Fast Network Reconfiguration Method with SOP Considering Random Output of Distributed Generation

Abstract

Power outages in non-faulted zones caused by system failures significantly reduce the reliability of distribution networks. To address this issue, this paper proposes a fault self-healing technique based on the integration of soft open points (SOPs) and network reconfiguration. A mathematical model for power restoration is developed. The model incorporates SOP operational constraints and the stochastic output of photovoltaic (PV) distributed generation. And this formulation enables the determination of the optimal network reconfiguration strategy and enhances the restoration capability. The study first analyzes the operational principles of SOPs and formulates corresponding constraints based on their voltage support and power flow regulation capabilities. The stochastic nature of PV power output is then modeled and integrated into the restoration model to enhance its practical applicability. This restoration model is further reformulated as a second-order cone programming (SOCP) problem to enable efficient computation of the optimal network configuration. The proposed method is simulated and validated in MATLAB R2019a. Results demonstrate that combining the SOP with the reconfiguration strategy achieves a 100% load restoration rate. This represents a significant improvement compared to traditional network reconfiguration methods. Furthermore, the second-order cone programming (SOCP) transformation ensures computational efficiency. The proposed approach effectively enhances both the fault recovery capability and operational reliability of distribution networks with high penetration of renewable energy.

1. Introduction

The structure of the global power system is undergoing significant transformation. The large-scale integration of clean energy sources has exposed the limitations of traditional power systems in terms of flexibility and economic efficiency. Additionally, the proliferation of electric vehicles (EVs) and distributed energy resources (DERs) has intensified these challenges. As a result, the development of new power systems is crucial. They help ensure power quality and enhance system reliability. Moreover, they enable self-healing capabilities when faults occur. The new power system is designed to be clean, intelligent, and flexible. It enables two-way interaction between the grid and users. As a result, energy utilization efficiency can be significantly improved [1].

As an important part of the power system, the distribution network directly interfaces with end users and profoundly affects their experience. In the context of the new power system, smart distribution grids should not only fulfill the basic task of energy distribution. They should also integrate additional functions such as energy collection, transmission, and storage. Smart distribution grids adopt advanced technologies such as distributed energy resources, energy storage systems, and static var compensators. These technologies enhance economic efficiency and improve system reliability, and they also increase the operational flexibility of the grid. Combined with automation technology, smart distribution grids have made significant progress in resource utilization and user participation [2]. To ensure stable power quality, it is crucial to strengthen the self-healing capability of smart distribution grids.

Self-healing of faults is a crucial feature of smart distribution grids and plays a key role in enhancing the reliability of power supply. When a system fault occurs, the grid first detects and isolates the fault. However, following fault isolation, parts of the system may still experience power loss due to the absence of a power supply source. In modern distribution grids, the high penetration of DERs enables faster fault detection and response [3,4,5,6,7]. Nevertheless, the voltage drops caused by a fault may result in the disconnection of DERs from the grid. The final step in fault self-healing technology involves restoring power to the non-faulted areas. This process is essential for ensuring the reliability of the system’s power supply and preventing the waste of energy from distributed power sources.

Network reconfiguration is a widely employed method for power supply restoration, which traditionally relies on tie switch and sectionalized switch in distribution networks. By controlling the switching layout and topology, network reconfiguration can restore power to non-faulted areas [8,9]. This makes network reconfiguration a crucial measure for ensuring the reliability of distribution networks. Significant progress has been made in the development of fault reconfiguration models for traditional distribution networks. These models include active distribution network fault reconfiguration approaches that incorporate restoration methods, such as tie switch. These approaches are then transformed into solvable mixed-integer linear programming (MILP) models [10]. Additionally, robust islanding restoration models have been developed that account for uncertainties in wind turbine outputs and the use of electric vehicles as emergency dispatch resources. These models are solved using dyadic theory [11]. In addition, the modeling process involves multiple types of constraints, including power flow nonlinearity, switching states, and voltage security. As a result, the problem is formulated as a large-scale Mixed-Integer Nonlinear Programming (MINLP) model. This type of problem is characterized by high computational complexity, and conventional optimization methods often fail to obtain the global optimal solution. With advancements in power electronics technology, electronic devices are increasingly integrated into distribution networks. This integration makes energy use more efficient and improves control capabilities. It also enables the intelligent and informatized operation of distribution networks.

As a soft open point device that can replace traditional tie switch, SOP has a trend control capability that traditional tie switch cannot provide. It greatly enhances the flexibility and controllability of the distribution network and has attracted wide attention at home and abroad [12,13]. One study [14] proposed an active distribution network power restoration model based on SOP and transformed it into a SOCP model for solving. However, the model treats distributed energy sources such as PV systems as constant output and does not consider the stochastic nature of PV generation. Studies in the literature [15] demonstrated the use of SOP to quickly transfer the remaining load on a faulted feeder to restore power to non-faulted section loads. Although several studies have explored the application of SOP in power supply restoration, most of them do not consider the stochastic variability of distributed power sources, such as PV generation. This variability has a significant impact on the load restoration rate in power supply restoration models.

The coordination of soft open points (SOPs) and network reconfiguration effectively enhances power flow regulation capability in distribution networks, expands the scope of power supply restoration, and improves the utilization level of renewable energy. However, the stochastic volatility of distributed generation outputs, such as photovoltaic (PV) systems, poses a serious challenge to the robustness of reconfiguration strategies [16,17]. In practical operation, this volatility can easily cause node voltage violations, leading to the failure of predetermined restoration strategies and directly threatening power supply reliability.

