Collaborative Scheduling Framework for Post-Disaster Restoration: Integrating Electric Vehicles and Traffic Dynamics in Waterlogging Scenarios

Abstract

Frequent and severe waterlogging caused by climate change poses significant challenges to urban infrastructure systems, particularly transportation networks (TNs) and distribution networks (DNs), necessitating efficient restoration strategies. This study proposes a collaborative scheduling framework for post-disaster restoration in waterlogging scenarios, addressing the impact of waterlogging on both transportation and distribution systems. The method integrates electric vehicles (EVs), mobile power sources (MPSs), and repair crews (RCs) into a unified optimization model, leveraging an improved semi-dynamic traffic assignment (SDTA) model that accounts for temporal variations in road accessibility due to water depth. Simulation results based on the modified IEEE 33-node distribution network and SiouxFalls 35-node transportation network demonstrate the framework’s ability to optimize resource allocation under real-world conditions. Compared to conventional methods, the proposed approach reduces system load loss by more than 30%.

1. Introduction

Frequent waterlogging disasters have caused severe equipment damage and service interruptions in distribution networks recently [1,2,3]. At the same time, the increasing adoption of EVs has introduced the possibility of enhancing the resilience of distribution networks through emergency power support via EV grid connection (vehicle to grid, V2G) [4]. Both EVs and MPSs (such as mobile generators and mobile energy storage) offer flexibility in power supply in terms of both space and time. They can reach areas beyond the service range of traditional distributed generators (DGs), based on disaster scenarios and load demands [5,6]. Meanwhile, RCs can restore the functionality of damaged nodes and lines [7,8,9]. Additionally, the operational optimization of DGs [10,11] and microgrid formation [12,13] can achieve fault isolation and emergency power support. Recent studies on multi-energy system restoration [14] and microgrid frequency regulation [15] have advanced resource coordination strategies. Considering the spatiotemporal coordination between different flexible resources and realizing unified coordination and allocation of resources is of great significance for fully utilizing limited restoration resources and improving post-disaster restoration efficiency.

Unlike natural disasters such as typhoons and heatwaves, waterlogging-induced road water accumulation and traffic congestion severely disrupt the dispatch processes of these mobile resources [16]. EVs, serving as a significant mobile power supplement to distribution networks, belong to user-side resources. The presence of secure and reasonable dispatch instructions will directly influence users’ decisions regarding participation in restoration dispatch operations. Consequently, comprehensive investigations into waterlogging evolution patterns and road traffic load distributions, coupled with establishing post-disaster transportation network operation models, constitute critical foundations for ensuring the rationality of multi-resource coordinated dispatch strategies. Recent advancements in power system post-disaster restoration have increasingly focused on coupled optimization of transportation and power systems. Reference [17] investigated the dynamic interactions between transportation networks and power grids in terms of traffic congestion and load distributions, proposing a joint optimization model based on convex relaxation and optimization-based bound tightening to optimize EV charging routes and grid operations. However, this model treats EV origin–destination pairs as predefined inputs, limiting its applicability to post-disaster dispatch scenarios. Reference [18] addressed post-disaster road damage constraints on load restoration through a hierarchical dispatch strategy integrating EV routing, road repair sequencing, and load restoration, effectively improving critical load supply duration and restoration efficiency. Nevertheless, this approach overlooks multi-resource coordination mechanisms. Adopting a model predictive control framework, Reference [19] incorporated real-time traffic flow data into post-disaster dispatch models to coordinate mobile power sources, energy storage systems, and repair crews. However, the rapid temporal variations in traffic flows render this model incapable of predictive traffic information modeling throughout the dispatch horizon, compromising scheduling strategy stability. Reference [20] proposed a multi-rate co-optimization strategy addressing dynamic traffic flow impacts on EV charging/discharging, employing rolling horizon optimization to coordinate EV dispatch with renewable energy integration. This approach still suffers from fixed origin–destination assumptions in EV routing. Reference [21] systematically analyzed the combined impacts of renewable energy (wind/solar) generation uncertainties, EV charging demand stochasticity, and transportation network constraints on power restoration. By employing Wasserstein generative adversarial networks with gradient penalty for renewable output simulation within a multi-objective optimization framework, the study achieved coordinated dispatch of renewables, mobile resources, and repair crews. However, its direct incorporation of time-varying road congestion coefficients as parameters fails to account for disaster-induced congestion mechanisms. Reference [14] developed a two-stage stochastic programming framework with progress hedging algorithm for multi-energy system restoration. A linearized thermal network via auxiliary flow variables reduced computational burden under multi-source uncertainties. Reference [15] proposed a hierarchical model predictive control (MPC) with event-triggered scheme for microgrid frequency regulation under phasor measurement unit (PMU) faults and communication intermittency. Extended load frequency control (LFC) via uncertain matrix improved robustness. These studies provide theoretical guidance for expanding the diversity of flexible resources and enhancing frequency stability during post-disaster recovery processes [14,15].

In summary, existing post-disaster restoration strategies for power distribution systems primarily focus on line fault scenarios caused by typhoon disasters, with limited research addressing the impacts of waterlogging disasters on transportation networks and traffic flow distribution. Current traffic network operation models mainly focus on EV scheduling under normal power system operations, assuming fixed origin–destination pairs and requiring only path selection decisions, which fail to meet the flexible scheduling needs of EVs in post-disaster scenarios. Although existing semi-dynamic traffic assignment (SDTA) models assume the existence of an independent system operator to achieve socially optimal traffic flow distribution (Wardrop system optimization), such idealized operational modes are difficult to realize in practice. To address these gaps, this paper proposes a post-disaster coordinated scheduling method for electric vehicles and flexible power distribution resources under waterlogging disasters, employing an improved SDTA model for power–transportation coupled system modeling. To address the limitation of existing models that require predefined EV origin–destination pairs as inputs, this paper formulates EV scheduling as a vehicle routing problem and integrates it as a complementary solution to dispatchable MPSs. The main contributions are as follows:

• Incorporating time-varying road waterlogging and congestion impacts into flexible resource scheduling decisions for waterlogging disasters, ensuring the rationality of scheduling strategies.
• Establishing a coordinated multi-resource scheduling model integrating dispatchable EVs, MPSs, and RCs to achieve synergistic collaboration and full utilization of various resources.
• Enhancing the SDTA model by considering waterlogging impacts on road networks and implementing time-segmented solutions to ensure traffic flow in each period remains unaffected by subsequent road conditions, aligning with real-world scenarios.

