As the adoption of electric vehicle cars accelerates globally, the planning and deployment of supporting charging infrastructure, particularly in regions with harsh climates, has become a critical challenge. This article proposes a comprehensive framework for planning electric vehicle car charging stations in low-temperature environments. The model integrates the degradation mechanisms of electric vehicle car batteries under cold conditions with coupled transportation and power distribution network dynamics to optimize station siting and capacity sizing, aiming to minimize societal costs and enhance service reliability.
1. Introduction and Background
The rapid proliferation of electric vehicle cars presents a significant opportunity for decarbonizing the transportation sector. However, the performance and energy demand of an electric vehicle car are highly sensitive to ambient temperature. In low-temperature regions, the driving range of an electric vehicle car is substantially reduced due to increased cabin heating demands and, more critically, the accelerated degradation and reduced efficiency of lithium-ion batteries. This phenomenon exacerbates “range anxiety” and leads to concentrated, high-peak charging demands, straining existing infrastructure. Traditional charging station planning models often overlook these temperature-dependent behavioral and physical changes, leading to suboptimal infrastructure that fails under winter conditions. This necessitates a new planning paradigm that explicitly models the “vehicle-road-network” coupling under low-temperature stress. An accurate model must capture the intricate interplay between the electrochemical state of the electric vehicle car battery, the driver’s routing decisions in a congested urban network, and the resulting spatiotemporal load on the power grid. This article addresses this gap by developing a holistic planning model that incorporates a reduced-order semi-empirical model for battery capacity fade, a dynamic traffic assignment model, and a queuing-theoretic approach for station capacity sizing, all optimized to minimize the total societal cost encompassing user time, infrastructure investment, and network operational expenses.
2. Electric Vehicle Car Modeling Under Low-Temperature Conditions
Accurate modeling of the individual electric vehicle car is the foundation for predicting aggregate charging demand. Two primary effects dominate in cold climates: the reduction in available battery capacity and the increase in energy consumption per kilometer traveled.
2.1 Low-Temperature Battery Capacity Attenuation Model
The usable capacity of an electric vehicle car battery declines at low temperatures. To precisely quantify this, a Reduced-Order Semi-Empirical Model (ROSEM) is employed, which couples three major degradation mechanisms: Solid Electrolyte Interphase (SEI) film growth, Loss of Active Material (LAM), and lithium plating. The coupled reaction currents are given by:
$$i_{\text{SEI}}(t) = nF k_{\text{SEI}} C_s(R_n, t) \exp\left(-\frac{0.5F}{RT}\eta_{\text{SEI}}\right)$$
$$i_{\text{LAM}}(t) = k_{\text{LAM}} \left( \frac{\sigma_{h,\text{max}} – \sigma_{h,\text{min}}}{\sigma_{\text{yield}}} \right)^{\frac{1}{m}} \quad \text{with} \quad \sigma_h = \frac{\sigma_r(R_n) + 2\sigma_t(R_n)}{3}$$
$$i_{\text{Li}}(t) = k_{\text{Li}} A \varepsilon_{s,n} \delta_n \Phi\left( \eta_m + U^{\text{ref}}_n + \frac{R_{\text{SEI}} R_n I_B}{3A \varepsilon_{s,n} \delta_n} – \phi_{\text{onset}} \right)$$
where \(F\) is Faraday’s constant, \(R_n\) is the particle radius, \(R\) is the ideal gas constant, \(T\) is temperature, \(k_{\text{SEI}}\), \(k_{\text{LAM}}\), \(k_{\text{Li}}\) are rate constants, and other terms represent overpotentials, stresses, and material properties. The total capacity loss \(Q_{\text{loss}}\) is the integrated sum of these reactions:
