Emergency Power Supply Restoration Strategy for Distribution Network Considering Support of Microgrids with High-Dimensional Dynamic Correlations

Abstract

With the rapid development of renewable energy, microgrids are becoming more and more essential in distribution networks. However, uncertainties brought by new energy sources have posed great challenges to the energy safety and stability of distribution networks, especially in the process of fault restoration. To address these issues, this paper investigates an emergency power supply recovery strategy for distribution networks by considering the support capability of microgrids and the high-dimensional dynamic correlations of uncertainty. Firstly, the structure of distribution networks including microgrids is analyzed. Then, the high-dimensional dynamic vine copula model is proposed to model the joint output uncertainty process. Next, island partition and operation models are established, including the objective function and constraint conditions, and the model solving methods are also presented. Finally, an example analysis is conducted to verify the effectiveness of the proposed strategy. The results show that the proposed strategy is able to deal with the uncertainties brought by renewable energy sources in distribution networks with microgrid support, improving energy safety and stability. This research provides valuable insights for the development of emergency power supply strategies in distribution networks.

1. Introduction

In recent years, power outages caused by natural disasters and malicious attacks have become more frequent, resulting in huge economic losses and serious social impacts [1]. For the distribution system, with the rapid development of new energy technologies [2], distributed generation (DG) such as wind power and photovoltaic power generation is increasingly connected to the distribution network, which makes the structure of the distribution network larger and the power flow more complex, presenting huge challenges to the troubleshooting of the distribution network [3]. The traditional passive waiting time for fault elimination is up to dozens of hours, the use of fault reconstruction technology can significantly shorten the outage time [4], making the rapid repair technology of the distribution network more and more important [5,6]. The distribution network response to extreme faults is a multi-stage decision process, including fault diagnosis and isolation, islanding operation, and maintenance and restoration. When a fault occurs in multiple areas (distribution system and superior grid) and the superior grid is unable to supply power to the distribution network, the distribution network needs to integrate and coordinate the generation resources among them to form smaller distribution islands and guarantee the restoration of power supply for important loads. Reasonable islanding partition and emergency power supply restoration are important to reducing distribution network outage time and improving distribution network power supply reliability [7,8,9].

The islanding partition of the distribution network is a prerequisite for emergency power supply restoration. The islanding partition and operation of the distribution network not only helps to maintain the operation of important loads, but also effectively prevents further expansion of faults and causes chain failures [10]. The literature [11] initially classified distribution islands based on the supply paths of DGs and nodes, and proposes an islanding topology adjustment strategy that takes into account demand-side response and static security constraints. The literature [12] proposes a multi-objective model for islanding considering demand response by taking into account the importance of user loads, controllability and uncontrollability, and demand response. The literature [13] proposed a distribution system islanding partition strategy for restoring critical loads based on virtual power flow to establish the connectivity and radiative constraints of distribution islands. The literature [14] introduces a rooted tree model with DG distribution network, constructs a mathematical model of the islanding problem, and generalizes a new set of heuristic rules for islanding, and proposes a new islanding search method called "shrink circle method". A new islanding search method is proposed.The literature [15] proposed a distribution system resilience enhancement strategy integrating islanding partition, fault area identification, fault isolation, and emergency power restoration. The literature [16] constructed two objective functions, the average number of layers and the average degree value, based on the characteristics of the distribution network, and proposes a method for determining the priority of branches and nodes to solve the multi-objective optimization problem of islanding partition. The literature [17] considered multiple attributes of the load and user preferences in the islanding partition to reflect the comprehensive importance of the load, overcoming the shortcomings of considering only a single attribute of the load. The literature [18] had created resilient islanded microgrids to withstand unexpected local disturbances and power mismatches by islanding partition with the goal of maximizing the load capacity in the islands. The literature [19] used the improved harmony search algorithm to find the best line switch position in the distribution system to reconfigure the distribution network. On this basis, literature [20] combined with bacterial foraging algorithm established a multi-objective optimization model, which can maximize the number of safe operation buses while restoring power supply. Reference [21] further considered the soft open points (SOPs) existing in the distribution network and established a deterministic model to achieve the best coordination between SOP and distributed resources in the distribution network, which can quickly restore the power supply to key loads. In addition, reference [22] introduced emergency power vehicles and proposed a coordinated recovery optimization method for distribution network reconfiguration under the condition of fully considering the uncertainty of emergency power vehicles (EPVs), which can make full use of the recovery capacity of EPV and utility grid.

In the stage of distribution network fault recovery, it is still necessary to make full use of various resources of the distribution network, such as energy router [23], mobile energy storage [24], and DGs, so as to achieve as much load power supply recovery as possible. The literature [25] improves the adaptability of the distribution network in extreme weather by pre-placing the mobile power supply before the accident and dynamically dispatching it after the accident. The literature [26] further considered the characteristics of the traffic network and integrated the mobile resource model into the distribution system recovery model to jointly optimize the scheduling of DG and mobile resources. The [27] literature considers the microgrid access to the distribution network so that more security factors are considered for fault recovery, and establishes a fault recovery model for distribution network with microgrids by taking the minimum network loss, the minimum number of switching actions and the minimum amount of lost power as the objective function, and combines the improved binary particle swarm algorithm and genetic algorithm for model solving. The literature [28] proposed a self-healing fault reconstruction strategy for the island emergency power grid based on the Floyd modified particle swarm optimization (MPSO) algorithm, aiming at the minimum path cost of mobile resources and the maximum recovery of important loads it discussed the key impact of DG startup sequence on the recovery path. The literature [29] considered a variety of distributed resources, such as photovoltaic, mess, fixed energy storage, integrated energy system (IES), and diesel generator (DEG), and described the coupling principle between electric energy, transportation, and natural gas based on the time–space state model of mess and the time series model of IES. The multi-stage restoration strategy of distribution system is proposed to enhance the flexibility of the system. The literature [30] further considered the variability and scarcity of power generation resources in the microgrid after a fault, took the reserve capacity status as the limiting factor of the recovery problem, and proposed an elastic oriented hierarchical service recovery method. In [31], considering the power support function of mobile energy storage, the mobile energy storage configuration node is dynamically optimized to restore the load to the greatest extent. Reference [32] proposed a power supply recovery strategy considering precise load control to solve the problem of insufficient power supply capacity of distribution network in the case of serious fault. The literature [33] used a variety of distributed generation and energy storage systems to quickly restore important loads, and based on this, proposed a post disaster power supply restoration model for active distribution networks aimed at reducing the economic losses caused by power outages from extreme events and improving the resilience of distribution networks. Reference [34] established a multi-energy flow network model with three aspects of electrical and thermal linkage, and proposed a distribution network fault recovery strategy considering the synergy of multiple energy sources.

The above literature describes DG output as an adjustable amount between the minimum and maximum output, without considering the uncertainty of some DG output. However, with the access of new energy generation such as wind power and photovoltaic, the emergency power restoration of distribution networks must consider the uncertainty of the output of these intermittent generation resources to avoid the failure of restoration strategies due to the randomness of their output. The literature [35] integrated the metrics of maximizing self-sufficiency and maximizing islanding partition success rate in islanding partition, and considers the uncertainty of renewable energy generation and the uncertainty in load distribution, based on which the backtracking search optimization algorithm, probabilistic power flow method, and graph theory are combined to achieve islanding and emergency power restoration. The literature [36] combined the Markov state-space method and the Monte Carlo simulation method for solving the uncertainty islanding partition problem. The literature [37] similarly considered the uncertainty of renewable energy and demand, transformed the uncertainty constraints into deterministic constraints based on robust discrete optimization theory, and constructed a mixed integer linear programming islanding partition and emergency power restoration model. The literature [38] used a minimal risk criterion to measure risk for different island partitioning scenarios and developed an islanding partition scheme that minimizes the risk of failure of island operations in uncertain environments. The literature [39] determined the composition of each initial island based on the Latin hypercubic hyper sampling method, deterministic tree knapsack algorithm, and mathematical statistical methods, and guaranteed the optimality, safety, and economy of DG island operation by using the optimization of random optimal power flow based on the consideration of island reconfiguration. The literature [40] proposed an active output control strategy that accounts for prediction errors for intermittent renewable energy generation to ensure that the operation strategy of distribution islanding is still feasible when there is a significant error in the active output prediction of wind and solar generation.