To address this issue, the uncertainty of distributed generation output is explicitly considered in the reconfiguration model proposed in this paper. A normal distribution probability model is adopted to characterize the randomness of PV output, thereby enhancing the robustness of the model [18]. Meanwhile, the second-order cone programming (SOCP) method is applied to transform and solve the reconstruction model. This approach ensures global optimality while significantly improving computational speed and numerical stability. As a result, the proposed method provides both fast and reliable decision support for fault restoration in distribution networks with high penetration of renewable energy.

Reference [19] proposes a comprehensive method based on stochastic programming to address the uncertainty in probabilistic photovoltaic forecasting. Besides stochastic programming, robust optimization represents another mainstream and powerful framework. Robust optimization constructs uncertainty sets to seek solutions that remain feasible even under worst-case scenarios. Its advantage lies in requiring less information about probability distributions and producing more conservative and reliable strategies. For the load restoration problem assisted by network reconfiguration, a robust optimization method based on the H-infinity norm and polyhedral uncertainty modeling is proposed in [20]. Although robust optimization offers a valuable alternative perspective, its computational efficiency is relatively low, and the required solving time is longer. Prolonged computation may lead to greater economic losses caused by fault-induced power outages. Therefore, the restoration model in this paper is primarily developed within a stochastic programming framework.

2. Related Work

An optimization model is proposed for fault recovery in distribution networks by integrating a Soft Open Point (SOP) and photovoltaic (PV) uncertainty modeling. First, a load restoration model is developed that accounts for PV output uncertainty, thereby enhancing the model’s adaptability under high-penetration renewable scenarios. A mathematical modeling framework is designed by incorporating SOP-based power flow control, which fully exploits its active and reactive power regulation capabilities. Furthermore, the complex MINLP model is reformulated into an equivalent Second-Order Cone Programming (SOCP) problem. This transformation allows the use of convex optimization techniques to improve computational efficiency and ensure solution feasibility. The proposed approach introduces a novel control strategy at the device level. In addition, it improves the fault self-healing capability through systematic model formulation and efficient algorithm design. Experimental simulations evaluate the impact of PV output stochasticity on the power supply restoration model. It also compares the advantages of combining SOP with network reconfiguration versus traditional network reconfiguration methods.

1.

Research Gaps:

Based on the literature review above, current research still exhibits the following limitations: Existing restoration strategies fail to adequately account for the impact of high-penetration photovoltaic output uncertainty. There is a lack of efficient solution algorithms.

2.

Research Objectives:

To address the above research gaps, this study aims to achieve the following goals: A robust fault restoration model considering PV output uncertainty is constructed. The original nonconvex MINLP model is transformed into a tractable convex optimization model via second-order cone relaxation

3. Materials and Methods

3.1. SOP Function and Control Modes

3.1.1. SOP Topology and Functions

The SOP, an advanced power electronic distribution device, has been introduced to replace traditional tie switch. It is designed to enable flexible control of active power transfer between feeders while also providing reactive power support. Its application has significantly enhanced the operational flexibility and controllability of distribution systems.

The functionalities of the SOP are realized through fully controlled power electronic devices. Common implementation methods include back-to-back voltage source converters, as well as other approaches such as unified power flow controllers and modular multilevel converters [21,22]. Among these, the back-to-back voltage source converter is the most widely used due to its high control flexibility, fast response speed, and excellent scalability. This topology is formed by connecting two converters through a DC capacitor, and is also known as the back-to-back voltage source converter, as shown in Figure 1.

To enhance feeder connectivity, a converter is connected in parallel on the DC capacitor side. The configuration of the SOP connection to the grid is illustrated in Figure 2.

3.1.2. SOP Control Modes

The SOP adopts distinct control modes under normal and fault conditions. During normal operation, VSC1 operates in UdcQ mode to stabilize the DC voltage and facilitate power exchange with VSC2. VSC2 operates in PQ mode to meet the power requirements of the distribution network.

After a fault occurs, the control strategy of the SOP is switched. The VSC on the fault side enters Vf mode to provide voltage support for the de-energized loads. The VSC on the non-fault side remains in UdcQ mode to ensure a stable power supply to the faulted section, as shown in Figure 3.

3.1.3. Power Restoration Process Using SOP

The SOP possesses flexible power flow regulation capabilities. It can respond rapidly during faults and switch control modes to ensure voltage stability in non-faulted sections. This enables quick restoration of de-energized loads.

Taking a low-resistance grounding fault in a 10 kV distribution network as an example, the restoration process is as follows: Immediately upon fault occurrence, the SOP responds instantly. The VSC on the fault side switches to Vf mode to provide voltage support. The VSC on the non-fault side remains in PQ mode to continue supplying power. This prevents the spread of the fault impact, as shown in Figure 4.

After the protection device completes fault location and isolates the faulted section, the SOP continues to maintain its current control mode. It restores the power supply to the downstream loads from the fault point, as shown in Figure 5.

To achieve millisecond-level rapid response, the SOP studied in this paper remains online during normal operation. It performs functions such as power flow optimization and power quality improvement. Both its converters and control system are maintained in a hot-standby state.

When a fault occurs, the SOP can swiftly switch its control mode, for example, from PQ mode to Vf mode. No cold-start process is required, significantly improving the efficiency of fault recovery.

3.2. Power Restoration Model of the New Distribution Network with SOP

To ensure model tractability and clarity of conclusions, the mathematical model in this study is based on the following fundamental assumptions: A constant power load model is adopted. A radially operated distribution network topology is considered. A single permanent fault scenario is assumed, where the fault has been accurately isolated.

The optimization objectives and constraints of the model are established within the above assumption framework. The conclusions are applicable for evaluating such typical scenarios.

