As the adoption of electric cars accelerates globally, particularly in regions like China where EV policies promote widespread use, the integration of these vehicles into power systems presents both opportunities and challenges. Electric cars, as mobile energy storage units, can enhance grid flexibility and support renewable energy integration. However, their uncontrolled charging patterns, especially during peak periods, often lead to distribution network congestion, jeopardizing grid stability. In China, the rapid expansion of EV infrastructure has intensified these concerns, necessitating innovative management strategies. This article explores a robust optimization framework that leverages dynamic tariffs and the spatiotemporal flexibility of electric cars to manage congestion in distribution networks under uncertain conditions, such as fluctuations in renewable generation and load demand.
The coupling between transportation and power networks is critical, as electric cars move between charging stations, influencing both systems. Unlike conventional loads, electric cars exhibit temporal and spatial flexibility, meaning their charging schedules and locations can be adjusted based on incentives like dynamic tariffs. These tariffs, derived from distribution locational marginal pricing principles, reflect real-time congestion levels and guide EV aggregators to optimize charging behavior. In uncertain environments, where wind, solar, and load forecasts are prone to errors, robust optimization ensures that congestion management strategies remain effective under worst-case scenarios. This approach not only mitigates congestion but also enhances the economic efficiency and reliability of distribution networks, supporting the sustainable growth of China’s EV ecosystem.
To model the behavior of electric cars in transportation networks, a linear dynamic traffic assignment (DTA) framework is employed. This captures the evolution of EV flows on roads and at fast charging stations (FCS), considering congestion, queuing, and state of charge (SOC) dynamics. The transportation network is represented as a graph $G = \{A, N\}$, where $a \in A$ denotes road segments and $n \in N$ represents nodes, including FCS locations. For each origin-destination (O-D) pair $i$, the EV flow can be described as:
$$f_{m,d}^{i,t} = \sum_{j \in P_i} u_{a^*,j}^t, \quad \forall i, \forall t$$
where $f_{m,d}^{i,t}$ is the total number of electric cars for O-D pair $i$ at time $t$, $u_{a^*,j}^t$ is the inflow to the first segment of path $j$, and $P_i$ is the set of paths for O-D pair $i$. The conservation of flows ensures that the total EVs entering a path equal those exiting over time:
$$\sum_{t \in T} u_{a^*,j}^t = \sum_{t \in T} v_{a^>,j}^t, \quad \forall j$$
Here, $v_{a^>,j}^t$ represents the outflow from the last segment of path $j$, and $T$ is the optimization horizon.
On individual road segments, the dynamics of electric car queues and total vehicles are modeled linearly to account for congestion delays. For a segment $a$ on path $j$, the number of queued EVs $x_{a,q}^{j,t}$ evolves as:
$$x_{a,q}^{j,t} – x_{a,q}^{j,t-1} = u_{a,j}^{t-t_a^0} – v_{a,j}^t, \quad \forall a, \forall j \in P_a, \forall t$$
where $t_a^0$ is the free-flow travel time, $u_{a,j}^{t-t_a^0}$ is the inflow at time $t – t_a^0$, and $v_{a,j}^t$ is the outflow. The total EVs on the segment $x_{a,\text{tot}}^{j,t}$ changes as:
$$x_{a,\text{tot}}^{j,t} – x_{a,\text{tot}}^{j,t-1} = u_{a,j}^t – v_{a,j}^t, \quad \forall a, \forall j \in P_a, \forall t$$
Capacity constraints ensure that the number of electric cars does not exceed limits:
$$\sum_{j \in P_a} v_{a,j}^t \leq \bar{N}_v^a, \quad \forall a, \forall t$$
At FCS nodes, electric cars can charge, discharge, or bypass the station. The inflow $v_{a^+,j}^t$ from the previous segment splits into those entering the FCS $u_{y,\text{fcs}}^{j,t}$ and those bypassing $u_{y,\text{free}}^{j,t}$:
$$v_{a^+,j}^t = u_{y,\text{fcs}}^{j,t} + u_{y,\text{free}}^{j,t}, \quad \forall j \in P_y, \forall y, \forall t$$
Similarly, the outflow $u_{a^-,j}^t$ to the next segment includes EVs finishing charging/discharging $v_{y,\text{cd}}^{j,t}$ and bypassing EVs $u_{y,\text{free}}^{j,t}$:
$$u_{a^-,j}^t = v_{y,\text{cd}}^{j,t} + u_{y,\text{free}}^{j,t}, \quad \forall j \in P_y, \forall y, \forall t$$
Queue dynamics at FCS are captured by:
$$x_{y,q}^{j,t} – x_{y,q}^{j,t-1} = u_{y,\text{fcs}}^{j,t} – u_{y,\text{cd}}^{j,t}, \quad \forall j \in P_y, \forall y, \forall t$$
$$x_{y,\text{qcd}}^{j,t} – x_{y,\text{qcd}}^{j,t-1} = u_{y,\text{cd}}^{j,t} – v_{y,\text{cd}}^{j,t} + \sum_{l \in L_y} p_l \eta \Delta t x_{l,j,\text{qcd}}^t, \quad \forall j \in P_y, \forall y, \forall t$$
where $p_l$ is the charging/discharging power of virtual charger $l$, $\eta$ is efficiency, and $\Delta t$ is the time step.