Generally speaking, the islanding partition and operation process of the distribution network needs to fully utilize the power generation capacity of DG or other adjustable resources to achieve emergency power supply restoration for important loads. The literature [41] fully considered the output characteristics of small hydro power stations, gas power stations, energy storage systems, wind farms and photovoltaic power stations, and constructed a rolling optimization model for the real-time operation of distribution islands considering AC power flow constraints, radiation-like constraints, and steady-state security constraints of the distribution network. The literature [12] proposed an integrated reconfiguration and islanding partition fault recovery method by making full use of resources such as multiple types of DG, flexible loads, and energy storage, and considered the black-start capability of distributed power sources and energy storage, as well as fault recovery time and overhaul sequence. The literature [42] proposed a multi-objective model for islanding partition considering demand response, which considers the importance, controllability, uncontrollability, and the demand response of the user load. The literature [43] utilized the regulation of flexible loads and grid structures in the distribution network, combined with the output of DG, to achieve more reasonable and efficient power supply islanding partition The literature [44] divided the active distribution network into multiple islands powered by DGs or DGs in concert with energy storage by controlling the switching states of switchgear, DGs, and energy storage to maximize the equivalent restored load. Reference [45] proposed a two-stage optimal islanding partition and emergency power supply restoration method that integrates photovoltaic, energy storage, and electric vehicles.

The above literature studies the problem of emergency power supply restoration after distribution network failure from various angles, which is of great significance to ensure the safe and stable operation of distribution networks and bulk grid. However, the existing studies still have three shortcomings as follows:

First, although some distribution network emergency power restoration methods consider the uncertainty of new energy sources in the modeling process, there is little in the literature which considers the correlation of different new energy stations. In the distribution network, the geographical distance between different nodes is limited, and the power output of the same type of new energy (wind power or PV) shows strong correlation characteristics, i.e., the magnitude and change trend of the power output of the same type of new energy are very similar at the same moment. If the correlation of the same type of new energy output is not considered, it is easy to cause the new energy forecast interval to be too large, which results in too many controllable resources in the distribution network to cope with the new energy uncertainty and, therefore, reduction the system economy.

Secondly, most of the existing literature uses DG units for emergency support of the distribution network, and rarely consider microgrids for active support of the distribution network. Using microgrids as an effective means of accommodating distributed power sources and connecting DG units into the distribution network through microgrids can reduce the impact on the security and stability of the distribution network. It is necessary to study emergency power supply recovery strategies for distribution networks that consider active support from microgrid clusters.

Thirdly, the existing distribution network emergency power restoration models are mostly based on the new energy and load data before the failure and the new energy and load forecast data within a fixed period of time (e.g., 24 h) after the failure to make the islanding partition and operation scheme. However, fault repair time is stochastic in nature, and fault recovery may be achieved within a short period of time (e.g., within 24 h) or remain unrealized for a long period of time (e.g., outside of 24 h). Therefore, it is not always optimal to use predicted data within a fixed time period to develop an islanding partition and operation scheme.

In view of the above problems, this paper proposes an emergency power supply restoration strategy for the distribution network considering support of microgrids with high-dimensional dynamic correlations. First, uncertainty modeling of new energy sources taking into account high-dimensional dynamic correlations is performed; second, a microgrids active support distribution network emergency power restoration model is constructed using a rolling optimization approach; and the finally, the effectiveness of the proposed method is verified by example analysis.

2. Structure of Distribution Network Including Microgrids and Uncertainty Modeling

2.1. Structure of Distribution Network including Microgrids

The distribution network structure, including a microgrid group, is a small-scale power system consisting of multiple microgrids that form a network of mutual support and joint operation. The typical topology structure of a distribution system network composed of six microgrids is shown in Figure 1. Each microgrid contains various distributed generation resources, such as micro gas turbines, wind power generation, and photovoltaic power generation as well as loads. In this structure, each microgrid can operate independently or in conjunction with other microgrids to ensure a more reliable and stable operation of the entire distribution network. In the event of natural disasters, emergencies, and other unforeseen situations, microgrids can operate autonomously to provide basic energy needs for local residents and act as an emergency power source for critical loads in the distribution network, ensuring a reliable power supply for important loads.

2.2. Uncertainty Modeling of New Energy Considering High-Dimensional Dynamic Correlations

In a distribution network, the geographical distance between different nodes is limited, and the output of the same type of new energy (such as wind power or photovoltaic) shows strong correlation characteristics. This means that at the same time, the magnitude and changing trend of the same type of new energy output are very similar. For example, the output data of two typical wind power systems in the same region are shown in Figure 2, which depicts the output curves of adjacent wind power. As shown in the figure, the change in trend of the output of two wind power systems in the same area is highly similar most of the time. However, new energy output has an element of uncertainty. If the correlation of the same type of new energy output is not considered, it can easily result in a wide range of new energy prediction, which can lead to too many controllable resources in the distribution network, adversely affecting the overall system economy. Furthermore, the correlation of the output of the same type of new energy is not fixed. Therefore, this paper takes wind power as an example and introduces dynamic copula theory and vine copula theory to establish a new energy uncertainty model that considers high-dimensional dynamic correlation.

2.2.1. Dynamic Copula Theory and Vine Copula Theory

Dynamic copula functions can be divided into elliptic and Archimedean groups. The elliptic families consist of dynamic Gaussian copula functions and dynamic t-copula functions, while the Archimedean families include dynamic Gumbel copula, dynamic Clayton copula, and dynamic Frank copula functions. The main difference between these two types of dynamic copula functions lies in whether their tail correlation is symmetrical. The dynamic copula function of the elliptic family generally has a symmetric tail correlation, while the dynamic copula function of the Archimedes family usually has an asymmetric tail correlation. Specifically, the dynamic Clayton copula of Archimedes has an asymmetric tail correlation, where there is a high correlation in the lower tail and a low correlation in the upper tail, which effectively describes the joint force dependence of multiple wind farms near the origin. Relying on the characteristics of actual wind power data collected in this paper, the dynamic Clayton copula function is used to establish a joint distribution model as follows:

$$C_C(u,v)=\max [{(u^{-{\theta }_{C.t}}+v^{-{\theta }_{C.t}}-1)}^{-1/{\theta }_{C.t}},0]$$

(1)

where u and v are marginal distribution functions, and ${\theta }_{C.t}$ is the dynamic correlation coefficient.

For a certain variety of Copula functions, the distribution characteristics are determined by the correlation coefficient inside the function. Therefore, the key step in establishing the copula model is obtaining the correlation coefficient, which is typically achieved through maximum likelihood estimation methods. While the correlation coefficient of the static copula function is a fixed value, the dynamic copula function introduces an evolutionary equation which changes the calculation of the correlation coefficient to the computation of the parameters of the evolutionary equation. Specifically, the evolutionary equation expression for the dynamic Clayton copula function is:

$${\theta }_{C.t}=\tilde{\Lambda }(\omega +\beta {\theta }_{t-1}+\alpha \frac{1}{10}{\displaystyle \sum _{j=1}^M|u_{t-j}-v_{t-j}|})$$

(2)

where $\omega$, $\beta$, and $\alpha$ are parameters to be estimated. Among them, $\omega$ is an invariant parameter, $\beta$ controls the progressive relationship of the correlation coefficient, and $\alpha$ controls the impact of the first M sets of input sequences on the correlation coefficient. The function $\tilde{\Lambda }(x)\equiv (1-e^{-x})/(1+e^{-x})$ is a logistic function, which is introduced to ensure that the correlation coefficient is between −1 and 1.

The evolutionary equation transforms the static $\theta$ into a dynamic ${\theta }_{C.t}$, and the corresponding likelihood function shifts from being a function of the correlation coefficient $\theta$ to a function of the evolutionary equation coefficients, $\omega$, $\beta$, and $\alpha$. Solving this likelihood function gives the maximum likelihood estimation values of $\omega$, $\beta$, and $\alpha$, thus obtaining a precise dynamic copula model.

The pair-copula theory serves as the foundation of the vine copula theory, which involves decomposing the joint density function of multiple variables in pairs into a series of copula functions and edge distribution functions based on a specific topological structure. With high-dimensional joint distributions, there are multiple topologies for pair-copula decomposition; therefore, vine copula theory is specifically used to describe these topologies. C-vine and D-vine are the two most widely used types of vine structures, which are suitable for different types of data combinations. When the correlation of one variable in the data with all other variables is particularly strong and can serve as a guide variable, C-vine is appropriate. On the other hand, when the variables have some relationships, but others are independent, they can be described by D-vine. Based on the collected wind power data in this study, the D-vine structure is utilized for modeling, and the D-vine topology diagram based on the paper’s data characteristics is illustrated in Figure 3.

