State evaluation method based on combination of combined empowerment and weighted rank sum ratio

Abstract

The invention relates to the technical field of power electronics, and discloses a state evaluation method based on combination of combined empowerment and weighted rank sum ratio, which is used for evaluating and analyzing the running state of a vacuum circuit breaker in a switch cabinet. In the evaluation method, four state indexes of vacuum characteristics, mechanical characteristics, electrical characteristics and temperature characteristics of the vacuum circuit breaker are selected for comprehensive evaluation. In order to improve the accuracy of evaluation, the invention provides a comprehensive evaluation method based on combination of combined weighting and weighted rank sum ratio. According to the method, a single weight method can be prevented from being influenced by subjective or objective factors, the evaluation model is simple and reliable, the possibility is provided for simultaneously detecting the states of a plurality of vacuum circuit breakers, and meanwhile, the results can be basically kept consistent with expert evaluation results. The detection means does not make up for the disadvantage that the measurement data is not utilized perfectly in the previous evaluation method, the measurement data information is utilized to a greater extent, and meanwhile, early warning can be carried out when the normal operation of the circuit breaker is not influenced by the fault.

Description

Technical Field

[0001] The present invention relates to the technical field of power electronics, and in particular to a state assessment method based on the combination of combined weighting and weighted rank sum ratio. Background Art

[0002] The KYN series switchgear is a widespread and widespread technology. However, due to the high currents and voltages constantly present within the switchgear's interior, strong electromagnetic fields contribute to a constant stream of operational defects and safety hazards, resulting in a wide variety of faults. As a crucial component of the power system, the vacuum circuit breaker plays a crucial role. Its fully sealed contact structure protects against interference from moisture, dust, and harmful gases, ensuring operational reliability. However, if the operating characteristics of the vacuum circuit breaker deviate from the permitted range, it directly impacts the overall operational safety and stability of the switchgear. Every year, thousands of major accidents are caused nationwide by switchgear circuit breaker failures, resulting in significant economic losses to the power system and the stability of social needs. Traditional circuit breaker data analysis underutilizes relevant data, leading to significant data waste and incomplete analysis results, often leading to misjudgments and omissions. Summary of the Invention

[0003] The purpose of the present invention is to provide a state assessment method based on the combination of combined weighting and weighted rank sum ratio, which evaluates the operating state of the vacuum circuit breaker based on the following four characteristic indicators, which include vacuum characteristics, mechanical characteristics, electrical characteristics and temperature characteristics. The numerical values of the detected vacuum circuit breaker are fully utilized. Not only the influence of subjective weights but also the influence of objective weights are considered, and the weights of each indicator are combined and comprehensively considered. The vacuum circuit breakers are graded and sorted in combination with the weighted rank sum ratio method, and the status of each vacuum circuit breaker is finally determined. According to the operating status of the vacuum circuit breaker, a corresponding maintenance plan can be formulated.

[0004] The purpose of the present invention can be achieved through the following technical solutions:

[0005] A state assessment method based on combining combined weighting and weighted rank sum ratio includes the following steps:

[0006] According to the vacuum characteristics, mechanical characteristics, electrical characteristics and temperature characteristic indicators of the vacuum circuit breaker, it is layered according to the corresponding influencing factors to form A, B and C layers;

[0007] The analytic hierarchy process is used to determine the subjective weight values of the status indicators;

[0008] Calculate the ranking weight of the C layer in the condition assessment system; perform consistency testing on the corresponding indicators of A, B, and C based on the consistency test index BQ to determine the rationality of the constructed judgment matrix;

[0009] Combining the data presentation with statistical indicators can yield objective weights; furthermore, the objective weight values of the status indicators at each level can be determined;

[0010] Normalize the operating status indicators of the vacuum circuit breaker to obtain relevant data. After all the data are normalized, a standardized matrix is obtained. The characteristic proportion of each evaluation object under a certain indicator is calculated. Then, the corresponding entropy value is calculated. The weight of each indicator is obtained using the formula, and the corresponding objective weight vector is calculated.

[0011] Determine the combined weight value, use the formula to obtain the combined weight value between each indicator, and finally obtain the combined weight vector of each indicator in the C layer;

[0012] The weighted rank ratio method is used to conduct a comprehensive evaluation of the assessment object.

