Express Charging Lanes for Electric Vehicle Evacuation

Abstract

The long-distance evacuation of electric vehicles (EVs) presents significant challenges for disaster management owing to their limited driving range and constrained charging infrastructure. EVs with long charging times can create negative externalities for all other EVs waiting in queues, especially during evacuation scenarios. This study investigates the use of express charging lanes to reduce overall evacuation delays. We propose optimization models to optimize the allocation of charging plugs for express and regular charging, considering both user-equilibrium and system-optimal scenarios. To account for heterogeneities and uncertainties, such as stochastic EV arrival patterns and variable charging demands, we further develop numerical simulation models to quantify the delay distribution. We found that separating EVs with lower charging demand had the potential to minimize total system delay. The proposed models identified the optimal level of express charging plug allocation to minimize the total charging delay without centralized enforcement of traffic distribution. In addition, the models could enable government agencies to estimate the required charging resources to fulfill an evacuation within a given time window. The insights generated by the proposed theoretical models were validated using agent-based simulation, in which uncertainties could be flexibly represented.

Introduction

The increasing trend of transportation electrification has introduced new complexities to emergency evacuation planning ( 1 ). Compared with internal combustion engine vehicles, battery electric vehicles require significantly longer charging times to complete long-distance travel given their limited battery capacity and charging power ( 2 , 3 ). These characteristics can lead to significant congestion at charging stations (CSs), undermining evacuation efficiency and safety. Therefore, an effective emergency charging management strategy for electric vehicle (EV) users is needed to ensure societal resilience and well-being.

In this paper, we focus on investigating the potential value of adding “express” charging lanes in relation to charging delay. Although the concept of express lanes has already been adopted in various domains, such as supermarket checkouts, highway transportation, airport boarding, banking services, and the healthcare industry, the rules and motivations of express lanes can differ dramatically across domains. For example, highway transportation may adopt express lanes to encourage ride sharing (e.g., high-occupancy vehicles [HOV] lanes [ 4 ]), whereas the healthcare industry implements express lanes (e.g., priority scheduling [ 5 ]) to ensure the timely treatment of high-risk patients. The motivation for adopting express lanes for EV charging is two-fold. First, the timely evacuation of all vehicles is critical, and long-charging EVs can create major bottlenecks. Separating EVs with different charging demands in multiqueues could limit this negative externality to those EVs with lower charging demands. Second, during evacuation scenarios, it would be beneficial to minimize charging demand given the limited charging resources. Although it is impractical for charging operators to enforce a maximum charging demand, adopting express lanes would encourage EV drivers to only charge the energy amount needed for their evacuation.

There has been a noticeable amount of literature on EV evacuation. Babaei and Wong analyzed the main challenges in the use of EVs during disaster scenarios and identified that a lack of charging infrastructure is one of the most significant ongoing issues ( 6 ). Several studies have focused on the planning of charging infrastructure to improve resilience in extreme events. For example, Zhang et al. proposed a multiobjective optimization model to plan flood-resilient CSs to maximize charging convenience while minimizing the impacts of flood hazards ( 7 ). Tang et al. optimized the placement of mobile CSs in addition to departure and charging scheduling to support EV evacuation ( 8 ). Lu proposed a two-stage stochastic program to optimize EV charging infrastructure allocation considering uncertain evacuation times and hurricane severity ( 9 ). Although addressing questions such as optimal station locations, network coverage, and capacity expansion, these studies do not examine the operational strategies of CSs, particularly under high-stress scenarios like emergency evacuations.

From an evacuation planning perspective, given the limited charging infrastructure, Li et al. studied when to evacuate and which route and CSs an EV should use to efficiently evacuate ( 10 ). They assigned CSs proportionally to meet charging demand. Purba et al. solved an evacuation route planning problem using a branch-and-price matheuristic algorithm based on column generation ( 11 ). MacDonald et al. used queueing models and microscopic traffic simulation to evaluate the impacts of EVs during mass evacuations ( 12 ). Instead of relying on passenger EVs for evacuation, Zhang and Zhang investigated the potential efficiencies, equity, and economic advantages of coordinated electric buses ( 13 ). However, the strategic operation of CSs (such as incorporating express lanes) to optimize the charging delay was not considered in the aforementioned literature.

For EV charging during normal scenarios, researchers have actively investigated novel management strategies to reduce waiting times. For example, Tian et al. ( 14 ) and Ammous et al. ( 15 ) strategically directed EV fleets to CSs to minimize the waiting time, charging costs, or both. However, these studies required centralized control and full information of EV travel trajectories, which is not realistic for passenger EVs. We identified only a few studies on traffic distribution to CSs, considering user equilibrium (UE) instead of system-optimal (SO) assignment. For example, some studies have used pricing mechanisms to optimize social welfare (including waiting time) and/or profits without enforcing EV drivers’ decisions ( 16 ). However, pricing mechanisms are not suitable during emergencies since they can create significant social justice issues (e.g., populations with higher incomes being assigned higher priority to charge and evacuate). Other studies have explored priority-based models to schedule charging times. For example, Xu et al. proposed a “less laxity and longer remaining processing time” principle to minimize the total costs to charge a fleet of EVs ( 17 ). However, this setting is not applicable in evacuation scenarios as EVs are trying to evacuate as soon as possible and should not provide charging laxity voluntarily. In addition, literature on the daily operation of CSs typically considers balanced supply and demand scenarios, which are not applicable when a surge of evacuation traffic is likely to overload the CS service capacity for an extended period of time.

In the priority-queue literature, separating “fast” from “slow” customers has been explored in several service settings that are analogous to the trade-offs faced with express charging. From a theoretical standpoint, dedicating one server to short jobs and another to long jobs has been shown to reduce delay under certain conditions ( 18 ). Empirical studies in retail support this finding. An item-count threshold for express checkouts can reduce average wait time when demand variability and load are high ( 19 , 20 ). Priority lanes have also been studied on highways. Menendez and Daganzo showed that a reserved HOV lane can smooth bottleneck flow and raise overall throughput if utilization and merge friction are managed ( 4 ). A recent fast-charging study by Kakillioglu et al. extended the idea to EV infrastructure ( 21 ). It developed a feedback-control scheme for everyday station operations, which reserves “express” chargers for drivers seeking a quick top-up (or paying a premium) to maintain a preset delay advantage, while the standard queue remains stable. However, Kakillioglu et al. did not study the impacts of express lanes on the total waiting delay nor on the non-pricing-based resource allocation strategies to achieve total delay minimization.