Based on the topology structure, the joint density function of the D-vine structure can be decomposed as

$$\begin{array}{l}f(x_1,x_2,x_3,x_4,)={\displaystyle \prod _{k=1}^{4}f(x_k)}{\displaystyle \prod _{j=1}^{n-1}{\displaystyle \prod _{i=1}^{n-j}}}\\ c_{i,j+i|i+1,\dots ,i+j-1}(F(x_i|x_{i+1},\dots ,x_{i+j-1},),F(x_{j+i}|x_{i+1},\dots ,x_{i+j-1},))\end{array}$$

(3)

where $F(x_i)$ is the probability distribution function of the probability density function $f(x_i)$.

As can be observed from Figure 3, the D-vine constructs the copula function by pairing the predicted values and prediction errors of Wind Farm 1 and Wind Farm 2, then links the four sets of data via their predicted values.

2.2.2. Modeling of High-Dimensional Dynamic Vine Copula

In this paper, the maximum likelihood estimation method (MLK) is employed to generate the kernel density estimation function. The core concept of the MLK method is to substitute the non-parametric kernel density estimation function of each variable for its edge distribution function, and the expression of the kernel density estimation method is given by

$$f(w_i)=\frac{1}{T}{\displaystyle \sum _{t=1}^T{K}_h(w_i-x_t)}$$

(4)

where $K_h$ is the kernel function, T is the sequence capacity, w_i represents the ith input sequence, and x_t represents the original data sequence at time t. For the static copula function, the likelihood function is obtained by substituting the edge distribution function $f(x_i)$, and its correlation coefficient can be determined by locating the extreme point of the likelihood function. The expression of the likelihood function is shown as

$$L(\theta )={\displaystyle \sum \ln c(f(x_1),f(x_2),f(x_3),f(x_4))}$$

(5)

$$\widehat{\theta }=\arg \max L(\theta )$$

(6)

where $\widehat{\theta }$ is the static correlation coefficient to be determined. By finding $\widehat{\theta }$, the corresponding copula model can be identified.

The kernel density estimation function is generated by the MLK method, and the corresponding kernel density estimation function maps the actual values of the predicted output and prediction error series to probability values, thereby achieving input sequence standardization while maintaining the original sequence’s volatility and correlation. The specific steps for constructing a high-dimensional dynamic vine copula model using the kernel density estimation function and input sequence are shown in Figure 4.

The specific steps are as follows:

Step 1: Read the original data sequences $x_1$, $x_2$, $x_3$, $x_4$, and calculate the kernel density estimation function.

Step 2: The original data sequences are scalarized to obtain the input sequences $w_1$,$w_2$, $w_3$, $w_4$.

Step 3: Substitute the input sequences $w_1$, $w_2$, $w_3$, $w_4$ into the copula function equation of each vine node to establish the mapping relationship between the dynamic correlation coefficient and the copula function.

Step 4: Based on the copula function established in step 3, construct a likelihood function based on the dynamic correlation coefficient.

Step 5: Substitute the input sequences $w_1$, $w_2$, $w_3$, $w_4$ into the evolution equation to establish the mapping relationship between $\alpha$, $\beta$, $\omega$ and the correlation coefficient ${\theta }_{C.t}$.

Step 6: Substitute the evolution equation into the likelihood function to construct the likelihood function $L(\theta )={\displaystyle \sum \ln c(f(x_1),f(x_2),f(x_3),f(x_4))}$ based on the parameters $\alpha$, $\beta$, $\omega$ of the evolution equation.

Step 7: Use MKL to calculate the maximum likelihood estimate LogL and the corresponding parameters of the evolution equation $\alpha$, $\beta$, $\omega$.

Step 8: The calculated $\alpha$, $\beta$, $\omega$ and the input sequences $w_1$, $w_2$, $w_3$, $w_4$ are substituted into the evolution equation to calculate the dynamic correlation coefficient sequences ${\theta }_{C.t}$ for each vine node.

Step 9: Calculate the copula function of each vine node based on the correlation coefficient sequences ${\theta }_{C.t}$ of each vine node and generate the copula joint distribution model.

2.2.3. Discrete Convolution Method Based on Copula Function

For the joint distribution density function of two continuous functions, the functional expression that translates into the probability density of the sum of the two variables is as follows:

$$f_{\alpha +\beta }(a)={\displaystyle {\int }_{\underset{\_}{a}}^{\overline{a}}{f}_{\alpha ,\beta }(x,a-x)dx},a\in [\underset{\_}{\alpha }+\underset{\_}{\beta },\overline{\alpha }+\overline{\beta }]$$

(7)

where $f_{\alpha +\beta }$ is the probability density function of sum and $f_{\alpha ,\beta }$ is the joint distribution density function.

For the joint distribution based on the copula function, this method is used in this paper to transform the high-dimensional dynamic vine copula model into a probability density function of the prediction error sum, and the convolution expression based on the copula function is as follows:

$$f_{\alpha +\beta }(a)={\displaystyle {\int }_{\underset{\_}{a}}^{\overline{a}}c(F_{\alpha }(x),F_{\beta }(a-x))f_{\alpha }(x)f_{\beta }(a-x)dx}$$

(8)

where $c(F_{\alpha },F_{\beta })$ is the copula density function of the variables $\alpha$ and $\beta$. To facilitate computer calculations, this expression is discretized to obtain the final discrete convolution equation:

$$P_{\alpha +\beta }(i)={\displaystyle {\sum }_{j=i_{\underset{\_}{a}}}^{i_{\overline{a}}}c({\displaystyle {\sum }_{m=0}^j{P}_{\alpha }(m)},{\displaystyle {\sum }_{n=0}^{i-j}{P}_{\beta }(n)})P_{\alpha }(j)P_{\beta }(i-j)}$$

(9)

2.2.4. Joint Output Uncertainty Modeling Process

According to the method and modeling process described above, the specific process of transforming the original data sequences into the prediction value uncertainty interval is shown in Figure 5.

The specific steps are as follows:

Step 1: Transform the input sequences into a high-dimensional dynamic vine copula model according to the high-dimensional dynamic vine copula modeling method described above.

Step 2: Substitute the predicted data into the copula model to obtain a two-dimensional error-joint error distribution function.

Step 3: Using the discrete convolution method, the error-joint error distribution function is transformed into a probability density function of the sum of errors.

Step 4: Generate confidence intervals for the sum of prediction errors with different probabilities based on the probability density function of the sum of errors.

Step 5: The confidence interval of the prediction error sum is superimposed on the predicted value to obtain the uncertainty interval of the predicted value, i.e., the model output.

3. Islanding Partition and Operation Model of Distribution Network

3.1. Objective Function

To better illustrate the model, first explain the variable subscripts and their corresponding sets in this paper. The subscript i represents node i; the subscript ij represents the line ij; the subscript k represents island k; the subscript t represents the time period t. Node i, line ij, island k, and time period t may belong to the set in

Table 1. The set which the subscript belongs to.

Subscript	Set Symbol	Set Meaning
Node i	N	Node set of distribution network
	Nk	The set of distribution network nodes contained in island k
	NS	The set of distribution network nodes connected to the main network
	NMG	The set of nodes containing microgrids in the distribution network
Line ij	E	The set of lines in the distribution network
	Ek	The set of distribution network lines contained in the island k
	EF	The set of faulty lines in the distribution network
	F	The set of contact switches for the distribution network
Island k	$\mathsf{\Omega }$	The set of divided island
Period t	T	Solution period for rolling optimization

.

The islanding partition and operation of the distribution network is a complex nonlinear optimization problem that requires simultaneous consideration of numerous objectives and constraints. In this paper, the following principles are considered when determining the islanding partition and operation scheme:

(1)

Important load priority principle: In order to ensure the stability and reliability of the power grid, it is necessary to classify the loads according to their importance and give them different weights. When a fault occurs, the power supply of important loads should be ensured first to guarantee the normal operation of the grid.

(2)

Maximum load principle: When assigning loads to individual islands, it should be ensured that the total load matches the supply capacity of the DGs in the islands in order to make full use of the DGs’ capacity and reduce the loads that lose power. At the same time, it is also necessary to avoid overloading of DGs.

(3)

The principle of minimum network loss: In order to ensure the economic operation of the island and supply as much power as possible to the load, the active loss of the island should be reduced as much as possible when developing the islanding plan.

(4)

The principle of the minimum number of switches: In the process of islanding partition, the line segmentation switches as well as the contact switches should be minimized.

