Safety Assessment of Loop Closing in Active Distribution Networks Based on Probabilistic Power Flow

Abstract

To investigate the security issues of loop-closing operations in medium–low-voltage distribution networks under the influence of stochastic fluctuations from distributed generators (DGs) and loads, probabilistic power flow is introduced for analyzing loop-closing currents in active distribution networks. A novel method combining Latin Hypercube Sampling (LHS) and the Gram–Charlier (GC) series, termed the LHS-GC method, is proposed to calculate the probability distribution of loop-closing currents. By modeling DGs and loads as random variables, their cumulants are efficiently obtained through LHS. Based on a linearized formulation of loop-closing current equations, the cumulants of loop-closing currents are calculated, ultimately reconstructing the probability distribution function of loop-closing currents in active distribution networks. Subsequently, a security assessment framework for loop-closing operations is established using the probability distribution of loop-closing currents. This framework provides a quantitative evaluation from two dimensions: preliminary loop-closing success rate and the severity of current limit violations, offering data-driven decision support for loop-closing operations. Taking the IEEE 34-node distribution network as an example for feeder loop-closing current assessment, the proposed LHS-GC method achieves results with less than 4% deviation from simulation values in terms of cumulative probability distribution of loop-closing currents and safety assessment metrics. Under a sampling scale of 500 points, the computational time is 0.76 s, demonstrating its efficiency and reliability. These outcomes provide actionable references for decision-making support in loop-closing operations of active distribution networks.

1. Introduction

In recent years, ensuring power-supply reliability has emerged as a critical user demand, driven by economic development. As the power-supply network on the user side, the distribution network in China typically adopts the “closed-loop design, open-loop operation” mode, with a radial network structure [1,2]. When the distribution network lines need to handle faults or undergo maintenance, ring switching can achieve load transfer without interrupting the power supply [3,4]. However, the “connect-first, disconnect-later” strategy modifies the distribution network’s topology and power flow. During the loop connection process, steady-state circulating currents and inrush currents may occur, leading to protection malfunctions and line overloads, which in turn threaten the safe and stable operation of the power system [5,6]. Therefore, a method is needed to safely assess the current generated during the loop closure.

In the field of looped connections in classical distribution networks, relevant research has formed a relatively comprehensive theoretical system. In terms of equivalent models for looped connections in distribution networks, reference [7] established a 10 kV distribution network equivalent model considering the equivalent of the main network, feeder load, and transformer. Reference [8] studied the calculation methods for loop current using the superposition theorem and Thevenin’s theorem, and analyzed the key factors affecting the magnitude of the current. Reference [9] constructed a typical equivalent network for loops, and designed three methods for calculating loop current in medium- and low-voltage distribution networks, taking into account the complexity of calculation and accuracy.

However, with the widespread adoption of renewable energy, photovoltaic (PV) power generation systems are playing an increasingly critical role in distribution networks. The inherent randomness and volatility of their output power significantly complicate the calculation of loop-closing currents in distribution networks [10,11]. Currently, theoretical research on loop-closing current assessment techniques remains relatively limited, with many studies focusing on power flow optimization and computation in active distribution networks. For instance, reference [12] proposed a nonparametric quasi-Monte Carlo (MC) method based on uniform experimental design for efficient probabilistic power flow calculations. This approach introduces a hybrid discrepancy metric to enhance result accuracy while reducing computational burden. Reference [13] developed a high-order Markov chain-based modeling framework for PV output and load characterization in probabilistic power flow, improving computational efficiency and accuracy through spatio-temporally correlated scenarios generated via joint probability distributions and inverse transform strategies. These studies reveal the potential of integrating probabilistic power flow methodologies into loop-closing current assessment, offering valuable insights for decision support in loop-closing operations. The calculation methods for probabilistic power flow can be categorized into three main classes: the simulation method, analytical method, and approximate method [14]. The simulation method [15,16], represented by the MC method, simulates various uncertain factors through large-scale random sampling. When the sampling is sufficient, the accuracy is high and the applicability is broad, but the computational efficiency is low in complex systems. The approximation method [17,18] directly describes the probabilistic statistical properties of the output random variables, but the computational load increases significantly when solving models with a large number of nodes. The analysis method [19,20] linearizes uncertain variables to obtain the probability distribution of the output random variables, demonstrating excellent performance in computational efficiency.

