with extreme heat—as well as detailed aging, maintenance, and repair processes to enable lifecycle resilience assessment.

(2)

Integration of multi-energy coupling in port energy systems.

Shore power systems, natural gas, hydrogen, and thermal networks can be included to build a comprehensive port energy hub, enabling joint optimization of energy cost, carbon emissions, and resilience metrics under a unified framework.

(3)

Intelligent and digital implementation for real-world applications.

Real-time measurements, digital twins, reinforcement learning, and multi-agent coordination can be integrated to develop an online adaptive decision-making system; in addition, communication and transportation constraints can be considered to form a deployable visualization platform for port operators.