$$Q_{\text{SEI}} = \int_0^t i_{\text{SEI}}(t) dt, \quad Q_{\text{LAM}} = \int_0^t i_{\text{LAM}}(t) dt, \quad Q_{\text{Li}} = \int_0^t i_{\text{Li}}(t) dt$$
$$Q_{\text{loss}} = Q_{\text{SEI}} + Q_{\text{LAM}} + Q_{\text{Li}}$$
The effective battery capacity \(C_T\) at a low temperature \(T\) is then:
$$C_T = C_b (1 – Q_{\text{loss}})$$
where \(C_b\) is the nominal capacity at 25°C. A key parameter set for the ROSEM is summarized below:
| Parameter | Symbol | Value |
|---|---|---|
| Faraday Constant | \(F\) | 96,487 C/mol |
| SEI Reaction Constant | \(k_{\text{SEI}}\) | 15.55 s⁻¹/² |
| LAM Pre-factor | \(k_{\text{LAM}}\) | 1.61 |
| Li Plating Pre-factor | \(k_{\text{Li}}\) | 4.20 |
| Yield Strength | \(\sigma_{\text{yield}}\) | 787.41 MPa |
2.2 Energy Consumption Model for an Electric Vehicle Car
The energy consumed per kilometer for an electric vehicle car is affected by traffic conditions (road saturation \(\gamma\)) and ambient temperature. The consumption \(P_{ij,\gamma}\) on road segment \(ij\) under saturation \(\gamma\) is modeled as a polynomial function of speed \(V_{ij,\gamma}\):
$$P_{ij,\gamma} = \sum_{y=-1}^{2} a_{\gamma} (V_{ij,\gamma})^y$$
The temperature correction factor \(k_T\) scales the base consumption at 25°C (\(Q_{25}\)) to the consumption at temperature \(T\) (\(Q_T\)):
$$Q_T = \sum_{u=0}^{5} b_{\gamma} (1.8T + 32)^u, \quad k_T = \frac{Q_T}{Q_{25}}$$
The combined energy consumption rate \(\omega_{ij,\gamma,T}\) is therefore:
$$\omega_{ij,\gamma,T} = k_T \cdot P_{ij,\gamma}$$
2.3 Electric Vehicle Car Driving Range
The instantaneous driving range \(D_T\) for an electric vehicle car under low-temperature conditions is the ratio of its available battery capacity to the consumption rate:
$$D_T = \frac{C_T}{\omega_{ij,\gamma,T}}$$
This equation clearly shows the dual penalty of cold weather: reduced \(C_T\) and increased \(\omega_{ij,\gamma,T}\), leading to a significantly shorter range and consequently more frequent charging needs for the electric vehicle car.
3. Coupled Road-Network and Charging Demand Prediction
Predicting where and when an electric vehicle car driver will seek to charge requires modeling their movement through a congested urban network. The road network is represented as a graph with nodes and links. The impedance \(W_{ij}\) of a road link \(ij\) is a piecewise function of traffic saturation \(\gamma\):
$$W_{ij} =
\begin{cases}
T_{ij,L1} + T_{i,o1}, & 0 < \gamma \leq 0.6 \\
T_{ij,L1} + T_{i,o2}, & 0.6 < \gamma \leq 1.0 \\
T_{ij,L2} + T_{i,o2}, & 1.0 < \gamma \leq 2.0
\end{cases}$$
where \(T_{ij,L1}, T_{ij,L2}\) are link travel times under different congestion levels, and \(T_{i,o1}, T_{i,o2}\) are node delay penalties.
The coupling between the transportation network (layer \(G_1\)) and the power distribution network (layer \(G_2\)) is defined by an inter-layer edge matrix \(W_{G_1G_2}\), where an element \(\delta_{ij}=1\) if transportation node \(i\) coincides with a potential charging station location at grid node \(j\).
The state of each electric vehicle car (location, battery State of Charge – SOC) is updated dynamically using a fixed-step method within a trip-chain framework. Monte Carlo simulation is employed to generate a large population of electric vehicle cars (categorized as private cars, taxis, and service vehicles) with stochastic initial SOC, start times, and destinations. For each electric vehicle car, the Dijkstra algorithm finds the shortest path based on real-time impedances \(W_{ij}\). The electric vehicle car’s SOC is decremented according to \(\omega_{ij,\gamma,T}\). When the SOC falls below a driver-anxiety threshold, the electric vehicle car is routed to the nearest eligible charging station within its remaining range. This process yields the spatiotemporal distribution of charging demand. The aggregated daily load profile shows pronounced peaks during morning and evening rush hours, with magnitude heavily dependent on temperature.
