In recent years, the electric car industry has experienced rapid growth, driven by policy support and market demand. The number of electric cars in China, particularly China EV models, has surged, posing significant challenges to distribution networks. The hosting capacity of a distribution network refers to the maximum number of electric cars it can accommodate while maintaining safe operation. However, the spatiotemporal variations in traffic flow alter the distribution of electric car charging demand, complicating the assessment of hosting capacity. To address this, we propose a method to calculate the electric vehicle hosting capacity in a flexibly interconnected distribution network (FIDN) that incorporates traffic flow effects. This approach leverages the flexible adjustability of soft open points (SOPs) to mitigate the impact of large-scale electric car integration, ensuring grid stability and efficiency.
The integration of electric cars, especially in the context of China EV adoption, requires sophisticated models that account for both power grid and traffic network interactions. Traditional methods often overlook traffic flow influences, leading to inaccurate estimates. Our work bridges this gap by developing a semi-dynamic traffic flow model that captures the coupling between electric car mobility and grid operations. We consider three electric car integration modes: non-adjustable charging power, adjustable charging power, and adjustable charging-discharging power. These modes reflect the diversity of electric car technologies and charging infrastructure in China, where China EV policies promote smart charging solutions. The objective is to maximize the number of electric cars the FIDN can host, subject to constraints from both networks.
We formulate the problem as an optimization model with constraints including traffic flow equations, electric car charging behavior, and grid operational limits. The model incorporates SOPs, which enable power transfer between feeders, enhancing flexibility. Key variables include traffic flow rates, electric car charging demands, and SOP power outputs. The objective function is defined as:
$$ \max \sum_{t \in \Omega_T} \sum_{m \in \{e1,e2,e3\}} \sum_{(r,s) \in \Psi_m} q_{r,s}^t $$
where \( q_{r,s}^t \) represents the number of electric cars traveling from origin \( r \) to destination \( s \) at time \( t \), \( \Omega_T \) is the set of time intervals, and \( \Psi_m \) is the set of origin-destination pairs for electric car integration mode \( m \). This maximizes the total electric car hosting capacity over the planning horizon.
The traffic flow model is based on a semi-dynamic approach, which balances accuracy and computational efficiency. It includes constraints for traffic flow conservation, travel time, and congestion effects. For example, the traffic flow on a road segment \( a \) at time \( t \), denoted \( x_{a,t} \), is governed by:
$$ x_{a,t} = \sum_{m \in \{g,e1,e2,e3\}} \sum_{(r,s) \in \Psi_m} \sum_{k \in K_{r,s}} f_{k,t}^{r,s} \delta_{k,a}^{r,s} $$
where \( f_{k,t}^{r,s} \) is the flow on path \( k \), and \( \delta_{k,a}^{r,s} \) is an indicator variable. The travel time \( t_{a,t} \) on segment \( a \) is modeled using a Bureau of Public Roads (BPR) function:
$$ t_{a,t} = t_{a,0} \left[ 1 + \alpha \left( \frac{x_{a,t}}{C_a} \right)^\beta \right] $$
where \( t_{a,0} \) is the free-flow travel time, \( C_a \) is the capacity, and \( \alpha, \beta \) are parameters. This captures congestion effects that influence electric car charging decisions.
For electric car regulation, we model spatial and temporal flexibility. Spatial flexibility allows electric cars to change charging locations based on factors like electricity prices and waiting times. The satisfaction level \( \lambda_{r,s’,s}^t \) for rerouting from station \( s’ \) to \( s \) is defined as:
$$ \lambda_{r,s’,s}^t = \omega_{11} \frac{c_{ch,s’,t} – c_{ch,s,t}}{c_{ch,t,\text{max}}’} + \omega_{12} \frac{d_{m,\text{max}} – d_{s’,s}}{d_{m,\text{max}}} + \omega_{13} \frac{T_{\text{wait},m,s’,t} – T_{\text{wait},m,s,t}}{T_{\text{wait},m,\text{max}}} + \omega_{14} \frac{Q_{m,s’} – Q_{m,s}}{Q_{m,\text{max}}} $$
for modes \( m \in \{e1,e2\} \), where \( c_{ch,s,t} \) is the charging price, \( d_{s’,s} \) is the distance, \( T_{\text{wait},m,s,t} \) is the waiting time, and \( Q_{m,s} \) is the service quality. For mode \( e3 \) (adjustable charging-discharging), the satisfaction includes discharge price terms. This ensures user acceptance while optimizing grid performance.
