The increasing frequency and intensity of extreme weather events, driven by global climate change, pose a significant and growing threat to the reliability and resilience of modern power distribution networks. Typhoons, in particular, can inflict widespread structural damage, leading to large-scale, protracted power outages that cripple communities and economies. Traditional post-disaster recovery strategies, which predominantly rely on manual repair crews and physical hardening of infrastructure, are increasingly inadequate. These approaches often suffer from limited coverage, delayed response times, and inherent vulnerability to cascading failures, struggling to meet the urgent demand for rapid restoration in the aftermath of a major disaster.
The proliferation of distributed energy resources (DERs) within distribution networks offers a transformative pathway for enhancing grid resilience. The integration of battery electric car fleets, coupled with distributed wind and photovoltaic (PV) generation, presents a dynamic and flexible resource pool for post-disaster recovery. Battery electric cars, with their mobile energy storage capability through Vehicle-to-Grid (V2G) technology, can be strategically dispatched to provide emergency power to critical loads. Similarly, distributed wind turbines and PV panels can form resilient microgrids to sustain local power supply. However, effectively coordinating these heterogeneous resources—each with distinct spatial, temporal, and operational constraints—under the uncertain and adverse conditions following an extreme event remains a formidable challenge. This paper addresses the critical gaps in multi-resource coordination and responsive power supply for post-disaster recovery by proposing a novel, integrated strategy.
The core innovation of the proposed strategy lies in a two-stage framework. First, it introduces a proactive pre-scheduling mechanism for battery electric cars. Recognizing that post-disaster road network degradation and communication disruptions can severely hinder spontaneous participation, a pre-emptive scheduling model is developed. This model incorporates an owner compensation scheme designed to incentivize participation by fairly remunerating owners for both the energy provided and the logistical challenges undertaken, thereby securing a committed fleet of mobile power sources before the disaster’s full impact is realized.
Second, it establishes a holistic post-disaster multi-resource coordinated repair and restoration model. This model synergistically schedules the pre-committed battery electric car fleet, available wind and PV generation, and mobile repair crews. The objective is to minimize the total socio-economic cost of the restoration process, which includes resource compensation, crew dispatch costs, and the penalty for unsupplied load. The optimization is subject to a comprehensive set of constraints encompassing the spatiotemporal dynamics of the battery electric cars, the uncertain output profiles of renewable resources, the operational independence and routing of repair crews, and essential power flow limits within the partially energized network.
Framework of the Proposed Restoration Strategy
The proposed strategy is structured as a sequential decision-making framework designed to maximize restoration efficiency and economic viability. The overall architecture is depicted below and consists of three core interconnected modules.
Stage 1: Proactive Pre-Scheduling of Battery Electric Cars. Prior to the anticipated landfall or impact of an extreme event, this module is activated. It targets regions assessed as high-risk for power outages. A compensation model is used to engage private battery electric car owners, locking in their commitment to report to designated V2G access points post-disaster. This pre-scheduling accounts for anticipated travel times and road conditions, ensuring a predictable and available fleet of mobile storage units when they are most needed.
Stage 2: Modeling Post-Disaster Renewable Generation. Concurrently, models for distributed wind turbines and PV systems are formulated to quantify their available power output in the disaster-altered environment. These models incorporate post-event weather data (e.g., wind speed, solar irradiance, cloud cover) to accurately capture the time-varying and uncertain generation potential of these resources during the restoration window.
Stage 3: Integrated Multi-Resource Recovery Optimization. Following the disaster, this central module is executed. It integrates the pre-scheduled battery electric car fleet, the characterized renewable generation from wind and PV, and the available repair crews into a unified optimization model. The model simultaneously determines the optimal dispatch of battery electric car power, the scheduling of repair crew routes and tasks, and the utilization of renewable generation, all while respecting network physics. The goal is to minimize the total cost of restoration, leading to a faster, more cost-effective recovery of the distribution network.
Mathematical Modeling of Multi-Resource Systems
Battery Electric Car Pre-Scheduling Model
Typhoons and similar disasters typically cause severe traffic network paralysis, reduced road capacity, and disrupted communications. These factors introduce high uncertainty and spatial costs for battery electric car owners, potentially deterring voluntary participation in post-disaster recovery if left to a purely real-time, post-event dispatch. To overcome this, a pre-scheduling mechanism is proposed for regions forecasted to experience faults.