According to the above principles, the objective function of distribution network islanding partition and rolling optimization operation in this paper is established as

$$\min {\displaystyle \sum }_{k\in \mathsf{\Omega }}{\sum }_{i\in N_k}{\sum }_{t\in T}{\alpha }_{i,k}{P}_{i,k,t}^{\mathrm{s}}+{\mu }_{\mathrm{loss}}{\sum }_{k\in \mathsf{\Omega }}{\sum }_{ij\in E_k}{\sum }_{t\in T}{R}_{ij}{I}_{ij,k,t}^2+{\mu }_{\mathrm{switch}}{\sum }_{t\in T}\left[{\sum }_{ij\in E}\left(1-y_{ij,t}\right)+{\sum }_{ij\in F}{y}_{ij,t}\right]$$

(10)

where $T=\left\{t,t+\mathsf{\Delta }T,\dots ,t+{\tau }_f\mathsf{\Delta }T\right\}$ denotes the solution period of rolling optimization, where ${\tau }_f$ is the step length of rolling optimization, i.e., the number of scheduling periods of the optimization model, and $\mathsf{\Delta }T$ denotes the scheduling time window; $E$ denotes the set of lines in the entire distribution network; ${\alpha }_{i,k}$ denotes the weight value assigned to the load of node i in island k, which may be set by the relevant government department based on factors such as the size of the load, the economic benefits created, and the impact on law and order and people’s lives; $P_{i,k,t}^{\mathrm{s}}$ denotes the active load to be removed at time t by node i in island k; ${\mu }_{\mathrm{loss}}$ denotes the weight of the net loss term; ${\mu }_{\mathrm{switch}}$ denotes the weight of the switch count term; $R_{ij}$ denotes the resistance of line ij; $I_{ij,k,t}$ denotes the current flowing in the line ij; $y_{ij}$ taking a value of 1, that the line ij cast, taking a value of 0, that the line ij disconnected.

In the first term of Equation (10), the load is given a weight value ${\alpha }_{i,k}$ that drives the important load to be classified into islands, safeguarding the supply of important loads, ${\alpha }_{i,k}{P}_{i,k,t}^{\mathrm{s}}$ guaranteeing minimum load shedding, and satisfying the maximum load principle in the process of islanding operation. In the second term of Equation (10), $R_{ij}{I}_{ij,k,t}^2$ is the line loss, in line with the principle of minimum network loss. In the third term of Equation (10), if the line’s sectional switch operates or the contact switch operates, it will bring a penalty, in line with the principle of minimum number of switches.

After failure of the distribution network and bulk grid, the distribution system may not be able to use the bulk grid for fault recovery. Due to the randomness and lag of troubleshooting time, each island in the distribution network needs to operate independently on the basis of islanding partition, and it is difficult to accurately judge the time when the distribution islands need to operate independently. The DG output of wind power and PV types has uncertainty, and the short-term forecast accuracy of their output is higher than the long-term forecast, and the DG output with short-term forecast can reduce the uncertainty of dispatch. Therefore, this paper adopts a rolling optimization method to gradually determine the distribution islanding operation scheme for each scheduling period (assuming that the interval between adjacent scheduling periods is $\mathsf{\Delta }T$), i.e., multiple time steps are considered each time the scheduling is performed, but only the scheduling scheme for the latter time step is issued, and when the next scheduling cycle comes, the above process is repeated for rolling optimization to achieve feedback correction. The scheduling period can be set according to the actual situation, and is generally taken as $\mathsf{\Delta }T=$ 15 min. The scheduling scheme for the tth scheduling period is obtained by solving the joint optimization model for the period $T=\left\{t, t+\mathsf{\Delta }T, \cdots , t+{\tau }_f\mathsf{\Delta }T\right\}$.

3.2. Constraint Conditions

In the rolling optimization model, due to the fluctuation of new energy and load, the islanding partition solutions obtained from different scheduling periods may be different, and the islanding partition results may also be different even for adjacent scheduling periods. Constant changes in the islanding partition scheme will result in frequent switching actions, which is not conducive to extending the switch life and will reduce customer satisfaction with electricity. Therefore, this paper stipulates that assuming islanding partition is performed during the scheduling period t, the islanding partition scheme can only be reformulated when the changes in new energy and load exceed any threshold shown in Equation (11). In other words, after islanding partition in the dispatching period t, if the change of new energy and load in the next $K\mathsf{\Delta }T$ dispatching periods does not exceed the threshold shown in Equation (11), the islanding partition scheme for time period t will be followed. It is also worth stating that while the islanding partition scheme may remain the same for a period of time, the DG output within an island still needs to be optimized for each dispatch period.

$$\left\{\begin{array}{c}{\sum }_{i\in N_k}\left|P_{i,k,t}^{\mathrm{L}}-P_{i,k,t_1}^{\mathrm{L}}\right|>{\sigma }^{\mathrm{L}} \forall k\in \Omega \\ {\sum }_{i\in N_k}\left|P_{i,k,t}^{\mathrm{wind}}-P_{i,k,t_1}^{\mathrm{wind}}\right|>{\sigma }^{\mathrm{wind}} \forall k\in \Omega \\ {\sum }_{i\in N_k}\left|P_{i,k,t}^{\mathrm{pv}}-P_{i,k,t_1}^{\mathrm{pv}}\right|>{\sigma }^{\mathrm{pv}} \forall k\in \Omega \end{array}\right.$$

(11)

where, $t_1$ denotes the scheduling period for which a new islanding scheme is calculated; $t>t_1$ denotes the scheduling period after $t_1$; $P_{i,k,t}^{\mathrm{L}}$ denotes the active power of the load at time t at node i in island k; $P_{i,k,t}^{\mathrm{wind}}$ denotes the wind power active power at time t for node i in island k; $P_{i,k,t}^{\mathrm{pv}}$ denotes the PV active power at time t for node i in island k; and ${\sigma }^{\mathrm{L}}$, ${\sigma }^{\mathrm{wind}}$, ${\sigma }^{\mathrm{pv}}$ are the specified threshold values respectively.

The switching states of the distribution network islanding partition model need to satisfy the constraints:

$${\displaystyle \sum }_{k\in \mathsf{\Omega }}{y}_{i,k,t}=1 \forall t\in T, i\in N_{\mathrm{MG}}$$

(12)

$$y_{ij,k,t}⩽y_{i,k,t} \forall t\in T,ij\in E,k\in \mathsf{\Omega }$$

(13)

$$y_{ij,k,t}⩽y_{j,k,t} \forall t\in T,ij\in E,k\in \mathsf{\Omega }$$

(14)

where, $y_{i,k,t}$ and $y_{ij,k,t}$ are denoted as the state of node i and line ij in distribution island k. When the value is 1, it means that node i and line ij are put into operation, and when the value is 0, it means that node i and line ij are disconnected. Equation (12) denotes that the DG in the distribution network must be divided to a certain distribution island. Equations (13) and (14) denote that if the distribution line ij is classified to island k, the nodes at its two ends must also be classified to the same island.

To ensure the radial topology of the distribution network, ${\phi }_{ij,k,t}$ is chosen as the directional variable to represent the line power flow direction, and the topological constraint is

$$0⩽P_{ij,k,t}⩽M_{\mathrm{e}}{\phi }_{ij,k,t} \forall ij\in E,t\in T,k\in \mathsf{\Omega }$$

(15)

$$0⩽Q_{ij,k,t}⩽M_{\mathrm{e}}{\phi }_{iij,k,t} \forall ij\in E,t\in T,k\in \mathsf{\Omega }$$

(16)

$${\phi }_{ij,k,t}\ge 0 \forall i\in N\backslash N_{\mathrm{S}},t\in T,k\in \mathsf{\Omega }$$

(17)

$${\phi }_{ij,k,t}=0 \forall i\in N_{\mathrm{S}},t\in T,k\in \mathsf{\Omega }$$

(18)

$${\phi }_{ij,k,t}+{\phi }_{ji,k,t}=1 \forall ij\in E\backslash E_{\mathrm{F}},t\in T,k\in \mathsf{\Omega }$$

(19)

$${\phi }_{ij,k,t}+{\phi }_{ji,k,t}=0 \forall ij\in E_{\mathrm{F}},t\in T,k\in \mathsf{\Omega }$$

(20)

$${\displaystyle \sum }_{j:ij\in E}{\phi }_{ji,k,t}⩽1 \forall i\in N,t\in T,k\in \mathsf{\Omega }$$

(21)

where $N\backslash N_{\mathrm{S}}$ denotes the set excluding the set $N_{\mathrm{S}}$ in the set $N$; $E\backslash E_{\mathrm{F}}$ denotes the set excluding the set $E_{\mathrm{F}}$ in the set $E$; $M_{\mathrm{e}}$ denotes a sufficiently large parameter; ${\phi }_{ij,k,t}$ denotes the flow of power at time t from node i to node j, a value of 1 denotes that power flows from node i to node j, and a value of 0 denotes that no power flows through the line; $j:ij\in E$ denotes a node j belonging to the set of lines E; and $P_{ij,k,t}$ and $Q_{ij,k,t}$ denote the active and reactive power flowing through the line ij, respectively.