3.2.1. SOP Mathematical Model

Despite the high efficiency of SOP converters in power flow regulation, active power losses are still inevitably incurred within the converters when large-scale load transfers induce significant power flows. To address this, the loss coefficient of power transmission must be incorporated into the modeling process [16]. SOP controls active and reactive power. the DC capacitor between the two converters provides isolation, ensuring that reactive power outputs on both sides do not interfere with each other. During power transmission, only the capacity and active power balance constraints need to be considered. Since the fault recovery model is constructed after a fault occurs, only the SOP mathematical model under the V_dcQ-Vf control mode is relevant [23,24]. The corresponding constraint expressions are as follows:

1.

SOP Active Power Balance Constraint

$$P_i^{SOP}+P_j^{SOP}+P_i^{SOP,L}+P_j^{SOP,L}=0$$

(1)

$$P_i^{SOP,L}\le A_i^{SOP}\sqrt{{(P_i^{SOP})}^2+{(Q_i^{SOP})}^2}$$

(2)

$$P_j^{SOP,L}\le A_j^{SOP}\sqrt{{(P_j^{SOP})}^2+{(Q_j^{SOP})}^2}$$

(3)

where $P_i^{SOP}$ and $P_j^{SOP,L}$ represent the active power input at the m-node and the power loss, respectively. n denotes the n-node on the opposite side, while $A_i^{SOP}$ represents the power loss coefficient of the converter at that node i.

2.

SOP Capacity Constraint

$$\sqrt{{(P^{SOP})}^2+{(Q^{SOP})}^2}=S^{SOP}\le S_{\max }^{SOP}$$

(4)

where S_max^SOP represents the maximum transmission capacity of SOP. $P^{SOP}$ and $Q^{SOP}$ represent the active and reactive power transmitted by the SOP, respectively. $S^{SOP}$ denotes the apparent power transmitted by the SOP.

3.

Voltage Upper and Lower Limit Constraints

$$U_m^{SOP}\in \left[U_{\min },U_{\max }\right]$$

(5)

where U_min and U_max denote the lower and upper voltage limits at the node, respectively, with a range of [0.95, 1.05]. The voltage at the fault-side converter of the SOP must comply with these limits. $U_m^{SOP}$ indicates the voltage magnitude on the faulted side of the SOP.

3.2.2. Modeling the Randomness of Photovoltaic Distributed Generation

In modern distribution networks, the power output of PV distributed generation is inherently random and cannot be represented as a constant current source. During daylight hours, generation levels are high due to prolonged sunlight exposure, while at night, output diminishes significantly due to the absence of sunlight. Generation also fluctuates between sunny and rainy days. Furthermore, when the PV source operates in islanding mode due to system changes, it can be approximated as a voltage source. Therefore, modeling PV generation as a simple constant source is insufficient. The stochastic nature of PV power generation must be accounted for to ensure accurate modeling.

The uncertainty of photovoltaic (PV) output is primarily reflected in its forecast error. Macroscopically, PV output follows a deterministic diurnal pattern, and its predicted value (expected value μ_pv) can be derived as a time-varying curve from historical data and weather forecasts. However, at any given moment, a deviation exists between the actual output and the forecasted value, which constitutes the forecast error. This error is mainly caused by numerous independent random factors, such as random cloud cover, measurement noise, and model inaccuracies. Therefore, probabilistic modeling is required.

In probabilistic modeling, the randomness of PV output is typically characterized by its forecast error. Existing studies often employ different probability distributions for this purpose: The Beta distribution is suitable for normalized PV output (within [0, 1]), as it flexibly captures skewness and naturally satisfies the boundary condition that output is non-negative and does not exceed the rated capacity. However, its mathematical complexity makes it difficult to integrate into optimization models. The Weibull distribution, though mainly used for wind speed/wind power modeling due to its ability to represent wind stochasticity, is occasionally adopted for PV output.

However, its mathematical complexity makes it difficult to integrate into optimization models. The Weibull distribution is mainly used for wind speed and wind power modeling. This preference is due to its ability to represent wind stochasticity accurately. It is occasionally adopted for PV output modeling as well. Nevertheless, its forced application for PV output may introduce biases. It can also add unnecessary computational burden.

Furthermore, the uncertainty distribution of PV is supported by the central limit theorem. PV forecast errors result from the combined effect of multiple independent random factors, and their distribution tends to be normal [25]. Moreover, the normal distribution is mathematically simple, easily integrated with optimization models, and computationally efficient. This aligns well with the rapid response required in power system fault restoration. Therefore, this paper uses the normal distribution to model PV uncertainty [26,27].

An hourly time resolution is adopted. Within each optimization period (1 h), PV output and its uncertainty are treated as constant. This granularity is suitable for distribution network restoration and planning studies, as it effectively captures the main variation trends of PV output while maintaining the computational efficiency of the optimization model.

For the active power output P_PV of photovoltaic generation, the probability density function f(PV) is expressed as

$$f(P_{PV})={\displaystyle \frac{1}{\sqrt{2\pi }{\sigma }_{PV}}}\exp \left(-{\displaystyle \frac{{(P_{PV}-{\mu }_{PV})}^2}{2{\sigma }_{PV}^2}}\right)$$

(6)

In the equation, μ_PV represents the predicted power output of the photovoltaic source, while σ_PV denotes the standard deviation of the random error in power generation. The reactive power output of the photovoltaic system is assumed to vary in accordance with the active power, based on the power factor relationship.