To enhance the willingness of distributed battery electric car resources to participate, an owner profit model is constructed. This model synergistically combines energy supply compensation and a mileage-based travel compensation to create a strong economic incentive. The profit \( R_m \) for owner \( m \) who is pre-scheduled to provide power at node \( i \) is given by:
$$ R_m = b_1 \cdot T_{i,m}^{\text{stay}} \cdot P^{\text{ev}} + C_m $$
where \( b_1 \) is the unit compensation rate for energy supply (e.g., $/kWh), \( T_{i,m}^{\text{stay}} \) is the total time the battery electric car \( m \) stays at the designated access node \( i \). This duration includes travel time to the node, potential waiting time, and the actual power supply duration. \( P^{\text{ev}} \) is the rated discharge power of the battery electric car, and \( C_m \) is the travel compensation paid to owner \( m \), which is a function of the distance from their initial location to the assigned node \( i \). This compensation structure explicitly rewards owners for both their energy contribution and their logistical effort, making participation attractive despite post-disaster hardships.
Wind Turbine Generation Model
Distributed wind turbines can serve as crucial compensation power sources during the network restoration phase due to their potential for rapid response and islanded operation. The power output of a wind turbine is primarily a nonlinear function of wind speed. The piecewise power curve is described as follows:
$$ P_t^{\text{wind}} = \begin{cases}
0, & v_t < v_{\text{in}} \text{ or } v_t \geq v_{\text{out}} \\
0.5 \rho A_w v_t^3 C, & v_{\text{in}} \leq v_t < v_d \\
P_{\text{max}}, & v_d \leq v_t < v_{\text{out}}
\end{cases} $$
Here, \( P_t^{\text{wind}} \) is the output power at time \( t \), \( \rho \) is air density, \( A_w \) is the swept area of the turbine blades, and \( v_t \) is the wind speed at time \( t \). The parameters \( v_{\text{in}} \), \( v_{\text{out}} \), and \( v_d \) are the cut-in, cut-out, and rated wind speeds, respectively, and \( P_{\text{max}} \) is the rated power. \( C \) is the power coefficient, a function of the tip-speed ratio \( \lambda \) and blade pitch angle \( \theta \), often approximated by:
$$ C = 0.22 \left( \frac{116}{\lambda_i} – 0.4\theta – 5 \right) e^{-\frac{12.5}{\lambda_i}} $$
where \( \frac{1}{\lambda_i} = \frac{1}{\lambda + 0.08\theta} – \frac{0.035}{\theta^3 + 1} \) and \( \lambda = \frac{\omega R}{v_t} \), with \( \omega \) being the rotor speed and \( R \) the rotor radius.
To incentivize wind farm operators to support restoration, a generation compensation model is established. The total compensation \( R_x^{\text{wind}} \) for wind turbine \( x \) over the restoration period \( T_1 \) is:
$$ R_x^{\text{wind}} = \sum_{t=0}^{T_1} b_2 \cdot P_{x,t}^{\text{wind}} $$
where \( b_2 \) is the compensation rate for wind power (typically higher than the normal feed-in tariff to reflect its emergency value) and \( P_{x,t}^{\text{wind}} \) is the output of turbine \( x \) at time \( t \).
Photovoltaic Generation Model
PV generation output is highly dependent on solar irradiance and ambient temperature. The hourly output power \( P_t^{\text{v}} \) can be modeled based on standard test conditions (STC):
$$ P_t^{\text{v}} = P_s \cdot \frac{G_t}{G_s} \cdot \left[ 1 + \gamma (T_t – T_d) \right] $$
where \( P_s \) is the rated power under STC, \( G_t \) and \( G_s \) are the actual and STC solar irradiance, respectively. \( \gamma \) is the temperature coefficient of power (typically negative for PV modules), \( T_t \) is the module temperature at time \( t \), and \( T_d \) is the temperature at STC (25°C). The module temperature \( T_t \) is itself influenced by irradiance and ambient conditions, often estimated by an empirical relation: \( T_t = \lambda_2 G_t + \lambda_3 v_t + \lambda_4 \), where \( v_t \) is wind speed and \( \lambda_2, \lambda_3, \lambda_4 \) are empirical coefficients.
Similar to wind, a compensation model for PV generation during restoration is defined. The compensation \( R_y^{\text{v}} \) for PV system \( y \) over period \( T_2 \) is:
$$ R_y^{\text{v}} = \sum_{t=0}^{T_2} b_3 \cdot P_{y,t}^{\text{v}} $$
where \( b_3 \) is the PV compensation rate and \( P_{y,t}^{\text{v}} \) is the output of system \( y \) at time \( t \).
