The rapid proliferation of distributed renewable energy sources has introduced significant volatility and uncertainty into power system operations. To maintain grid stability and efficiently integrate this variable generation, harnessing the flexibility of demand-side resources has become imperative. Among these resources, the battery EV car stands out as a highly flexible, mobile load. Its charging behavior, particularly in high-stakes environments like highway corridors, is critical for balancing system loads and accommodating renewable energy fluctuations. With the global fleet of battery-powered cars expanding exponentially, the aggregated, uncontrolled charging demand poses substantial challenges to grid infrastructure due to its inherent randomness and spatiotemporal concentration. Therefore, developing effective mechanisms to guide the charging decisions of battery EV car drivers is a pivotal research frontier for enabling a sustainable energy transition.
Demand Response (DR) refers to the modification of electricity consumption patterns by end-users in response to market signals, incentives, or direct instructions. For a battery EV car, this typically manifests in two ways: Vehicle-to-Grid (V2G) services, where the car feeds power back to the grid, and price-based mechanisms, where charging behavior is shaped by electricity tariffs. While V2G offers valuable grid services, this study focuses on the latter, more immediately scalable approach. Time-of-Use (TOU) pricing and dynamic discounts have proven effective in shifting residential charging. However, the context of a long-distance highway journey for a battery EV car is fundamentally different. The driver cannot simply defer all charging; they must make en-route decisions that trade off travel time, queuing time at charging stations, and charging cost. This paper addresses this complex decision-making process from a systemic perspective, proposing a hierarchical framework where a Charging Station Operator (CSO) designs location-specific electricity price discounts to optimally guide the collective behavior of a battery EV car fleet, thereby maximizing its own profit while alleviating station congestion and smoothing grid impact.
Modeling Individual Battery EV Car Charging Strategy on a Highway
To analyze the decision-making of a single battery EV car driver, we first construct a model for the highway charging strategy under static traffic conditions. The core idea is to transform the sequential decision problem—where to charge and how much—into a pathfinding problem on an expanded state-space graph. We make the following foundational assumptions: 1) We focus on en-route charging decisions for battery-powered cars traveling significant distances on a highway, where V2G is not considered. 2) For a given origin-destination pair on a highway with limited exits, the route choice is often constrained; thus, we model the decision of whether and how much to charge at sequential stations along a largely predetermined path.
State-of-Charge Layered Road Network Model
The highway with its charging stations is modeled as a directed, weighted graph \( R = (V_R, E_R, F_R) \). The innovation lies in defining the node set \( V_R \). Each node \( v \) is a two-dimensional vector representing the state of a battery EV car: its location at a specific charging station and its discrete State of Charge (SOC) level. Formally, \( v = (s_v, c_v) \), where \( s_v \in \{1, 2, …, S\} \) denotes the charging station index along the route, and \( c_v \in \{c_{v,dn}, c_{v,dn}+\Delta c, …, c_{v,up}\} \) represents the discretized SOC. Here, \( c_{v,up} \) and \( c_{v,dn} \) are the upper and lower SOC bounds (e.g., 80% and 20%), and \( \Delta c \) is the discrete SOC step (e.g., 10%). This creates a layered network where horizontal movement between stations decreases SOC, and vertical “movement” at a station increases SOC through charging.
The edge set \( E_R \) defines possible transitions. An edge \( (v, w) \in E_R \) and its weight \( f(v, w) \in F_R \) represent the cost (in generalized time) for a battery-powered car to move from state \( v \) to state \( w \). There are two fundamental edge types:
- Travel Edges (\( E_{road,R} \)): Corresponding to driving to the next station without charging. If at node \( v=(s_v, c_v) \), the car can move to node \( v’=(s_v+1, c_v – \Delta c) \), assuming a standardized energy consumption between stations. The weight is purely the travel time:
$$ f(v, v’) = t_{road, v,v’} = t_{arrive}(s_v+1) – t_{arrive}(s_v) $$ - Charging Edges (\( E_{cha,R} \)): Corresponding to charging at the current station. From node \( v=(s_v, c_v) \), the car can move to node \( v”=(s_v, c_{v”}) \), where \( c_{v”} > c_v \). The weight amalgamates charging time, queuing time, and the monetary cost converted into equivalent time via the user’s value of time:
$$ f(v, v”) = \frac{c_{v”} – c_v}{r_{s_v}} + t_{q, s_v} + \kappa \cdot p_{s_v} \cdot (c_{v”} – c_v) $$
Here, \( r_{s_v} \) is the charging power rate at station \( s_v \), \( t_{q, s_v} \) is the perceived queuing time, \( p_{s_v} \) is the electricity price, and \( \kappa \) is a conversion factor translating monetary cost into time cost (e.g., the inverse of the user’s hourly wage).
