Equivalent Circuit Model based Electric Vehicle Evacuation with Mobile Charging Stations

Abstract: The increasing penetration of electric vehicles (EVs) introduces new challenges for emergency evacuation planning due to limited driving range, long charging times, and constrained charging infrastructure, particularly under disaster induced disruptions. This paper proposes a novel optimization based evacuation framework for EVs using Equivalent Circuit Models (ECMs) to jointly address routing, charging, and congestion management. By leveraging electrical analogies, traffic flow is modeled as electrical current, travel time as resistance, and driving range as voltage, enabling the use of Kirchhoff laws to enforce flow balance and energy feasibility constraints. The proposed controllable ECM incorporates binary switches to regulate route selection and explicitly models charging delays and range replenishment at both Fixed Charging Stations (FCSs) and Mobile Charging Stations (MCSs). The resulting formulation leads to an integer programming problem that determines optimal evacuation routes, charging durations, and the placement and number of MCSs to minimize evacuation time. The framework is extended to multiple origin destination pairs using the principle of superposition and supports fairness aware performance metrics, including worst case, average, and variance based evacuation times. Simulation studies on large scale transportation networks in California demonstrate that the proposed approach significantly improves evacuation efficiency and robustness, particularly in scenarios with limited charging access, highlighting the critical role of MCSs in EV based emergency evacuations.

• The paper develops a Kirchhoff-law-based optimization framework that jointly selects EV evacuation routes, enforces driving-range limits, determines charging durations, and deploys mobile charging stations under road and service-capacity constraints.
• Simulations on Anaheim and Mariposa networks show meaningful benefits over greedy deployment, including 36–44% evacuation-time improvements in Aimsun and effective support when fixed chargers are unavailable or saturated.
• The framework improves charger utilization and fairness across evacuation groups, but its free-flow, deterministic, simultaneous-departure, and unsplittable-route assumptions limit accuracy under congestion and evolving disaster conditions.

Overview and motivation

This paper develops an optimization framework for emergency evacuation of electric vehicles (EVs) built on an equivalent circuit model (ECM) representation of the transportation network. The motivating concern is that EV-specific constraints—limited driving range, long charging times, and charging infrastructure that may be disabled or congested during disasters—are not handled by conventional evacuation planning, which typically assumes unconstrained refueling. The authors extend their prior ECM-based evacuation work (2606.03149) in three directions: incorporation of mobile charging stations (MCSs), a new voltage-based formulation of driving-range depletion via Kirchhoff's voltage law (KVL), and computation of optimal charging durations at each charging stop.

ECM formulation

The core modeling device maps traffic quantities onto circuit variables: vehicle flow corresponds to electrical current, free-flow travel time to resistance, and vehicle concentration on a link to voltage drop. Under the assumption that link flows remain within free-flow capacity, the fundamental diagram relation reduces to an Ohm's-law-like expression, $f_{ij} = (k_{ij}d_{ij})/T_{ij}$, and nodal flow conservation maps to Kirchhoff's current law (KCL). Binary switch variables $S_{ij} \in \{0,1\}$ are attached to each branch, making the model "controllable": the optimizer decides which links carry flow.

Two additional circuits encode energy feasibility. Each road edge is represented as a voltage source whose magnitude equals the edge length; applying KVL around loops containing each edge yields constraints of the form $S_{ij}(r_i - r_j - d_{ij}) = 0$, where $r_i$ is the remaining driving range at node $i$. Charging stations are handled by introducing artificial nodes connected through parallel edges: one edge represents bypassing the charger, the other encodes net range gained ($\bar{d}_{ch} = w^{ch} d_{ch} - d_{ij}$) and total time spent ($\bar{T}_{ch} = w^{ch} T_{ch} + T_{ij}$), where $w^{ch}$ is an integer number of charging intervals. This construction jointly determines route selection and charging duration.

The resulting problem $\mathbb{P}^{FCS}$ minimizes total evacuation time subject to KCL flow balance, KVL range constraints, road free-flow capacities, and fixed charging station (FCS) service rates. Under a no-path-splitting assumption (all vehicles of an origin–destination pair follow a single unidirectional route), the paper proves that the activated switches define a unique path between each origin and destination—a structural property that also rules out cyclic branches. The proof relies on degree arguments for non-branching chains and is straightforward but depends critically on the no-splitting and steady-state assumptions.

Incorporating mobile charging stations

MCSs are modeled as deployable units on candidate edges, with three distinguishing capabilities relative to FCSs: (i) multiple units can be collocated to scale service rate, (ii) they can supplement saturated FCS sites, and (iii) they can be placed on edges without any existing infrastructure. Integer variables $v^{F\text{-}MCS}$ and $S_{ij} \in \{0,1\}$0 count MCS units supporting FCS edges and standalone edges respectively, with capacity constraints $S_{ij} \in \{0,1\}$1 and $S_{ij} \in \{0,1\}$2, plus a budget constraint on the total fleet size $S_{ij} \in \{0,1\}$3. An optional penalty term $S_{ij} \in \{0,1\}$4 trades off evacuation time against the number of deployed units. The unique-path proposition extends directly to this setting.

