Three-Phase Unbalanced Distribution Network Restoration Considering EV Charging Station Phase Scheduling

Abstract

This paper investigates post-disaster restoration and stable operation of three-phase unbalanced distribution networks with high penetration of distributed generation and electric-vehicle integration. A two-stage hierarchical restoration and coordinated dispatch strategy is developed to improve restoration feasibility and reliability while reducing post-fault restoration cost. In Stage I, island partitioning and load prioritization are carried out using DFS and BFS to maximize the restoration of critical loads. In Stage II, without reducing the Stage I restoration level, branch switch statuses and coordinated outputs of DG-EV-BESS are jointly optimized under three-phase unbalanced power flow, radiality, and other operational constraints. The objective minimizes the total restoration cost, including network losses, switching operations, three-phase voltage-unbalance penalties, and customer compensation. Simulation results on a modified IEEE 33-bus system demonstrate that the proposed method reduces restoration cost and improves the system restoration ratio. In addition, phase-wise EV dispatch provides targeted compensation for weak-phase power deficits and improves post-fault inter-phase operating quality. The results validate the effectiveness of the proposed strategy.

1. Introduction

Against the background of the dual-carbon (carbon peaking and carbon neutrality) targets and the development of new-type power systems, distribution networks are undergoing a profound transition from passive, unidirectional operation to active, bidirectional interaction. Distributed generation (DG), such as photovoltaic (PV) and wind power, together with battery energy storage systems (BESS) and electric vehicles (EVs), is rapidly penetrating distribution networks [1]. While this transition increases flexibility, it also introduces new challenges, including power-flow reversal, aggravated three-phase unbalance, and tighter operational constraints. In particular, the temporal and spatial aggregation of EV charging loads may further deteriorate power quality and exacerbate phase unbalance, making traditional protection and fault-recovery schemes—based on conventional assumptions—increasingly unsuitable for practical operation [2,3].

Existing studies on distribution-network self-healing and fault restoration have mainly focused on reconfiguration strategies under renewable uncertainty and load variability. For example, Ref. [4] proposes a two-stage optimal scheduling method to enhance the self-healing capability of distribution networks by minimizing load reduction under emergency conditions, Dan et al. [5] developed a graph deep reinforcement learning-based restoration method for active distribution networks, Wang et al. [6] proposed a restoration model considering forecast uncertainty and ref. [7] considers the impact of distribution generation and load uncertainties on the reliability of active distribution networks. These studies improved restoration performance under uncertain operating conditions; however, they did not explicitly consider the support capability of EV charging stations with vehicle-to-grid (V2G) functionality during post-fault recovery.

To further enhance restoration flexibility, recent studies have incorporated EVs into distribution-network restoration. Zheng et al. [8] studied V2G charging-station siting and resilient network reconfiguration under disasters. Du et al. [9] analyzed the impact of EVs on post-disaster power-supply restoration in urban distribution systems. Su et al. [10] investigated critical-load restoration in coupled power-distribution and traffic networks considering spatio-temporal EV scheduling. These studies demonstrate that EVs can provide valuable support during restoration. However, most of them model EVs either as aggregated flexible loads or storage resources and do not explicitly address the three-phase unbalance caused by faults and EV charging/discharging behavior. Although ref. [11] considered joint EV scheduling and network reconfiguration for fault recovery, the interaction between EV participation and three-phase unbalanced network operation was not fully captured.

Meanwhile, three-phase optimization models have been increasingly adopted to improve the physical accuracy of active distribution-network analysis. Wang et al. [12] developed a parametric linear relaxation model for operation optimization under three-phase unbalanced conditions, while Chowdhury et al. [13] proposed an SOCP-based three-phase optimal power-flow model for active distribution networks. In addition, Li et al. [14] explored the use of EV charging stations for self-healing and optimal operation in unbalanced distribution networks. Nevertheless, many existing restoration studies still rely on single-phase equivalent models or simplified balanced assumptions. As a result, they cannot accurately represent post-fault phase imbalance, phase-specific support requirements, or the coordinated effects of wind/PV/storage/EV dispatch in unbalanced networks [15,16,17,18,19].

Therefore, a remaining research gap lies in developing a restoration framework that simultaneously considers three-phase unbalanced operation after faults, coordinated network reconfiguration, source–load–storage scheduling, and phase-specific EV charging/discharging flexibility. To address this issue, this paper proposes a hierarchical post-fault restoration and coordinated scheduling method for three-phase unbalanced distribution networks. In the first stage, the objective is to restore as many critical loads as possible. In the second stage, the objective is to minimize the overall recovery cost. The proposed model integrates island partitioning, network reconfiguration, and coordinated source–load scheduling under three-phase unbalanced power-flow constraints, thereby improving post-fault restoration capability and operational economy.

The remainder of this paper is organized as follows. Section 2 introduces the related models of distributed generators, battery energy storage systems, and EV charging stations. Section 3 presents the Stage I fault restoration model, in which island partitioning and restoration decisions are carried out with the objective of maximizing critical-load restoration. Section 4 develops the Stage II optimization model to achieve network reconfiguration and coordinated scheduling of DGs, BESS units, and EV charging stations while minimizing the overall restoration cost under operational constraints. Section 5 provides case studies based on the modified IEEE 33-bus distribution system to verify the effectiveness of the proposed method. Finally, Section 6 concludes the paper and discusses possible future research directions.