It is worth stating that if Equation (11) is not satisfied, i.e., there is no need to develop a new islanding partition scheme, the switching state constraint and the radial topology constraint of the distribution network will be naturally satisfied and there is no need to consider the constraints of Equations (12)–(21) in the rolling optimization.

To ensure the stable operation of the islanded system, the total load in the island should not exceed the generation capacity of the DG in the island, i.e.,

$${\displaystyle \sum }_{i\in N_k}{P}_{i,k,t}^{\mathrm{DG}}\ge {\displaystyle \sum }_{i\in N_k}{P}_{i,k,t}^{\mathrm{L}} \forall t\in T,k\in \mathsf{\Omega }$$

(22)

where $P_{i,k,t}^{\mathrm{DG}}$ denotes the active power generation capacity of DG at time t in island k; $P_{i,k,t}^{\mathrm{L}}$ denotes the active power of the load at time t at node i in island k.

Due to the extensive access of new energy sources such as wind power and PV in the distribution network, there may be uncertainties in $P_{i,k,t}^{\mathrm{DG}}$. The uncertainty treatment of $P_{i,k,t}^{\mathrm{DG}}$ will be described in Section 4.

The island rolling optimal operation needs to satisfy the safety constraint that line power flow and node voltage magnitudes cannot cross the limits, as shown in Equations (23) and (24).

$$-S_{ij}{y}_{ij,k,t}\le P_{ij,k,t}\le S_{ij}{y}_{ij,k,t} \forall k\in \mathsf{\Omega },ij\in E_k,t\in T$$

(23)

$$U_{min}{y}_{i,k,t}\le U_{i,k,t}\le U_{max}{y}_{i,k,t}\forall k\in \mathsf{\Omega },i\in N_k,t\in T$$

(24)

where $S_{ij}$ denotes the maximum apparent power allowed to flow through the line ij, which can be determined by the line’s thermal stability, dynamic stability conditions and insulation level; and$U_{i,k,t}$ denotes the node voltage; $U_{min}$ and $U_{max}$ denote the minimum and maximum voltage amplitude of the allowed nodes.

Since the above distribution islanding topology is radial, a distribution system power flow model in the form of second-order cone relaxation can be established based on the branch power flow model to describe the voltage and power relationship in Equations (23) and (24), i.e.,

$$U_{i,k,t}^2{I}_{ij,k,t}^2⩾P_{ij,k,t}^2+Q_{ij,k,t}^2 \forall k\in \mathsf{\Omega },ij\in E,t\in T$$

(25)

$$P_{i,k,t}^{\mathrm{DG}}-P_{i,k,t}^{\mathrm{L}}+P_{i,k,t}^{\mathrm{S}}={\sum }_{j:ij\in E}{P}_{ij,k,t}-{\sum }_{m:mi\in E}\left(P_{mi,k,t}-R_{mi}{I}_{mi,k,t}^2\right) \forall k\in \mathsf{\Omega },i\in N,t\in T$$

(26)

$$Q_{i,k,t}^{\mathrm{DG}}-Q_{i,k,t}^{\mathrm{L}}+Q_{i,k,t}^{\mathrm{s}}={\sum }_{j:ij\in E}{Q}_{ij,k,t}-{\sum }_{m:mi\in E}\left(Q_{mi,k,t}-X_{mi}{I}_{mi,k,t}^2\right) \forall k\in \mathsf{\Omega },i\in N,t\in T$$

(27)

$$U_{i,k,t}^2-U_{j,k,t}^2⩾2\left(R_{ij}{P}_{ij,k,t}+X_{ij}{Q}_{ij,k,t}\right)+\left(R_{ij}^2+X_{ij}^2\right)I_{ij,k,t}^2-M_{\mathrm{v}}\left(1-y_{ij,k,t}\right) \forall k\in \mathsf{\Omega },ij\in E,t\in T$$

(28)

$$U_{i,k,t}^2-U_{j,k,t}^2\le 2\left(R_{ij}{P}_{ij,k,t}+X_{ij}{Q}_{ij,k,t}\right)+\left(R_{ij}^2+X_{ij}^2\right)I_{ij,k,t}^2+M_{\mathrm{v}}\left(1-y_{ij,k,t}\right) \forall k\in \mathsf{\Omega },ij\in E,t\in T$$

(29)

where $P_{i,k,t}^{\mathrm{DG}}$ and $Q_{i,k,t}^{\mathrm{DG}}$ denote the DG active and reactive power output of node i at time t, respectively; $P_{i,k,t}^{\mathrm{S}}$ and $Q_{i,k,t}^{\mathrm{s}}$ denote the active and reactive power of the load removed by node i at time t, respectively; $P_{i,k,t}^{\mathrm{L}}$ and $Q_{i,k,t}^{\mathrm{L}}$ denote the active and reactive demand of the load at time t at node i, respectively; $R_{ij}$ and $X_{ij}$ denote the resistance and reactance of line ij; $M_{\mathrm{v}}$ is a sufficiently large constant to relax the constraint Equations (28) and (29) when the line is in the disconnected state, and its value can be taken as the square of the maximum allowable voltage amplitude $U_{max}$ in order to make the above constraint more compact. Equations (26) and (27) denote the nodal active and reactive power balance in the distribution system, respectively; Equation (25) is an expression for the relationship between distribution line power flow, current magnitude squared, and voltage magnitude squared in the form of second-order cone relaxation, which is exact in radial distribution networks.

In this paper, DG in the distribution network considers diesel generators, energy storage, and wind and solar power station. The three DGs are modeled separately below.

The active power output adjustment range and speed of the diesel generator are shown in Equations (30) and (32), respectively, that is, the active power of the diesel generator cannot exceed the allowed upper and lower limits, and the active power adjustment range between adjacent dispatching periods should be within the range that the climbing rate can reach. The reactive power output adjustment range and speed of the diesel generator are shown in Equations (31) and (33), respectively.

$$P_{i,k}^{\min }⩽P_{i,k,t}^{\mathrm{d}}⩽P_{i,k}^{\max } \forall k\in \mathsf{\Omega },i\in N_{\mathrm{d}},t\in T$$

(30)

$$Q_{i,k}^{\min }⩽Q_{i,k,t}^{\mathrm{d}}⩽Q_{i,k}^{\max } \forall k\in \mathsf{\Omega },i\in N_{\mathrm{d}},t\in T$$

(31)

$$\left|P_{i,k,t}^{\mathrm{d}}-P_{i,k,t-1}^{\mathrm{d}}\right|⩽R_{i,k}^{\mathrm{p}}\mathsf{\Delta }T \forall k\in \mathsf{\Omega },i\in N_{\mathrm{d}},t\in T$$

(32)

$$\left|Q_{i,k,t}^{\mathrm{d}}-Q_{i,k,t-1}^{\mathrm{d}}\right|⩽R_{i,k}^{\mathrm{q}}\mathsf{\Delta }T \forall k\in \mathsf{\Omega },i\in N_{\mathrm{d}},t\in T$$

(33)

where $N_{\mathrm{d}}$ is the set of diesel generator nodes; $P_{i,k,t}^{\mathrm{d}}$ and $Q_{i,k,t}^{\mathrm{d}}$ are the active and reactive power output of diesel generator i, respectively; $P_{i,k}^{\min }$, $P_{i,k}^{\max }$ and $Q_{i,k}^{\min }$, $Q_{i,k}^{\max }$ are the minimum and maximum values of active and reactive power output of diesel generator i, respectively; $R_{i,k}^{\mathrm{p}}$ and $R_{i,k}^{\mathrm{q}}$ are the active and reactive climbing rates of diesel generator i, respectively.

