In recent years, the rapid adoption of electric vehicles (EVs) has transformed the transportation sector, offering a promising pathway to reduce carbon emissions and fossil fuel dependence. As a global leader in EV deployment, China has witnessed exponential growth in electric vehicle sales, driven by supportive policies and technological advancements. The integration of China EV into power systems presents both opportunities and challenges for grid stability. Specifically, the uncontrolled charging of large-scale electric vehicle fleets can lead to distribution network congestion, jeopardizing the safe operation of power infrastructure. Unlike conventional loads, electric vehicle loads exhibit temporal and spatial flexibility, allowing for coordinated charging and discharging schedules that can alleviate grid stress. However, the inherent uncertainties in renewable energy generation and load demand complicate congestion management strategies. In this paper, I propose a robust optimization framework that leverages dynamic tariffs and the spatiotemporal flexibility of electric vehicles to address distribution network congestion under uncertainty. This approach not only enhances grid reliability but also promotes the efficient integration of China EV into the energy ecosystem.
The proliferation of electric vehicles in China has underscored the need for innovative grid management solutions. Electric vehicle charging patterns, if unmanaged, can cause localized congestion in distribution lines, especially during peak hours. Moreover, the variability of wind and solar power outputs, coupled with fluctuating loads, introduces additional layers of uncertainty. To tackle these issues, I develop a model that incorporates dynamic tariffs—a pricing mechanism reflecting real-time grid conditions—to guide electric vehicle charging behavior. By optimizing the spatiotemporal distribution of electric vehicle loads, this method mitigates congestion risks while accommodating uncertainties. The core of this work lies in a robust optimization formulation that ensures reliable performance even under worst-case scenarios of renewable generation and demand fluctuations. Through extensive simulations, I demonstrate the efficacy of this approach in maintaining grid stability and supporting the sustainable growth of China EV adoption.
The modeling of electric vehicle mobility in transportation networks is critical for capturing their impact on power systems. I represent the transportation network as a graph comprising arcs (roads) and nodes (junctions or charging stations), denoted as $G = \{A, N\}$, where $a \in A$ represents arcs and $n \in N$ represents nodes. For each origin-destination (O-D) pair $i$, the electric vehicle flow dynamics are described by linear equations that account for congestion, charging station queues, and state-of-charge (SOC) variations. The total electric vehicle demand for O-D pair $i$ at time $t$ is given by:
$$f_{m,d}^{i,t} = \sum_{j \in P_i} u_{a^*,j,t}, \quad \forall i, \forall t$$
where $f_{m,d}^{i,t}$ is the aggregate electric vehicle demand, and $u_{a^*,j,t}$ denotes the inflow into the first arc of path $j$. The conservation of electric vehicles across arcs is enforced through:
$$x_{a,q}^{j,t} – x_{a,q}^{j,t-1} = u_{a,j,t-t_a^0} – v_{a,j,t}, \quad \forall a, \forall j \in P_a, \forall t$$
Here, $x_{a,q}^{j,t}$ represents the number of electric vehicles in congestion queues, $u_{a,j,t-t_a^0}$ is the inflow, $v_{a,j,t}$ is the outflow, and $t_a^0$ is the free-flow travel time. The capacity constraints for arcs ensure that the electric vehicle count does not exceed limits:
$$\sum_{j \in P_a} v_{a,j,t} \leq \bar{N}_v^a, \quad \forall a, \forall t$$
At charging stations (e.g., fast charging stations or FCS nodes), electric vehicles can charge or discharge, influencing their SOC. The dynamics at FCS nodes are modeled as:
