A comprehensive evaluation method of electric power quality based on improved grey correlation analysis

. Comprehensive evaluation of electric power quality is essential to the implementation of effective power quality control. In traditional evaluation methods, the weight of indicators is fixed and lacks flexibility. Aiming at this problem, this paper has proposed a power quality evaluation method based on improved grey correlation method. This method uses improved analytic hierarchy process (IAHP) to calculate subjective weights and introduces the improved entropy weight method (IEWM) that is modified using variable weight theory which calculates variable objective weights. The proposed method also uses intergrated weighting to combine the subjective and objective weights calculated above, obtaining multi-dimensional indicator weights. Finally, improved gray correlation analysis (IGCA), improved through employing ideal solution theory and logarithmic information aggregation method based on barrel theory, is implemented in this method to comprehensively evaluate electric power quality.


Introduction
With the development of electric power systems, especially the large increase in nonlinearity, and the capacity and quantity impact loads, power quality problems have become increasingly prominent [1]. The grid connection of distributed power sources has an impact on electric power quality, which may cause problems such as frequency deviation, voltage fluctuation, voltage flicker, voltage unbalance, harmonic distortion, and DC injection [2]. The seven existing electric power quality standards limited by electric power regulations are: voltage deviation, frequency deviation, voltage three-phase unbalance, voltage fluctuation and flicker, harmonics, interharmonics, temporary overvoltage and transient overvoltage [3]. How to achieve a more scientific and comprehensive evaluation of power quality is of great significance to the research on comprehensive assessment of electric power quality, and is crucial to the revision of power prices, power quality management and control in the power market environment [4].
The current evaluation methods of electric power quality include fuzzy mathematics method based on indicator characteristics, set pair analysis method [5], probability statistics and vector algebra method [6]; subjective weighting method and objective weighting method based on weight characteristics [7], intergrated weighting method [8]; matter element analysis method based on model characteristics [9], radar chart method [10], gray correlation analysis method (GCA) [11]. Among them, Z. Cheng [5] proposed an evaluation method based on set pair analysis and variable fuzzy sets, and introduced the two-tuple linguistic method to conduct secondary evaluation within the evaluation level. Y. Xiong [6] improved the traditional analytic hierarchy process (AHP), which can quickly find the indicator items that affect the consistency to determine the reasonable weight. There are many power quality evaluation methods and weight determination methods, and each has its own advantages and disadvantages. It is necessary to select an appropriate method to evaluate electric power quality according to actual needs.
In this paper, the IGCA, modified through employing ideal solution theory and logarithmic information aggregation method based on barrel theory, has been proposed to evaluate the power quality of the distribution network, and combined with the intergrated weighting method to obtain the weights. Furthermore, the subjective and objective weights are obtained by IAHP and IEWM to obtain the evaluation results more scientifically and reasonably. This paper uses the IGCA combined with the intergrated weighting method, in which the subjective weight is obtained by using IAHP; the objective weight is obtained by using IEWM and the variable weight theory to form a variable objective weight; finally, intergrated weighting is used to calculate the combined weight and comprehensively evaluate the power quality. The specific process is shown in Figure 1.

Indicators
In this paper, seven indicators including voltage deviation X1, voltage flicker X2, voltage fluctuation X3, harmonic distortion X4, three-phase unbalance X5, frequency deviation X6 and power supply reliability X7 are used as the first-level indicators of the power quality evaluation indicator set. The first-level indicator measurement value and the abnormal duration of the indicator are used as the 13 second-level indicators (there is only one secondlevel indicator under power supply reliability), the average voltage deviation x1(%) and the deviation duration x2(s) , flicker level x3(%), flicker duration x4(s), average fluctuation range x5(%), fluctuation duration x6(s), total harmonic distortion x7(%), harmonic duration x8(s), unbalance degree x9(%), unbalance duration x10(s), average deviation x11(Hz), deviation duration x12(s), voltage sag interruption time x13(s). The judgment of indicator abnormality is based on the national standards of each first-level indicator. If the value exceeds the standard value, it is judged that the indicator is abnormal. The national standards for each indicator (take 10kV distribution network as an example) are as follows: 1) The voltage deviation range in the table is the range value of single-phase voltage. 2) In the table, the subscript re represents the measured value, and the subscript N represents the rated value.

