A Section Location Method of Single-Phase Short-Circuit Faults for Distribution Networks Containing Distributed Generators Based on Fusion Fault Confidence of Short-Circuit Current Vectors

Abstract

To ensure safe and stable operation, accurate fault localization within active distribution networks is required, and this has attracted much attention. Influenced by many factors such as the control strategy, control performance, initial state of the distributed generators, and distribution network topology, it is still difficult to reliably locate complex and variable single-phase short-circuit faults relying only on a single feature quantity, while localization methods incorporating intelligent algorithms are affected by the choice of a priori samples and the fact that the solution process is a black-box model. To address this challenge, in this work, an expression for the single-phase short-circuit current vector of a distribution network containing distributed generators is derived, and the differences in magnitude and phase angle of the short-circuit current vectors upstream and downstream of the fault point are analyzed. Based on measurement theory, a fault confidence distribution function that reacts to the relative size of the current magnitude difference and phase angle difference is established, and the fusion fault confidence of the short-circuit current vector is constructed with the help of evidence theory. Finally, a method of locating single-phase short-circuit faults in distribution networks that contain distributed generators is proposed. The simulation results show that the ratio of the fusion fault confidence of the short-circuit current vector between faulted and non-faulted sections under the influence of different distributed generator capacities, fault locations, and transition resistances differ significantly. The proposed single-phase short-circuit fault localization method is both adaptive and physically interpretable and has clear boundaries, sound sensitivity, and engineering practicability.

1. Introduction

1.1. Background

With the increasingly severe global energy crisis and environmental pollution, the proportion of photovoltaic, wind turbine, and energy storage systems connected to distribution networks in the form of distributed generators (DGs) is increasing year by year, and distribution networks are gradually changing into active distribution networks [1,2]. Unlike synchronous motors, where the rotor is directly connected to the grid, distributed generation uses a converter as a grid connection interface or excitation control loop. The external characteristics of DGs under grid faults are subject to the control response of converters and have the characteristics of controllability, polymorphism, and high-order strong coupling [3]. In particular, the converter has weak anti-interference ability and is prone to blocking by over-current or low voltage under grid faults, which not only makes the fault response of a DG more complicated but may even aggravate the power imbalance during grid faults. Therefore, in order to ensure the safe and stable operation of an active distribution network, accurate fault localization within active distribution networks has attracted much attention.

With the increase in distributed generation penetration, the influence of the distinct electromagnetic transient behavior of converters on grid fault characteristics is increasing. In particular, single-phase short-circuit faults have the highest probability of occurrence, but their fault characteristics are affected by many factors, such as neutral grounding, DG control strategy, fault transition resistance, etc. Therefore, many scholars have studied the response characteristics of DGs under grid faults. Reference [4] analyzed the contribution of DGs to fault current and pointed out that the fault current of a DG is less than that of synchronous generators and generally does not exceed two times its rated value. References [5,6] determined that the short-circuit current depends on the DG’s operating state and the degree of voltage drop. In view of the asymmetric nature of faults, references [7,8] also pointed out that a DG only provides positive sequence components under positive and negative sequence separation control, which can be equivalent to a positive-sequence-controlled current source. In addition to the characteristics of industrial frequency, reference [9] considered two strategies consisting of excitation control and crowbar input, respectively, and deduced that the short-circuit current of a DG has an attenuation component and a DC component. The existing research fully demonstrates that the addition of DGs causes a fundamental change in the fault characteristics of a distribution network. Therefore, if operators of an active distribution network employ the single-phase short-circuit fault location method employed in traditional distribution networks, there may be a problem of insufficient sensitivity [10,11,12,13,14].

1.2. Current Statu of Research

For this reason, researchers have proposed improved fault localization methods, including improved physical characterization and intelligent algorithms, as shown in

Table 1. Current state of research on single-phase faults in active distribution networks.

Research Object	Research Idea	Insufficient
Conventional Distribution Network	Physical Characterization Methods [10,11,12,13]	The large number of distributed generators changes the fault characteristics, meaning these methods cannot be directly applied.
	Artificial Intelligence Methods [14]
Active Distribution Network	Physical Characterization Methods [15,16,17,18,19,20,21]	Setting the threshold value relies on simulation or experience, the adjustment process is complicated with the increased penetration of distributed generators, and the sensitivity is lacking.
	Artificial Intelligence Methods [22,23,24,25,26,27]	Influenced by the choice of a priori samples and the solution process, which is s black-box model that lacks physical interpretability.

. Regarding improving physical characterization, reference [15] proposed a fault localization method for an active distribution network based on band-limited transient currents that only requires current measurements to achieve fast and accurate fault location. Reference [16] improved the impedance method, which has high accuracy and robustness and can withstand inevitable load fluctuations. Reference [17] analyzed the transient model of small current grounding faults and proposed a method for determining the fault section using the relationship between the initial transient of the zero-sequence current and the faulty phase voltage just after the occurrence of faults. Reference [18] utilized the initial mobile wave arrival moment at multiple ends to establish a fault branch search matrix and determine the fault branch according to the change characteristics of the matrix elements. The above methods often rely on simulation or experience to set a certain fixed threshold to establish fault location criteria. However, with the continuous access of distributed generators to the distribution network, the thresholds need to be re-adjusted several times, and the adjustment process is also more complicated. Single-feature-based fault localization methods can have insufficient sensitivity when the fault transition resistance is large, and there is an urgent need to improve the adaptive capability under different scenarios. Reference [19] proposed using converters to provide low-frequency symmetrical three-phase voltage to locate the fault section, but this requires the addition of soft open points. References [20,21] used a synchronized phasor measurement unit to extract the voltage and current distribution characteristics of a feeder; a single-phase fault localization method based on the characteristic entropy and power increment direction was proposed, effectively reducing the search area. The research cited above solves the problem of the effect of the DG response on the sensitivity of single-phase fault localization. However, the achieved location sensitivity and speed depend on the placement of additional equipment, such as soft open points and synchronized phasor measurement units, and the efficiency of the optimization algorithm.

With the gradual maturity of artificial intelligence algorithms, some researchers have also applied them to single-phase fault localization in active distribution networks [22]. In [23], the fruit fly optimization algorithm was applied to fault localization, and the node current magnitude matrix was discussed in different regions with a faster convergence speed. Reference [24] used fuzzy Petri nets for modeling and verified that the proposed fault localization model is characterized by simple reasoning and fast diagnosis through example data. However, the latter approach is difficult to widely apply in large-scale active distribution networks due to the limitation of computational efficiency. Reference [25] utilized the likelihood ratio form of the Bayesian probability formula to determine the maximum possible interval of occurrence of faults and showed that this approach has good real-time performance. Reference [26] used a biostatic binary particle swarm optimization algorithm to address the above shortcomings and maintain effectiveness despite partial distortion of fault information. Reference [27] applied a combination of wavelet transform and information entropy to extract three-phase voltage features, which they used to establish a fault type recognition network based on SVM that can quickly and accurately recognize various fault types and is not affected by transition resistance and fault location. However, localization methods that synthesize multiple fault characteristics with the help of intelligent algorithms are affected by the use of a priori samples, and the solution process is a black-box model. A fault localization method that is both adaptive to localization and physically interpretable is lacking.