$$v_{a^+,j,t} = u_{y,\text{fcs}}^{j,t} + u_{y,\text{free}}^{j,t}, \quad \forall j \in P_y, \forall y, \forall t$$
$$u_{a^-,j,t} = v_{y,\text{cd}}^{j,t} + u_{y,\text{free}}^{j,t}, \quad \forall j \in P_y, \forall y, \forall t$$
where $u_{y,\text{fcs}}^{j,t}$ and $v_{y,\text{cd}}^{j,t}$ represent electric vehicles entering and leaving charging/discharging queues, respectively. The SOC evolution for electric vehicles is crucial for ensuring sufficient energy for trips and grid interactions. The SOC dynamics on arcs are given by:
$$e_{x,q}^{a,j,t} – e_{x,q}^{a,j,t-1} = e_{u}^{a,j,t-t_a^0} – e_{v}^{a,j,t} – c_{\text{tra}}^a u_{a,j,t-t_a^0}, \quad \forall j$$
where $e_{x,q}^{a,j,t}$ is the aggregate SOC in queues, $c_{\text{tra}}^a$ is the energy consumption per unit distance, and $e_{u}^{a,j,t-t_a^0}$ and $e_{v}^{a,j,t}$ are SOC inflows and outflows. Similarly, at charging stations, the SOC changes due to charging/discharging actions:
$$e_{x,\text{qcd}}^{j,t} – e_{x,\text{qcd}}^{j,t-1} = e_{u,\text{cd}}^{j,t} – e_{v,\text{cd}}^{j,t} + \sum_{l \in L_y} p_l \eta \Delta t x_{\text{qcd}}^{l,j,t}, \quad \forall j \in P_y, \forall t$$
Here, $p_l$ is the charging/discharging power of virtual charger $l$, $\eta$ is the efficiency, and $\Delta t$ is the time interval. These equations ensure that the electric vehicle behavior is accurately captured, facilitating effective coordination with the power grid.
To address uncertainties in distribution networks, I model wind power output $P_{\text{WT}}^r$, photovoltaic output $P_{\text{PV}}^r$, and conventional load $P_{\text{Con}}^r$ using box uncertainty sets. The uncertain parameters are defined as:
$$P_{\text{WT}}^r = \bar{P}_{\text{WT}}^r + \Delta P_{\text{WT}}^r$$
$$P_{\text{PV}}^r = \bar{P}_{\text{PV}}^r + \Delta P_{\text{PV}}^r$$
$$P_{\text{Con}}^r = \bar{P}_{\text{Con}}^r + \Delta P_{\text{Con}}^r$$
where $\bar{P}_{\text{WT}}^r$, $\bar{P}_{\text{PV}}^r$, and $\bar{P}_{\text{Con}}^r$ are forecasted values, and $\Delta P_{\text{WT}}^r$, $\Delta P_{\text{PV}}^r$, and $\Delta P_{\text{Con}}^r$ are perturbations bounded by:
$$\Delta P_{\text{WT}}^r \in [-\xi_{\text{WT}}^r, \xi_{\text{WT}}^r]$$
$$\Delta P_{\text{PV}}^r \in [-\xi_{\text{PV}}^r, \xi_{\text{PV}}^r]$$
$$\Delta P_{\text{Con}}^r \in [-\xi_{\text{Con}}^r, \xi_{\text{Con}}^r]$$
I introduce robust control parameters $\Gamma_{\text{WT}}^r, \Gamma_{\text{PV}}^r, \Gamma_{\text{Con}}^r \in [0,1]$ to adjust the conservatism of the model, satisfying:
$$\left| \frac{P_*^r – \bar{P}_*^r}{\xi_*^r} \right| \leq \Gamma_*^r$$
For the distribution network, I employ a linearized AC power flow model to represent power flows and voltage constraints. The power balance at node $r$ is expressed as:
$$\sum_{r \in c(s)} P_{sr}^l + P_r^N – \sum_{k \in c(r)} P_{rk}^l = P_r^D, \quad \forall r$$
where $P_{sr}^l$ is the active power flow from node $s$ to $r$, $P_r^N$ is the net injection (including renewables), and $P_r^D$ is the demand (comprising conventional load and electric vehicle load). Incorporating uncertainties, the robust counterpart of the power balance equation for worst-case scenarios becomes:
$$\sum_{r \in c(s)} P_{sr}^l + (\bar{P}_{\text{WT}}^r – \Gamma_{\text{WT}}^r \xi_{\text{WT}}^r) + (\bar{P}_{\text{PV}}^r – \Gamma_{\text{PV}}^r \xi_{\text{PV}}^r) – \sum_{k \in c(r)} P_{rk}^l = (\bar{P}_{\text{Con}}^r + \Gamma_{\text{Con}}^r \xi_{\text{Con}}^r) + P_{\text{EV}}^r, \quad \forall r$$
Line flow and voltage constraints are enforced to prevent congestion:
$$-f_{sr}^l \leq P_{sr}^l \leq f_{sr}^l, \quad \forall l, (\lambda^-,\lambda^+)$$
$$V_s^{\min} \leq V_s \leq V_s^{\max}, \quad \forall s, (\gamma^-,\gamma^+)$$
where $f_{sr}^l$ is the flow limit, $V_s^{\min}$ and $V_s^{\max}$ are voltage bounds, and $\lambda^+$, $\lambda^-$, $\gamma^+$, $\gamma^-$ are dual variables representing marginal prices for congestion and voltage violations.