Post-Disaster Multi-Resource Coordinated Repair and Restoration Model
Objective Function
The objective of the restoration model is to minimize the total socio-economic cost \( C^{\text{grid}} \) associated with the recovery process. This comprehensive cost includes five major components:
$$ \min C^{\text{grid}} = \min \left( C^{\text{ev}} + C^{\text{new}} + C^{\text{crew}} + C^{\text{loss}} \right) = \min \left( C^{\text{ev}} + (C^{\text{wind}} + C^{\text{v}}) + C^{\text{crew}} + C^{\text{loss}} \right) $$
- Battery Electric Car Supply Cost (\( C^{\text{ev}} \)): The sum of profits paid to all participating battery electric car owners.
$$ C^{\text{ev}} = \sum_{m=1}^{L} R_m = \sum_{m=1}^{L} \left( b_1 \cdot T_{i,m}^{\text{stay}} \cdot P^{\text{ev}} + C_m \right) $$
where \( L \) is the total number of scheduled battery electric cars. - New Energy Generation Cost (\( C^{\text{new}} \)): The sum of compensation paid to wind (\( C^{\text{wind}} \)) and PV (\( C^{\text{v}} \)) operators.
$$ C^{\text{wind}} = \sum_{x=1}^{n^{\text{wind}}} R_x^{\text{wind}} = \sum_{x=1}^{n^{\text{wind}}} \sum_{t=0}^{T_1} b_2 \cdot P_{x,t}^{\text{wind}} $$
$$ C^{\text{v}} = \sum_{y=1}^{n^{\text{v}}} R_y^{\text{v}} = \sum_{y=1}^{n^{\text{v}}} \sum_{t=0}^{T_2} b_3 \cdot P_{y,t}^{\text{v}} $$
where \( n^{\text{wind}} \) and \( n^{\text{v}} \) are the numbers of participating wind turbines and PV systems. - Repair Crew Dispatch Cost (\( C^{\text{crew}} \)): The cost associated with mobilizing repair crews, modeled as proportional to the total travel distance.
$$ C^{\text{crew}} = b_4 \cdot \sum_{i=1}^{N} \sum_{j=1}^{N} \sum_{n=1}^{O} y_{i,j,n}^{\text{R}} \cdot d_{i,j} $$
where \( b_4 \) is the cost per unit distance, \( y_{i,j,n}^{\text{R}} \) is a binary decision variable indicating whether crew \( n \) travels from node \( i \) to node \( j \), \( d_{i,j} \) is the distance, \( N \) is the total number of nodes, and \( O \) is the number of repair crews. - Cost of Lost Load (\( C^{\text{loss}} \)): The economic penalty for unsupplied load, representing the societal impact of the outage.
$$ C^{\text{loss}} = b_5 \cdot \sum_{t=0}^{T} \sum_{i=1}^{N} z_{i,t} \cdot P_{i,t}^{\text{loss}} $$
where \( b_5 \) is the value of lost load (VOLL), \( T \) is the total restoration horizon, \( z_{i,t} \) is a binary fault state indicator for node \( i \) at time \( t \) (1 if faulted, 0 if restored), and \( P_{i,t}^{\text{loss}} \) is the unsupplied load at node \( i \) at time \( t \).
Constraints
The optimization is subject to a comprehensive set of operational and logical constraints.
1. Battery Electric Car Constraints:
- Energy Balance: A battery electric car must retain sufficient energy for its return trip after providing power.
$$ T_{k,m}^{\text{stay}} \leq \frac{c_{k,m}^{\text{ev}} – c_{k,m}^{\text{back}}}{P^{\text{ev}}} $$
where \( c_{k,m}^{\text{ev}} \) and \( c_{k,m}^{\text{back}} \) are the battery energy levels upon arrival and required departure at node \( k \). - Power Limit: The aggregate power from battery electric cars at a node cannot exceed local capacity.
$$ P^{\text{ev}} \leq P_{k,t}^{\text{ev}} \cdot \eta_{k,t}^{\text{ev}} $$
where \( P_{k,t}^{\text{ev}} \) is the maximum allowable discharge power at node \( k \) and \( \eta_{k,t}^{\text{ev}} \) is an efficiency/participation factor. - Spatiotemporal Logic: Constraints ensure each battery electric car arrives, stays, and leaves its assigned node with correct sequencing.
$$ \sum_{t} f_{k,m,t}^{\text{ev}} = \sum_{t} l_{k,m,t}^{\text{ev}} = 1 $$
where \( f_{k,m,t}^{\text{ev}} \) and \( l_{k,m,t}^{\text{ev}} \) are binary variables indicating arrival and departure events. - Time Continuity: The travel and service times must be sequentially consistent.
