The rapid development of battery electric vehicles (BEVs) is a cornerstone of the global energy transition towards decarbonization. However, the large-scale integration of battery electric vehicles into existing distribution networks presents significant operational challenges. Traditional alternating current (AC) distribution systems, designed for relatively predictable and balanced loads, face substantial stress from the concentrated, high-power, and often single-phase charging demands of battery electric vehicles. This can lead to frequent violations of thermal limits on lines and transformers, voltage dips, and exacerbated three-phase imbalance, compromising power quality and system security.
Direct current (DC) distribution technology offers a compelling solution. By converting strategic parts of the network to DC operation, capacity can be increased, line losses reduced, and power quality enhanced. This paper investigates the synergistic planning of AC-DC hybrid distribution networks alongside the strategic upgrade of existing slow AC charging points to high-power DC fast charging stations.
The core challenge lies in making optimal, long-term investment decisions—such as which AC lines to convert to DC, where to install Voltage Source Converters (VSCs) and local converters, and when to upgrade charging facilities—under deep uncertainty surrounding the growth of battery electric vehicle adoption and the intermittent output of distributed generation (DG).
To address this, we formulate a multi-stage stochastic programming (MSSP) model for the expansion planning of AC-DC hybrid distribution networks. The model minimizes total expected investment and operational costs while ensuring secure operation under a wide range of future scenarios. A key feature is the explicit modeling of three-phase unbalanced power flow to accurately capture the impact of single-phase battery electric vehicle charging. The stochastic problem is solved efficiently using a Stochastic Dual Dynamic Integer Programming (SDDIP) algorithm. Case studies on modified test systems validate the economic efficiency, operational robustness, and scalability of the proposed planning framework.
1. Deterministic Expansion Planning Model for AC-DC Hybrid Distribution Networks
We consider an existing medium-voltage AC distribution network equipped with DG units and AC slow charging stations for battery electric vehicles. The planning objective is to determine an optimal upgrade strategy over a multi-year horizon. The decision variables include: converting specific AC lines to bipolar DC lines, upgrading AC slow charging stations to DC fast charging stations (FCS), and installing VSCs (at the interface between AC and DC sections) and local converters (at DC nodes to serve three-phase loads).
1.1 Notation and Network Structure
The network consists of a set of nodes $\mathcal{N} = \{0,1,…,N\}$ (node $0$ is the substation) and a set of lines $\mathcal{L} \subseteq \mathcal{N} \times \mathcal{N}$. The planning horizon spans $T$ stages, denoted by $\Omega = \{1,…,T\}$. Let $\Phi = \{a,b,c\}$ represent the three phases.
For node $i \in \mathcal{N}$, its phase-$\varphi$ voltage is $U_{i,t}^{\varphi}$, and its phase-$\varphi$ load is $P_{j,t}^{L,\varphi}+jQ_{j,t}^{L,\varphi}$. For an AC line $(i,j)$, its phase-$\varphi$ power flow is $P_{ij,t}^{\varphi}+jQ_{ij,t}^{\varphi}$ with impedance $r_{ij}^{\varphi}+jx_{ij}^{\varphi}$. For a DC line $(i,j)$, its power flow is $P_{ij,t}$ with resistance $r_{ij}$. The set of nodes with DG is $\mathcal{N}^{DG} \subset \mathcal{N}$, with output $P_{j,t}^{DG,\varphi}+jQ_{j,t}^{DG,\varphi}$. The set of nodes with charging stations is $\mathcal{N}^{CS} \subset \mathcal{N}$.