The sets \( E_{road,R} \) and \( E_{cha,R} \) are disjoint and their union forms the complete edge set for the model. The composition of edge weights is summarized in the table below.
| Edge Type | Weight Components | Formula |
|---|---|---|
| \( (v, v’) \in E_{road,R} \) | Road Travel Time | \( t_{road,v,v’} \) |
| \( (v, v”) \in E_{cha,R} \) | Charging Time | \( \frac{c_{v”} – c_v}{r_{s_v}} \) |
| Queuing Time | \( t_{q, s_v} \) | |
| Monetary Cost (Time-Equivalent) | \( \kappa \cdot p_{s_v} \cdot (c_{v”} – c_v) \) |
This SOC-layered network model elegantly captures the core trade-offs for a battery EV car driver: expending time to travel, versus expending time and money to recharge. Finding the optimal charging strategy from an origin node \( o \) (e.g., highway entry with high SOC) to a destination node \( p \) (e.g., highway exit with a minimum required SOC) is equivalent to finding the shortest path on this graph, where path length is the total generalized time cost.
Boundary Existence in Charging Decisions Under Continuous Discount Variation
A key insight from this modeling approach is the nature of the driver’s response to changes in charging station prices. Consider a simplified three-station highway segment (A, B, C) where a battery EV car needs to gain a total of \( R_{SOC} \) units of charge to complete its trip. The driver can choose to charge entirely at B, entirely at C, or split between B and C. The objective is to minimize total cost, which includes travel time, charging time, queuing time, and converted monetary cost. Let \( \alpha_B \) and \( \alpha_C \) be the discount factors (where 1 means no discount) applied to the base price at stations B and C, respectively.
The generalized time cost for each strategy can be expressed. For strategy 1 (charge only at B):
$$ EVC_1 = K + t_{q,B} + \frac{R_{SOC}}{r_B} + \kappa \cdot \alpha_B \cdot p_B \cdot R_{SOC} $$
where \( K \) encompasses constant travel times. For strategy 2 (charge only at C):
$$ EVC_2 = K’ + t_{q,C} + \frac{R_{SOC}}{r_C} + \kappa \cdot \alpha_C \cdot p_C \cdot R_{SOC} $$
For strategy 3 (split charging), the cost function is linear in the amount charged at B, denoted \( r \):
$$ EVC_3(r) = K” + t_{q,B} + t_{q,C} + \frac{r}{r_B} + \frac{R_{SOC}-r}{r_C} + \kappa [ \alpha_B \cdot p_B \cdot r + \alpha_C \cdot p_C \cdot (R_{SOC}-r) ] $$
Analysis shows that strategy 3 is never optimal when prices are linear and queues are independent of the charging amount for a single user. The optimal choice jumps directly from “charge all at B” to “charge all at C” when the following threshold condition is met:
$$ \alpha_C = \frac{t_{\kappa, B}}{t_{\kappa, C}} \alpha_B + \frac{t_{q,C} – t_{q,B}}{R_{SOC} \cdot t_{\kappa, C}} $$
where \( t_{\kappa, B} = \kappa \cdot p_B \) and \( t_{\kappa, C} = \kappa \cdot p_C \) represent the time-equivalent cost per unit charge. This proves that under static conditions, the optimal charging strategy for a battery EV car changes abruptly at a specific linear boundary in the \( (\alpha_B, \alpha_C) \) discount space, rather than varying continuously. This threshold behavior is crucial for the CSO’s discount optimization problem.