To address this challenge, in this work, we deduce the short-circuit current vector upstream and downstream of the fault point under different conditions. By analyzing the magnitude difference and phase angle difference of the current vectors at both ends of the fault section and the non-fault section, the magnitude difference and phase difference measure of the current in the section are constructed as the characteristic quantities of a single-phase short-circuit fault. Then, the fault confidence assignment function of the magnitude difference measure and phase difference measure of the current in the section is established. Based on evidence theory, a localization method of single-phase short-circuit faults for distribution networks containing DGs based on the fusion fault confidence of the short-circuit current vector is proposed. Theoretical analysis and example results show that the proposed method balances protection adaptability, physical interpretability, and engineering practicality. It does not need an investment in additional equipment, which improves its economic benefits; it avoids the sensitivity limitation of threshold comparison methods by comparing fusion fault confidence of the short-circuit current vector between sections; it does not need fault sample training, and the fault localization process has a clear physical meaning. Therefore, the proposed localization method of single-phase short-circuit faults for distribution networks contained distributed generation based on fusion fault confidence of the short-circuit current vector makes up for the shortcomings of the existing methods such as the sensitivity being easily affected by the DG output and fault transition resistances, and the fault characteristics being difficult to derive from the localization results.

2. Short-Circuit Current Vector Analysis for Active Distribution Networks

2.1. Fault Point Is Located between the Bus and a DG

According to the positional relationship between the fault point and the point of common coupling (PCC), single-phase short-circuit faults of active distribution networks can be divided into three cases: the fault is located between the bus and a DG, between DGs, and between a DG and the load. The capacitive charging and discharging process to the ground is short under distribution network faults, and the influence of the fault transition process is minimal, so the capacitive reactance to the ground of the AC line can be neglected. Therefore, we refer to the existing studies on fault analysis and utilize the resistance and reactance series model to equate the AC line. The resistance grounding model is used as the fault model. According to the positive sequence equivalent rule, the composite sequence network of single-phase short-circuit faults incorporating all three cases can be established as shown in Figure 1. In Figure 1, $e_{\mathrm{S}}$ is the equivalent potential vector of the superior grid; $Z_{\mathrm{L}1}$ is the equivalent impedance of the AC line from the bus to the PCC for DG₁ (PCC₁); $\alpha$ it is the ratio of the distance from the fault point f₁ to the bus to the distance from the bus to PCC₁; $Z_{\mathrm{L}2}$ is the equivalent impedance of the AC line between PCC₁ and the PCC for DG₂ (PCC₂); $\beta$ is the ratio of the distance from fault point f₂ to PCC₁ to the distance from PCC₁ to PCC₂; $Z_{\mathrm{L}3}$ is the equivalent impedance of the AC line from PCC₂ to the end of the feeder; $\chi$ is the ratio of the distance from fault point f₃ to PCC₂ to the distance from PCC₂ to the end of the feeder; $Z_{\mathrm{L}}$ is the load equivalent impedance; $Z_{\mathrm{f}i}=Z_{2i}+Z_{0i}+3R_{\mathrm{f}i}$ is the additional impedance, $Z_{2i}$ and $Z_{0i}$ are the equivalent impedances of negative sequence network and zero sequence network of the single-phase short-circuit fault, respectively; $R_{\mathrm{f}i}$ is the fault transition resistance; and $i=1~3$ represents a single-phase short-circuit fault located between the bus and a DG, between DGs, and between a DG and the load, respectively.

In Figure 1, when the single-phase short-circuit fault occurs between the bus and a DG, the current flowing head of the feeder is the short-circuit current provided by the superior grid. The current flowing through the upstream of PCC₁ is the sum of the short-circuit current vectors of DG₁ and DG₂. Because the capacity of the superior power grid is much larger than that of the distribution network, the bus voltage remains constant during the fault. With the bus to the load as the reference direction, the current vector at the head of the feeder can be expressed as follows:

$$I_{\mathrm{up}.1}=\frac{\sqrt{E_{\mathrm{S}}^2+U_{\mathrm{f}1}^2-2E_{\mathrm{S}}{U}_{\mathrm{f}1}\cos {\delta }_{\mathrm{f}1}}}{\alpha \left|Z_{\mathrm{L}1}\right|}{\mathrm{e}}^{\mathrm{j}\left(\arctan \frac{-U_{\mathrm{f}1}\sin {\delta }_{\mathrm{f}1}}{E_{\mathrm{S}}-U_{\mathrm{f}1}\cos {\delta }_{\mathrm{f}1}}-\phi \right)}$$

(1)

where $I_{\mathrm{up}.1}$ is the current vector at the head of the feeder when a single-phase short circuit fault occurs between the bus and a DG; $E_{\mathrm{S}}$ is the equivalent potential magnitude of the superior grid; $U_{\mathrm{f}1}$ and ${\delta }_{\mathrm{f}1}$ are the voltage magnitude and phase angle at the additional impedance, respectively; and $\left|Z_{\mathrm{L}1}\right|$ and $\phi$ are the magnitude and phase angle of the equivalent impedance of the AC line from the bus to PCC₁, respectively.