In general queueing literature, multiclass and priority queueing systems have been extensively studied. For example, Rothkopf and Rech conceptually discussed key trade-offs between multiqueue and combined queue systems and concluded that combined queues may not always be preferred ( 22 ). One of the cited advantages of a multiqueue system is that it allows segregation of jobs by service time (i.e., express lanes), which can help reduce service time variability. Empirical studies have also tried to quantify the impacts of express lanes in different domains. Sleptchenko et al. proposed an exact solution approach for the state probabilities of a multiclass queue with preemptive priorities ( 23 ). Heterogeneous queueing lanes were not considered. Kwak investigated the effect of express checkouts in retail stores through simulation and found that when the arrival rate is high and the regular service time is long, the average queue length can be reduced by having express checkout lanes ( 19 ). Kalló and Koltai proposed both analytical (i.e., M/G/1 and M/G/k) and simulation approaches to understand the optimal limit to qualify for the express lines ( 20 ). Henprasert et al. proposed a tandem queueing model to model the shopping and checkout process at retail stores with express lanes ( 24 ). A simulation platform was proposed to evaluate the performance. However, these results do not easily generalize to EV evacuation owing to the unique assumptions of retail store settings.

Whereas multiclass and priority queueing systems have been extensively studied, the application of express lanes to EV evacuation scenarios—particularly under surge conditions with oversaturated flow rates—remains limited. Moreover, the notions of UE and SO have never been discussed in priority-queue literature. To the best of the authors’ knowledge, this is the first attempt to adopt the concept of priority queues to CS operations to support EV evacuation considering both SO and UE flow distributions. We propose a new modeling framework to capture the heterogeneity in charging demand, charging locations, and charging lane types, as well as using rigorous mathematical modeling and derivations to study the potential impacts of express lanes on evacuation delay under both UE and SO traffic patterns. Therefore, the main contribution of our work is to investigate the impacts of express lanes for EV evacuation, considering UE/SO traffic distribution, heterogeneous energy needs, charging resource allocation, and express-lane qualification threshold.

The remainder of this paper is organized as follows: Section 2 summarizes the problem settings and specific research questions. Section 3 presents the methodologies for both SO and UE. Section 4 discusses the results to generate insights, and Section 5 concludes the paper with a summary and suggestions for future extensions to the research.

Problem Statement

In this paper, we consider a set of CSs, $\mathcal{J}$ , serving an evacuation corridor (i.e., one origin–destination pair), where $\mathcal{I}$ is the set of EV types, characterized by the level of initial state of charge (SOC); and $D_i$ is the total EV evacuation traffic flow $D_i$ for each type of EV. EVs with the energy capacity to finish the evacuation trip without en route charging were excluded from $D_i$ . We assume that each EV prefers to charge once, at most, as multiple charging stops could result in significant delays owing to the additional time spent pulling in and out of CSs. Evacuees have the flexibility to decide which CS/plug they will use.

To enhance system-level evacuation efficiency, each CS has the option to dedicate a portion of charging plugs to EVs with a charging demand less than a certain threshold, $\zeta$ . The qualifying EVs have the option to be queued in a separate lane, denoted as “express lane” in this paper. We assume that the express lanes and regular lanes have the same charging power per plug and have the same charging prices per unit of energy charged, since we did not want to introduce price and charging speed discrimination during emergencies. We assume that both types of lane have an unbounded queueing capacity that can accommodate the evacuation traffic. The EV owners choose their refueling plans and charging locations/lanes to minimize their evacuation delay. Since this paper focuses on delay at CSs, we assume a constant travel time and en route energy consumption per unit of travel distance, independent of traffic flow. This assumption is plausible since EVs still make up a relatively small portion of passenger vehicles, especially in evacuation traffic, and therefore would not have a significant influence on the travel time.

Figure 1 illustrates a toy evacuation corridor. EVs depart the origin, traverse the corridor, and arrive at the destination. Each station, CS₁ to CS₃, offers a “regular” (R) queue (blue plugs) and an “express” (E) queue (orange, dashed-line plugs). Purple cars represent high-SOC (i.e., low energy demand) EVs and yellow cars represent low-SOC (high energy demand) EVs. We can see that low-SOC EVs are not eligible for express lanes and can only queue at the regular lanes in this example (Figure 1). In addition, CS₃ is considered infeasible for low-SOC EVs because their initial energy is not sufficient for them to reach that particular CS, thus, CS₃ can only serve high-SOC EVs.

Figure 1.

Schematic of the evacuation corridor showing origin, destination, charging stations (CS₁ to CS₃) with regular (R) and express (E) plugs, and representative high- and low-SOC EVs.

In this paper, we aim to answer the following research questions:

• Will express lanes reduce the total delay at CSs?

• How much service capacity should be allocated to express lanes?

• How much greater is the total charging delay under UE traffic distribution compared with the SO solution when express lanes are implemented?

• How many plugs are sufficient to ensure a timely evacuation within a desired time window?

• What is the optimal SOC cutoff threshold to qualify for using express lanes?

Methodology

In this section, we first model the express lanes in deterministic scenarios, where the arrival and charging demands are deterministic (i.e., D). We assume that EV drivers know (or can predict) the expected waiting time when they make their charging decision and will optimally select the most efficient CSs/lanes for them. As a result, theoretical queueing models based on D/D/k are adopted for modeling both the regular and express lanes, where $k$ is the number of charging plugs for each lane type. We then extend our analyses to consider the uncertainties of EV arrival patterns following Poisson distributions, and EV charging demand following a truncated normal distribution or uniform distribution. M/G/k queueing models can be implemented in a simulation framework to evaluate the impacts of uncertainties ( 20 ).

