CN106251027B - Electric load probability density Forecasting Methodology based on fuzzy support vector quantile estimate - Google Patents

Abstract

The invention relates to a power load probability density prediction method based on fuzzy support vector quantile regression. First, the daily maximum load data and average temperature data before the prediction date are collected, and a training set and a test set are constructed using historical data. Then, use the training set to obtain the Lagrangian multipliers and support vector subscripts of the fuzzy support vector quantile regression model, and establish the fuzzy support vector quantile regression prediction model according to the obtained model parameter values, and transfer the test set Substitute into the model to get the predicted value. Finally, using the predicted values obtained under different quantile points, and using kernel density estimation to realize the probability density prediction of daily maximum load. The invention can effectively reduce forecasting errors, improve power load forecasting accuracy, achieve good forecasting effects, and provide a relatively reliable basis for power system dispatching departments to adjust power consumption plans and optimize generator output.

Description

Electric Load Probability Density Forecasting Method Based on Fuzzy Support Vector Quantile Regression

technical field

The invention belongs to the field of power load prediction combining statistical methods and intelligent calculations, and mainly relates to a power load probability density prediction method based on fuzzy support vector quantile regression.

Background technique

Power system load forecasting is based on the historical data of power load, economy, society, weather, etc., to explore the influence of the change law of the historical data of power load on the future load, and to seek the internal relationship between the power load and various related factors, so as to predict the future. Scientific prediction of power load. Accurate power system load forecasting is of great significance for power system dispatching, power consumption, planning, making power purchase plans, and arranging operation modes. This also requires power system researchers to propose more effective methods to improve prediction accuracy.

In recent years, with the rapid development of smart grid and the increase of uncertain factors, and the change of power load is restricted by many factors such as system operating characteristics, social factors and natural conditions, load forecasting requires a large amount of data, and these data It is difficult to guarantee accuracy and reliability; even when the resulting data is accurate, there are uncertainties. For example, temperature factors may factor in load changes. Therefore, the power system requires a more intelligent method to improve the accuracy of power load forecasting. In addition, in order to improve the safety and economy of power grid operation and improve the quality of power supply, the requirements for load forecasting accuracy of power system operation scheduling are getting higher and higher.

In recent years, the traditional field of power system load forecasting only considers the historical load and factors affecting load forecasting, but fails to deal with uncertain influencing factors. The main shortcomings are:

(1) There are many factors affecting power load forecasting, including date, meteorological factors, temperature factors, etc. How these factors affect load accuracy is uncertain or ambiguous. However, traditional power load forecasting methods do not preprocess these uncertain factors. That is to say, the historical data used in the prediction are all definite values, but the formation of these historical data has certain accidental factors, ignoring the uncertainty of historical data;

(2) The traditional power load forecasting method only gives point forecasting results or interval forecasting results, which cannot accurately describe the volatility of power loads, and does not give the probability density distribution of the forecasting results obtained.

Contents of the invention

The purpose of the present invention is to overcome above-mentioned deficiencies in the prior art, propose a kind of electric load probability density prediction method based on fuzzy support vector quantile regression, in order to carry out fuzzy treatment to average temperature factor, and greatly reduce influence factor Uncertainty, so as to effectively reduce the forecast error, improve the accuracy of power load forecast, and provide a more reliable basis for the power system dispatching department to adjust the power plan and optimize the output of the generator set.

A kind of electric load probability density prediction method based on fuzzy support vector quantile regression of the present invention is characterized in that it is carried out as follows:

Step 1. Collect and determine the daily maximum load data L′ and average temperature data W′=(w ₁ ′,w ₂ ′,…,w _j ′,…,w′ _N ); w′ _j is the first The average temperature of j days; 1≤j≤N, N is the total number of days;

Step 2. Normalize the daily maximum load data L′ to obtain normalized daily maximum load data L=(l ₁ ,l ₂ ,...,l _j ,...,l _N ); l _j is the normalized daily maximum load on day j;

Step 3. Fuzzy the average temperature data W′ using fuzzy theory to obtain fuzzy average temperature data W=(w ₁ ,w ₂ ,...,w _j ,...,w _N ), where w _j is Average temperature after fuzzing on day j;