In the case of a single-phase short-circuit fault, the active distribution network is asymmetric, and there are negative- and zero-sequence components of the voltage of the PCC for DG. However, the connected transformer for the DG is usually not grounded, so there is no zero-sequence current on the DG side, but there is a negative-sequence current path [28]. The superposition of positive- and negative-sequence currents on the DG side may exceed the converter capacity and threaten DG safety, so the DG usually needs to suppress its negative-sequence current under the single-phase short-circuit fault and provide reactive current to support the grid voltage. The short-circuit current of DG under grid fault usually reaches the maximum limit, so according to the control equation of the low-voltage ride-through for photovoltaic [29], the short-circuit current vectors of DG₁ and DG₂ can be expressed, respectively, as:

$$\left\{\begin{array}{l}{I}_{\mathrm{DG}1}=I_{\max 1}{\mathrm{e}}^{\mathrm{j}\left({\delta }_1-{\lambda }_1\right)}\\ I_{\mathrm{DG}2}=I_{\max 2}{\mathrm{e}}^{\mathrm{j}\left({\delta }_2-{\lambda }_2\right)}\end{array}\right.$$

(2)

where $I_{\mathrm{DG}1}$ and $I_{\mathrm{DG}2}$ are the short-circuit current vectors of DG₁ and DG₂, respectively; $I_{\max 1}$ and $I_{\max 2}$ are the maximum allowable current magnitude of DG₁ and DG₂, respectively; ${\delta }_1$ and ${\delta }_2$ are the voltage phase angles of PCC₁ and PCC₂, respectively; and ${\lambda }_1$ and ${\lambda }_2$ are the angles at which the short-circuit current vectors of DG₁ and DG₂ lag behind the voltage vectors of PCC₁ and PCC₂, respectively, calculated according to the following:

$$\left\{\begin{array}{l}{\lambda }_1=\arctan \frac{K\left(U_{\mathrm{DG}1}-0.9U_{\mathrm{N}}\right)I_{\mathrm{DG}1.\mathrm{N}}/U_{\mathrm{N}}}{\sqrt{I_{\max 1}^2-{\left[K\left(U_{\mathrm{DG}1}-0.9U_{\mathrm{N}}\right)I_{\mathrm{DG}1.\mathrm{N}}\right]}^2}}\\ {\lambda }_2=\arctan \frac{K\left(U_{\mathrm{DG}2}-0.9U_{\mathrm{N}}\right)I_{\mathrm{DG}2.\mathrm{N}}/U_{\mathrm{N}}}{\sqrt{I_{\max 2}^2-{\left[K\left(U_{\mathrm{DG}2}-0.9U_{\mathrm{N}}\right)I_{\mathrm{DG}2.\mathrm{N}}\right]}^2}}\end{array}\right.$$

(3)

where $K$ is the reactive current coefficient of DG, which is determined according to the grid code; $U_{\mathrm{DG}1}$ and $U_{\mathrm{DG}2}$ is the voltage magnitude of PCC₁ and PCC₂, respectively; $U_{\mathrm{N}}$ is the rated voltage of the distribution network; and $I_{\mathrm{DG}1.\mathrm{N}}$ and $I_{\mathrm{DG}2.\mathrm{N}}$ are the rated currents of DG₁ and DG₂, respectively.

According to Kirchhoff’s law, the relationship between the voltage vectors of PCC₁ and PCC₂ and the voltage vector at the additional impedance can be established as follows:

$$\left\{\begin{array}{l}{U}_{\mathrm{DG}1}=U_{\mathrm{DG}2}+I_{\mathrm{DG}2}{Z}_{\mathrm{L}2}\\ U_{\mathrm{f}1}=U_{\mathrm{DG}1}+\left(I_{\mathrm{DG}1}+I_{\mathrm{DG}2}\right)\left(1-\alpha \right)Z_{\mathrm{L}1}\end{array}\right.$$

(4)

where $U_{\mathrm{DG}1}$ and $U_{\mathrm{DG}2}$ are the voltage vectors of PCC₁ and PCC₂, respectively; and $U_{\mathrm{f}1}$ are the voltage vectors at the additional impedance.

Combining Equations (2)–(4), full-rank nonhomogeneous linear equations can be constructed for the voltage magnitudes and phase angles of PCC₁ and PCC₂ based on the equal vector magnitudes and phase angles. Therefore, the voltage magnitudes and phase angles of PCC₁ and PCC₂ can be uniquely represented by the voltage magnitude and phase angle at the additional impedance. The current vector upstream of PCC₁ can be expressed as a function of the voltage magnitude and phase angle at the additional impedance as follows:

$$I_{\mathrm{do}.1}={\mathrm{e}}^{\mathrm{j}\arctan \frac{I_{\max 1}\sin \left({\delta }_1-{\lambda }_1\right)+I_{\max 2}\sin \left({\delta }_2-{\lambda }_2\right)}{I_{\max 1}\cos \left({\delta }_1-{\lambda }_1\right)+I_{\max 2}\cos \left({\delta }_2-{\lambda }_2\right)}}\times \sqrt{I_{\max 1}^2+I_{\max 2}^2+2I_{\max 1}{I}_{\max 2}\cos \left({\delta }_1-{\lambda }_1-{\delta }_2+{\lambda }_2\right)}$$

(5)

where $I_{\mathrm{do}.1}$ is the current vector measured upstream of PCC₁ when a single-phase short-circuit fault occurs between the bus and a DG.

2.2. Fault Point Is Located between DGs

When the fault point is located between DGs, the current vector flowing downstream of PCC₁ is the sum of the short-circuit current of the superior grid and DG₁, and the current flowing upstream of PCC₂ is the opposite short-circuit current vector of DG₂. Referring to Equations (2) and (3), the current vector upstream of PCC₂ can be expressed as:

$$I_{\mathrm{do}.2}=I_{\max 2}{\mathrm{e}}^{\mathrm{j}\left({\delta }_2-\arctan \frac{K\left(U_{\mathrm{DG}2}-0.9U_{\mathrm{N}}\right)I_{\mathrm{DG}2.\mathrm{N}}}{U_{\mathrm{N}}\sqrt{I_{\max 2}^2-{\left[K\left(U_{\mathrm{DG}2}-0.9U_{\mathrm{N}}\right)I_{\mathrm{DG}2.\mathrm{N}}\right]}^2}}\right)}$$

(6)

where $I_{\mathrm{do}.2}$ is the current vector measured upstream of PCC₂ when a single-phase short-circuit fault occurs between DGs.

Since the phase angle of the short-circuit current of DG₂ depends on the voltage magnitude and phase angle of PCC₂, according to Kirchhoff’s law, the voltage vector of PCC₂ can also be expressed by the voltage vector at the additional impedance as:

$$U_{\mathrm{DG}2}=U_{\mathrm{f}2}\cos {\delta }_{\mathrm{f}2}+\mathrm{j}{U}_{\mathrm{f}2}\sin {\delta }_{\mathrm{f}2}+I_{\max 2}\left(1-\beta \right)\times \left|Z_{\mathrm{L}2}\right|{\mathrm{e}}^{\mathrm{j}\left({\delta }_2-\arctan \frac{K\left(U_{\mathrm{DG}2}-0.9U_{\mathrm{N}}\right)I_{\mathrm{DG}2.\mathrm{N}}}{U_{\mathrm{N}}\sqrt{I_{\max 2}^2-{\left[K\left(U_{\mathrm{DG}2}-0.9U_{\mathrm{N}}\right)I_{\mathrm{DG}2.\mathrm{N}}\right]}^2}}+\phi \right)}$$

(7)

where $U_{\mathrm{f}2}$ and ${\delta }_{\mathrm{f}2}$ are the voltage magnitude and phase angle at the additional impedance; and $\left|Z_{\mathrm{L}2}\right|$ is the magnitude of the equivalent impedance of the AC line between PCC₁ and PCC₂.