The remainder of this paper is organized as follows. Section 2 describes the modeling framework, including the improved semi-dynamic traffic network model under waterlogging conditions, power grid cmponent failure model, the collaborative scheduling models for EVs, MPS, and RCs and presents the proposed optimization methodology and the integration of traffic and power network constraints. Section 3 validates the effectiveness of the proposed approach through simulations using the modified IEEE 33-node distribution network coupled with SiouxFalls 35-node transportation network, analyzing the restoration process and comparing it with alternative strategies. Finally, Section 4 concludes the paper with key findings, highlights the contributions, and suggests directions for future work.

2. Materials and Methods

This chapter develops a post-disaster scheduling model for EVs, MPSs, and RCs under waterlogging disasters. By constructing a semi-dynamic traffic network operation model that accounts for waterlogging impacts on road conditions and vehicle routing decisions, it provides data support for mobile resource scheduling strategies. Through the coupling of EV scheduling, MPS dispatching, RC allocation, and distribution system operation models, this approach achieves optimal resource allocation and coordinated operations, ultimately enhancing the overall efficiency and effectiveness of distribution systems in responding to waterlogging disasters.

2.1. Semi-Dynamic Traffic Assignment Model for Waterlogging Scenarios

We model the traffic network as a directed graph composed of traffic nodes and roads as $G_T=\left(V_T,A\right)$, in which $G_T$ represents the traffic network model, $V_T$ represents the set of traffic network nodes, and $A$ represents the set of roads, with roads denoted by $a\in A$.

Reference [22] introduced a vehicle speed model under waterlogged road conditions:

$$v_{\mathrm{T}}={\displaystyle \frac{v_{0,\mathrm{T}}}{2}}\tanh \left({\displaystyle \frac{\alpha -h}{\beta }}\right)+{\displaystyle \frac{v_{0,\mathrm{T}}}{2}}$$

(1)

In the equation, $v_{\mathrm{T}}$ represents the vehicle speed under waterlogged conditions, $v_{0,\mathrm{T}}$ denotes the road’s designed speed, $h$ is the water depth, $\alpha$ represents half of the critical water depth (beyond which vehicles are prohibited from passing), and $\beta$ is the attenuation coefficient.

Reference [23] established a simplified relationship between road travel time and congestion levels, widely adopted in the literature:

$$t_a=t_a^0\left[1+0.15{\left({\displaystyle \frac{x_a}{c_a}}\right)}^4\right]$$

(2)

In the equation, $t_a$ and $t_a^0$ denote the travel time under congestion and free-flow conditions for road a, respectively, $x_a$ is the traffic flow on road a, and $c_a$ represents the road capacity.

Let $x_{a,t}$ denote the traffic flow on road a at step t, where t∈T, and $T=\{1,\dots ,T^{\mathrm{total}}\}$ is the set of time periods. Let $h_{a,t}$ represent the accumulated water depth on road a at time step t. By synthesizing Equations (1) and (2), the driving time and speed on road a are formulated as [22,23]:

$$t_{a,t}=2t_a^0\left[1+0.15{\left({\displaystyle \frac{x_{a,t}}{c_a}}\right)}^4\right]{\left[1+\tanh \left({\displaystyle \frac{\alpha -h_{a,t}}{\beta }}\right)\right]}^{-1},\forall a\in A,\forall t\in T$$

(3)

In the equation, the decision variable $t_{a,t}$ represents the travel time on road a during time period t. The parameters are defined as follows:

$t_a^0$: The free-flow travel time on road a, which represents the time to travel without congestion or waterlogging.

$c_a$: The capacity of road a.

$h_{a,t}$: The accumulated water depth on road a during time period t. It is assumed that, without considering municipal drainage, the natural drainage depth of water and time satisfy the equation $h_{a,t}=h_a^0·e^{-{\theta }_{w}t}$, and in this paper, ${\theta }_w=0.01$.

$\alpha$: Half of the critical water depth, beyond which vehicles are prohibited from passing. In this paper, it is taken as a specific value, say $\alpha =15\mathrm{cm}$.

$\beta$: The attenuation coefficient, taken as a specific value in this model, say $\beta =4$.

Equation (3) represents a variant of the Bureau of Public Roads (BPR) function [24] under road waterlogging conditions, which is used to describe the impact of traffic congestion and waterlogging on road travel time.

The SDTA model adopted in this paper is as follows:

$$x_{a,t}={\displaystyle \sum _{w\in W}{\displaystyle \sum _{k\in K_w}{f}_{w,k,t}{\chi }_{a,w,k}}},\forall a$$

(4)

$${\displaystyle \sum _{k\in K_w}{f}_{w,k,t}}=f_{w,t}^{\mathrm{mod}},\forall w\in W,\forall t$$

(5)

$$0\le f_{w,k,t}\perp (t_{w,k,t}-{\tau }_w)\ge 0$$

(6)

In the equation, the following definitions are provided:

$W$: The set of origin–destination points;

$K_w$: The set of paths between origin–destination point w;

$f_{w,k,t}$: The traffic flow on path k between origin–destination point w during time step t;

${\chi }_{a,w,k}$: A 0–1 variable. ${\chi }_{a,w,k}=1$ indicates road a is part of path k between origin-destination point w.

Equations (4) and (5) define the decision variables $x_{a,t}$ and $f_{w,k,t}$, respectively. Equation (6) represents the Wardrop User Equilibrium (UE) in its complementarity relaxation form, which states that the travel time on all paths used between the origin–destination points is equal and corresponds to the shortest travel time. In other words, users will choose the path that minimizes their travel time. Equations (3)–(6) represent the Wardrop UE problem, which can be solved by transforming it into the optimality conditions of a convex traffic assignment problem [25]:

$$\min {\displaystyle \sum _{a\in A}{\displaystyle {\int }_0^{x_{a,t}}{t}_{a,t}(x)dx}}$$