Step 4. Select the daily maximum load value of the first d days before the i-th day and the average temperature after blurring As the input vector of the training sample on the i-th day, that is Select the daily maximum load on the i-th day As the target output value of the training sample on the i-th day, that is So as to get the training sample set i=1,2,...,N ^train , N ^train is the total number of training samples;

Step 5. Select the daily maximum load value of the first d days before the kth day and the average temperature after blurring As the input vector of the test sample on the kth day, that is Select the daily maximum load on the kth day As the target output value of the test sample on the kth day, that is To get the test sample set k=1,2,...,N ^test , N ^test is the total number of test samples;

Step 6, input the d-dimensional row vector of the i-th row As the i-th nonlinear component x _i and the i-th linear component u _i of the training input variable respectively, the one-dimensional actual output value of the i-th row in the training set As the i-th actual output value y _i , the fuzzy support vector quantile regression model shown in formula (1) is established as:

In formula (1), T is the transposition; w _i represents the average temperature factor after fuzzification on the i-th day; τ _r represents the r-th quantile point, and τ _r ∈ (0,1), r=1,2 ,…,N _τ ; N _τ represents the number of quantile points; Represents the parameter vector under the rth quantile point τ _r , C is the penalty parameter, is the coefficient vector under the rth quantile point τ _r , φ(·) represents the nonlinear mapping function, denotes the test function; and has: in, Denotes the threshold at the rth quantile τ _r and has: Among them, α _i and Represents the i-th optimal Lagrange multiplier; K is the kernel matrix in the input space, and has:

K(x _i , x _v )=φ(x _i ) ^T φ(x _v ); v∈I; I is the subscript set of the obtained support vector

Step 7, according to described fuzzy support vector quantile regression model, introduce slack variable and construct Lagrange function and solve formula (1), obtain the parameter vector under the rth quantile point τ _r as shown in formula (2) threshold and coefficient vector

In formula (2), α, α ^* is the optimal Lagrange multiplier vector, y={y _i |i∈I}; the design matrix is

Step 8, the parameter vector under the rth quantile point τ _r threshold and coefficient vector Substituting into the fuzzy support vector quantile regression prediction model; and the d-dimensional row vector of the kth row input in the test set The k-th nonlinear component x _k and the k-th linear component u _k are used as test input variables, so that the output value of the k-th row under the r-th quantile point τ _r in the test set can be obtained by using formula (3)

In formula (3), K _k represents the kth row vector of kernel matrix K;

Step 9. Use kernel density estimation to achieve probability density prediction of daily maximum loads:

Step 9.1, let the prediction result under the r-th quantile point τ _r on the k-th day be: In order to obtain the forecast results under all quantile points on the kth day Then get the test output value of the daily maximum load

Step 9.2, using formula (4) to obtain the value of the probability density function corresponding to the wth quantile point τ _w on the kth day

In formula (4), z _{k, w} represent the prediction results under the wth quantile point τ _w on the kth day; the number of quantile points w=1,2,...,N _τ ; h is the window width, K ₁ (η) is the Epanechnikov kernel function, and has: Among them, |η≤1|;

Step 9.2, utilize the principle of thumb shown in formula (5) to calculate the optimal window width h ^* of Epanechnikov kernel function K ₁ (η):

h ^* ＝1.06s _X N _τ ^-1/5 (5)

In formula (5), s _X is the standard deviation of the test output value of the daily maximum load;

Step 9.3: According to the Epanechnikov kernel function K ₁ (η) and the optimal window width h ^* , obtain the probability density prediction result of the daily maximum load.

The electric load probability density prediction method of the present invention is also characterized in that the fuzzy processing in the step 3 is carried out as follows:

According to the membership degree function of low temperature shown in formula (6), the membership degree function of medium temperature shown in formula (7) and the high temperature membership function shown in formula (8), the average temperature w' of the jth day _j is divided into low-temperature data, medium-temperature data or high-temperature data, and the fuzzy set to which the average temperature w′ _j of the j-th day belongs is obtained; thus, the fuzzy average temperature data W==(w ₁ ,w ₂ ,…,w _j , ..., w _N );

The membership function of the low temperature is:

The membership function of the middle temperature is:

The membership function of the high temperature is:

In formula (6), formula (7) and formula (8), e<g<f<m<n<p is satisfied.

