The rapid global proliferation of battery EV cars presents a dual challenge for modern power systems: an immense strain on grid infrastructure due to concentrated, uncertain charging loads, and a transformative opportunity to harness their distributed energy storage capacity. These vehicles are evolving from passive, unidirectional loads into active, bidirectional flexible resources capable of providing grid services through Vehicle-to-Grid (V2G) technology. Effectively mobilizing this vast, spatially and temporally distributed flexibility is crucial for enhancing grid stability, integrating renewable energy, and alleviating congestion. However, the core challenge lies in accurately quantifying and visualizing this aggregated flexibility potential, which is intrinsically governed by the dynamic travel patterns of users. This paper focuses on characterizing the spatial-temporal flexible operation region of aggregated battery EV cars, explicitly modeling their movement within dynamic traffic flows and their interactive coupling with the distribution network.
Existing research on V2G primarily focuses on collaborative scheduling within coupled power-transportation systems, using price signals to guide traffic flow and meet system-wide objectives like cost minimization or peak shaving. While valuable, this approach often treats flexibility as an implicit outcome rather than explicitly characterizing its bounds. Other studies assess the schedulable capacity of battery EV car clusters but frequently rely on simplified, static traffic assignment models. These models, which assume instantaneous vehicle redistribution over coarse time intervals, fail to capture the precise temporal dynamics of traffic flow, such as travel delays and queueing at charging stations, leading to an inaccurate estimation of real-time flexible capacity. Crucially, many models prioritize system-wide optimality, potentially overlooking the individual cost fairness for users—a key factor influencing real-world participation in V2G programs. Therefore, a gap exists in precisely delineating the feasible, user-equilibrium-driven flexible operation region of aggregated battery EV cars, considering high-fidelity dynamic traffic simulation.
To address this, we propose a novel framework for characterizing the Aggregated Spatial-Temporal Flexibility Region of battery EV cars under dynamic traffic flows. Our contribution is threefold. First, we establish a high-fidelity Dynamic Traffic Assignment (DTA) model integrated with V2G behavior. This model captures vehicle propagation with explicit time delays and enforces a dynamic User Equilibrium (UE) principle, ensuring that no traveler can reduce their perceived cost (travel time, charging/V2G expense with incentives) by unilaterally changing their route. Second, we develop a method to characterize the flexible operation region (FOR) of aggregated battery EV car charging/discharging power at bi-directional charging stations. By employing a rotating hyperplane technique and convex hull fitting, we visualize the multi-dimensional feasible region of net load power, representing the full spectrum of achievable grid support. Third, we couple this DTA model with a second-order cone programming (SOCP) based optimal power flow model for the distribution network, creating an integrated power-transportation nexus where price incentives (in the form of charging/discharging coupons) guide flows to explore flexibility limits.
1. Dynamic Power-Transportation Coupling Model with V2G Interaction
The integrated system comprises a transportation network and a distribution network, coupled through bi-directional charging stations (CS/V2G). The model coordinates the spatio-temporal behavior of battery EV car fleets with grid operational constraints.
1.1 Dynamic Traffic Flow Model with V2G Integration
The transportation network is defined as a graph $G_T = (N_T, A_T)$, with node set $N_T$, link set $A_T$, origin set $R \subset N_T$, and destination set $S \subset N_T$. Travel demand $q_{rs}(t)$ exists from origin $r \in R$ to destination $s \in S$ at time $t$. Each Origin-Destination (O-D) pair $rs$ has a set of feasible paths $K_{rs}$. Users are categorized into three types: regular users, users with charging needs, and users willing to provide V2G service, choosing paths $K_{rs}^o$, $K_{rs}^e$, and $K_{rs}^v$, respectively. The proportions of charging and V2G vehicles are $\lambda$ and $\varphi$. Key variables include $w_{a}^{rs,k}(t)$, $o_{a}^{rs,k}(t)$, and $h_{a}^{rs,k}(t)$, representing the inflow, outflow, and number of vehicles on link $a$ at time $t$ from O-D pair $rs$ on path $k$. Aggregated link flows are:
$$
w_a(t) = \sum_{rs}\sum_{k} w_{a}^{rs,k}(t), \quad o_a(t) = \sum_{rs}\sum_{k} o_{a}^{rs,k}(t), \quad h_a(t) = \sum_{rs}\sum_{k} h_{a}^{rs,k}(t).
