The evolution of modern power distribution networks is increasingly shaped by the dual integration of large-scale, intermittent distributed renewable energy resources and the burgeoning fleet of electric vehicle cars. This paradigm shift introduces significant operational uncertainties, pushing conventional network infrastructures and control strategies to their limits. AC/DC hybrid distribution networks have emerged as a pivotal topology to enhance power supply reliability and renewable energy accommodation. Within this context, advanced flexible resources like Soft Open Points (SOPs) and Distributed Energy Storage Systems (DESS) are critical for ensuring stable and economical operation. SOPs provide real-time, continuous, and precise power flow control in the spatial dimension, while DESS units offer temporal energy shifting capabilities. Concurrently, the widespread adoption of the electric vehicle car presents not merely a challenging load but a substantial flexible resource that can be orchestrated through sophisticated demand response mechanisms. Therefore, a coordinated planning framework that synergistically leverages the spatial control of SOPs, the temporal flexibility of DESS, and the responsive capacity of electric vehicle car charging loads is essential for the future-proof development of resilient and efficient distribution grids.
Addressing the inherent uncertainties in renewable generation and load, especially from electric vehicle cars, requires moving beyond deterministic or simple stochastic planning approaches. This work proposes a novel two-stage robust optimization model for the joint planning of SOPs and DESS in AC/DC hybrid distribution networks, explicitly incorporating price-based demand response from electric vehicle car users. The model’s core objective is to identify the optimal installation locations and capacities of SOPs and DESS that minimize the total annualized cost while remaining feasible under the worst-case realization of uncertain scenario probabilities. This ensures a robust planning solution against a spectrum of possible future operating conditions.
Modeling Framework and Mathematical Formulation
The proposed planning methodology is built upon a two-stage robust optimization structure. The first-stage decisions, made before the uncertainty is realized, involve the integer choices of where and how much capacity to install for SOPs and DESS. The second-stage operational decisions, such as power dispatch, charging schedules for DESS and electric vehicle cars, and SOP setpoints, are adaptive to the realized scenario. The uncertainty is modeled through a set of plausible daily operation scenarios for renewable output and load, with their probabilities allowed to vary within a prescribed uncertainty set, thus searching for the worst-case probability distribution.
1. Uncertainty Modeling via Scenario Clustering
To capture the variability of distributed photovoltaic (PV), wind turbine (WT), and base load, historical data is processed using the K-means clustering algorithm. This generates a set of $K$ representative daily profiles, $\hat{s}_i$, each with an initial nominal probability $\hat{p}_i$. To account for errors in probability estimation and extreme events, a combined $l_1$-norm and $l_\infty$-norm uncertainty set $\mathcal{P}$ is constructed around the nominal distribution:
$$
\mathcal{P} = \left\{ p \in \mathbb{R}^K \left|
\begin{array}{l}
\|p – \hat{p}\|_1 \leq \varepsilon_1 \\
\|p – \hat{p}\|_\infty \leq \varepsilon_\infty \\
p_i \geq 0, \quad \sum_{i=1}^{K} p_i = 1
\end{array}
\right. \right\}
$$
The budget parameters $\varepsilon_1$ and $\varepsilon_\infty$ control the model’s conservativeness, allowing a tunable trade-off between cost efficiency and robustness.
2. Two-Stage Robust Optimization Model
The overall objective is to minimize the total annual cost, which includes investment, operation, and penalty costs. The compact form of the model is:
$$
\begin{aligned}
& \min_{x} \quad c^T x + \max_{p \in \mathcal{P}} \min_{y \in \Omega(x,p)} \left[ d^T y + w^T \left( \sum_{i=1}^{K} p_i \hat{s}_i \right) \right] \\
& \text{s.t.} \quad A x \leq a, \quad B x = b
\end{aligned}
$$
Here, $x$ denotes first-stage investment variables (SOP/DESS installation binaries and capacities), and $y$ represents second-stage operational variables. The set $\Omega(x,p)$ defines the feasible operating region given investment $x$ and scenario probability $p$.