2. Wind–PV–Storage and Load Modeling

2.1. Wind Power Output Model

For a wind turbine generator (WTG), the active power output is typically described by a piecewise power curve as a function of wind speed [20].

$$P_{\mathrm{WT}}(t)=\left\{\begin{array}{ll}0,\hfill & v(t)<v_{\mathrm{in}}\hfill \\ P_{\mathrm{WT}}^{\mathrm{r}}{\displaystyle \frac{v{(t)}^3-v_{\mathrm{in}}^3}{v_{\mathrm{r}}^3-v_{\mathrm{in}}^3}},\hfill & v_{\mathrm{in}}\le v(t)<v_{\mathrm{r}}\hfill \\ P_{\mathrm{WT}}^{\mathrm{r}},\hfill & v_{\mathrm{r}}\le v(t)\le v_{\mathrm{out}}\hfill \\ 0,\hfill & v(t)>v_{\mathrm{out}}\hfill \end{array}\right.$$

(1)

where $P_{\mathrm{WT}}(t)$ denotes the predicted active power output of the wind turbine at time t; $P_{\mathrm{WT}}^{\mathrm{r}}$ is the rated capacity (rated power) of the wind turbine; $v(t)$ is the wind speed at time t; $v_{\mathrm{in}}$ is the cut-in wind speed; $v_{\mathrm{out}}$ is the cut-out wind speed; and $v_{\mathrm{r}}$ is the rated wind speed.

2.2. PV Power Output Model

Solar irradiance is a key factor affecting the power output of a photovoltaic generator (PVG). Based on the photovoltaic effect of solar cells, and by incorporating an empirical photoelectric conversion equation together with the corresponding empirical coefficients, the PV system’s power output can be calculated from the input irradiance data as follows [21]:

$$P_{\mathrm{PV}}(t)=\eta AG$$

(2)

where $P_{\mathrm{PV}}\left(t\right)$ denotes the PV power output at time $t$; $\eta$ is the rated photoelectric conversion efficiency; $A$ is the surface area of the PV array; and $G$ is the solar irradiance.

2.3. Output Models of Wind–Storage and PV–Storage Systems

The charging/discharging model of the wind–storage and PV–storage systems is tightly coupled with the wind/PV power outputs, and the charging model is given by (3):

$$P_{\mathrm{ch}}={\eta }_{\mathrm{ch}}·(P_{\mathrm{PV}}+P_{\mathrm{WT}}-P_{\mathrm{load}})$$

(3)

Similarly, the discharging model is given by (4):

$$P_{\mathrm{dis}}={\eta }_{\mathrm{dis}}·(P_{\mathrm{load}}-P_{\mathrm{PV}}-P_{\mathrm{WT}})$$

(4)

where $P_{\mathrm{ch}}$ is the charging power of the BESS; ${\eta }_{\mathrm{ch}}$ is the charging efficiency of the BESS; $P_{\mathrm{dis}}$ is the discharging power of the BESS; ${\eta }_{\mathrm{dis}}$ is the discharging efficiency of the BESS; $P_{\mathrm{PV}}$ is the PV power output; $P_{\mathrm{WT}}$ is the wind power output; and $P_{\mathrm{load}}$ is the load demand.

The energy stored in the BESS is given by (5):

$$E_{\mathrm{st}}(t)=E_{\mathrm{st}}(t-1)+(P_{\mathrm{ch}}-P_{\mathrm{dis}})·\Delta t$$

(5)

where $E_{\mathrm{st}}(t)$ denotes the energy stored in the BESS at time $t$, and $\Delta t$ is the time step length. The BESS is subject to maximum charging and discharging rate limits; i.e., the charging/discharging power is bounded in each time interval. Therefore, the charging/discharging constraints of the BESS are given by (6) and (7) [21]:

$$0\le P_{\mathrm{ch}}\le P_{\mathrm{ch}\max }$$

(6)

$$0\le P_{\mathrm{dis}}\le P_{\mathrm{dis}\max }$$

(7)

Considering the energy capacity of the energy storage system, the stored energy is bounded by the following minimum and maximum energy constraints [20]:

$$E_{\min }\le E_{\mathrm{st}}\le E_{\max }$$

(8)

where $P_{\mathrm{ch}\max }$ is the maximum charging power of the BESS; $P_{\mathrm{dis}\max }$ is the maximum discharging power of the BESS; $E_{\min }$ and $E_{\max }$ are the minimum and maximum energy capacities of the BESS, respectively.

2.4. Joint Power Output Model of the Energy Storage System and Wind/PV Generation

Since the active power outputs of the photovoltaic generator (PVG) and wind turbine generator (WTG) are highly sensitive to environmental conditions such as solar irradiance and wind speed, they exhibit pronounced stochastic fluctuations. To mitigate the adverse impacts of such variability on system operation, distributed generation (DG) is co-located and coupled with energy storage units to form a wind–PV–storage hybrid generation structure, which is used to smooth DG output uncertainty and maintain the stable operation of the islanded system. Furthermore, based on the actual output levels of the PVG, WTG, and the energy storage device in each dispatch interval, the aggregated (joint) total power output of the wind/PV sources and the storage unit at each time instant after a distribution-network fault can be obtained.

$$P_{\mathrm{tot}}=P_{\mathrm{PV}}+P_{\mathrm{WT}}+P_{\mathrm{dis}}-P_{\mathrm{ch}}$$

(9)

where $P_{\mathrm{tot}}$ denotes the total aggregated (joint) power output of the system.