1.2 Objective Function
The objective is to minimize the total net present cost, comprising investment cost $C^{\text{INV}}$ and operational cost $C^{\text{OPE}}$:
$$ \min \; C^{\text{INV}} + C^{\text{OPE}} $$
Investment Cost: Includes the cost of building DC lines ($C^{\text{LN}}$), fast charging stations ($C^{\text{FCS}}$), VSCs ($C^{\text{VSC}}$), local converters ($C^{\text{LC}}$), and the cost of potentially removing VSCs ($C^{\text{RE}}$).
$$ C^{\text{INV}} = C^{\text{LN}} + C^{\text{FCS}} + C^{\text{VSC}} + C^{\text{LC}} + C^{\text{RE}} $$
$$ C^{\text{LN}} = \sum_{t \in \Omega} \frac{1}{(1+d)^t} \sum_{(i,j) \in \mathcal{L}} \chi^{\text{LN}} \lambda_{ij,t}^{\text{LN}} $$
$$ C^{\text{FCS}} = \sum_{t \in \Omega} \frac{1}{(1+d)^t} \sum_{j \in \mathcal{N}^{CS}} \chi^{\text{FCS}} n_{j}^{\text{ch,DC,max}} \lambda_{j,t}^{\text{FCS}} $$
$$ C^{\text{VSC}} = \sum_{t \in \Omega} \frac{1}{(1+d)^t} \sum_{(i,j) \in \mathcal{L}} \chi^{\text{VSC}} S_{ij,t}^{\text{VSC,rated}} \lambda_{ij,t}^{\text{VSC}} $$
$$ C^{\text{LC}} = \sum_{t \in \Omega} \frac{1}{(1+d)^t} \sum_{j \in \mathcal{N}} \chi^{\text{LC}} S_{j,t}^{\text{LC,rated}} \lambda_{j,t}^{\text{LC}} $$
$$ C^{\text{RE}} = \sum_{t \in \Omega} \sum_{(i,j) \in \mathcal{L}} \omega \psi_{ij,t}^{\text{VSC}} $$
where $\lambda_{(\cdot),t}^{(\cdot)}$ are binary investment decision variables, $\chi^{(\cdot)}$ are unit costs, $d$ is the discount rate, $n_{j}^{\text{ch,DC,max}}$ is the number of new DC chargers, $S^{(\cdot),\text{rated}}$ are rated powers, and $\psi_{ij,t}^{\text{VSC}}$ is the binary removal decision variable for a VSC.
Operational Cost: Includes maintenance cost $C^{\text{MAIN}}$ for operating assets and a penalty cost $C^{\text{COM}}$ for unsatisfied battery electric vehicle charging demand.
$$ C^{\text{OPE}} = C^{\text{MAIN}} + C^{\text{COM}} $$
$$ C^{\text{MAIN}} = \sum_{t \in \Omega} \frac{1}{(1+d)^t} \left[ \sum_{(i,j) \in \mathcal{L}} \gamma^{\text{LN}} \kappa_{ij,t}^{\text{LN}} + \sum_{j \in \mathcal{N}^{CS}} \gamma^{\text{FCS}} \kappa_{j,t}^{\text{FCS}} + \sum_{(i,j) \in \mathcal{L}} \gamma^{\text{VSC}} S_{ij,t}^{\text{VSC,rated}} \kappa_{ij,t}^{\text{VSC}} + \sum_{j \in \mathcal{N}} \gamma^{\text{LC}} S_{j,t}^{\text{LC,rated}} \kappa_{j,t}^{\text{LC}} \right] $$
$$ C^{\text{COM}} = \sum_{t \in \Omega} \frac{1}{(1+d)^t} \sum_{j \in \mathcal{N}^{CS}} \xi^{\text{COM}} \left( P_{j,t}^{\text{ch}} – \alpha_{j,t} \sum_{\varphi \in \Phi} P_{j,t}^{\text{SCS},\varphi} – \beta_{j,t} P_{j,t}^{\text{FCS}} \right)^+ $$
Here, $\kappa_{(\cdot),t}^{(\cdot)}$ are binary operational status variables, $\gamma^{(\cdot)}$ are unit maintenance costs, $\xi^{\text{COM}}$ is the penalty coefficient, $P_{j,t}^{\text{ch}}$ is the aggregated battery electric vehicle charging demand, and $\alpha_{j,t}, \beta_{j,t}$ are binary node status variables (AC/DC).
1.3 Operational and Planning Constraints
The model must enforce physical laws, operational limits, and logical relationships between planning decisions and network topology.