(7)

$$\mathrm{s}.\mathrm{t}.(3)–(5)$$

For handling the strong nonlinearity in the objective function and the BPR function, several methods have been proposed in the literature, including piecewise linearization [17,26], McCormick Envelope [17,27], binary expansion [28], and convex hull relaxation [17]. In this paper, we adopt the convex hull relaxation method as described in Ref. [17]. The optimization problem is solved for each time step. Let the initial travel demand for origin–destination point w during time period t be denoted as $f_{w,t}^0$. For the first time step, the actual travel demand is equal to the initial travel demand, i.e., $f_{w,1}^{\mathrm{mod}}=f_{w,1}^0$. For subsequent time steps, the actual travel demand is calculated using the following equation:

$$t_{w,k,t}={\displaystyle \sum _{a\in A}{t}_{a,t}·{\chi }_{a,w,k}},\forall k\in K_w,\forall w$$

(8)

$$r_{w,t}={\displaystyle \sum _{k\in K_w}\frac{f_{w,k,t}{t}_{w,k,t}}{\Delta t}},\forall w,\forall t$$

(9)

$$f_{w,t}^{\mathrm{mod}}=f_{w,t}^0+{\displaystyle \frac{1}{2}}{r}_{w,t-1},\forall t\in T/\left\{1\right\},\forall w$$

(10)

In the equation:

$t_{w,k,t}$: The travel time on path k between origin–destination point w during time step t;

$r_{w,t}$: The remaining traffic flow at time step t, which is the number of vehicles that have not yet reached their destination by the end of time period t. This is assuming that the departure times of vehicles between origin–destination point w during time step t are uniformly distributed;

$\Delta t$: The time step length.

The optimization problem is solved to obtain the road traffic situation for different time periods under waterlogging disaster. Since the number of dispatchable resources in the power grid is much smaller than the number of private vehicles, and these resources can be directed to avoid congested routes through scheduling instructions, the impact of dispatchable grid resources on road congestion is neglected in this paper.

2.2. Power Grid Component Failure Model

Waterlogging disasters and their associated hazards, such as typhoons, are primary causes of failures in distribution network lines and nodes. Typhoons, particularly in coastal regions, often accompany waterlogging disasters, as heavy rainfall induced by typhoons is a common trigger for waterlogging. This section establishes a failure rate model for distribution network components by considering the impacts of waterlogging and typhoons.

The effects of waterlogging and typhoons on distribution network lines primarily manifest in the mechanical stress exerted by typhoon winds on conductors and towers, leading to damage. The failure rates of conductors and towers can be modeled as log-normal distributions dependent on typhoon wind speed [29]:

$$P_{\mathrm{line}}(v)=\left\{\begin{array}{l}1,v\ge v_{\mathrm{line},2}\\ \Phi \left({\displaystyle \frac{1}{{\beta }_1}}\ln ({\displaystyle \frac{v}{\overline{v_1}}})\right),v_{\mathrm{line},1}\le v<v_{\mathrm{line},2}\\ P_{\mathrm{line},\min },v<v_{\mathrm{line},1}\end{array}\right.$$

(11)

$$P_{\mathrm{tow}}(v)=\left\{\begin{array}{l}1,v\ge v_{\mathrm{tow},2}\\ \Phi \left({\displaystyle \frac{1}{{\beta }_2}}\ln ({\displaystyle \frac{v}{\overline{v_2}}})\right),v_{\mathrm{tow},1}\le v<v_{\mathrm{tow},2}\\ 0,v<v_{\mathrm{tow},1}\end{array}\right.$$

(12)

In the equation:

$P_{\mathrm{line}}$ and $P_{\mathrm{tow}}$: Failure probabilities of the conductor and tower, respectively;

$v$: Typhoon wind speed;

$P_{\mathrm{line},\min }$: Baseline failure probability of the conductor under normal operating conditions;

$\Phi$: Standard normal distribution function;

$v_{\mathrm{line},1}$ and $v_{\mathrm{tow},1}$: The typhoon wind speed at which the failure rate of the conductor or tower sharply increases;

$v_{\mathrm{line},2}$ and $v_{\mathrm{tow},2}$: The maximum typhoon wind speed that the conductor or tower can withstand;

$\overline{v_1}$ and $\overline{v_2}$: Mean values of the natural logarithm of wind speeds causing abrupt increases in conductor and tower failure rates, derived from historical disaster data;

${\beta }_1$ and ${\beta }_2$: Standard deviations of the natural logarithm of wind speeds causing conductor and tower failures, obtained from statistical analyses.

For an overhead line to operate normally, both its conductor and all supporting towers must remain functional:

$$P_{\mathrm{L},(i,j)}=1-(1-P_{\mathrm{line},(i,j)}){\displaystyle \prod _{k=1}^{m_{(i,j)}}(1-}{P}_{\mathrm{tow},(i,j),k}),\forall (i,j)\in L$$

(13)

In the equation:

L: The set of lines in the distribution network;

$P_{\mathrm{L},(i,j)}$: Failure probability of line $(i,j)$;

$P_{\mathrm{line},(i,j)}$: Conductor failure probability of line $(i,j)$;

$P_{\mathrm{tow},(i,j),k}$: Failure probability of the k-th tower on line $(i,j)$;

$m_{(i,j)}$: Number of towers installed on line $(i,j)$.

Waterlogging and typhoons also impact distribution network nodes, such as substations and distribution rooms, where equipment like transformers, circuit breakers, and switchgear may fail due to submersion. The failure probability of a node is related to its water depth and can be expressed as [29]:

$$P_{\mathrm{N}}(d)=\left\{\begin{array}{l}1,d\ge d_{\mathrm{N},2}\\ \Phi \left({\displaystyle \frac{1}{{\beta }_3}}\ln ({\displaystyle \frac{d}{\overline{d}}})\right),d_{\mathrm{N},1}\le d<d_{\mathrm{N},2}\\ 0,d<d_{\mathrm{N},1}\end{array}\right.$$

(14)

In the equation:

$P_{\mathrm{N}}$: Failure probability of the node;

$d$: Water depth at the node;

$d_{\mathrm{N},1}$: The water depth at which the failure rate of the node sharply increases;

$d_{\mathrm{N},2}$: The water depth at which the probability of the node remaining functional becomes negligible;

$\overline{d}$: Mean value of the natural logarithm of critical water depths causing node failures, derived from experimental or historical data;

${\beta }_3$: Standard deviation of the natural logarithm of water depths causing node failure.

Analyzing component failure rates under waterlogging disasters is essential for generating realistic post-disaster scenarios and optimizing restoration strategies. In the case study of this paper, fault scenarios were generated according to the aforementioned models. Since fault scenarios are treated as known parameters in post-disaster restoration problems, this paper does not elaborate on the process of generating fault scenarios via Monte Carlo simulation.