IAHP calculates subjective weights
AHP is a simple subjective weighting method, but traditional AHP needs to check the consistency of the judgment matrix. In practical applications, the judgment matrix is mostly adjusted based on experience to meet the consistency test, which is blind and often fails to succeed at first try. IAHP is based on the general analytic hierarchy process, adding an optimal transfer matrix to make it easier to meet the consistency requirements. The calculation steps of IAHP are as follows: (1) Determine the target and evaluate the set of factors The research goal is the comprehensive evaluation of the electric power quality of the distribution network. The evaluation indicators are shown in Table 1.
(2) Construct a judgment matrix A is the target, ui is the evaluation factor, ui ∈U (i=1, 2, 3, ..., n). uij represents the relative importance value of ui to uj (j=1, 2, 3, ..., n), and the judgment matrix W is obtained according to the meaning of the above symbols, which is called the A-U judgment matrix.
(3) Using the scale expansion method CE to improve the judgment matrix The scale expansion method arranges n schemes in an sequence, in which their importance are undiminished, such as X1≥X2≥...≥Xn. Compare the importance of scheme Xi with Xi+1 and determine its weight (i=1, 2, 3, ..., n-1). The value of other elements in the matrix can be passed through the transitivity of importance. The judgment matrix constructed by the CE algorithm is W * , and its ranking vector is (t1t2…tn-1, t2t3…tn-1,…, tn-1,1) T .
Since the matrix W * satisfies wij=1/ wij, and any two rows of the matrix W * are proportional, which means W * is a positive and reciprocal matrix with complete consistency and automatically meets the consistency requirements. Therefore, the weight distribution can be obtained directly through the eigenvector of the maximum eigenvalue of W * , without the need for consistency check.
(4) Calculate subjective weight Calculate the eigenvector corresponding to the largest eigenvalue of W * by matrix operation, and normalize it to the weight distribution of the target: Where wi is the subjective weight value of each indicator of power quality calculated by the IAHP method.

IEWM calculates objective weights
The traditional EWM determines the indicator weight according to the degree of difference between the observed values of the indicators. The greater the degree of difference, the greater the indicator weight. This article combines variable weight theory and information entropy weighting method to construct variable objective weights. The specific calculation is as follows: (1) Determine the constant entropy weight of the indicators Define three variables i∈ [1, m], j∈ [1, n], k∈ [1, s], m, n, s are the number of monitoring points, the number of power quality indicators and the number of power quality levels, and i, j and k are all integers. In this paper, the power quality is divided into five levels: excellent, good, medium, poor, and inferior, so the value of s is 5; the constant entropy weight calculation needs to be performed on all secondary indicators, so the value of n is 13. The constant entropy weight is calculated according to Eq.4-Eq.6: Where rjk ' is the fuzzy membership value of the evaluation indicator j to the power quality level k; fjk is the fuzzy mapping between the power indicator set and the power evaluation set; ej is the information entropy value; vj is the constant entropy weight value, and it satisfies (2) Determine the indicators' variable entropy weight If the indicator observation values of s monitoring points are evenly distributed, that is, According to Eq.4 to Eq.6 calculation, the constant entropy weight value vj of the indicator j is very small. At this time, if the actual observation value of the indicator j deviates from the normal value seriously, it may cause the overall power quality to be judged as normal when the single power problem needs to be investigated, leading to a misjudgment phenomenon.
In actual engineering, when the observed value of a certain indicator j exceeds the value specified by the national standard, the weight of the indicator should be increased. Based on this, this paper introduces the variable weight theory to process the constant entropy weight, and realizes the information entropy variable weight to highlight the influence of bad indicators on the comprehensive evaluation results. Denote the power quality level "medium" as J, and the membership of the level "medium" and above as rj ' , introduce the relatively bad parameters shown in equation Eq.10, and realize the variable entropy weight according to equation Eq.11.
In the equation, the smaller rj ' means the worse the indicator, and the larger the variable weight coefficient xj a-1 ; a is the performance balance correction coefficient, which is taken as 0.2 in this paper. The fuzzy membership of indicator j to the five power quality levels from good to bad is recorded as rj1 '~ rj5 ' , and that rj ' =max(rj1 ' , rj2 ' , rj3 ' ), suppose the observed value of indicator j changes from grade "medium" to grade" poor" and the observed values of other indicators remain unchanged, then rj3 ' decreases and rj4 ' increases, the denominator decreases in Eq.10, the numerator increases, xj decreases, and xj a-1 increases , vj * in Eq.9 is always less than 1 and the numerator and denominator increase by the same value, so vj * , the entropy weight value of the indicator j, increases.

