The deep implementation of the dual-carbon strategy has positioned electric vehicles as a pivotal pathway for carbon emission reduction in the transportation sector. In recent years, the global fleet of electric vehicles has experienced rapid growth, presenting both significant challenges and opportunities for urban power grids. On one hand, large-scale, uncoordinated charging of electric vehicles can impose substantial pressure on grid stability, increasing operational risks for the power system. On the other hand, the rational utilization of electric vehicle battery resources can effectively regulate the supply-demand balance of the grid, promote the integration of clean energy, and create additional economic value for vehicle owners. Consequently, the optimal scheduling of electric vehicle charging and discharging activities has become a critical area of research. To achieve large-scale, orderly participation of electric vehicles in grid services, establishing a stable, efficient, and mutually beneficial matching mechanism between electric vehicle users and charging stations, while coordinating the needs of the grid, is an urgent problem requiring a solution.
As the penetration rate of electric vehicles in daily transportation continues to rise, their bi-directional charging and discharging (V2G) capabilities demonstrate immense value in regulating grid fluctuations and enhancing the utilization efficiency of renewable energy. This potential drives sustained growth in their associated economic and environmental benefits. Existing research on electric vehicle charging and discharging predominantly focuses on objectives such as minimizing grid load variance or maximizing aggregator profits, utilizing time-of-use pricing to guide behavior. However, these studies often overlook the role of charging stations as independent, profit-driven entities connecting the grid and electric vehicles. Furthermore, there is a notable gap in research concerning the coordinated scheduling and stable matching between a large fleet of electric vehicles and multiple charging stations for bi-directional energy exchange.
To address this, we investigate the optimization problem for large-scale electric vehicle charging and discharging. From the perspective of maximizing charging station revenue, and considering electric vehicle user travel plans, we construct a charging and discharging optimization model to calculate the respective benefits for users and stations. Accounting for the bilateral benefits, we then design a many-to-one stable matching model between electric vehicles and charging stations. An improved deferred acceptance algorithm is proposed to solve this matching problem. Numerical experiments based on a real urban road network are conducted, followed by sensitivity analysis on grid incentive strategies and station competition. Our work aims to provide a foundational framework for the efficient and sustainable integration of massive electric vehicle fleets into future smart grid operations.
Problem Description and System Modeling
We consider an urban transportation network represented by a graph $G = (N, A)$, where $N$ is the set of nodes and $A$ is the set of arcs. Within this network, there exists a fleet of electric vehicles, denoted by the set $J$. Simultaneously, there are multiple charging stations equipped with bi-directional chargers, represented by the set $I$.
Each electric vehicle user $j \in J$ commutes from an origin node $n_j^o$ to a destination node $n_j^d$. Since the user typically stays at the destination for an extended period, they can park their electric vehicle at a charging station $i$ located at node $n_i$ to participate in scheduled charging and discharging activities, thereby obtaining monetary rewards. The charging station, based on the vehicle’s arrival time $t_j^{a,i}$, departure time $t_j^{l,i}$, and minimum required state-of-charge (SOC) $e_j^{min}$, schedules its charging and discharging power flow to earn a profit from the price difference between grid tariffs and user tariffs. Under the assumption of perfect information, both electric vehicle users and charging stations aim to maximize their own benefits, determining the optimal matching pairs and the detailed charging/discharging schedules.
The schematic illustrates the matching process. For instance, electric vehicle EV1 travels from home to charging station 1, parks and plugs in, then walks to workplace 1. The station schedules its bi-directional energy flow. In the afternoon, the user returns to station 1 to drive home. Electric vehicle EV3, valuing the more favorable price differential at station 1, is willing to tolerate a longer walking distance to workplace 2, thus choosing to match with station 1. This scenario encapsulates the core trade-off between user profit (from energy arbitrage minus detour cost) and station profit.