[0013] As a further solution of the present invention: in the process of determining the subjective weight value, the relevant factors that affect the evaluation object are listed, and the judgment matrix B is constructed as follows: zk )m×m; All elements in the judgment matrix B satisfy b zz =1,b zk =1 / b kz (k,z=1,2....,m), where b zk The meaning of is the ratio of the importance between the zth indicator and the kth indicator.

[0014] As a further solution of the present invention: when BQ<0.1, the total ranking result of the set level meets the consistency requirement.

[0015] As a further solution of the present invention: further comprising the following steps:

[0016] The numerical values of the relevant status indicators of the six vacuum circuit breakers were obtained, and a 6×12 original data matrix was obtained;

[0017] Rank the data in the matrix according to the order of the columns to obtain the rank matrix R; and calculate wRSR i The values of are arranged in ascending order, and are required to correspond to the serial number of the vacuum circuit breaker in the first column; wRSR value is the dependent variable, and P robit The linear regression equation is obtained for the independent variable, and finally the fitted value of wRSR is obtained.

[0018] As a further solution of the present invention: further comprising the following steps:

[0019] The P of each vacuum circuit breaker robit After substituting the value into the regression equation, the critical value of δwRSR can be obtained. Then, the critical value can be compared with the δwRSR value of each vacuum circuit breaker to classify it and obtain the classification result.

[0020] As a further solution of the present invention: the construction of judgment matrices of indicators at different levels:

[0021] Since B1-C and B4-C in the target layer have only one indicator, the judgment matrix constructed in the C layer is a first-order matrix with one element;

[0022] However, AB, B2-C, and B3-C do not have unique indicators, so the following judgment matrix is constructed as shown below:

[0023]

[0024]

[0025] As a further solution of the present invention: the calculation process of the characteristic proportion of each evaluation object under a certain indicator is:

[0026] Combined weight value (W i ) is calculated as follows, where α i is the subjective weight vector, β i is the objective weight vector;

[0027]

[0028] As a further solution of the present invention: the calculation process of the objective weight vector:

[0029] The number of evaluation objects selected is n, and each evaluation object selects m identical evaluation indicators. Then xij in the following formula represents the jth evaluation indicator corresponding to the i-th sample, from which the sample feature proportion pij can be calculated as:

[0030]

[0031] Then calculate the entropy value (e j ), where the entropy value calculation formula corresponding to the j-th indicator is as follows, where ln() is the logarithm of the natural logarithm with the constant e as the base:

[0032]

[0033] Then the coefficient of difference (g i ) is calculated as shown in the formula:

[0034] g i =1-e j

[0035] Finally, the weight of each indicator (β i ):

[0036]

[0037] As a further solution of the present invention: the weighted rank ratio method is as follows:

[0038] Ranking: construct the original data of the selected n evaluation objects and the corresponding m evaluation indicators into an n×m matrix, and rank each evaluation object and system to obtain the rank matrix R;

[0039] Then calculate the weighted rank sum ratio (wRSR) as shown in the formula:

[0040]

[0041] Where R ij is the rank of the element in row i and column j, W j is the weight of the j-th indicator.

[0042] Determine the probability unit (P robit ): Prepare the frequency distribution table of wRSR; determine the average rank of each group of wRSR Calculate the frequency p using the formula i , according to the cumulative frequency, query the reference table and find the corresponding probability unit P robit value;

[0043]

[0044] Calculate the linear regression equation: Calculate the fitted value of wRSR by the linear regression equation, and further use the least squares method to find the coefficient of the regression equation:

[0045]

[0046] In the formula, a and b are the coefficients corresponding to the linear equation. is the estimated value of the i-th item of wRSR;

[0047] The ranking is based on the probability unit P under each ranking situation. robit The assessment objects are graded and classified according to the optimal grading principle.

[0048] As a further solution of the present invention: robit The linear regression equation obtained for the independent variable is as follows:

[0049]

[0050] The resulting rank matrix R is:

[0051]

[0052] Beneficial effects of the present invention:

[0053] This patent evaluates the operating status of vacuum circuit breakers based on the following four characteristic indicators: vacuum characteristics, mechanical characteristics, electrical characteristics, and temperature characteristics. The detected values of the vacuum circuit breakers are fully utilized. Not only the influence of subjective weights but also the influence of objective weights are considered, and the weights of each indicator are combined for comprehensive consideration. The vacuum circuit breakers are ranked and sorted using the weighted rank sum ratio method, ultimately determining the status of each vacuum circuit breaker. A corresponding maintenance plan can be formulated based on the operating status of the vacuum circuit breaker.

