As the adoption of electric vehicles (EVs) continues to rise, the need to accurately assess the hosting capacity of distribution networks becomes critical. This paper presents a comprehensive approach to calculating the EV hosting capacity in flexible interconnected distribution networks (FIDN) while considering the impact of traffic flow. The method integrates semi-dynamic traffic flow models, EV regulation strategies, and advanced optimization techniques to ensure optimal network operation and maximum EV integration.
1. Introduction
The rapid growth of EVs, with China’s reaching 15.52 million by the end of 2023, has posed significant challenges to distribution network stability . The hosting capacity, defined as the maximum number of EVs a distribution network can accommodate without violating operational constraints, is influenced by spatio-temporal traffic flow variations, which alter charging demand patterns .
Existing studies often overlook the coupling between power and transportation networks, leading to inaccurate capacity assessments. For instance, assume no traffic flow impact, while most research focuses on radial distribution networks, limiting scalability . This study addresses these gaps by integrating soft open point (SOP) flexibility and traffic flow dynamics to enhance FIDN adaptability to EVs.
2. EV Regulation Model Considering Traffic Flow
2.1 Semi-Dynamic Traffic Flow Model
Semi-dynamic traffic flow models strike a balance between accuracy and computational efficiency, suitable for hourly EV hosting capacity calculations . The model describes using the Bureau of Public Roads function:\(t_{a,t} = t_a^0 \left[1 + 0.15 \left(\frac{x_{a,t}}{c_a}\right)^4\right] \quad \forall a \in \Omega_R, t \in \Omega_T \tag{1}\) where \(t_{a,t}\) is the travel time on link a at time t, \(t_a^0\) is the free-flow travel time, \(x_{a,t}\) is the traffic flow, and \(c_a\) is the link capacity .
2.2 EV Regulation Framework
The EV regulation model considers three integration modes: fixed charging power (\(e_1\)), adjustable charging power (\(e_2\)), and bidirectional charging-discharging power (\(e_3\)) . Key components include:
- Charging Location Regulation: Prices guide EVs to optimal charging stations, modifying origin-destination (O-D) demands:\(q_t^{r,s} = q_t^{r,s,0} + \sum_{s’ \in \Omega_{TE}/\{s\}} \left(q_t^{r,s’,s} – q_t^{r,s,s’}\right) \tag{2}\) where \(q_t^{r,s}\) is the adjusted demand from origin r to destination s, and \(q_t^{r,s’,s}\) is the demand transferred from \(s’\) to s .
- Satisfaction Constraint: EVs only adjust location if satisfaction \(\lambda_t^{r,s’,s} \geq \lambda_{\text{min}}\), where:\(\lambda_t^{r,s’,s} = \begin{cases} \omega_{11}\frac{c_{ch,s’,t}-c_{ch,s,t}}{c_{ch,t,\text{max}}’} + \omega_{12}\frac{d_{m,\text{max}}-d_{s’,s}}{d_{m,\text{max}}} + \dots & d_{s’,s} \leq d_{m,\text{max}} \\ 0 & d_{s’,s} > d_{m,\text{max}} \end{cases} \tag{3}\)\(\omega_{ij}\) are weight coefficients, \(c_{ch,s,t}\) is the charging price, and \(d_{s’,s}\) is the distance between stations .
- Charging/Discharging Power Constraints: For mode \(e_3\), power limits are:\(0 \leq P_{i,t}^{\text{EVC},e_3,\text{ch}} \leq \gamma_{i,t}^{\text{EVC},e_3} Y_{i,t}^{e_3} P_{i,\text{EVC,max}} \tag{4}\)\(0 \leq P_{i,t}^{\text{EVC},e_3,\text{dis}} \leq (1 – \gamma_{i,t}^{\text{EVC},e_3}) Y_{i,t}^{e_3} P_{i,\text{EVC,max}} \tag{5}\) where \(\gamma_{i,t}^{\text{EVC},e_3}\) is a binary variable for charging/discharging state, and \(Y_{i,t}^{e_3}\) is the number of EVs at station i .
3. FIDN EV Hosting Capacity Calculation Model
3.1 Objective Function
The model maximizes the total number of accommodated EVs:\(\max \sum_{t \in \Omega_T} \sum_{m \in \{e_1,e_2,e_3\}} \sum_{(r,s) \in \Psi^m} q_t^{r,s} \tag{6}\) subject to power network and traffic flow constraints .
3.2 Key Constraints
- Power Flow Constraints:\(v_{i,t} – v_{j,t} = 2(R_{ij}P_{ij,t} + X_{ij}Q_{ij,t}) – (R_{ij}^2 + X_{ij}^2)l_{ij,t} \tag{7}\) where \(v_{i,t}\) is the voltage squared at node i, \(P_{ij,t}/Q_{ij,t}\) are active/reactive powers, and \(l_{ij,t}\) is the line current squared .
- SOP Operation Constraints:\(P_{p,t}^{\text{SOP,p}} + P_{p,t}^{\text{SOP,o}} + P_{p,t}^{\text{SOP,L,p}} + P_{p,t}^{\text{SOP,L,o}} = 0 \tag{8}\)\((P_{p,t}^{\text{SOP}})^2 + (Q_{p,t}^{\text{SOP}})^2 \leq (S_p^{\text{SOP}})^2 \tag{9}\) where \(P_{p,t}^{\text{SOP}}\)/\(Q_{p,t}^{\text{SOP}}\) are SOP active/reactive powers, and \(S_p^{\text{SOP}}\) is the SOP capacity .