Mathematical Model Formulation
Symbols and Assumptions
The key parameters and variables used in our models are summarized in the table below.
| Symbol | Description | Type |
|---|---|---|
| $N, A$ | Set of nodes and arcs in the road network. | Parameter |
| $I, J$ | Set of charging stations and electric vehicles. | Parameter |
| $K, K_s, K_g, K_p$ | Set of time slots, and subsets for peak, off-peak, and flat periods. | Parameter |
| $d_{n, n’}$ | Distance between nodes $n$ and $n’$. | Parameter |
| $r_{ij}, v_j$ | Detour distance for EV $j$ to station $i$ and its unit cost. | Parameter |
| $g_k^c, g_k^d$ | Grid charging and discharging price at time $k$. | Parameter |
| $s_{ik}^c, s_{ik}^d$ | Station $i$’s charging/discharging price offered to users at time $k$. | Parameter |
| $e_j^0, e_j^{min}, E_j$ | Initial, minimum required, and maximum battery capacity of EV $j$. | Parameter |
| $P, \alpha, \beta$ | Charging power rate, charging efficiency, discharging efficiency. | Parameter |
| $T$ | Duration of each time slot. | Parameter |
| $h_i$ | Number of chargers at station $i$. | Parameter |
| $u$ | Battery degradation cost per unit of discharging time. | Parameter |
| $R_{ij}$ | Net revenue for EV $j$ if matched with station $i$. | Variable |
| $B_{ij}$ | Revenue for station $i$ from serving EV $j$. | Variable |
| $BF_{ij}$ | Grid peak-shaving benefit from EV $j$ at station $i$. | Variable |
| $x_{ij}$ | Binary matching variable (1 if EV $j$ matched with station $i$). | Variable |
| $y_{jk}^c, y_{jk}^d$ | Binary scheduling variables for charging/discharging of EV $j$ at time $k$. | Variable |
To make the problem tractable while retaining its essence, we adopt the following assumptions: 1) Perfect information is available to all parties. 2) All chargers at a station have identical power ratings. 3) All agents are fully rational. 4) The grid provides uniform energy prices to all stations. 5) Each charger can serve at most one electric vehicle per day. 6) The travel cost between the matched station and the user’s workplace is distance-dependent and is treated as a fixed parameter in the matching phase.
Charging Station-Centric Charging/Discharging Optimization Model
For a specific charging station $i$ and a specific electric vehicle $j$ that has chosen to park there, the station acts as an independent profit center. Its objective is to schedule the electric vehicle’s charging ($y_{jk}^c$) and discharging ($y_{jk}^d$) actions during the parking interval $[t_j^{a,i}, t_j^{l,i}]$ to maximize its own profit from the price arbitrage between the user tariff and the grid tariff. The optimization model for station $i$ serving EV $j$ is formulated as follows:
Objective: Maximize station profit from this single electric vehicle.
$$Max \, B_{ij} = \sum_{k \in K} \left[ \alpha P y_{jk}^{c} (s_{ik}^{c} – g_k^{c}) + \beta P y_{jk}^{d} (g_k^{d} – s_{ik}^{d}) \right] \frac{T}{60}$$
Subject to:
1. Destination SOC Requirement: The electric vehicle must have sufficient energy upon departure.
$$e_j^{0} + \sum_{k < t_j^{l,i}} \left[ \alpha P y_{jk}^{c} – \beta P y_{jk}^{d} \right] \frac{T}{60} \geq e_j^{min}$$
2. Battery Capacity Limits: The SOC must remain within safe operating bounds at all times.
$$0.2E_j \leq e_j^{0} + \sum_{k’ \in [t_j^{a,i}, k]} \left[ \alpha P y_{jk’}^{c} – \beta P y_{jk’}^{d} \right] \frac{T}{60} \leq E_j, \quad \forall k \in [t_j^{a,i}, t_j^{l,i}]$$
3. Activity Window: Charging/discharging only occurs during the parking period.
$$y_{jk}^{c} = y_{jk}^{d} = 0, \quad \forall k \notin [t_j^{a,i}, t_j^{l,i}]$$
4. Grid-Friendly Scheduling: Charging is restricted to off-peak grid periods, and discharging to peak periods. This aligns with common grid incentive structures.
$$y_{jk}^{d} = 0, \quad \forall k \in K_g \cup K_p$$
$$y_{jk}^{c} = 0, \quad \forall k \in K_s$$
5. Mutual Exclusivity: The electric vehicle cannot charge and discharge simultaneously.
$$y_{jk}^{c} + y_{jk}^{d} \leq 1, \quad \forall k \in [t_j^{a,i}, t_j^{l,i}]$$
6. Binary Decision Variables:
$$y_{jk}^{c}, y_{jk}^{d} \in \{0, 1\}, \quad \forall k \in K$$
This model is a binary integer program. For a given pair $(i, j)$ and known parameters, it can be solved efficiently using commercial solvers like Gurobi or CPLEX to obtain the optimal schedule and the resultant station profit $B_{ij}$.