$$
Flow Conservation and Propagation: The model enforces dynamic flow conservation and propagation constraints. The inflow at origins equals the travel demand:
$$
\sum_{a \in F(r)} \sum_{k \in K_{rs}^o} w_{a}^{rs,k}(t) = (1-\lambda-\varphi) q_{rs}(t), \quad \sum_{a \in F(r)} \sum_{k \in K_{rs}^e} w_{a}^{rs,k}(t) = \lambda q_{rs}(t), \quad \sum_{a \in F(r)} \sum_{k \in K_{rs}^v} w_{a}^{rs,k}(t) = \varphi q_{rs}(t),
$$
where $F(r)$ is the set of links starting at node $r$. The number of vehicles on a link evolves as:
$$
h_{a}^{rs,k}(t+1) = h_{a}^{rs,k}(t) + w_{a}^{rs,k}(t) – o_{a}^{rs,k}(t).
$$
The outflow from a link enters the next link or reaches the destination after the link’s free-flow travel time $\tau_a$:
$$
h_{a}^{rs,k}(t) = \sum_{b \in k^+} [h_{b}^{rs,k}(t+\tau_a) – h_{b}^{rs,k}(t+\tau_a-1)] + [M^{rs,k}(t+\tau_a) – M^{rs,k}(t+\tau_a-1)],
$$
where $k^+$ denotes subsequent links on path $k$, and $M^{rs,k}(t)$ is the cumulative number of vehicles from O-D pair $rs$ on path $k$ that have arrived at the destination by time $t$.
Link Travel Time and Cost: The link set $A_T$ is partitioned into regular links $A_O$, charging links $A_E$, V2G links $A_V$, and dummy links $A_D$ (for station bypass). Travel times are modeled using a modified Bureau of Public Roads (BPR) function for regular links and incorporate service/waiting times for charging/V2G links:
$$
t_a(w_a(t)) = \tau_a \left[ 1 + 0.15 \left( \frac{w_a(t)}{c_a^{cap}} \right)^4 \right], \quad a \in A_O,
$$
$$
t_a(h_a(t)) = \tau_a + t_{ch} + t_{ch}^{max} \left( \frac{h_a(t+1)}{c_a^{ch}} \right)^3, \quad a \in A_E,
$$
$$
t_a(h_a(t)) = \tau_a + t_{v2g} + t_{v2g}^{max} \left( \frac{h_a(t+1)}{c_a^{v2g}} \right)^3, \quad a \in A_V,
$$
where $c_a^{cap}$, $c_a^{ch}$, $c_a^{v2g}$ are road, charger, and V2G port capacities; $t_{ch}$, $t_{v2g}$ are average service times; $t_{ch}^{max}$, $t_{v2g}^{max}$ are maximum acceptable waiting times. Dummy link travel time is zero. The generalized cost for using a link includes time cost and monetary cost/subsidy:
$$
g_a(t) = \theta_t \cdot t_a(\cdot), \quad a \in A_O \cup A_D,
$$
$$
g_a(t) = \theta_t \cdot t_a(\cdot) + [\omega_a^{ch}(t) – \psi_a^{ch}(t)] \cdot p^{ch}, \quad a \in A_E,
$$
$$
g_a(t) = \theta_t \cdot t_a(\cdot) – [\omega_a^{v2g}(t) + \psi_a^{v2g}(t)] \cdot p^{v2g}, \quad a \in A_V,
$$
where $\theta_t$ is the value of time, $\omega_a^{ch}(t)$ is the charging electricity price, $\psi_a^{ch}(t)$ is a charging coupon subsidy (price discount), $\omega_a^{v2g}(t)$ is the V2G discharge price, $\psi_a^{v2g}(t)$ is a one-time V2G participation subsidy, and $p^{ch}$, $p^{v2g}$ are average charging and V2G power rates. The path cost $g^{rs,k}(t)$ is the sum of costs of all links constituting that path.
