Corrigendum to “Fault Location of Active Distribution Networks Based on the Golden Section Method”

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Introduction
With the development of society, more and more DERs are connected to the distribution network, which leads to more clean energy applications and brings huge economic benefits [1][2][3]. e integration of distributed generation (DG) and its synchronous operation with power grid is one of the challenges we now face [4][5][6]. In this case, the supervision and management of power grid with self-healing function become more complex. Fast fault location is an important task to restore power supply when the distribution network fails. Although technology is developing rapidly, due to the increasing expectations of users, the fault location technology of distribution network still needs improvement.
At present, the fault location methods for active distribution network are based on traveling wave method, artificial intelligence method, and impedance method. Traveling wave analysis is widely used. In [7][8][9][10], the traveling wave fault location method was introduced, but this method requires a higher sampling frequency and has many links, which has some impacts on the overall reliability. In [11], the evaluation function for the characteristics of active distribution network was redefined, and a suitable immune algorithm was proposed. In [12], a radial basis function neural network model was proposed based on the fault information uploaded by the distribution network monitoring and data acquisition system. e improved algorithm is applied to fault location of active distribution network. In [13], a fault location method based on wavelet fuzzy neural network for postfault transient and steady-state measurements is proposed. Although the abovementioned artificial intelligence-based fault location method has high performance, it requires a lot of data to learn, and its accuracy is affected by the database. In [14][15][16][17], fault location is achieved by collecting synchronous current and voltage vectors. is method needs to measure synchronous current and voltage information, thus increasing equipment investment. Impedance-based fault location method is one of the current research methods and has achieved good results in applications [18][19][20][21]. However, this method cannot solve the problem of fault location in the active distribution network. In [22,23], the effect of DG on the distribution network was simulated using a synchronous machine, but only one DER was considered, and the effect of different DER penetration levels was not analyzed [24]. e fault location method proposed in this paper combines the golden section method and the trapezoidal iteration technique to analyze the fault location of the active distribution network under different operating conditions [25]. is method eliminates the potential relationship between the accuracy of fault location and search step size. In addition, this method only needs to collect the synchronous current vector of the DER, which reduces the equipment investment. It improves the accuracy of fault location and reduces costs.

Minimum Fault Reactance
e method proposed in this paper is to use the minimum fault reactance method [26]. Assuming that the fault points of the analysis line segment are random, the fault reactance equation can be obtained by calculating the fault reactance through the analysis circuit [27]. Calculate the fault reactance at each assumed fault point. Repeat this process on all line segments. Most line faults have impedance characteristics, so the fault location can be solved by solving the minimum fault reactance value. In the distribution network shown in Figure 1, there is a line fault point between node n and n + 1. Because the fault line segment is unknown in advance, therefore, the first line section of the substation is considered first.
As shown in Figure 2, by controlling the closing and opening of switches S 1 , S 2 , S 3, and S 4 , the active distribution network can obtain different fault types. A generalized expression of the fault point voltage V f is obtained by analyzing Figure 2 [28].
where [Z abc ] is the impedance matrix between nodes n and n + 1, [Y abc ] is the admittance matrix, [U] is the third-order identity matrix, [V f n ] is the fault voltage vector of node n, [I f n ] is the fault current vector, x is the normalized fault distance which is the ratio of the fault distance of the line to its length.
Considering the influence of the downstream current, use formula (2) to calculate the fault current: Generalized expression of fault reactance obtained by formulas (1) and (2): where V fm is the voltage vector of fault phase m and V fn is the voltage vector of fault phase n. WhenS − q m , S − q n is 1, it indicates that the phase has a fault; otherwise, it is 0. Im is the imaginary part solve operation. Table 1 shows the measured voltage and current vectors for solving the reactance of different fault types.