Electric Vehicle and Charging Station Stable Matching Model
The core of the large-scale coordination problem is determining which electric vehicle parks at which station. This decision is based on the calculated benefits for both parties.
1. Electric Vehicle User’s Net Revenue:
If electric vehicle $j$ matches with station $i$, its net revenue $R_{ij}$ consists of the energy arbitrage revenue minus the detour cost and the battery degradation cost.
- Detour Cost: $C_{ij} = v_j \cdot r_{ij}$, where $r_{ij} = d_{n_j^o, n_i} + d_{n_i, n_j^d} – d_{n_j^o, n_j^d}$.
- Degradation Cost: $D_{ij} = u T \sum_{k} y_{jk}^{d}$.
- Energy Revenue: Calculated from the optimal schedule: $Rev_{ij} = \sum_{k} \left[ \beta P y_{jk}^{d} s_{ik}^{d} – \alpha P y_{jk}^{c} s_{ik}^{c} \right] \frac{T}{60}$.
Thus, the user’s net revenue is:
$$R_{ij} = Rev_{ij} – C_{ij} – D_{ij}$$
2. Charging Station’s Revenue:
This is directly the output $B_{ij}$ from the station-centric optimization model solved above.
$$B_{ij} = \sum_{k \in K} \left[ \alpha P y_{jk}^{c} (s_{ik}^{c} – g_k^{c}) + \beta P y_{jk}^{d} (g_k^{d} – s_{ik}^{d}) \right] \frac{T}{60}$$
3. Grid Peak-Shaving Benefit:
We quantify one key grid benefit as the total discharge energy during peak periods, which helps reduce grid stress.
$$BF_{ij} = \beta P \sum_{k \in K_s \cap [t_j^{a,i}, t_j^{l,i}]} y_{jk}^{d}$$
4. The Stable Matching Model:
With $R_{ij}$ and $B_{ij}$ precomputed for all potential $(i, j)$ pairs, the matching problem can be defined. Each electric vehicle $j$ has a preference list over stations $i$, ranked in descending order of $R_{ij}$. Each charging station $i$ has a preference list over electric vehicles $j$, ranked in descending order of $B_{ij}$. The station has a capacity $h_i$ (number of chargers). We seek a stable, many-to-one matching that respects these preferences. A matching is stable if there is no blocking pair: an electric vehicle $j$ and a station $i$ that are not matched to each other but both prefer each other to their current matches (or being unmatched).
We can formulate this as an integer program with stability constraints:
$$Max \, Z = \sum_{i \in I} \sum_{j \in J} (R_{ij} + B_{ij}) x_{ij}$$
Subject to:
$$\sum_{i \in I} x_{ij} \leq 1, \quad \forall j \in J$$
$$\sum_{j \in J} x_{ij} \leq h_i, \quad \forall i \in I$$
$$x_{ij} + \sum_{j’: B_{ij’} > B_{ij}} x_{ij’} + \sum_{i’: R_{i’j} > R_{ij}} x_{i’j} \geq 1, \quad \forall i \in I, j \in J$$
$$x_{ij} \in \{0, 1\}, \quad \forall i \in I, j \in J$$
The third constraint is the stability condition. It ensures that for any potentially blocking pair $(i, j)$, at least one of them is already matched to a partner they prefer at least as much, thus preventing the block.
Solution Methodology: An Improved Deferred Acceptance Algorithm
Directly solving the integer programming model with stability constraints can be computationally challenging for large-scale instances involving thousands of electric vehicles and stations. Therefore, we employ a matching theory approach. The classic Gale-Shapley deferred acceptance algorithm can find a stable matching. However, for large urban networks, the initial preference lists can be very long, leading to inefficient computations.