Intergrated weighting method to calculate comprehensive weights
The subjective weighting method can be combined with the intention of the decision maker to determine the weight, but it will ignore the internal connection of the data itself, and the objectivity is poor, while the objective weighting method cannot reflect the importance of the decision maker to different indicators, and there is a possibility that the weight and actual indicators are opposite. Therefore, when assigning weights to indicators, an intergrated weighting method is adopted, which can not only take into account subjective human factors, but also does not ignore the inherent statistical laws between the indicator measurement data.
Intergrated weighting method aims at finding the balance point of subjective and objective weighting method, that is, the Nash equilibrium point, so as to obtain a more scientific and reasonable comprehensive weight of each indicator. The specific process is as follows: The weights of the indicators obtained by IAHP and IEWM are recorded as w1 and w2 respectively, and then the combined weight vector is constructed by linear combination. The equation is as follows: In the equation, a1 and a2 are the linear coefficients of the main and objective weights, respectively. According to the intergrated weighting theory, the Nash equilibrium point is where the deviation between the subjective and objective weights and the combined weight is the smallest. The equation is: The optimal solution sought through the above equation is the optimal combination weight of subjective and objective weights intergrated. Under the optimal first derivative condition of the above equation: The weight coefficients a1 and a2 obtained by the above equation are normalized, and the equation is as follows: To determine the comprehensive intergrated weight, the equation is as follows: 5 Comprehensive evaluation based on IGCA

Principle of traditional GCA
The basic idea of traditional GCA is to judge whether the correlation between different data sequences is close according to the similarity of the curve geometry of the data sequence. In multi-indicator comprehensive evaluation problems, traditional GCA constructs the positive ideal sequence and calculates the correlation degree between each evaluation object and the positive ideal sequence. The greater the correlation degree, the more similar the positive ideal sequence and the better the evaluation result. In the multi-indicator comprehensive evaluation In Eq.17, m is the number of monitoring points in the system, n is the number of indicators, rj + represents the positive ideal solution of the indicator, rij * is the number of indicator intervals after normalization, ρ is the introduced resolution coefficient and ρ∈[0,1].