Golden Section Method
e primary task of the fault location method based on the minimum fault reactance value is to find the minimum reactance. e previous methods usually use a constant step length of Δm or a second derivative to solve the problem [29]. e golden section method proposed in this paper selects the test point by the principle of symmetry and keeping the reduction ratio [30]. By comparing the function values of the test points, the search interval is shortened to reach the minimum interval of the minimum point. e accuracy and speed of the search method are related to the step length. e golden section method takes fewer test points to determine the minimum fault reactance value, which improves the accuracy and speed of calculation. e golden section method uses the golden section rule to select two symmetrical points in the search interval and then compares the function values at two points to determine a new search interval. In the fault location problem t ∈ [a, b], it indicates the change of the line fault distance. Select two golden points t 1 and t 2 in the interval [a, b] and then use the golden section method to search the interval. e first search point t 1 takes 0.618 of [a, b], and the second search point t 2 takes the symmetric point of t 1 , that is, 0.382 of [a, b], as shown in Figure 3.
Calculate the values of X f(t 1 ) , X f(t 2 ) and compare the sizes of the two values. If Conversely, if X f(t 1 ) > X f(t 2 ) , then t 2 is the effective point, the interval [t 1 , b] is removed, the interval [a, t 1 ] is reserved, and the new interval [a 1 , b 1 ] � [a, t 1 ]. Judged whether the new interval is smaller than the allowable error ε. If the allowable error is not reached, the above search process is continued in the new interval until the search interval [a i , b i ] is smaller than the allowable error ε: e search is stopped when the allowable error is reached, and finally the fault distance is determined according to the interval:

Establishing the Minimum Fault Reactance
Model. e golden section method is used to model and analyze the fault location problem, and the fault location is transformed into the optimal solution problem. e optimization formula is as follows: e objective function is given by the absolute value of formula (4), x is the normalized fault distance, and this function is used to solve the fault location. Since the faulted line is unknown, it is necessary to apply this method in each line of the distribution network.

Analysis Process.
e flow diagram of the analysis method proposed in the paper is shown in Figure 4, which consists of four steps. e data processing for each step is as follows: Step 1. Upload data: Upload data of the active distribution network system, such as distribution system topology, load, and line electrical parameters. In addition, there are the voltage and current vectors of the substation at fault time obtained by the Fourier transform and the synchronous current vector of the DER.
Step 2. Fault location: Calculate the fault current according to formula (2), and then calculate the fault reactance using formula (4).
] is the prefault current of node n. Formula (7) is used to calculate the fault distance of the line. e fault current is defined as   Step 3. Calculation of downstream fault current: Because the feeder of the active distribution network is radial, the traditional iterative method for power flow analysis has poor convergence in the active distribution network. erefore, the improved trapezoidal iteration method is used to calculate the fault current of the radial power distribution system. As shown in Figure 5, this figure illustrates the fault distance in the circuit which is m and then analyzes the downstream circuit of the fault point. Assume that the system load is constant impedance during the fault and DER is the current source. e current is measured by the installed synchronous current device, and the downstream fault current value is obtained by an improved trapezoidal iteration method.
is method has two processes: forward and backward [31].    In the forward process, calculate the voltage vector [32] from the source node to the downstream node, as shown in the following formula: In the backward process, the current vector is calculated from the load node to the source node by using the voltage and circuit model calculated and derived from the most recent forward calculation, as shown in the following formula: e voltage and current vectors in formulas (9) and (10) where [Y f n+1 ] is the admittance matrix of the parallel components. When DER is connected, current [I f n+1 ] is provided by the corresponding synchronous current measuring device: Iterative search until the voltage error at the reference node is less than the specified T bfs : Finally, when convergence is reached, the current at the reference node is taken as the fault current of the downstream circuit.
Step 4. Update the voltage and current vector of the downstream circuit: e steps 2 and 3 are used to analyze and calculate the line sections of the system. At this time, the calculation accuracy is improved by calculating the voltage and current of the downstream circuit nodes: where [I n+1 ′ ] is the line current of node n + 1 without considering other branches, [I n+1 ] is the current at node n + 1 when considering other branches, and [I L n+1 ] is the sum of the currents of the connection branches at node n + 1. For the node containing the DER branch, current [I n+1 ] is where [I DER ] is the current flowing from DER to node n + 1; this current is calculated by the step 3. If the fault is within the analyzed line section, the expression for the fault distance is where L i is the length of the analyzed line i and X i is the normalized fault length of the last lineL n , the value of which is calculated by formula (7).

Simulation Modeling.
Using PSCAD to build an IEEE 34-node test feeder system with different types of wires and loads, the working voltage level is 30 kV and the total length of the line is 58 km. It is changed by adding three DER with a penetration level of 10% in the system; the changed IEEE 34-node test feeder system is shown in Figure 6. e system is simulated and analyzed. Table 2 presents a summary of the tests realized. When the system fails, its fault data are obtained and the fault distance is calculated by the golden section method.

Analysis of Results.
e accuracy of the method is verified by analyzing the following scenarios. Considering the error between the calculated result and the actual value, the simulation result is evaluated by the following formula: where d is the calculated fault distance, d rel is the actual fault distance, and d tol is the total length of the line.