$$ T_{j,m}^{\text{ev}} = T_{i,m}^{\text{ev}} + T_{i,m}^{\text{ev,stay}} + T_{i,j,m}^{\text{ev,d}} $$
$$ (1 – y_{i,j,m}^{\text{ev}}) M^{\text{ev}} \leq T_{i,m}^{\text{ev}} + T_{i,m}^{\text{ev,stay}} + T_{i,j,m}^{\text{ev,d}} $$
where \( T^{\text{ev}} \) terms represent arrival, service, and travel times, \( y_{i,j,m}^{\text{ev}} \) is the routing binary variable, and \( M^{\text{ev}} \) is a large constant.
2. Wind and PV Generation Constraints:
- Output Limits: Generation must be within zero and their available/rated capacity.
$$ 0 \leq P_{x,t}^{\text{wind}} \leq P_{\text{max}}^{\text{wind}} $$
$$ 0 \leq P_{y,t}^{\text{v}} \leq P_{\text{max}}^{\text{v}} $$
3. Repair Crew Constraints:
- Single Crew per Fault: Each fault location is assigned to exactly one repair crew.
$$ \sum_{n=1}^{O} x_{i,n}^{\text{R}} = 1, \quad \forall i \in \mathcal{F} $$
where \( x_{i,n}^{\text{R}} \) indicates if crew \( n \) is assigned to fault \( i \), and \( \mathcal{F} \) is the set of fault locations. - Flow Conservation: A crew’s route forms a continuous path.
$$ \sum_{i} y_{i,j,n}^{\text{R}} = x_{j,n}^{\text{R}}, \quad \sum_{j} y_{i,j,n}^{\text{R}} = x_{i,n}^{\text{R}} $$ - Time Continuity for Crews: Similar to battery electric cars, crew travel and repair times must be consistent.
$$ T_{j,n}^{\text{R}} = T_{i,n}^{\text{R}} + T_{i,n}^{\text{R,stay}} + T_{i,j,n}^{\text{R,d}} $$ - Complete Restoration: All identified faults must be repaired.
$$ \sum_{i=1}^{N} \sum_{n=1}^{O} x_{i,n}^{\text{R}} = n^{\text{fault}} $$
where \( n^{\text{fault}} \) is the total number of faults.
4. Power Flow and Network Constraints:
- Node Power Balance: Using a linearized DistFlow approximation for radial networks.
$$ P_{i,t}^{\text{in}} + P_{i,t}^{\text{source}} – P_{i,t}^{\text{loss}} – P_{i,t}^{\text{load}} = 0 $$
$$ Q_{i,t}^{\text{in}} + Q_{i,t}^{\text{source}} – Q_{i,t}^{\text{loss}} – Q_{i,t}^{\text{load}} = 0 $$
where \( P^{\text{in}}/Q^{\text{in}} \) are line flow powers, \( P^{\text{source}}/Q^{\text{source}} \) are injections from battery electric cars, renewables, or the main grid, and \( P^{\text{load}}/Q^{\text{load}} \) are loads. - Line Flow Limits:
$$ (P_{i,t}^{\text{in}})^2 + (Q_{i,t}^{\text{in}})^2 \leq (S_{l}^{\text{max}})^2 $$ - Voltage Limits:
$$ V_{\text{min}} \leq V_{i,t} \leq V_{\text{max}} $$
Case Study and Simulation Analysis
To validate the proposed strategy, a simulation was conducted based on the impact of Typhoon Saola (2023) on the distribution network of a coastal city, referred to as X City. The network topology is adapted from the IEEE 33-node test system, with fault locations mapped according to actual damage reports from the event.
Simulation Setup and Parameters
The post-disaster restoration window was set to 24 hours. The network had 13 faulted line sections. Three V2G access points were established at specific nodes. One wind turbine (200 kW max) and one PV system (200 kW max) were connected at designated nodes. Five repair crews were available, and a pool of 330 private battery electric cars was considered for pre-scheduling. Post-disaster travel times between nodes were calculated using Dijkstra’s algorithm on a degraded road network model within ArcGIS. Realistic post-typhoon weather data (wind speed, solar irradiance) for the 24-hour period was incorporated. Key simulation parameters are summarized below.