1.3.1 Network Topology Constraints
Binary variables define the network state: $\alpha_{i,t}=1$ if node $i$ is an AC node; $\beta_{i,t}=1$ if node $i$ is a DC-only node; $\delta_{ij,t}=1$ if line $(i,j)$ is an AC line.
$$ \sum_{(i,j) \in \mathcal{L}} \delta_{ij,t} \leq \alpha_{i,t} \leq \sum_{(i,j) \in \mathcal{L}} \delta_{ij,t} + \sum_{(j,i) \in \mathcal{L}} \delta_{ji,t}, \quad \forall i,t $$
$$ \alpha_{i,t} + \beta_{i,t} = 1, \quad \forall i,t $$
Coupling with operational variables:
$$ \kappa_{ij,t}^{\text{LN}} = 1 – \delta_{ij,t}, \quad \forall (i,j),t $$
$$ \beta_{j,t} \leq \kappa_{j,t}^{\text{FCS}} \leq (1+\beta_{j,t})/2, \quad \forall j,t $$
$$ \kappa_{ij,t}^{\text{VSC}} = \alpha_{i,t} (1 – \delta_{ij,t}), \quad \forall (i,j),t $$
$$ \beta_{j,t} \leq \kappa_{j,t}^{\text{LC}} \leq (1+\beta_{j,t})/2, \quad \forall j,t $$
1.3.2 Investment and Operational Logic Constraints
An asset can be invested in only once, and it can only operate if it has been invested in by that stage.
$$ \sum_{t \in \Omega} \lambda_{ij,t}^{\text{LN}} \leq 1, \; \sum_{t \in \Omega} \lambda_{ij,t}^{\text{VSC}} \leq 1, \; \sum_{t \in \Omega} \lambda_{j,t}^{\text{FCS}} \leq 1, \; \sum_{t \in \Omega} \lambda_{j,t}^{\text{LC}} \leq 1 $$
$$ \kappa_{ij,t}^{\text{LN}} \leq \sum_{w=1}^{t} \lambda_{ij,w}^{\text{LN}}, \; \kappa_{ij,t}^{\text{VSC}} \leq \sum_{w=1}^{t} \lambda_{ij,w}^{\text{VSC}}, \; \kappa_{j,t}^{\text{FCS}} \leq \sum_{w=1}^{t} \lambda_{j,w}^{\text{FCS}}, \; \kappa_{j,t}^{\text{LC}} \leq \sum_{w=1}^{t} \lambda_{j,w}^{\text{LC}} $$
1.3.3 Battery Electric Vehicle Charging Station Constraints
The charging power provided depends on the number of active chargers.
$$ P_{j,t}^{\text{FCS}} = n_{j,t}^{\text{ch}} \eta^{\text{ch,DC}} P^{\text{ch,DC}}, \quad \forall j \in \mathcal{N}^{CS}, t $$
$$ P_{j,t}^{\text{SCS},\varphi} = n_{j,t}^{\text{ch}} \eta^{\text{ch,AC}} P^{\text{ch,AC}}, \quad \forall j \in \mathcal{N}^{CS}, \varphi, t $$
$$ 0 \leq n_{j,t}^{\text{ch}} \leq n_{j}^{\text{ch,AC,max}} \quad \text{or} \quad 0 \leq n_{j,t}^{\text{ch}} \leq n_{j}^{\text{ch,DC,max}} $$
The aggregated battery electric vehicle charging demand $P_{j,t}^{\text{ch}}$ at node $j$ is the sum of individual charging profiles, modeled based on stochastic arrival/departure times and state-of-charge.