The detailed annual cost components are:
- Investment Cost ($C_{\text{inv}}$): Annuitized cost of SOP and DESS equipment.
- Operation & Maintenance Cost ($C_{\text{om}}^{\text{SOP}}, C_{\text{om}}^{\text{DESS}}$): Proportional to usage and installed capacity.
- Energy Purchase Cost ($C_{\text{grid}}$): Cost of power bought from the upstream grid at time-varying prices $\rho_t^{\text{buy}}$.
- Load Shedding & Curtailment Penalty ($C_{\text{shed}}, C_{\text{curt}}$): Penalties for involuntary load reduction and renewable energy waste.
- Network Loss Cost ($C_{\text{loss}}$): Cost associated with losses in lines and SOP converters.
- Energy Sales Revenue ($R_{\text{sell}}$): Revenue from selling power to residential users and electric vehicle car charging stations.
The total cost is: $\min \left[ C_{\text{inv}} + C_{\text{om}}^{\text{SOP}} + \max_{p \in \mathcal{P}} \min_{y} ( C_{\text{om}}^{\text{DESS}} + C_{\text{grid}} + C_{\text{shed}} + C_{\text{curt}} + C_{\text{loss}} – R_{\text{sell}} ) \right]$.
3. Electric Vehicle Car Demand Response Model
A critical component is modeling the elasticity of electric vehicle car charging demand in response to real-time pricing (RTP) signals. The demand change is characterized by a self- and cross-elasticity matrix $\epsilon_{tj}$. For a charging station at node $i$, the adjusted power $P_{i,t}^{\text{EV}}$ at time $t$ after demand response is:
$$
P_{i,t}^{\text{EV}} = P_{i,t}^{\text{EV,0}} + \sum_{j=1}^{T} P_{i,t}^{\text{EV,0}} \cdot \epsilon_{tj} \cdot \left( \frac{\rho_j^{\text{RTP}} – \rho_j^{\text{EV,0}}}{\rho_j^{\text{EV,0}}} \right)
$$
where $P_{i,t}^{\text{EV,0}}$ and $\rho_j^{\text{EV,0}}$ are the baseline charging power and price. Constraints ensure user benefits and practical limits:
$$
\begin{aligned}
& (1 – r_{\text{tot}}) \sum_t P_{i,t}^{\text{EV,0}} \leq \sum_t P_{i,t}^{\text{EV}} \leq (1 + r_{\text{tot}}) \sum_t P_{i,t}^{\text{EV,0}} \\
& (1 – r_t) P_{i,t}^{\text{EV,0}} \leq P_{i,t}^{\text{EV}} \leq (1 + r_t) P_{i,t}^{\text{EV,0}} \\
& \sum_t \sum_{i \in \Omega_{\text{EV}}} \rho_t^{\text{RTP}} P_{i,t}^{\text{EV}} \leq \sum_t \sum_{i \in \Omega_{\text{EV}}} \rho_t^{\text{EV,0}} P_{i,t}^{\text{EV,0}} \\
& \rho_t^{\text{RTP,min}} \leq \rho_t^{\text{RTP}} \leq \rho_t^{\text{RTP,max}}
\end{aligned}
$$
The bilinear term $\rho_t^{\text{RTP}} P_{i,t}^{\text{EV}}$ is linearized using the McCormick envelope technique to maintain model tractability.
4. SOP and DESS Operational Constraints
SOP Constraints: A back-to-back voltage source converter-based SOP model is used. For an SOP installed between nodes $i$ and $j$, its operation must satisfy:
$$
\begin{aligned}
& P_{i,t}^{\text{SOP}} + P_{j,t}^{\text{SOP}} + P_{i,t}^{\text{loss}} + P_{j,t}^{\text{loss}} = 0 \\
& \sqrt{(P_{i,t}^{\text{SOP}})^2 + (Q_{i,t}^{\text{SOP}})^2} \leq S_{ij,\text{rated}}^{\text{SOP}} \\
& P_{k,t}^{\text{loss}} = \eta_k^{\text{SOP}} \sqrt{(P_{k,t}^{\text{SOP}})^2 + (Q_{k,t}^{\text{SOP}})^2}, \quad k \in \{i, j\}
\end{aligned}
$$
The non-convex loss equation and capacity constraint are converted into second-order cone (SOC) constraints using relaxation techniques.