4. Low-Temperature Charging Station Planning Model
The planning model determines the optimal location and number of chargers for each station to serve the predicted low-temperature demand while minimizing total societal cost.
4.1 Queuing Model for Charging Station Capacity
An M/M/S/N mixed queuing model is adopted for each charging station \(c\), where \(S = s_c\) is the number of chargers and \(N\) is the maximum system capacity (including those in queue). The maximum queue length is related to the number of chargers: \(N_c = s_c + s_c / \sigma\). The service rate \(\mu_c\) is temperature-dependent due to reduced charging efficiency \(\vartheta(T)\):
$$\mu_c = \frac{X_c \vartheta(T)}{C_b}$$
where \(X_c\) is the average charging power. The average waiting time \(W_{c,q}\) for an electric vehicle car is a function of the arrival rate \(\lambda_c\), service rate \(\mu_c\), number of servers \(s_c\), and system capacity \(N_c\). The effective service intensity must be constrained to ensure service quality.
4.2 Optimization Model Formulation
The objective is to minimize the total annual societal cost \(C_0\), which balances infrastructure investment, grid operation, road network congestion, and user time costs through weighting coefficients \(\alpha\) and \(\beta\) (\(\alpha + \beta = 2\)):
$$\min C_0 = \alpha (C_{\text{JS}} + C_{\text{O}} + C_{\text{DN}} + C_{\text{TN}}) + \beta C_{\text{SJ}}$$
4.2.1 Cost Components
1. Charging Station Construction Cost (\(C_{\text{JS}}\)):
$$C_{\text{JS}} = (M_s + \theta M_s + D_s) \cdot s_c \cdot \frac{r_0 (1+r_0)^z}{(1+r_0)^z – 1}$$
where \(M_s\) is charger unit cost, \(\theta\) is ancillary equipment coefficient, \(D_s\) is civil engineering cost, \(r_0\) is discount rate, and \(z\) is depreciation period.
2. Charging Station Operation Cost (\(C_{\text{O}}\)):
$$C_{\text{O}} = U(M_s) \cdot (M_s + \theta M_s + D_s) \cdot s_c$$
where \(U(M_s)\) is the annual operational cost factor.
3. Distribution Network Operation Cost (\(C_{\text{DN}}\)):
$$C_{\text{DN}} = C_{\text{loss}} + C_{\text{buy}}$$
$$C_{\text{loss}} = 365 \sum_{t=1}^{24} \sum_{i=1}^{N} \sum_{j \in u(i)} c_l I_{ij}^2 R_{ij} \Delta t$$
$$C_{\text{buy}} = 365 \sum_{t=1}^{24} c_{\text{buy}} P_{\text{buy}}(t)$$
This includes power loss cost and cost of purchasing electricity from the main grid.
4. Road Network Congestion Cost (\(C_{\text{TN}}\)):
$$C_{\text{TN}} = 365 \sum_{t=1}^{24} y(t) \sum_{l \in L} x_l \tau_l$$
where \(x_l\) is traffic flow on link \(l\), \(\tau_l\) is travel time, and \(y(t)\) is a time-varying cost coefficient.
5. Electric Vehicle Car User Time Cost (\(C_{\text{SJ}}\)):
$$C_{\text{SJ}} = 365 \sum_{i=1}^{N} \left( W_{c,q} + \frac{2d_c \xi}{\upsilon_d} \right) C_d N_{c,d} K_{c,d} / \varepsilon_c$$
This cost includes queuing time \(W_{c,q}\) at the station and the detour travel time to reach it, weighted by the value of time \(C_d\) for user type \(d\).
4.2.2 Key Constraints
- Service Intensity: \(\frac{\lambda_c}{s_c \mu_c} \left[1 – \frac{1}{s_c! s_c^{N_c-s_c}} \left(\frac{\lambda_c}{\mu_c}\right)^{N_c} P_{c,0}\right] < 1, \quad \forall c \in B\)
- Service Rejection Rate: \(\frac{1}{s_c! s_c^{N_c-s_c}} \left(\frac{\lambda_c}{\mu_c}\right)^{N_c} P_{c,0} \leq 0.05, \quad \forall c \in B\)
- Road Capacity: \(x_l \leq x_{l,\text{max}}, \quad \forall l\)
4.2.3 Solution Algorithm
A Genetic Algorithm (GA) is utilized to solve this mixed-integer nonlinear optimization problem. The decision variables include the locations of candidate stations and the number of chargers \(s_c\) at each. The fitness function evaluates the total cost \(C_0\) for a given configuration based on the demand input from the coupled simulation. The algorithm iteratively selects, crosses, and mutates solution populations to converge towards the minimum-cost planning scheme.
