As one of the most important attributes of fault information, the time of fault occurrence and power restoration can be extracted from the data recorded in the original ledger. This time information includes year, quarter, month, day, and hour. of this time point, which can be used for statistical analysis of fault time properties in different depths in all aspects, and also indirectly to obtain information such as fault duration to test the self-healing, repairability, and timeliness of maintenance personnel after the damage occurred in this grid. As one of the most important attributes of fault information, the time of fault occurrence and restoration of power can be extracted from the data recorded in the original ledger. The time information includes year, quarter, month, day, and hour, which can be used to analyze the temporal characteristics of the fault in different depths, and also to indirectly obtain information such as the duration of the fault, in order to test the self-healing, repairability, and timeliness of maintenance after the damage of the power grid here.

The cause of the outage is one of the most important risk factors affecting the results of the grid fault risk assessment, which is mainly extracted from the data recorded in the original ledger to describe the responsibility and cause of the grid outage.

The fault level mainly refers to three levels extracted from the data recorded in the original ledger based on the severity of the fault, i.e., I, II, and III level faults. This fault attribute represents the severity of the consequences of the fault that is important in risk assessment, i.e., the severity of the fault occurrence commonly used in the original risk assessment.

In this study, in order to improve the correctness of the algorithm and to obtain the most realistic complementary data that better fits the possibility of the system, we use the idea of clustering to select similar samples when selecting data samples. The smaller the distance, the higher the similarity. The formula for calculating the “distance” between the two samples is: d = ( a 1 ( x i ) − a 1 ( x f ) ) 2 + ... + ( a n ( x i ) − a n ( x j ) ) 2 , (1-1) where a n ( x i ) refers to the sample x i on the attribute a n on which the value is taken.

The ROUSTIDA method is based on the identifiable matrix in the rough set as an algorithm to reflect the invisible inner law embodied in the information system as much as possible, and its basic goal is to keep the incomplete object as much as possible with the same property law of similar objects in the original system, i.e., the value difference is as small as possible.

Let the information table system be S = 〈 U , A , V , f 〉 , A = { a i | i = 1,2 , ... , m } be the set of attributes, U = { x 1 , x 2 , ... , x n } is the thesis domain and a i ( x j ) is the sample x j on the attribute a i which is the value of the sample on the attribute and M ( i , j ) denotes the value of the sample after the expansion matrix in the i row j elements of the column, then the expanded identifiable matrix M can be defined as: M ( i , j ) = { a k | a k ∈ A , a k ( x i ) ≠ a k ( x j ) , a k ( x i ) ≠ ∗ , a k ( x j ) ≠ ∗ } , (1-2) where i , j = 1,2 , ... , n and " ∗ " indicates omitted values.

Let the initial information system be S 0 , the set of objects is { x 0 i } , and the corresponding extended difference matrix is M 0 , the object x i , the set of missing attributes M A S i , the set of undifferentiated objects N S i , and the information system S the set of missing objects of M O S are defined as: N S i = { j | M ( i , j ) = φ , i ≠ j , j = 1,2 , ... , n } , (1-3) M O S = { i | M A S i ≠ φ , i = 1,2 , ... , n } . (1-4)

First, for randomly selected data, a simple discretization is performed.

1) Fault duration a1 [0 ∼ 1.5, 1.5 ∼ 5.0, 5.0 ∼ 15.0, and 15.0–506.4]

Since the fault frequency varies with the size of the selected sample base range, this study chooses to use the combination of the equal distance and equal frequency method and draws on the idea of “adjacent attribute value midpoint” to process the original data against the division of the original frequency range.

Let the equidistance division method be based on the number of intervals K , the value domain of the numerical property interval [ X min , X max ] is divided into intervals whose widths are all δ = ( X max − X min ) / K of K . The right end of the breakpoint on the interval can be expressed as L i ′ = X min + i δ ( i = 0 , ... , K ) .

Similarly, the equal frequency method divides the value domain of the numerical property interval [ X min , X max ] into an equal number of objects in the interval K . The right end of the breakpoint on the attribute is obtained as L p ″ .