Incorporating Dynamic Traffic Flow and an Enhanced Solution Algorithm
The static model assumes constant travel times. To reflect reality, we extend the model to a time-dependent weighted graph \( G = [V, E, F(t)] \), where edge weights (primarily travel times \( t_{road} \)) are functions of time, updated at discrete intervals \( \Delta t \). The weight function is piecewise constant:
$$ F(t) = F_k, \quad \text{for } t \in [t_k, t_{k+1}), \ t_k = k \cdot \Delta t $$
Finding the shortest path on this time-dependent, state-expanded graph requires an enhanced algorithm. The classic Dijkstra algorithm must be modified to handle the fact that the cost of traversing an edge depends on the arrival time at its start node. The modified algorithm maintains, for each node, not only the best-known cumulative cost but also the arrival time at that node. When exploring edges from a node, the algorithm evaluates the weight based on the current arrival time. This ensures the path minimizes the total generalized time cost in a dynamic traffic environment. The computational efficiency of this approach is high, transforming a complex combinatorial charging decision problem into a polynomial-time pathfinding problem, making it suitable for real-time navigation suggestions for a battery EV car.
Modeling Cluster Behavior and the CSO Discount Optimization Problem
The decision of one battery-powered car affects the queuing time at a station, which in turn influences the decisions of subsequent cars. To manage this systemic interaction, we propose a socio-physical-information system framework involving four agents: the Traffic Highway Operator (THO), the Charging Station Operator (CSO), an Information Exchange Center (IEC), and the population of battery EV car users.
The IEC acts as a neutral coordinator. It receives real-time traffic data from the THO and real-time queuing data & discount offers from the CSO. When a battery EV car user requests a charging navigation plan, the IEC computes the optimal path (using the model and algorithm above) and provides a charging reservation at the recommended station(s). This reservation updates the station’s virtual queue. The CSO aims to maximize profit. It purchases electricity from the grid at a real-time price \( \rho_t \) and sells it to users at a posted price \( \zeta \), which can be discounted station-specifically by a factor \( \alpha_h \). Upon receiving reservation information from the IEC, the CSO solves an optimization problem to determine the discount vector \( \boldsymbol{\alpha} = \{\alpha_1, \alpha_2, …, \alpha_H\} \) for the next period, balancing the attraction of more customers against the reduction in profit margin.
Dynamic Queuing Model for Charging Stations
To accurately simulate the cluster effect, a dynamic event-driven queuing model is essential. We model the arrival of battery EV cars at the highway entrance as a Poisson process with rate \( \lambda \). Each charging station maintains three logical queues:
- Queue 1: Cars currently charging.
- Queue 2: Cars physically waiting at the station.
- Queue 3: A virtual queue of cars with active reservations from the IEC that are en-route.
Events are categorized as Request (\(Rq\)), Arrival (\(Ar\)), Start charging (\(St\)), and Leave (\(Le\)). The state of station \( h \) at time \( T_i \) is defined by Boolean variables \( A_h(T_i) \), \( B_h(T_i) \), \( C_h(T_i) \), \( D_h(T_i) \) indicating whether an event of each type occurs, and queue lengths \( N_{q1,h}(T_i) \), \( N_{q2,h}(T_i) \), \( N_{q3,h}(T_i) \). The dynamics between consecutive event times \( [T_i, T_{i+1}] \) are governed by:
$$ N_{q1,h}(T_{i+1}) = N_{q1,h}(T_i) + C_h(T_i) – D_h(T_i) $$
$$ N_{q2,h}(T_{i+1}) = N_{q2,h}(T_i) + B_h(T_i) – C_h(T_i) $$
$$ N_{q3,h}(T_{i+1}) = N_{q3,h}(T_i) + A_h(T_i) – B_h(T_i) $$
The perceived queuing time \( t_{q,h} \) for a new reservation is estimated based on the current queues and charging rates. For a battery-powered car needing \( \Delta c \) charge at station \( h \) with \( M_h \) chargers, the queuing delay can be approximated as the time until a charger is free plus the time to complete the charging of cars ahead in the queue.