Compared with the prior art, the beneficial effects of the present invention are reflected in:

1, the present invention combines traditional support vector regression method and quantile regression method and introduces fuzzy membership function to obtain fuzzy support vector quantile regression method, adopts support vector regression method to solve nonlinear problems, and quantile The regression method uses the information provided by the regressor variable to estimate the conditional quantile of the response variable, and obtains the influence of the explanatory variable on the response variable at different quantile points, and can accurately describe the variation range and conditional distribution shape of the explanatory variable on the response variable. In this way, the combination of the two methods can obtain a complete probability distribution of the future load, which is convenient for decision makers to make scientific decisions.

2. The present invention mainly considers the influence of the average temperature on the daily maximum load forecast. Because the average temperature factor is fuzzy, the fuzzy theory is introduced and the average temperature is fuzzified by the degree of membership function to improve the prediction accuracy, and different The probability density prediction results under the quantile point prove the effectiveness and accuracy of the method.

3. The present invention not only effectively reduces the prediction error and improves the prediction accuracy of electric load, but also can accurately describe the fluctuation of electric load and provide the probability density prediction curve. This provides a more reliable basis for the power system dispatching department to adjust the power plan and optimize the output of the generator set.

Description of drawings

Fig. 1 is the overall flowchart of the method of the present invention;

Fig. 2 is a detailed flow chart of the inventive method;

Fig. 3 is a probability density diagram of the method of the present invention.

Detailed ways

In the implementation process, a power load probability density forecasting method based on fuzzy support vector quantile regression mainly considers the influence of average temperature on power load forecasting. The flowchart is shown in Figure 1, and proceed as follows:

Step 1. There are many factors affecting power load forecasting. Through research and analysis, it is concluded that the average temperature factor has a greater impact on the power load forecasting results;

Step 1.1, the present invention selects the data of the global load forecasting competition that EUNITE network organizes to test, and this data comprises the load data of the time interval of 48 hours every day in 1997-1998 (corresponding to a load point every half hour), and 1997-1998 Average daily temperature data for the year. This data is complete data. And predict the daily maximum load data for 31 days in January 1999;

Step 1.2. Collect and determine the daily maximum load data L' and average temperature data W'=(w' ₁ ,w' ₂ ,...,w' _j ,...,w' _N ); w' _j is the first The average temperature of j days; 1≤j≤N, N is the total number of days;

Step 2. In order to avoid calculation saturation during the calculation process, normalize the daily maximum load data.

Normalize the daily maximum load data L′ to obtain the normalized daily maximum load data L=(l ₁ ,l ₂ ,…,l _j ,…,l _N ); l _j is the Daily maximum load after normalization, wherein the average temperature factor is determined according to the weather forecast of the meteorological department, and the daily maximum load data is determined according to the data provided by the power company;

Step 3. Since the average temperature factor is uncertain or fuzzy in real life, the fuzzy theory is used to establish a membership function to fuzzify the average temperature factor.

For the average temperature data W′, use the fuzzy theory to carry out fuzzy processing, and obtain the average temperature data after fuzzy W=(w ₁ ,w ₂ ,...,w _j ,...,w _N ), w _j is the fuzzy data of the jth day After the average temperature; Among them, the fuzzy treatment is carried out according to the following process:

According to the low temperature membership function shown in formula (6), the middle temperature membership function shown in formula (7) and the high temperature membership function shown in formula (8), the average temperature w′ _j of day j is divided into For low temperature data, medium temperature data or high temperature data, get the fuzzy set to which the average temperature w′ _j of the jth day belongs; thus get the fuzzy average temperature data W==(w ₁ ,w ₂ ,…,w _j ,…, w _N );

The membership function for low temperature is:

The membership function of medium temperature is:

The membership function for high temperature is:

E<g<f<m<n<p is satisfied in formula (6), formula (7) and formula (8), and the values of these variables are determined according to specific conditions. Among them, e ∈ [-10, -2], g ∈ [-3, 3], f ∈ [5, 12], m ∈ [10, 16] n ∈ [17, 25], p ∈ [30, 40 ]. According to the data selected in this paper, the values of the variables here are: e=-5, g=0, f=10, m=15, n=20, p=35.