[0054] The present invention is mainly used to evaluate and analyze the operating status of vacuum circuit breakers in switch cabinets. In this evaluation method, four status indicators of vacuum circuit breakers, namely vacuum characteristics, mechanical characteristics, electrical characteristics and temperature characteristics, are selected for comprehensive evaluation. In order to improve the accuracy of the evaluation, we propose a comprehensive evaluation method based on the combination of combined weighting and weighted rank sum ratio (WRSR). At the same time, the application process of this method and the analysis through field examples are given in this article. This method can avoid the influence of subjective or objective factors on the single weight method. The evaluation model is simple and reliable, which makes it possible to test the status of multiple vacuum circuit breakers at the same time, and is basically consistent with the results of expert evaluation. This detection method not only makes up for the shortcomings of the previous evaluation method in the imperfect use of measurement data, but also makes full use of the measurement data information. At the same time, it can also issue an early warning before the fault affects the normal operation of the circuit breaker. BRIEF DESCRIPTION OF THE DRAWINGS

[0055] In order to more clearly illustrate the technical solutions in the embodiments of the present invention, the following briefly introduces the drawings required for use in the description of the embodiments. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without creative work.

[0056] Figure 1 This is a flow chart of a state assessment method combining combined weighting and weighted rank sum ratio provided by the present invention. DETAILED DESCRIPTION

[0057] In order to enable those skilled in the art to better understand the solutions of the present invention, the technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the embodiments described are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts should fall within the scope of protection of the present invention.

[0058] like Figure 1 As shown, an embodiment of the present invention is a state assessment method that combines combined weighting and weighted rank sum ratio. This patent evaluates the operating state of the vacuum circuit breaker based on the following four characteristic indicators, which include vacuum characteristics, mechanical characteristics, electrical characteristics, and temperature characteristics. Make full use of the numerical values of the detected vacuum circuit breaker. Not only the influence of subjective weights is taken into account, but also the influence of objective weights is taken into account, and the weights of each indicator are combined and considered comprehensively. The vacuum circuit breakers are graded and sorted in combination with the weighted rank sum ratio method, and the status of each vacuum circuit breaker is finally determined. The corresponding maintenance plan can be formulated according to the operating status of the vacuum circuit breaker.

[0059] To achieve the above-mentioned purpose, the present invention proposes a combined evaluation method, specifically, the four characteristic state indicators corresponding to the vacuum circuit breaker state evaluation are layered according to the corresponding influencing factors to form a state evaluation system hierarchy diagram, wherein layer A is the target layer, layer B is the project layer, and layer C is the indicator layer, and the weights of the characteristic indicators are determined. The present invention mainly adopts the hierarchical analysis method (AHP) to determine the subjective weight values of the state indicators. First, the evaluation factor set of the evaluation object must be determined; secondly, the judgment matrix B = (b zk )m×m, which is also the most critical step in the hierarchical analysis method. The elements in the judgment matrix B all meet b zz =1,b zk =1 / b kz (k,z=1,2....,m), experts determined b zk The meaning of is the ratio of the importance of the zth indicator to the kth indicator. The relative importance of each layer of indicators in the vacuum circuit breaker condition assessment system needs to be compared pairwise using the 9-scaling method, and their importance is marked. The meaning of the marking is shown in Table 1. The eigenvector of the maximum eigenroot of the judgment matrix corresponding to each layer in the condition assessment system is calculated. These vectors can be used to obtain the ranking weight of the bottom-level indicators:

[0060] α j =[0.483; 0.027; 0.031; 0.027; 0.021; 0.022; 0.017; 0.126; 0.049; 0.031; 0.077; 0.088]

[0061] Table 1 Judgment matrix scale table

[0062]

[0063] The evaluation factors used are Figure 1 Based on the relevant indicators in the target layer, expert opinions, and the characteristics of vacuum circuit breakers, judgment matrices for indicators at different levels were constructed. Since B1-C and B4-C in the target layer only have unique indicators, the judgment matrix constructed for the C layer is a first-order matrix with one element. However, AB, B2-C, and B3-C do not have unique indicators, so the following judgment matrices were constructed, as shown in Formulas 1, 2, and 3.

