Equivalent Circuit Model based Electric Vehicle Evacuation with Mobile Charging Stations

Technical Summary: Equivalent Circuit Model–based Electric Vehicle Evacuation with Mobile Charging Stations

Problem Statement
The increasing adoption of Electric Vehicles (EVs) presents unique challenges for emergency evacuation planning, particularly under disaster-induced disruptions. Unlike conventional gasoline vehicles, EVs are constrained by limited driving ranges, longer refueling times, and a reliance on Fixed Charging Stations (FCSs). In disaster scenarios, FCSs may be disabled due to power grid failures or physical damage, and the high ratio of EVs to chargers can lead to severe congestion and stranded vehicles. Existing evacuation models often assume fully operational charging infrastructure or fail to jointly optimize routing, charging decisions, and the deployment of Mobile Charging Stations (MCSs). This paper addresses the need for an optimization framework that accounts for battery state-of-charge (SoC), charging delays, and the strategic placement of MCSs to ensure energy-feasible evacuation routes.

Methodology
The authors propose a novel optimization-based framework utilizing an Equivalent Circuit Model (ECM) to model the evacuation problem. The methodology leverages analogies between electrical engineering and transportation systems:

• Traffic Flow as Current: Vehicle flow is modeled as electrical current.
• Travel Time as Resistance: The impedance of a road link is represented by travel time.
• Driving Range as Voltage: The vehicle's remaining driving range is modeled as voltage.
• Charging as Voltage Sources: Charging stations act as sources that replenish voltage (range).

The framework formulates the evacuation problem as an Integer Programming (IP) problem. Key components include:

1. Controllable ECM: The model incorporates binary switches to regulate route selection (activating or blocking specific links) and integer variables to determine charging durations at both FCSs and MCSs.
2. Kirchhoff's Laws:
   • Kirchhoff's Current Law (KCL): Enforces flow balance at network nodes, ensuring that the number of vehicles entering a node equals the number leaving (accounting for switches).
   • Kirchhoff's Voltage Law (KVL): Enforces energy feasibility. By treating range loss as voltage drop across resistors (travel time) and range gain as voltage sources (charging), the model ensures that the remaining "voltage" (range) never drops below zero along the selected path.
3. Superposition Principle: To handle multiple Origin-Destination (OD) pairs, the framework applies the principle of superposition. It solves for individual OD pairs independently to determine optimal paths and charging strategies, then aggregates the flows to ensure they do not exceed the free-flow capacity of road links or the service rates of charging stations.
4. Optimization Objectives: The framework supports various performance metrics, including minimizing the worst-case evacuation time ($J_{max}$), average evacuation time ($J_{avg}$), and a weighted combination that includes variance to promote fairness among evacuees ($J_{avg+\Delta}$).

Key Contributions
The paper identifies four primary contributions:

1. Controllable ECM Extension: Development of a controllable ECM for traffic networks that integrates switching elements for active traffic flow regulation and explicitly models charging delays and range replenishment.
2. Voltage-Based Range Modeling: A novel analogy using Kirchhoff's Voltage Law to model the loss of driving range along evacuation routes, a feature not previously addressed in ECM-based traffic models.
3. Integrated MCS Deployment: The incorporation of MCSs into the ECM framework to determine the optimal placement, number, and charging duration of mobile units to minimize total evacuation time.
4. Multi-OD Fairness Framework: An extension of the model to multiple OD pairs using superposition, supporting fairness-aware metrics (worst-case, average, and variance-based) to prevent significant disparities in evacuation times.

Results
The proposed framework was validated through simulation studies on large-scale transportation networks in California: the urban Anaheim network and the rural Mariposa network.

• Anaheim Network (Urban): Simulations with 8 OD pairs and 13,400 EVs demonstrated that the $J_{avg}$ cost function minimized average evacuation times, while $J_{avg+\Delta}$ significantly reduced the deviation in evacuation times (by ~8.8% compared to $J_{max}$ at specific initial ranges). Sensitivity analysis showed that as input flow increases, the network utilizes more edges, and the number of required MCSs increases sharply to prevent congestion once fixed station capacity is exceeded.
• Mariposa Network (Rural): In a scenario simulating a wildfire evacuation where downtown FCSs were deactivated, the ECM-based optimization deployed 6 MCSs strategically to enable on-route charging. Compared to a baseline "greedy" algorithm that placed MCSs only at the nearest locations (causing saturation and queuing), the proposed method distributed MCSs to mitigate both road congestion and charging queues.
• High-Fidelity Validation: The optimization results were validated using Aimsun micro-simulation. For shorter routes, the error between the ECM model and the microscopic simulation was below 6.67%. For longer routes, errors increased to approximately 13.84%, attributed to the optimization model's simplification of traffic control elements and speed variations.
• Performance Improvement: In the Mariposa case, the ECM-based approach improved evacuation times by 36% to 43% for several OD pairs compared to the baseline algorithm, primarily by reducing queuing and leveraging the initial SoC of vehicles for more efficient MCS deployment.

Significance
The paper claims that the proposed ECM-based framework offers a unified and physically interpretable approach to EV evacuation planning. By explicitly modeling energy feasibility through voltage-based constraints and integrating the flexibility of Mobile Charging Stations, the method significantly improves evacuation efficiency and robustness, particularly in scenarios with limited initial battery charge or disrupted fixed infrastructure. The study highlights that MCSs are critical for emergency evacuations, as they can dynamically alleviate bottlenecks that fixed infrastructure cannot address during disasters. The framework provides a scalable tool for planners to jointly optimize routing, charging duration, and mobile resource deployment to ensure safe and timely evacuations.