Step 4. Build a training set: select the daily maximum load value of the first d days of the i-th day and the average temperature after blurring As the input vector of the training sample on the i-th day, that is Select the daily maximum load on the i-th day As the target output value of the training sample on the i-th day, that is So as to get the training sample set i=1,2,...,N ^train , N ^train is the total number of training samples;

Step 5. Build a test set: select the daily maximum load value of the first d days of the kth day and the average temperature after blurring As the input vector of the test sample on the kth day, that is Select the daily maximum load on the kth day As the target output value of the test sample on the kth day, that is To get the test sample set k=1,2,...,N ^test , N ^test is the total number of test samples;

Step 6. Input the d-dimensional row vector of the i-th row As the i-th nonlinear component x _i and the i-th linear component u _i of the training input variable respectively, the one-dimensional actual output value of the i-th row in the training set As the i-th actual output value y _i , the fuzzy support vector quantile regression model shown in formula (1) is established as:

In formula (1), T is the transposition; w _i represents the average temperature factor after fuzzification on the i-th day; τ _r represents the r-th quantile point, and τ _r ∈ (0,1), r=1,2 ,…,N _τ ; N _τ represents the number of quantile points; Represents the parameter vector under the rth quantile point τ _r , C is the penalty parameter, is the coefficient vector under the rth quantile point τ _r , φ( ) represents the nonlinear mapping function, denotes the test function; and has: in, Denotes the threshold at the rth quantile τ _r and has: Among them, α _i and Represents the i-th optimal Lagrange multiplier; K is the kernel matrix in the input space, and has:

K(x _i , x _v )=φ(x _i ) ^T φ(x _v ); v∈I; I is the subscript set of the obtained support vector

Step 7. Solve parameter values: According to the fuzzy support vector quantile regression model, introduce slack variables to construct the Lagrange function and solve formula (1) to obtain the rth quantile point τ _r as shown in formula (2) parameter vector threshold and coefficient vector

In formula (2), α, α ^* is the optimal Lagrange multiplier vector, y={y _i |i∈I}; the design matrix is

Step 8. Substitute the obtained parameter values into the model and solve the test target output value:

The parameter vector under the rth quantile point τ _r threshold and coefficient vector Substitute into the fuzzy support vector quantile regression prediction model; and input the d-dimensional row vector of the kth row in the test set The k-th nonlinear component x _k and the k-th linear component u _k are used as test input variables, so that the output value of the k-th row under the r-th quantile point τ _r in the test set can be obtained by using formula (3)

In formula (3), K _k represents the kth row vector of kernel matrix K;

Step 9. Use kernel density estimation to predict the probability density of the daily maximum load: the flow chart is shown in Figure 2;

Step 9.1, Solve the test target output value under different quantile points: Let the prediction result under the rth quantile point τ _r on the kth day be: In order to obtain the forecast results under all quantile points on the kth day Then get the test output value of the daily maximum load

Step 9.2. Calculate the value of the probability density function: use formula (4) to obtain the value of the probability density function corresponding to the wth quantile point τ _w on the kth day

In formula (4), z _{k, w} represent the prediction results under the wth quantile point τ _w on the kth day; the number of quantile points w=1,2,...,N _τ ; h is the window width, K ₁ (η) is the Epanechnikov kernel function, and has: Among them, |η≤1|;

Step 9.2. Solve the optimal window width: In the research of kernel density estimation method, window width selection is a very important issue in the local smoothness of the probability density prediction function. The optimal window width h ^* of the Epanechnikov kernel function K ₁ (η) is calculated using the thumb principle shown in formula (5):

h ^* ＝1.06s _X N _τ ^-1/5 (5)

In formula (5), s _X is the standard deviation of the test output value of the daily maximum load, and then the optimal window width is obtained based on the standard of 1.06 times the sample standard deviation. The thumb principle is to directly obtain the optimal window width according to the sample standard deviation through fixed standards.

Step 9.3: According to the Epanechnikov kernel function K ₁ (η) and the optimal window width h ^* , obtain the probability density prediction result of the daily maximum load.

Step 9.4, the probability density curve is shown in Figure 3: Figure 3 shows the probability density distribution of the predicted values on the 1st day, the 6th day, the 11th day, the 16th day, the 21st day, and the 26th day. It can be seen from the figure that the predicted values all appear at the mode of the probability density prediction curve with a relatively high probability.