The energy storage system shall satisfy the charge/discharge power constraint and the state of charge constraint as shown in Equations (34) to (43).

$$P_{i,k,t}^{\mathrm{es}}=P_{i,k,t}^{\mathrm{dis}}-P_{i,k,t}^{\mathrm{ch}} \forall k\in \mathsf{\Omega },i\in N_{\mathrm{es}},t\in T$$

(34)

$$Q_{i,k,t}^{\mathrm{es}}=Q_{i,k,t}^{\mathrm{dis}}-Q_{i,k,t}^{\mathrm{ch}} \forall k\in \mathsf{\Omega },i\in N_{\mathrm{es}},t\in T$$

(35)

$$P_{i,k,\min }^{\mathrm{dis}}{\beta }_{i,k,t}^{\mathrm{dis}}⩽P_{i,k,t}^{\mathrm{dis}}⩽P_{i,k,\max }^{\mathrm{dis}}{\beta }_{i,k,t}^{\mathrm{dis}} \forall k\in \mathsf{\Omega },i\in N_{\mathrm{es}},t\in T$$

(36)

$$P_{i,k,\min }^{\mathrm{ch}}{\beta }_{i,k,t}^{\mathrm{ch}}⩽P_{i,k,t}^{\mathrm{ch}}⩽P_{i,k,\max }^{\mathrm{ch}}{\beta }_{i,k,t}^{\mathrm{ch}} \forall k\in \mathsf{\Omega },i\in N_{\mathrm{es}},t\in T$$

(37)

$$Q_{i,k,\min }^{\mathrm{dis}}{\beta }_{i,k,t}^{\mathrm{dis}}⩽Q_{i,k,t}^{\mathrm{dis}}⩽Q_{i,k,\max }^{\mathrm{dis}}{\beta }_{i,k,t}^{\mathrm{dis}} \forall k\in \mathsf{\Omega },i\in N_{\mathrm{es}},t\in T$$

(38)

$$Q_{i,k,\min }^{\mathrm{ch}}{\beta }_{i,k,t}^{\mathrm{ch}}⩽Q_{i,k,t}^{\mathrm{ch}}⩽Q_{i,k,\max }^{\mathrm{ch}}{\beta }_{i,k,t}^{\mathrm{ch}} \forall k\in \mathsf{\Omega },i\in N_{\mathrm{es}},t\in T$$

(39)

$${\beta }_{i,k,t}^{\mathrm{ch}}+{\beta }_{i,k,t}^{\mathrm{ch}}⩽1 \forall k\in \mathsf{\Omega },i\in N_{\mathrm{es}},t\in T$$

(40)

$${\varrho }_{i,k,t}^{\mathrm{es}}={\varrho }_{i,k,t-1}^{\mathrm{es}}-\left({\eta }_{i,k}^{\mathrm{dis}}{P}_{i,k,t}^{\mathrm{dis}}-{\eta }_{i,k}^{\mathrm{ch}}{P}_{i,k,t}^{\mathrm{ch}}\right)\mathsf{\Delta }T \forall k\in \mathsf{\Omega },i\in N_{\mathrm{es}},t\in T$$

(41)

$${\varrho }_{i,k,min}^{\mathrm{es}}⩽{\varrho }_{i,k,t}^{\mathrm{es}}⩽{\varrho }_{i,k,max}^{\mathrm{es}} \forall k\in \mathsf{\Omega },i\in N_{\mathrm{es}},t\in T$$

(42)

$${\varrho }_{i,k,t}^{\mathrm{es}}={\varrho }_{i,k,t+T}^{\mathrm{es}} \forall k\in \mathsf{\Omega },i\in N_{\mathrm{es}},t\in T$$

(43)

where $N_{\mathrm{es}}$ is the set of energy storage system; $P_{i,k,t}^{\mathrm{dis}}$, $P_{i,k,t}^{\mathrm{ch}}$ and $Q_{i,k,t}^{\mathrm{dis}}$, $Q_{i,k,t}^{\mathrm{ch}}$ are the discharging, charging active and reactive power of energy storage system i, respectively; A and B are 0–1 variables characterizing the discharging and charging states of energy storage system i, respectively, and their values of 1 indicate that energy storage system g is in the discharging and charging states at scheduling time t, the two variables are mutually exclusive, i.e., only one of them can take 1 during the same scheduling time t; $P_{i,k,\min }^{\mathrm{dis}}$, $P_{i,k,\max }^{\mathrm{dis}}$, $P_{i,k,\min }^{\mathrm{ch}}$, $P_{i,k,\max }^{\mathrm{ch}}$ and $Q_{i,k,\min }^{\mathrm{dis}}$, $Q_{i,k,\max }^{\mathrm{dis}}$, $Q_{i,k,\min }^{\mathrm{ch}}$, $Q_{i,k,\max }^{\mathrm{ch}}$ are the minimum and maximum charge and discharge active and reactive power of energy storage system i, respectively; ${\varrho }_{i,t}^{\mathrm{es}}$ denotes the energy stored in energy storage system i; ${\eta }_{i,k}^{\mathrm{ch}}$ and ${\eta }_{i,k}^{\mathrm{dis}}$ are the charging and discharging efficiencies of the energy storage system i, respectively; ${\varrho }_{i,k,min}^{\mathrm{es}}$ and ${\varrho }_{i,k,max}^{\mathrm{es}}$ are the minimum and maximum values of energy that can be stored in energy storage system i, respectively. It is worth stating that Equation (41) represents the energy relationship of the energy storage system between adjacent dispatch periods; Equation (43) represents that the energy stored in the storage system is same at the beginning and end of the scheduling period.

The wind and solar power station output is uncertain, and its model and uncertainty treatment will be described in Section 4.

4. Model Solving Methods

The above distribution network islanding partition and operation model can be formulated as a standard second-order cone planning problem without considering the uncertainty of wind and solar power station output. Assuming that the dimension of the variable with solution is n, the n-dimensional second-order cone standard form is

$$C=\left\{\left(\mathit{u},t\right)\parallel \mathit{u}{\parallel }_2\le t,t\ge 0\right\}$$

(44)

where $\mathit{u}\in {\mathit{R}}^{n-1}$; $t\in \mathit{R}$.

The second-order cone feasible domain formed by the constraint is

$$\parallel \mathit{A}\mathit{x}+\mathit{b}{\parallel }_2\le {\mathit{c}}^{\prime }\mathit{x}+d$$

(45)

where the variable $\mathit{x}\in {\mathit{R}}^n$; real coefficients $\mathit{b}\in {\mathit{R}}^w$; $\mathit{c}\in {\mathit{R}}^n$; $d\in \mathit{R}$; matrix $\mathit{A}\in {\mathit{R}}^{w\times n}$; ${\mathit{c}}^{\prime }$ is the transpose of vector $\mathit{c}$.

Combined with the general model of second-order cone planning, the deterministic distribution network islanding partition and operation model can be expressed as

$$\left\{\begin{array}{c}minf\left(x\right)\\ \mathrm{s}.\mathrm{t}.\parallel Ax+b{\parallel }_2\le {\mathit{c}}^{\prime }x+d\\ g\left(x\right)=0\\ h\left(x\right)\le 0\end{array}\right.$$

(46)

In order to solve the optimization model containing the uncertainty of wind and solar power station output, the wind power and photovoltaics fluctuation is simulated in this paper using the scenario generation reduction method. The scene generation reduction method consists of two sub-processes, i.e., scene generation and scene reduction. In the scenario generation process, the power fluctuation values of wind and solar power station at each node are randomly simulated based on the uncertainty model of wind and solar power station, and they are combined into a set of simulated scenarios. By repeating the process several times, a more comprehensive description of the various fluctuations can be obtained. Scene reduction refers to the clustering analysis of the generated large number of scenes to get a few scenes that are more representative and more different from each other.

In the scene generation process, this paper uses Latin hypercube sampling to sample the new energy output samples from the new energy joint output probability density function obtained by the dynamic vine Copula method to generate a large number of wind and solar power station output scenes, and the sampling size is N.

The scene restoration process can be expressed in the following steps:

(1) Based on the above generated N wind and solar power station output scenes, set the number of clusters as K, and randomly select K cluster centers $C_i$, calculate the Euclidean distance between each pair of scenes and the cluster center, and its calculation formula is

$$d\left(X,C_i\right)=\sqrt{{\sum }_{j=1}^M{\left(X_j-C_{ij}\right)}^2}$$

(47)

where X data denotes objects; $C_i$ denotes the ith clustering center; M denotes the dimension of the data object; $X_j$ and $C_{ij}$ denote the jth attribute values of $X$ and $C_i$, respectively.

(2) Divide each scene into the nearest clustering centers.

(3) Calculate the average of the data from each cluster center as the new cluster center and proceed to the next iteration.

(4) Repeat steps 2–3, when the set number of iterations is reached or the clustering center no longer changes, the clustering is completed, and the scene is reduced to K.

