The rapid proliferation of electric vehicles (EVs) in China, with over 15.52 million units registered by the end of 2023, has intensified the need for accurate assessment of distribution network capabilities. This study addresses the critical challenge of evaluating the electric vehicle hosting capacity (EVHC) in flexibly interconnected distribution networks (FIDNs) while accounting for the dynamic influence of traffic flow patterns. We develop a comprehensive optimization framework that synergistically models power distribution systems and transportation networks, incorporating advanced regulation mechanisms for diverse electric vehicle integration modes and leveraging the flexible power control capabilities of soft open points (SOPs). Our methodology represents a significant advancement in managing the complex interdependencies between energy systems and transportation infrastructures in the context of China’s evolving electric vehicle ecosystem.
The integration of large-scale electric vehicle charging infrastructure creates substantial challenges for distribution network operations, particularly when considering the spatial and temporal variations in charging demand driven by traffic flow dynamics. Traditional approaches to electric vehicle hosting capacity assessment often neglect these critical transportation network influences, leading to overestimated capacity values and potential operational risks. Our research bridges this gap by introducing a semi-dynamic traffic flow model that captures the temporal evolution of traffic conditions while maintaining computational tractability for planning applications. This approach enables more realistic modeling of electric vehicle charging behavior and its impact on distribution network constraints.
We formulate the electric vehicle hosting capacity calculation as a maximization problem, where the objective function aims to determine the maximum number of electric vehicles that can be accommodated while satisfying both distribution network operational constraints and transportation network feasibility conditions. The mathematical formulation incorporates three distinct electric vehicle integration modes: unregulated charging, regulated charging, and bidirectional power flow capability. This differentiation allows for more granular representation of the diverse electric vehicle technologies and charging infrastructures present in China’s evolving electric vehicle market.
The core optimization problem is expressed through the following objective function:
$$\max \sum_{t \in \Omega_T} \sum_{m \in \{e1,e2,e3\}} \sum_{(r,s) \in \Psi_m} q_{r,s}^t$$
where $q_{r,s}^t$ represents the number of electric vehicles traveling from origin $r$ to charging station $s$ during time period $t$, $\Omega_T$ denotes the set of time intervals, and $\Psi_m$ defines the set of origin-destination pairs for electric vehicle integration mode $m$.
The transportation network component employs a semi-dynamic traffic assignment model that captures the temporal coupling between consecutive time periods while maintaining computational efficiency. The model incorporates the following fundamental relationships:
$$x_{a,t} = \sum_{m \in \{g,e1,e2,e3\}} \sum_{(r,s) \in \Psi_m} \sum_{k \in K_{r,s}} f_{k,t}^{r,s} \delta_{k,a}^{r,s} \quad \forall a \in \Omega_R, \forall t \in \Omega_T$$
where $x_{a,t}$ represents the traffic flow on road segment $a$ during time period $t$, $f_{k,t}^{r,s}$ denotes the flow of vehicles on path $k$ from origin $r$ to destination $s$, and $\delta_{k,a}^{r,s}$ indicates the incidence relationship between path $k$ and road segment $a$.
The travel time on each road segment is modeled using a modified Bureau of Public Roads (BPR) function:
$$t_{a,t} = t_{a,0} \left[ 1 + \alpha \left( \frac{x_{a,t}}{C_a} \right)^\beta \right] \quad \forall a \in \Omega_R, \forall t \in \Omega_T$$
where $t_{a,0}$ represents the free-flow travel time, $C_a$ denotes the capacity of road segment $a$, and $\alpha$, $\beta$ are calibration parameters.
The electric vehicle regulation model incorporates spatial and temporal flexibility through price signals and user behavior considerations. The charging and discharging price constraints are defined as:
$$
\begin{aligned}
c^{\text{EV,ch,min}} &\leq c_{\text{ch},s,t} \leq c^{\text{EV,ch,max}} \quad \forall s \in \Omega_{TE}, \forall t \in \Omega_T \\
c^{\text{EV,dis,min}} &\leq c_{\text{dis},s,t} \leq c^{\text{EV,dis,max}} \quad \forall s \in \Omega_{TE}, \forall t \in \Omega_T
\end{aligned}
$$
The user satisfaction model for charging location adjustment incorporates multiple factors including electricity prices, travel distances, waiting times, and service quality:
$$
\lambda_{t}^{r,s’,s} =
\begin{cases}
\sum_{i=1}^{4} \omega_{1i} \cdot \frac{\Delta \text{Factor}_i}{\text{Factor}_{i,\text{max}}} & \text{if } d_{s’,s} \leq d_{m,\text{max}} \\
0 & \text{otherwise}
\end{cases}
$$
where $\omega_{1i}$ represents weighting coefficients and $\Delta \text{Factor}_i$ denotes the normalized differences in various influencing factors.