Improvement of the GCA
(1) Adjust the resolution coefficient In traditional methods, the resolution coefficient is generally 0.5, which has limitations in practical applications. This paper adjusts the resolution coefficient based on the integrity of the indicator set and the anti-interference. First, rewrite Eq.17 into Eq.18 and Eq.19.
For the monitoring value of the secondary indicator j at any monitoring point i, the number of solvable indicator intervals is rij * . The integrity of the indicator set is reflected in the correlation coefficient γ(ri + , rij * ). The larger the adjustment range, the better, and the antiinterference performance is reflected in the variation of the indicator interval number rij * to the correlation coefficient γ(ri + , rij * ) The smaller the impact, the better. In the denominator of Eq.19, |ri + -rij *| /Δmax∈[0,1] , therefore, the greater the value of ρ, the stronger the integrity of the indicator set but the resistance to interference is worse. On the contrary, the worse the integrity but the stronger the anti-interference. Accordingly, when the range of |ri + -rij *| /Δmax is smaller, the value of ρ should be increased to improve the integrity of the indicator set, otherwise, the value ρ must be decreased. The actual value of resolution ρ is carried out according to Eq.20 and Eq.21.
(2) Introduce the ideal solution to improve the correlation equation. The traditional gray correlation method is used to calculate the gray correlation degree between the evaluation object and the positive ideal solution to make the ranking decision. It can only reflect the closeness of the overall trend of the data, and cannot directly reflect the distance between the indicators. It has shortcomings in comprehensive decision-making. The ideal solution method uses the Euclidean distance between the indicator and the ideal solution to make a decision sequence, which can make up for the deficiency of the gray correlation method. In the equation, γ(rj -, rij * ) represents the correlation coefficient between the monitored value of indicator j at monitoring point i and the negative ideal solution of indicator j. The gray correlation between the indicator monitoring set of the monitoring point i (corresponding to the indicator interval number ri1 *~ rin * ) and the positive ideal solution set r1 +~ rn + and the negative ideal solution set r1 -~ rn -The degree is recorded as εi + and εi -, and its Euclidean distance is recorded as ζi + and ζi -. From the definition, the larger the value of εi + and ζi -, the better the overall indicator set's evaluation result; similarly, the larger the value of εiand ζi + , the worse the overall indicator se's evaluation result.
(3) Design information aggregation coefficient In the traditional GCA, when the correlation degree equation is linearly aggregated in consideration of the indicator difference, the numerator of Eq.24 is adjusted to Eq.26. In the equation, ωjs is the comprehensive weight value calculated by Eq.16. Based on the exponential aggregation method, this article aims to implement the penalty for low indicators, and introduces the logarithmic method based on the barrel theory to aggregate indicators. In the equation, Ijc is the health score of indicator j and Ijc∈ [1,5]. The logarithmic function amplifies the short-board effect when bad indicators collapse, and can reflect the "punishment" effect on short-board indicators.

The evaluation steps of IGCA
The improved evaluation method achieves effective correlation calculation by designing dynamic resolution coefficients; by calculating the correlation between each indicator in the sequence, it better characterizes the change trend and correlation of the two sequence curves, and can achieve better results. The specific implementation process of the excellent evaluation effect is as follows.
(1) Construct the original decision matrix Assume that there are m monitoring points in the system, and a total of n evaluation indicators are monitored. The jth indicator of the monitoring point i is labeled rij, and i∈ [1,m], j∈ [1,n]. Ri=(ri1, ri2,...,rin) T constitutes an observation sequence, and the observation sequence of m monitoring points constitutes the original decision matrix R=(rij) m×n.
(2) Construct a dimensionless decision matrix In the original decision matrix, indicators with different meanings are not the same in dimension and order of magnitude. In order to ensure data comparability, the original decision matrix needs to be standardized, and two standardized methods are used according to the type of indicators. The original indicator data can generally be divided into two types: benefit-based and cost-based. The larger the value of the "benefit-based" indicator, the better, and the smaller the value of the "cost-based" indicator, the better. Respectively adopt normalized expressions: The standardized calculation of rij * , the larger the value, the better the indicator, and there is rij * ∈[0,1], the new decision matrix is denoted as R * =(rij * ) m×n.
( (6) Forming decision criteria Introduce the linear coefficients α1 and α2 to construct a comprehensive gray correlation degree: In the equation, α1, α2∈[0,1] and α1+α2=1. When α1=1, it is the traditional grey relational evaluation method; when α1=0, it is the traditional ideal solution evaluation method; when α1 ∈(0,1) is the comprehensive evaluation method, this paper assumes α1=0.5 and α2=0.5. Based on the comprehensive correlation value, the decision criterion Si= Di + /( Di + + Di -) of the plan ITM Web of Conferences 47, 03013 (2022) CCCAR2022 https://doi.org/10.1051/itmconf/20224703013 to be evaluated is further calculated, and the level boundary vector is compared with the Si value of the evaluation indicator set to achieve the power quality level. Evaluation, the larger Si, the better the power quality.