(5) Count the number of scenes in the central cluster of each cluster as its weight value ${\rho }_i$.

The scenario generation reduction method is used to generate the K typical scenarios of wind and solar power station output as ${\mathsf{\Psi }}_{\mathrm{typ}}$. At the same time, in order to ensure that the fault recovery strategy has good resistance to fluctuations, two extreme scenarios are added to the typical scenario set ${\mathsf{\Psi }}_{\mathrm{typ}}$, namely, the “maximum wind power output − minimum PV output” scenario and the “minimum wind power output − maximum PV output” scenario, and the expanded scenario set is ${\mathsf{\Psi }}_{\mathrm{com}}$. The expression of ${\mathsf{\Psi }}_{\mathrm{com}}$ is

$${\mathsf{\Psi }}_{\mathrm{com}}={\mathsf{\Psi }}_{\mathrm{typ}}\cup \left\{\left(P_{i,max}^{\mathrm{wind}},P_{i,min}^{\mathrm{pv}}\right),\left(P_{i,min}^{\mathrm{wind}},P_{i,max}^{\mathrm{pv}}\right)\right\}$$

(48)

After the extreme conditions are added, the distribution network islanding partition and operation model and the distribution network fault recovery model considering the new energy uncertainty can be expressed according to the summary form of the deterministic optimization problem in Equation (46) as

$$\left\{\begin{array}{c}min{\displaystyle {\displaystyle \sum }_{s\in {\mathsf{\Psi }}_{\mathrm{com}}}}{\rho }_{s}f\left(\mathit{x},{\mathit{y}}_s\right)\\ \begin{array}{c}\mathrm{s}.\mathrm{t}. \parallel \mathit{A}\mathit{x}+\mathit{b}{\parallel }_2\le {\mathit{c}}^{\prime }\mathit{x}+d, \forall s\in {\mathsf{\Psi }}_{\mathrm{com}}\\ g\left(\mathit{x},{\mathit{y}}_s\right)=0, \forall s\in {\mathsf{\Psi }}_{\mathrm{com}}\end{array}\\ h\left(\mathit{x},{\mathit{y}}_s\right)\le 0, \forall s\in {\mathsf{\Psi }}_{\mathrm{com}}\end{array}\right.$$

(49)

The optimization model in Equation (49) can be solved using the well-established commercial software CPLEX 12.10.

5. Example Analysis

5.1. Example System

In this paper, an improved IEEE 33-node distribution network configured with MG is used to verify the feasibility and effectiveness of the proposed distribution network islanding partition and operation strategy, and the improved IEEE 33-node distribution network is shown in Figure 1, which includes nine DGs, two wind farms, and two photovoltaics. The capacities of DG1 to DG9 are 700 kW, 700 kW, 200 kW, 400 kW, 400 kW, 150 kW, 150 kW, 300 kW, and 300 kW, respectively. The capacities of Wind Farm 1 and Wind Farm 2 are both 175 kW, and the capacities of Photovoltaic 1 and Photovoltaic 2 are also 175 kW.

According to the importance of each node load in the distribution network, this paper classifies the load into primary load, secondary load, and tertiary load, and the weight coefficient and located node of the three types of loads are shown in

Table 2. Weight coefficient and located node of the three types of loads.

Weight Coefficient	Located Node
100	4, 8, 9, 13, 20, 21, 31
10	6, 7, 12, 14, 16, 17, 22, 23, 24, 25, 26, 32
1	0, 1, 2, 3, 5, 10, 11, 15, 18, 19, 27, 28, 29, 30

.

5.2. Analysis of Uncertainty Modeling of New Energy

In this paper, the output power and forecast data of two adjacent wind farms as well as PV stations in a central China site are used as an example to establish the uncertainty model of joint output power. Since the output correlation of wind power and PV is low, the output uncertainty models of wind power and PV are constructed separately. Taking wind power as an example for illustration, the synchronous measurement data and forecast data for the first 30 days of May 2016 are selected to construct a high-dimensional dynamic vine copula model to determine the uncertainty interval of the forecast value on day 31. The data density is one point every 5 min, for a total of 8928 data points over 8 days. Among them, 31 days’ forecast data and the first 30 days’ measured data are the uncertainty model inputs, and the measured data on the 31st day are used to test the fitting effect of the uncertainty model. The input sequences are substituted into the likelihood function equation and the likelihood function is solved iteratively to obtain the parameters of the evolution equation for each vine node, as shown in

Table 3. Parameters of the evolution equation.

Vine Node	α	β	γ
φ12	0.0010	0.0008	0.3232
φ23	0.6544	−4.3160	0.3642
φ34	−1.7109	−0.0920	−0.4688
φ13|2	0.0013	0.0005	0.2192
φ24|3	0.3447	−2.0146	0.8001
φ14|23	−0.6772	−1.0989	−0.1115

.

The copula function equation of each vine node can be calculated from the dynamic correlation coefficient of vine nodes, and the high-dimensional dynamic vine copula model can be obtained according to the joint distribution formulas. Using the discrete convolution method based on copula function, the high-dimensional model can be reduced to two dimensions. According to the strong correlation characteristic of the predicted values of two wind farms and the correlation of the predicted values and prediction errors of the same wind farm, this paper adopts the D-Vine structure for modeling. D-Vine constructs copula function by pairing the predicted values and prediction errors of Wind Farm 1 and Wind Farm 2, and then relates the four sets of data together by the predicted values of both. The uncertainty interval is generated based on the probability density function of the sum of prediction errors, and the model output is obtained as shown in Figure 6.

On this basis, the uncertainty of wind power is described by scene generation and scene reduction. Set the number of scene generation to 500 and set the number of scene restoration to 5. Figure 7 shows the scene generation results for Wind Farm 1 and Figure 8 shows the scene restoration results for Wind Farm 1. Figure 9 shows the scene generation results for Wind Farm 2 and Figure 10 shows the scene restoration results for Wind Farm 2.

5.3. Analysis of Emergency Power Supply Restoration

Assume that the distribution system is in an extreme fault situation, i.e., the upstream substation exit breaker trips, the distribution system is disconnected from the higher power grid, and line s28 is faulty. Taking the output data of wind farms and PV stations in a certain time period as an example, the proposed emergency power supply restoration model is solved, and the obtained islanding partition results are shown in Figure 11.

As can be seen from the figure, all MGs and significant loads are divided into two distribution silos, where silo 1 contains three MGs and thirteen loads, and silo 2 contains three MGs and twelve loads, as shown in

Table 4. MG and load nodes contained in islands 1 and 2, respectively.

Island	Included MGs	Included Load Nodes
island 1	MG 1, MG 3, MG 5	2–7, 20–26
island 2	MG 2, MG 4, MG 6	8–9, 12–17, 29–32

.

The proposed autonomous operation strategy does not require a pre-determined overall scheduling time, and the operation strategy for each scheduling period is updated in real-time on a rolling basis. We considered a 15-min period, dividing the 24-h day into 96 periods. Based on this, we analyze the 1st island, with two wind farm outputs, five DG outputs, and the total load demand within the 96 time periods as shown in Figure 12.

It can be seen from the figure that the output of WF 1+WF 2 gradually increases in the first 29 periods and peaks at 190 MW in the 29th period; WF 1+WF 2 output remains at a high level in the 30th to 39th periods, maintaining between 123 MW and 282 MW. In the 40th to 59th periods, WF 1+WF 2 output fluctuates with a minimum of 111 MW and a maximum of 294 MW, but the overall trend is decreasing; In the 60th to 71st periods, the output of WF 1+WF 2 gradually decreases to a minimum of 198 MW. In the 72nd to 79th periods, WF 1+WF 2 output fluctuates, but the overall trend decreases. In the 80th to 86th periods, WF 1+WF 2 output drops sharply to a minimum of 108 MW. A slight recovery of WF 1+WF 2 output in the 87th to 91st periods, but still very low. In the last period, the output of WF 1+WF 2 is stable at about 83 MW.