Deterministic Queueing Models

In this section, we formulate the queueing models with deterministic arrival patterns and charging demands for SO and UE scenarios. The key notation is summarized in Table 1.

Table 1.

Glossary of Main Symbols

Variables
$f_{i,j,l}$	Traffic flow of EV type $i$ charged at station $j$ , lane $l$
${\alpha }_{j,l}$	The percentage of plugs at station $j$ assigned to lane $l$
Intermediate quantities
$w_{j,l}$	Total charging delay (including waiting and charging times) at station $j$ , lane $l$
$d_{j,l}\left(t\right)$	Perceived charging delay at CS $j$ and lane type $l$ at time $t$
${\tau }_i\left(t\right)$	Minimum delay for EV type $i$ at time $t$ for all available charging lanes
$z_{j,l}$	Auxiliary variable used for linearization in complementarity conditions
${\overline{t}}_{j,l}$	Queue dissipation time for lane $l$ at station $j$ without pooling plugs
$\Delta t_j$	Time gap between lane-dissipation events at station $j$
${\overline{\lambda }}_{j,l}$	Mean arrival rate at station $j$ , lane $l$
${\overline{\mu }}_{j,l}$	Saturated departure rate before pooling
${\overline{\overline{\mu }}}_{j,l}$	Saturated departure rate after pooling
Sets
$\mathcal{I}$	Set of EV types (by initial SOC)
$\mathcal{J}$	Set of charging stations (CSs)
${\mathcal{J}}_i$	CSs reachable by an EV of type $i$
$\mathcal{L}$	Set of lane types (regular, express)
${\mathcal{L}}_{i,j}$	Lanes eligible for EV type $i$ at station $j$
Parameters
$T$	Evacuation time window
$D_i$	Total evacuation demand of EV type $i$
$E_{i,j}$	Energy required by type $i$ if charging at station $j$ (kWh)
$\overline{E}$	Upper bound of stochastic charging-energy demand (kWh)
$\underset{¯}{E}$	Lower bound of stochastic charging-energy demand (kWh)
$P_{j,l}$	Charging power per plug on lane $l$ at station $j$ (kW)
${\alpha }_{0,l}$	Minimum plug share reserved for lane $l$
$\zeta$	Energy-demand threshold qualifying an EV for express lanes (kWh)
$t_j$	Travel time from origin to station $j$
$N$	Number of plugs available at a station
$k$	Number of plugs per lane in generic D/D/ $k$ queues

Note: EV = electric vehicle; CS = charging station; SOC = state of charge.

System Optima

In an SO scenario, charging operators dictate the distribution of traffic flow to each CS and each lane type, in addition to the percentage of charging plugs being used for express lanes, to optimize the evacuation efficiency. The objective is to minimize the total waiting time, which is formulated as Equation 1.

$$\underset{\mathit{f},\mathbf{\alpha },\mathit{w}\ge 0}{\min }\sum _{j\in \mathcal{J},l\in }^{\mathcal{L}}{w}_{j,l}$$

(1)

In relation to constraints, first, we needed to ensure evacuation traffic-flow conservation for each type of EV, as shown in Constraint 2.

$$\sum _{j\in \mathcal{J},l\in \mathcal{L}}{f}_{i,j,l}=D_i,\forall i\in \mathcal{I}$$

(2)

Second, the percentage of charging plugs allocated to each lane type should total 1 (Equation 3). It was not possible to allocate no charging resource to either type of lane, otherwise the problem would degenerate into a uniform CS operation problem. Therefore, we added Constraint 4 to ensure that lane type $l$ received a minimum ${\alpha }_{0,l}$ fraction of plugs.

$$\sum _{l\in \mathcal{L}}{\alpha }_{j,l}=1,\forall j\in \mathcal{J}$$

(3)

$${\alpha }_{j,l}\ge {\alpha }_{0,l},\forall j\in \mathcal{J}$$

(4)

Third, we needed to formulate the relationship between the waiting time and our decision variables, $f$ and $\alpha$ . To do that, we calculated the arrival and departure rates of the charging queues. We assume that the government evacuation order requires all residents in the affected area (i.e., the origin) to be evacuated within a time window, $T$ . The travel time from the origin to each CS is $t_j$ . Therefore, for $t\in [t_j,t_j+T]$ , the arrival rate for station $j$ lane type $l$ can be expressed as Equation 5.

$${\overline{\lambda }}_{j,l}=\sum _{i\in \mathcal{I}}{f}_{i,j,l},\forall j\in \mathcal{J},l\in \mathcal{L}$$

(5)

The departure rates depend on whether any EVs that have the priority to be charged at CS $j$ and lane $l$ still need to be served at that lane or not. This is because when the express lanes are not being utilized by high-type (i.e., low charging demand) EVs, both charging lanes should be made available to any EVs, regardless of their charging demand, to ensure a timely evacuation. The queue dissipation time, ${\overline{t}}_{j,l}$ , at each lane type can be calculated by Equation 6, where the nominator ( $\sum _{i\in \mathcal{I}}{f}_{i,j,l}T$ ) is the total number of EVs arriving at station $j$ lane $l$ , and ${\overline{\mu }}_{j,l}$ is the saturated departure rate before the queue dissipates.

$${\overline{t}}_{j,l}=\frac{\sum _{i\in \mathcal{I}}{f}_{i,j,l}T}{\min ({\overline{\mu }}_{j,l},\sum _{i\in \mathcal{I}}{f}_{i,j,l})}+t_j,\forall j\in \mathcal{J},l\in \mathcal{L}$$

(6)