1.3.4 Three-Phase Decoupled Linearized AC Power Flow
To model imbalance from single-phase battery electric vehicle charging and unbalanced loads, a three-phase linearized DistFlow model is used, relaxed with big-M constraints for planning.
$$ -M(1-\delta_{ij,t}) \leq U_{j,t}^{\varphi} – U_{i,t}^{\varphi} – \frac{r_{ij}^{\varphi} P_{ij,t}^{\varphi} + x_{ij}^{\varphi} Q_{ij,t}^{\varphi}}{U_0} \leq M(1-\delta_{ij,t}), \quad \forall (i,j),\varphi,t $$
$$ \sum_{i \in \delta(j)} P_{ij,t}^{\varphi} – \sum_{k \in \phi(j)} P_{jk,t}^{\varphi} = \alpha_{j,t} \left( P_{j,t}^{L,\varphi} + P_{j,t}^{\text{SCS},\varphi} – P_{j,t}^{DG,\varphi} \right), \quad \forall j,\varphi,t $$
$$ U_{j}^{\text{AC,min}} \alpha_{j,t} \leq U_{j,t}^{\varphi} \leq U_{j}^{\text{AC,max}} \alpha_{j,t}, \quad \forall j,\varphi,t $$
$$ -P_{ij}^{\text{AC,max}} \delta_{ij,t} \leq P_{ij,t}^{\varphi} \leq P_{ij}^{\text{AC,max}} \delta_{ij,t}, \quad \forall (i,j),\varphi,t $$
Similar constraints apply for reactive power $Q_{ij,t}^{\varphi}$.
1.3.5 DC Linear Power Flow
A linear DC power flow model is used for the DC sections.
$$ -M \delta_{ij,t} \leq U_{j,t} – U_{i,t} – \frac{r_{ij} P_{ij,t}}{U_0} \leq M \delta_{ij,t}, \quad \forall (i,j),t $$
$$ \sum_{i \in \delta(j)} P_{ij,t} – \sum_{k \in \phi(j)} P_{jk,t} = \beta_{j,t} \left( P_{j,t}^{\text{FCS}} \kappa_{j,t}^{\text{FCS}} + P_{j,t}^{L} \kappa_{j,t}^{\text{LC}} \right), \quad \forall j,t $$
$$ U_{j}^{\text{DC,min}} \beta_{j,t} \leq U_{j,t} \leq U_{j}^{\text{DC,max}} \beta_{j,t}, \quad \forall j,t $$
$$ -P_{ij}^{\text{DC,max}} (1-\delta_{ij,t}) \leq P_{ij,t} \leq P_{ij}^{\text{DC,max}} (1-\delta_{ij,t}), \quad \forall (i,j),t $$
1.3.6 VSC and Local Converter Models
VSCs connect AC and DC subsystems. Their operation couples the AC and DC power flows with conversion losses.
$$ \sum_{\varphi \in \Phi} \Re(S_{ij,t}^{\text{VSC},\varphi}) + P_{ij,t}^{\text{VSC,loss}} = P_{ij,t}^{\text{VSC}}, \quad \forall (i,j),t $$
$$ P_{ij,t}^{\text{VSC,loss}} = a_0 + a_1 \sum_{\varphi \in \Phi} |S_{ij,t}^{\text{VSC},\varphi}|, \quad \forall (i,j),t $$
$$ -M(1-\kappa_{ij,t}^{\text{VSC}}) \leq P_{ij,t}^{\text{VSC},\varphi} – \Re(S_{ij,t}^{\text{VSC},\varphi}) \leq M(1-\kappa_{ij,t}^{\text{VSC}}), \quad \forall (i,j),\varphi,t $$
Similar constraints model the local converter, which serves three-phase loads at DC nodes: $S_{j,t}^{\text{LC},\varphi} = P_{j,t}^{DG,\varphi}+jQ_{j,t}^{DG,\varphi} – (P_{j,t}^{L,\varphi}+jQ_{j,t}^{L,\varphi})$.
1.3.7 DG Operational Constraints
$$ |Q_{j,t}^{DG,\varphi}| \leq P_{j,t}^{DG,\varphi} \tan(\theta_j^{DG}), \quad \forall j \in \mathcal{N}^{DG}, \varphi, t $$
$$ (P_{j,t}^{DG,\varphi})^2 + (Q_{j,t}^{DG,\varphi})^2 \leq (S_{j}^{DG,\text{rated}})^2, \quad \forall j \in \mathcal{N}^{DG}, \varphi, t $$
The complete deterministic model is a Mixed-Integer Linear Programming (MILP) problem.