The cumulant method (CM) is one of the mainstream analytical methods for probabilistic power flow. However, in medium- and low-voltage distribution networks, the probability distributions of many input variables are often unknown. Traditional CM inevitably introduces computational errors due to its reliance on approximated probability distribution functions [21]. To address this limitation, reference [22] improved CM by incorporating maximum entropy-based probability density function approximation, significantly enhancing its accuracy. Building on these advancements, this study focuses on refining the application of CM in loop-closing current assessment. By integrating simulation and analytical approaches, we propose a hybrid method for calculating the probability distribution of loop-closing currents. This approach reduces cumulant calculation errors while maintaining computational efficiency. The technical novelty of the method is summarized as follows:

(1)

The Integration of LHS and GC Series: The combination of LHS and GC series for loop-closing current probability distribution calculation in active distribution networks retains the flexibility of simulation methods and the efficiency of analytical approaches.

(2)

The Direct Processing of Discrete Sampling Points: LHS is employed to handle discrete sampling points with minimal dependence on predefined input variable distributions. Correlations between sampling points are mitigated through ranking strategies, yielding more accurate cumulants for input variables.

(3)

Linear Relationship Between Loop Currents and Nodal Power Injections: By establishing a linear relationship between loop-closing currents and nodal power injections, convolution operations are transformed into algebraic cumulant calculations, significantly enhancing computational efficiency.

(4)

Two-Dimensional Safety Assessment Framework: A quantitative framework incorporating preliminary loop-closing success rate and the severity of current limit violations is developed, providing comprehensive and precise quantitative support for operational decision making.

(5)

The proposed method is fundamentally based on linearized power flow equations, making it inherently applicable to distribution networks of diverse scales and configurations. Its effectiveness has been validated through a case study on the IEEE 34-node system, demonstrating robust potential for real-world engineering applications.

2. Calculation of Loop Current in Active Distribution Networks

2.1. Calculation of Steady-State Current in a Closed Loop

Taking the typical active distribution network combined-loop operation shown in Figure 1 as an example, the calculation theory and method of combined-loop steady-state current are studied.

In the figure, the distribution network operates in an open loop. Two main transformers T1 and T2 are, respectively, configured at 110 kV busbar; the distributed power supplies DG1 and DG2 are, respectively, connected at 10 kV busbar; the outlet of the feeder is controlled by circuit breakers QF1 and QF2, respectively; and the feeder is electrically connected through connection switch QF3. In terms of load distribution, S_aj and S_bj, respectively, represent the power load of the two feeders, and S_af and S_bf, respectively, represent the power load of the two sides of the connection switch.

The steady-state current analysis of the ring network is shown in Figure 2. Figure 2a is the network after the closure of the interconnection switch QF3, where ${\dot{I}}_{\mathrm{a}}^{\prime }$ and ${\dot{I}}_{\mathrm{b}}^{\prime }$ are the steady-state currents of the two feeders after the ring is closed, and ${\dot{I}}_{\mathrm{c}}$ is the circulating current generated by the closure of the ring; Figure 2b is the network before the ring is closed, where ${\dot{I}}_{\mathrm{a}}$ and ${\dot{I}}_{\mathrm{b}}$ are the initial currents of the two feeders before the ring is closed; Figure 2c is the equivalent circuit of the closed-ring network, simplified according to Thevenin’s theorem, where the equivalent voltage source ${\dot{U}}_{\mathrm{oc}}$ represents the voltage phasor difference across the interconnection switch before the ring closure, and the equivalent impedance Z_eq is the sum of the impedances of the ring network.

According to the reference [7], the mesh current in a loop can be represented as:

$${\dot{I}}_{\mathrm{c}}={\displaystyle \frac{{\dot{U}}_{\mathrm{oc}}}{\sqrt{3}{Z}_{\mathrm{eq}}}}={\displaystyle \frac{{\dot{U}}_{\mathrm{a}}-{\dot{U}}_{\mathrm{b}}}{\sqrt{3}{Z}_{\mathrm{eq}}}}$$

(1)

According to the superposition theorem, when the tie switch QF3 is closed to form a looped network, the steady-state current in the feeders can be decomposed into two independent components: the initial current components ${\dot{I}}_{\mathrm{a}}^{\prime }$ and ${\dot{I}}_{\mathrm{b}}^{\prime }$ from the original open-loop operation of each feeder, and the circulating current component ${\dot{I}}_{\mathrm{c}}$ generated by the potential difference at the loop-closing point. By vectorially superimposing these two components, the steady-state current phasors of the feeders under looped operation can be determined as follows:

$$\left\{\begin{array}{c}{\dot{I}}_{\mathrm{a}}^{\prime }={\dot{I}}_{\mathrm{a}}+{\dot{I}}_{\mathrm{c}}\\ {\dot{I}}_{\mathrm{b}}^{\prime }={\dot{I}}_{\mathrm{b}}-{\dot{I}}_{\mathrm{c}}\end{array}\right.$$

(2)