5. Case Study Analysis and Discussion
A case study based on a modified urban network with 29 nodes and 49 links, divided into residential, commercial, and office zones, is presented. A fleet of 13,000 electric vehicle cars (private, taxi, service) is simulated.
5.1 Charging Demand Under Low Temperature
The simulation predicts significantly higher charging demand at -15°C compared to 25°C. The number of electric vehicle cars requiring charging daily increases, and the load profile peaks are more pronounced. The following table summarizes the impact:
| Ambient Temperature | Total Daily Charging Load | Load Relative to 25°C | Peak Load (MW) |
|---|---|---|---|
| 25 °C | 289.61 MWh | 100% | 25.61 |
| -15 °C | 390.93 MWh | 134.98% | 33.10 |
| -25 °C | 402.46 MWh | 138.97% | 36.87 |
The spatial distribution of demand also shifts, concentrating more around commercial and major residential traffic intersections during peak hours, directly influencing optimal charging station placement for the electric vehicle car fleet.
5.2 Optimized Charging Station Planning Results
For the -15°C scenario, the optimization model suggests deploying 7 charging stations. The table below shows a sample of the planning result for one station, illustrating how costs escalate with decreasing temperature.
| Temperature (°C) | Battery Capacity Fade (%) | Number of Chargers (\(s_c\)) | Total Annual Societal Cost for Station 1 (Million) |
|---|---|---|---|
| 25 | 0.62 | 16 | 0.415 |
| -15 | 16.74 | 18 | 0.481 |
| -25 | 26.11 | 20 | 0.567 |
| -35 | 38.81 | 22 | 0.677 |
The optimized layout places stations strategically at high-demand nodes identified by the coupled simulation, often at the interface between different functional zones to serve inter-zonal electric vehicle car trips effectively.
5.3 Algorithm Performance and Comparative Analysis
The proposed GA was compared against Particle Swarm Optimization (PSO) and Grey Wolf Optimizer (GWO). The GA achieved the lowest societal cost for a comparable number of total chargers, demonstrating its effectiveness in balancing the multi-faceted cost components inherent in electric vehicle car infrastructure planning.
| Optimization Algorithm | Total Chargers in Network | Total Annual Societal Cost (Million) |
|---|---|---|
| Genetic Algorithm (GA) | 114 | 3.170 |
| Particle Swarm Optimization (PSO) | 113 | 3.252 |
| Grey Wolf Optimizer (GWO) | 114 | 3.190 |
Furthermore, sensitivity analysis on the weighting coefficients \(\alpha\) and \(\beta\) reveals the trade-off between infrastructure/grid costs and user time costs, providing planners with a Pareto front of solutions based on policy priorities.
6. Conclusion
This article presents a holistic framework for planning electric vehicle car charging infrastructure in low-temperature climates. The core contribution is the tight integration of a physics-based model for electric vehicle car battery performance degradation under cold stress with a dynamic, coupled transportation and power network model. This integration allows for a realistic prediction of the spatiotemporal charging demand surge caused by reduced electric vehicle car range and increased energy consumption. By formulating the siting and sizing problem as a societal cost minimization problem that includes a queuing-theoretic capacity model, the proposed method generates plans that are robust to winter conditions. The results demonstrate that ignoring low-temperature effects leads to under-provisioning of charging infrastructure, which would result in excessive wait times for the electric vehicle car user and increased grid stress. The model provides a practical tool for urban planners and utility companies to future-proof electric vehicle car charging networks against climatic challenges, ensuring reliable service, enhancing user satisfaction, and promoting the sustainable adoption of electric vehicle cars in all geographical regions. Future work will integrate real-time weather data and traffic management strategies to develop dynamic operational policies for the planned charging stations.