2.5. Time-Varying Load Model and Load Priority Classification

2.5.1. Load Time-Variation Model

The nodal loads in a distribution network are influenced by factors such as customer electricity-consumption behavior, meteorological conditions, and operating/production conditions, and thus exhibit pronounced intra-day time-varying characteristics. In this paper, based on typical operating experience of a practical distribution network, a 24-point daily load profile is specified a priori. The time-varying load at each node is obtained by proportionally scaling the nodal base load. Consequently, integrating the load over the corresponding time interval yields the electricity demand of each node during that period, which is given by [21]:

$$L_k(t)={\displaystyle {\int }_t^{t+1}{f}_k}(x)dx,t=0,1,\dots ,23$$

(10)

where $L_k(t)$ denotes the electricity demand of node k at time t, and $f_k(x)$ is the load-profile function of node k.

2.5.2. Load Priority Classification

Under islanded operation and source–load imbalance scenarios, the available capacities of DG, EVs, and the BESS are jointly constrained by feeder thermal limits and voltage constraints. As a result, after a fault, the system is generally unable to restore all loads across the entire network simultaneously. In line with the widely adopted operational principle of prioritizing critical loads in practical distribution systems, this paper classifies nodal loads into three priority sets according to their importance: Class-I, Class-II, and Class-III loads [21]. Class-I loads include critical customers such as hospitals and important substations; Class-II loads correspond to other important commercial and industrial customers; the remaining nodes are categorized as Class-III loads, which can be treated as interruptible loads. Class-I loads must be supplied at 100%, whereas the restoration priority of Class-II and Class-III loads is determined by their time-varying importance. Accordingly, minimum restoration factors are specified for different load classes [20] as follows:

$${\alpha }_{L_k}=\left\{\begin{array}{ll}1,\hfill & L_k=1,\hfill \\ 0.90,\hfill & L_k=2,\hfill \\ 0,\hfill & L_k=3.\hfill \end{array}\right.$$

(11)

where ${\alpha }_{L_k}$ denotes the minimum restoration factor for load class, and $L_k$ indicates the load priority class of node k. The overall priority-based restoration constraint is given by [22]:

$${\lambda }_k(t)+s_k(t)\ge {\alpha }_{L_k}{z}_k(t)$$

(12)

where ${\lambda }_k(t)\in \left[0,1\right]$ denotes the load restoration ratio of node k during period t (${\lambda }_k(t)=1$ indicates full supply and ${\lambda }_k(t)=0$ indicates complete curtailment; here ${\lambda }_k(t)$ represents the aggregated restoration ratio over the three phases); $s_k(t)$ is a slack variable; and $z_k(t)$ is a binary energization indicator, equal to 1 if node k is energized at period t and 0 otherwise. Furthermore, a weighted penalty term is introduced:

$$J={\displaystyle \sum _{t=1}{\displaystyle \sum _{k=1}{w}_{L_k}}}{s}_k(t),w_1>w_2>w_3$$

(13)

where J denotes the load-related penalty cost, and $w_{L_k}$ is the penalty weight associated with the load class.

By assigning monotonically decreasing weights to the slack variables of different load classes, the proposed formulation ensures that, following a fault, Class-I loads are supplied with the highest priority, while Class-II loads are restored as much as possible subject to system constraints; Class-III loads are treated as the primary load-shedding candidates. In this way, the dispatch adheres to the principle of critical-load priority and tiered restoration.

2.6. EV Charging Station and V2G Model

With the development of vehicle–grid interaction technologies, EV charging stations are no longer merely passive loads; rather, subject to meeting users’ mobility requirements, they can feed power back to the distribution network via vehicle-to-grid (V2G) operation [23].

(1) To reflect the rated capacity limits of the charging piles and to prevent the unreasonable operating condition of simultaneous charging and discharging within the same time interval, the following constraints are imposed:

The charging power constraint [2] is given by:

$$0\le P_{k,\phi ,t}^{\mathrm{ch}}\le u_{k,t}^{\mathrm{ch}}{P}_{k,\phi }^{\mathrm{chmax}}$$

(14)

The discharging power constraint [2] is given by:

$$0\le P_{k,\phi ,t}^{\mathrm{dis}}\le u_{k,t}^{\mathrm{dis}}{P}_{k,\phi }^{\mathrm{dismax}}$$

(15)

The mutually exclusive charging/discharging constraint [2] is given by:

$$u_{k,t}^{\mathrm{ch}}+u_{k,t}^{\mathrm{dis}}\le 1$$

(16)

where $P_{k,\phi ,t}^{\mathrm{ch}}$ and $P_{k,\phi ,t}^{\mathrm{dis}}$ denote the charging and discharging power, respectively, of node k on phase $\phi$ during period t; $u_{k,t}^{\mathrm{ch}}$ and $u_{k,t}^{\mathrm{dis}}$ are the corresponding binary charging and discharging status variables; and $P_{k,\phi }^{\mathrm{chmax}}$ and $P_{k,\phi }^{\mathrm{dismax}}$ represent the rated charging and discharging capacities, respectively, of node k on phase $\phi$ during period t.