- EV State of Charge (SOC) Constraints:\(S_{k,t}^{\text{EV,SOC,min},s} = S_{k,t}^{\text{EV,SOC,ini},r} – S^{\text{EV,SOC,1}} \sum_{a \in \Omega_a} l_a \delta_{k,a}^{r,s} \tag{10}\) where \(S_{k,t}^{\text{EV,SOC}}\) is the SOC of EV k, \(S^{\text{EV,SOC,1}}\) is the SOC loss per km, and \(l_a\) is the link length .
4. Model Transformation and Solution Algorithm
4.1 Non-Convex Constraint Relaxation
- Big-M Method: Converts logical constraints like Eq. (3) into linear form:\(\begin{cases} \lambda_{\text{min}} – M(1-\gamma_{r,s,s’,t}) \leq \lambda_{r,s,s’,t} \leq \lambda_{\text{min}} + M\gamma_{r,s,s’,t} – \varepsilon \\ -M(1-\gamma_{r,s,s’,t}) \leq q_t^{r,s,s’} \leq M\gamma_{r,s,s’,t} \end{cases} \tag{11}\) with M as a large constant and \(\gamma_{r,s,s’,t}\) as a binary variable .
- Quadratic Convex Envelope Relaxation: Approximates quartic terms in traffic flow models:\((x_{a,t}^2)^2 \leq \omega_{a,t}^{x^4} \leq (x_{a,t,\text{min}}^2 + x_{a,t,\text{max}}^2)\omega_{a,t}^{x^2} – x_{a,t,\text{min}}^2x_{a,t,\text{max}}^2 \tag{12}\) where \(\omega_{a,t}^{x^2}/\omega_{a,t}^{x^4}\) are auxiliary variables .
- Second-Order Cone Relaxation (SOCR): Transforms power flow constraints into SOCR form:\(\left\| \begin{bmatrix} 2P_{ij,t} \\ 2Q_{ij,t} \\ l_{ij,t} – v_{i,t} \end{bmatrix} \right\|_2 \leq l_{ij,t} + v_{i,t} \tag{13}\) ensuring convexity for efficient solving .
4.2 Nested Tightening Relaxation Algorithm
The algorithm employs a double-layer framework:
- Inner Layer (Sequential Bound Tightening, SBT): Tightens variable bounds to reduce relaxation gaps for quadratic terms:\(x_{a,t,\text{max}}^{(u)} = \min \left\{x_{a,t}^{(u-1)} + \alpha(x_{a,t,\text{max}}^{(u-1)} – x_{a,t,\text{min}}^{(u-1)}), x_{a,t,\text{max}}^{(u-1)}\right\} \tag{14}\) with \(\alpha\) as the tightening factor and u as the iteration index .
- Outer Layer (Increasingly Tight Linear Cut, ITLC): Adds linear cuts to reduce SOCR gaps:\(l_{ij,t} \leq \frac{(P_{ij,t}^{(n-1)})^2 + (Q_{ij,t}^{(n-1)})^2}{v_{i,t}^{(n-1)}} \tag{15}\) where n is the ITLC iteration index .
5. Case Studies
5.1 Modified 24-Node FIDN and 29-Node Transportation Network
Settings:
- FIDN load: 37.48 MW, SOP capacity: 6 MV·A
- EV parameters: Rated capacity 50.1 kWh, SOC range 0.1–1.0
- Traffic flow constraints: \(T_{\text{TTI,max}} = 1.4\) .
Results:
- EV hosting capacity: 7,095 vehicles
- System costs: 58.089 万元， 0.306 万元 .
| Cost Type | Cost (10³ CNY) | Key Indicator | Result |
|---|---|---|---|
| Power purchase | 58.089 | Average voltage dev. | 0.049 p.u. |
| Network loss | 0.306 | Line load rate | 31.87% |
| Total | 58.395 | Substation load rate | 78.49% |
Table 1. FIDN Operation Costs and Indices (Case 1) .
5.2 Impact of Traffic Flow Constraints
- Without traffic flow consideration: Capacity = 7,280 vehicles (2.61% higher than with traffic flow) .
- Varying \(T_{\text{TTI,max}}\): As \(T_{\text{TTI,max}}\) decreases from 1.5 to 1.1, capacity drops due to stricter traffic smoothness requirements (Table 2).
| \(T_{\text{TTI,max}}\) | EV Hosting Capacity | Total Cost (10³ CNY) | Line Load Rate (%) |
|---|---|---|---|
| 1.1 | 6,315 | 535.88 | 30.47 |
| 1.4 | 7,095 | 583.95 | 31.87 |
| 1.6 | 7,115 | 585.69 | 31.89 |
Table 2. Effect of \(T_{\text{TTI,max}}\) on Hosting Capacity .
5.3 SOP Capacity Impact
- Increasing SOP capacity from 0 to 6 MV·A boosts capacity by 15.65% (Table 3).
- SOP enables power redistribution, reducing substation overload risks during EV charging peaks .
| SOP Capacity (MV·A) | EV Hosting Capacity | Substation Load Rate (%) |
|---|---|---|
| 0 | 6,135 | 75.21 |
| 6 | 7,095 | 78.49 |
| 10 | 7,185 | 79.48 |
Table 3. Impact of SOP Capacity on Hosting Capacity .
6. Conclusion
This study presents a novel method to calculate EV hosting capacity in FIDN considering traffic flow. Key findings include:
- Traffic flow significantly influences EV charging demand distribution, with unaccounted traffic flow leading to 2.61–11.46% overestimated capacity .
- SOPs enhance hosting capacity by 15.65% through inter-feeder power transfer and voltage support .
- The nested tightening relaxation algorithm reduces relaxation gaps to <0.4%, ensuring accurate solutions .
Future work will focus on short-time scale traffic-electrical interactions and uncertainty management in EV hosting capacity calculations.