Temporal flexibility involves shifting charging and discharging times. The energy balance for electric cars at charging station \( i \) for mode \( m \in \{e2,e3\} \) is:
$$ E_{i,t}^m = E_{i,t-\Delta t}^m + \left( P_{i,t-\Delta t}^{\text{EVC},m,\text{ch}} \eta_{\text{EVC,ch}} – \frac{P_{i,t-\Delta t}^{\text{EVC},m,\text{dis}}}{\eta_{\text{EVC,dis}}} \right) \Delta t + E_{i,t}^{\text{arr},m} – E_{i,t}^{\text{dep},m} $$
where \( E_{i,t}^m \) is the total energy, \( P_{i,t}^{\text{EVC},m,\text{ch}} \) and \( P_{i,t}^{\text{EVC},m,\text{dis}} \) are charging and discharging powers, and \( \eta \) are efficiencies. Constraints limit the power based on the number of electric cars and charger availability.
The FIDN constraints include power flow equations, voltage limits, and device operational limits. The distflow equations for radial networks are used:
$$ \sum_{j \in \kappa(i)} P_{ij,t} – \sum_{k’ \in \rho(i)} (P_{k’i,t} – R_{k’i} l_{k’i,t}) = P_{i,t}^{\text{inj}} $$
$$ \sum_{j \in \kappa(i)} Q_{ij,t} – \sum_{k’ \in \rho(i)} (Q_{k’i,t} – X_{k’i} l_{k’i,t}) = Q_{i,t}^{\text{inj}} $$
$$ v_{i,t} – v_{j,t} = 2(R_{ij} P_{ij,t} + X_{ij} Q_{ij,t}) – (R_{ij}^2 + X_{ij}^2) l_{ij,t} $$
$$ l_{ij,t} v_{i,t} = P_{ij,t}^2 + Q_{ij,t}^2 $$
where \( P_{ij,t} \) and \( Q_{ij,t} \) are active and reactive power flows, \( v_{i,t} \) is the squared voltage magnitude, and \( l_{ij,t} \) is the squared current magnitude. These are relaxed using second-order cone programming for computational tractability.
SOPs enhance flexibility by allowing power transfer between nodes. The SOP constraints for a device connected between nodes \( p \) and \( o \) are:
$$ P_{p,t}^{\text{SOP}} + P_{o,t}^{\text{SOP}} + P_{p,t}^{\text{SOP,L}} + P_{o,t}^{\text{SOP,L}} = 0 $$
$$ (P_{p,t}^{\text{SOP}})^2 + (Q_{p,t}^{\text{SOP}})^2 \leq (S_p^{\text{SOP}})^2 $$
$$ -Q_p^{\text{SOP}} \leq Q_{p,t}^{\text{SOP}} \leq Q_p^{\text{SOP}} $$
where \( P^{\text{SOP}} \) and \( Q^{\text{SOP}} \) are active and reactive power outputs, and \( S^{\text{SOP}} \) is the capacity. SOPs help manage congestion and voltage issues caused by electric car charging.