Effect of Fault Resistance.
When analyzing the effect of fault resistance on the method, different fault types are considered, as shown in Figure 7. e results show that under the influence of fault resistance, the maximum error of fault location is 0.16%, and this error occurs during single-phase ground fault. In the 58 km test feeder, the average fault distance calculate error of this method is about 25 m. is shows that the method has high accuracy of fault location.
In the figure, L-g is a single-phase ground fault, LL is a two-phase short-circuit fault, LL-g is a two-phase ground fault, LLL is a three-phase short-circuit fault, and LLL-g is a three-phase ground fault.

Effect of Fault Distance.
e results of Figures 8 and 9 show the effect of fault distance on the proposed method.
It can be concluded from the analysis that the error increases slightly with increasing fault distance, as shown in Figures 8 and 9, and as the fault reactance value increases, this effect will be amplified, but still less than 0.14%. In the 58 km test feeder, the error is less than 80 m. erefore, it is observed from the above that the behavior of the method is little affected by the fault distance.

Effect of Load Variation.
Since the load in the active distribution network system varies greatly, this will have some effects on the system. erefore, it is necessary to analyze the effect of load variations on this method. As shown in Figure 10, the results show that the accuracy of the method is related to the value of the load. Because the load variation compensation method is not used in the simulation, the fault distance calculated by this method is smaller than the actual fault distance when the load level is low. is  Figure 6: IEEE 34-node system.   Mathematical Problems in Engineering phenomenon occurs because the load conditions used by the proposed method are higher than the actual load conditions of the system, so the impedance observed by the proposed method from the substation is smaller. In contrast, for high load levels, the method calculates a fault distance that is higher than the actual fault distance. In this scenario, the error caused by the load variation is less than 0.18%. In the 58 km test feeder, the error is less than 104 m. is method still has good performance.

Effect of DER Penetration
Level. Add three DERs with 10% penetration to the IEEE 34-node test feeder model. Analyze the effect of DER penetration level on fault location of active distribution networks. As shown in Figure 11, with the increase of DER penetration level, the calculated fault distance will be greater than the actual fault distance. is phenomenon is due to the increase in DER penetration, which leads to an increase in the error in the calculation of the current from the DER to the fault point. In this paper, an improved trapezoidal iteration method is used to calculate the current value, which reduces the error in calculating the current value. us, the error value is less than 0.15% in this scenario. In the 58 km test feeder, the error is less than 87 m. e method proposed in this paper still has high accuracy of fault location.

Comparison Test.
e two methods that have been implemented are compared with the proposed analytical methods: Nunes's method [21] and Bedoya's method [27]. Nunes's method and Bedoya's method are formulated considering only one DER connected into network with a DER penetration level of 10%. us, for the comparison of the proposed method with these methods, the two DERs were disconnected. ese methods were validated and compared using scenario 1, which evaluates the fault resistance effect. Test results are presented using the average error obtained for each fault type, as shown in Figure 12.
e test results show the better accuracy of our method when compared with the method proposed by Nunes and the method proposed by Bedoya. e average error of the method proposed by Nunes is 2% and the average error of the method proposed by Bedoya is 1.8%. However, the average error of the method proposed by this paper is 0.6%.
Finally, choose a test case to compare these two methods with the method proposed in this paper. Single-phase faults with resistance of 50 Ω. Figure 13 allows to understand why the Nunes's method presents a low performance: it depends on the fault resistance and the location of the fault. e figure also shows that the performance of the proposed method is better than Bedoya's method. e above is expected because Mathematical Problems in Engineering the method proposed by Bedoya is formulated using symmetrical components, and the unbalanced system is not considered in the formula derivation.

Conclusions
In this paper, a method for fault location of active distribution network is proposed. is method uses the golden section method to search for the minimum fault reactance value and converts the fault location problem into a search minimum problem. Compared with the traditional fixed step search method, the golden section method proposed in this paper improves the search speed. According to the different loads of the nodes, an improved trapezoidal iterative power flow calculation method is proposed in this paper. is method improves the accuracy of power flow calculation and fault location. e simulation results show that this method has a high efficiency and robustness under different fault conditions and operating conditions. is method only needs to measure the synchronous current vector of DER and does not need to measure the voltage vector. e fault location is realized by a simple method, which shows the potential of the method in practical application.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.