2.2. Calculation of Loop Impulse Current

The combined-loop operation may cause a large transient current shock, and the influence of the combined-loop shock current on the safety of the power grid needs to be considered. According to reference [10], the loop impulse current instantaneous values i_M and the maximum rms values I_M are:

$$i_{\mathrm{M}}=\sqrt{2}\left(1+e^{-{\displaystyle \frac{0.01}{T_{\mathrm{a}}}}}\right)I_{\mathrm{c}}=\sqrt{2}{k}_{\mathrm{m}}{I}_{\mathrm{c}}$$

(3)

$$I_{\mathrm{M}}=I_{\mathrm{c}}\sqrt{1+2{(k_{\mathrm{m}}-1)}^2}$$

(4)

where I_c is the rms value of the circulating current; T_a is the attenuation time constant of the resultant loop shock current; and T_a = L_eq/R_eq.

As can be seen from Equations (1)–(4), the amplitude of the combined-loop steady-state current and the shock current is determined by the voltage difference on both sides of the break of the link switch, and decreases as the equivalent impedance of the combined-loop network increases. According to Equation (4), the effective values of the maximum shock currents of the feeders at both ends of the combined ring are calculated as follows:

$$\left\{\begin{array}{c}{I}_{\mathrm{Ma}}=I_{\mathrm{a}}+I_{\mathrm{c}}\sqrt{1+2{(k_{\mathrm{m}}-1)}^2}\\ I_{\mathrm{Mb}}=I_{\mathrm{b}}+I_{\mathrm{c}}\sqrt{1+2{(k_{\mathrm{m}}-1)}^2}\end{array}\right.$$

(5)

Through power flow calculations at the reference node, the voltage magnitudes U_i and phase angles δ_i (i = 1, 2, …, n) of all nodes in the pre-loop-closing network can be obtained. Let $U=[U_1,{\delta }_1,\dots ,U_{\mathrm{a}},{\delta }_{\mathrm{a}},U_{\mathrm{b}},{\delta }_{\mathrm{b}},\dots ,U_{\mathrm{n}},{\delta }_{\mathrm{n}}]$ denote the state variables of the closed-loop network, and $I=[{I^{\prime }}_{\mathrm{a}},{I^{\prime }}_{\mathrm{b}},I_{\mathrm{Ma}},I_{\mathrm{Mb}}]$ represent the effective values of the steady-state currents and inrush currents on both sides of the feeders after loop closing. The loop-closing current equations can then be expressed as:

$$I=G(U)$$

(6)

In engineering practice, the inrush coefficient k_m is typically selected within the range of 1.8–1.9 [23,24]. For conservative design purposes, k_m = 1.9 is adopted. Substituting this value into Equation (5), the expanded form of Equation (6) can be derived as:

$$\left\{\begin{array}{l}{\dot{I}}_{\mathrm{a}}^{\prime }=\left|{\dot{I}}_{\mathrm{a}}+{\dot{I}}_{\mathrm{c}}\right|=g_1(U)\hfill \\ {\dot{I}}_{\mathrm{b}}^{\prime }=\left|{\dot{I}}_{\mathrm{a}}-{\dot{I}}_{\mathrm{c}}\right|=g_2(U)\hfill \\ I_{\mathrm{Ma}}=I_{\mathrm{a}}+1.62I_{\mathrm{c}}=g_3(U)\hfill \\ I_{\mathrm{Mb}}=I_{\mathrm{b}}+1.62I_{\mathrm{c}}=g_4(U)\hfill \end{array}\right.$$

(7)

3. Establishment of System Probability Model

3.1. Power Probability Model of Distributed Power Supply

Distributed household photovoltaic has developed rapidly, and photovoltaic power generation systems will become an important form of renewable energy power generation in low- and medium-voltage distribution networks. The increasing penetration of distributed PV systems, characterized by their randomness and output power variability, has significantly intensified the challenges in performing power flow calculations prior to loop-closing operations. Therefore, not only should the load fluctuation be considered in the calculation of combined loop current, but a probability model for photovoltaic power generation should also be constructed. The illumination intensity r can be approximately regarded as a Beta distribution over a finite period of time, and its probability density function is [13]:

$$f(r)={\displaystyle \frac{\mathrm{\Gamma }(\alpha +\beta )}{\mathrm{\Gamma }(\alpha )\mathrm{\Gamma }(\beta )}}{({\displaystyle \frac{r}{r_{\max }}})}^{\alpha -1}{(1-{\displaystyle \frac{r}{r_{\max }}})}^{\beta -1}$$

(8)

where r_max is the maximum illumination intensity, α and β are the two shape parameters of the Beta distribution, and $\mathrm{\Gamma }$ is the Gamma function.