For DG 1, there is no output power in time periods 1–9, it gradually increased in time periods 10–20, stayed stable at 237 kW in time periods 21–39, rose sharply in time periods 40–43, is 0 in time periods 44–59, stayed around 400 kW in time periods 60–86, started to fall in time periods 87–94, and stayed at 237 kW in time periods 95–96. For DG 3, there was no output power in time periods 1–5, it gradually increased in time periods 5–9, stabilized at 50 kW in time periods 10–20, had no output power in time periods 21–39, suddenly increased in time periods 40–49, was more unstable in time periods 50–53, remained at about 90 kW in time periods 54–59, had no output power in time periods 60–79, gradually increased in time periods 80–85, fluctuated in time periods 86–91, and had no output power in time periods 92–96. For DG 4, the output power gradually increased in time periods 1–4, drops sharply to 0 in time periods 5–10, had no output power in time periods 11–19, remained between 162 kW and 228 kW in time periods 20–39, gradually increased in time periods 40–49, remained around 228 kW in time periods 50–59, fluctuated between 150 kW and 218 kW in time periods 60–71, and the output power increased again during 72–75 time periods, reached a peak of 253 kW during 75 time period, and showed a decreasing trend during 76–79 time periods, fluctuating between 170 kW and 189 kW, with no output power during 80–89 time periods and 0 during 90–96 time periods. For both DG 6 and DG 7, the outputs fluctuated significantly. DG 6 increased dramatically to 96 kW between time periods 1–2, then dropped rapidly to 0, followed by constant fluctuations; DG 7 reached a peak of 110 kW in period 1, then gradually dropped to 0, stabilized at 110 kW during time periods 33–39, dropped abruptly to 0 during time periods 40–41, stabilized at 110 kW during time periods 51–59, started decreasing again during time periods 62–70, rose rapidly again during time periods 71–80, fluctuated significantly during time periods 81–87, and dropped rapidly to 0 during time periods 88–96.

Next, the node voltages of each node of the distribution island in time period 4, time period 16, time period 33, and time period 56 are analyzed, and the node voltages of the distribution island in the four time periods are shown in Figure 13, Figure 14, Figure 15 and Figure 16.

Further, the node voltages of each node of the distribution island from time period 1 to 96 were analyzed, and the node voltage expectations and distribution intervals of the 96 time periods of the distribution island are shown in Figure 17.

On the whole, the overall voltage expectation of each node in the distribution island is relatively stable in 96 time periods, with the average value concentrated between 0.96 and 1.002, which meets the voltage stability requirements of the distribution system, but there are some nodes with slightly higher or lower voltage expectation than this range. The voltage distribution interval reflects the variation range of voltage measurements at different nodes, where the smaller the voltage distribution interval is, the smaller the voltage fluctuation is, and the more stable the grid operation is. Therefore, it can be seen from the data that the voltage distribution interval of node 4 is 0, which indicates that the voltage of this node is very stable; while the voltage distribution intervals of nodes 20 and 21 are large, which indicates that the voltage of these two nodes fluctuates. There may be some degree of variation in voltage expectation at the same node during different time periods, which is related to the electrical load. For example, the relatively low voltage expectation of node 7 during different time periods is due to the higher electrical load carried by this node during these time periods.

5.4. Influence of Uncertainty Models to Island Operation

In this paper, dynamic vine copula is used to describe the joint output uncertainty of wind farms. In order to analyze the impact of different wind farm joint output uncertainty models on islanding operation, dynamic copula model, vine copula model, and normal error model were selected as comparative analysis tools. Among them, the model in this paper is a multi-wind farm joint output model based on dynamic vine copula, the confidence interval of this model considers the time-varying characteristics, and the joint output curve of multi-wind farm based on dynamic vine copula model is shown in Figure 6. Comparison Example 1 is a joint output model for multiple wind farms based on vine copula, the confidence interval of this model has no time-varying characteristics, and the joint output curve of multiple wind farms based on the vine copula model is shown in Figure 18. Comparison Example 2 is based on the dynamic copula simple superposition model, this model does not take into account the relevant characteristics of multiple wind farms; the joint output curve of multiple wind farms based on the dynamic Copula simple superposition model is shown in Figure 19. Comparison Example 3 is based on the normal error model, the model default wind power prediction error in line with the normal distribution, and the normal distribution parameters obey the statistical characteristics of this wind farm prediction error, based on the normal error model of the joint output curve of multiple wind farms is shown in Figure 20.

It can be seen that the dynamic vine copula model has the smallest uncertainty interval and the smallest error relative to several other models. The vine copula model does not take into account the time-varying characteristics of multiple wind farms, so the confidence intervals obtained are larger and the errors are larger. The dynamic copula model is a more flexible model than the vine copula model, but the dynamic copula simple superposition model used in this paper does not take into account the correlation characteristics of multiple wind farms, so the confidence intervals obtained are larger than those of the dynamic vine copula model, and the errors are correspondingly larger. The normal error model is a model based on normal distribution, which assumes that the wind power prediction error conforms to the normal distribution, and the normal distribution parameters obey the statistical characteristics of the prediction error of this wind farm, and there are large differences with the actual wind power distribution characteristics, thus the error is the largest.

The dynamic copula model, the vine copula model, and the normal error model for the two wind farm outputs, the five DG outputs, and the total load demand curves within the island are shown in Figure 21, Figure 22 and Figure 23.

The total objective function values of several models for 96 time periods are shown in

Table 5. The objective function values of several models.

Model	Dynamic Vine Copula Model	Dynamic Vine Copula Model	Vine Copula Model	Normal Error Model
Objective function value	1	1.08	1.12	1.23

, where the objective function value obtained from the dynamic vine copula model is used as the baseline value.

It can be seen that the total objective function values of the dynamic vine copula model, the dynamic copula model, the vine copula model, and the normal error model increase sequentially over 96 time periods. Specifically, since the dynamic vine copula model takes into account the time-varying characteristics of the joint output of wind farm, the model is able to describe the joint output of multiple wind farms more accurately than other models, thus minimizing the value of the objective function. However, the values of the objective function obtained from the dynamic copula model, the vine copula model and the normal error model increase sequentially. This indicates that as the uncertainty interval of the model becomes larger, the performance of the model gradually decreases, and more running costs are required to cope with the wind power uncertainty. In summary, it can be concluded that the dynamic vine copula model is a model that can accurately and reliably describe the joint output of multiple wind farms with a smaller uncertainty interval compared to other models, which is important for reducing system operating costs.

5.5. Comparison with Existed Method

The literature [46] established an active distribution network reconfiguration model based on the actual distribution network model. The model considered the randomness of distributed generation and the impact of load power, as well as the impact of power grid structure in the power supply area on load standby power. The Latin hypercube sampling method is used to generate the initial sample data, and the Cholesky method is used to sort the samples to obtain the power flow distribution of the distribution network under operating conditions. Before and after fault reconstruction, the power flow capacity is shown in

Table 6. Comparison of power supply capacity before and after reconstruction.

	Original Load	Total Supply Capability	Available Supply Capacity
Before	82.74 MW	123.08 MW	40.34 MW
After		132.96 MW	50.23 MW

.

The total power supply capacity (TSC) represents the maximum total power supply capacity of the distribution network under the “N−1” safety criterion. The available power supply capacity (ASC) is the difference between the maximum power supply capacity and the existing load, which can better reflect the changes in the actual power supply capacity after the reconstruction of distribution network. According to the results in the Table 6, before reconstruction, the maximum total power supply capacity of the distribution network was 123.08 MW, and the maximum available power supply capacity was 40.34 MW. After the reconstruction, the total power supply capacity increased by 9.88 MW to 132.96 MW, and the maximum available power supply capacity increased by 9.89 MW to 50.23 MW. This means that more loads can be satisfied, improving the reliability and availability of the distribution network.

Without considering the reconfiguration of the distribution network, connecting to DGs can improve the power supply capacity of the system. When DGs are connected to the reconstructed system, the available power supply capacity will be further improved on the original basis. However, the objective function in the fault reconstruction model proposed in the literature only considers the maximization of power supply capacity and does not fully utilize the support capacity of DGs, leaving room for further optimization. In contrast, the emergency power supply recovery strategy proposed in this paper for distribution network fully considers the priority of critical loads, and deeply explores and utilizes the active support ability of microgrids, which can achieve higher efficiency and be more suitable for the actual power supply of modern distribution networks.

6. Conclusions

Considering the high-dimensional dynamic correlation of microgrid support, a new emergency power recovery strategy for the distribution network is proposed. In this paper, a reliable and accurate model of combined output of multiple wind farms is established by fully combining the characteristics of minimum uncertainty interval and minimum error of dynamic vine copula model. On this basis, compared with the traditional fault reconstruction algorithm, this paper considers the principle of priority power supply for important loads after fault, establishes the optimization objective function, and introduces the model solving algorithm. The simulation example shows that the strategy can deal with the uncertainty caused by renewable energy in the distribution network supported by microgrid, so as to improve the security and stability of energy and provide effective suggestions for the formulation of emergency power supply strategy of distribution network.