(2) Station-level charging power constraint [22].

$${\displaystyle \sum _{\phi \in a,b,c}{P}_{k,\phi ,t}^{\mathrm{ch}}}\le P_{k,t}^{\mathrm{base}}$$

(17)

where $P_{k,t}^{\mathrm{base}}$ denotes the desired charging power of the EV charging station during period t, which is consistent with the users’ charging-demand profile.

(3) State-of-charge constraints [22].

$$E_{k,t+1}=E_{k,t}+{\eta }_{\mathrm{EV}}^{\mathrm{ch}}\Delta t{\displaystyle \sum _{\phi }{P}_{k,\phi ,t}^{\mathrm{ch}}}-{\displaystyle \frac{\Delta t}{{\eta }_{\mathrm{EV}}^{\mathrm{dis}}}}{\displaystyle \sum _{\phi }{P}_{k,\phi ,t}^{\mathrm{dis}}}$$

(18)

where $E_{k,t}$ denotes the equivalent stored energy of the EV charging station at time t.

In this paper, the forecasting models for wind power, PV generation, and nodal loads are specified at the resource level in terms of station-level aggregated total power. To ensure consistency with the subsequent three-phase unbalanced power-flow formulation and restoration optimization model, it is assumed that—except for phase-specific dispatch of EV charging stations—other conventional DG units and loads are connected in a three-phase balanced manner. That is, the total nodal active/reactive power is evenly allocated among phases A, B, and C.

3. Island Partitioning Considering Wind/PV/Storage Integration and Time-Varying Load

After a fault occurs in the distribution network, the faulted branches are isolated, and the original radial topology is partitioned into several electrical islands. Considering the wind/PV–storage output models and the intraday load variation characteristics presented in the previous section, for each dispatch interval t, the available generation capacity at each node on phases A, B, and C is first determined based on the forecasted outputs of PV generation, wind power, and energy storage. On this basis, island partitioning and source–load allocation are optimized to prioritize the restoration of critical loads within each island, subject to operational constraints including voltage limits, line-capacity limits, and three-phase power-balance constraints. If the combined output from wind/PV–storage resources and V2G support within a given island is insufficient, lower-priority loads are curtailed to ensure the secure and reliable operation of the islanded system.

3.1. Objective Function

During the post-fault island partitioning stage, the objective is to restore as much critical load as possible within each island over all dispatch intervals while allowing the curtailment of lower-priority loads when necessary. Accordingly, the objective function is formulated as:

$$F_1={\displaystyle \sum _{t=1}^T{\displaystyle \sum _{d\in {\mathrm{\Omega }}_{\mathrm{D}}}{\displaystyle \sum _{k\in D_d}{\displaystyle \sum _{\phi \in \{a,b,c\}}{\sigma }_k}}}}{\rho }_{k,\phi ,t}{L}_{k,\phi ,t}$$

(19)

where $F_1$ is the objective function, which maximizes the total restored energy of critical loads over the entire fault-restoration horizon by summing, across all islands, all nodes, and all phases, the amount of critical-load energy restored; ${\sigma }_k$ is the importance (priority) coefficient of node k; ${\rho }_{k,\phi ,t}\in \left[0,1\right]$ is the restoration factor of node k on phase $\phi \in \left\{a,b,c\right\}$ during period t; $L_{k,\phi ,t}$ is the corresponding three-phase base load; ${\mathrm{\Omega }}_{\mathrm{D}}$ denotes the set of all islands; $D_d$ is the set of nodes in island d; and T is the total number of time periods during the fault-restoration process.

3.2. Constraints

The three-phase network operation constraints are mainly formulated with reference to existing three-phase unbalanced distribution-network power-flow and branch-flow models, and are further adapted to the fault-restoration scenario considered in this paper.

(1) Three-phase active power adequacy constraint for each island

To ensure source-load balance within each island and the feasibility of the restoration scheme, the following active-power adequacy constraint should be satisfied:

$${\displaystyle \sum _{k\in D_d}{P}_{k,\phi ,t}^{\sup }}\ge {\displaystyle \sum _{k\in D_d}{\rho }_{k,\phi ,t}}{L}_{k,\phi ,t}$$

(20)

where $P_{k,\phi ,t}^{\sup }$ denotes the net active power available for supply at node k on phase $\phi$ during period t.

(2) Three-phase nodal voltage constraints

To ensure that the system operates within safe voltage limits, we introduce the three-phase nodal voltage constraint:

$$V_{\min }\le V_{k,\phi ,t}\le V_{\max }$$

(21)

where $V_{k,\phi ,t}$ denotes the phase-to-neutral voltage magnitude at node k on phase $\phi$ during period t; and $V_{\min }$ and $V_{\max }$ are the lower and upper voltage limits, respectively.

(3) Three-phase nodal power-balance constraints

In distribution network restoration, it is essential to ensure that the active and reactive power at each node are balanced to maintain system stability. These constraints are critical for ensuring that the system can achieve steady-state operation after a fault.