In practice, the shape parameters of the Beta distribution can be determined by statistical analysis, and two shape parameters in the Beta distribution are calculated from the expectations μ and variance σ of the given time-series data [11]:

$$\left\{\begin{array}{l}\alpha =\mu \left[{\displaystyle \frac{\mu (1-\mu )}{{\sigma }^2}}-1\right]\hfill \\ \beta =(1-\mu )\left[{\displaystyle \frac{\mu (1-\mu )}{{\sigma }^2}}-1\right]\hfill \end{array}\right.$$

(9)

The power probability characteristic of the photovoltaic power supply satisfies the following requirements:

$$P_{\mathrm{V}}(r)=r\cdot A\cdot \eta$$

(10)

where P_V is the total output power; A is the total area of the photovoltaic panel, and η is the photoelectric conversion efficiency.

It can be seen from Equation (10) that the output power of the photovoltaic system has a positive correlation with the illumination. In combination with Equation (8), the probability density function of the light intensity and the photovoltaic output force can be derived:

$$f(P_{\mathrm{V}})={\displaystyle \frac{\mathrm{\Gamma }(\alpha +\beta )}{\mathrm{\Gamma }(\alpha )\mathrm{\Gamma }(\beta )}}{({\displaystyle \frac{P}{{\mathrm{V}}_{\max }}})}^{\alpha -1}{(1-{\displaystyle \frac{P}{{\mathrm{V}}_{\max }}})}^{\beta -1}$$

(11)

where P_Vmax is the maximum power of photovoltaic power generation, which meets P_Vmax = r_maxAη.

In practice, through the regulation of the control system, photovoltaic power generation units can achieve stable control of the power factor, so they can be modeled as PQ nodes in the load flow calculation of the distribution network. Thus, under the known power factor angle of φ, the reactive output power of the photovoltaic array can be obtained as Q_V [14].

$$Q_{\mathrm{V}}=P_{\mathrm{V}}\tan \phi$$

(12)

After applying PQ node equivalent processing to the photovoltaic power source, it can be regarded as a load to derive the Thevenin equivalent impedance expression Z_Veq:

$$Z_{\mathrm{veq}}=-{\displaystyle \frac{P_{\mathrm{V}}{U}_{\mathrm{V}}^2}{P_{\mathrm{V}}^2+Q_{\mathrm{V}}^2}}-\mathrm{j}{\displaystyle \frac{Q_{\mathrm{V}}{U}_{\mathrm{V}}^2}{P_{\mathrm{V}}^2+Q_{\mathrm{V}}^2}}$$

(13)

3.2. Load Power Probability Model

Using a normal distribution to approximate the uncertainty of reactive power load, assuming the corresponding parameters for active power and reactive power are μ_P, σ_P and μ_Q, σ_Q, the probability density functions of active power P_L and reactive power Q_L can be expressed as [20]:

$$\left\{\begin{array}{l}f(P_{\mathrm{L}})={\displaystyle \frac{1}{\sqrt{2\pi }{\sigma }_P}}\exp \left[-{\displaystyle \frac{\left(P_{\mathrm{L}}-{\mu }_P\right)}{2{\sigma }_P^2}}\right]\hfill \\ f(Q_{\mathrm{L}})={\displaystyle \frac{1}{\sqrt{2\pi }{\sigma }_Q}}\exp \left[-{\displaystyle \frac{\left(Q_{\mathrm{L}}-{\mu }_Q\right)}{2{\sigma }_Q^2}}\right]\hfill \end{array}\right.$$

(14)

4. Solution of Cumulants for Discrete Input Variables

4.1. Latin Hypercube Sampling

When calculating the joint probability of the load considering the randomness of photovoltaic power generation, it is necessary to establish a method based on stochastic power flow calculations. LHS can effectively reflect the overall distribution of random variables through sampling values. This method divides the range of each variable equally and samples uniformly within each interval, thereby improving the accuracy of the simulation. Therefore, compared to MC sampling, it can obtain more accurate sample mean and variance values with fewer sampling numbers.

LHS mainly has two steps of sampling and arrangement. The sampling step ensures that the sample distribution area can be sufficiently collected, assuming X₁, X₂, …, and X_p are p input random variables, where the cumulative probability distribution function of X_p is:

$$Y_{\mathrm{p}}=F_{\mathrm{p}}(X_{\mathrm{p}})$$

(15)

The sampling scale is N, the value-taking interval (0,1) of Yp is divided into N intervals, and the length of each interval is 1/N. By setting the sample point Y_p at the center of each interval, the i-th sample value of the corresponding input random variable X_p can be obtained from the inverse function:

$$X_{\mathrm{pi}}=F_{\mathrm{p}}^{-1}({\displaystyle \frac{i-0.5}{N}})$$

(16)

The sampling matrix of each input random variable is 1 × N, and all p input random variables are processed to form an initial sampling matrix of p × N, so that the sampling step is finished, and the next order is carried out. The sorting step randomly arranges the position of each element in each row, reduces the correlation between each random variable of the initial sampling matrix, and obtains the final sampling matrix. Calculations of the order of origin moments ${\alpha }_{\mathrm{p}}^{\left(\mathrm{v}\right)}$ and cumulants ${\gamma }_{\mathrm{p}}^{\left(\mathrm{v}\right)}$ of the input random variable X_p are then performed based on the final sampling matrix.