2.3. EV Scheduling Model

When a heavy rain and waterlogging disaster is predicted to occur, electric vehicles (EVs) are notified through broadcasts and other means to perform emergency evacuation scheduling. The EVs that are already on the road are guided to nearby evacuation stations. After the disaster, price incentives are used to encourage EVs at evacuation stations to participate in post-disaster emergency restoration. The proportion of EVs responding to emergency evacuation and participating in post-disaster restoration is influenced by the incentive price and the user’s perception threshold for the incentives [30]. In the post-disaster scheduling phase, the number of EVs participating in the restoration scheduling and their initial state are considered known.

EV scheduling must meet spatiotemporal constraints, quantity constraints, and operational (capacity) constraints:

$$0\le {\displaystyle \sum _{n\in N_{\mathrm{V}2\mathrm{G}}}{K}_{n,e,t}}\le 1,\forall e\in N_{\mathrm{EV}},\forall t$$

(15)

$${\displaystyle \sum _{t=1}^{T_{n,e}^{\mathrm{Tra}}}{K}_{n,e,t}}=0,\forall n\in N_{\mathrm{V}2\mathrm{G}},\forall e$$

(16)

$$T_{n,e}^{\mathrm{Tra}}=\min \left\{t+t_{n,e,t}^{\min }|t\in T\right\},\forall n,\forall e$$

(17)

$${\displaystyle \sum _{t=T_{n,e}^{\mathrm{Tra}}+1}^{T^{\mathrm{total}}}{K}_{n,e,t}}=\left(T^{\mathrm{total}}-T_{n,e}^{\mathrm{Tra}}\right)K_{n,e,t},\forall t\in \left\{T_{n,e}^{\mathrm{Tra}}+1,\dots ,T^{\mathrm{total}}\right\},\forall n,\forall e$$

(18)

$$0\le {\displaystyle \sum _{e\in N_{\mathrm{EV}}}{K}_{n,e,t}}\le N_{n,\mathrm{V}2\mathrm{G}},\forall n,\forall t$$

(19)

$$0\le p_{n,e,t}\le K_{n,e,t}·P_e^{\max },\forall n,\forall e,\forall t$$

(20)

$$0\le q_{n,e,t}\le K_{n,e,t}·Q_e^{\max },\forall n,\forall e,\forall t$$

(21)

$$0\le {\displaystyle \sum _{t\in T}{p}_{n,e,t}\Delta t}\le S_e^{\mathrm{ini}}-S_{n,e},\forall n,\forall e$$

(22)

In the equation:

$N_{\mathrm{V}2\mathrm{G}}$ and $N_{\mathrm{EV}}$: The set of V2G stations and the set of EVs, respectively;

$K_{n,e,t}$: A 0–1 variable indicating whether the e-th EV arrives at the n-th V2G station at time period t. If the EV arrives, $K_{n,e,t}=1$; otherwise, $K_{n,e,t}=0$;

$T_{n,e}^{\mathrm{Tra}}$: The shortest time required for the e-th EV to travel from its initial parking location to the n-th V2G station;

$t_{n,e,t}^{\min }$: The shortest travel time from the initial parking location to the n-th V2G station at time step t, which can be calculated using the Dijkstra algorithm;

$N_{n,\mathrm{V}2\mathrm{G}}$: The maximum number of EVs that the n-th V2G station can accommodate;

$p_{n,e,t}$ and $q_{n,e,t}$: The active and reactive discharging power of the e-th EV at the n-th V2G station during time period t;

$P_e^{\max }$ and $Q_e^{\max }$: The maximum active and reactive discharging power for the e-th EV;

$S_e^{\mathrm{ini}}$: The initial remaining battery charge of the e-th EV;

$S_{n,e}$: The amount of energy consumed by the e-th EV to travel from its initial position to the n-th V2G station.

Equation (15) indicates that each EV cannot be at two stations simultaneously. Equation (16) represents the EV arrival time constraint. Equation (18) states that once an EV arrives at the target V2G station during the post-disaster scheduling process, it will no longer change locations. Equation (19) ensures that a V2G station cannot accommodate more EVs than its capacity. Equations (20) and (21) constrain the EVs to only discharge at the V2G stations they are connected to. Equation (22) represents the EV battery charge constraint.

2.4. MPS Dispatching Model

MPS, due to its spatial and temporal flexibility in power supply, can quickly provide power support for critical loads after a disaster. The controllable MPS interface power sources mainly include diesel generators, combined cooling, heating, and power systems, and energy storage units. Different types of MPSs have slightly different operational characteristics. This paper considers the MPS scheduling under time-varying road conditions and constructs the MPS model based on the operational characteristics of mobile energy storage:

$$0\le {\displaystyle \sum _{i\in B}{K}_{i,g,t}}\le 1,\forall g\in N_{\mathrm{MPS}},\forall t$$

(23)

$${\displaystyle \sum _{t=1}^{T_{i,g}^{\mathrm{Tra}}}{K}_{i,g,t}}=0,\forall i\in B,\forall g$$

(24)

$$T_{i,g}^{\mathrm{Tra}}=\min \left\{t+t_{i,g,t}^{\min }|t\in T\right\},\forall i,\forall g$$

(25)

$${\displaystyle \sum _{t=T_{i,g}^{\mathrm{Tra}}+1}^{T^{\mathrm{total}}}{K}_{i,g,t}}=\left(T^{\mathrm{total}}-T_{i,g}^{\mathrm{Tra}}\right)K_{i,g,t},\forall t\in \left\{T_{i,g}^{\mathrm{Tra}}+1,\dots ,T^{\mathrm{total}}\right\},\forall i,\forall g$$

(26)

$$0\le p_{i,g,t}\le K_{i,g,t}·P_g^{\max },\forall i,\forall g,\forall t$$

(27)

$$0\le q_{i,g,t}\le K_{i,g,t}·Q_g^{\max },\forall i,\forall g,\forall t$$

(28)

$$0\le {\displaystyle \sum _{t\in T}{p}_{i,g,t}\Delta t}\le {\eta }_g(1-{\sigma }_g)\left(E_g^{\mathrm{ini}}-E_g^{\min }\right),\forall i,\forall g$$

(29)

In the equation:

$B$ and $N_{\mathrm{MPS}}$: Set of grid nodes and set of MPS nodes;

$K_{i,g,t}$: A 0–1 variable indicating whether the g-th MPS arrives at node i during time period t;

$T_{i,g}^{\mathrm{Tra}}$: The shortest time required for the g-th MPS to travel from its initial deployment position to node i;

$t_{i,g,t}^{\min }$: The shortest travel time for the g-th MPS from its initial deployment position to node i during time period t;

$p_{i,g,t}$ and $q_{i,g,t}$: The active and reactive discharge power of the g-th MPS at node i during time period t;

$P_g^{\max }$ and $Q_g^{\max }$: The maximum active and reactive discharge power of the g-th MPS;

$E_g^{\mathrm{ini}}$ and $E_g^{\min }$: The initial energy and minimum allowable energy of the g-th MPS;

${\eta }_g$ and ${\sigma }_g$: The discharge efficiency and self-discharge rate of the g-th MPS.