Let the equal distance and equal frequency correspond to the first P . The breakpoints at the right end of the interval are L ′ p and L p ″ , then the new breakpoint is L p = L p ′ 2 + L p ′ ′ 2 2 . Then, the first P range of the first interval is changed to [ L p − 1 , L p ] .

This study constructs an optimization model for the allocation of emergency repair resources in the producer–consumer community with the objective of minimizing the total potential fault loss. The objective function is as follows: C F = W 1 ∑ i = 1 n S i L i S ∑ + W 2 ∑ i = 1 n L i V i , (2-1)

where W₁ and W₂ are the weight values of the fault risk classification and fault repair response capability, respectively. S_i represents the fault subline classification of each producer and consumer, L_i represents the distance between the repair station and each producer, and V_i represents the speed between the repair station and each producer.

In this study, two aspects, frequency and time, as well as percentage indicators, are selected to deal with the quantification of fault risk in the producer–consumer community in a certain way. Based on the fault attribute categories derived from the collation and with reference to the regulations, such as the “Power Supply Reliability Regulations for Power Supply Systems,” and combined with the assessment process, indicators such as the number of fault occurrences, the percentage of fault attributes, the average outage time of faults, and the probability of faults are selected.

In order to evaluate the risk of outage in the producer–consumer community, the risk calculation method presented in this study is to establish a fault benchmark risk index system, as shown in Figure 1 quantify the probability and consequence of fault occurrence to obtain the fault probability and consequence values, which initially reflect the risk value of fault outage and then consider the influence of the remaining fault attributes and other factors on the probability of fault occurrence and the consequence of fault, and perform the corresponding calculation to obtain the integrated fault probability value and integrated fault consequence value. The calculation is based on these values.

Based on the concepts related to risk assessment, the formula for the failure risk value can be derived: Failure risk value  =  integrated failure probability value  ×  integrated failure consequence value .

Since the risk of failure in the producer–consumer community does not depend only on itself but also is influenced by factors related to the geographical area of the failure and the weather of the failure, etc., and the influence of the remaining attributes of the producer–consumer community indicator system on the main risk indicators is introducedas follows:

1) Number of failures, n

Within the statistical data information, the number of failures under different fault attributes is indicated by n, and the statistical unit is [times]. For example, the number of failures in different regions, the number of failures at different times, and the number of failures in different devices, etc.

2) Probability of failure properties, λ

Within the statistical data message, the proportion of the number of counts of the lower-level attributes of a fault attribute value to the total number of this fault attribute, is expressed as λ , and the statistical units are expressed as decimal points. For example, in the fault attribute weather factor, the occurrence of high winds as a percentage of all fault causes is weather conditions. λ = n n ∑ . (2-2)

3) Average power outage time for faults, S A I D I

Within the statistical data information, the average number of hours of outages for fault outages, in statistical units [h/time], is expressed in S A I D I . T is the fault outage time. N is the number of fault outages. For example, the average number of hours of fault outages in spring for urban faults in a region is as follows: S A I D I = T N . (2-3)

Based on the known geographical latitude and longitude information of the producer–consumer distribution, the distance between the producer–consumer and the ideal repair station is first calculated by applying the inverse half-positive sagittal function, then the time required to reach the repair location is calculated by considering the actual fault repair process, and finally the fault repair response capability is evaluated according to the required time.

1) Calculation of the distance between the producer and the ideal repair station

For any two points on the sphere, the semi-positive vector of the circular center angle can be calculated by the following equation: hav ( d r ) = hav ( φ 2 − φ 1 ) + cos ( φ 1 ) cos ( φ 2 ) hav ( λ 2 − λ 1 ) ,

where hav is an abbreviation for semi-positive vector function, d is the distance between two points, r is the radius of the ball, θ 1 θ 2 is the latitude of point1 and the latitude of point2, measured in radians, and λ 1 λ 2 is the longitude of the point1 and the longitude of the point2, measured in radians.

d can be solved by applying the inverse semi-sine function or by using the inverse sine function: d = r archav ( h ) = 2 r ⁡ arcsin ( h ) , h = hav d r .