Dynamic User Equilibrium (DUE) Conditions: We adopt the Wardropian principle extended to dynamic conditions. At equilibrium, for each O-D pair and user type, the cost on all used paths at their departure time is equal and minimal, while unused paths have equal or higher cost. This is formulated as a nonlinear complementarity problem:
$$
0 \le w_{a}^{rs,k}(t) \perp [g^{rs,k}(t) – c_{rs}^{min,type}(t)] \ge 0, \quad \forall a \in F(r), k \in K_{rs}^{type}, type \in \{o, e, v\},
$$
where $c_{rs}^{min,type}(t)$ is the minimum travel cost for user type at time $t$, and $\perp$ indicates that at least one of the two inequalities must hold as an equality. This condition ensures cost fairness among users.
1.2 Distribution Network Optimal Power Flow Model
The distribution network is modeled using a branch flow formulation with SOCP relaxation for convexity and computational efficiency. For each line $(i,j)$ and time $t$:
$$
\sum_{m: j \to m} P_{jm}(t) = P_{j}^{is}(t) – \sum_{i: i \to j} (P_{ij}(t) – r_{ij} l_{ij}(t)),
$$
$$
\sum_{m: j \to m} Q_{jm}(t) = Q_{j}^{is}(t) – \sum_{i: i \to j} (Q_{ij}(t) – x_{ij} l_{ij}(t)),
$$
$$
v_j(t) = v_i(t) – 2(r_{ij}P_{ij}(t) + x_{ij}Q_{ij}(t)) + (r_{ij}^2 + x_{ij}^2)l_{ij}(t),
$$
where $P_{ij}/Q_{ij}$ are active/reactive power flows, $r_{ij}/x_{ij}$ are line resistance/reactance, $l_{ij}(t)=|I_{ij}(t)|^2$, $v_i(t)=|V_i(t)|^2$, and $P_{j}^{is}/Q_{j}^{is}$ are net active/reactive power injections at node $j$. The SOCP relaxation is applied:
$$
\left\| \begin{matrix} 2P_{ij}(t) \\ 2Q_{ij}(t) \\ l_{ij}(t) – v_i(t) \end{matrix} \right\|_2 \le l_{ij}(t) + v_i(t).
$$
The nodal power balance accounts for various sources and loads, including battery EV car stations:
$$
\begin{aligned}
P_{i}^{is}(t) &= P_{i}^{gen}(t) + P_{i}^{DG}(t) + P_{i}^{PV}(t) – P_{i}^{load}(t) – \eta_i^a (P_{i,a}^{ch}(t) – P_{i,a}^{v2g}(t)), \\
Q_{i}^{is}(t) &= Q_{i}^{gen}(t) + Q_{i}^{DG}(t) + Q_{i}^{PV}(t) – Q_{i}^{load}(t),
\end{aligned}
$$
where $\eta_i^a$ is an indicator linking distribution node $i$ to transportation link $a$ hosting a CS/V2G station. The battery EV car charging and V2G power at a station connected to link $a$ are:
$$
P_{i,a}^{ch}(t) = h_a(t) \cdot p^{ch}, \quad P_{i,a}^{v2g}(t) = h_a(t) \cdot p^{v2g}.
$$
Additional constraints include voltage and current limits $( \underline{V}_i \le |V_i(t)| \le \overline{V}_i, |I_{ij}(t)| \le \overline{I}_{ij} )$, and capacity limits for utility grid, distributed generators (DG), and photovoltaic (PV) inverters.
2. Characterization of the Aggregated Flexibility Region
The flexible operation region (FOR) of aggregated battery EV cars defines the feasible set of combined charging/discharging (net load) power at specified charging stations over a given time horizon. Characterizing this region involves finding its boundaries in the multi-dimensional power space.