The state of charge (SOC) for electric cars is crucial for planning. The aggregate SOC on road segments evolves as:
$$e_{x,q}^{a,j,t} – e_{x,q}^{a,j,t-1} = e_{u}^{a,j,t-t_a^0} – e_{v}^{a,j,t} – c_{\text{tra}}^a u_{a,j}^{t-t_a^0}, \quad \forall j$$
where $c_{\text{tra}}^a$ is the energy consumption per unit distance, and $e_{u}^{a,j,t-t_a^0}$ and $e_{v}^{a,j,t}$ represent the SOC of inflows and outflows. At FCS, the SOC dynamics include charging/discharging effects:
$$e_{a^+,v}^{j,t} = e_{u,\text{fcs}}^{j,t} + e_{\text{free}}^{j,t}, \quad \forall j \in P_y, \forall t$$
$$e_{a^-,u}^{j,t} = e_{v,\text{cd}}^{j,t} + e_{\text{free}}^{j,t}, \quad \forall j \in P_y, \forall t$$
$$e_{x,q}^{y,j,t} – e_{x,q}^{y,j,t-1} = e_{u,\text{fcs}}^{j,t} – e_{u,\text{cd}}^{j,t}, \quad \forall j \in P_y, \forall t$$
$$e_{x,\text{qcd}}^{j,t} – e_{x,\text{qcd}}^{j,t-1} = e_{u,\text{cd}}^{j,t} – e_{v,\text{cd}}^{j,t} + \sum_{l \in L_y} p_l \eta \Delta t x_{l,j,\text{qcd}}^t, \quad \forall j \in P_y, \forall y, \forall t$$
Bounds on SOC ensure battery health, with individual EV SOC constrained between $0.2$ and $1$.
In the distribution network, uncertainties from wind, solar, and load are modeled using robust optimization. Let $P_r^{\text{WT}}$, $P_r^{\text{PV}}$, and $P_r^{\text{Con}}$ denote the wind power, solar power, and conventional load at node $r$, respectively. They are expressed as:
$$P_r^{\text{WT}} = \bar{P}_r^{\text{WT}} + \Delta P_r^{\text{WT}}$$
$$P_r^{\text{PV}} = \bar{P}_r^{\text{PV}} + \Delta P_r^{\text{PV}}$$
$$P_r^{\text{Con}} = \bar{P}_r^{\text{Con}} + \Delta P_r^{\text{Con}}$$
where $\bar{P}_r^{*}$ are forecasted values, and $\Delta P_r^{*}$ are uncertainties bounded by:
$$\Delta P_r^{\text{WT}} \in [-\xi_r^{\text{WT}}, \xi_r^{\text{WT}}]$$
$$\Delta P_r^{\text{PV}} \in [-\xi_r^{\text{PV}}, \xi_r^{\text{PV}}]$$
$$\Delta P_r^{\text{Con}} \in [-\xi_r^{\text{Con}}, \xi_r^{\text{Con}}]$$
Robust control parameters $\Gamma_r^{*} \in [0,1]$ adjust the conservatism:
$$\left| \frac{P_r^{*} – \bar{P}_r^{*}}{\xi_r^{*}} \right| \leq \Gamma_r^{*}$$
For congestion management, the distribution system operator (DSO) uses a linearized AC power flow model. The power balance at node $r$ is:
$$\sum_{r \in c(s)} P_{sr}^l + P_r^{\text{WT}} + P_r^{\text{PV}} – \sum_{k \in c(r)} P_{rk}^l = P_r^{\text{Con}} + P_r^{\text{EV}}$$
Under uncertainty, this becomes:
$$\sum_{r \in c(s)} P_{sr}^l + (\bar{P}_r^{\text{WT}} – \Gamma_r^{\text{WT}} \xi_r^{\text{WT}}) + (\bar{P}_r^{\text{PV}} – \Gamma_r^{\text{PV}} \xi_r^{\text{PV}}) – \sum_{k \in c(r)} P_{rk}^l = (\bar{P}_r^{\text{Con}} + \Gamma_r^{\text{Con}} \xi_r^{\text{Con}}) + P_r^{\text{EV}}, \quad \forall r$$
Line flow and voltage constraints are:
$$-f_{sr}^l \leq P_{sr}^l \leq f_{sr}^l, \quad \forall l, (\lambda^-,\lambda^+)$$
$$V_s^{\min} \leq V_s \leq V_s^{\max}, \quad \forall s, (\gamma^-,\gamma^+)$$
where $f_{sr}^l$ is the flow limit, and $V_s^{\min}$ and $V_s^{\max}$ are voltage bounds. The dual variables $\lambda^+$, $\lambda^-$, $\gamma^+$, and $\gamma^-$ represent marginal prices for constraints.