Combining Equations (6) and (7), full-rank nonhomogeneous linear equations can be constructed for the voltage magnitude and phase angle of PCC₂. Therefore, when the fault point is located between DGs, the current magnitude measured upstream of PCC₂, i.e., at the tail of the fault section, is constant. At the same time, the phase angle depends on the voltage magnitude and phase angle at the additional impedance. The current measured at the head of the fault section can be expressed as:

$$I_{\mathrm{up}.2}=I_{\mathrm{up}.2}{\mathrm{e}}^{\mathrm{j}{\delta }_{\mathrm{up}.2}}$$

(8)

where $I_{\mathrm{up}.2}$ is the current vector flowing downstream of PCC₁ when a single-phase short-circuit fault occurs between DGs. Its magnitude $I_{\mathrm{up}.2}$ and phase angle ${\delta }_{\mathrm{up}.2}$ are calculated according to the following:

$$\left\{\begin{array}{l}{I}_{\mathrm{up}.2}=\sqrt{\begin{array}{l}{\left(\frac{\sqrt{E_{\mathrm{S}}^2+U_{\mathrm{DG}1}^2-2E_{\mathrm{S}}{U}_{\mathrm{DG}1}\cos {\delta }_1}}{\alpha \left|Z_{\mathrm{L}1}\right|}\right)}^2+{\left(I_{\max 1}\right)}^2\\ +2\frac{\sqrt{E_{\mathrm{S}}^2+U_{\mathrm{DG}1}^2-2E_{\mathrm{S}}{U}_{\mathrm{DG}1}\cos {\delta }_1}}{\alpha \left|Z_{\mathrm{L}1}\right|}{I}_{\max 1}\cos \left(p-q\right)\end{array}}\\ {\delta }_{\mathrm{up}.2}=\frac{\frac{\sqrt{E_{\mathrm{S}}^2+U_{\mathrm{DG}1}^2-2E_{\mathrm{S}}{U}_{\mathrm{DG}1}\cos {\delta }_1}}{\alpha \left|Z_{\mathrm{L}1}\right|/\sin p}+I_{\max 1}\sin q}{\frac{\sqrt{E_{\mathrm{S}}^2+U_{\mathrm{DG}1}^2-2E_{\mathrm{S}}{U}_{\mathrm{DG}1}\cos {\delta }_1}}{\alpha \left|Z_{\mathrm{L}1}\right|/\cos p}+I_{\max 1}\cos q}\end{array}\right.$$

(9)

where the parameters p and q are calculated according to the following:

$$\left\{\begin{array}{l}p=\arctan \frac{-U_{\mathrm{DG}1}\sin {\delta }_1}{E_{\mathrm{S}}-U_{\mathrm{DG}1}\cos {\delta }_1}-\phi \\ q={\delta }_1-\arctan \frac{K\left(U_{\mathrm{DG}1}-0.9U_{\mathrm{N}}\right)I_{\mathrm{DG}1.\mathrm{N}}/U_{\mathrm{N}}}{\sqrt{I_{\max 2}^2-{\left[K\left(U_{\mathrm{DG}1}-0.9U_{\mathrm{N}}\right)I_{\mathrm{DG}1.\mathrm{N}}\right]}^2}}\end{array}\right.$$

(10)

From Equation (8), it can be seen that the current magnitude and phase angle flowing downstream of PCC₁ are related to the voltage magnitude and phase angle of PCC₁. The relationship between the voltage vector of PCC₁ and the voltage vector at the additional impedance can be expressed as:

$$U_{\mathrm{DG}1}=I_{\mathrm{up}.2}{Z}_{\mathrm{L}1}+U_{\mathrm{f}2}$$

(11)

where $U_{\mathrm{f}2}$ is the voltage vector at the additional impedance.

Combining Equations (8) and (11) yields full-rank nonhomogeneous linear equations for the voltage magnitude and phase angle of PCC₁. Therefore, the voltage magnitude and phase angle of PCC₁ can be uniquely represented by the voltage magnitude and phase angle at the additional impedance. The current magnitude and phase angle flowing downstream of PCC₁ depend on the voltage magnitude and phase angle at the additional impedance.

3. Characteristic Quantities of Single-Phase Short-Circuit Faults

3.1. Measurement of Current Magnitude Difference

When the fault point is located between the bus and a DG, the current flowing through the head of the feeder is the short-circuit current provided by the superior grid, and the short-circuit current increases with the decrease in voltage at the additional impedance. The current flowing upstream of PCC1 is the short-circuit current provided by the DGs, which does not exceed the algebraic sum of the maximum allowable current of DGs downstream of the fault point. Because the short-circuit current of the superior grid is generally several times the rated current, and the short-circuit current of a DG does not exceed 1.5 times the rated current under the current-limiting controller [30], therefore, the current magnitude upstream of the fault point is significantly larger than that downstream, and the magnitude difference between the two decreases with the increase in DG capacity downstream of the fault point. When the fault point is located between DGs, it can be seen from Equations (6) and (8) that the current upstream of the fault point is the short-circuit current of the superior grid and DG1, and the current downstream of the fault point is the short-circuit current provided by the DGs downstream of the fault point. Due to the short-circuit current of the superior grid, the current magnitude upstream of the fault point is also much larger than the current magnitude downstream, as shown in Figure 2. When the fault point is located between a DG and the load, because the load impedance is much larger than the line impedance [31], the current flowing downstream of the fault point is much smaller than the short-circuit current measured upstream of the fault point.

Whether a single-phase short-circuit fault occurs between the bus and a DG, between DGs, or between a DG and the load due to the difference between the short-circuit current of DGs and synchronous generators, there is a phenomenon of limited magnitude. The magnitude of the current vector measured at the end of the fault section is always smaller than the current magnitude measured at the head of the fault section. The current magnitude difference between the two ends of the section is the absolute value of the current magnitude difference between the head and the end of the section. When an internal fault occurs, the current magnitude difference between the two ends of the section is greater than zero. Because the AC line in the section is short, the capacitive current to the ground in the section is small, and the current magnitude difference between the two ends of the non-fault section is almost zero. Using Max-Min standardization to eliminate the influence of dimension, the measurement of the current magnitude difference can be constructed as follows [32]:

$$F_I^j=\frac{I_j-I_{\min }}{I_{\max }-I_{\min }}$$

(12)

where $F_I^j$ is the measurement of the current magnitude difference of section j; $I_j$ is the current magnitude difference between the two ends of section j; and $I_{\max }$ and $I_{\min }$ are the maximum and minimum values of the current magnitude difference between the two ends of all sections in the historical monitoring data, respectively, and are constants.