[0064]

[0065]

[0066] Each level of indicators requires consistency testing to determine whether the constructed judgment matrix is reasonable. The corresponding indicators of A, B, and C are tested for consistency based on the consistency test indicator BQ to determine the rationality of the constructed judgment matrix. When BQ<0.1, the total ranking result of the set level meets the consistency requirements. Determine the objective weight value. The objective weight can be obtained by combining the presentation form of the data with the statistical indicators. The objective weight value of the status indicator of each level is further determined. All data in the evaluation matrix need to be normalized. Among the 12 indicators of the C layer, except for the opening speed and closing speed, all indicators are cost-based indicators, and the opening and closing speeds are economic indicators. The normalization formula for economic indicators is shown in Formula 4, and the normalization formula for cost-based indicators is shown in Formula 5. In the formula, the maximum value of the sample data is x max , the minimum value of the sample data is x min . .

[0067]

[0068] The operating status indicators of the vacuum circuit breaker are normalized as shown in Table 2 to obtain Table 3. After all data are normalized, a standardized matrix is obtained, and the characteristic proportion of each evaluation object under a certain indicator is calculated.

[0069] Table 2 Vacuum circuit breaker operating status index values

[0070] Vacuum Break Vacuum Break Vacuum Break Vacuum Break Vacuum Break Vacuum Break

[0071]

[0072] Table 3 Normalization results of vacuum circuit breaker status index data

[0073]

[0074]

[0075] Then calculate the corresponding entropy value and use the formula to get the weight of each indicator. Calculate the feature proportion in step 5 using formula 6. Select the number of evaluation objects as n, and each evaluation object selects m identical evaluation indicators. Then, xij in the following formula represents the jth evaluation indicator corresponding to the i-th sample. From this, the sample feature proportion pij can be calculated as:

[0076]

[0077] Then calculate the entropy value. The entropy value calculation formula of the j-th indicator is as follows 7.

[0078]

[0079] Next, the difference coefficient is calculated, as shown in Formula 8.

[0080] g i =1-e j (8)

[0081] Finally, use Formula 9 to calculate the weight of each indicator.

[0082]

[0083] The corresponding objective weight vector can be calculated as:

[0084] β j =[0.060; 0.068; 0.085; 0.096; 0.127; 0.101; 0.103; 0.090; 0.077; 0.063; 0.070; 0.060]

[0085] Formula 10 is calculated based on the combined weight value as follows.

[0086]

[0087] Determine the combined weight value, use formula 10 to get the combined weight value of each indicator, and finally get the combined weight vector of each indicator in layer C:

[0088] W j =[0.413; 0.026; 0.037; 0.037; 0.038; 0.032; 0.025; 0.160; 0.053; 0.028; 0.077; 0.075]

[0089] Use the weighted rank ratio method to comprehensively evaluate the evaluation objects. (1). Rank. According to the assumptions in step 4, the original data of the selected n evaluation objects and the corresponding m evaluation indicators are constructed into an n×m matrix. Each evaluation object and system is ranked to obtain the rank matrix R. Rank the economic indicators from small to large, and the cost indicators from large to small. Under the same indicator, when the data is the same, write the average rank. (2). Then calculate the weighted rank sum ratio (wRSR). Since the weights of the evaluation indicators are different, the calculation of the weighted rank sum ratio is shown in Formula 11.

[0090]

[0091] Where R ij is the rank of the element in row i and column j, W j is the weight of the jth indicator. Generally, when there are a large number of evaluation objects, it is necessary to find the distribution law of wRSR and use it to classify the evaluation objects. (3). Determine the probability unit (P robit ). First, compile the frequency distribution table of wRSR; then determine the average rank of each group of wRSR Next, use Formula 12 to calculate the frequency p i , according to the cumulative frequency, query the reference table and find the corresponding probability unit P robit value.

[0092]

[0093] (4) Calculate the linear regression equation. This allows us to obtain the fitted value of wRSR, and calculate the coefficients a and b of the regression equation using the least squares method.