4.2. Latin Hypercube Sampling for Discrete Data

The LHS requires the cumulative probability distribution function Y_p = F_p(X_p) of the random variable X_p as input. However, in engineering applications, many of the operating parameters and historical data of variables are often discrete, making it impossible to directly obtain the cumulative probability distribution function of these discrete data. Therefore, a method is needed to determine semi-variates using LHS based on discrete data.

For a subsample set of the random variable X_p, when it contains N discrete observations, it can be represented as {x_p1, x_p2, x_p3, …, x_pN}. Sort the subsample set of each input random variable in ascending order and calculate the empirical distribution function of X_p:

$$H_{\mathrm{p}}(X_{\mathrm{p}})=\left\{\begin{array}{l}0, x_{\mathrm{p}}<{x^{\prime }}_{\mathrm{p}1}\hfill \\ {\displaystyle \frac{k}{n}},{x^{\prime }}_{\mathrm{pk}}\le x_{\mathrm{p}}\le {x^{\prime }}_{\mathrm{p}\left(\mathrm{k}+1\right)},k=1,2,\dots ,N-1\hfill \\ 1,x_{\mathrm{p}}\ge {x^{\prime }}_{\mathrm{p}\left(\mathrm{N}+1\right)}\hfill \end{array}\right.$$

(17)

According to the empirical distribution function H_p(X_p), a sample set of X_p $\{{x^{″}}_{\mathrm{p}1},{x^{″}}_{\mathrm{p}2},{x^{″}}_{\mathrm{p}3},\dots ,{x^{″}}_{\mathrm{pN}}\}$ is obtained using LHS, and the moments of different orders of X_p are calculated, respectively:

$${\alpha }_{\mathrm{p}}^{\left(\mathrm{v}\right)}={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N{x}_{\mathrm{pk}}^{\left(\mathrm{v}\right)}},k=1,2,\dots ,n$$

(18)

Then, based on the relationship between cumulants and the moment about the origin, we obtain the cumulants of each order for X_p.

5. Probabilistic Calculation of Loop-Closing Currents and Line Losses

5.1. Linearization of the Loop Current Equation

Let $W=[P_1,Q_1,\dots ,P_n,Q_n]$ be the injection power of each node in the ring network, and the matrix form of the node power equation is:

$$W=F(U)$$

(19)

The state variable U of the closed-loop network system can be decomposed into the expected value U₀ and the disturbance amount ΔU, which satisfies the relationship W₀ = F(U₀) with the expected value of the node injection power W₀. Expanding Equation (19) according to the Taylor series and ignoring higher-order terms yields:

$$\Delta W=J_0\Delta U$$

(20)

where J₀ is the Jacobian matrix, defined as:

$$J_0={\left.{\displaystyle \frac{\partial F(U)}{\partial U}}\right|}_{U=U_0}$$

(21)

Similarly, the loop current equation can also be subjected to similar linearization processing, resulting in:

$$\Delta I=G_0\Delta U=G_0{J}_0^{-1}\Delta W$$

(22)

where G₀ is the coefficient matrix, defined as:

$$G_0={\left.{\displaystyle \frac{\partial G(U)}{\partial U}}\right|}_{U=U_0}$$

(23)

Equation (22) shows that the combined-loop current can be expressed as a linear combination of the variable ΔW, providing a theoretical basis for obtaining the probability distribution of the combined-loop current based on cumulants.