Example
Based on the data of 3 monitoring points in a 10kV power distribution system, the power quality assessment is carried out. A1~ A5 are five power quality grades from good to bad, and the indicator numbers and units are consistent with those in Table 1. Count the monitoring data collected in a certain week, and fill in Table 2 with the 95% probability as the original indicator data. Based on the above method, the comprehensive weight is calculated as Table  3.
In Table 3,ωi is the subjective weight value, vi1 is the constant entropy weight objective weight value, vi2B1 , vi2B2 And vi2B3 are the objective weight values of the information entropy of the indicator set introduced with the variable weight theory, and ai is the final comprehensive weight value. The variable weight objective weights of the three groups of indicators to be evaluated are shown in Figure 2.   Comparing the three curves of constant entropy weight, B1 variable weight, and B3 variable weight in Figure 2, it can be seen that the three curves have a better closeness, indicating that the information entropy weight considering the variable weight theory can still effectively reflect the characteristics of the indicator monitoring data. For monitoring point B3, the monitoring value of indicator x2 is at the "poor" level and the quality of the indicator is poor, while the monitoring value of indicator x5 is at the "excellent" level and the quality of the indicator is good. The deviation between the variable weight curve of the monitoring point B2 and the constant entropy weight curve is relatively large. This is because the quality of the two indicators x2 and x5 in the monitoring value of B2 is poor and the quality of other indicators are good.  Comparing Eq.38 and Eq.39, it can be seen that ΔG1 < ΔG2, which shows that the correlation degree distribution interval obtained by selecting the ρ value according to the method in this paper is relatively large, and the interference of the bad values of the observation sequence on the evaluation results can be better suppressed.
Calculate the positive and negative gray correlation degree εi + and εi -, and the positive and negative ideal solution ζi + and ζi -, according to Eq.22-Eq.25. Then, the comprehensive gray ITM Web of Conferences 47, 03013 (2022) CCCAR2022 https://doi.org/10.1051/itmconf/20224703013 correlation degree is calculated by Eq.33 and Eq.34, and the final decision criterion Si is calculated. Using the traditional GCA and the evaluation method proposed in this paper, respectively, the evaluation results of the calculation examples are shown in Table 4. The two evaluation schemes in Table 4 are: the traditional GCA and the IGCA proposed in this paper.  Table 4, it can be seen that in the data of monitoring point B2, indicator x2 (voltage deviation duration) and indicator x8 (harmonic duration) are at the level "bad", and the comprehensive evaluation result of power quality should tend to the level "bad" ". In the comprehensive evaluation results of monitoring point B2, the result from traditional GCA is "poor", and the result from IGCA is "bad". The main reason for the difference between the results of the two methods is the linear information aggregation method adopted by the traditional GCA. Because the weights of indicator x2 and indicator x8 are low, their bad influence is concealed by other high-weight good indicatores, resulting in a better evaluation value. Using the same method in this article, the B2 indicator x2 monitoring value is modified to 6.0, the influence of the bad indicators is weakened, and the final grade evaluation value becomes "poor", and the evaluation result is consistent with the traditional GCA. Comparing the evaluation results of monitoring point B2 under the two schemes, it can be seen that the method proposed in this paper can better reflect the punishment for a single bad indicator, and the final evaluation results obtained are more in line with the needs of engineering applications.

Conclusion
In the traditional comprehensive evaluation of power quality, insufficient attention is paid to low-weight indicators, which leads to inaccurate power quality evaluation under "bad" conditions. To address the above problems, this paper first introduces the intergrated weighting method, uses IAHP to determine subjective weights, uses IEWM to determine objective weights, and comprehensively considers subjective and objective weights to combine weights for various indicators, which is more scientific and reasonable. Among them, by adding the optimal transfer matrix as improvement for AHP, combined with the variable weight theory to improve the traditional EWM, giving higher objective weight to the bad indicators, and increasing the importance of the bad indicators in the decision-making plan. Next, the ideal solution method is introduced to improve the traditional gray correlation method and enhance the ability of the evaluation method to distinguish different sets of indicators to be evaluated. Finally, the logarithmic polymerization method based on the barrel theory was used to replace the traditional linear polymerization method, and the correlation calculation equation of the gray correlation method was adjusted to improve the ability to ITM Web of Conferences 47, 03013 (2022) CCCAR2022 https://doi.org/10.1051/itmconf/20224703013 distinguish "bad" indicators and achieve an evaluation that is more in line with the power quality evaluation target result.