3.2.3. The Power Restoration Model of a New Distribution Network with SOP

The core of the power restoration problem in distribution networks lies in analyzing the system’s operating state after fault isolation. This paper develops a model based on a single power flow section. During the fault recovery process, SOP can provide reactive power to optimize voltage distribution and reduce active power losses. The objective of this model is to maximize active power load recovery at nodes while minimizing system losses. The mathematical formulation is as follows:

$$\min f={\eta }_L{f}_L-{\eta }_D{f}_D$$

(7)

$$f_D={\displaystyle \sum _{i\in {\Omega }_n}{\lambda }_i{P}_i^D}$$

(8)

$$f_L={\displaystyle \sum _{ij\in {\Omega }_b}{r}_{ij}{I}_{ij}^2}+{\displaystyle \sum _{ij\in {\Omega }_b}{P}_i^{SOP,L}}$$

(9)

In the equation, η_L and η_D are the weight coefficients of system loss f_L and system load recovery f_D, respectively. They represent the importance of the load, with η_D >> η_L. Ω_n is the set of all nodes in the system, and Ω_b is the set of all branches in the system. r_ij is the line resistance on branch ij, and I_ij is the current flowing through branch ij. $P_i^D$ is the active power consumed by the load at node i, λ_i is the load recovery coefficient at node i (where recovery equals 1 and non-recovery equals 0), and $P_i^{SOP,L}$ is the power loss of the SOP connected at node i.

In addition to the SOP capacity and active power balance constraints, the system also includes power flow, operational constraints and spanning tree constraint.

• Power Balance Constraint:

$${\displaystyle \sum _{k\in v(j)}{\alpha }_{jk}{P}_{jk,t}-}{\displaystyle \sum _{i\in u(j)}(}{\alpha }_{ij}{P}_{ij,t}-{\alpha }_{ij}{I}_{ij,t}^2{r}_{ij})=P_{j,t}$$

(10)

$${\displaystyle \sum _{k\in v(j)}{\alpha }_{jk}{Q}_{jk,t}-}{\displaystyle \sum _{i\in u(j)}(}{\alpha }_{ij}{Q}_{ij,t}-{\alpha }_{ij}{I}_{ij,t}^2{x}_{ij})=Q_{j,t}$$

(11)

$$U_i^2-U_j^2+(r_{ij}^2+x_{ij}^2)I_{ij}^2-2(r_{ij}{P}_{ij}+x_{ij}{Q}_{ij})=0$$

(12)

$$I_{ij}^2{U}_i^2=P_{ij}^2+Q_{ij}^2$$

(13)

$$P_{j,t}=P_{j,t}^{DG}+P_{j,t}^{SOP}-{\lambda }_j{P}_{j,t}^D$$

(14)

$$Q_{j,t}=Q_{j,t}^{DG}+Q_{j,t}^{SOP}-{\lambda }_j{Q}_{j,t}^D$$

(15)

where α_ij represents the operating state of branch ij, where 1 indicates the branch is closed, and 0 indicates the branch is open. P_ij,t and Q_ij,t represent the active and reactive power flowing through branch ij at time t, respectively. r_ij and x_ij are the equivalent resistance and reactance of the line, respectively. P_j,t and Q_i,t are the net active and reactive power injected at node j at time t, respectively. $P_{j,t}^D$ and $Q_{j,t}^D$ are the active and reactive power consumed by the load at node j, respectively. $P_{j,t}^{SOP}$ and $Q_{j,t}^{SOP}$ are the active and reactive power injected by the distributed generation at node j, respectively. $I_{ij,t}$ represents the current flowing through branch ij at time t; $U_i$ denotes the voltage magnitude at node i.

2.

System Operating Constraints:

$$U_{\min }^2\le U_i^2\le U_{\max }^2$$

(16)

$$I_{ij}^2\le I_{\max }^2$$

(17)

where U_min and U_max are the minimum and maximum allowable node voltages in the system, and I_max is the maximum allowable current flow in the line.

3.

SOP operating constraints: Equations (1)–(5).

During fault recovery, the SOP on the disconnected side operates in the Vf mode. Therefore, the converter on the faulted side must satisfy specific voltage requirements:

$$U_{i,t}^{SOP}\ge U_{set}$$

(18)

where $U_{i,t}^{SOP}$ is the voltage of the converter connected to the fault-side node i, and U_set is the pre-set voltage lower limit on the fault side to prevent a significant voltage drop.

To ensure that the reconfigured distribution network satisfies actual operational requirements, its topology must remain radial. That is, no closed loops are allowed in the system. The distribution network is originally radial in structure. During reconfiguration, especially within the faulted feeder, actions such as disconnecting faulted branches and closing tie switches are often performed. However, these operations must not create any loops.

To address this, a constraint is added to the reconfiguration model to limit the combination of closed switches. This constraint ensures a loop-free structure and maintains the radiality requirement of single power supply and unique power flow path. A spanning tree condition is therefore established to prevent the formation of mesh structures after reconfiguration.

$${\displaystyle \sum _{i,j}{f}_{ij}=n-1}$$

(19)

where $f_{ij}$ is all the connected branches in the model. n refers to the total number of nodes in the system.

4. Second-Order Cone Programming (SOCP)

The power restoration model for the active distribution network with intelligent SOP is reformulated as a SOCP model. The original model is a complex mixed-integer nonlinear model, which suffers from low solution efficiency and difficulties in finding the optimal solution. The nonlinearity primarily originates from the terms involving $I_{ij,t}^2{r}_{ij}$ and $U_i^2$ in the power flow Equations (10)–(13). These terms represent products or squares of decision variables, resulting in a nonconvex problem that is difficult to solve efficiently for a global optimum. To improve solution efficiency and reliability, second-order cone relaxation is employed in this paper. The nonlinear model is transformed into a mixed-integer second-order cone programming model.

In contrast, the SOCP model offers advantages such as convexity, smoothness, closedness, and symmetry. Its solution process is efficient and stable, ensuring the attainment of a global optimal solution. Compared to other nonlinear programming models, SOCP provides a simpler search space. It also has a broad range of applications, particularly in areas such as distribution network optimization and resource allocation [28,29].

The complex fault recovery model can also be reformulated into a SOCP model, as follows:

$$\min \left\{c^{T}x\left|Ax=b,x\in K\right.\right\}$$

(20)

where $Ax=b$ represents the set of linear constraints, while $x\in K$ indicates the set of nonlinear cone constraints.