[0094]

[0095] In the formula, a and b are the coefficients corresponding to the linear equation. (5). Sorting by grading. The evaluation objects are graded and classified according to the best grading principle. The grading level is determined by experts based on the relevant data and then the corresponding technology is determined. The higher the evaluation level, the better the accuracy of the evaluation object. Finally, the relevant data of the field example are analyzed. Combined with the status index values of the six vacuum circuit breakers provided in Table 2, the 12×6 matrix composed of all its data is transposed to obtain the 6×12 original data matrix. The data in the matrix are ranked by column, and the resulting rank matrix R is obtained.

[0096]

[0097] The weighted rank sum ratio of each vacuum circuit breaker is obtained according to formula 11, and then the corresponding probability unit is determined. The results are shown in Table 4.

[0098] Table 4 wRSR value distribution table

[0099]

[0100] And wRSR will be calculated i The values of are arranged in ascending order and are required to correspond to the serial number of the vacuum circuit breaker in the first column. robit The linear regression equation is obtained for the independent variable, and finally the fitted value of wRSR is obtained.

[0101]

[0102] This allows the fitting value of wRSR to be calculated. The evaluation objects are then ranked. To maximize consistency with regulations related to vacuum circuit breakers, this paper adopts a four-level classification system, with levels 1, 2, 3, and 4 corresponding to the critical, abnormal, caution, and normal states of vacuum circuit breakers, respectively.

[0103] Table 5 Critical value table for grading

[0104]

[0105] The P of each vacuum circuit breaker robit After substituting the value into the regression equation, the critical value of δwRSR can be obtained. Then, the critical value can be compared with the δwRSR value of each vacuum circuit breaker to classify it.

[0106] Table 6. Sorting results by file

[0107]

[0108] The P of each vacuum circuit breaker robit Substituting the values into the regression equation yields the critical δwRSR value. This critical value is then compared with the δwRSR value of each vacuum circuit breaker to categorize them and obtain the classification results. The final result indicates that vacuum circuit breaker No. 3 is assessed as "normal," while vacuum circuit breakers Nos. 4 and 5 are assessed as "caution," and vacuum circuit breakers Nos. 1, 2, and 6 are assessed as "abnormal." Expert evaluations of the six circuit breakers demonstrate the accuracy of this assessment method.

[0109] Table 7 Comparison of evaluation results of the two methods

[0110]

[0111] 7. Table 7 in Step 11 shows that the expert assessed circuit breaker 6 as being in a "caution state." The measured three-phase opening and closing asynchrony of circuit breaker 6 is slightly larger than that of other circuit breakers, but within the normal range. However, to prevent further deterioration of the circuit breaker's state, appropriate monitoring of the circuit breaker is required.

[0112] The above is a detailed description of an embodiment of the present invention. However, the content described is only a preferred embodiment of the present invention and should not be considered to limit the scope of the present invention. All equivalent changes and improvements made within the scope of the present invention should still fall within the scope of the patent coverage of the present invention.

Claims

1. A state assessment method based on the combination of combined weighting and weighted rank sum ratio, characterized in that: The following steps are involved: According to the vacuum characteristics, mechanical characteristics, electrical characteristics and temperature characteristic indicators of the vacuum circuit breaker, it is layered according to the corresponding influencing factors to form A, B and C layers; The analytic hierarchy process is used to determine the subjective weight values of the status indicators; Calculate the ranking weight of the C layer in the condition assessment system; According to the consistency test index BQ, the corresponding indicators of A, B, and C are tested for consistency to determine the rationality of the constructed judgment matrix; Combining the data presentation with statistical indicators can yield objective weights; furthermore, the objective weight values of the status indicators at each level can be determined; Normalize the operating status indicators of the vacuum circuit breaker to obtain relevant data. After all the data are normalized, a standardized matrix is obtained. The characteristic proportion of each evaluation object under a certain indicator is calculated. Then, the corresponding entropy value is calculated. The weight of each indicator is obtained using the formula, and the corresponding objective weight vector is calculated. Determine the combined weight value, use the formula to obtain the combined weight value between each indicator, and finally obtain the combined weight vector of each indicator in the C layer; The weighted rank ratio method is used to conduct a comprehensive evaluation of the assessment object.