2. Multi-Stage Stochastic Expansion Planning Model
The deterministic model assumes perfect foresight. In reality, key drivers like battery electric vehicle charging demand and DG output are uncertain. A multi-stage stochastic framework is essential to make adaptive decisions that are robust across a range of plausible futures.
2.1 Modeling Uncertainties
Battery Electric Vehicle Uncertainty: The charging demand of a battery electric vehicle is derived from its stochastic driving pattern. Arrival time $h^{\text{arr}}$, departure time $h^{\text{dep}}$, and initial state-of-charge (SOC) $\varsigma^{\text{arr}}$ are modeled as random variables.
Arrival/departure times are modeled by inhomogeneous Poisson processes with time-varying rate parameters $\theta_k(h)$. The initial SOC is modeled as a normal random variable: $\varsigma^{\text{arr}} \sim \mathcal{N}(\mu^{\text{SOC}}, (\sigma^{\text{SOC}})^2)$. Monte Carlo simulation is used to generate scenarios of aggregated charging profiles $P_{j,t}^{\text{ch}}$.
Distributed Generation Uncertainty: The active power output of DG (e.g., solar PV) is modeled as a normal random variable for each phase:
$$ P_{j,t}^{DG,\varphi} \sim \mathcal{N}(\mu_{j,t}^{DG,\varphi}, (\sigma_{j,t}^{DG,\varphi})^2) $$
Scenarios are generated via Monte Carlo simulation and clustered into representative profiles (e.g., high, medium, low).
2.2 Multi-Stage Stochastic Programming Formulation
The planning decisions are made sequentially as uncertainty is revealed over time. Let $\boldsymbol{\xi}_t$ represent the vector of random parameters (DG output, battery electric vehicle demand) in stage $t$. A scenario tree represents the evolution of uncertainty. The MSSP problem has a nested formulation:
$$ \min_{\mathbf{x}_1 \in \mathcal{X}_1} \left( \mathbf{c}_1^T \mathbf{x}_1 + \mathbb{E}_{\boldsymbol{\xi}_1} \left[ \min_{\mathbf{x}_2 \in \mathcal{X}_2(\mathbf{x}_1, \boldsymbol{\xi}_1)} \left( \mathbf{c}_2^T \mathbf{x}_2 + \mathbb{E}_{\boldsymbol{\xi}_2} \left[ … + \mathbb{E}_{\boldsymbol{\xi}_{T-1}} \left[ \min_{\mathbf{x}_T \in \mathcal{X}_T(\mathbf{x}_{T-1}, \boldsymbol{\xi}_{T-1})} \mathbf{c}_T^T \mathbf{x}_T \right] \right] \right) \right] \right) $$
where $\mathbf{x}_t$ are the stage-$t$ decision variables (both integer and continuous), and $\mathcal{X}_t$ is the feasible set defined by constraints coupling stages $t-1$ and $t$ and those specific to stage $t$ under realization $\boldsymbol{\xi}_{t-1}$.
To adapt the deterministic constraints for MSSP, we introduce cumulative investment variables $\nu_{ij,t}^{\text{LN}} = \sum_{w=1}^{t} \lambda_{ij,w}^{\text{LN}}$, etc. This allows the coupling constraints to be written recursively:
$$ \nu_{ij,t}^{\text{LN}} = \nu_{ij,t-1}^{\text{LN}} + \lambda_{ij,t}^{\text{LN}}, \quad \kappa_{ij,t}^{\text{LN}} \leq \nu_{ij,t}^{\text{LN}}, \quad \nu_{ij,T}^{\text{LN}} \leq 1 $$
This ensures the “non-anticipativity” property: decisions at stage $t$ can only depend on information available up to $t$.