Equations (23)–(26) represent the MPS scheduling constraints, which indicate the uniqueness of MPS arrival at nodes, the arrival time, and the condition that the MPS does not change position after connecting to the target node. Equations (27)–(29) are the MPS operational constraints, which represent the limitations on the MPS power and energy capacity.

2.5. RC Allocation Model

This paper considers multi-period, time-varying road conditions for multi-RC collaborative scheduling, which is modeled as a vehicle scheduling problem for multiple logistics centers.

$${\displaystyle \sum _{i_c\in N_{\mathrm{F}}}{x}_{f,i_c,t}}\le 1,\forall f\in F,\forall t$$

(30)

$$\left\{\begin{array}{c}{x}_{f,i_{\mathrm{SP}},1}=1\\ x_{f,i_{\mathrm{DP}},T^{\mathrm{total}}}=1\end{array}\right.,\forall f$$

(31)

$${\displaystyle \sum _{\tau =t+1}^{\min \left\{t+t_{i_c,j_c,t}^{\min },T^{\mathrm{total}}\right\}}{x}_{f,j_c,t}\le \left(1-x_{f,i_c,t}\right)}·\min \left\{t_{i_c,j_c,t}^{\min },T^{\mathrm{total}}-t\right\},\forall \left(i_c,j_c\right)\in W_{\mathrm{F}},\forall f,\forall t$$

(32)

$$h_{f,i_c,t}\le {\displaystyle \frac{{\displaystyle \sum _{t=1}^t{x}_{f,i_c,t}}}{T_{i_c}^{\mathrm{Rep}}}},\forall i_c\in N_{\mathrm{BC}},\forall f,\forall t$$

(33)

$$h_{f,i_c,t-1}\le h_{f,i_c,t},\forall i_c\in N_{\mathrm{BC}},\forall t\in T/\left\{1\right\},\forall f$$

(34)

$${\displaystyle \sum _{f\in F}{h}_{f,i_c,t}}\le 1,\forall i_c\in N_{\mathrm{BC}},\forall t$$

(35)

$${\rho }_{i_c,t}\le {\displaystyle \sum _{f\in F}{h}_{f,i_c,t}},\forall i_c\in N_{\mathrm{BC}},\forall t$$

(36)

In the equation:

F: The set of RCs;

$N_{\mathrm{F}}$ and $W_{\mathrm{F}}$: The sets of nodes that the RC can potentially reach, and the set of start and end points, $W_{\mathrm{F}}=\left\{\left(i_c,j_c\right)|\forall i_c\in N_{\mathrm{F}},\forall j_c\in N_{\mathrm{F}},i_c\ne j_c\right\}$;

$x_{f,i_c,t}$: A 0–1 variable, $x_{f,i_c,t}=1$ indicates that RC f is located at the node where the damaged component i_c is during time period t;

$i_{\mathrm{SP}}$ and $i_{\mathrm{DP}}$: The deployment and destination nodes of the RC;

$t_{i_c,j_c,t}^{\min }$: The shortest travel time from the node where the damaged component i_c located to the node where component j_c located during time period t;

$h_{f,i_c,t}$: A 0–1 variable, $h_{f,i_c,t}=1$ indicates that RC f has repaired the damaged component i_c during time period t;

$T_{i_c}^{\mathrm{Rep}}$: The time required to repair the damaged component i_c;

$N_{\mathrm{BC}}\subseteq N_{\mathrm{F}}$: The set of damaged components;

${\rho }_{i_c,t}$: a 0–1 variable, ${\rho }_{i_c,t}=1$ indicates that the damaged component i_c has been powered during time period t.

Equation (30) represents that each RC can only be at one location at any given time. Equation (31) represents that the RC, starting from the deployment node, must eventually dock at the destination node. Equation (32) indicates that if RC f is located at the node where the damaged component i_c is during time period t, it must pass through the node of component i_c after at least a certain amount of time $t_{i_c,j_c,t}^{\min }$. Equation (33) indicates that the RC must work at the repair node for a certain amount of time $T_{i_c}^{\mathrm{Rep}}$ in order for the component to be repaired. Equation (34) states that a component can only be repaired once. Equation (35) restricts that the repair of a component must be carried out by a single RC. Equation (36) indicates that the component can only be powered after it has been repaired.

2.6. Distribution System Operation Model

The emergency restoration process needs to meet the actual operational requirements of the distribution network, including constraints related to radial topology, power generation output and line capacity, power balance, and safe operation.

Let the available sets of nodes, lines, and substations during time period t be represented as $B_t^{\mathrm{Ava}}$, $L_t^{\mathrm{Ava}}$ and $S_t^{\mathrm{Ava}}$, respectively. Then, the radial topology constraint can be expressed as [31]:

$${\rho }_{+,(i,j),t}+{\rho }_{-,(i,j),t}={\rho }_{(i,j),t},\forall (i,j)\in L_t^{\mathrm{Ava}}$$

(37)

$${\displaystyle \sum _{j:(j,i)\in L_t^{\mathrm{Ava}}}{\rho }_{+,(i,j),t}}+{\displaystyle \sum _{j:(i,j)\in L_t^{\mathrm{Ava}}}{\rho }_{-,(i,j),t}}\le {\displaystyle \frac{\left|B_t^{\mathrm{Ava}}\right|-\left|S_t^{\mathrm{Ava}}\right|}{\left|B_t^{\mathrm{Ava}}\right|-\left|S_t^{\mathrm{Ava}}\right|+1}},\forall i\in B_t^{\mathrm{Ava}}\backslash S_t^{\mathrm{Ava}}$$