Multiple origin–destination pairs are handled by superposition: each od-pair receives its own copy of the ECM with its own switch variables, and aggregate flows $S_{ij} \in \{0,1\}$5 must satisfy shared capacity constraints. A slack variable $S_{ij} \in \{0,1\}$6 enables worst-case ($S_{ij} \in \{0,1\}$7), average ($S_{ij} \in \{0,1\}$8), and deviation-based ($S_{ij} \in \{0,1\}$9) objectives, combinable as a weighted sum to promote fairness across evacuee groups.

Simulation results

The framework is evaluated on two California networks using Gurobi 12.0:

• Anaheim (urban): 416 nodes, 914 edges, 8 od-pairs, ~13,400 EVs at 420 vehicles/hour per od-pair, DC fast chargers at 80 km/h charging rate, 20 MCSs available.
• Mariposa (rural): 632 nodes, 1065 edges, 7 od-pairs, ~420 EVs at 60 vehicles/hour, downtown FCSs assumed deactivated, 20 MCSs at 200 km/h.

Key findings include:

Result	Magnitude
Average-time reduction from $S_{ij}(r_i - r_j - d_{ij}) = 0$0 vs. other objectives (Anaheim, $S_{ij}(r_i - r_j - d_{ij}) = 0$1 km)	~4.5%
Deviation reduction from $S_{ij}(r_i - r_j - d_{ij}) = 0$2 vs. $S_{ij}(r_i - r_j - d_{ij}) = 0$3 / $S_{ij}(r_i - r_j - d_{ij}) = 0$4 (Anaheim, $S_{ij}(r_i - r_j - d_{ij}) = 0$5 km)	~8.8% / ~29%
Average-time reduction from $S_{ij}(r_i - r_j - d_{ij}) = 0$6 vs. $S_{ij}(r_i - r_j - d_{ij}) = 0$7 (Mariposa, $S_{ij}(r_i - r_j - d_{ij}) = 0$8 km)	~68%
Evacuation-time improvement over greedy baseline in Aimsun (od-pairs 3–6)	36–44%

Notably, worst-case evacuation times are identical across all cost functions in both networks—the authors attribute this to inherent physical limitations of the network topology rather than the objective choice, which is an honest concession about what optimization can achieve. In Mariposa, where all nearby FCSs are disabled, the optimizer deploys 6 MCSs en route for a single od-pair whose initial range cannot reach any FCS, underscoring the location-independence advantage. A sensitivity study shows that below roughly 20 vehicles/hour input flow, existing FCSs suffice in Anaheim; above this threshold, required MCS counts rise sharply. Comparing against a greedy baseline that places MCSs at nearest candidate locations, the baseline saturates two sites with utilization ratios exceeding one (queue buildup), while the ECM-based deployment keeps all utilization ratios below one.

High-fidelity validation

The Mariposa scenario was replicated in Aimsun microsimulation, including EV battery tracking, queuing at chargers, and MCS battery depletion via a custom API module. Validation against the flow-based model shows errors below 6.67% for short-route od-pairs but 13.84–22.34% for longer routes, reflecting traffic control elements and speed variations absent from the macroscopic model. Against the greedy baseline implemented in Aimsun, the ECM approach improves evacuation times by 1.7–43.51% depending on od-pair, with the largest gains on routes where targeted MCS placement eliminates charging queues.

Limitations and open questions

Several assumptions constrain applicability. Flows are restricted to free-flow conditions, so congestion effects on travel time are not captured within the optimization itself; the authors acknowledge this and list dynamic congestion models as future work. All evacuees are assumed to depart simultaneously, network parameters and disaster impacts are deterministic and known in advance, and each od-group follows a single unsplittable path—all simplifications relative to real evacuations. The gap between flow-model predictions and microscopic simulation (up to 22.34%) indicates the objective values should be interpreted as planning estimates rather than precise forecasts. Open questions left by the paper include how stochastic disaster evolution and real-time MCS reallocation would alter optimal deployments, and how power-grid coupling (MCS state-of-charge, grid outages) could be integrated into the integer program.

Conclusion

The paper contributes a physically interpretable, Kirchhoff-law-based integer programming formulation that jointly optimizes EV evacuation routing, charging durations, and MCS placement under road and charging capacity constraints. Its distinctive elements are the KVL-based driving-range representation and the scalable superposition treatment of multiple od-pairs with fairness-aware objectives. Validation on urban and rural California networks, including microsimulation, demonstrates concrete improvements over greedy baselines—particularly when initial state-of-charge is low or fixed infrastructure is disrupted—while candidly identifying the steady-state, no-congestion assumptions as the principal boundary of the method's validity.