The dynamic tariff $R_t$ is derived from these dual variables, providing a price signal that reflects grid congestion:
$$R_t = (\lambda_t^+ – \lambda_t^-) + (\gamma_t^+ – \gamma_t^-)$$
This dynamic tariff is used by electric vehicle aggregators (EVAs) to optimize their charging schedules, thereby alleviating congestion. The distribution system operator (DSO) solves a robust optimization problem to minimize the total cost of electric vehicle travel time and charging/discharging, subject to grid constraints. The objective function for the DSO is:
$$\min \left( \sum_m (C_T^m + C_{C\&D}^m) \right), \quad m \in A_m$$
where $C_T^m$ is the travel time cost for EVA $m$, and $C_{C\&D}^m$ is the charging/discharging cost. The travel time cost is computed as:
$$C_T^m = w \sum_{i \in i_m} \left( \sum_{j,a,t} x_{a,\text{tot}}^{j,t} + \sum_{j,y,t} (x_{y,q}^{j,t} + x_{y,\text{qcd}}^{j,t}) \right)$$
with $w$ being the unit time cost. The charging/discharging cost includes a price sensitivity coefficient $\beta$ to ensure unique solutions:
$$C_{C\&D}^m = (c_t + \beta \sum_m p_{m,t}) \sum_m p_{m,t}$$
Here, $c_t$ is the electricity price, and $p_{m,t} = p_{m,c}^{y,t} + p_{m,d}^{y,t}$ represents the net power of electric vehicles, where charging and discharging powers are defined as:
$$p_{m,c}^{y,t} = \max \left( \sum_{i \in P_y, l \in L_y} p_l x_{\text{qcd}}^{l,i,t}, 0 \right)$$
$$p_{m,d}^{y,t} = \min \left( \sum_{i \in P_y, l \in L_y} p_l x_{y,\text{qcd}}^{l,i,t}, 0 \right)$$
Upon receiving the dynamic tariff, each EVA independently optimizes its electric vehicle schedules to minimize costs, considering the updated energy prices. The EVA’s objective function is similar but incorporates the dynamic tariff:
$$\min (C_T^m + C_{C\&D}^m), \quad \forall m \in A_m$$
$$C_{C\&D}^m = (c_t + \beta \cdot p_{m,t} + R_t) \cdot p_{m,t}$$
This hierarchical optimization ensures that electric vehicle loads are spatially and temporally shifted to avoid congestion, even under uncertainty.