(38)

$${\displaystyle \sum _{i\in S_t^{\mathrm{Ava}}}\left[{\displaystyle \sum _{j:(j,i)\in L_t^{\mathrm{Ava}}}{\rho }_{+,(i,j),t}}+{\displaystyle \sum _{j:(i,j)\in L_t^{\mathrm{Ava}}}{\rho }_{-,(i,j),t}}\right]}\le {\displaystyle \frac{\left|B_t^{\mathrm{Ava}}\right|-\left|S_t^{\mathrm{Ava}}\right|}{\left|B_t^{\mathrm{Ava}}\right|-\left|S_t^{\mathrm{Ava}}\right|+1}}$$

(39)

$${\rho }_{+,(i,j),t},{\rho }_{-,(i,j),t}\ge 0,\forall (i,j)\in L_t^{\mathrm{Ava}}$$

(40)

In the equation:

${\rho }_{(i,j),t}$: A 0–1 variable indicating the line’s energized status, where ${\rho }_{(i,j),t}=1$ means the line is energized during time period t;

${\rho }_{+,(i,j),t}$, ${\rho }_{-,(i,j),t}$: Continuous auxiliary variables;

$\left| · \right|$: The cardinality of the set.

Equations (37)–(40) are radiality constraints for distribution networks based on graph maximum density. Equation (38) limits the total flow entering or leaving any non-substation node to enforce acyclicity in the contracted network. Equation (39) restricts the total flow absorbed by merged substation nodes, ensuring each connected component contains at most one substation.

Substations, DGs, EVs, MPS, and other components can only be connected after the node fault is repaired,

$$\left\{\begin{array}{c}0\le p_{\mathrm{S},i,t}\le {\rho }_{i,t}·P_{\mathrm{S},i}^{\max }\\ 0\le q_{\mathrm{S},i,t}\le {\rho }_{i,t}·Q_{\mathrm{S},i}^{\max }\end{array}\right.,\forall i,\forall t$$

(41)

$$\left\{\begin{array}{c}0\le p_{\mathrm{DG},i,t}\le {\rho }_{i,t}·P_{\mathrm{DG},i}^{\max }\\ 0\le q_{\mathrm{DG},i,t}\le {\rho }_{i,t}·Q_{\mathrm{DG},i}^{\max }\end{array}\right.,\forall i,\forall t$$

(42)

$$\left\{\begin{array}{c}0\le p_{n,e,t}\le {\rho }_{i_n,t}·P_e^{\max }\\ 0\le q_{n,e,t}\le {\rho }_{i_n,t}·Q_e^{\max }\end{array}\right.,\forall n,\forall e,\forall t$$

(43)

$$\left\{\begin{array}{c}0\le p_{i,g,t}\le {\rho }_{i,t}·P_g^{\max }\\ 0\le q_{i,g,t}\le {\rho }_{i,t}·Q_g^{\max }\end{array}\right.,\forall i,\forall g,\forall t$$

(44)

In the equation:

$p_{\mathrm{S},i,t}$, $q_{\mathrm{S},i,t}$, $P_{\mathrm{S},i}^{\max }$, $Q_{\mathrm{S},i}^{\max }$: The active and reactive power injected by the substation and the maximum values at node i during time period t, respectively;

$p_{\mathrm{DG},i,t}$, $q_{\mathrm{DG},i,t}$, $P_{\mathrm{DG},i}^{\max }$, $Q_{\mathrm{DG},i}^{\max }$: The active and reactive power output of the DG and their maximum values at node i during time period t, respectively;

$i_n$: The node where the V2G station n is located.

Only energized lines can transmit power, and they must satisfy the line capacity constraint.

$$\left\{\begin{array}{c}-{\rho }_{(i,j),t}·P_{(i,j)}^{\max }\le P_{(i,j),t}\le {\rho }_{(i,j),t}·P_{(i,j)}^{\max }\\ -{\rho }_{(i,j),t}·Q_{(i,j)}^{\max }\le Q_{(i,j),t}\le {\rho }_{(i,j),t}·Q_{(i,j)}^{\max }\end{array}\right.,\forall (i,j)\in L$$

(45)

In the equation:

$P_{(i,j),t}$, $Q_{(i,j),t}$, $P_{(i,j)}^{\max }$, $Q_{(i,j)}^{\max }$: The active and reactive power values on the line $(i,j)$ during time period t and the maximum active and reactive power capacities of the line $(i,j)$.

The power balance and safe operation constraints are as follows:

$$\left\{\begin{array}{c}{p}_{i,t}={\displaystyle \sum _{(i,j)\in L}{P}_{(i,j),t}-}{\displaystyle \sum _{(j,i)\in L}{P}_{(j,i),t}}+{\rho }_{\mathrm{load},i,t}·p_{\mathrm{load},i,t}\\ q_{i,t}={\displaystyle \sum _{(i,j)\in L}{Q}_{(i,j),t}-}{\displaystyle \sum _{(j,i)\in L}{Q}_{(j,i),t}}+{\rho }_{\mathrm{load},i,t}·q_{\mathrm{load},i,t}\end{array}\right.,\forall i,\forall t$$

(46)

$${\rho }_{\mathrm{load},i,t}\le {\rho }_{i,t},\forall i,\forall t$$

(47)

$$\left\{\begin{array}{c}{p}_{i,t}=p_{\mathrm{S},i,t}+p_{\mathrm{DG},i,t}+{\displaystyle \sum _{i_n=i,e\in N_{\mathrm{EV}}}{p}_{n,e,t}}+{\displaystyle \sum _{g\in N_{\mathrm{MPS}}}{p}_{i,g,t}}\\ q_{i,t}=q_{\mathrm{S},i,t}+q_{\mathrm{DG},i,t}+{\displaystyle \sum _{i_n=i,e\in N_{\mathrm{EV}}}{q}_{n,e,t}}+{\displaystyle \sum _{g\in N_{\mathrm{MPS}}}{q}_{i,g,t}}\end{array}\right.,\forall i,\forall t$$

(48)

$$-(1-{\rho }_{(i,j),t})·M\le V_{j,t}+{\displaystyle \frac{r_{(i,j)}{P}_{(i,j),t}+x_{(i,j)}{Q}_{(i,j),t}}{V_0}}-V_{i,t}\le (1-{\rho }_{(i,j),t})·M,\forall (i,j)\in L,\forall t$$

(49)

$$1-\epsilon \le {\displaystyle \frac{V_{i,t}}{V_0}}\le 1+\epsilon$$

(50)

In the equation:

$p_{i,t}$, $q_{i,t}$: The active and reactive power injected by the power source at node i during time period t;

$p_{\mathrm{load},i,t}$, $q_{\mathrm{load},i,t}$: The active and reactive power demands of the load at node i during time period t;

${\rho }_{\mathrm{load},i,t}$: The connection status of the load at node i during time period t, assuming that a load switch is installed;

$V_{i,t}$, $V_0$: The voltage at node i during time period t and the system’s rated voltage value;

$\epsilon$: The allowed maximum voltage deviation, which is taken as $\epsilon =0.1$ in this paper.