Claims (2)

1. A power load probability density prediction method based on fuzzy support vector quantile regression is characterized by comprising the following steps:

step 1, collecting and determining day maximum load data L 'before the predicted day and average temperature data W ═ W'₁,w′₂,…,w′_j,…,w′_N)；w′_jAverage temperature on day j; j is more than or equal to 1 and less than or equal to N, and N is the total days;

step 2, carrying out normalization processing on the daily maximum load data L' to obtain normalized daily maximum loadLoad data L ═ L₁,l₂,…,l_j,…,l_N)；l_jNormalized postday maximum load for day j;

step 3, fuzzifying the average temperature data W' by adopting a fuzzy theory to obtain fuzzy average temperature data W ═ (W ═₁,w₂,…,w_j,…,w_N)，w_jMean temperature after blur for day j;

step 4, selecting the day maximum load value of d days before the ith dayAnd the mean temperature after blurringAs input vectors for day i training samples, i.e.Selecting the day maximum load of the ith dayAs target output values for day i training samples, i.e.Thereby obtaining a training sample seti＝1,2,…,N^train，N^trainThe total number of training samples;

step 5, selecting the day maximum load value d days before the k dayAnd the mean temperature after blurringAs day k testInput vectors of samples, i.e.Selecting the day maximum load of the k dayAs target output value of the day k test sample, i.e.Thereby obtaining a test sample setk＝1,2,…,N^test，N^testThe total number of the test samples is;

step 6, inputting the d-dimensional row vector of the ith rowI-th nonlinear components x as training input variables respectively_iAnd the ith linear component u_iAnd one-dimensional actual output value of the ith row in the training setAs the ith actual output value y_iEstablishing a fuzzy support vector quantile regression model shown as the formula (1) as follows:

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in the formula (1), T is transposition; w is a_iRepresents the average temperature factor after fuzzification on day i; tau is_rDenotes the r-th quantile and_r∈(0,1)，r＝1,2,…,N_τ；N_τrepresenting the number of quantiles;denotes the r-th quantile_rParameter vector ofAnd C is a penalty parameter,is the r-th quantile_rThe lower coefficient vector, phi (-) represents a non-linear mapping function,representing a checking function; and comprises the following components:wherein, denotes the r-th quantile_rThe following thresholds, in combination:wherein alpha is_iAndrepresents the ith optimal Lagrange multiplier; k is a kernel matrix in the input space and has:

K(x_i,x_v)＝φ(x_i)^Tφ(x_v) (ii) a v is an element of I; i is the subscript set of the resulting support vectors

Step 7, according to the fuzzy support vector quantile regression model, introducing a relaxation variable to construct a Lagrange function and solving the formula (1) to obtain the gamma quantile point tau shown in the formula (2)_rParameter vector ofThreshold valueSum coefficient vector

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in the formula (2), α^*For the optimal Lagrange multiplier vector, y ═ y_iI belongs to I; design the matrix as

Step 8, dividing the r th quantile point tau_rParameter vector ofThreshold valueSum coefficient vectorSubstituting the fuzzy support vector quantile regression prediction model; and inputting the d-dimensional row vector of the k line in the test setKth nonlinear component x as a test input variable_kAnd the k-th linear component u_kThereby obtaining the r-th quantile τ in the test set using equation (3)_rOutput value of line k below

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In the formula (3), K_kA K-th row vector representing the kernel matrix K;

and 9, realizing the probability density prediction of daily maximum load by using the kernel density estimation:

step 9.1, let the kth quantile τ on day k_rThe following predicted results are:thereby obtaining the prediction results of all the quantiles on the k dayFurther obtaining the test output value of daily maximum load

Step 9.2, obtaining w quantile tau at day k by using the formula (4)_wCorresponding probability density function value

<mrow> <msub> <mover> <mi>f</mi> <mo>^</mo> </mover> <mi>h</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>z</mi> <mrow> <mi>k</mi> <mo>,</mo> <mi>w</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <msub> <mi>N</mi> <mi>&amp;tau;</mi> </msub> <mi>h</mi> </mrow> </mfrac> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>r</mi> <mo>=</mo> <mn>1</mn> </mrow> <msub> <mi>N</mi> <mi>&amp;tau;</mi> </msub> </munderover> <msub> <mi>K</mi> <mn>1</mn> </msub> <mfrac> <mrow> <msub> <mi>z</mi> <mrow> <mi>k</mi> <mo>,</mo> <mi>w</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>z</mi> <mrow> <mi>k</mi> <mo>,</mo> <mi>r</mi> </mrow> </msub> </mrow> <mi>h</mi> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow>