Two-Stage Charging Strategy Model Based on CSO Optimal Discounts
The interaction between the CSO and the cluster of battery EV cars is modeled as a two-stage process. In the upper level, the CSO sets discounts to maximize its total profit over a planning horizon \( T \), anticipating the collective response of users. In the lower level, each individual battery EV car user, upon receiving the posted discounts \( \boldsymbol{\alpha} \) and real-time queuing estimates \( t_{q,h}(t) \) from the IEC, chooses a path and charging plan to minimize its own total cost.
CSO’s Profit Maximization Problem (Upper Level):
$$ \max_{\boldsymbol{\alpha}} \sum_{t \in T} \sum_{h \in C_s} \left[ \zeta \cdot \alpha_h \cdot \sum_{j \in N_{q1,h}(t)} (c’_{j,h,t} – c_{j,h,t}) – \rho_t \cdot \sum_{j \in N_{q1,h}(t)} (c’_{j,h,t} – c_{j,h,t}) \right] $$
subject to: \( \alpha_{min} \leq \alpha_h \leq 1 \), and the lower-level user equilibrium constraints. Here, \( (c’_{j,h,t} – c_{j,h,t}) \) is the energy delivered to car \( j \) at station \( h \) in period \( t \).
Battery EV Car User’s Cost Minimization Problem (Lower Level):
For a given user \( i \), the problem is to choose a path \( \eta \) from its origin to destination and a set of charging decisions \( \beta_{h} \in \{0,1\} \) (whether to charge at station \( h \)) and amounts \( \Delta c_{i,h} \) to:
$$ \min_{\eta, \beta_{h}, \Delta c_{i,h}} \left[ \sum_{(m,n) \in \eta} t_{road}^{mn}(t) + \sum_{h} \beta_{h} \cdot \left( t_{q,h}(t) + \frac{\Delta c_{i,h}}{r_h} + \kappa \cdot \zeta \cdot \alpha_h \cdot \Delta c_{i,h} \right) \right] $$
subject to: SOC dynamics and a final SOC requirement.
The bilevel problem is solved using a metaheuristic approach. Given the threshold behavior of users (proven in Section 1.3), a Tabu Search (TS) algorithm is well-suited for the upper-level discount optimization. The TS algorithm explores the discount space \( \boldsymbol{\alpha} \). For each candidate discount vector proposed by TS, the lower-level user equilibrium is simulated: a large number of battery EV car trips are generated via Monte Carlo simulation, and each trip’s optimal charging strategy is determined using the enhanced dynamic Dijkstra algorithm on the SOC-layered network, incorporating updated queuing times from the dynamic queue model. The CSO’s profit is then calculated from the resulting charging load distribution. The TS algorithm iteratively seeks the discount vector that maximizes this profit, using a tabu list to avoid cycling and escape local optima.
Case Study and Quantitative Analysis
A case study based on a real highway corridor in Eastern China with 7 geographical nodes (6 being charging stations, A to G) was conducted. The SOC was discretized into levels from 20% to 80% with a 10% step, creating a 28-node layered network for a typical trip from A to G. Data included 15-minute interval traffic speeds and station utilization over a 3-hour peak period (12:00-15:00). The value of time \( \kappa \) was set based on a local hourly wage benchmark.
Validation of the SOC-Layered Network Model
The proposed model’s effectiveness was validated against a classic Mixed-Integer Linear Programming (MILP) formulation for the charging strategy problem. Three scenarios were tested: (S1) with queuing times, (S2) without queuing times, and (S3) with queuing times and a 20% discount offered at Station F. The results, shown in the table below, confirm that the SOC-layered model with the enhanced Dijkstra algorithm yields identical or nearly identical optimal charging station choices with a deviation in total cost of less than 1%, while achieving a computational speedup of two orders of magnitude. This demonstrates its accuracy and superior efficiency for real-time application.