The saturated departure rate, ${\overline{\mu }}_{j,l}$ , depends on the mix of EV flows that have arrived at CS $j$ and lane type $l$ , as well as their service time, $\frac{E_{i,j}}{P_{j,l}}$ . For example, if more high charging demand EVs arrive at the regular lane, the saturated departure rate should be lower. Mathematically, ${\overline{\mu }}_{j,l}$ can be calculated by Equation 7, where $N{\alpha }_{j,l}$ is number of charging plugs for lane type $l$ ; and $\frac{\sum _{i\in \mathcal{I}}{f}_{i,j,l}}{\sum _{i\in \mathcal{I}}{f}_{i,j,l}\frac{E_{i,j}}{P_{j,l}}}$ is inverse of the average service time needed per vehicle queueing at station $j$ lane $l$ , which is the average departure rate per plug.

$${\overline{\mu }}_{j,l}=N{\alpha }_{j,l}\frac{\sum _{i\in \mathcal{I}}{f}_{i,j,l}}{\sum _{i\in \mathcal{I}}{f}_{i,j,l}\frac{E_{i,j}}{P_{j,l}}},\forall j\in \mathcal{J},l\in \mathcal{L}$$

(7)

Note that the system cannot be optimal when queues at the regular lanes have dispersed at some point while the express lanes at the same CS still have queues. This is because high-type EVs can freely switch to regular lanes to save total waiting time. In addition, since we considered evacuation scenarios comprising a surge of evacuation traffic, the regular lanes, $l\prime$ , will always be operating at a saturated departure rate—otherwise, more high-type EVs could use the regular lanes without causing additional queueing, and reduce the queue length at the express lanes, $l″$ .

By substituting ${\overline{\mu }}_{j,l}$ in Equation 6 using Equation 7, we have the queue dissipation times as a function of $f$ and $\alpha$ ,

$${\overline{t}}_{j,l\prime }=\frac{T}{N{\alpha }_{j,l\prime }}\sum _{i\in \mathcal{I}}{f}_{i,j,l\prime }\frac{E_{i,j}}{P_{j,l\prime }}+t_j,\forall j\in \mathcal{J}$$

(8)

$${\overline{t}}_{j,l″}=\max (\frac{T}{N{\alpha }_{j,l}}\sum _{i\in \mathcal{I}}{f}_{i,j,l″}\frac{E_{i,j}}{P_{j,l″}}+t_j,T+t_j),\forall j\in \mathcal{J}$$

(9)

We assume that once lane $l″$ has fully dissipated its queue, all EVs in the other lane, $l\prime$ , can access the entire set of charging plugs. This assumption is particularly justifiable in EV evacuation scenarios, where it is critical to fully utilize any available charging capacity to minimize delay, especially once the originally prioritized group of EVs has been served. As a result, the aggregate departure rate of CS $j$ at lane $l\prime$ after the lane $l″$ queue has dissipated can be calculated in Equation 10 until the queue in lane $l\prime$ is dissipated (i.e., $t\in [{\overline{t}}_{j,l″},{\overline{t}}_{j,l″}+\Delta t_j]$ ).

$${\overline{\overline{\mu }}}_{j,l\prime }=N\frac{\sum _{i\in \mathcal{I}}{f}_{i,j,l\prime }}{\sum _{i\in \mathcal{I}}{f}_{i,j,l\prime }\frac{E_{i,j}}{P_{j,l\prime }}},\forall j\in \mathcal{J}$$

(10)

where $\Delta t_j$ is difference between the queue dissipation times of both lane types and can be formulated in Equation 11, where the nominator is how many EVs are still waiting at lane type $l\prime$ at time ${\overline{t}}_{j,l″}$ .

$$\Delta t_j=\frac{{\overline{\lambda }}_{j,l\prime }T-{\overline{\mu }}_{j,l\prime }({\overline{t}}_{j,l″}-t_j)}{{\overline{\overline{\mu }}}_{j,l\prime }},\forall j\in \mathcal{J}$$

(11)

The total delay at lane $l″$ , whose queue dissipates first, can be calculated using Equation 12.

$$w_{j,l″}\ge \max (\frac{1}{2}{T}^2(\frac{{\overline{\lambda }}_{j,l^{″}}}{{\overline{\mu }}_{j,l^{″}}}-1){\overline{\lambda }}_{j,l^{″}},0),\forall j\in \mathcal{J}$$

(12)

The total delay at lane $l\prime$ , whose queue cannot dissipate before lane $l″$ , can be calculated using Equation 13. Note that when both lanes dissipate queues at the same time, $\Delta t_j=0$ and the queues at both lanes will follow the same form.

$$w_{j,l\prime }\ge \frac{1}{2}{T}^2(\frac{{\overline{\lambda }}_{j,l\prime }}{{\overline{\mu }}_{j,l\prime }}-1){\overline{\lambda }}_{j,l\prime }-\frac{1}{2}\Delta t_j^2(\frac{{\overline{\overline{\mu }}}_{j,l\prime }}{{\overline{\mu }}_{j,l\prime }}-1){\overline{\overline{\mu }}}_{j,l\prime }$$

(13)

Last but not least, we can impose restrictions on certain lane types, $l$ , at CS $j$ being accessed by certain EV types, $i$ . For example, if low-type EVs (Type $1$ EVs) cannot access express lanes (Lane Type $2$ ) because of the high charging demand, $E_{i,j}$ , we can specify that the corresponding flow equals zero. In addition, some CSs are infeasible for EV type $i$ owing to them being too far away from either the origin or destination, we therefore need to restrict the traffic flow to zero here as well. In a general form, we add Constraint 14 to ensure charging lane eligibility and CS feasibility.

$$f_{i,j,l}=0,\forall i\in \mathcal{I},j\in \mathcal{J}/{\mathcal{J}}_i,l\in \mathcal{L}\mathrm{or}j\in {\mathcal{J}}_i,l\in \mathcal{L}/{\mathcal{L}}_{i,j}$$

(14)

The SO optimization problem is summarized in Model 15:

$$\underset{\mathit{f},\mathbf{\alpha },\mathit{w}\ge \mathit{0}}{\min }(\mathrm{Equation}1)$$

(15a)

$$\mathrm{s}.\mathrm{t}.(\mathrm{Equations}2\mathrm{to}14)$$

(15b)

