Calculation Method for Electric Vehicle Hosting Capacity in Flexibly Interconnected Distribution Networks Considering Traffic Flow

Abstract

The temporal-spatial variation of traffic flow significantly influences the distribution of electric vehicle (EV) charging demand, thereby affecting the hosting capacity of distribution networks. This study proposes a novel calculation method for evaluating the EV hosting capacity in flexibly interconnected distribution networks (FIDNs) by integrating traffic flow dynamics. The method leverages the flexible regulation capability of soft open points (SOPs) to mitigate the impact of large-scale EV integration. Through semi-dynamic traffic flow modeling, EV regulation frameworks, and a nested tightening relaxation algorithm, the research aims to enhance the accuracy of hosting capacity assessment in power-transportation coupled systems.

1. Introduction

1.1 Research Background

As of 2023, the global stock of electric vehicles has reached 15.52 million, with a year-on-year growth of 48.5% . The uncoordinated integration of EVs into distribution networks poses challenges to grid stability, necessitating precise evaluation of EV hosting capacity—the maximum number of EVs a distribution network can accommodate without violating operational constraints .

Existing studies often overlook the coupling between energy flow and traffic flow, leading to inaccuracies in hosting capacity calculations. For instance, 文献 [2,8,11] assumes EVs are unaffected by traffic congestion, which may mispredict EV arrival times at charging stations . Additionally, most models are tailored for radial distribution networks, limiting their applicability to FIDNs with SOPs .

1.2 Research Significance

FIDNs with SOPs enable flexible power flow control and reactive power compensation, supporting higher EV penetration . By integrating traffic flow dynamics, the proposed method addresses two key limitations:

Spatial-temporal coupling: Traffic congestion affects EV charging location choices and arrival times, impacting grid load distribution .
Network flexibility: SOPs facilitate inter-feeder power transfer, alleviating overloads caused by concentrated EV charging .

2. Semi-Dynamic Traffic Flow Model for EV Regulation

2.1 Model Framework

The semi-dynamic traffic flow model balances accuracy and computational efficiency, suitable for time scales of 15–90 minutes . It describes traffic flow dynamics by coupling vehicle flows with EV charging behaviors, as expressed in Equations (1)–(3):

\(\begin{cases} c_{\text{EV,ch,min}} \leq c_{\text{ch,s,t}} \leq c_{\text{EV,ch,max}} & \forall s \in \Omega_{\text{TE}}, \forall t \in \Omega_{\text{T}} \\ c_{\text{EV,dis,min}} \leq c_{\text{dis,s,t}} \leq c_{\text{EV,dis,max}} & \forall s \in \Omega_{\text{TE}}, \forall t \in \Omega_{\text{T}} \end{cases}\) where \(c_{\text{ch,s,t}}\) and \(c_{\text{dis,s,t}}\) are charging and discharging prices at station s and time t; \(\Omega_{\text{TE}}\) and \(\Omega_{\text{T}}\) denote charging station and time slot sets, respectively .

2.2 EV Charging Location Regulation

The model adjusts charging prices to guide EVs to less congested stations, modifying the origin-destination (O-D) demand \(q_{t}^{r,s}\) as:\(q_{t}^{r,s} = q_{t}^{r,s,o} + \sum_{\substack{s’ \in \Omega_{\text{TE}} / \{s\} \\ \forall (r,s’) \in \Psi^{m}}} (q_{t}^{r,s’,s} – q_{t}^{r,s,s’})\) where \(q_{t}^{r,s,o}\) is the original O-D demand, and \(q_{t}^{r,s’,s}\) is the demand transferred from station \(s’\) to s . User satisfaction \(\lambda_{t}^{r,s’,s}\) must exceed a threshold \(\lambda_{\text{min}}\) for EVs to switch stations:\(\begin{cases} q_{t}^{r,s’,s} \geq 0 & \text{if } \lambda_{t}^{r,s’,s} \geq \lambda_{\text{min}} \\ q_{t}^{r,s’,s} = 0 & \text{otherwise} \end{cases}\) with \(\lambda_{t}^{r,s’,s}\) dependent on price differences, distance, waiting time, and service quality .

3. EV Regulation Model in FIDNs

3.1 Charging and Discharging Constraints

EVs are categorized into three integration modes: fixed charging power (\(e_1\)), adjustable charging power (\(e_2\)), and bidirectional charging-discharging (\(e_3\)) . The state of charge (SOC) at arrival must satisfy:\(S_{k,t}^{\text{EV,SOC,ini,s}} = S_{k,t}^{\text{EV,SOC,ini,r}} – S^{\text{EV,SOC,1}} \sum_{a \in \Omega_{R}} l_a \delta_{k,a}^{r,s}\)\(S_{k,t}^{\text{EV,SOC,ini,s}} \geq S^{\text{EV,SOC,min}}\) where \(S_{k,t}^{\text{EV,SOC,ini,s}}\) is the SOC at station s, \(S^{\text{EV,SOC,1}}\) is the SOC loss per unit distance, and \(l_a\) is the length of road segment a –.