5.2. Loop-closing Current Probability Density Function

Apply the LHS method to process discrete data to obtain the cumulants $\Delta W_{\mathrm{P}}^{\left(\mathrm{k}\right)}$ and $\Delta W_{\mathrm{L}}^{\left(\mathrm{k}\right)}$ for photovoltaic power generation and load. Due to the additivity of cumulants, by summing the cumulants of each node’s load power $\Delta W_{\mathrm{L}}^{\left(\mathrm{k}\right)}$ and the cumulant of the photovoltaic output $\Delta W_{\mathrm{P}}^{\left(\mathrm{k}\right)}$, we obtain the cumulative amount of node injection power ${\Delta }_W^{\left(\mathrm{k}\right)}$.

$${\Delta }_W^{\left(\mathrm{k}\right)}=\Delta W_{\mathrm{L}}^{\left(\mathrm{k}\right)}+\Delta W_{\mathrm{P}}^{\left(\mathrm{k}\right)}$$

(24)

Refer to the linear properties of the cumulant and combine with Equation (22) to obtain the cumulant of the combined ring current $\Delta I^{(k)}$.

$$\Delta I^{(k)}={\left(G_0{J}_0^{-1}\right)}^k\Delta W^{(k)}$$

(25)

After obtaining the cumulants of each order of the circular current, the cumulative probability distribution function F(x) can be constructed using the Gram–Charlier series expansion method, as follows [22]:

$$F(x)=\Psi (x)+{\displaystyle \frac{c_1}{1!}}{\Psi }^{\prime }(x)+{\displaystyle \frac{c_2}{2!}}{{\Psi }^{\prime }}^{\prime }(x)+\dots$$

(26)

where $\Psi (x)$ is the cumulative distribution function of the standard normal distribution; the coefficients c_i can be obtained using the semi-invariants of each order.

5.3. Probability Function of Power Loss in Distribution Networks

Power loss is a critical comprehensive indicator for evaluating the economic and technical performance of power grid operation. The integration of DGs into distribution networks significantly complicates power loss calculations. To reduce computational complexity, power loss can be defined as the difference between injected power and output power. The cumulants of different orders for line losses in the distribution network can be calculated as follows:

$$\Delta P_{\mathrm{loss}}^{(k)}=P_{\mathrm{s}}^{(k)}+{\displaystyle \sum _{i=1}^g{P}_{i\_\mathrm{DG}}^{(k)}}-{\displaystyle \sum _{j=1}^h{P}_{j\_\mathrm{L}}^{(k)}}$$

(27)

where $\Delta P_{\mathrm{loss}}^{(k)}$ is the k-th order cumulant of distribution network power loss; $P_{\mathrm{s}}^{(k)}$ is the k-th order cumulant of the output power from the main power source; ${\displaystyle \sum _{i=1}^g{P}_{i\_\mathrm{DG}}^{(k)}}$ is the sum of k-th order cumulants from g distributed generators; ${\displaystyle \sum _{j=1}^h{P}_{j\_\mathrm{L}}^{(k)}}$ is the sum of k-th order cumulants from h loads.

6. Safety Assessment Indicators for Distribution Network Ring Connection

6.1. Initial Success Rate of Loop Closure

The conditions for the safe loop operation of the distribution network are (1) the steady-state current of each feeder after looping does not exceed its maximum carrying capacity; (2) the impact current of each feeder during the closing loop is below the fast trip action threshold of the protective device.

Let F₁(x), F₂(x), F₃(x), and F₄(x) be the cumulative distribution functions of the combined-loop currents ${I^{\prime }}_{\mathrm{a}}$, ${I^{\prime }}_{\mathrm{b}}$, I_Ma, and I_Mb, respectively. The current carrying limits of the two feeder lines are I_max.a and I_max.b, and the settings for the current instantaneous protection are I_setI;.a and I_setI.b. Then, the probability of the combined-loop current exceeding the safety threshold can be expressed as [25]:

$$\left\{\begin{array}{l}{P}_1=P({\dot{I}}_{\mathrm{a}}^{\prime }\ge I_{\max .\mathrm{a}})=1-F_1(I_{\max .\mathrm{a}})\hfill \\ P_2=P({\dot{I}}_{\mathrm{a}}^{\prime }\ge I_{\max .\mathrm{b}})=1-F_2(I_{\max .\mathrm{b}})\hfill \\ P_3=P({\dot{I}}_{\mathrm{Ma}}\ge I_{\mathrm{set}.\mathrm{a}})=1-F_3(I_{\mathrm{set}.\mathrm{a}})\hfill \\ P_4=P({\dot{I}}_{\mathrm{Mb}}\ge I_{\mathrm{set}.\mathrm{b}})=1-F_4(I_{\mathrm{set}.\mathrm{b}})\hfill \end{array}\right.$$

(28)

The preliminary success rate of grid connection is quantitatively described to assess the impact of distributed power sources on exceeding the grid connection current limit, providing data support for grid connection decisions. The expression for the preliminary success rate P is as follows:

$$P=(1-P_1)(1-P_2)(1-P_3)(1-P_4)$$

(29)

If P is greater than 95%, this operation is defined as a preliminary successful closure.