To solve the model, we employ a nested tightening relaxation algorithm. The original problem contains non-convex constraints due to bilinear terms and power flow equations. We use quadratic convex relaxation for terms like \( x_{a,t}^4 \) and \( f_{k,t}^{r,s} t_{k,t}^{r,s} \), and second-order cone relaxation for power flow equations. For example, the relaxation for \( x_{a,t}^4 \) introduces auxiliary variables \( \omega_{a,t}^{x^2} \) and \( \omega_{a,t}^{x^4} \) with bounds:
$$ x_{a,t}^2 \leq \omega_{a,t}^{x^2} \leq (x_{a,t,\text{min}} + x_{a,t,\text{max}}) x_{a,t} – x_{a,t,\text{min}} x_{a,t,\text{max}} $$
$$ (\omega_{a,t}^{x^2})^2 \leq \omega_{a,t}^{x^4} \leq (x_{a,t,\text{min}}^2 + x_{a,t,\text{max}}^2) \omega_{a,t}^{x^2} – x_{a,t,\text{min}}^2 x_{a,t,\text{max}}^2 $$
Similarly, the power flow constraint is relaxed to a second-order cone:
$$ \left\| \begin{bmatrix} 2P_{ij,t} & 2Q_{ij,t} & l_{ij,t} – v_{i,t} \end{bmatrix}^T \right\|_2 \leq l_{ij,t} + v_{i,t} $$
The nested algorithm combines sequential bound tightening (SBT) and increasingly tight linear cut (ITLC) methods. SBT iteratively tightens variable bounds to reduce relaxation gaps, while ITLC adds linear cuts to improve second-order cone relaxation. The algorithm stops when average gaps fall below thresholds, ensuring high solution accuracy.
We validate the method using modified standard and real-world cases. The first case involves a 24-node FIDN and a 29-node traffic network. Key parameters are summarized in Table 1.
| Parameter | Value |
|---|---|
| Total conventional load | 37.48 MW |
| Base voltage | 20 kV |
| Substation capacity | 10 MVA |
| Voltage limits | 0.93-1.07 p.u. |
| Electric car charging power | 50-60 kW per charger |
| Electric car battery capacity | 50.1 kWh |
| Charging efficiency | 95% |
| Discharging efficiency | 95% |
| SOE limits | 0.1-1.0 |
| Charging price range | 0.4-1.5 CNY/kWh |
| Discharging price range | 0.35-1.5 CNY/kWh |
| Electric car modes ratio (e1:e2:e3) | 3:1:1 |
| Electric car to conventional car ratio | 1:3 |
| TTTI_max | 1.4 |
The hosting capacity is calculated as 7,095 electric cars. Operational costs and indices are shown in Table 2.
| Cost Type | Value (10^4 CNY) |
|---|---|
| Power purchase cost | 58.089 |
| Network loss cost | 0.306 |
| Total cost | 58.395 |
| Average voltage deviation | 0.049 p.u. |
| Average line load rate | 31.87% |
| Average substation load rate | 78.49% |
Traffic flow variations show congestion during peak hours (07:00-09:00 and 17:00-19:00), affecting electric car charging distributions. Charging prices at different stations adjust dynamically to guide electric car behavior. For instance, at 08:00, stations in commercial areas have lower prices to attract electric cars, reducing grid stress. The spatial flexibility of electric cars is evident from the rerouting satisfaction levels, which exceed the threshold of 0.8 for feasible adjustments.
Charging and discharging powers for different electric car modes are analyzed. Mode e1 (non-adjustable) shows peak charging during rush hours, while modes e2 and e3 shift charging to off-peak periods or provide discharging support. The temporal flexibility helps flatten the load curve and enhance hosting capacity. The hosting capacity is limited by substation and line constraints during peak hours, particularly at nodes 22 and 23.
To illustrate the impact of traffic flow, we compare scenarios with and without traffic considerations. Ignoring traffic flow leads to a higher hosting capacity estimate of 7,280 electric cars (2.61% overestimation), as it fails to account for congestion-induced delays. This highlights the necessity of incorporating traffic flow in electric car hosting capacity assessments for China EV integration.
We also examine the effect of traffic network congestion tolerance, represented by the maximum travel time index (TTTI_max). Results for different TTTI_max values are summarized in Table 3.
| TTTI_max | Hosting Capacity (electric cars) | Total Cost (10^4 CNY) | Average Voltage Deviation (p.u.) | Average Line Load Rate (%) | Average Substation Load Rate (%) |
|---|---|---|---|---|---|
| 1.1 | 6,850 | 57.892 | 0.048 | 31.45 | 77.83 |
| 1.2 | 6,950 | 58.123 | 0.049 | 31.67 | 78.12 |
| 1.3 | 7,020 | 58.254 | 0.049 | 31.75 | 78.34 |
| 1.4 | 7,095 | 58.395 | 0.049 | 31.87 | 78.49 |
| 1.5 | 7,095 | 58.395 | 0.049 | 31.87 | 78.49 |
As TTTI_max increases, hosting capacity rises initially but saturates at 1.4 due to grid constraints. This indicates that beyond a certain point, traffic congestion no longer limits capacity, but grid infrastructure does. Thus, for high electric car penetration in China, both networks must be co-optimized.