From Equation (12), it can be seen that the measurement of the current magnitude difference is only related to its current magnitude difference. The current magnitude difference measurement of the fault section is greater than that of a non-fault section, and the measurement of the current magnitude difference can reflect the location of the single-phase short-circuit fault in the active distribution network.

3.2. Measurement of Current Phase Angle Difference

According to Equation (1), when the fault point is located between the bus and a DG, the phase angle of the current vector measured at the head of the fault section over the voltage vector is:

$${\varphi }_{\mathrm{up}}^1=\arctan \frac{-U_{\mathrm{f}1}\sin {\delta }_{\mathrm{f}1}}{E_{\mathrm{S}}-U_{\mathrm{f}1}\cos {\delta }_{\mathrm{f}1}}-\phi$$

(13)

where ${\varphi }_{\mathrm{up}}^1$ is the current phase angle at the head of the fault section when the single-phase short-circuit fault occurs between the bus and a DG.

The phase angle of the current vector ahead of the voltage vector measured at the end of the fault section is:

$${\varphi }_{\mathrm{do}}^1=-{\delta }_1-\arctan \frac{I_{\max 1}\sin \left({\delta }_1-{\lambda }_1\right)+I_{\max 2}\sin \left({\delta }_2-{\lambda }_2\right)}{I_{\max 1}\cos \left({\delta }_1-{\lambda }_1\right)+I_{\max 2}\cos \left({\delta }_2-{\lambda }_2\right)}$$

(14)

where ${\varphi }_{\mathrm{do}}^1$ is the current phase angle at the end of the fault section when the single-phase short-circuit fault occurs between the bus and a DG.

The current phase angle at the end of the fault section depends only on the voltage at the additional impedance. The current direction is from the load to the bus, opposite the reference. Hence, the current phase angle is positive. The current phase angle at the head of the fault section is related to the voltage at the additional impedance and the line impedance angle, and the current direction is the same as the reference direction. Regardless of whether the fault point is located between DGs or between a DG and the load, the current vector of the head of the fault section lags behind the voltage vector. However, the difference between the two is that when a single-phase short-circuit fault occurs between DGs, the current vector measured at the end is ahead of the voltage vector, while when a single-phase short-circuit fault occurs between a DG and the load, the current vector measured at the end of the fault section, i.e., the load side, is almost zero.

Regardless of the location of the fault and the PCC for a DG, the current vector measured at the end of the fault section is always in the opposite direction to the reference direction, and the current phase angle varies with the voltage drop. The maximum current phase angle is obtained when the reactive current provided by a DG increases to the maximum value according to the voltage drop. Therefore, the current phase angle difference between the two ends of the section is the absolute value of the current phase angle measured at the end over the current phase angle at the head end. The current phase angle difference between the two ends of the section is significantly greater than zero when the section is faulted, and the current phase angle difference between the two ends of the non-faulted section is almost zero. From this, the measurement of the current phase angle difference can be constructed as follows:

$$F_{\eta }^j=\frac{{\eta }_j-{\eta }_{\min }}{{\eta }_{\max }-{\eta }_{\min }}$$

(15)

where $F_{\eta }^j$ is the measurement of the current phase angle difference of section j; ${\eta }_j$ is the current phase angle difference between the two ends of section j; and ${\eta }_{\max }$ and ${\eta }_{\min }$ are the maximum and minimum values of the current phase angle difference between the two ends of all sections in the historical monitoring data, respectively, and are constants.

From Equation (15), it can be seen that the measurement of the current phase angle difference is only related to its current phase angle difference. The measurement of the current phase angle difference of the fault section is larger than that of a non-fault section, so the measurement of the current phase angle difference can also reflect the location of the single-phase short circuit fault in the active distribution network. The closer it is to 1, the more likely it is to be a fault section, and the closer it is to 0, the more likely it is to be a non-fault section.

4. Principle of Single-Phase Short-Circuit Fault Location

4.1. Fault Confidence Derived from Measurement of Current Magnitude Difference and Phase Angle Difference

Fault confidence refers to the degree of confidence in supporting a section as a fault section when a certain characteristic quantity is used as evidence. The difference between fault and non-fault sections under a single-phase short-circuit fault is defined as the horizontal confidence distribution function. The difference between current and previous fault sections is defined as the vertical confidence distribution function. The product of the two can be used as the fault confidence of a certain piece of evidence. A proportional function is introduced to describe the ratio of the measurement of the current magnitude difference of a section to the measurement of the current magnitude difference of all sections, thus reflecting the horizontal comparison of the fault probability of each section under a single-phase short-circuit fault. The larger the ratio of the measurement of the current magnitude difference of a section, the more likely it is to be a fault section. According to evidence theory, a segment with a tiny percentage of the measurement of the current magnitude difference cannot be evidence of fault segment location, and its percentage value should reflect the uncertainty in confidence. Therefore, the horizontal confidence distribution function of the measurement of the current magnitude difference is established as follows:

$${\mu }_I(j)=\left\{\begin{array}{l}{F}_I^j,F_I^j>F_I^{\mathrm{set}}\\ 0,F_I^j\le F_I^{\mathrm{set}}\end{array}\right.$$

(16)

where ${\mu }_I(j)$ and $F_I^{\mathrm{set}}$ are the horizontal confidence for section j and the minimum threshold when the measurement of current magnitude difference is used as evidence, respectively.

Considering that the current magnitude difference decreases as the DG capacity downstream of the fault point increases, sections with different measurements of current magnitude difference may have the same horizontal confidence. However, the confidence of a single-phase short-circuit fault occurring in the section with a small measurement of current magnitude difference is lower than that of the section with a large measurement of current magnitude difference. For this reason, the vertical confidence distribution function is used to redistribute the confidence based on the original horizontal confidence, making the measurement of the current magnitude difference more reliable as evidence of single-phase short-circuit faults. In this paper, the maximum value of the measurement of the current magnitude difference is used as a reference. The larger the current magnitude difference, the more significant the limitation of the short-circuit current magnitude of DGs downstream of the fault point, and the more reliable the single-phase short-circuit fault localization result. The vertical confidence distribution function when using the measurement of the current magnitude difference as evidence is [33]:

$$k_I=\left\{\begin{array}{l}1,b_I\le F_I^{\max }\le 1\\ \frac{F_I^{\max }}{b_I-a_I}-\frac{a_I}{b_I-a_I},a_I<F_I^{\max }<b_I\\ 0,0\le F_I^{\max }\le a_I\end{array}\right.$$

(17)

where $k_I$ is the vertical confidence distribution function when using the measurement of the current magnitude difference as evidence; $F_I^{\max }$ is the maximum value of the measurement of the current magnitude difference of all sections; and the parameter $a_I$ and $b_I$ are the fuzzy boundaries of the vertical confidence distribution function when using the measurement of the current magnitude difference as evidence.