The distribution network operational constraints encompass power flow equations, voltage limits, thermal capacity restrictions, and equipment operational boundaries. The DistFlow equations form the foundation of the power system model:
$$
\begin{aligned}
&\sum_{j \in \kappa(i)} P_{ij,t} – \sum_{k’ \in \rho(i)} (P_{k’i,t} – R_{k’i}l_{k’i,t}) = P_{i,t}^{\text{inj}} \\
&\sum_{j \in \kappa(i)} Q_{ij,t} – \sum_{k’ \in \rho(i)} (Q_{k’i,t} – X_{k’i}l_{k’i,t}) = Q_{i,t}^{\text{inj}} \\
&v_{i,t} – v_{j,t} = 2(R_{ij}P_{ij,t} + X_{ij}Q_{ij,t}) – (R_{ij}^2 + X_{ij}^2)l_{ij,t} \\
&l_{ij,t}v_{i,t} – P_{ij,t}^2 – Q_{ij,t}^2 = 0
\end{aligned}
$$
The power injection at each node accounts for various sources and loads:
$$
\begin{aligned}
P_{i,t}^{\text{inj}} = &P_{i,t}^S – \sum_{m \in \{e1,e2,e3\}} P_{i,t}^{\text{EVC},m,\text{ch}} + P_{i,t}^{\text{EVC},e3,\text{dis}} + P_{i,t}^{\text{DG}} \\
&- P_{i,t}^{\text{ESS,ch}} + P_{i,t}^{\text{ESS,dis}} – P_{i,t}^N + P_{i,t}^{\text{SOP}}
\end{aligned}
$$
The SOP operational constraints facilitate power transfer between different feeders:
$$
\begin{aligned}
&P_{p,t}^{\text{SOP}} + P_{o,t}^{\text{SOP}} + P_{p,t}^{\text{SOP,L}} + P_{o,t}^{\text{SOP,L}} = 0 \\
&(P_{p,t}^{\text{SOP}})^2 + (Q_{p,t}^{\text{SOP}})^2 \leq (S_p^{\text{SOP}})^2 \\
&-Q_p^{\text{SOP}} \leq Q_{p,t}^{\text{SOP}} \leq Q_p^{\text{SOP}}
\end{aligned}
$$
To address the non-convexities in the original formulation, we employ a combination of relaxation techniques and customized solution algorithms. The second-order cone relaxation transforms the power flow equations into a computationally tractable form:
$$\left\| \begin{bmatrix} 2P_{ij,t} & 2Q_{ij,t} & l_{ij,t} – v_{i,t} \end{bmatrix}^\top \right\|_2 \leq l_{ij,t} + v_{i,t}$$
The quadratic convex relaxation handles the nonlinear terms in the traffic flow model through envelope constraints:
$$
\begin{aligned}
&x_{a,t}^2 \leq \omega_{a,t}^{x^2} \leq (x_{a,t,\text{min}} + x_{a,t,\text{max}})x_{a,t} – x_{a,t,\text{min}}x_{a,t,\text{max}} \\
&(\omega_{a,t}^{x^2})^2 \leq \omega_{a,t}^{x^4} \leq (x_{a,t,\text{min}}^2 + x_{a,t,\text{max}}^2)\omega_{a,t}^{x^2} – x_{a,t,\text{min}}^2x_{a,t,\text{max}}^2
\end{aligned}
$$
We develop a nested tightening relaxation algorithm that iteratively refines the solution quality through sequential bound tightening and increasingly tight linear cuts. The algorithm progresses until the relaxation gaps fall below specified thresholds:
$$
\begin{aligned}
&\chi_{\text{QC,avg}} = \frac{\chi_{\text{QC,total}}}{|\Omega_T| \left( |\Omega_R| + \sum_{m} \sum_{(r,s) \in \Psi_m} |K_{r,s}| \right)} \leq \varepsilon_{\text{QC,avg}} \\
&\chi_{\text{SOCR,avg}} = \frac{\sum_{t \in \Omega_T} \sum_{ij \in \Omega_L} \chi_{ij,t}}{|\Omega_T| \cdot |\Omega_L|} \leq \varepsilon_{\text{SOCR,avg}}
\end{aligned}
$$
Our experimental evaluation employs two test cases: a modified standard 24-node FIDN with 29-node transportation network and a practical 56-node distribution system with 45-node transportation network from Fujian Province, China. The parameter configurations for these test cases are summarized in the following tables:
| Parameter | Value |
|---|---|
| Total Conventional Load | 37.48 MW |
| Base Voltage | 20 kV |
| Substation Capacity | 10 MVA |
| Voltage Limits | 0.93-1.07 p.u. |
| Charging Station Locations | Nodes 17, 18, 2, 11, 5, 10 |
| Charger Power Ratings | 50 kW, 50 kW, 50 kW, 50 kW, 60 kW, 60 kW |
| Photovoltaic Capacity | Multiple nodes with varying capacities |
| Energy Storage Systems | Nodes 4, 5, 7, 12 (1 MWh each) |
| SOP Installations | 6 locations (1 MVA each) |
| Parameter | Value |
|---|---|
| Battery Capacity | 50.1 kWh |
| Charging/Discharging Efficiency | 95% |
| State of Charge Range | 10%-100% |
| Charging Price Range | 0.4-1.5 CNY/kWh |
| Discharging Price Range | 0.35-1.5 CNY/kWh |
| EV Integration Mode Ratio | 3:1:1 (Unregulated:Regulated:Bidirectional) |
| EV to Conventional Vehicle Ratio | 1:3 |
The optimization results demonstrate that the proposed methodology achieves an electric vehicle hosting capacity of 7,095 vehicles for Test Case 1 while maintaining all operational constraints. The comprehensive analysis reveals several key insights regarding the interaction between transportation networks and distribution systems in the context of China’s expanding electric vehicle infrastructure.