2. A state assessment method based on combining combined weighting and weighted rank sum ratio according to claim 1, characterized in that: The process of determining the subjective weight value is to list the relevant factors that affect the evaluation object and construct the judgment matrix B = (b zk )m×m; All elements in the judgment matrix B satisfy b zz =1,b zk =1 / b kz (k,z=1,2....,m), where b zk The meaning of is the ratio of the importance between the zth indicator and the kth indicator.

3. The state assessment method based on the combination of combined weighting and weighted rank sum ratio according to claim 1 is characterized in that: When BQ<0.1, the total ranking result of the set level meets the consistency requirement.

4. The state assessment method based on the combination of combined weighting and weighted rank sum ratio according to claim 1 is characterized in that: The following steps are also included: The numerical values of the relevant status indicators of the six vacuum circuit breakers were obtained, and a 6×12 original data matrix was obtained; Rank the data in the matrix according to the order of the columns to obtain the rank matrix R; And wRSR will be calculated i The values of are arranged in ascending order, and are required to correspond to the serial number of the vacuum circuit breaker in the first column; wRSR value is the dependent variable, and P robit The linear regression equation is obtained for the independent variable, and finally the fitted value of wRSR is obtained.

5. The state assessment method based on the combination of combined weighting and weighted rank sum ratio according to claim 4 is characterized in that: The following steps are also included: Each vacuum circuit breaker's P robit After substituting the value into the regression equation, the critical value of δwRSR can be obtained. Then, the critical value can be compared with the δwRSR value of each vacuum circuit breaker to classify it and obtain the classification result.

6. A state assessment method based on combining combined weighting and weighted rank sum ratio according to claim 1, characterized in that: Construction of judgment matrix of indicators at different levels: Since B1-C and B4-C in the target layer have only one indicator, the judgment matrix constructed in the C layer is a first-order matrix with one element; However, AB, B2-C, and B3-C do not have unique indicators, so the following judgment matrix is constructed as shown below:

7. The state assessment method based on the combination of combined weighting and weighted rank sum ratio according to claim 1 is characterized in that: The calculation process of the characteristic proportion of each evaluation object under a certain indicator is: Combined weight value (W i ) is calculated as follows, where α i is the subjective weight vector, β i is the objective weight vector; 8. The state assessment method based on the combination of combined weighting and weighted rank sum ratio according to claim 7 is characterized in that: The calculation process of the objective weight vector: The number of evaluation objects selected is n, and each evaluation object selects m identical evaluation indicators. Then xij in the following formula represents the jth evaluation indicator corresponding to the i-th sample, from which the sample feature proportion pij can be calculated as: Then calculate the entropy value (e j ), where the entropy value calculation formula corresponding to the j-th indicator is as follows, where ln() is the logarithm of the natural logarithm with the constant e as the base: Then the coefficient of difference (g i ) is calculated as shown in the formula: g i =1-e j Finally, the weight of each indicator (β i ):

9. The state assessment method based on the combination of combined weighting and weighted rank sum ratio according to claim 1 is characterized in that: The weighted rank ratio method is as follows: Ranking: construct the original data of the selected n evaluation objects and the corresponding m evaluation indicators into an n×m matrix, and rank each evaluation object and system to obtain the rank matrix R; Then calculate the weighted rank sum ratio (wRSR) as shown in the formula: Where R ij is the rank of the element in row i and column j, W j is the weight of the j-th indicator. Determine the probability unit (P robit ): Prepare the frequency distribution table of wRSR; determine the average rank of each group of wRSR Calculate the frequency p using the formula i , according to the cumulative frequency, query the reference table and find the corresponding probability unit P robit value; Calculate the linear regression equation: Calculate the fitted value of wRSR by the linear regression equation, and further use the least squares method to find the coefficient of the regression equation: In the formula, a and b are the coefficients corresponding to the linear equation. is the estimated value of the i-th item of wRSR; The ranking is based on the probability unit P under each ranking situation. robit The assessment objects are graded and classified according to the optimal grading principle.

10. The state assessment method based on the combination of combined weighting and weighted rank sum ratio according to claim 1, characterized in that: P robit The linear regression equation obtained for the independent variable is as follows: The resulting rank matrix R is:

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