| Scenario | Metric | Classic MILP Model | SOC-Layered Network Model |
|---|---|---|---|
| S1 (With Queue) | Charging Station Choice | D | D |
| Total Generalized Time (s) | 226.72 | 224.96 | |
| Solution Time (s) | 1.96 | 0.01 | |
| S2 (No Queue) | Charging Station Choice | C, D | C, D |
| Total Generalized Time (s) | 205.72 | 204.75 | |
| Solution Time (s) | 1.87 | 0.01 | |
| S3 (Queue + 20% Disc. at F) | Charging Station Choice | F | F |
| Total Generalized Time (s) | 222.38 | 221.11 | |
| Solution Time (s) | 1.80 | 0.01 |
Analysis of Discount Influence on Strategy Boundaries
Expanding on the theoretical proof, a Monte Carlo simulation was performed on the dynamic 7-station network, varying the discounts at three key stations (C, D, and F). The resulting optimal strategies for a representative battery EV car trip were plotted in the 3D discount space \( (\alpha_C, \alpha_D, \alpha_F) \). The visualization confirmed the existence of distinct, contiguous regions where a specific charging strategy (e.g., “charge only at D”, “charge at C and F”) is optimal. The boundaries between these regions are complex, multi-faceted surfaces. Crucially, the transition between strategies remains abrupt, validating the threshold behavior in a more complex, dynamic setting. This non-continuous response is a fundamental characteristic that the CSO’s optimization model must account for.
Performance of the CSO Discount Guidance Strategy
The full two-stage framework was simulated for a cluster of 500 battery EV cars arriving according to a Poisson process during the study period. Three market regimes were compared:
- Regime 1 (Real-Time Pricing): The CSO charges users the real-time grid electricity price \( \rho_t \).
- Regime 2 (Flat Pricing): The CSO charges a fixed, non-discounted retail price \( \zeta \).
- Regime 3 (Proposed Discount Pricing): The CSO sets an optimized discount vector \( \boldsymbol{\alpha} \) on the fixed price \( \zeta \) using the Tabu Search algorithm.
The results are summarized in the following table. Regime 1 (real-time price) leads to highly uneven station utilization, as battery-powered cars flock to stations during low-price periods, causing significant load differences and long, variable queues. Regime 2 (flat price) improves balance and reduces average queue times by removing price-based attraction. However, Regime 3 (optimized discount) achieves the best outcome from the CSO’s perspective and for system balance: it increases the CSO’s profit compared to the flat price regime by strategically offering discounts to shift load away from congested stations or to stations where grid power is cheaper. This simultaneously reduces the maximum load difference between stations by approximately 15% compared to the flat price regime, while keeping average user queue times virtually unchanged. The discount mechanism successfully guides the battery EV car cluster to a charging distribution that is more profitable for the operator and more balanced for the network.
| Market Regime | CSO Total Profit | Avg. Queuing Time per Battery EV Car | Max-Min Station Load Difference |
|---|---|---|---|
| 1. Real-Time Price | N/A (Pass-through) | 13.99 min | 210 kWh |
| 2. Flat Price (No Discount) | 176.43 | 11.26 min | 200 kWh |
| 3. Optimized Discount | 181.35 | 11.35 min | 170 kWh |
Conclusion
This study has developed a comprehensive framework for analyzing and optimizing the charging behavior of battery EV car clusters on highways. By introducing a State-of-Charge layered road network model, the complex charging decision problem for a single battery-powered car is transformed into a computationally efficient pathfinding task. The analysis formally proved the existence of decision boundaries, showing that a driver’s optimal strategy changes abruptly at specific price discount thresholds rather than continuously. This insight is critical for designing effective price-based control mechanisms.
Extending to the cluster level, a socio-physical-information system architecture integrating a Traffic Highway Operator, a Charging Station Operator, an Information Exchange Center, and users was proposed. Within this framework, a two-stage model was formulated where the CSO optimizes station-specific electricity price discounts to maximize profit, accounting for the aggregated equilibrium response of battery EV car users who minimize their own travel, queuing, and charging costs. The solution employs an enhanced dynamic Dijkstra algorithm for user decisions and a Tabu Search metaheuristic for the CSO’s discount optimization.
Simulation results based on real highway data validate the model’s accuracy and efficiency. More importantly, they demonstrate that the proposed optimized discount strategy outperforms both real-time pricing and simple flat pricing regimes. It successfully increases the CSO’s revenue while significantly flattening the load profile across different highway charging stations and maintaining low average queuing times for the battery EV car users. This work provides a practical, market-based tool for managing the growing charging demand from battery-powered cars on critical transportation corridors, contributing to both grid stability and improved user experience.