| Cost Category | Value (10,000 CNY) | Performance Indicator | Value |
|---|---|---|---|
| Power Purchase Cost | 58.089 | Average Voltage Deviation | 0.049 p.u. |
| Network Loss Cost | 0.306 | Average Line Loading Rate | 31.87% |
| Total Operational Cost | 58.395 | Average Substation Loading Rate | 78.49% |
The temporal and spatial analysis of traffic flow patterns reveals significant congestion during peak hours (07:00-09:00 and 17:00-19:00), which directly influences electric vehicle charging behavior and distribution network loading conditions. The dynamic pricing mechanism effectively guides electric vehicles to charging stations with sufficient capacity and lower congestion, demonstrating the value of coordinated optimization across interdependent infrastructures.
To quantify the impact of traffic flow considerations on electric vehicle hosting capacity assessment, we compare our approach with traditional methods that neglect transportation network constraints. The results clearly demonstrate the necessity of integrated modeling:
| Scenario | EV Hosting Capacity | Deviation from Proposed Method |
|---|---|---|
| Proposed Method (with traffic flow) | 7,095 vehicles | Reference |
| Traditional Method (without traffic flow) | 7,280 vehicles | +2.61% |
| Strict Traffic Conditions (TTTImax=1.1) | 6,832 vehicles | -3.71% |
| Relaxed Traffic Conditions (TTTImax=1.6) | 7,095 vehicles | 0.00% |
The analysis of SOP integration demonstrates its significant role in enhancing electric vehicle hosting capacity through flexible power flow control and reactive power support. The progressive increase in SOP capacity yields diminishing returns beyond certain thresholds due to other limiting factors such as substation and line capacities:
| SOP Capacity (MVA) | EV Hosting Capacity | Improvement over No SOP |
|---|---|---|
| 0 | 6,135 vehicles | Reference |
| 3 | 6,748 vehicles | +9.99% |
| 6 | 7,095 vehicles | +15.65% |
| 9 | 7,095 vehicles | +15.65% |
The composition of the vehicle fleet significantly influences the achievable electric vehicle hosting capacity, as demonstrated by sensitivity analysis across different penetration scenarios:
| Scenario | Conventional Vehicle Ratio | EV Integration Mode Ratio | EV Hosting Capacity |
|---|---|---|---|
| 1 | 0.75 | 3:1:1 | 7,095 vehicles |
| 2 | 0.60 | 3:1:1 | 7,215 vehicles |
| 3 | 0.45 | 3:1:1 | 7,280 vehicles |
| 4 | 0.75 | 3:0:0 | 4,832 vehicles |
| 5 | 0.75 | 3:1:0 | 6,542 vehicles |
The validation using the practical test case from Fujian Province confirms the scalability and applicability of our methodology to real-world systems. The results demonstrate consistent performance patterns, with the proposed approach achieving an electric vehicle hosting capacity of 1,615 vehicles while maintaining stable system operation.
The computational performance of the nested tightening relaxation algorithm proves essential for obtaining high-quality solutions with manageable computation times. The algorithm effectively reduces both quadratic convex and second-order cone relaxation gaps below the specified thresholds of 0.4% within reasonable iteration counts:
$$
\begin{aligned}
&\chi_{\text{QC,avg}}^{\text{final}} = 0.35\% \\
&\chi_{\text{SOCR,avg}}^{\text{final}} = 0.36\% \\
&\text{Computation Time} = 1,489 \text{ seconds}
\end{aligned}
$$
Comparative analysis with alternative solution approaches demonstrates the superiority of our nested tightening strategy. The single-layer tightening algorithm requires 2,154 seconds to achieve similar solution quality, highlighting the efficiency gains of our proposed methodology.
In conclusion, this research establishes a comprehensive framework for evaluating electric vehicle hosting capacity in flexibly interconnected distribution networks while explicitly considering traffic flow influences. The integrated modeling approach, advanced optimization techniques, and customized solution algorithm provide power system planners with valuable tools for managing the complex challenges associated with China’s rapidly growing electric vehicle adoption. Future work will focus on incorporating short-term traffic flow dynamics and multiple uncertainty sources to further enhance the practical applicability of the proposed methodology in supporting China’s transition to sustainable transportation and energy systems.
The successful implementation of our approach demonstrates significant potential for improving the planning and operation of urban energy systems facing substantial electric vehicle integration. By accurately capturing the interactions between transportation networks and power distribution systems, our methodology enables more reliable assessment of infrastructure capabilities and more effective utilization of flexible resources such as SOPs and controllable electric vehicle charging. These advancements contribute substantially to the development of resilient and efficient urban energy systems that can support China’s ambitious goals for electric vehicle deployment and sustainable urban mobility.