We propose an improved algorithm that iteratively relaxes a detour distance constraint to prune preference lists and expedite the search for a stable matching:
Step 1: Precomputation. For all electric vehicle $j$ and station $i$ pairs, compute $R_{ij}$, $C_{ij}$, $D_{ij}$, and $B_{ij}$ by solving the charging/discharging optimization model.
Step 2: Preference List Construction. Each electric vehicle $j$ ranks all stations $i$ in descending order of $R_{ij}$ to form its preference list $PL_j$. Each station $i$ ranks all electric vehicles $j$ in descending order of $B_{ij}$ to form its preference list $PL_i$.
Step 3: Constraint Initialization. Impose a strict detour distance constraint $r_{ij} \leq \delta$. For each electric vehicle $j$, remove from $PL_j$ any station $i$ where $r_{ij} > \delta$ or $R_{ij} \leq 0$.
Step 4: Proposal Phase. Each unmatched electric vehicle $j$ proposes to its most preferred station $i$ remaining in $PL_j$.
Step 5: Review & Rejection Phase. Each station $i$ reviews all proposals it receives. It tentatively accepts the top $h_i$ proposers based on its own preference list $PL_i$ and rejects the rest. If a station that has already filled its $h_i$ slots receives a new proposal from an electric vehicle $j’$ that it prefers over a currently tentatively accepted electric vehicle $j$, it rejects $j$ and accepts $j’$.
Step 6: Iteration. Rejected electric vehicles propose to their next preferred station. Steps 4 and 5 repeat until no electric vehicle can make a new proposal (its list is exhausted) or all stations’ lists are finalized.
Step 7: Constraint Relaxation and Search. Gradually increase the detour distance limit $\delta$ (e.g., $\delta = \delta + \Delta$). For the new $\delta$, update the preference lists by adding previously excluded stations that now satisfy $r_{ij} \leq \delta$. Electric vehicles that are unmatched or that now have more attractive options due to the relaxed constraint may trigger new proposals. The algorithm (Steps 4-6) runs again with the updated lists, potentially forming new tentative matches while maintaining stability with respect to all currently considered pairs.
Step 8: Termination. The process terminates when increasing $\delta$ no longer results in any changes to the stable matching outcome. The final matching is returned.
This method efficiently finds a stable matching by focusing the search on geographically plausible matches first, significantly reducing the initial problem size.
Numerical Experiments and Analysis
We conduct numerical experiments using a real-world road network from the Jiangbei District in Chongqing, China. The network’s major arteries form the basis for distance calculations.
Experimental Setup and Key Metrics
The simulation time horizon is one day, discretized into 96 time slots of 15 minutes each ($T=15$). Time-of-use tariffs are defined based on local regulations: off-peak (00:00-08:00), flat (08:00-11:00, 17:00-20:00, 22:00-24:00), and peak (11:00-17:00, 20:00-22:00). The charging/discharging price parameters for stations and the grid are set as follows:
| Period | Grid Discharge Price ($g_k^d$) | Grid Charge Price ($g_k^c$) | Station Discharge Price ($s_{ik}^d$) | Station Charge Price ($s_{ik}^c$) |
|---|---|---|---|---|
| Peak | 1.5 | 1.3 | 1.2 | 1.8 |
| Flat | 0.6 | 0.6 | 0.9 | 0.9 |
| Off-Peak | 0.4 | 0.6 | – | – |
Electric vehicle travel OD (Origin-Destination) data is generated randomly based on statistical distributions from national travel surveys. Key parameters: charging/discharging power $P=7$ kW, efficiencies $\alpha=0.95$, $\beta=0.90$, battery degradation cost $u=0.01$ ¥/kWh. We test various scales by changing the number of electric vehicles $|J|$, stations $|I|$, and total chargers $\sum h_i$.
We evaluate performance using four key metrics:
- Electric Vehicle User Matching Rate ($\theta$): The proportion of electric vehicles successfully matched to a station. $\theta = (\sum_{i,j} x_{ij}) / |J|$.
- Charger Utilization Rate ($\omega$): The proportion of chargers matched with an electric vehicle. $\omega = (\sum_{i,j} x_{ij}) / (\sum_i h_i)$.
- Average EV User Revenue ($AveR$): The average net revenue of matched electric vehicle users. $AveR = (\sum_{i,j} R_{ij} x_{ij}) / (\sum_{i,j} x_{ij})$.