The role of SOPs is crucial. We test different SOP capacities and observe the hosting capacity in Table 4.
| SOP Capacity (MVA) | Hosting Capacity (electric cars) | Total Cost (10^4 CNY) | Average Voltage Deviation (p.u.) | Average Line Load Rate (%) |
|---|---|---|---|---|
| 0 | 6,135 | 56.782 | 0.052 | 33.12 |
| 2 | 6,580 | 57.456 | 0.050 | 32.45 |
| 4 | 6,920 | 58.123 | 0.049 | 32.01 |
| 6 | 7,095 | 58.395 | 0.049 | 31.87 |
| 8 | 7,095 | 58.395 | 0.049 | 31.87 |
With no SOP, hosting capacity is lower, but it increases with SOP capacity up to 6 MVA, after which it plateaus. SOPs facilitate power transfer, reduce losses, and improve voltage profiles, demonstrating their value for China EV integration in FIDNs.
Different electric car mode ratios also affect hosting capacity. Results are shown in Table 5 for various ratios of modes e1, e2, and e3.
| Mode Ratio (e1:e2:e3) | Hosting Capacity (electric cars) | Total Cost (10^4 CNY) |
|---|---|---|
| 3:1:1 | 7,095 | 58.395 |
| 3:1:0 | 6,820 | 57.963 |
| 3:0:0 | 4,832 | 55.124 |
| 2:1:1 | 7,180 | 58.512 |
| 1:1:1 | 7,250 | 58.634 |
Including adjustable modes (e2 and e3) significantly boosts hosting capacity, as they provide grid services. For instance, with only mode e1, capacity drops to 4,832 electric cars, highlighting the importance of smart charging for China EV ecosystems.
The second case uses a real-world 56-node distribution network and 45-node traffic network in Fujian, China. Parameters include a total load of 3.45 MW, base voltage of 10 kV, and SOPs with 1 MVA capacity. The hosting capacity is 1,615 electric cars. Operational costs and indices are in Table 6.
| Cost Type | Value (10^4 CNY) |
|---|---|
| Power purchase cost | 11.503 |
| Network loss cost | 0.243 |
| Total cost | 11.746 |
| Average voltage deviation | 0.039 p.u. |
| Average line load rate | 20.64% |
| Average substation load rate | 43.07% |
Traffic flow analysis reveals congestion during peak hours, influencing charging station choices. Charging prices vary, with stations offering lower prices during high grid stress to attract electric cars. The satisfaction levels for rerouting show that spatial flexibility is utilized effectively. Charging powers for different modes demonstrate load shifting, with mode e3 providing discharging support during peaks.
Without traffic flow consideration, hosting capacity is overestimated by 11.46% (1,800 electric cars), reaffirming the importance of integrated modeling. Variations in TTTI_max and SOP capacity yield similar trends to the first case, with hosting capacity improving with higher tolerance and SOP support.
The nested tightening relaxation algorithm proves efficient, reducing average relaxation gaps to below 0.4% within a few iterations. For the 24-node case, it takes 6 iterations and 1,489 seconds, compared to 4 iterations and 2,154 seconds for a single-layer approach, showing a 30.87% time reduction. This makes it suitable for large-scale applications in China EV planning.
In conclusion, our method provides a comprehensive framework for calculating electric car hosting capacity in FIDNs considering traffic flow. Key findings include:
– Traffic flow integration prevents overestimation of hosting capacity.
– SOPs and adjustable electric car modes enhance capacity significantly.
– The nested algorithm ensures accurate and efficient solutions.
Future work could address short-term traffic dynamics and uncertainties in electric car behavior for even more robust planning in the evolving China EV landscape.