Therefore, the fault confidence distribution function and uncertainty confidence when the measurement of the current magnitude difference are used as evidence are:

$$m_I\left(j\right)={\mu }_I(j)·k_I$$

(18)

$$m_I\left(U\right)=1-{\displaystyle \sum _{j=1}^n{\mu }_I(j)·k_I}$$

(19)

where $m_I\left(j\right)$ is the fault confidence distribution function for the measurement of the current magnitude difference; $m_I\left(U\right)$ is the uncertainty confidence when the measurement of the current magnitude difference is used as evidence; and n is the number of sections.

Similarly, by rewriting the measurement of the current magnitude difference in Equations (16)–(19) as the measurement of the current phase angle difference, we can obtain the horizontal confidence distribution function, vertical confidence distribution function, fault confidence distribution function, and uncertainty confidence when the measurement of the current phase angle difference is used as evidence.

4.2. Fault Location Based on Fusion Fault Confidence of Short-Circuit Current Vector

The flow of the proposed localization method of single-phase short-circuit faults for distribution networks containing distributed generators based on the fusion fault confidence of the short-circuit current vector is shown in Figure 3. When a fault occurs in the distribution network, and the relay protection completes the selection of the fault line, the fault localization method is activated, the voltage and current of the feeder terminal unit(FTU) are collected, the measurement of current magnitude difference and phase angle difference of each section are calculated according to Equations (12) and (15), respectively, and the fault confidence and uncertainty confidence of each section under two kinds of evidence of measurement—current magnitude difference and phase angle difference— is calculated. The fusion fault confidence of the short-circuit current vector for each section can then be calculated based on Dempster’s evidence rule:

$$m(j)=\frac{m_I(j)m_{\eta }(j)+m_I(j)m_{\eta }(U)+m_I(U)m_{\eta }(j)}{1-{\displaystyle \sum _{x\ne y}^{\begin{array}{l}x=1,2\dots n\\ y=1,2\dots n\end{array}}{m}_I(x)m_{\eta }(y)}}$$

(20)

where $m(j)$ is the fusion fault confidence of the short-circuit current vector of section j; and $m_{\eta }(j)$ is the fault confidence of section j when the measurement of current phase angle difference is used as evidence.

Furthermore, the fusion uncertainty confidence can be calculated as follows:

$$m\left(U\right)=1-{\displaystyle \sum _{j=1}^{n}m(j)}$$

(21)

where $m\left(U\right)$ is the fusion uncertainty confidence.

Next, the fusion fault confidence of the short-circuit current vector for each section and the fusion uncertainty confidence are compared. If there is a section with the maximum fusion fault confidence and a fusion uncertainty confidence of less than 0.1, then the section is determined to be the fault section.

Power system relay protection pursues simplicity and reliability. The proposed method calculates the fusion fault confidence of the short-circuit current vector only once by collecting the current magnitude difference and phase angle difference of each section without repeated iterations. Therefore, even if multiple branch lines exist in the active distribution network, locating the faulty section at once is also available, effectively improving localization efficiency. In addition, the proposed method requires no training of fault samples, so the fault localization computation is small. Therefore, the proposed method can meet the requirements of distribution networks of being simple in principle and having a high feasibility of fault localization. In addition, the proposed method does not restrict the distribution network topology. Branch lines are usually equipped with current collection devices, which can form a section and thus continue to apply the proposed method.

5. Case Study

Distribution networks are generally radial and accompanied by DG T-connections, so the simulation system shown in Figure 4 is built to verify the correctness of the proposed fault location method [34]. A 110 ± 8 × 1.25%/10 kV transformer neutral is grounded through a small 10 Ω resistor and five groups of capacitors are connected in parallel at the bus terminals, with the capacity of a single group being 0.5 MVar. The positive-sequence unit parameter of the feeder line and the negative-sequence unit parameter are $r_1=0.031 \Omega /\mathrm{km}$, $l_1=0.096 \mathrm{mH}/\mathrm{km}$, and $c_1=0.338 \mu \mathrm{F}/\mathrm{km}$, and the zero-sequence unit parameters are $r_0=0.234 \Omega /\mathrm{km}$, $l_0=0.355 \mathrm{mH}/\mathrm{km}$, and $c_0=0.265 \mu \mathrm{F}/\mathrm{km}$. The end of the feeder is connected to the load by a 10 kV/0.4 kV transformer. The load and DG parameters are shown in

Table 2. Parameters of load and DGs.

Feeder Number	Feeder Length/km	Load Capacity/MW	DG Power/MW	DG Location/km
1	12	7	1.5, 2, 2.5	2, 4, 8
2	10	6	2, 2	5, 7.5
3	8	10	3.5	3
4	9	8	4	6

. FTUs are installed at the heads of the feeders, upstream and downstream of the PCC for the DGs, and at the end of the feeder to collect current and voltage.

We use the example of a phase A ground fault for testing purposes. The accuracy of the proposed method is demonstrated by correctly localizing the faulted section under different scenarios of fault location, transition resistance, and DG capacity.

5.1. Different Fault Locations

Let the single-phase short-circuit fault occur at Feeder 2 at distances of from 1 to 9 km, and the transition resistance be 10 Ω. The fault confidence of measurements of the current magnitude differences and phase angle differences of the FTU9~10, FTU11~12, and FTU13~14 composition sections can then be obtained, as shown in

Table 3. Measurement of current magnitude difference and phase angle difference under different fault locations.

Fault Location/km	2.5	4.5	6.5	8.5
Magnitude difference measurement	0.814	0.085	0.139	0.859
	0.107	0.804	0.090	0.060
	0.079	0.111	0.771	0.081
Phase angle difference measurement	0.793	0.055	0.229	0.875
	0.097	0.819	0.144	0.067
	0.110	0.126	0.627	0.058

. Combined with Equations (20) and (21), the fusion fault confidence of the short-circuit current vector and the fusion uncertainty confidence under a single-phase short-circuit fault can be obtained, as shown in

Table 4. Fusion fault confidence of short-circuit current vector and fusion uncertainty confidence under different fault locations.