SOCP models necessitate that the objective function remains linear throughout the computation process. Consequently, in the context of power restoration models, SOCP relaxation is employed to transform constraints involving products into second-order cone constraints. The core principle of second-order cone relaxation lies in the introduction of intermediate variables to replace nonlinear terms. Additionally, the original equality constraints are relaxed into inequality constraints. The specific implementation is described as follows:

New variables $I_{ij}$ and $U_i$ are introduced to represent the square of branch current $I_{ij}^2$ and the square of nodal voltage $U_i^2$, respectively. This substitution linearizes the quadratic terms in the original power flow equations. The modified variables are substituted into Equations (10)–(13). This leads to linearized power balance constraints, which correspond to Equations (21)–(24) in the paper.

$${\displaystyle \sum _{k\in v(j)}{\alpha }_{jk}{P}_{jk,t}-}{\displaystyle \sum _{i\in u(j)}(}{\alpha }_{ij}{P}_{ij,t}-{\alpha }_{ij}{I}_{ij,t}{r}_{ij})=P_{j,t}$$

(21)

$${\displaystyle \sum _{k\in v(j)}{\alpha }_{jk}{Q}_{jk,t}-}{\displaystyle \sum _{i\in u(j)}(}{\alpha }_{ij}{Q}_{ij,t}-{\alpha }_{ij}{I}_{ij,t}{x}_{ij})=Q_{j,t}$$

(22)

$$U_i-U_j+(r_{ij}^2+x_{ij}^2)I_{ij}-2(r_{ij}{P}_{ij}+x_{ij}{Q}_{ij})=0$$

(23)

$$I_{ij}{U}_i=P_{ij}^2+Q_{ij}^2$$

(24)

Then, the SOP operation constraint Equations (2)–(5) are transformed into Equations (25)–(27) by rotating cone constraint. Finally, the second-order cone constraint relaxation is performed on the system power flow constraint Equation (24), and Equation (28) is obtained.

$${(P_i^{SOP})}^2+{(Q_i^{SOP})}^2\le 2{\displaystyle \frac{P_i^{SOP,L}}{\sqrt{2}{A}_i^{SOP}}}{\displaystyle \frac{P_i^{SOP,L}}{\sqrt{2}{A}_i^{SOP}}}$$

(25)

$${(P_j^{SOP})}^2+{(Q_j^{SOP})}^2\le 2{\displaystyle \frac{P_j^{SOP,L}}{\sqrt{2}{A}_j^{SOP}}}{\displaystyle \frac{P_j^{SOP,L}}{\sqrt{2}{A}_j^{SOP}}}$$

(26)

$${(P^{SOP})}^2+{(Q^{SOP})}^2\le 2{\displaystyle \frac{S_{\max }^{SOP}}{\sqrt{2}}}{\displaystyle \frac{S_{\max }^{SOP}}{\sqrt{2}}}$$

(27)

$${‖{\left[2P_{ij} 2Q_{ij} i_{ij}-u_i\right]}^T‖}_2\le i_{ij}+u_i$$

(28)

After second-order cone relaxation, the nonlinear optimization model is converted to a SOCP model. Current computational tools can compute this model efficiently. However, the constraints after relaxation expand the feasible region of the original power system model. The solution obtained from the relaxed model must satisfy the constraints of the original problem. Therefore, the relaxation gap, denoted by Equation (29), must be examined. If the gap is sufficiently small—for instance, smaller than a predefined threshold g—the relaxation is considered exact. Under this condition, the solution is deemed feasible for the original problem.

$$g={‖i_{ij}-{\displaystyle \frac{P_{ij}^2+Q_{ij}^2}{u_i}}‖}_{\infty }$$

(29)

To highlight the advantages of the second-order cone programming (SOCP) method adopted in this paper, this section compares it with two mainstream optimization approaches: (1) exact solution via mixed-integer nonlinear programming (MINLP), and (2) approximate solution via mixed-integer linear programming (MILP).

All simulations are performed in the MATLAB environment. The optimization models are formulated using the YALMIP modeling toolbox. The high-performance commercial mathematical optimization solver Gurobi is employed for solution. The numerical computations are conducted on a hardware platform with an Intel Core i7-12700H CPU @ 2.70 GHz and 16 GB RAM (Intel Corp., Santa Clara, CA, USA).

Under the same fault scenario in the IEEE 33-node system, the optimization results and computational performance of the three methods are compared in

Table 1. Comparison of Optimization Methods.

Optimistic Method	Load Recovery Ratio (%)	Computing Time (s)	Duality Gap (%)
MINLP	-	>3600	N/A
MILP	98.55	15.2	1.45
SOCP	100.00	8.7	0.00

.

As can be seen from the table, the original MINLP model fails to converge to any feasible solution within a one-hour computation time. This highlights the computational complexity of directly solving nonconvex problems. In contrast, the SOCP method proposed in this paper obtains a globally optimal solution within seconds. Its computational efficiency is significantly superior. The MILP method introduces model errors due to its linear approximations. This leads to a lower load restoration rate compared to the SOCP approach. Although MILP exhibits relatively fast solution speed, it remains slower than the SOCP method. These results demonstrate that the proposed SOCP method outperforms the MILP approximation in both computational speed and solution quality.