To validate the proposed approach, I conduct simulations on a coupled transportation-distribution network comprising 20 transportation nodes and 17 distribution nodes. The transportation network includes multiple charging stations (F1 to F8) connected to corresponding distribution nodes. Key parameters for the electric vehicle travel demands and network constraints are summarized in the following tables.
| Origin | Destination | Electric Vehicle Demand | Initial SOC |
|---|---|---|---|
| T9 | T1 | 5,000 | 2,000 |
| T4 | T10 | 5,000 | 2,000 |
| Line | Flow Limit (kW) |
|---|---|
| E5–F1 (L1) | 800 |
| E5–F2 (L2) | 300 |
| E5–F3 (L3) | 750 |
| E5–F4 (L4) | 350 |
| Arc | Capacity | Free-Flow Time (h) |
|---|---|---|
| T1–T2 | 200 | 5 |
| T2–T5 | 79 | 2 |
| T5–T6 | 82 | 2 |
| T6–T10 | 200 | 4 |
| T9–T10 | 91 | 3 |
| T9–T12 | 90 | 3 |
| T3–F5 | 180 | 1 |
| T6–F6 | 170 | 2 |
| T5–F2 | 135 | 3 |
| T8–F1 | 140 | 3 |
| T9–F4 | 132 | 3 |
| T9–F3 | 138 | 3 |
| T11–F7 | 175 | 1 |
| T10–F8 | 182 | 2 |
In the deterministic case, without considering uncertainties, the dynamic tariff successfully guides electric vehicle charging to avoid line flow violations. For instance, the power flows on lines L1 to L4 remain within limits throughout the optimization horizon. The obtained dynamic tariff values for charging stations F2 and F3 are shown in the following table, highlighting higher tariffs during congested periods.
| Time | F2 | F3 |
|---|---|---|
| 10:00 | 0.3973 | 0.3407 |
| 11:00 | 0.3869 | 0.3303 |
| 12:00 | 0.3710 | 0.3232 |
| 13:00 | 0.3544 | 0.3128 |
| 14:00 | 0.3325 | 0.2891 |
| 15:00 | 0.3096 | 0.2654 |
| 16:00 | 0.2822 | 0.2362 |
| 17:00 | 0.2630 | 0.2205 |
| 18:00 | 0.2378 | 0.2024 |
| 19:00 | 0.2070 | 0.1672 |
However, when uncertainties in wind, solar, and load are considered (e.g., with a 10% perturbation), the deterministic strategy may fail to prevent congestion. In contrast, the robust optimization approach ensures congestion management even under worst-case scenarios. The robust dynamic tariffs are higher during peak periods, as shown below, encouraging electric vehicles to shift their loads.
| Time | F2 | F3 |
|---|---|---|
| 10:00 | 0.7738 | 0.6521 |
| 11:00 | 0.7634 | 0.6417 |
| 12:00 | 0.7475 | 0.6346 |
| 13:00 | 0.7310 | 0.6242 |
| 14:00 | 0.7090 | 0.6005 |
| 15:00 | 0.6861 | 0.5767 |
| 16:00 | 0.6588 | 0.5476 |
| 17:00 | 0.6395 | 0.5319 |
| 18:00 | 0.6143 | 0.5137 |
| 19:00 | 0.6035 | 0.4785 |
The impact of robust control parameters on the total cost is analyzed by varying $\Gamma_*^r$ from 0 to 1. As $\Gamma_*^r$ increases, the model becomes more conservative, leading to higher costs due to elevated dynamic tariffs. For example, with a 10% uncertainty range, the total cost rises from approximately $4,537.4$ (deterministic) to $5,846.2$ (robust with $\Gamma_*^r=1$). This trade-off between robustness and optimality is essential for practical implementations, especially in regions with high China EV penetration.
In conclusion, the integration of electric vehicles into distribution networks requires advanced congestion management strategies that account for uncertainties. The proposed robust optimization framework, leveraging dynamic tariffs and spatiotemporal flexibility of electric vehicles, effectively mitigates congestion risks while supporting the growth of China EV. By adapting electric vehicle charging patterns in response to grid conditions, this approach enhances the resilience of power systems and promotes sustainable transportation. Future work could explore real-time implementations and market mechanisms to further optimize electric vehicle-grid interactions.