In the formula (4), z_k,wW quantile τ representing day k_wThe prediction results of (1); number of quantile points w is 1,2, …, N_τ(ii) a h is window width, K₁(η) is an Epanechnikov kernel function and has:wherein | η is less than or equal to 1 |;

step 9.3, calculating the Epanechnikov kernel function K by using the thumb principle shown in the formula (5)₁(η) optimum window width h^*：

h^*＝1.06s_XN_τ ^-1/5(5)

In the formula (5), s_XStandard deviation of the test output value for daily maximum load;

step 9.4, according to Epanechnikov kernel function K₁(η) and the optimum window width h^*And obtaining the probability density prediction result of the daily maximum load.

2. The method of predicting the probability density of an electric power load according to claim 1, wherein the blurring process in the step 3 is performed as follows:

the average temperature w 'of the j-th day is calculated from the membership function for low temperature represented by the formula (6), the membership function for medium temperature represented by the formula (7) and the membership function for high temperature represented by the formula (8)'_jDividing the data into low temperature data, medium temperature data or high temperature data to obtain the average temperature w 'of the j day'_jThe fuzzy set to which it belongs; the fuzzy average temperature data W ═ W (W) is obtained₁,w₂,…,w_j,…,w_N)；

The low temperature membership function is:

<mrow> <msub> <mi>w</mi> <mi>j</mi> </msub> <mo>=</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <mn>0</mn> <mo>,</mo> </mrow> </mtd> <mtd> <mrow> <msubsup> <mi>w</mi> <mi>j</mi> <mo>&amp;prime;</mo> </msubsup> <mo>&gt;</mo> <mi>f</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mrow> <mi>f</mi> <mo>-</mo> <msubsup> <mi>w</mi> <mi>j</mi> <mo>&amp;prime;</mo> </msubsup> </mrow> <mrow> <mi>f</mi> <mo>-</mo> <mi>e</mi> </mrow> </mfrac> <mo>,</mo> </mrow> </mtd> <mtd> <mrow> <mi>e</mi> <mo>&amp;le;</mo> <msubsup> <mi>w</mi> <mi>j</mi> <mo>&amp;prime;</mo> </msubsup> <mo>&lt;</mo> <mi>f</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>1</mn> <mo>,</mo> </mrow> </mtd> <mtd> <mrow> <msubsup> <mi>w</mi> <mi>j</mi> <mo>&amp;prime;</mo> </msubsup> <mo>&lt;</mo> <mi>e</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow>

the membership function of the intermediate temperature is as follows:

the membership function for high temperature is:

<mrow> <msub> <mi>w</mi> <mi>j</mi> </msub> <mo>=</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <mn>0</mn> <mo>,</mo> </mrow> </mtd> <mtd> <mrow> <msubsup> <mi>w</mi> <mi>j</mi> <mo>&amp;prime;</mo> </msubsup> <mo>&lt;</mo> <mi>n</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mrow> <msubsup> <mi>w</mi> <mi>j</mi> <mo>&amp;prime;</mo> </msubsup> <mo>-</mo> <mi>n</mi> </mrow> <mrow> <mi>p</mi> <mo>-</mo> <mi>n</mi> </mrow> </mfrac> <mo>,</mo> </mrow> </mtd> <mtd> <mrow> <mi>n</mi> <mo>&amp;le;</mo> <msubsup> <mi>w</mi> <mi>j</mi> <mo>&amp;prime;</mo> </msubsup> <mo>&amp;le;</mo> <mi>p</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>1</mn> <mo>,</mo> </mrow> </mtd> <mtd> <mrow> <msubsup> <mi>w</mi> <mi>j</mi> <mo>&amp;prime;</mo> </msubsup> <mo>&gt;</mo> <mi>p</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow>

in the formula (6), the formula (7) and the formula (8), e < g < f < m < n < p, and e ∈ 10, -2, g ∈ 3, f ∈ 5, 12, m ∈ [10, 16], n ∈ [17, 25], p ∈ [30, 40] are satisfied.