- Price of Anarchy for the Grid ($POA$): Measures the system efficiency loss due to stable matching versus a globally optimal matching (maximizing $R_{ij}+B_{ij}$ without stability constraints). Let $FS = \sum_{i,j} BF_{ij} x_{ij}^s$ be the grid benefit under the stable match $x^s$, and $FG = \sum_{i,j} BF_{ij} x_{ij}^g$ under the globally optimal match $x^g$. Then $POA = FS / FG$. A $POA$ close to 1 indicates minimal efficiency loss.
Results for Different Problem Scales
The table below presents the results for different combinations of electric vehicle fleet size and charging infrastructure.
| $|J|$ (EVs) | $|I|$ (Stations) | $\sum h_i$ (Chargers) | $\theta$ (Match Rate) | $\omega$ (Utilization) | $AveR$ | $POA$ |
|---|---|---|---|---|---|---|
| 20 | 5 | 30 | 0.60 | 0.40 | 29.85 | 0.74 |
| 20 | 10 | 60 | 0.65 | 0.22 | 34.30 | 0.71 |
| 20 | 10 | 90 | 0.75 | 0.17 | 32.43 | 0.83 |
| 40 | 5 | 30 | 0.65 | 0.87 | 39.47 | 0.68 |
| 40 | 10 | 60 | 0.70 | 0.47 | 41.22 | 0.59 |
| 40 | 10 | 90 | 0.75 | 0.33 | 44.85 | 0.76 |
| 60 | 5 | 30 | 0.48 | 0.97 | 51.67 | 0.63 |
| 60 | 10 | 60 | 0.68 | 0.68 | 40.82 | 0.57 |
| 60 | 10 | 90 | 0.75 | 0.50 | 45.52 | 0.69 |
| 80 | 5 | 30 | 0.38 | 1.00 | 58.24 | 0.49 |
| 80 | 10 | 60 | 0.66 | 0.88 | 49.65 | 0.42 |
| 80 | 10 | 90 | 0.74 | 0.66 | 51.39 | 0.53 |
| 100 | 5 | 30 | 0.30 | 1.00 | 61.17 | 0.41 |
| 100 | 10 | 60 | 0.59 | 0.98 | 51.21 | 0.37 |
| 100 | 10 | 90 | 0.73 | 0.81 | 54.95 | 0.49 |
Analysis:
- Matching Rate ($\theta$) and Utilization ($\omega$): As the total number of chargers increases, the electric vehicle matching rate generally increases because more resources are available. Conversely, the charger utilization rate decreases because the pool of unused chargers grows larger. For a fixed number of chargers, as the number of electric vehicles increases, the matching rate may decrease due to intensified competition for the limited spots, while utilization reaches 100% when demand outstrips supply.
- Average User Revenue ($AveR$): This metric does not monotonically increase with fleet size. With scarce chargers, stations selectively match with electric vehicles that offer them the highest profit $B_{ij}$, which often also correlates with higher $R_{ij}$ for the user (e.g., vehicles with longer parking times for more arbitrage). Thus, $AveR$ can be high with few chargers and many electric vehicles. With abundant chargers, more electric vehicles with moderate revenue get matched, potentially lowering the average.
- Price of Anarchy ($POA$): The trend in $POA$ closely follows $AveR$. A higher $AveR$ typically indicates that matched electric vehicles provide greater discharge services (higher $BF_{ij}$), which also benefits the grid. Therefore, when stable matching leads to partnerships with high user revenue, the efficiency loss for the overall system (from a grid peak-shaving perspective) is smaller. Scenarios with intense competition (many electric vehicles, few chargers) show lower $POA$, indicating a greater divergence between individual stable choices and the global grid optimum.
Sensitivity Analysis: Impact of Grid Incentive Strategies
To encourage electric vehicle participation, the grid operator can implement incentive strategies. We analyze two:
- Strategy 1 (Discharge Volume Incentive): Provides a subsidy $\rho_1$ (¥/kWh) based on the total discharge energy of the electric vehicle.
- Strategy 2 (Detour Compensation): Provides a subsidy $\rho_2$ (¥/km) based on the user’s incurred detour distance.