2.3 Solution via Stochastic Dual Dynamic Integer Programming (SDDIP)
Solving the large-scale MSSP MILP directly is intractable. The SDDIP algorithm decomposes the problem by scenarios and iteratively approximates the “cost-to-go” function (the expected future cost of current decisions) using Benders-type cuts. The main steps are:
1. Sampling: Generate $S$ sample paths (scenarios) from the scenario tree.
2. Forward Pass: For each scenario $s$ and stage $t$, solve a subproblem where the future cost function $Q_{t+1}(\mathbf{x}_t)$ is approximated by a piecewise linear function $\overline{Q}_{t+1}^m(\mathbf{x}_t)$ constructed from cuts generated in previous iterations $l < m$.
3. Backward Pass: Starting from the last stage $T$, compute new tight cuts for the approximate cost function $\overline{Q}_t^m$ at each node of the scenario tree. These cuts are added to the subproblems in the next iteration to improve the approximation.
4. Termination: The algorithm stops when the lower bound (from the first-stage problem) stabilizes within a specified tolerance $\epsilon$.
SDDIP efficiently handles the combinatorial complexity of integer decisions under uncertainty, making the multi-stage planning of AC-DC networks with battery electric vehicle infrastructure computationally feasible.
3. Case Studies and Analysis
The proposed framework is tested on a modified IEEE 33-node distribution system and a larger IEEE 141-node system to demonstrate its effectiveness, economic benefit, and scalability.
3.1 Test System and Parameters
The modified IEEE 33-node system operates at 12.66 kV (AC) and ±10 kV (DC). It has DG at 4 nodes and 3 existing AC slow charging stations. The planning horizon is 15 years, divided into 3 stages (5 years each). Key parameters are summarized below.
| Parameter | Value |
|---|---|
| Line Investment/Maintenance Cost | 200,000 / 20,000 ¥/km |
| DC Charger (with converter) Investment/Maintenance Cost | 80,000 / 4,000 ¥/unit |
| VSC Investment/Maintenance Cost | 1,000 / 50 ¥/kVA |
| Local Converter Investment/Maintenance Cost | 1,000 / 50 ¥/kVA |
| Unsatisfied BEV Demand Penalty ($\xi^{\text{COM}}$) | 1.2 ¥/kWh |
| Discount Rate ($d$) | 0.03 |
| AC/DC Charging Power ($P^{\text{ch,AC}}$ / $P^{\text{ch,DC}}$) | 6.6 kW / 44 kW |
Uncertainty scenarios for DG output and battery electric vehicle demand are generated and clustered into “High”, “Medium”, and “Low” profiles for each stage, forming a 9-leaf scenario tree.
3.2 Analysis of Multi-Stage vs. Two-Stage Planning
We compare the proposed MSSP approach with a Two-Stage Stochastic Programming (TSSP) approach. In TSSP, all investment decisions for the entire horizon must be made in the first stage, before any uncertainty is resolved in the second “operational” stage. MSSP allows decisions to be adapted at each stage based on the realized uncertainty path.
Key Findings:
- Flexibility: MSSP produces a decision tree of expansion plans. For example, in Stage 1, both methods upgrade two critical charging stations to DC fast charging due to high expected penalty costs. In Stage 2, the specific lines to convert to DC depend on the realized load/DG scenario in MSSP, whereas TSSP commits to a single fixed plan.
- Economy: The adaptive nature of MSSP leads to lower total expected costs. The table below compares costs for two different scenario tree realizations (Case 1 & 2). MSSP consistently outperforms TSSP in stages 2 and 3 by avoiding over-investment or under-investment.
| Method | Stage | Investment Cost (k¥) | Operation & Penalty Cost (k¥) | Total Stage Cost (k¥) |
|---|---|---|---|---|
| MSSP (Case 1) | 1 | 188.27 | 184.02 | 372.29 |
| 2 | 359.97 | 59.77 | 419.74 | |
| 3 | 298.92 | 103.20 | 402.12 | |
| Total | – | 847.16 | 346.99 | 1,194.15 |
| TSSP | 1 | 188.27 | 184.02 | 372.29 |
| 2 | 467.13 | 83.64 | 550.77 | |
| 3 | 373.98 | 136.67 | 510.65 | |
| Total | – | 1,029.38 | 404.33 | 1,433.71 |
The upgrade of charging stations effectively mitigates the penalty cost from unsatisfied battery electric vehicle demand as penetration grows.