DESS Constraints: The storage operation is governed by:
$$
\begin{aligned}
& E_{i,t}^{\text{DESS}} = E_{i,t-1}^{\text{DESS}} + \left( \eta_i^{\text{cha}} P_{i,t}^{\text{cha}} – \frac{P_{i,t}^{\text{dis}}}{\eta_i^{\text{dis}}} \right) \Delta t \\
& 0 \leq P_{i,t}^{\text{cha}} \leq u_{i,t}^{\text{cha}} P_{i,\text{rated}}^{\text{DESS}}, \quad 0 \leq P_{i,t}^{\text{dis}} \leq u_{i,t}^{\text{dis}} P_{i,\text{rated}}^{\text{DESS}} \\
& u_{i,t}^{\text{cha}} + u_{i,t}^{\text{dis}} \leq 1, \quad E_{i,0}^{\text{DESS}} = E_{i,T}^{\text{DESS}} \\
& \eta_i^{\text{min}} E_{i,\text{rated}}^{\text{DESS}} \leq E_{i,t}^{\text{DESS}} \leq \eta_i^{\text{max}} E_{i,\text{rated}}^{\text{DESS}}
\end{aligned}
$$
where $u_{i,t}^{\text{cha/dis}}$ are binary variables indicating charge/discharge states, linearized using big-M methods.
5. AC/DC Hybrid Network Power Flow Constraints
The DistFlow model is employed for both AC and DC subsystems. For an AC branch $(i,j)$:
$$
\begin{aligned}
& \sum_{k} P_{jk,t} = P_{ij,t} – r_{ij} l_{ij,t} + P_{j,t}^{\text{inj}} \\
& v_{j,t} = v_{i,t} – 2(r_{ij}P_{ij,t} + x_{ij}Q_{ij,t}) + (r_{ij}^2 + x_{ij}^2) l_{ij,t} \\
& l_{ij,t} v_{i,t} \geq P_{ij,t}^2 + Q_{ij,t}^2 \quad \text{(Relaxed to SOC constraint)}
\end{aligned}
$$
For a DC branch, the reactive power terms are omitted. $P_{j,t}^{\text{inj}}$ represents the net power injection at node $j$, encompassing generation from PV, WT, discharge from DESS and electric vehicle car stations, and load.
Solution Methodology: The i-C&CG Algorithm
The two-stage robust model with mixed-integer variables and SOC constraints is solved using an enhanced version of the Column-and-Constraint Generation (C&CG) algorithm, known as the inexact C&CG (i-C&CG). It iterates between a Master Problem (MP) and a Subproblem (SP).
- Master Problem: Given a set of identified worst-case scenarios, it optimizes the first-stage investment decisions $x$ and generates a lower bound (LB) for the original problem.
- Subproblem: For a fixed first-stage decision $\bar{x}$, it finds the worst-case probability distribution $p^*$ within $\mathcal{P}$ that maximizes the minimum operational cost, providing an upper bound (UB).
A key challenge in the SP is the bilinear term involving $p$ and dual variables from the inner minimization. To handle this, the continuous probability variables $p_i$ are discretized using binary expansion:
$$
p_i = p_i^{\text{min}} + \alpha \sum_{k=k_{\text{min}}^i}^{k_{\text{max}}^i} 2^k \delta_{i,k}
$$
where $\delta_{i,k}$ are binary variables. This allows the SP to be reformulated as a mixed-integer second-order cone program (MISOCP). The i-C&CG algorithm improves computational efficiency by allowing the MP to be solved with a moderate relative gap in early iterations and employing a backtracking mechanism, significantly reducing solution time for large-scale problems compared to standard C&CG.