These subsidies directly increase the user’s net revenue: $R_{ij}^{new} = R_{ij} + \rho_1 (\beta P \sum y_{jk}^d T/60) + \rho_2 \cdot r_{ij}$.
The results for a scenario with 60 electric vehicles under different charger capacities are shown below.
| $\rho_1$ (¥/kWh) | 60 EVs, 30 Chargers | 60 EVs, 60 Chargers | 60 EVs, 90 Chargers | ||||||
|---|---|---|---|---|---|---|---|---|---|
| $\theta$ | $AveR$ | $POA$ | $\theta$ | $AveR$ | $POA$ | $\theta$ | $AveR$ | $POA$ | |
| 0.0 | 0.48 | 51.67 | 0.63 | 0.68 | 40.82 | 0.57 | 0.75 | 45.52 | 0.69 |
| 0.2 | 0.50 | 59.82 | 0.64 | 0.72 | 47.10 | 0.56 | 0.78 | 53.16 | 0.70 |
| 0.4 | 0.50 | 73.91 | 0.64 | 0.77 | 53.28 | 0.57 | 0.87 | 62.98 | 0.68 |
| 0.6 | 0.50 | 86.13 | 0.62 | 0.83 | 61.55 | 0.55 | 0.98 | 69.56 | 0.67 |
| 0.8 | 0.50 | 98.92 | 0.61 | 0.93 | 65.96 | 0.55 | 1.00 | 76.49 | 0.65 |
| 1.0 | 0.50 | 104.6 | 0.60 | 0.98 | 71.97 | 0.53 | 1.00 | 83.81 | 0.64 |
Strategy 1 Analysis: Increasing the discharge subsidy $\rho_1$ significantly boosts the average revenue $AveR$ for matched electric vehicles and improves the matching rate $\theta$, eventually reaching 100% when the subsidy is high enough. However, the $POA$ remains relatively stable or slightly decreases. This is because Strategy 1 uniformly increases the attractiveness of discharging for all electric vehicles at all stations but does not fundamentally alter the relative ranking in their preference lists (if $R_{ij} > R_{i’j}$ before, it likely remains so after adding a proportional subsidy). Thus, the stable matching outcome and its systemic efficiency relative to the global optimum are largely preserved, just at a higher participation level.
| $\rho_2$ (¥/km) | 60 EVs, 30 Chargers | 60 EVs, 60 Chargers | 60 EVs, 90 Chargers | ||||||
|---|---|---|---|---|---|---|---|---|---|
| $\theta$ | $AveR$ | $POA$ | $\theta$ | $AveR$ | $POA$ | $\theta$ | $AveR$ | $POA$ | |
| 0.0 | 0.48 | 51.67 | 0.63 | 0.68 | 40.82 | 0.57 | 0.75 | 45.52 | 0.69 |
| 0.6 | 0.50 | 49.15 | 0.65 | 0.78 | 39.66 | 0.60 | 0.80 | 43.76 | 0.71 |
| 1.2 | 0.50 | 49.43 | 0.69 | 0.83 | 37.04 | 0.66 | 0.88 | 42.75 | 0.77 |
| 1.8 | 0.50 | 47.92 | 0.61 | 0.91 | 37.85 | 0.58 | 0.95 | 40.40 | 0.71 |
| 2.4 | 0.50 | 43.80 | 0.59 | 1.00 | 35.94 | 0.54 | 1.00 | 38.66 | 0.65 |
| 3.0 | 0.50 | 40.96 | 0.53 | 1.00 | 33.68 | 0.49 | 1.00 | 36.18 | 0.58 |
Strategy 2 Analysis: The detour subsidy $\rho_2$ effectively improves the matching rate $\theta$ by making distant stations economically viable. However, it reduces the average net energy revenue $AveR$ because it encourages matching based on subsidy rather than pure energy arbitrage potential. The impact on $POA$ is non-monotonic: it first increases and then decreases. Initially, relaxing the geographic constraint allows some electric vehicles with high discharge potential but previously prohibitive detour costs to match, improving system efficiency ($POA$ increases). However, as $\rho_2$ grows very large, the subsidy dominates user decisions, potentially leading to matches where electric vehicles choose stations based on proximity rather than optimal discharge schedules, thereby reducing the overall grid benefit and causing $POA$ to fall.