Fault Location/km	2.5	4.5	6.5	8.5
FTU9~10	0.973	0.951	0.034	0.233
FTU11~12	0.020	0.038	0.944	0.204
FTU13~14	0.007	0.011	0.022	0.467
Fusion uncertainty confidence	0.000	0.000	0.000	0.096

.

As shown in Table 4, when a single-phase short-circuit fault occurs on Feeder 2 at 2.5 km, the fusion fault confidence of the short-circuit current vector of the section from FTU9 to FTU10 is 0.973; the fusion fault confidence of the non-fault sections is 0.020 and 0.007, respectively; and the fusion uncertainty reliability is 0. Thus, it can be judged that the fault point is located between FTU9 and FTU10, i.e., between the head of Feeder 2 and 5 km, and the fault location is accurate. Similarly, when a single-phase short-circuit fault occurs at 4.5 km, 6.5 km, and 8.5 km at Feeder 2, the fault can be accurately located based on the fusion fault confidence of the short-circuit current vector. In most cases, the fusion fault confidence of the short-circuit current vector of the fault section is greater than 0.9, while the fusion fault confidence of the non-fault section is not greater than 0.01. The boundary between the non-fault section and the fault section is clear. The method proposed in this paper can accurately locate the fault section with high reliability.

5.2. Different Transition Resistance

Let the fault point be located at 2.5 km from Feeder 2, and change the transition resistance to 1 Ω, 10 Ω, 100 Ω, 500 Ω, and 1 kΩ, respectively, to obtain the measurement of the current magnitude difference and phase angle difference of each section under a single-phase short circuit fault, as shown in

Table 5. Measurement of current magnitude difference and phase angle difference under different transition resistances.

Transition Resistors/Ω	1	10	100	500	1000
Magnitude difference measurement	0.882	0.863	0.401	0.585	0.854
	0.058	0.072	0.312	0.233	0.060
	0.060	0.066	0.287	0.183	0.086
Phase angle difference measurement	0.893	0.706	0.525	0.420	0.859
	0.058	0.126	0.173	0.190	0.056
	0.049	0.169	0.302	0.390	0.085

. Based on the data contained in Table 5, the fusion fault confidence of the short-circuit current vector and the fusion uncertainty confidence with different transition resistances can be calculated, as shown in

Table 6. Fusion fault confidence of short-circuit current vector and fusion uncertainty confidence under different transition resistances.

Transition Resistors/Ω	1	10	100	500	1000
FTU9~10	0.740	0.982	0.938	0.572	0.665
FTU11~12	0.198	0.013	0.043	0.298	0.236
FTU13~14	0.061	0.004	0.016	0.130	0.099
Fusion uncertainty confidence	0.001	0.001	0.003	0.000	0.000

.

As shown in Table 6, the fusion fault confidence of the short-circuit current vector of the fault section composed of FTU9~10 reaches 0.740, 0.982, 0.938, 0.572, and 0.665, which are significantly higher than that of the non-fault sections, and the fusion uncertainty confidence varies little and is close to 0. The fusion fault confidence can accurately discriminate the fault section. Although the values of current magnitude difference and phase angle difference of the section are affected by the transition resistance, which makes the measurement of current magnitude difference and phase angle difference change under different transition resistances, the measurements of current magnitude difference and phase angle difference of the fault section are always greater than those of the non-fault sections. Since the fusion fault confidence reflects the difference between the magnitude and phase angle of the short-circuit current vectors upstream and downstream of the fault point rather than the numerical difference, the proposed method can effectively avoid the loss of sensitivity due to the change in transition resistance.

5.3. Different DG Capacity

Let the fault point be located at 2.5 km from Feeder 2, and change DG4 capacity to 0.5 MW, 1 MW, 2 MW, 4 MW, and 6 MW, respectively. By the measurement of the current magnitude difference and phase angle difference of each section in the single-phase short-circuit fault line, the fusion fault confidence of the short-circuit current vector and the fusion uncertainty confidence under different DG capacities can be obtained, as shown in

Table 7. Measurement of current magnitude difference and phase angle difference under different DG capacities.

DG Capacity/MW		0.5	1.0	2.0	4.0	6.0
Magnitude difference measurement	FTU9~10	0.854	0.890	0.860	0.853	0.854
	FTU11~12	0.060	0.049	0.069	0.058	0.060
	FTU13~14	0.086	0.061	0.072	0.088	0.086
Phase angle difference measurement	FTU9~10	0.859	0.844	0.849	0.849	0.859
	FTU11~12	0.056	0.044	0.038	0.058	0.056
	FTU13~14	0.085	0.112	0.113	0.093	0.085

and

Table 8. Fusion fault confidence of short-circuit current vector and fusion uncertainty confidence under different DG capacities.

DG Capacity/MW	0.5	1.0	2.0	4.0	6.0
FTU9~10	0.966	0.975	0.968	0.967	0.974
FTU11~12	0.022	0.014	0.019	0.021	0.017
FTU13~14	0.012	0.009	0.012	0.012	0.001
Fusion uncertainty confidence	0.000	0.002	0.001	0.000	0.008

.

As shown in Table 8, the fusion fault confidence of the fault section fluctuates from 0.96 to 0.98, while the fusion fault confidence of the short-circuit current vector of the non-fault sections is always less than 0.1. The fusion fault confidence of the fault section is always significantly greater than that of the non-fault sections, and can all reliably locate the fault. Using the magnitude difference and phase angle difference of the current vector of the section, the fusion fault confidence-based localization criterion is constructed with the help of evidence theory, and its localization accuracy is not affected by the fault location and DG capacities. It has better resistance to fault transition resistance.

5.4. Sensitivity Analysis of the Proposed Method

The proposed method compares the fusion fault confidence of the short-circuit current vector of each section and determines that the largest one is the faulty section. Therefore, we take the ratio of the fusion fault confidence of the short-circuit current vector of the faulty section to the maximum value of the fusion fault confidence of the short-circuit current vector in the non-fault sections as the sensitivity. The larger the ratio, the more significant the difference between the faulted and non-faulted sections under the proposed method and thus, the higher the sensitivity. The sensitivities of the proposed method under the influence of different fault locations, fault transition resistances, and distributed generator capacity are shown in

Table 9. Sensitivity at different fault locations.

Fault Location/km	2.5	4.5	6.5	8.5
Sensitivity	48.65	25.03	27.76	2.00

,

Table 10. Sensitivity at different fault transition resistances.

Transition Resistors/Ω	1	10	100	500	1000
Sensitivity	3.74	75.46	21.81	1.92	2.82

and

Table 11. Sensitivity at different distributed generator capacities.

DG Capacity/MW	0.5	1.0	2.0	4.0	6.0
Sensitivity	43.91	69.64	50.95	46.05	57.29

.