The active power-balance constraint is given by [14]:

$$P_{k,\phi ,t}-V_{k,\phi ,t}{\displaystyle \sum _{j=1}^n{V}_{j,\phi ,t}}[G_{kj,\phi }\cos ({\delta }_{k,\phi ,t}-{\delta }_{j,\phi ,t})+B_{kj,\phi }\sin ({\delta }_{k,\phi ,t}-{\delta }_{j,\phi ,t})]=0$$

(22)

The reactive power-balance constraint is given by [14]:

$$Q_{k,\phi ,t}-V_{k,\phi ,t}{\displaystyle \sum _{j=1}^n{V}_{j,\phi ,t}}[G_{kj,\phi }\sin ({\delta }_{k,\phi ,t}-{\delta }_{j,\phi ,t})-B_{kj,\phi }\cos ({\delta }_{k,\phi ,t}-{\delta }_{j,\phi ,t})]=0$$

(23)

where $P_{k,\phi ,t}$ and $Q_{k,\phi ,t}$ are the injected active and reactive power at node k on phase $\phi$ during period t (generation is positive and load is negative); ${\delta }_{k,\phi ,t}$ is the corresponding voltage phase angle; and $G_{kj,\phi }$ and $B_{kj,\phi }$ are the admittance-matrix elements between nodes k and j on phase $\phi$.

(4) Three-phase DistFlow branch constraints [24,25,26]

The voltage-drop equation is given by:

$$V_{j,\phi ,t}^2=V_{k,\phi ,t}^2-2(r_{kj}{P}_{kj,\phi ,t}+x_{kj}{Q}_{kj,\phi ,t})+(r_{kj}^2+x_{kj}^2)I_{kj,\phi ,t}^2$$

(24)

The power–current constraint is given by:

$$P_{kj,\phi ,t}^2+Q_{kj,\phi ,t}^2\le I_{kj,\phi ,t}^2{V}_{\min }^2$$

(25)

where $P_{kj,\phi ,t}$, $Q_{kj,\phi ,t}$, and $I_{kj,\phi ,t}$ denote the active power, reactive power, and branch current of line on phase $\phi$ during period t, respectively; and $r_{kj}$ and $x_{kj}$ are the branch resistance and reactance, respectively.

(5) Three-phase branch-capacity constraints and outage constraints for faulted branches

To ensure that the system operates within its maximum power limits, we introduce the branch capacity constraint. This constraint limits the power flow through each branch, ensuring that the grid does not exceed its capacity during the restoration process. This constraint is expressed by the following equation [2]:

$$P_{kj,\phi ,t}^2+Q_{kj,\phi ,t}^2\le {(S_{kj,\phi }^{\max })}^2$$

(26)

In addition to the branch capacity constraint, the outage constraint plays a crucial role in the process. This constraint ensures that certain branches can be out of service during faults, thereby maintaining the stability of the grid. The specific outage constraint is given by the following equation:

$$P_{kj,\phi ,t}=0,Q_{kj,\phi ,t}=0,I_{kj,\phi ,t}^2=0$$

(27)

where $S_{kj,\phi }^{\max }$ denotes the apparent-power capacity limit of branch $\left(k,j\right)$ on phase $\phi$.

(6) Three-phase voltage unbalance constraint

Since this study focuses on a three-phase unbalanced distribution network, excessively large post-fault three-phase voltage deviations in certain islands may severely degrade power-supply quality and may even trigger protection maloperation. Therefore, the three-phase voltage unbalance at critical nodes is constrained.

The average three-phase voltage at node k during period t is given by:

$${\overline{V}}_{k,t}=1/3(V_{k,a,t}+V_{k,b,t}+V_{k,c,t})$$

(28)

The relative deviation of phase $\phi$ is defined as:

$$e_{k,\phi ,t}={\displaystyle \frac{V_{k,\phi ,t}-{\overline{V}}_{k,t}}{{\overline{V}}_{k,t}}}$$

(29)

Given the allowable upper limit on three-phase voltage unbalance (according to the national standard, the voltage unbalance in distribution networks is required to be within ±2%), the corresponding constraint is given by:

$$-{\epsilon }_{\mathrm{unb}}\le e_{k,\phi ,t}\le {\epsilon }_{\mathrm{unb}}$$

(30)

4. Distribution Network Reconfiguration and Optimal Operation

To address post-disaster self-healing and optimal operation of three-phase unbalanced distribution networks, this paper proposes a two-stage post-fault self-healing operation strategy for unbalanced distribution systems with integrated wind/PV generation, energy storage, and EV charging, building upon the island-partitioning framework developed in the previous section. In Stage I, while accounting for the time-varying output characteristics of wind/PV–storage resources and EVs, island partitioning is performed via a combined depth-first search and breadth-first search (DFS + BFS) traversal based on fault locations and load-priority classes. This procedure identifies grid-connected (main-grid) islands and energized autonomous islands with local supply capability from DG, EVs, and the BESS, thereby restoring as much load as possible across different priority levels [4,27,28]. In Stage II, without degrading the load-restoration level achieved in Stage I, the branch switching statuses and the coordinated outputs of DG–EV–BESS are jointly optimized under three-phase unbalanced power-flow constraints. The objective is to minimize the overall cost, including line losses, switching operations, three-phase voltage unbalance, and EV charging deviations, thereby enabling post-disaster reconfiguration and optimal operation of the unbalanced distribution network. The proposed strategy is illustrated in Figure 1.