6.2. Excessive Ring Current Limit

Although the initial success rate of the loop closure can indicate a high probability of meeting the requirements for loop-closure current, certain loop-closure currents with lower probabilities may cause serious violation issues. Therefore, it is necessary to define the maximum violation rate of the loop-closure current η_M and the average violation rate η_A as follows [25]:

$${\eta }_{\mathrm{M}}=\left({\displaystyle \frac{I_{\mathrm{F}}}{I_{\mathrm{Lmax}}}}-1\right)\times 100\%$$

(30)

$${\eta }_{\mathrm{A}}=\left({\displaystyle \frac{{\displaystyle {\int }_{I_{\mathrm{Lmax}}}^{I_{\mathrm{F}}}x\cdot f(x)\mathrm{d}x}}{{\displaystyle {\int }_{I_{\mathrm{Lmax}}}^{I_{\mathrm{F}}}f(x)\mathrm{d}x}}}-1\right)\times 100\%$$

(31)

where I_F is the current value corresponding to the cumulative probability distribution of the line current at 99.9%, I_Lmax is the maximum current-carrying capacity of the line, and f(x) is the probability density function of the line current.

Since the power system has a certain overload capacity, it is considered that the safe operation of the distribution network requires that the η_M does not exceed 10% and η_A does not exceed 5%. If a preliminary loop closing is unsuccessful but the exceeding current rate meets the requirements, then the part of the current that does not meet the preliminary loop-closing success rate is determined to be the safe current. In summary, using preliminary loop-closing success rate, maximum exceeding rate, and average exceeding rate as quantitative standards for assessing the safety of loop-closing operations provides data references for the operators’ loop-closing decisions.

In summary, the process of the LHS-GC method based on LHS and Gram–Charlier series is shown in Figure 3. The arrows in Figure 3 indicate the next step.

7. Example Verification

This paper uses the IEEE 34-node distribution network system as an example to verify the proposed method. The standard system is simplified to a single voltage level by removing transformers and voltage regulators from the lines. Due to the text length constraints in the main body, the line impedance data of IEEE 34 nodes and the normal distribution data of loads are presented in Appendix A and Appendix B. Based on the simplified distribution network, photovoltaic power sources are configured using solar radiation intensity samples from a region in South China as input data. The system topology and an example of the tie line are shown in Figure 4. In Figure 4, the black solid line represents the existing distribution line, and the blue dashed line represents the tie line that has not been closed.

Based on the sample data of solar radiation intensity within a day, the shape parameters of the Beta distribution are calculated to be α = 0.679 and β = 1.778, with a maximum solar radiation intensity of 1.134 kW/m², as shown in Figure 5. Take the reference capacity S_B as 1 MVA and the reference voltage U_B as 24.9 kV.

Based on variations in PV integration nodes, loop-closing line locations, and load standard deviations, two cases are defined as shown in

Table 1. Configuration of loop-closing operating conditions.

Case	PV Integration Nodes	Loop-Closing Line Locations	Load Standard Deviation
1	23, 29	28–34	30%
2	6, 23	6–13	20%

. These cases are selected to analyze the safety of loop-closing operations and power losses.

7.1. Probabilistic Calculation of Loop-Closing Currents and Power Losses

This case study uses loop-closing currents obtained from MATLAB/SIMULINK simulations as a reference to compare the results of three methods: MC method, traditional cumulant method, and the proposed LHS-GC method, with a sampling size of 500 for both MC and LHS. The cumulative probability distributions of loop-closing currents for Case 1 and Case 2 are illustrated in Figure 6 and Figure 7, respectively.

As shown in Figure 6 and Figure 7, compared to the MC method and traditional CM, the proposed LHS-GC method achieves higher accuracy, with its cumulative probability distribution curves aligning more closely with simulation results. Due to limitations in sampling scale, the traditional CM demonstrates higher precision than the MC method under the current configuration.

Figure 8 illustrates the deviations of the three methods from simulation values at the 90% cumulative probability level. Taking Case 1 as an example, the calculation errors of the LHS-GC method for steady-state currents and inrush currents in the two feeders are 3.49%, 6.33%, 3.4%, and 3.11% lower than those of the MC method, and 1.72%, 3.1%, 3.09%, and 1.8% lower than those of the CM, respectively. Furthermore, the current calculation errors of the LHS-GC method are consistently below 3%, confirming its superior performance in approximating the cumulative probability distribution function of feeder loop-closing currents. These results validate the reliability of the LHS-GC method for loop-closing safety assessments.

Table 2. Comparison of computation time.

Computational Methods	Single Cycle Time/s	Total Duration/s
MC method	0.0019	2.43
CM	-	0.15
LHS-GC method	0.0001	0.76

compares the computation times of the methods. The LHS-GC method requires shorter single-cycle time and total duration, achieving a 68% speed improvement compared to the MC method. Under identical sampling scales, it exhibits significant speed advantages. Therefore, compared to the traditional CM, the LHS-GC method achieves higher accuracy with only a minor sacrifice in computational efficiency, making it a practical tool for real-time operator decision making.