3.3 Operational Benefits: Voltage Profile and Three-Phase Imbalance
We compare the operational outcome of our AC-DC hybrid plan against a traditional plan that only expands AC charging stations within the original AC network.
Voltage Profile: Under balanced load conditions, the AC-DC hybrid network maintains all nodal voltages within the safe limits [0.95, 1.05] p.u. throughout all stages. In contrast, the traditional AC expansion leads to severe voltage drops below 0.95 p.u. in later stages due to increased loading from battery electric vehicles and general load growth.
Three-Phase Voltage Unbalance Factor (VUF): We define the VUF for phase $\varphi$ at node $i$ as:
$$ \gamma_{i,t}^{\varphi} = \frac{|U_{i,t}^{\varphi} – U_{i,t}^{\text{avr}}|}{U_{i,t}^{\text{avr}}} \times 100\% $$
where $U_{i,t}^{\text{avr}}$ is the average voltage. The impact is analyzed under different levels of load unbalance.
| Method | Load Unbalance | Avg. VUF – Stage 1 | Avg. VUF – Stage 2 | Avg. VUF – Stage 3 |
|---|---|---|---|---|
| Proposed AC-DC Hybrid | 0% (Balanced) | 0.15% | 0.00% | 0.00% |
| 10% | 0.18% | 0.05% | 0.05% | |
| 20% | 0.25% | 0.12% | 0.11% | |
| Traditional AC Expansion | 0% (Balanced) | 0.15% | 0.85% | 1.20% |
| 10% | 0.45% | 1.95% | 2.80% | |
| 20% | 0.80% | 3.10% | 4.25% |
The results are striking. In the proposed method, once charging stations are upgraded to DC (Stages 2 & 3), the single-phase charging distortion is eliminated, reducing VUF to near zero under balanced loads. Furthermore, VSCs help mitigate imbalance from unbalanced loads, keeping VUF well below the typical 2% limit. The traditional method fails to control the VUF, which escalates dangerously with both battery electric vehicle charging and load unbalance.
3.4 Scalability Analysis on a Larger System
The methodology is applied to a modified IEEE 141-node system with 10 charging stations and 24 DG nodes. The table below shows the total cost and computation time for different planning horizons and numbers of scenarios, confirming the scalability of the SDDIP-based approach.
| Number of Stages | Number of Scenarios | Total Cost (k¥) | Computation Time (seconds) | Convergence Error |
|---|---|---|---|---|
| 3 | 8 | 8,165.92 | 1,981.78 | 0.18% |
| 4 | 16 | 10,849.08 | 4,194.94 | 0.19% |
| 5 | 32 | 11,919.83 | 7,613.28 | 0.18% |
4. Conclusion
This paper presents a comprehensive multi-stage stochastic expansion planning framework for transforming traditional AC distribution networks into AC-DC hybrid systems to accommodate the rising demand from battery electric vehicles. The model co-optimizes investments in DC line conversion, fast charging station upgrades, and power electronic converters, while explicitly accounting for three-phase unbalanced operation. By incorporating uncertainties in battery electric vehicle adoption and renewable generation through a multi-stage stochastic programming approach solved with SDDIP, the method yields flexible, adaptive, and economically superior plans compared to static or two-stage alternatives.
Case studies demonstrate that the proposed planning strategy not only ensures economic efficiency by dynamically adapting to revealed uncertainties but also guarantees secure operation by maintaining voltage profiles and significantly suppressing three-phase imbalance—a critical issue exacerbated by single-phase battery electric vehicle charging. The successful application to a large 141-node system confirms the computational scalability of the framework. This work provides system planners with a robust tool for designing future-proof, power-quality-aware, and cost-effective distribution infrastructure for the era of electrified transportation.