5. Experiment and Analysis

5.1. Analysis of the Impact of Photovoltaic Power Output Randomness

The IEEE 33-bus system is a typical radial network composed of a main feeder with successive branches. It is well-suited for studies on distribution network faults and reconfiguration. Therefore, the proposed reconfiguration method is validated by constructing a mathematical model of the standard IEEE 33-bus distribution system. The study begins by verifying the negative impact of the stochastic nature of PV generation on the effectiveness of network reconfiguration. The system operates at 12.66 kV, with a total active load of 3715 kW and a total reactive load of 2300 kVar, which are the standard load values for IEEE 33-bus systems. After fault isolation, the loads at buses 3–18 and 23–33 were completely disconnected, resulting in a loss of 3255 kW of active load. A photovoltaic generator with a capacity of 1 MVA was installed at bus 30. During the fault period, if the PV output is assumed to be constant, it is set to the rated value with a power factor of 0.95. The standard deviation σ_pv of the random error is set to 10% of the forecasted value, corresponding to 50 kW. The actual output follows a normal distribution, denoted as P_PV~N(500,50²) kW. In probabilistic power flow calculations, Monte Carlo simulation is employed with 500 random samples. This approach is used to evaluate the statistical performance of the strategy. If the randomness of the PV output is taken into account, the PV power is set to 500 kW, the power factor is set to 0.95, and the standard deviation of the PV output and the loads is set to 10% of the rated power. As shown in Figure 6, a sample size of 500 is used in the probabilistic tidal current calculation and the confidence level is set to 99.21%. In this example, the PV capacity is set to 500 kW, which is mainly used to verify the influence of photovoltaic randomness on the recovery model. In the actual scenario, the photovoltaic penetration rate may be much higher than this value.

Since the recovery of power supply to the load is the primary objective of the model calculation, the system load recovery coefficient ηD under the fault condition should be much larger than the system loss coefficient ηL, where ηD is set to 100 and ηL is set to 1. Different scenarios are set up for the simulation analysis and the results are shown in

Table 2. Comparison of Network Reconfiguration Without SOP Considering the Randomness of PV Generation.

Case	Load Recovery Ratio (No Randomness)	Load Recovery Ratio (Randomness)	Outage Nodes
Case 1	95.34%	87.12%	17
Case 2	90.89%	90.89%	17

.

Case 1: The PV power output is set to the rated value without considering the randomness of PV power generation. The network is reconfigured using the tie switch to restore the power supply to the non-faulty area. This is shown in Figure 7.

Case 2: Consider the randomness of PV power generation. Reconfigure the network using the tie switch to restore power to the non-faulted area. As shown in Figure 8.

This paper compares the reconfigured network topologies obtained for two scenarios through simulation analysis: one that does not account for PV stochasticity, and another that incorporates it. The adaptability of the reconfigured networks to PV stochasticity in each scenario is evaluated through comparative analysis.

The results indicate that the theoretical load recovery rate in Case 1, which does not account for the stochastic nature of PV generation, is 95.34%. However, PV distributed generation is inherently random and fluctuating. It means that network reconfiguration without considering this stochasticity fails to meet actual demand. Specifically, after the fault, the voltages at nodes 3, 4, 5, and 18 do not satisfy the required standards, and node 17 experiences a complete blackout. As a result, the actual load recovery rate is only 87.12%. In Case 2, which incorporates the stochastic nature of PV generation, both the theoretical and actual load recovery rates improve to 94.89%. This represents an improvement over Case 1, highlighting the necessity of considering PV output randomness in practical engineering applications.

Figure 9 demonstrates the impact of PV generation stochasticity on the recovery of system node voltages. It can be observed that the lower voltage limits at some nodes fall below 0.95 p.u., which is lower than the allowable minimum. To evaluate the robustness of Case 1 (the reconfiguration strategy ignoring PV uncertainty), its effectiveness under actual PV fluctuations is analyzed in

Table 3. Voltage Variation in Nodes in Scheme 1 Network Reconfiguration Model Before and After Considering PV Randomness.

Case	Whether to Consider Randomness	Nodes Exceeding the Voltage Upper Limit	Nodes Exceeding the Voltage Lower Limit
Case 1	No	None	None
Case 1	Yes	None	2, 22

. This strategy maintains voltage stability when PV output remains constant. However, when PV output fluctuates randomly, voltage quality deteriorates and leads to violations of nodal voltage limits, as shown in Table 2. These results indicate that the strategy cannot adapt to PV output variations and exhibits poor robustness. If the power restoration model fails to consider the stochastic nature of PV generation output, actual load recovery may fall short of the expected value. This can lead to voltage quality issues at certain nodes, undermining power supply reliability, causing economic losses, and posing security risks.

The instability of PV distributed generation arises from the stochastic variation in solar radiation and its diurnal cycle. Peak output typically occurs around noon on sunny days, while output is minimal at night or during adverse weather conditions. This variability significantly reduces the adaptability of the reconfigured network to PV generation. At any given time, a fixed network reconfiguration topology may lead to fluctuations in power supply resilience due to the changing output of PV generation.

To mitigate the effects of disturbances, system configurations must include operating margins, which themselves can reduce power utilization efficiency. SOP can modulate the tidal current, respond quickly to PV fluctuations, and adjust power allocation to improve energy utilization efficiency. In the following section, the effectiveness of integrating SOP with network reconfiguration models is verified.

5.2. SOP’s Impact on the Power Restoration Effect of the Reconfiguration Case

This study evaluates the improvements in power restoration capability provided by the SOP. It further verifies the effectiveness of the network reconfiguration methodology and compares the SOP with a traditional tie switch using the IEEE 33-bus distribution system. In this analysis, the SOP replaces the tie switch TS1 and operates with a loss factor of 0.02. It allows a permissible voltage fluctuation range of ±5% and has a capacity of 2 MW. The minimum unit voltage value on the faulted side is set to 1.0. Three groups of photovoltaic power generation systems are connected to the IEEE 33-bus test example. The location and capacity are shown in

Table 4. Configuration parameters of distributed power supply.