Case Study and Analysis
The proposed model is validated on a modified IEEE 69-node AC/DC hybrid test system. Key system parameters are shown below.
| Equipment | Location (Node) | Rated Capacity (kW) |
|---|---|---|
| Wind Turbine | 34, 35, 67 | 1000, 1000, 800 |
| Photovoltaic | 32, 45, 46, 65 | 1200, 1000, 1200, 1000 |
| Electric Vehicle Car Station | 15, 25, 27, 63 | 150, 500, 900, 900 |
Five planning schemes are compared to evaluate the economic and operational benefits:
- Case 1: Base case with no SOP or DESS.
- Case 2: SOP planning only.
- Case 3: DESS planning only (max 6 units).
- Case 4: Joint SOP & DESS planning (max 3 each) with electric vehicle car demand response.
- Case 5: Same as Case 4 but without electric vehicle car demand response.
The annual economic results are summarized in the following table.
| Case | Total Net Revenue (10^4 CNY) | Investment Cost (10^4 CNY) | Grid Purchase Cost (10^4 CNY) | Load Shed Cost (10^4 CNY) | Loss Cost (10^4 CNY) |
|---|---|---|---|---|---|
| 1 | 1583.05 | 0.00 | 409.29 | 329.50 | 63.33 |
| 2 | 2030.52 | 11.41 | 426.59 | 0.00 | 23.98 |
| 3 | 1719.58 | 65.60 | 347.02 | 220.15 | 66.93 |
| 4 | 2040.32 | 63.86 | 355.63 | 0.00 | 26.82 |
| 5 | 2032.96 | 61.81 | 372.29 | 0.00 | 28.45 |
The final installation results for Cases 4 and 5 are:
| Case | DESS (Node, MWh/MW) | SOP (Line, MVA) |
|---|---|---|
| 4 | (2, 2.7/0.5), (46, 2.7/0.5), (64, 2.4/0.5) | (46-19, 0.32), (35-61, 0.64) |
| 5 | (2, 2.7/0.5), (46, 2.7/0.5), (65, 2.1/0.45) | (46-19, 0.32), (35-61, 0.64) |
Key Findings from the Case Study:
- Superiority of Joint Planning: Case 4 (joint planning with DR) achieves the highest net revenue. It combines SOP’s spatial power flow control to eliminate voltage violations and load shedding with DESS’s temporal energy shifting to reduce peak purchases and integrate renewables, demonstrating clear synergy.
- Value of Electric Vehicle Car Demand Response: Comparing Case 4 and Case 5 shows that integrating price-responsive electric vehicle car charging provides significant economic benefit (increase of 73,600 CNY annually). It flattens the net load profile, reduces peak grid purchases, and influences the optimal sizing and siting of DESS.
- Robustness and Solution Quality: The proposed robust model was compared with a stochastic programming model and a traditional “box” robust model. While stochastic programming yielded a slightly higher revenue under typical scenarios, the proposed model exhibited significantly better performance (higher revenue, lower curtailment) under tested extreme scenarios, confirming its balanced robustness. The i-C&CG algorithm provided a substantial computational speed advantage over standard C&CG.
- Model Accuracy: The relaxations (second-order cone, McCormick) introduced for tractability were verified to have negligible errors in the final solution, confirming the model’s practical applicability.
Conclusion
This paper presents a comprehensive two-stage distributionally robust optimization framework for the coordinated planning of Soft Open Points and Distributed Energy Storage Systems in AC/DC hybrid distribution networks. By explicitly modeling the price elasticity of electric vehicle car charging demand and constructing an uncertainty set for scenario probabilities, the method delivers planning decisions that are both economically efficient and robust against a wide range of uncertain future operating conditions. The case study validates that the joint planning of SOP and DESS, coupled with the active management of electric vehicle car flexibility, maximizes network hosting capacity, ensures voltage security, minimizes renewable curtailment, and reduces overall system costs. The solution approach, based on the i-C&CG algorithm and convex relaxations, proves to be computationally efficient for realistic-sized systems. Future work will focus on integrating more dynamic models of electric vehicle car user behavior and investigating adaptive algorithms for candidate SOP location screening.