Substitute to get d = 2 r ⁡ arcsin ( hav ( φ 2 − φ 1 ) + cos ( φ 1 ) cos ( φ 2 ) hav ( λ 2 − λ 1 ) ) , = 2 r ⁡ arcsin ( sin 2 ( φ 2 − φ 1 2 ) + cos ( φ 1 ) cos ( φ 2 ) sin 2 ( λ 2 − λ 1 2 ) ) .

2) Calculation of the time required to rush to the repair site

Considering the fault repair process, the road congestion coefficient and road curvature coefficient can be set to a and λ , respectively, and the average speed of the vehicle in which the repair personnel are traveling is v and i is the repair station and j for the production and elimination of the person, which can be obtained in the time to the repair point T i j that can be calculated according to the following formula. T i j = ( d i j ⋅ a ⋅ λ ) / v .

3) The calculation of the maximum time to repair the road

Assuming that the total time specified by the power company to reach the emergency repair site at the latest is T max . The maximum time to reach the urban area is 40 min, i.e., T max = 40 mins , and the maximum time to reach the suburban area is 60 min, i.e., T max = 60 mins . The time spent by the service center in processing work orders, the time spent by the dispatching layer in locating and handling faults, the time spent in preparing the repair vehicles before departure, and the time spent in searching for repair locations are set at t 1 , t 2 , t 3 , and t 4 . The time required for the repair station to reach the feeder is T n . Therefore, the formula for T n can be expressed by the following equation: T n = T max − t 1 − t 2 − t 3 − t 4 .

Time constraint T i j < T n .

The aforementioned equation is a time constraint on the time required to travel from the repair station to the point of failure.

The latitude and longitude of the repair station are greater than the minimum latitude and longitude of the producer and less than the maximum latitude and longitude.

The more uniformly the initial population is distributed in the search space, the better it is for improving the efficiency of the algorithm in finding the best solution and the accuracy of the solution. The initial gray wolf population of the traditional gray wolf optimization algorithm is randomly generated, which means it cannot guarantee that the initial population of individuals is uniformly distributed in the solution space. Chaotic sequences have better randomness, regularity, and traversal characteristics. Compared with other mappings, the tent mapping can generate a more balanced distribution of sequences, so the tent mapping is used to initialize the gray wolf population (Tian et al., 2021).

Tent mapping expressions x t + 1 = {       x t u     0 ≤ x t < u , 1 − x t 1 − u   u ≤ x t ≤ 1. (3-1)

When u = 0.5 , the most uniform distribution sequence can be produced, and the distribution density at this time is not sensitive to parameter changes. This belongs to the most typical tent mapping. At this time, the aforementioned equation can be changed to: x t + 1 = {     2 x t       0 ≤ x t < 0.5 , 2 ( 1 − x t )   0.5 ≤ x t ≤ 1. (3-2)

Based on the tent mapping, the population X is represented as: X = X min + x t ⋅ ( X max − X min ) , (3-3)

where X max and X min are the X upper and lower bounds of the search.

In the GWO algorithm, the individuals of the population follow α , β , δ . The algorithm will have the disadvantages of slow convergence speed and low accuracy in finding the best, so the cooperative competition mechanism is introduced, and before the prey is rounded up, all wolves except α , β , δ , each wolf of x ' i k will randomly choose another wolf x k j for communication, and if x ' i k location is better than x k j , then x k j move closer to x ' i k ; x k j move according to Eqn. 3-6, i.e., x ' i k move away from x k j and then x ' i k move according to Eqn. 3-7; otherwise they perform the opposite operation. x ′ j k = x j k + r a n d ⋅ ( x i k − x i k ) , (3-4) x ′ i k = x i k + r a n d ⋅ ( x i k − x i k ) , (3-5) where x ' i k and x k j are, respectively, the cooperative competition mechanism after the first i th individual and j ^th position of the first individual.

In order to meet the requirements of the GWO algorithm in different iteration cycles for finding the best and to improve the convergence speed and the accuracy of finding the best, adaptive inertia weights are introduced in the position update ω : ω = ω min + ( ω max − ω min ) max _ i t e r − t max _ i t e r , (3-6) where ω max and ω min are the upper and lower bounds of the weighting coefficients, respectively. t is the number of iterations. max _ i t e r is the maximum number of iterations.