User Equilibria

Centrally coordinating the flow distribution of all EVs is neither scalable nor practical given the decentralized nature of evacuation behavior. Next, we consider that EV drivers can select any CSs and lanes that are eligible and feasible to minimize their waiting time. Since we assume that high-type EVs can access both types of lane whereas low-type EVs can only access regular lanes, the equilibrium conditions need to be specified for different EV types based on their feasible CSs and eligible lanes. Mathematically, we can express the equilibrium conditions as complementarity conditions (Equation 16).

$$\begin{array}{c}\multicolumn{1}{c}{0\le d_{j,l}(t+t_j)-{\tau }_i\left(t\right)\perp f_{i,j,l}\ge 0,}\\ \multicolumn{1}{c}{\forall t\in [0,T],i\in \mathcal{I},j\in {\mathcal{J}}_i,l\in {\mathcal{L}}_{i,j},}\end{array}$$

(16)

where $d_{j,l}(t+t_j)$ is perceived waiting time at CS $j$ , lane $l$ , at time $t+t_j$ when EVs depart the origin at time $t$ . Note that instead of considering the current waiting time at time $t$ , EVs need to consider the travel time, $t_j$ , to reach that CS and predict the expected waiting time when they arrive at CS $j$ based on current arrival and departure rates. ${\tau }_i\left(t\right)$ is minimum waiting time for all available charging lanes for EV type $i$ if they depart from the origin at time $t$ .

Since the current arrival and departure rates are time invariant for $t\in [0,T]1$ (the arrival rate is linear because we assume the traffic flow is time invariant within a given period. The departure rate is time invariant because it is either the saturation flow rate when there is queue formed in a lane or it is equal to the arrival rate, which is time invariant), the perceived waiting time, $d_{j,l}(t+t_j)$ , can be calculated in Equation 17, which is the travel time, $t_j$ , plus the clearing time for all previous EVs departing before time $t$ minus the arrival time, $t+t_j$ . Note that Equation 17 is a linear function of $t$ in our problem settings.

$$\begin{array}{c}\multicolumn{1}{c}{d_{j,l}(t+t_j)=\max [t_j+\frac{\sum _i{f}_{i,j,l}t}{{\overline{\mu }}_{j,l}}-(t+t_j),0]}\\ \multicolumn{1}{c}{=\max [(\frac{\sum _{i\in \mathcal{I}}{f}_{i,j,l}\frac{E_{i,j}}{P_{j,l}}}{N{\alpha }_{j,l}}-1)t,0],}\\ \multicolumn{1}{c}{\forall t\in [0,T],j\in \mathcal{J},l\in \mathcal{L}}\end{array}$$

(17)

Since the arrival and departure flow rates from CS $j$ lane $l$ is constant over the duration $[t_j,t_j+T]$ , ${\tau }_i\left(t\right)$ is linear function of $t$ , denoted as ${\tau }_i\left(t\right)={\tau }_{i}t$ .

Based on Equations 16 and 17, the UE condition can be written as Equation 18, which is independent of time $t$ . Note that Equation 18 is not a standard complementarity condition because of the max operator. To reformulate it into a standard form, we introduce an intermediate variable $z_{j,l}=\max [(\frac{\sum _{i\in \mathcal{I}}{f}_{i,j,l}\frac{E_{i,j}}{P_{j,l}}}{N{\alpha }_{j,l}}-1),0]$ , so that Equation 18 can be further represented in a complementarity condition (Equation 19). In such a case, the UE, given $\alpha$ , can be solved by optimizing the objective function (Equation 1) subject to Equations 2 to 14, and complimentary conditions (19a–19b). We note that other linear reformulations of Equation 18 are available but may require modification of the objective function.

$$\begin{array}{c}\multicolumn{1}{c}{0\le \max [(\frac{\sum _{i\in \mathcal{I}}{f}_{i,j,l}\frac{E_{i,j}}{P_{j,l}}}{N{\alpha }_{j,l}}-1),0]-{\tau }_i\perp f_{i,j,l}\ge 0,}\\ \multicolumn{1}{c}{i\in \mathcal{I},j\in {\mathcal{J}}_i,l\in {\mathcal{L}}_{i,j}}\end{array}$$

(18)

$$0\le z_{j,l}-{\tau }_i\perp f_{i,j,l}\ge 0,i\in \mathcal{I},j\in {\mathcal{J}}_i,l\in {\mathcal{L}}_{i,j}$$

(19a)

$$0\le z_{j,l}-(\frac{\sum _{i\in \mathcal{I}}{f}_{i,j,l}\frac{E_{i,j}}{P_{j,l}}}{N{\alpha }_{j,l}}-1)\perp z_{j,l}\ge 0,j\in \mathcal{J},l\in \mathcal{L}$$

(19b)

Simulation-Based Stochastic Queueing Models

In the section covering deterministic queueing models, we derived theoretical models for both system optima and user equilibria, assuming deterministic EV arrival and charging patterns. To further investigate the impact of randomness on the CS management strategies and charging delay, we consider random EV departure patterns from the origin and random charging energy demand in this section. Although rigorous mathematical modeling of UE in a stochastic environment is challenging, the impacts of randomness can be easily quantified by agent-based simulation. We adopted Python/SimPy as our simulation platform. The resource is charging plugs, whereas the requests made by EVs are based on their preference (i.e., shorter waiting time) and eligibility. We assume that EVs depart from the origin, follow a Poisson process with an expected flow rate, $D_i$ . The charging demand follows either a truncated normal distribution or a uniform distribution between the lower and upper bounds $[\underset{¯}{E},\overline{E}]$ .

The simulation flowchart is summarized in Figure 2. At the top level, an outer loop iterates across SOC-eligibility thresholds, $\zeta$ , and assesses the express-lane eligibility of EVs. For each $\zeta$ , we then perform $R$ random simulations (the inner loop). A single iteration corresponds to one complete evacuation from time 0 to $T$ . Concretely, each iteration proceeds as follows:

1. Environment setup: Instantiate each CS, allocate express and regular plug resources according to the power split, and reset all performance metrics.