The DSO’s objective is to minimize the total cost for electric cars, including travel time and charging/discharging expenses, while ensuring network constraints. For EV aggregator $m$, the cost 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:
$$C_T^m = w \sum_{i \in i_m} \left( \sum_{j,a,t} x_{a,\text{tot}}^{j,t} + \sum_{j,y,t} (x_{y,q}^{j,t} + x_{y,\text{qcd}}^{j,t}) \right)$$
and $C_{C\&D}^m$ is the charging/discharging cost with a price sensitivity coefficient $\beta$:
$$C_{C\&D}^m = (c_t + \beta \sum_m p_{m,t}) \sum_m p_{m,t}$$
Here, $p_{m,t} = p_{m,c}^{y,t} + p_{m,d}^{y,t}$, with charging and discharging powers defined as:
$$p_{m,c}^{y,t} = \max \left( \sum_{i \in P_y, l \in L_y} p_l x_{l,i,\text{qcd}}^t, 0 \right)$$
$$p_{m,d}^{y,t} = \min \left( \sum_{i \in P_y, l \in L_y} p_l x_{l,i,\text{qcd}}^t, 0 \right)$$
The dynamic tariff $R_t$ is derived from dual variables:
$$R_t = (\lambda_t^+ – \lambda_t^-) + (\gamma_t^+ – \gamma_t^-)$$
This tariff guides EV aggregators to adjust their schedules, reducing congestion.
On the aggregator side, each EVA optimizes its electric car operations based on the received dynamic tariff. The objective for EVA $m$ is:
$$\min (C_T^m + C_{C\&D}^m), \quad \forall m \in A_m$$
with the updated charging cost:
$$C_{C\&D}^m = (c_t + \beta \cdot p_{m,t} + R_t) \cdot p_{m,t}$$
Constraints include all earlier EV flow and SOC equations. This decentralized approach ensures that electric cars in China EV networks respond efficiently to congestion signals, leveraging their spatiotemporal flexibility.
To validate the model, consider a case study with a 20-node transportation network and a 17-node distribution network. Key parameters for electric car travel are summarized in the table below:
| Origin | Destination | EV Demand | Total SOC |
|---|---|---|---|
| T9 | T1 | 5000 | 2000 |
| T4 | T10 | 5000 | 2000 |
Distribution line limits are:
| Line | Limit (kW) |
|---|---|
| E5-F1 (L1) | 800 |
| E5-F2 (L2) | 300 |
| E5-F3 (L3) | 750 |
| E5-F4 (L4) | 350 |
Road parameters include capacity $\bar{N}_v^a$ and free-flow time $t_a^0$, e.g., for segment T1-T2, $\bar{N}_v^a = 200$ and $t_a^0 = 5$ hours. Assuming uncertainties of ±10% in wind, solar, and load, with $\Gamma_r^{*} = 1$ for the worst case, the robust optimization yields dynamic tariffs that prevent congestion. For instance, at charging station F2, the DT during peak hours (e.g., 10:00) is calculated as:
$$R_t = 0.7738 \text{ $/kWh}$$
compared to a deterministic value of $0.3973 $/kWh. This higher DT incentivizes electric cars to shift charging to non-congested times and locations.
The evolution of SOC and power flows demonstrates the effectiveness of this approach. For example, the power flow on line L2 under robust optimization remains within limits, whereas deterministic strategies may fail under uncertainties. The total cost for EV operations under robust optimization is $5846.2, higher than the deterministic $4537.4, but ensures reliability. The table below shows DT values for F2 and F3 over time:
| 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 |
Sensitivity analysis on the robust control parameter $\Gamma_r^{*}$ reveals that as $\Gamma_r^{*}$ increases from 0 to 1, the total cost rises, reflecting a trade-off between robustness and optimality. For example, with a 10% uncertainty range, the cost increases approximately 20% at $\Gamma_r^{*} = 1$ compared to $\Gamma_r^{*} = 0$. This underscores the importance of tuning $\Gamma_r^{*}$ based on risk tolerance in China EV deployments.
In conclusion, the integration of dynamic tariffs and robust optimization provides a powerful tool for managing distribution network congestion exacerbated by electric cars. By accounting for uncertainties in renewable generation and load, this approach ensures that electric cars in China EV networks can be dynamically routed and scheduled to maintain grid stability. Future work could explore real-time implementations and machine learning enhancements for uncertainty prediction, further solidifying the role of electric cars in sustainable energy systems.