Under both operating cases, the power losses of the entire network before and after loop closing are shown in Figure 9. For Case 1, the power losses showed no significant variation between pre- and post-loop-closing operations. In contrast, for Case 2, the power losses decreased after loop closing compared to pre-loop-closing levels, contributing to the economic operation of the distribution network. The proposed LHS-GC method accurately captures the variations in power losses, exhibiting errors within 8% compared to simulation values at the 90% cumulative probability level. This demonstrates its capability to provide a reliable reference for assessing the economic and technical performance of post-loop-closing distribution networks.

7.2. Loop-Closing Safety Assessment

Safety assessments for loop-closing operations are conducted using the cumulative probability distribution curves of feeder loop-closing currents calculated by the LHS-GC method. Based on the 3-sigma principle of load normal distribution, the maximum allowable current-carrying capacity of branches is determined. The instantaneous current protection settings are derived from the maximum short-circuit current at the end of the protected line during a fault. The calculated results for the maximum allowable current-carrying capacity and protection settings under Case 1 and Case 2 are summarized in

Table 3. The current indicator values of Case 1 and Case 2.

Feeder	Maximum Allowable Current-Carrying Capacity/A	Instantaneous Current Protection Settings/A
26–28	3.24	13.21
32–34	2.62	36.50
4–6	12.93	143.94
11–13	12.73	100.68

.

The safety assessment results of loop-closing currents are presented in

Table 4. Assessment indices of loop-closing safety under Case 1.

Loop Current	Preliminary Closure Probability		Maximum Overrun Rate		Average Exceedance Rate
	Simulated Value	LHS-GC	Simulated Value	LHS-GC	Simulated Value	LHS-GC
26–28 Steady-state current	94.66%	95.92%	12.43%	13.12%	4.37%	4.01%
32–34 Steady-state current			13.29%	14.46%	4.89%	3.94%
26–28 impact current			—	—	—	—
32–34 impact current			—	—	—	—

and

Table 5. Assessment indices of loop-closing safety under Case 2.

Loop Current	Preliminary Closure Probability		Maximum Overrun Rate		Average Exceedance Rate
	Simulated Value	LHS-GC	Simulated Value	LHS-GC	Simulated Value	LHS-GC
4–6 Steady-state current	91.35%	93.16%	15.33%	16.86%	5.11%	5.97%
11–13 Steady-state current			14.72%	15.31%	5.25%	5.92%
4–6 impact current			—	—	—	—
11–13 impact current			—	—	—	—

. In Case 1, the proposed LHS-GC method exhibits errors of 0.69% and 1.17% in the maximum violation rates for feeders 26–28 and 32–34 compared to simulation values, respectively. The errors in average exceedance rates are 0.36% and 0.95%, respectively. In Case 2, the errors of maximum violation rates and average exceedance rates calculated by LHS-GC for feeders 4–6 and 11–13 are 1.53%, 0.59%, and 0.86%, 0.67%, respectively, all within 2%. Additionally, the preliminary loop-closing success rates calculated by LHS-GC for both cases closely align with simulation values, exhibiting errors of 1.26% and 1.81%.

In summary, an analysis of these two cases demonstrates that the LHS-GC method can effectively assess the safety of loop-closing operations in active distribution networks, even with limited samples. The entire process—from calculating the cumulative probability distribution of loop-closing currents to deriving safety assessment metrics—requires only 0.76 s, meeting the timeliness requirements for engineering applications.

8. Conclusions

To address the challenges in the safety assessment of loop-closing operations for active distribution networks, this paper proposes an LHS-GC-based framework for probabilistic loop-closing current analysis and safety evaluation. Validation on the IEEE 34-node system yields the following conclusions:

(1)

Compared to the MC method, the LHS-GC method significantly improves both accuracy and computational speed under identical sampling scales. Compared to the traditional CM, it achieves higher accuracy while maintaining comparable speed. With a sampling size of 500, the method completes calculations in 0.76 s with probability distribution errors below 4%.

(2)

Under the two-dimensional safety assessment framework (preliminary success rate and severity of current violations), the LHS-GC method achieves errors within 3% for all metrics. The total computation time of 0.76 s fulfills real-time operational demands for active distribution networks.

(3)

The LHS-GC method accurately captures the probabilistic distribution of loop-closing currents and safety metrics across diverse scenarios, including varying PV integration nodes, loop-closing line locations, and load standard deviations. Rooted in linearized power flow equations, the method is inherently adaptable and holds potential for application in distribution networks with various configurations.