2. Arrival generation: Schedule EV departures from the origin as a Poisson process of expected rate, $D_i$ , over the evacuation horizon, $T$ .

3. Energy sampling: For each EV, sample an initial SOC and compute its minimum required charging energy from the specified distribution on $[\underset{¯}{E},\overline{E}]$ .

4. Eligibility and routing: Each EV whose required energy is equal to or less than threshold, $E\le \zeta$ , is eligible for the express queue; otherwise it joins the regular queue. It then chooses the CS and lane with the shorter predicted wait time (UE decision).

5. Service and logging: Each EV waits for its turn, seizes a plug, charges for the duration $\frac{\mathrm{energy}\mathrm{demand}}{\mathrm{charging}\mathrm{power}}$ , and departs. We record its waiting and charging times.

6. Replication summary: Once the final EV has departs, compute and store performance metrics (mean delay, throughput) for that replication.

After all $R$ replications for a given $\zeta$ are complete, we calculate the per-run statistics to yield final performance metrics at that threshold. We then advance the outer loop to the next $\zeta$ and repeat. The simulation framework enables us to quantitatively assess the systematic effects of $\zeta$ and the impacts of stochasticity, such as arrival patterns and charging demand variability.

Figure 2.

Workflow of the EV charging simulation.

Results

Toy Examples

In the base case, an evacuation route with 100 DC fast-charging plugs of 60 kW charging power was considered. Without loss of generality, all charging plugs were assumed to be feasible for finishing the evacuation trips for both types of EV; otherwise, they were excluded from the analysis. The total flow rates of low- and high-type EVs were 100 and 500 vph, respectively, for an evacuation time window of $T=8$ h. This 8-h window represents a policy maker–defined target evacuation from the origin, and was used to determine the arrival rates and charging loads at CSs. The minimum required charging energy to finish their evacuation trip was 30 kWh for low-type EVs and 10 kWh for high-type EVs. Only high-type EVs were eligible for the express lanes. All EVs were assumed to charge only the minimum required energy to ensure a timely evacuation. The models were solved using IPOPT software ( 25 ).

First, the total charging delay was compared between the UE and SO scenarios. We considered different percentages of charging plug allocation to the regular and express lanes, where ${\alpha }_1$ is percentage of charging power allocated to the regular lane. The results are shown in Figure 3.

• When the percentage of charging power allocated to regular lanes was small (i.e., ${\alpha }_1<0.16$ ), the total delay decreased with the increase of ${\alpha }_1$ and the UE and SO led to identical solutions. The reason for this is that when ${\alpha }_1$ is too small, the regular lanes are overcongested whereas the express lanes are underutilized since only high-type EVs are eligible to use them. Therefore, the lower the ${\alpha }_1$ , the greater the underutilization of the express lanes and the higher the total delay. In addition, when ${\alpha }_1$ is low, all high-type EVs will use express lanes regardless of whether it is a UE or SO scenario.

• When ${\alpha }_1>0.16$ , the total delay increases with the increase in ${\alpha }_1$ . The difference in total delay between UE and SO increases when $\alpha \in [0.16,0.4]$ and then decreases when $\alpha \in [0.4,1]$ . This difference between UE and SO can be explained by the externality that UE behavior will cost the system. When we compared the flow to each lane (see Figure 4), UE resulted in more traffic at express lanes compared with SO. This is because each EV tries to minimize its waiting time in UE, and express lanes have relatively shorter waiting times. However, from an SO perspective, it is better that high-type EVs utilize both lanes so that the initial departure rate can be maximized. This is beneficial because if all the plugs are fully utilized, the final queue dissipation time is fixed. A higher initial departure rate reduces the total waiting time because EVs that arrive later will be less delayed.

• When ${\alpha }_1$ approaches 1, SO and UE produce an identical total delay because the problem degenerates to the case without express lanes. It was observed that once the express lanes were fully utilized (i.e., ${\alpha }_1>0.16$ ), adding express lanes to the CSs does not increase total charging delay under either UE or SO. In SO, adding express lanes led to less total charging delay, no matter how many charging plugs were allocated to the express lanes. In UE, when ${\alpha }_1$ was greater than 0.4, the total delay was the same as the no-express-lane scenario.

• Figure 3 also provides insights on the optimal operation of CSs. For example, to minimize the total delay, ${\alpha }_1$ should be set to around 0.16 for this numerical example. Note that when ${\alpha }_1=0.16$ , UE and SO will have the same traffic-flow pattern, therefore the charging operator can share the waiting time across EVs to allow them to make the best decision for themselves without dictating the charging flow distribution. However, we note that equity has not been considered in this study. When ${\alpha }_1=0.16$ , which means that the majority of charging plugs are used for express lanes, we are prioritizing charging resources for high-type EVs. Although this approach will encourage all EVs to charge the minimum amount during evacuation, some EVs may have to charge more energy owing to a lack of home charging opportunities. A valuable future research direction would be to consider a multiobjective optimization approach to balance considerations of total system efficiency and individual EV groups’ equity.

Second, the impact of the number of charging plugs on queue dissipation time was investigated. A key question for emergency management is how much charging resource will be sufficient to ensure the full evacuation of all EVs by a given time. To answer this question, we investigated the relationship between the number of plugs and the minimum time needed to dissipate all the queues in UE, given the optimal charging plug allocation. The results are shown in Figure 5: the 24-h horizon is to illustrate how many charging plugs are required to fully clear the queues within 24 h considering an 8-h arrival surge. In this example, it can be seen that approximately 45 charging plugs were required to clear all queues within 24 h. Whereas the arrival demand was generated over an 8-h evacuation window, charging service continued beyond the 8 h until all queues had cleared. Furthermore, from Figure 5 we can see that the marginal benefits of reducing queue dissipation time decreased with an increase in the number of plugs. Such information could be utilized to justify a reasonable level of charging plug investment to ensure that the marginal benefits of an additional unit of charging plugs can outperform the additional costs incurred.