Parameter Name	Numerical Value
Access Position	7	13	30
Capacity /kVA	300	200	200

. Details are presented in Figure 10. The stochastic nature of PV power generation is also considered. Under fault conditions, the PV power output is set to 500 kW with a power factor of 0.95, The standard deviation σ_pv of the random error is set to 10% of the forecasted value, corresponding to 50 kW. The actual output follows a normal distribution, denoted as P_pv~N(500, 50²) kW. In probabilistic power flow calculations, Monte Carlo simulation is employed with 500 random samples. This approach is used to evaluate the statistical performance of the strategy, and the confidence level is set at 99.21%.

In addition, although multiple SOPs can theoretically be deployed at different locations, this study considers only a single SOP node for the following reasons:

• The study focuses on the restoration of a single-feeder fault, where one SOP is sufficient to regulate power flow and support service restoration.
• Multiple SOPs require coordinated control, which increases the complexity of optimization and control algorithm design.
• In practical applications, SOPs are expensive and require proper site selection and control system integration. In this study, the SOP is introduced solely for its power flow regulation capability. A single SOP, once appropriately placed, can effectively support large-scale power restoration and flow regulation, thereby improving energy utilization efficiency.

Therefore, to ensure fault recovery while highlighting the advantages of SOP in power flow control and restoration efficiency, only one SOP is deployed in this study.

To highlight the advantages of the proposed SOP-assisted network reconfiguration method, several comparative cases are designed. These include: traditional tie-switch-based reconfiguration (case 1), SOP-only restoration (case 2), and conventional reconfiguration without considering stochastic variations.

A total of four cases are selected to conduct comparative experiments on service restoration performance. In all cases, the stochastic characteristics of photovoltaic (PV) generation are taken into account.

Case 1: Use the tie switch 8–21 for restoring power to the outage area.

Case 2: Use the SOP installed between nodes 8–21 for power restoration in the outage area.

Case 3: Apply the network reconfiguration scheme to analyze the load restoration plan and evaluate its restoration results.

Case 4: Combine the application of SOP with the network reconfiguration scheme to verify that SOP can improve the load restoration effect of the network reconfiguration.

The simulation results are obtained after analysis and calculation. Both Case 3 and Case 4 involve changes to the system’s topology. The configuration of Case 3 is illustrated in Figure 8, while the topology of Case 4 is depicted in Figure 11. The restoration results for all cases are summarized in

Table 5. Comparison of Power Restoration Results for Each Case.

Case	Total Restored Load/KW	Load Recovery Ratio/%	Partial Power Loss Node	Total Power Loss Node
Case 1	1322.1	40.6	3–5, 28, 32, 11–14	15–18, 23–26
Case 2	2001.3	61.5	16, 17, 25, 26	18
Case 3	2958.5	90.89	23	17
Case 4	3255	100	-	-

. Additionally,

Table 6. SOP Control Mode.

Case	SOP Converter	Control Mode	Parameters
Case 1	VSC1-21	P/kW	−1596.5
		Q/kWar	159
	VSC2-8	U	1.05
		f/Hz	50
Case 2	VSC1-21	P/kW	−1369.2
		Q/kVar	421
	VSC2-8	U	1.05
		f/Hz	50

presents the SOP control modes and the corresponding converter control parameters for Case 2 and Case 4.

By comparing the results of Case 1 and Case 2, the load recovery ratio of Case 2 is nearly 20% higher than that of Case 1. The number of partially or completely failed nodes is also relatively fewer. This confirms that SOP, compared to traditional tie switch, can restore system power supply more effectively. The SOP can more efficiently provide power to the needed areas.

In Case 3, compared to the first two cases, the load recovery ratio significantly increases to nearly 91%, with a substantial reduction in failed nodes. This proves the effectiveness of the reconfiguration scheme. Even considering the random output of the photovoltaic system, the power supply recovery to the non-faulted sections can still be achieved with a high load recovery ratio.

In Case 4, the network reconfiguration combined with SOP restores 100% of the power. This is in contrast to using only tie switch for network reconfiguration. It ensures that all node voltages remain within the set voltage range. This significantly improves the system’s power supply recovery capacity and enhances the self-healing ability of the system. Additionally, the relaxation deviation of the reconfiguration scheme is within the allowable range, ensuring accurate and effective computation.

By introducing SOP, the issue of full load recovery has been resolved. The system voltage level has been improved. All node voltages are maintained within the allowed range. This fully demonstrates the role of SOP in voltage support. Combined with the network reconfiguration scheme, the power flow control function of SOP is fully utilized. It achieves load recovery in non-faulted areas after a fault.

5.3. Sensitivity Analysis

To further verify the universality and robustness of the method proposed in this paper in practical applications. This paper conducts a sensitivity analysis on the key parameters of photovoltaic penetration rate and SOP capacity [30,31]. The basic scenarios are the same as those in Section 5.2.

Analysis of PV Penetration Impact:

While the SOP capacity is maintained at 2 MW, the total installed capacity of PV generation is adjusted to simulate different penetration levels (ranging from 20% to 80% in increments). For each penetration level, the load restoration ratio and the minimum node voltage under Scheme 4 (SOP combined with network reconfiguration) are computed. The results are presented in Figure 12.

As shown in Figure 12, when the PV penetration rate remains below 60%, the proposed method consistently achieves 100% load restoration while maintaining system voltage within the acceptable range (0.95–1.10 p.u.). This demonstrates that the model retains excellent restoration performance under medium to high PV penetration levels. However, when the penetration rate exceeds 60%, the load restoration ratio begins to decline, and the minimum node voltage exhibits a noticeable drop. This occurs due to increased volatility in PV output, which—under extreme scenarios such as sudden power reduction—leads to power deficits that exceed the regulation capacity of both the SOP and the system itself. It should be noted that, in current practice, PV penetration on individual feeders is typically limited to relatively low levels to avoid issues such as voltage violations, protection maloperations, and reverse power flow. Therefore, the proposed method demonstrates strong adaptability to both current and near-future scenarios with high shares of renewable energy, indicating considerable practical applicability.