2.1 Methodology for FOR Characterization
The FOR is a potentially non-convex polytope. We approximate it by its convex hull, which is the smallest convex set containing all feasible points. The characterization proceeds in two steps:
Step 1: Boundary Point Acquisition via Rotating Hyperplane. To find extreme points of the FOR, we solve a series of optimization problems. For a FOR defined by the net loads of $n$ stations $(P_1, P_2, …, P_n)$, we parameterize a supporting hyperplane with a unit normal vector $\mathbf{a}(\boldsymbol{\theta}) = (a_1, a_2, …, a_n)$ that explores all directions. For a 3D case $(P_x, P_y, P_z)$, the normal vector is defined by angles $\alpha$ (polar) and $\beta$ (azimuth): $a_1=\sin\alpha\sin\beta$, $a_2=\sin\alpha\cos\beta$, $a_3=\cos\alpha$. For each direction $\boldsymbol{\theta}^k$, we solve:
$$
\begin{aligned}
& \min_{\mathbf{x}} \quad d^k = \mathbf{a}(\boldsymbol{\theta}^k)^T \mathbf{P} \\
& \text{s.t.} \quad \text{Dynamic Power-Transportation Coupling Constraints (Section 1)}, \\
& \qquad \, F_{bs} = \sum_{t}\sum_{a \in A_E \cup A_V} \left[ p^{ch} \psi_a^{ch}(t) w_a(t) + \psi_a^{v2g}(t) w_a(t) \right] \Delta t \le F_{bs}^{max},
\end{aligned}
$$
where $\mathbf{x}$ includes all traffic and power flow variables, and $F_{bs}$ is the total coupon subsidy cost, bounded by a maximum $F_{bs}^{max}$ to explore different flexibility levels. Minimizing (or maximizing) the projection $\mathbf{a}^T\mathbf{P}$ pushes the solution to a boundary point of the feasible set along direction $-\mathbf{a}$. By discretizing the angles (e.g., $\alpha \in [0,\pi]$ and $\beta \in [0,2\pi)$ with steps of $45^\circ$) and solving the problem for each combination, we obtain a set of boundary points $\mathcal{V} = \{\mathbf{P}_{boundary}^1, \mathbf{P}_{boundary}^2, …\}$.
Step 2: Convex Hull Fitting. The convex hull $Conv(\mathcal{V})$ of the collected boundary points provides a piecewise-linear inner approximation of the true FOR. For visualization (2D or 3D), this hull can be plotted. For higher dimensions or for use in optimization models, the hull is described by its vertices and the linear inequalities defining its facets.
2.2 Optimal Economic Dispatch Validation Model
To validate the practicality and accuracy of the characterized FOR, we formulate an optimal economic dispatch problem for the coupled system, assuming a central coordinator.
$$
\min \quad F_{PDN} = F_{gen} + F_{DG} – F_{ch} + F_{v2g} + F_{bs}
$$
where:
$$
\begin{aligned}
F_{gen} &= \sum_t \sum_i \omega(t) P_i^{gen}(t) \Delta t, \quad F_{DG} = \sum_t \sum_i \omega^{DG}(t) P_i^{DG}(t) \Delta t, \\
F_{ch} &= \sum_t \sum_{a \in A_E} p^{ch} w_a(t) (\omega_a^{ch}(t) – \psi_a^{ch}(t) – \omega_{avg}^{ch}) \Delta t, \\
F_{v2g} &= \sum_t \sum_{a \in A_V} \left( p^{v2g} \omega_a^{v2g}(t) w_a(t) \Delta t + \psi_a^{v2g}(t) w_a(t) \right), \\
F_{bs} &= \sum_t \sum_{a \in A_E \cup A_V} \left( p^{ch} \psi_a^{ch}(t) w_a(t) \Delta t + \psi_a^{v2g}(t) w_a(t) \right).
\end{aligned}
$$
Here, $F_{ch}$ represents charging station profit (revenue minus energy cost and subsidy), and $F_{v2g}$ represents V2G compensation cost. The optimal solution point $\mathbf{P}_{opt}^*$ should lie within the characterized FOR. This validates that the FOR encompasses economically viable operating points and can inform system operators about the feasible range of battery EV car dispatch.
3. Case Study and Analysis
We demonstrate the proposed framework using a coupled system with a modified IEEE 33-bus distribution network and a Nguyen-Dupius 13-node transportation network. Three bi-directional charging stations (CS1, CS2, CS3) connect transportation nodes to distribution buses 22, 16, and 30, respectively. The simulation period is a 2-hour evening peak (17:00-19:00) discretized into 12 time intervals ($\Delta t = 10$ min). Key parameters are summarized below.
| Parameter | Description | Value |
|---|---|---|
| $\lambda$, $\varphi$ | Proportion of charging / V2G battery EV cars | 20%, 10% |
| $c_a^{ch}$, $c_a^{v2g}$ | Charger / V2G port capacity per station | 60, 40 |
| Average charging / V2G power | 60 kW, 40 kW | |
| $\theta_t$ | Value of travel time | 30 ¥/h |
| $\omega_a^{ch}(t)$ | Base charging price | 2 ¥/kWh |
| $\omega_a^{v2g}(t)$ | V2G discharge price | 3 ¥/kWh |
| $F_{bs}^{max}$ | Max total subsidy budget | Variable |
3.1 Visualization of the Flexibility Region
We characterize the 3D FOR for the net loads $(P_x, P_y, P_z)$ at CS1, CS2, and CS3 at a specific time $t^*=t_6$ (18:00), under different subsidy budgets $F_{bs}^{max}$. The rotating hyperplane method is applied with angular steps of $45^\circ$. The resulting boundary points are fitted with a convex hull. The volumes of the FOR for different subsidy caps are calculated and shown below.