Figure 3.

Comparing the minimum waiting times between user equilibrium and system optimal.

Figure 4.

Comparing the high-type electric vehicle (EV) flow distribution in user equilibrium (UE) and system optimal (SO).

Figure 5.

Impact of the number of plugs on queue dissipation time.

Third, the impacts of stochastic arrival on total waiting time were evaluated. Although it is widely acknowledged that random arrivals will increase the total delay, the extent to which randomness will affect the existence of express lanes remains unexplored. We considered two scenarios. The first scenario comprised 16% of the plugs being allocated to regular lanes, and the second scenario investigated 100% of the plugs being dedicated to regular lanes (i.e., no express lane). We considered the arrival pattern to follow a Poisson distribution with the expected arrival rates identical to the base case. We repeated the simulation 100 times in Python/SimPy to quantify the probability distribution of total charging delay. The results are shown in Figure 6. For both scenarios, total delay in the deterministic case was within the 25th- and 50th percentiles when the arrival rate followed a Poisson distribution. This was as expected because arrival randomness will lead to temporary overloads and queue buildup even if the average arrival rate equals the average service rate. When there were express lanes, the difference between the deterministic average waiting time and the median waiting time was larger compared with the case without express lanes. This is because without express lanes, all EVs will have to queue in one lane. Since the charging time is lengthy, the queue will build up quickly and persist; therefore, random arrivals will not cause a major difference to the average waiting time. On the other hand, with express lanes, the variance in charging time can be reduced because only EVs with lower charging demand will use the express lanes. The express lanes have a faster service rate, which results in a more balanced arrival and departure pattern. When arrivals are uncertain, there is a higher likelihood of the queue building up, which leads to a higher average waiting time. This should be taken into account, especially when designing a CS with express lanes.

Figure 6.

Impact of arrival uncertainties on average delay.

Fourth, we investigated the impacts of the eligibility threshold for express lanes on the total charging delay, considering the stochasticity and heterogeneity of EV energy demand and the arrival process. We assumed a Poisson distribution of EV arrivals with an average arrival rate equal to 600 EVs per hour. We assumed four probability distributions for energy demand, a truncated normal distribution (mean: 50 kWh, SD: 10 kWh), a truncated lognormal distribution (mean: 50 kWh, SD: 10 kWh), a uniform distribution between 20 and 80 kWh, and a bimodal distribution (mean: 30 kWh and 70 kWh, SD: 10 kWh). These four probability distributions were chosen to demonstrate the impact of the shape and variance of charging demand on charging delay. We considered a total of 300 charging plugs, 50% of which were prioritized for EVs with charging demand less than threshold $\zeta$ . The other parameters were kept the same as the base case. The results are shown in Figure 7.

• When the threshold, $\zeta$ , was set to the lower bound of the energy demand, no one had priority to charge at the express lane. On the contrary, when the threshold, $\zeta$ , was set to the upper bound of the energy demand, all EVs were eligible to use the express lane. These two extreme scenarios will lead to the same results theoretically because we allow all vehicles to use the express lane if it is not occupied by the “eligible” EVs. Figure 7 confirmed this conjecture numerically: the average waiting times between $\zeta =20$ and $80$ kWh were very similar.

• The optimal charging demand threshold for all distributions was around 40 to 50 kWh. This was slightly less than the mean value of the charging demand, which was 50 kWh. If the threshold, $\zeta$ , is set too high, too many EVs will be eligible to use the express lanes, and the average charging time at the express lane may still be high, which means that those EVs that have a minimum charging demand will still need to wait for those that have a higher charging demand. If the threshold, $\zeta$ , is set too low, very few EVs will be eligible to use the express lanes, in which case the capacity at the express lanes may be underutilized for the EVs with less demand. Once the express lanes are underutilized, all EVs can take advantage of that, and the benefits of the express lanes diminish. This is the fundamental trade-off when selecting a threshold, $\zeta$ , for the express lane.

• When comparing different distributions, we observed that the impact of an optimally selected threshold, $\zeta$ , on the average waiting time was more significant for the distribution with higher variance. For example, taking the uniform and the truncated normal distributions, the worst-case medians of the average waiting time for both distributions were around 3.3 h, whereas the best-case medians were 3.15 and 2.9 h for the normal and uniform distributions, respectively. The reason for this is that the uniform distribution had a larger SD ( $10\sqrt{3}$ compared with $10$ kWh for the normal distribution). It is more beneficial to separate different charging demands into different queues to prevent EVs with minimum charging demand from being blocked by high charging demand EVs.

Figure 7.

Impact of express-lane threshold on average waiting time: (a) normal distribution, (b) lognormal distribution, (c) uniform distribution, and (d) bimodal distribution.

Real-World Case Study

Although mass evacuations dominated by EV charging have not yet occurred (since the current EV adoption rate is still low), recent disaster events have clearly demonstrated the real-world infrastructural bottlenecks under evacuation or grid-stress conditions (e.g., 2017 Hurricane Irma [ 26 , 27 ] and 2021 Texas Winter Storm [ 28 , 29 ]). To demonstrate the applicability of the proposed express charging lane framework, we developed a real-world case study focused on long-distance EV evacuation from the city of Houston, TX. Houston has a history of large-scale evacuations from hurricanes. For instance, Hurricane Rita in 2005 prompted about 2.5 million evacuations from Southeastern Texas, leading to severe traffic congestion, fuel shortages, and over 100 evacuation-related deaths ( 30 ). Hurricane Ike in 2008 saw over 1 million coastal evacuations owing to different risk perceptions ( 31 ). We examined evacuation along two major corridors: I-45 (Houston to Dallas) and I-10 (Houston to San Antonio), both critical for outbound mobility during hurricane emergencies. A schematic overview of the modeled network, including origin–destination cities and charging infrastructure configurations, is shown in Figure 8.

Figure 8.

Electric vehicle (EV) evacuation corridors between Houston and Dallas (via I-45), and between Houston and San Antonio (via I-10).