Analysis of SOP Capacity Impact:

Based on the simulation setup described in Section 5.2, the PV penetration rate is fixed at 40% (with an installed capacity of approximately 1570 kVA), while the rated capacity of the SOP is varied from 1 MW to 3 MW in steps. For each capacity value, the load restoration ratio and system power loss under Scheme 4 (SOP combined with network reconfiguration) are calculated. The results are summarized in

Table 7. Effect of SOP capacity on fault recovery.

SOP Capacity/MVA	Active Load Recovery Amount/kW	Load Recovery Ratio/%	System Network Loss/kW
1	3078.855	83.10	340.12
2	3715.0	100	301.42
3	3715.0	100	213.33

.

As shown in Table 7, an increase in SOP capacity significantly enhances the power supply restoration capability. When the capacity is 1 MW, the power transfer capability to the de-energized area is limited. As a result, full load restoration cannot be achieved. When the capacity reaches 2 MW, 100% load restoration is achieved. A further increase to 3 MW does not improve the restoration performance. However, the system power loss is further reduced due to more flexible power flow control by the SOP. These results indicate the existence of a critical capacity value. Beyond this value, larger SOP investments do not contribute to better restoration. The main benefit then lies in the improvement of operational economy.

6. Conclusions

SOPs, as advanced power electronic devices, can provide flexible current regulation, reduce line losses, and improve system voltage quality. With the increasing popularity of distributed energy sources, SOPs can effectively solve the problem of limited consumption and accommodation of renewable energy sources in the distribution network, bringing great advantages to the overall performance and reliability of the distribution system.

Network reconfiguration optimizes energy distribution paths by adjusting the states of switches, thereby enhancing system reliability, reducing network losses, and improving power quality. This study integrates SOPs with network reconfiguration to achieve efficient power restoration in distribution networks. Compared to traditional tie switch, SOPs offer significant advantages, including simplified control, enhanced reliability, and an expanded range of power restoration.

This study examines four scenarios: network reconfiguration, direct use of tie switch, direct use of SOPs to supply power to non-faulted zones, and the combination of SOPs with network reconfiguration. The results validate the effectiveness of the network reconfiguration strategy and demonstrate that integrating SOPs with reconfiguration significantly expands the range of power restoration. The model incorporates the stochastic characteristics of distributed energy sources, such as photovoltaic output, to evaluate their impact on the power restoration capability of the network reconfiguration approach.

Finally, the study demonstrates the effectiveness of the SOP and network reconfiguration combined model in expanding the power restoration range.

7. Discussion

Simulation results show significant differences in load restoration rates among the cases: Case 4, which integrates the SOP and stochastic PV modeling into the network reconfiguration strategy, achieves 100% load restoration and powers all nodes. The integration of the SOP and consideration of PV fluctuations significantly improve the fault recovery performance. Figure 13 is the flow chart of this method.

This improvement can be attributed to two main reasons: Traditional reconfiguration relies on tie switches for interconnection, which cannot effectively regulate power flow direction. This often leads to limited restoration or uneven power distribution. In contrast, the SOP flexibly controls power flow and optimizes power distribution. It compensates for the shortcomings of traditional methods and enhances the power supply restoration capability.

The randomness and volatility of distributed generation, such as PV, are often ignored in traditional reconfiguration. This omission can easily cause power quality degradation and voltage limit violations. By modeling these fluctuations in advance, the power quality of the reconfigured system is improved. Moreover, the robustness and adaptability of the restoration strategy are enhanced.

Sensitivity analysis indicates that restoration performance declines when PV penetration exceeds a certain level. However, increasing the SOP capacity can maintain voltage stability and achieve full restoration.

In practical engineering applications, this method can effectively enhance the hosting capacity of distributed generation in scenarios with high renewable energy penetration. It also assists grid operators in developing more robust fault restoration strategies. The approach helps improve power supply reliability, reduce outage duration and affected areas, and ensure relatively stable electricity supply for customers even during faults.

Furthermore, the proposed method is applicable not only to the IEEE 33-node system but can also be extended to other typical distribution networks. For larger or structurally more complex systems, the same restoration objectives can be achieved by appropriately extending the spanning tree constraints, multi-feeder topology modeling, and operational constraints for multiple SOPs or diverse distributed energy resources. More efficient optimization algorithms can be incorporated to maintain computational performance. This provides a valuable reference for future research and development of self-healing capabilities in smart distribution networks with high penetration of renewable energy.

It should be noted that SOP deployment requires consideration of practical factors such as equipment cost, communication conditions, and control responsiveness. Although the initial investment in SOPs is relatively high, their value in improving power supply reliability and renewable energy integration can partially offset the cost. Regarding communication, the method mainly relies on simple real-time data such as power outputs and line statuses. The required data volume is small, and bandwidth demands are low, provided that communication remains stable and reliable. In terms of temporal response, an hourly scheduling resolution is adopted in this study. PV output is treated as constant within each time period, reducing the dependence on ultra-fast real-time communication and control. This feature enhances the engineering feasibility of the proposed approach.

Although this study has achieved positive results, several limitations remain, which also indicate directions for future research:

A constant power (PQ) load model and an ideal communication system are assumed in this study. Dynamic voltage-dependent characteristics of loads and potential delays or failures in communication systems are not considered. These simplifications may affect the practical applicability of the strategy under extreme conditions.

The current research is limited to single-feeder systems with a single SOP. Future work should extend the approach to coordinated control in multi-feeder systems with multiple SOPs. This extension would better reflect the complexity of real-world applications. However, it also introduces challenges related to cooperative control among multiple devices.