5.5. Comparison with Conventional Method

Existing distribution network fault localization usually uses impedance as the localization criterion. The simulation model and parameters are kept the same. A single-phase short-circuit fault with different locations and transition resistances occurs on Feeder 2, and the superiority of the proposed fault localization method is verified by comparing it with the impedance principle localization method. A single-phase short-circuit fault with a transition resistance of 1Ω is located at 2.5 km, 4.5 km, 6.5 km, and 8.5 km from Feeder 2, and the measured impedance and the corresponding section localization results are shown in

Table 12. Comparison at different fault sections.

Fault Point/km	Fault Section	Impedance Threshold Value/Ω	Measured Impedance/Ω	Localization Section
2.5	FTU9~10	0.16 + j0.15	1.11 + j0.18	Outside fault
4.5	FTU9~10	0.16 + j0.15	1.14 + j0.21	Outside fault
6.5	FTU11~12	0.08 + j0.08	1.19 + j0.24	Outside fault
8.5	FTU13~14	0.06 + j0.07	1.18 + j0. 32	Outside fault

.

As shown in Table 12, when the faults are located at 2.5 km, 4.5 km, 6.5 km, and 8.5 km from the feeder, the measured impedance of the FTU9~10, FTU11~12, and FTU13~14 are all larger than the impedance threshold value. All of the impedance localization principles incorrectly judge the fault section. The reason for the low correctness of the impedance localization principle is that the impedance principle is affected by the transition resistance and the additional impedance introduced by DGs. As seen in Section 5.1, the proposed fault location method can accurately locate the faulted section under the above fault conditions, avoiding the influence of the additional impedance introduced by DGs on the impedance principle location method. This proves that the proposed method has the advantage of high accuracy compared with existing impedance principle fault localization methods.

A single-phase short-circuit fault with transition resistances of 0.1 Ω, 1 Ω, 2 Ω, and 10 Ω is set to occur at 2.5 km from Feeder 2, and the simulation resuzlts are shown in

Table 13. Comparison under different transition resistances.

Transition Resistance/Ω	Fault Section	Impedance Threshold Value/Ω	Measured Impedance/Ω	Localization Section
0	FTU9~10	0.16 + j0.15	0.14 + j0.14	FTU9~10
1	FTU9~10		1.15 + j0.17	Outside fault
2	FTU9~10		2.19 + j0.26	Outside fault
10	FTU9~10		10.65 + j0.22	Outside fault

. Table 13 shows that when a metallic short circuit occurs, the impedance localization principle can correctly determine that the fault is located in FTU9~10. However, with an increase in transition resistance, the impedance localization principle cannot correctly locate the fault. As seen from Section 5.2, for different transition resistance sizes, the proposed fault location method can accurately determine the fault location; even when the transition resistance is as high as 1000 Ω, the proposed method can also correctly locate the fault. This proves that the proposed method has the advantage of being insensitive to changes in transition resistance, unlike existing fault localization methods.

5.6. Comparison with Physical Data Combination-Based Methods

Machine-learning-based fault localization methods’ high historical data requirements and limited physical interpretability have resulted in them being yet to be further demonstrated for application in real power grids. The drawbacks of traveling wave methods in distribution grids containing distributed generators have been discussed extensively. Therefore, we compare with the method of reference [35]. The method of reference [35] is based on the current deviation 2-parameter for fault localization in active distribution networks.

The simulation model and parameters remain unchanged from the reference. As

Table 14. The relationship between FTU node n and current deviation.

Fault distance: 2.5 km
Node number	9	10	12	14
Current deviation 2-parameter	0.457	0.855	1.512	2.505
Fault distance: 4.5 km
Node number	9	10	12	14
Current deviation 2-parameter	0.596	0.354	0.587	1.685
Fault distance: 6.5 km
Node number	9	10	12	14
Current deviation 2-parameter	1.655	0.682	0.655	1.324
Fault distance: 8.5 km
Node number	9	10	12	14
Current deviation 2-parameter	2.866	1.245	0.752	0.522

shows, it can be obtained that when the fault distance is 2.5 km, the current deviation of FTU 9 and FTU 10 are 0.457 and 0.855, respectively, which are the two nodes with the least current deviation, so it is judged that the fault interval is FTU 9~10, and the fault section localization is accurate. However, when the fault distance is 4.5 km, FTU11 and FTU12 are the two nodes with the least current deviation, and the fault interval is judged to be FTU11~12, but the real fault point occurs in FTU9~FTU10, and, therefore, the fault location is wrong. The reason for this is that when the fault point is close to the FTU, the node with the smallest current deviation can be judged accurately. However, its upstream and downstream neighboring nodes have current deviations of similar magnitude, and there is a risk of misjudgment. In contrast, as shown in Section 5.1, the method proposed in this paper can tolerate fault location changes. Even if the fault point is close to the FTU, there is no misjudgment of the fault section.

6. Conclusions

Addressing the challenge of single-phase short-circuit fault location in distribution networks containing distributed generators, we derive the expression of the single-phase short-circuit current vector in an active distribution network. Based on the characteristics that the magnitude difference and phase angle difference of the fault section are larger than those of non-fault sections, the fault location characteristic quantities of measurement of the current magnitude difference and phase angle difference are established, and then the fault location criterion based on the fusion fault confidence of the short-circuit current vector is constructed. The validation of 15 sets of different operating conditions shows that the proposed method has strong resistance to changes in fault location, transition resistance, and DG capacity; the ratio of the fusion fault confidence of short-circuit current vector between faulted and non-faulted sections is greater than 2, and the variation in the fusion uncertainty confidence is slight and close to 0.

Through a comparison as shown in

Table 15. Comparison of the proposed method with existing methods.

Method	Protection Adaptability	Physical Interpretability	Engineering Practicability
Impedance method	0	0	−1
Mobile wave method	−1	0	+1
Intelligent method	0	−1	0

Note: +1 means advantageous, 0 means fair, and −1 means disadvantageous.

, the proposed method is a method that balances protection adaptability, physical interpretability, and engineering practicability. It does not require high-precision mobile wave detection devices; by comparing the fusion fault confidence of the short-circuit current vector between sections, it avoids the sensitivity limitation of the impedance method; it does not require training of fault samples, and the fault localization process has a clear physical meaning. Therefore, the proposed method has a balance of advantages over existing methods.

The theoretical basis of the proposed method is the positive sequence equivalent rule, so the proposed method also applies to other short-circuit faults. However, there are significant differences in the variation patterns of electrical quantities before and after the break. Therefore, it is challenging to apply the proposed method to break faults. Further integration of non-industrial frequency component characteristics of short-circuit current is the future research direction.