We identified the total number of DC fast CSs with a rated output ≥50 kW located on each corridor. The I-45 corridor hosts 132 plugs, whereas I-10 has 87 plugs ( 32 , 33 ). The weighted average plug power was calculated as 272.4 kW on I-45 and 289.1 kW on I-10. For simulation, we assumed a mandatory evacuation order over an 8-h window ( $T=8$ h), with a uniform departure rate. EVs traveled either 240 mi (I-45) or 200 mi (I-10), requiring 72 and 60 kWh of energy, respectively, under a consumption rate of 0.30 kWh/mi. We categorized EVs into two flow classes: a low-SOC group requiring full charge and a high-SOC group needing only 20% of that energy. We evaluated the performance of the proposed express-lane framework in two evacuation scenarios: (i) a baseline scenario where only regular charging lanes were available, and (ii) an enhanced scenario in which each station provided both express and regular lanes with equal power allocation. Both scenarios were simulated under an 8-h mandatory evacuation window from Houston, involving 40,185 EVs distributed between two corridors. The SO solution aimed to minimize the total waiting and travel time by assigning EVs to specific CSs and lane types.

Table 2 summarizes the SO solutions of delay, EV volume, and travel time across both corridors under the two lane configurations. The results highlighted how the presence or absence of express lanes influenced delay patterns across both corridors. In the regular-lanes-only scenario, all vehicles used the same charging lanes regardless of their energy needs. This led to high congestion, particularly on the I-45 corridor, where over 232,000 vehicle-hours of charging delay occurred. Much of this charging delay was caused by high-demand EVs that required longer charging times and consequently slowed down throughput for others. I-10 also experienced substantial congestion, with over 92,000 vehicle-hours of charging delay. In total, the system saw 454,089 vehicle-hours of delay (including travel time and charging delay), and the average time each EV spent in the system was 11.30 h.

Table 2.

Comparison of Travel Time, Charging Delay, and Total Delay by Corridor and System Configuration

Corridor	EV count (type)	Travel time, h	Charging delay, h	Total delay, h
Regular lanes only
I-45	17,363 (high)	60,770	232,427	293,197
I-10	20,092 (high) + 2,730 (low)	68,466	92,426	160,892
Total	40,185	129,236	324,853	454,089
Avg. delay, h	na	na	na	11.30
With express lanes
I-45 regular	10,866 (high)	38,031	116,535	154,566
I-45 express	10,638 (low)	37,233	2,778	40,011
I-10 regular	9,227 (high)	27,681	101,998	129,679
I-10 express	9,454 (low)	28,362	4,832	33,194
Total	40,185	131,307	226,143	357,450
Avg. delay, h	na	na	na	8.90

Note: EV = electric vehicle; Avg. = average; na = not applicable.

The introduction of express lanes led to a more efficient evacuation. In this configuration, low-demand EVs were assigned to express lanes with faster turnover, whereas high-demand EVs used regular lanes. This separation helped reduce competition for charging and improved overall efficiency. Total system delay dropped to 357,450 vehicle-hours, and the average time per vehicle decreased to 8.90 h. The improvement was particularly evident on I-45, where there was a sharp reduction in EVs’ total delay. I-45 express lanes alone served over 10,600 EVs with only 2,778 vehicle-hours of delay. A similar effect was seen on I-10. The total travel time across the system was roughly equal in both scenarios, confirming that improvements in average time per EV were driven by reduced charging delay. These findings demonstrated that express-lane configurations can dramatically improve evacuation efficiency by minimizing the charging delay externality caused by high energy demand EVs. This was achieved without adding new chargers, solely through the smarter allocation of existing charging capacity.

Conclusion

In this paper, we have proposed both theoretical models and numerical simulations to better understand the potential impacts of express lanes on EV waiting time during evacuation. We considered both UE and SO traffic-flow patterns in deterministic and stochastic settings. Through an illustrative example, we have presented the reasoning behind the optimal allocation of charging plugs to express lanes, the minimum number of plugs to ensure a timely evacuation, and the impacts of uncertainties and express-lane thresholds on charging delay. We found that,

• The optimal charging policy is to allocate sufficient charging resources for high-type EVs to ensure the minimum waiting time for EVs that arrive later in the queues.

• The marginal benefits of the number of plugs on queue dissipation time may significantly reduce. A necessary number of plugs can be estimated using the proposed approach, given a desired evacuation time window.

• Uncertainties play a more significant role in estimating the impacts on waiting time at express lanes, where the arrival and departure rates are more balanced.

• The overall relationship between the express-lane threshold and average waiting time remains similar across different levels of charging demand variance. However, the benefit magnitude of an optimally selected express-lane threshold was higher for the case with higher charging demand variance.

This study could be extended in several research directions. First, we will consider the steady state of evacuation traffic flow over a given evacuation time window. This assumption could be relaxed to better understand potential time-varying evacuation behavior on system charging efficiency. Second, instead of optimizing the express-lane setting given the estimated traffic flow, it would be valuable to deploy an adaptive learning approach to dynamically change the allocation of charging plugs to different types of EV based on real-time queueing lengths and EV charging behaviors. Third, we only considered a single evacuation path. When multiple evacuation paths are available, the operation of CSs is not only going to affect the charging efficiency, but also the routing behavior and travel time. A valuable next step would be to research an integrated modeling framework considering both travel and charging delay. Although this paper focuses on the operational aspect of maximizing charging efficiency given the existing charging resources, strategically placing and identifying charging resources, including stationary chargers, to support emergency evacuation could be a valuable future research direction. Moreover, one future extension of our research could incorporate equity-aware analyses that jointly consider systemwide efficiency and distributional fairness. This includes examining how priority-based charging may disproportionately affect households with limited home charging opportunities and/or higher energy needs, and developing multiobjective formulations that balance total evacuation delay with equity metrics. Finally, a promising avenue for practical deployment would be to embed the proposed model into a lightweight decision-support tool with a user-friendly interface to assist emergency management agencies in real-time operational planning.