The value of ω changes dynamically with the number of iterations, allowing the particles to continuously balance between global search and local search. The early iteration of the algorithm ω is larger, which is beneficial to the global search of the algorithm; later iterations of ω gradually decrease, so that the gray wolf can better find whether there is a better solution around the prey while updating its position, thus improving the local searchability of the algorithm.

The improved position equation is shown in Eqn. 3-7—Eqn. 3-9. X 1 = ω ⋅ x α k − A ⋅ D α k , (3-7) X 2 = ω ⋅ x β k − A ⋅ D β k , (3-8) X 3 = ω ⋅ x δ k − A ⋅ D δ k . (3-9)

The flow chart is shown in Figure 2.

The data are collected from the distribution network fault data provided by a city power company in the south for the year 2020–2021. In total, three randomly distributed producers and consumers in a square producer–consumer community with the same length and width are selected as the study case of the repair resource allocation optimization algorithm. The fault risk classification of the producers and consumers is shown in Figure 3.

By preprocessing the historical failure data of three randomly distributed producers and consumers in a square producer–consumer community with both length and width of 3 km, the failure risk value calculation formula is used to finally obtain the failure risk-grading map of producers and consumers shown in the aforementioned figure. From the information in the aforementioned figure, it can be seen that the failure risk level can be roughly divided into three categories: 0–0.5, 0.5–1.5, and 1.5 or more, among which the number of the first failure risk-level producers and consumers is the largest.

The distance between each producer and consumer is calculated using the latitude and longitude transformation method in 0.3 Section 2, and then the information of each point is displayed on the two-dimensional coordinate plane, and the producer–consumer configuration scenario shown as follows is obtained by combining the producer–consumer failure risk-grading situation as shown in Figure 4.

The convergence curve of the optimization process, the distance map of the optimized robotic stationing point, the comparison of the results of different algorithms, and the information of the optimal solution are shown in Figures 5, 6, Tables 1, 2, respectively.

The aforementioned figure shows the iterative process of the optimization model. At the beginning of the iteration, the algorithm adaptation is large and the change in the value of the objective function fluctuates; as the iteration proceeds, the corresponding total potential fault loss gradually decreases. After about 200 iterations, the optimization model adaptation consistency variable converges to the same value, i.e., the global optimal solution is obtained.

To further illustrate the performance of the algorithm proposed in this article, the algorithm is compared with the particle swarm algorithm and the traditional gray wolf optimization algorithm, and each algorithm is performed 20 times independently, and then the average value is calculated, and the calculation results are shown in Table 1.

As can be seen from Table 1, the optimal solution of this algorithm is better than that of the traditional gray wolf optimization algorithm and particle swarm algorithm. Also, the algorithm is improved by introducing tent mapping chaos initialization, introducing a cooperative competition mechanism, and introducing adaptive inertia weights in generating the initial population. The comparison of the average number of iterations and the average time consumed by the three algorithms shows that this algorithm can take into account both the global and local searchability. It is not only better than the other two methods in terms of the number of convergence iterations, but also the average number of iterations is around 143, and the convergence time is shorter. Compared with the traditional gray wolf optimization algorithm, the speed is improved by 30.76%.

As can be seen in Table 2, when the objective function of minimizing the total potential failure loss is taken into account, the weight values of the failure risk classification and the repair response capability of the producer and consumer are set to 0.6 and 0.4, respectively, based on which the optimal stationing point coordinates obtained by solving the optimization model are 1.772 and 1.433, respectively. The total potential failure loss is 0.8496, and the distance between the repair stationing point and each producer and consumer is shown in Figure 6.

Comparing the average distance and potential loss of fault between the stationing point and the producer and consumer before and after optimization, as shown in Table 3, the average distance is reduced by 10.21% after optimization; the potential loss of fault is reduced by 17.38% mainly because the optimized deployment of repair resources makes the repair resources closer to the high-risk fault in the region.