3.2 Power Flow Constraints

For mode \(e_1\), the charging power at station i is:\(P_{i,t}^{\text{EVC},e_1,\text{ch}} = P_{i,\text{EVC,max}} Y_{i,t}^{e_1}\) where \(Y_{i,t}^{e_1}\) is the number of \(e_1\)-type EVs charging at i, and \(P_{i,\text{EVC,max}}\) is the maximum power per charger . For modes \(e_2\) and \(e_3\), the energy balance equation 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 of m-type EVs at i, and \(E_{i,t}^{\text{arr},m}/E_{i,t}^{\text{dep},m}\) are energies of arriving/departing EVs .

4. FIDN EV Hosting Capacity Calculation Model

4.1 Objective Function

The model maximizes the total number of accommodated EVs across all modes:\(\max \sum_{t \in \Omega_{\text{T}}} \sum_{m \in \{e_1,e_2,e_3\}} \sum_{(r,s) \in \Psi^{m}} q_{t}^{r,s}\) where \(q_{t}^{r,s}\) is the number of EVs traveling from r to s at time t .

4.2 Operational Constraints

Substation power limits:\(P^{\text{S,min}} \leq P_{b,t}^{\text{S}} \leq P^{\text{S,max}}, \quad Q^{\text{S,min}} \leq Q_{b,t}^{\text{S}} \leq Q^{\text{S,max}}\) where \(P_{b,t}^{\text{S}}/Q_{b,t}^{\text{S}}\) are active/reactive powers at substation b –.
Voltage and current limits:\(V_{i,\text{min}}^2 \leq v_{i,t} \leq V_{i,\text{max}}^2, \quad 0 \leq l_{ij,t} \leq I_{ij,\text{max}}^2\) where \(v_{i,t}\) is the squared voltage magnitude at node i, and \(l_{ij,t}\) is the squared current of line ij –.
SOP constraints:\(P_{p,t}^{\text{SOP}} + P_{o,t}^{\text{SOP}} + P_{p,t}^{\text{SOP,L}} + P_{o,t}^{\text{SOP,L}} = 0\)\((P_{p,t}^{\text{SOP}})^2 + (Q_{p,t}^{\text{SOP}})^2 \leq (S_p^{\text{SOP}})^2\) where \(S_p^{\text{SOP}}\) is the SOP capacity at node p .

5. Model Transformation and Solution Algorithm

5.1 Non-Convex Constraint Relaxation

Quadratic convex envelope relaxation: For the fourth-order term \(x_{a,t}^4\) in traffic flow equations, introduce auxiliary variables \(\omega_{a,t}^{x^4}\) and \(\omega_{a,t}^{x^2}\):\(x_{a,t}^2 \leq \omega_{a,t}^{x^2} \leq (x_{a,t,\text{min}} + x_{a,t,\text{max}})x_{a,t} – x_{a,t,\text{min}}x_{a,t,\text{max}}\)\((\omega_{a,t}^{x^2})^2 \leq \omega_{a,t}^{x^4} \leq (x_{a,t,\text{min}}^2 + x_{a,t,\text{max}}^2)\omega_{a,t}^{x^2} – x_{a,t,\text{min}}^2x_{a,t,\text{max}}^2\) –.
Second-order cone relaxation (SOCR): The power flow constraint \(l_{ij,t}v_{i,t} – P_{ij,t}^2 – Q_{ij,t}^2 = 0\) is converted to:\(\left\| \begin{bmatrix} 2P_{ij,t} \\ 2Q_{ij,t} \\ l_{ij,t} – v_{i,t} \end{bmatrix} \right\|_2 \leq l_{ij,t} + v_{i,t}\) .

5.2 Nested Tightening Relaxation Algorithm

The algorithm employs a dual-layer framework:

Inner layer: Sequential bound tightening (SBT) updates variable bounds to reduce quadratic convex relaxation gaps:\(x_{a,t,\text{max}}^{(u)} = \min \left\{ x_{a,t}^{(u-1)} + \alpha(x_{a,t,\text{max}}^{(u-1)} – x_{a,t,\text{min}}^{(u-1)}), x_{a,t,\text{max}}^{(u-1)} \right\}\)\(x_{a,t,\text{min}}^{(u)} = \max \left\{ x_{a,t}^{(u-1)} – \alpha(x_{a,t,\text{max}}^{(u-1)} – x_{a,t,\text{min}}^{(u-1)}), x_{a,t,\text{min}}^{(u-1)} \right\}\) where \(\alpha\) is the tightening factor .
Outer layer: Increasingly tight linear cut (ITLC) generates linear constraints to reduce SOCR gaps:\(l_{ij,t} \leq \frac{(P_{ij,t}^{(n-1)})^2 + (Q_{ij,t}^{(n-1)})^2}{v_{i,t}^{(n-1)}}\) .

6. Case Studies

6.1 Modified 24-Node FIDN Case

System Parameters:

FIDN: 24 nodes, 37.48 MW base load, 20 kV voltage level .
Traffic network: 29 nodes, EV-to-fuel vehicle ratio 1:3 .
SOPs: 6 MVA total capacity across 6 lines .

Results:

EV hosting capacity: 7,095 vehicles .
Operational costs: 购电成本 58.089 万元，网损成本 0.306 万元 (Table 1) .

Cost Type	Cost (10,000 CNY)	Index	Result
Power purchase	58.089	Avg. voltage deviation	0.049 p.u.
Network loss	0.306	Line load rate	31.87%
Total	58.395	Substation load rate	78.49%

Table 1. FIDN operational costs and indices (Case 1) .

6.2 Fujian Actual Case