4.1. Objective Function

In order to achieve the goal of reducing total costs in the second stage mentioned above, the Stage II optimization problem is formulated as follows. The objective function consists of the line-loss cost, switching-operation cost, three-phase voltage-unbalance penalty cost, and EV user-compensation cost in the period. The required formula is shown as follows:

$$F_2=\min {\displaystyle \sum _{t=1}^T(}{C}_{\mathrm{loss}}(t)+C_{\mathrm{sw}}(t)+C_{\mathrm{unb}}(t)+C_{\mathrm{ev}}(t))$$

(31)

$$C_{\mathrm{loss}}(t)={\gamma }_{e,t}\Delta t{\displaystyle \sum _{(k,j)\in \tau }{\displaystyle \sum _{\phi \in \{a,b,c\}}{r}_{kj}}}{I}_{kj,\phi ,t}^2$$

(32)

$$C_{\mathrm{sw}}(t)={\displaystyle \sum _{(k,j)\in \tau }{c}_{\mathrm{sw},kj}}{W}_{kj,t}$$

(33)

$$C_{\mathrm{unb}}(t)=c_{\mathrm{unb}}\Delta t{\displaystyle \sum _{k\in N_{\mathrm{K}}}{\displaystyle \sum _{\phi \in \{a,b,c\}}(}}{\displaystyle \frac{e_{k,\phi ,t}}{{\epsilon }_{\mathrm{unb}}}}{)}^2$$

(34)

$$C_{\mathrm{ev}}(t)=c_{\mathrm{ev}}\Delta t{\displaystyle \sum _{k\in N_{\mathrm{EV}}}(}{P}_{k,t}^{\mathrm{base}}-{\displaystyle \sum _{\phi \in \{a,b,c\}}{P}_{k,\phi ,t}^{\mathrm{ch}}}{)}^2$$

(35)

where $F_2$ denotes the total operating cost of the distribution network over the scheduling horizon; $C_{\mathrm{loss}}(t)$, $C_{\mathrm{sw}}(t)$, $C_{\mathrm{unb}}(t)$, and $C_{\mathrm{ev}}(t)$ denote, respectively, the line-loss cost, switching-operation cost, three-phase voltage-unbalance penalty cost, and EV user-compensation cost in period t; ${\gamma }_{e,t}$ is the electricity price in period t (CNY/kWh); $\Delta t$ is the time-step length; $\tau$ is the set of all candidate branches, including the original feeder sections and tie lines; $c_{\mathrm{sw},kj}$ is the equivalent operation-and-maintenance cost incurred by a single switching action (opening/closing) of branch switch $\left(k,j\right)$ (CNY/action); $W_{kj,t}$ is the switching-action indicator of branch $\left(k,j\right)$ between two consecutive periods, used to count whether the switch changes its status in period t; $c_{\mathrm{unb}}$ is the equivalent penalty price for voltage unbalance; $N_{\mathrm{K}}$ denotes the set of nodes; $c_{\mathrm{ev}}$ is the user-compensation price (CNY/kWh); and $N_{\mathrm{EV}}$ is the set of nodes equipped with EV charging stations.

4.2. Constraints

To ensure feasible post-fault network operation, the above objective is subject to the following topology and operating constraints.

(1) Branch–node consistency constraints.

$$Z_{kj,t}\le z_{k,t},Z_{kj,t}\le z_{j,t}$$

(36)

where $Z_{kj,t}\in \left\{0,1\right\}$ is the branch-switch status variable ($Z_{kj,t}=0$ indicates open and $Z_{kj,t}=1$ indicates closed); and $z_{k,t}\in \left\{0,1\right\}$ is the nodal energization status variable.

(2) Network connectivity constraints.

$$0\le f_{kj,t}\le (|N_k|-1)Z_{kj,t}$$

(37)

(3) Radiality constraints.

$${\displaystyle \sum _{(k,j)\in \tau }{Z}_{kj,t}}={\displaystyle \sum _{k\in N_k}{z}_{k,t}}-1$$

(38)

The three-phase nodal voltage constraints, power-balance constraints, and related constraints are given in (21)–(30).

5. Case Studies

5.1. Parameter Settings

The simulation model was developed by the authors in MATLAB R2023b based on a modified IEEE 33-bus distribution network. As shown in Figure 2, the test system in this paper is a modified IEEE 33-bus distribution network, with a base voltage of 12.66 kV. The network topology consists of 32 original branches and five tie lines, and the initial configuration is radial. PV generation units rated at 200 kW, 200 kW, and 300 kW, each co-located with an energy storage system, are connected at buses 7, 20, and 27, respectively. Wind generation units rated at 250 kW and 300 kW, each co-located with an energy storage system, are connected at buses 16 and 30, respectively. In addition, EV charging stations rated at 300 kW are installed on buses 24 and 29. Wind/PV–storage units are deployed on the renewable buses to smooth output fluctuations. The load-priority classes of all buses are summarized in

Table 1. Node load level.