2.7. Post-Disaster Scheduling Optimization Problem

The post-disaster scheduling optimization problem is formulated with the objective of maximizing the load supply throughout the entire restoration process:

$$\max {\displaystyle \sum _{t\in T}{\displaystyle \sum _{i\in B}{w}_{\mathrm{load},i}{\rho }_{\mathrm{load},i,t}·p_{\mathrm{load},i,t}\Delta t}}$$

(51)

In this equation: $w_{\mathrm{load},i}$ is the weight coefficient of the load at node i.

The post-disaster scheduling optimization problem can be modeled as a mixed-integer linear programming (MILP) problem with the objective function given by Equation (51) and constraints from Equations (15)–(50).

3. Results

This study uses the modified IEEE 33-node distribution network and SiouxFalls 35-node transportation network to form a power traffic coupling model and analyzes the impact of the traffic model and EV on the overall restoration scheduling decision under the specific waterlogging and coupling system damage scenarios. The case study proves the role of the proposed model in improving the resilience of distribution network under waterlogging disasters.

3.1. Traffic Flow Analysis Under Waterlogged Conditions

This section uses the modified SiouxFalls 35-node traffic network to analyze the impact of waterlogging on vehicle travel and road congestion. The adjacent traffic nodes in the traffic network are connected by two one-way lanes, and the road traffic direction is driving on the right side of the road. See

Table A1. Road parameters of the modified SiouxFalls 35-node transportation network.

From	To	Length/m	Capacity	FFS */km·h−1	From	To	Length/m	Capacity	FFS/km·h−1
1	3	1400	500	65	16	18	2000	500	65
1	33	1350	100	35	17	10	2332.38	500	65
2	6	1400	500	65	17	16	1200	200	40
2	31	1350	100	35	17	19	1400	500	65
3	1	1400	500	65	18	7	1200	200	40
3	4	1600	500	65	18	16	2000	500	65
3	34	1200	200	40	18	20	5738.66	500	65
4	3	1600	500	65	19	15	2000	500	65
4	5	1800	500	65	19	17	1400	500	65
4	11	2400	500	65	19	30	1400	500	65
5	4	1800	500	65	20	18	5738.66	500	65
5	6	2000	500	65	20	21	1999.62	500	65
5	9	1200	200	40	20	22	2536.88	500	65
6	2	1400	500	65	20	30	1369.36	200	40
6	5	2000	500	65	21	20	1999.62	500	65
6	8	1200	200	40	21	22	1600	500	65
7	8	2000	500	65	21	24	1800	500	65
7	18	1200	200	40	22	15	1200	200	40
8	6	1200	200	40	22	20	2536.88	500	65
8	7	2000	500	65	22	21	1600	500	65
8	16	1200	200	40	22	23	1800	500	65
8	25	1000	200	40	23	14	1200	200	40
9	5	1200	200	40	23	22	1800	500	65
9	10	1200	200	40	23	24	1600	500	65
9	25	1000	200	40	24	13	1600	500	65
10	9	1200	200	40	24	21	1800	500	65
10	11	1800	500	65	24	23	1600	500	65
10	16	2000	500	65	25	8	1000	200	40
10	17	2332.38	500	65	25	9	1000	200	40
10	26	1300	500	65	26	10	1300	500	65
11	4	2400	500	65	26	15	1300	500	65
11	10	1800	500	65	27	13	1350	100	35
11	12	1600	500	65	27	28	1350	100	35
11	35	1300	500	65	28	27	1350	100	35
12	11	1600	500	65	28	29	1350	100	35
12	29	1350	100	35	29	12	1350	100	35
12	34	1200	200	40	29	28	1350	100	35
13	24	1600	500	65	30	19	1400	500	65
13	27	1350	100	35	30	20	1369.36	200	40
14	15	1800	500	65	31	2	1350	100	35
14	23	1200	200	40	31	32	1350	100	35
14	35	1300	500	65	32	31	1350	100	35
15	14	1800	500	65	32	33	1350	100	35
15	19	2000	500	65	33	1	1350	100	35
15	22	1200	200	40	33	32	1350	100	35
15	26	1300	500	65	34	3	1200	200	40
16	8	1200	200	40	34	12	1200	200	40
16	10	2000	500	65	35	11	1300	500	65
16	17	1200	200	40	35	14	1300	500	65

* FFS: free-flow speed.

in Appendix A for detailed road parameters. The traffic network model and water accumulation area are shown in Figure 1. The roads with severe water accumulation and their initial water depth are listed in

Table 1. Severely waterlogged roads and their initial water depth.

From	To	Initial Water Depth/cm	From	To	Initial Water Depth/cm
4	5	20	16	17	20
5	4	20	17	16	20
4	11	40	15	19	70
11	4	40	19	15	70
10	16	100	15	22	40
16	10	100	22	15	40
10	17	100	11	10	20
17	10	100	18	20	20

, and the initial water depth of other roads is assumed to be 10 cm.

The SDTA model in Section 2.1 is used to analyze the road water accumulation and congestion within 6 h after the disaster. The time step is taken as 20 min and compared with the distribution of traffic flow without waterlogging. Figure 2 shows the distribution of traffic flow with waterlogging and without waterlogging when the time step t = 1, 9, and 18 (corresponding to the time period of 0–20 min, 160–180 min, and 340–360 min). We describe road traffic conditions based on the impact coefficient of congestion (ICC), $\left[1+0.15{\left(x_{a,t}/c_a\right)}^4\right]$ and the impact coefficient of water accumulation (ICW) on road travel time, $2\times {\left[1+\tanh \left(\alpha /\beta -h_{a,t}/\beta \right)\right]}^{-1}$.