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| \( b_1 \) (EV energy rate) | 0.800 $/kWh | \( b_2, b_3 \) (Wind/PV rate) | 1.500 $/kWh |
| \( b_4 \) (Crew travel cost) | 150.000 $/km | \( b_5 \) (Value of Lost Load) | 3.000 $/kWh |
| \( P^{\text{ev}} \) (EV discharge power) | 20.000 kW | EV Pool Size (\( L \)) | 330 |
| Number of Repair Crews (\( O \)) | 5 | Number of Faults (\( n^{\text{fault}} \)) | 13 |
| Wind Cut-in/Cut-out/Rated Speed | 3.5 / 25 / 10 m/s | PV Temperature Coefficient (\( \gamma \)) | -0.005 /°C |
Multi-Resource System Dispatch Results
The pre-scheduling model committed 255 out of the 330 battery electric cars to the restoration task. The vehicles were distributed among the three V2G access points based on anticipated need and travel feasibility. The profit for owners ranged up to $242, with a significant cluster earning between $60 and $150, effectively validating the incentive model’s ability to secure participation.
The output characteristics of the different resources over the 24-hour restoration horizon are shown in the table below, illustrating their complementary roles:
| Time Period | Dominant Resource | Key Characteristic |
|---|---|---|
| Hours 0-12 | Battery Electric Cars | Provided primary support, output dependent on the number of cars scheduled per access point. |
| Hours 12-24 | Wind Turbine | Became the main supply as EVs completed their service windows; output fluctuated with wind speed (12-15 m/s). |
| Hours 18-22 | PV System | Contributed significantly during peak daylight hours as solar irradiance recovered post-typhoon, reaching ~1600 kW. |
Scenario Comparison
Two scenarios were simulated to evaluate the benefit of the proposed pre-scheduling mechanism:
Scenario A (No Pre-scheduling): Battery electric cars are dispatched only after the disaster occurs, with no prior commitments.
Scenario B (With Pre-scheduling): The proposed strategy is implemented.
The results, solved using the Gurobi optimizer, demonstrate clear advantages for the proposed strategy. The repair schedule was significantly compressed. In Scenario A, the last crew finished its repairs after 16.48 hours. In Scenario B, with pre-positioned battery electric cars providing early power support to critical loads and enabling more efficient crew routing, the last crew finished after only 10.42 hours—a 37% reduction in total repair time.
The economic comparison is summarized in the following table:
| Cost Component | Scenario A: No Pre-scheduling ($) | Scenario B: With Pre-scheduling ($) | Change ($) |
|---|---|---|---|
| Battery Electric Car Cost (\( C^{\text{ev}} \)) | 20,052 | 33,224 | +13,172 |
| New Energy Cost (\( C^{\text{new}} \)) | 12,949 | 25,689 | +12,740 |
| Repair Crew Dispatch Cost (\( C^{\text{crew}} \)) | 6,258 | 5,309 | -949 |
| Cost of Lost Load (\( C^{\text{loss}} \)) | 132,226 | 90,632 | -41,594 |
| Total Restoration Cost (\( C^{\text{grid}} \)) | 171,485 | 158,675 | -12,810 |
The analysis reveals that while the direct compensation costs for battery electric cars and renewables increase in Scenario B (by $13,172 and $12,740 respectively), this investment yields substantial system-wide benefits. The faster, more coordinated recovery driven by pre-scheduled battery electric car support leads to a notable reduction in crew dispatch costs ($949 saved) and a dramatic reduction in the cost of lost load ($41,594 saved). The net effect is a 7.5% reduction in the total integrated restoration cost, saving $12,810. This conclusively demonstrates that the proposed strategy enhances both the speed and the economic efficiency of post-disaster recovery.
Conclusion
This paper has addressed the critical challenge of coordinating diverse resources for efficient distribution network restoration following extreme disasters. The proposed strategy, centered on a V2G pre-scheduling mechanism for battery electric cars, successfully integrates mobile energy storage, distributed renewable generation, and repair crews into a unified optimization framework.
The case study based on a Typhoon Saola scenario validated the strategy’s effectiveness. The pre-scheduling mechanism secured the commitment of 255 battery electric cars, which played a pivotal role in the early restoration phase. This coordination led to a 37% reduction in the time required to complete all physical repairs. Economically, although payments to resource owners increased, the massive reduction in the cost of unsupplied load and improved crew efficiency resulted in a net 7.5% decrease in total restoration costs.
The findings underscore the immense value of proactive, coordinated planning that leverages emerging flexible resources like battery electric cars. Future work will focus on incorporating more granular models of uncertainty, including dynamic fault scenarios, evolving traffic conditions, and the heterogeneous behavior of battery electric car owners, to further enhance the robustness and adaptability of the restoration strategy.