Load Priority Class	Node	Load Weighting Factor
Class-I load	2, 6, 19, 25, 30	100
Class-II load	5, 7, 14, 18, 24	10
Class-III load	Other nodes	1

, and the main parameter settings are listed in

Table 2. Main data parameters.

DG cost/[CNY·(kWh)−1]	0.66
Load-curtailment compensation cost/[CNY·(kWh)−1]	1.10
Three-phase unbalance penalty cost/CNY	0.21
Switching operation cost/[CNY·(operation)−1]	13
EV revenue/[CNY·(kWh)−1]	0.54

.

To validate the effectiveness and superiority of the proposed method, four simulation scenarios are designed for comparison.

Scenario 1: The proposed two-stage self-healing framework is applied while considering only wind and PV generation; EVs and energy storage systems are not included.

Scenario 2: Wind/PV generation and wind–storage/PV–storage systems are considered, and post-disaster restoration is carried out using the proposed method.

Scenario 3: EVs participate in V2G scheduling, and post-disaster restoration is performed on a three-phase balanced network using the proposed method.

Scenario 4: The proposed method is applied, and the charging stations are allowed to independently regulate the outputs of phases A, B, and C according to the phase voltage levels and load deficits, thereby fully exploiting the inter-phase regulation capability of EVs.

5.2. Results and Discussion

At 12:00, the distribution network is assumed to experience line outages on branches 15–16 and 19–20, as well as a phase-A single-line-to-ground fault on branch 28–29. The faulted branches are taken out of service, and simulations are conducted with a scheduling interval of 1 h. Since prioritized restoration of different load classes is a widely adopted performance-evaluation perspective in distribution-network fault restoration studies, the restoration results in the following scenarios are analyzed from the viewpoints of Class-I, Class-II, Class-III, and overall system restoration performance [20,21].

5.2.1. Results for Scenario 1

In Scenario 1, island partitioning and distribution-network reconfiguration are performed using only the DG units installed in the IEEE 33-bus system. The resulting post-restoration topology is shown in Figure 3.

As shown in Figure 3, the proposed method is able to isolate the faulted branches and partition the network into three islands. During the reconfiguration stage, the unserved loads can be restored by closing the tie lines, and the corresponding restoration performance is illustrated in Figure 4.

In the baseline case without EVs and without BESS, the system can supply all loads in full prior to the fault. Once the fault occurs, a portion of Class-III loads is curtailed to guarantee the supply of Class-I and Class-II critical loads, leading to a substantial drop in the overall restoration level. After network reconfiguration, 100% restoration of Class-I and Class-II loads can be ensured; however, some Class-III loads must remain curtailed for an extended period. Consequently, the total system restoration ratio can only stabilize at approximately 94.34%.

5.2.2. Results for Scenario 2

Scenario 2 extends Scenario 1 by incorporating wind–storage and PV–storage systems. For a fair comparison with Scenario 1, the same branch faults are assumed, and the same set of tie lines is used for reconfiguration.

Figure 5 shows the post-fault reconfigured topology in Scenario 2; it can be observed that the resulting network topology and island partitioning are essentially the same as those in Scenario 1.

The load restoration ratios are shown in Figure 6.

Compared with Scenario 1, Scenario 2 achieves higher restoration ratios for Class-III loads and for the overall system at the fault occurrence time, with improvements of 4.98% and 3.37%, respectively. This indicates that distributed energy storage can provide additional short-term support during the fault period and thereby mitigate the severity of load interruptions. After reconfiguration is completed, the unserved loads are mainly located on islands without local supply support. Owing to network connectivity limitations, the storage units can hardly further increase the proportion of restorable loads; consequently, the restoration ratios of each load class and the overall system remain essentially the same as those in Scenario 1.

5.2.3. Results for Scenario 3

In Scenario 3, the EV charging stations at buses 24 and 29 are equipped with V2G capability, allowing them to feed power back to the distribution network during faults and islanded operation. They participate in a three-phase balanced restoration process through the proposed dispatch model.

Considering the same fault locations and tie-line set as in Scenario 1, the system is reconfigured to form multiple energized islands jointly supported by distributed generation and EVs. Figure 7 illustrates the post-fault topology and island partitioning results for Scenario 3. Each island contains not only distributed energy resources such as PV and wind generation but also EV aggregations capable of exporting power to the grid side, thereby providing additional regulation capability for subsequent voltage support and load restoration.

The load restoration ratios are shown in Figure 8.

Compared with Scenarios 1 and 2, Scenario 3 leverages the V2G support capability of EVs. During the fault period, the EVs are dispatched in the discharging mode to provide additional active/reactive power support for the islands and constrained areas, thereby reducing the need to curtail lower-priority loads and mitigating the restoration-ratio drop at the fault occurrence time. Specifically, at the fault instant, Scenario 3 improves the Class-III load restoration ratio and the total load restoration ratio by 1.16% and 0.77%, respectively, compared with Scenario 2. After network reconfiguration is completed, the topology remains unchanged in subsequent periods, and the available supply is sufficient to meet the load demand. Consequently, the system enters a steady post-fault operating state, in which the restoration ratios of all load classes remain more stable than those in Scenarios 1 and 2.