Compared with Figure 2a,b, it can be seen that waterlogging disasters have a serious impact on road traffic, greatly changing the distribution of traffic flow in the traffic network, and causing serious congestion on some roads. It can be seen from the distribution of traffic flow in the comparison period t = 1, 9, and 18 that the distribution of traffic flow in the scene with waterlogging will gradually approach that in the scene without waterlogging with the progress of natural water recession.

3.2. Analysis of Power Distribution Network Restoration After Waterlogging Disasters

This section uses the modified IEEE 33-node distribution network coupled with SiouxFalls 35-node transportation network to analyze the role of EVs in post-disaster restoration and the impact of waterlogging on distribution network resource scheduling process. Node 1 in the distribution system is a substation node, and nodes 18 and 25 are equipped with controllable DGs. A V2G station is located at nodes 4, 7, 15, 22, 31, and 32 of the distribution network. The EV used for restoration can be connected to 20 charging piles in V2G station. The maximum charging and discharging power of charging piles is 7 kW. The system is equipped with two dispatchable MPSs and four RCs, which are deployed in depot 1 and depot 2 on average. The depots are located at nodes 7 and 35 of the traffic network, respectively. The time for RC to eliminate faults at nodes and lines is 80 min and 60 min, respectively. It is assumed that a total of 200 EVs can be dispatched after the disaster and have been parked at the refuge station before the disaster. The refuge stations are located at nodes 4, 14, 19, 28, 31, and 33 of the transportation network. As urban waterlogging often accompanies typhoons, this paper builds a power distribution network disaster scenario based on the wind and waterlogging disaster component vulnerability model in Section 2.2. The power–traffic coupling model and the damaged components of the power grid are shown in Figure 3, where a total of 4 node submersion faults and 8 line break faults occur.

The post-disaster restoration model of distribution network in Section 2.3, Section 2.4, Section 2.5, Section 2.6 and Section 2.7 is used to make scheduling decisions. The restoration resource scheduling process at each stage is shown in Figure 4a–d. The first stage: 0–20 min, the tie-line switch is closed to form multiple microgrids, which is supported by DGs; the MPSs and EVs drive to the target discharge node and V2G station, and the RCs drive to the damaged node or line. The second stage: 20–80 min, RCs carry out maintenance on damaged nodes or lines, and DGs, MPSs, and EVs provide power supply support for critical load nodes. The third stage: 80–100 min, repair the lines (1,2), (2,19), (23,24), and connect the distribution network with the substation node. The fourth stage: 100–180 min, the damaged nodes 2, 11, 17, and 27 of the distribution network were repaired, and all of the load was restored. The active power output of each V2G station, MPS, and DG during the restoration process is shown in Figure 5.

In order to verify the effect of the proposed method on improving the resilience of distribution network, four comparative schemes are set: (1) considering the impact of water accumulation on traffic congestion, the joint scheduling strategy of MPSs, RCs, and EVs (the method proposed in this paper); (2) considering the impact of water accumulation on traffic congestion, the scheduling strategy of MPSs and RCs; (3) not considering the impact of water accumulation on traffic congestion, the joint scheduling strategy of MPSs, RCs, and EVs; and (4) not considering water accumulation and traffic congestion, the joint scheduling strategy of MPSs, RCs, and EVs. The post-disaster load restoration curves of the four schemes are shown in Figure 6. The total load loss and repair duration of the system under each scheme are shown in

Table 2. Total system load loss and repair duration under each scheme.

Scheme	Load Loss/kWh	Restoration Duration/min
1	2566.01	180
2	3370.67	180
3	3464.76	200
4	16,202.86	680

, in which the calculation method of load size is the actual load demand multiplied by the normalized load weight.

It can be seen from Figure 6 and Table 2 that considering the road water accumulation, traffic flow model, and EV participation in post-disaster dispatching, the post-disaster restoration time of critical loads is shortened and the amount of system load loss is reduced. Compared with Scheme 2, Scheme 1 restored more loads before the completion of repair, and the load loss was reduced by 31.36%, which proved the role of EV in the process of post-disaster restoration. As an important supplement to the removable power supply of distribution network, EVs provided power support for critical loads in a short time after the disaster, and improved the load restoration rate. Compared with Scheme 3, Scheme 1 improves the scheduling speed of mobile resources and reduces the system load loss by 35.03%, which proves that post-disaster traffic congestion has an important impact on the load restoration process. Notably, Scheme 4 shows the worst performance with 16,202.86 kWh load loss and 680 min restoration duration, which is 6.3 times higher than Scheme 1 in terms of energy loss and 3.8 times in terms of restoration duration. This significant gap demonstrates that ignoring road and traffic conditions would severely delay repair processes and exacerbate load shedding under waterlogging disasters. To sum up, the flexible resource collaborative scheduling method of electric vehicles and distribution network considering the post-disaster traffic situation proposed in this paper improves the load restoration rate under waterlogging disaster, reduces the load loss of distribution network, and improves the resilience of distribution network.

4. Discussion

This paper proposes a post-disaster coordinated scheduling method for power distribution systems under waterlogging disasters, explicitly considering the impacts of waterlogged roads on traffic conditions. Case studies highlight the advantages of integrating traffic models under waterlogged road scenarios into the restoration decision framework, as well as the role of dispatchable EVs in enhancing distribution network resilience. Previous studies on power system resilience often neglected the influence of traffic congestion on mobile resource scheduling or simplistically treated it as a fixed parameter, while research on the resilience of power–transportation coupled systems with EVs typically assumed fixed origin–destination pairs for EVs, failing to address flexible scheduling requirements in post-disaster scenarios. The proposed model bridges waterlogging impacts on road networks with traffic accessibility constraints on vehicle routing, establishing a semi-dynamic traffic model under waterlogging disasters. By coupling this model with a multi-resource scheduling framework for power distribution systems through reasonable assumptions, it addresses critical gaps in existing literature and provides a practical solution for real-world challenges.

Future research will refine the coordinated multi-resource scheduling model by incorporating traffic flows generated during the scheduling process into the traffic model and integrating data from real-time monitoring devices to improve the accuracy and responsiveness of scheduling decisions.