| Subsidy Cap $F_{bs}^{max}$ (¥) | FOR Volume (MW³) | Volume Increase |
|---|---|---|
| 1,000 | 33.65 | – |
| 2,000 | 61.84 | 83.8% |
| 4,000 | 72.46 | 17.2% |
| 6,000 | 73.31 | 1.2% |
The results indicate that increasing the subsidy budget expands the FOR, meaning greater potential for spatial-temporal load shifting from battery EV cars. However, the marginal gain diminishes significantly after 2,000 ¥, suggesting a saturation point where most readily available flexibility has been mobilized. Further increases provide limited additional maneuverability, due to physical constraints like station capacity and the inherent trade-offs in the user equilibrium.
3.2 Validation via Optimal Economic Dispatch
We validate the FOR by solving the optimal economic dispatch (Section 2.2) with a specific grid need: to reserve 5 MW of load capacity at distribution bus 30 (connected to CS3) at time $t_6$. The system can choose from subsidy tiers (1,000 ¥, 2,000 ¥, 4,000 ¥). The optimization selects the 2,000 ¥ tier with an actual subsidy cost of 1,653.2 ¥. The optimal operating point $\mathbf{P}_{opt}^*$ is found to lie precisely on the boundary of the 2,000 ¥ FOR and is outside the 1,000 ¥ FOR. This confirms: 1) The FOR accurately represents feasible operating points; 2) The system correctly selects the minimum-cost subsidy tier that provides sufficient flexibility to meet the grid need; 3) The FOR can guide decision-making, showing operators the required “price” (subsidy) to achieve a specific dispatch target.
3.3 Comparison: Dynamic vs. Static Traffic Assignment
To highlight the importance of dynamic modeling, we compare the FOR obtained using our Dynamic Traffic Assignment (DTA) model against one using a Static Traffic Assignment (STA) model. The STA model aggregates the 2-hour demand into four 30-min intervals and assumes instantaneous flow propagation within each interval.
| Model | FOR Dimension | Approx. Range per axis (MW) | Key Limitation |
|---|---|---|---|
| Static TA | 2D Plane | ±2.4 | No temporal load shifting; total net load fixed. |
| Dynamic TA | 3D Volume | ±2.25 to ±4.29 | Captures both spatial and temporal flexibility. |
The STA model yields a 2D planar FOR because it cannot model the delayed arrival of vehicles at stations; the total number of battery EV cars being charged or discharging in a long time interval is fixed. Thus, it only captures flexibility from redistributing a fixed total power among stations (spatial shift). In contrast, the DTA model captures how incentives can make battery EV cars arrive earlier or later to charge/discharge, changing the total net load at a given instant (temporal shift). This results in a full 3D volume, offering a significantly larger and more accurate representation of the real flexibility potential inherent in mobile battery EV car fleets.
Conclusion
This paper presents a comprehensive framework for characterizing the aggregated spatial-temporal flexible operation region of battery EV cars, explicitly accounting for their dynamics within the transportation network. By integrating a high-fidelity dynamic user equilibrium traffic model with a distribution network OPF model, we can accurately trace how price incentives (coupons) guide vehicle flows and, consequently, the aggregate charging/discharging power at stations. The core methodology of rotating hyperplane optimization and convex hull fitting successfully visualizes the multi-dimensional feasible region of battery EV car flexibility. Case studies demonstrate that the characterized region is meaningful, saturates with increased subsidies, and effectively contains optimal economic dispatch points. Crucially, the comparison with static modeling underscores that dynamic traffic flow simulation is essential for unlocking and accurately quantifying the full temporal flexibility potential of mobile battery EV car resources. This characterized flexibility region provides a powerful tool for system operators and aggregators to assess, communicate, and exploit the grid-support potential of large-scale battery EV car deployment within a cyber-physical coupling system.