5.2.4. Results for Scenario 4

Scenario 4 considers post-fault three-phase unbalance and allows the EV charging stations to independently regulate the charging/discharging power on phases A, B, and C according to the phase voltage levels and load deficits, thereby fully exploiting the inter-phase regulation capability of EVs. Under the same fault conditions and tie-line configuration, the three-phase outputs of DG and EVs are coordinated within each island to suppress voltage and load imbalance. Figure 9 presents the post-fault reconfigured topology in Scenario 4, Figure 10 shows the load restoration ratios, and Figure 11 depicts the phase-specific dispatch power of EVs.

As can be observed from Figure 10 by comparing Scenarios 1–3, Scenario 4 achieves the highest Class-III load restoration ratio and the highest overall system restoration ratio at the fault instant, indicating that the proposed strategy exhibits strong resilience to fault disturbances in unbalanced distribution networks. In Figure 11, the three-phase EV power outputs at buses 24 and 29 exhibit pronounced, concentrated discharging spikes around the fault occurrence and key restoration periods, suggesting that the dispatch strategy rapidly releases EV energy when system support is required. Moreover, phase A dominates the discharging response, which more effectively alleviates the three-phase unbalance caused by the phase A fault on the branch and enhances post-fault restoration support capability.

The fault-restoration performance under different scenarios is compared in

Table 3. Comparison of fault recovery for different solutions.

Scenario	Total Comprehensive Restoration Cost/CNY	System Restoration Ratio at the Fault Instant/%	EV Revenue/CNY
Scenario 1	8792.50	86.92	0
Scenario 2	8598.65	89.53	0
Scenario 3	7245.62	90.14	2527.32
Scenario 4	7185.16	94.37	2474.84

, as summarized below:

From the above comparisons, it can be concluded that the overall operating performance is improved. On the one hand, at the fault instant, Scenario 4 achieves the highest total system restoration ratio among the four scenarios (94.37%), indicating that more load can be restored immediately after the fault and that the system exhibits stronger restoration capability. On the other hand, Scenario 4 yields the lowest overall cost, implying that the improved restoration performance does not incur additional economic burden; instead, it achieves a joint optimization of “higher restoration ratio and lower cost.”

In summary, enabling phase-specific EV dispatch not only alleviates the operational stress induced by three-phase unbalance but also improves restoration performance while reducing the overall restoration cost, highlighting the superiority of Scenario 4 in both fault-recovery effectiveness and economic efficiency. The proposed post-disaster hierarchical restoration and coordinated scheduling method for three-phase unbalanced distribution networks can reduce the total recovery cost and enhance system resilience at the fault occurrence time.

6. Conclusions

This paper investigates rapid post-fault service restoration and optimal operation of three-phase unbalanced distribution networks under wind/PV uncertainty. Building upon a two-stage post-disaster restoration paradigm, the V2G flexibility of EV charging stations and the coordinated support of battery energy storage systems (BESS) are further incorporated. A coordinated optimization model is developed that accounts for load-priority classification, island partitioning, and distribution network reconfiguration. Comparative case studies are conducted on a modified IEEE 33-bus system, and the main conclusions are summarized as follows:

(1) The proposed two-stage self-healing restoration framework enables prioritized load restoration while satisfying radiality requirements and three-phase operating constraints. During the fault period, coordinated source–load–storage support mitigates the drop in restoration level; after reconfiguration is completed, the system enters a stable post-fault operating state, with the restoration level remaining essentially flat in subsequent periods, demonstrating the stability and implementability of the proposed strategy.

(2) An EV charging-station participation model is established that explicitly considers V2G energy constraints, mutually exclusive charging/discharging operation, and reactive power support capability. By incorporating integrated wind/PV–storage–charging resources, EVs are extended beyond the conventional role of “aggregated storage/dispatchable load” to resilience resources that can provide support within the fault window, thereby enlarging the available support mechanisms and dispatch flexibility for post-disaster restoration.

(3) Case-study results demonstrate that phase-specific output regulation can provide targeted compensation for deficits in weak phases and further improve inter-phase operating quality after restoration while satisfying three-phase voltage constraints, which better aligns with the practical requirements of post-disaster restoration in unbalanced distribution networks.

(4) Introducing adjustable resources (BESS and EV-V2G) can effectively reduce the overall cost during the post-disaster restoration process and improve system operating economics while reducing the need to curtail lower-priority loads, thereby enhancing the overall performance indices.

(5) Future work may include the following aspects. First, the proposed restoration strategy can be extended to multi-day or seasonal scheduling scenarios, accounting for longer-term variations in loads and renewable generation. Second, the method can be validated on larger-scale and more complex distribution networks with multiple feeders and meshed structures to assess scalability and robustness. Third, additional sources of uncertainty, such as weather forecast errors, stochastic EV arrival and departure, and load behavior variability, can be incorporated to improve the robustness of the scheduling scheme. Fourth, multi-objective optimization considering reliability, economic cost, equipment lifetime, and power quality can be integrated into the restoration decision-making to enhance practical applicability. Finally, experimental validation or hardware-in-the-loop simulations can be performed to verify the feasibility and implementability of the proposed strategy in real-world distribution systems.