Sensitivity Analysis: Impact of Charging Station Competition
Charging stations, as profit-driven entities, may engage in competition for high-value electric vehicles. A station can offer a more favorable price (increase $s_{ik}^d$ or decrease $s_{ik}^c$) to attract a user, effectively transferring a portion of its profit $B_{ij}$ to the user, thereby increasing $R_{ij}$ and making itself more attractive. We model this by allowing stations to use a percentage of the revenue $B_{ij}$ they would earn from a desirable electric vehicle as a budget to outbid competitors. The results for a scenario with 60 electric vehicles are presented below.
| Revenue Transfer Ratio | 60 EVs, 30 Chargers | 60 EVs, 60 Chargers | 60 EVs, 90 Chargers | ||||||
|---|---|---|---|---|---|---|---|---|---|
| $\theta$ | $AveR$ | $POA$ | $\theta$ | $AveR$ | $POA$ | $\theta$ | $AveR$ | $POA$ | |
| 0% | 0.48 | 51.67 | 0.63 | 0.68 | 40.82 | 0.57 | 0.75 | 45.52 | 0.69 |
| 20% | 0.50 | 56.71 | 0.65 | 0.78 | 46.10 | 0.71 | 0.83 | 51.04 | 0.71 |
| 40% | 0.50 | 63.04 | 0.67 | 0.85 | 51.83 | 0.74 | 0.90 | 57.44 | 0.74 |
| 60% | 0.50 | 69.28 | 0.68 | 0.92 | 57.32 | 0.76 | 0.98 | 64.39 | 0.76 |
| 80% | 0.50 | 74.05 | 0.71 | 0.96 | 64.96 | 0.78 | 1.00 | 71.77 | 0.78 |
Analysis: As stations transfer a larger share of their potential revenue to attract electric vehicle users, the matching rate $\theta$ and the average user revenue $AveR$ both increase. This active competition benefits users and increases overall participation. Importantly, the $POA$ also increases steadily. This indicates that station competition, guided by their own profit motive ($B_{ij}$), indirectly steers the stable matching towards outcomes that are also better for the grid. Stations compete for electric vehicles that are profitable, which are often those with high discharge potential (high $BF_{ij}$). Therefore, such competition reduces the efficiency loss of the stable matching mechanism from the system perspective.
Conclusion
This study addresses the large-scale coordination problem for electric vehicle charging and discharging by integrating operational scheduling with strategic matching. We developed a two-stage framework: first, a charging station-centric optimization model determines the optimal charging/discharging schedule and calculates the bilateral benefits for any given electric vehicle-station pair; second, a many-to-one stable matching model pairs electric vehicles with stations based on their mutual preferences derived from these benefits. An improved deferred acceptance algorithm incorporating geographic constraints is designed for efficient solution.
The numerical experiments and sensitivity analysis yield several key insights:
- The scale of the system significantly impacts outcomes. A larger pool of chargers increases electric vehicle matching rates but decreases charger utilization. The average revenue for matched electric vehicle users and the systemic efficiency loss (Price of Anarchy) exhibit correlated trends, influenced by the selectivity of matches under resource constraints.
- Grid incentive strategies effectively boost participation but have different systemic impacts. A discharge volume-based subsidy increases user revenue and participation without drastically altering the stable matching structure or its efficiency. A detour compensation subsidy, while improving matching rates, can change user preference rankings, leading to a non-monotonic effect on system efficiency, which may first improve and then degrade as the subsidy becomes the dominant decision factor.
- Charging station competition, modeled as revenue transfer to users, proves beneficial for all parties. It increases the electric vehicle matching rate, raises average user revenue, and crucially, enhances the overall grid benefit from the matching, thereby reducing the efficiency loss inherent in the decentralized stable matching process.
Our work highlights the importance of considering charging stations as active profit-maximizing agents in the V2G ecosystem. The proposed stable matching framework provides a principled approach to achieve a fair and sustainable coordination mechanism for large-scale electric vehicle integration. Future research could incorporate uncertainty in user behavior, dynamic pricing, and network constraints into the matching model to further enhance its practicality and robustness.