CN106251027B - Electric load probability density Forecasting Methodology based on fuzzy support vector quantile estimate - Google Patents
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Abstract
The present invention relates to a kind of electric load probability density Forecasting Methodology based on fuzzy support vector quantile estimate, first, Collection and Forecast Day pervious Daily treatment cost data and average temperature data, and using historical data structure training set and test set.Then, the Lagrange multiplier and supporting vector subscript of fuzzy support vector quantile estimate model are obtained using training set, and fuzzy support vector quantile estimate prediction model is established according to obtained model parameter value, and test set substitution model is obtained into predicted value.Finally, using the predicted value under obtained different quantiles, and realize that the probability density of Daily treatment cost is predicted with Density Estimator.The present invention can be effectively reduced prediction error, improve load forecast precision, achieve good prediction effect, and adjust the more reliable foundations of offer such as electricity consumption plan, optimization generating set output for electric power system dispatching department.
Description
Technical Field
The invention belongs to the field of power load prediction combining a statistical method and intelligent calculation, and mainly relates to a power load probability density prediction method based on fuzzy support vector quantile regression.
Background
The load prediction of the power system is to search the influence of the change rule of the historical data of the power load on the future load according to the historical data of the power load, economy, society, weather and the like, and seek the internal relation between the power load and various relevant factors so as to scientifically predict the future power load. Accurate load prediction of the power system has important significance for power system scheduling, power utilization, planning, power purchase plan making, operation mode arrangement and the like. This also requires power system researchers to come up with more efficient methods to improve prediction accuracy.
In recent years, with the rapid development of smart grids and the increase of uncertain factors, and the change of power load is restricted by many factors such as system operation characteristics, social factors and natural conditions, a large amount of data is needed for load prediction, and the data are difficult to ensure accuracy and reliability; even if the resulting data is accurate, there is uncertainty. For example, temperature factors may cause load variations. Therefore, the power system requires a more intelligent method for improving the accuracy of power load prediction. In addition, in order to improve the safety and economy of grid operation and improve the quality of power supply, the demand of power system operation scheduling on load prediction accuracy is higher and higher.
In recent years, the conventional power system load prediction field only considers historical loads and factors influencing load prediction, and fails to deal with uncertain influencing factors. The main disadvantages are:
(1) factors affecting the power load prediction are many, including date, meteorological factors, temperature factors, and so on. How these factors affect the load accuracy is uncertain or ambiguous. The traditional power load prediction method does not preprocess the uncertain factors. The historical data adopted in prediction are all determined values, but the historical data form certain accidental factors and the uncertainty of the historical data is ignored;
(2) the traditional power load prediction method only gives a point prediction result or an interval prediction result, can not accurately depict the fluctuation of the power load, and does not give the probability density distribution of the obtained prediction result.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provides a power load probability density prediction method based on fuzzy support vector quantile regression so as to fuzzify an average temperature factor and greatly reduce the uncertainty of influencing factors, thereby effectively reducing prediction errors, improving the power load prediction precision and providing a reliable basis for a power system dispatching department to adjust a power utilization plan, optimize the output of a generator set and the like.
The invention relates to a power load probability density prediction method based on fuzzy support vector quantile regression, which is characterized by comprising the following steps of:
step 1, collecting and determining daily maximum load data L' before a prediction day and average temperature data W ═ W (W)1′,w2′,…,wj′,…,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, normalizing the daily maximum load data L' to obtain normalized daily maximum load data L ═ L1,l2,…,lj,…,lN)；ljNormalized 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,…,wj,…,wN)，wjMean 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,…,Ntrain，NtrainThe total number of training samples;
step 5, selecting the day maximum load value d days before the k dayAnd the mean temperature after blurringAs input vectors for the day k test 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,…,Ntest，NtestThe 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 respectivelyiAnd the ith linear component uiAnd one-dimensional actual output value of the ith row in the training setAs the ith actual output value yiEstablishing a fuzzy support vector quantile regression model shown as the formula (1) as follows:
in the formula (1), T is transposition; w is aiRepresents the average temperature factor after fuzzification on day i; tau isrDenotes the r-th quantile andr∈(0,1)，r＝1,2,…,Nτ；Nτrepresenting the number of quantiles;denotes the r-th quantilerA vector of parameters, C being a penalty parameter,is the r-th quantilerThe lower coefficient vector, phi (-) represents a non-linear mapping function,representing a checking function; and comprises the following components:wherein, denotes the r-th quantilerThe following thresholds, in combination:wherein alpha isiAndrepresents the ith optimal Lagrange multiplier; k is a kernel matrix in the input space and has:
K(xi,xv)＝φ(xi)Tφ(xv) (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
in the formula (2), α*For the optimal Lagrange multiplier vector, y ═ yiI belongs to I; design the matrix as
Step 8, dividing the r th quantile point taurParameter 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 variablekAnd the k-th linear component ukThereby obtaining the r-th quantile τ in the test set using equation (3)rOutput value of line k below
In the formula (3), KkA 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 krThe 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 the kth day by using the formula (4)w quantites tauwCorresponding probability density function value
In the formula (4), zk,wW quantile τ representing day kwThe prediction results of (1); number of quantile points w is 1,2, …, Nτ(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 9.2, calculating the Epanechnikov kernel function K by using the thumb principle shown in the formula (5)1(η) optimum window width h*：
h*＝1.06sXNτ -1/5(5)
In the formula (5), sXStandard deviation of the test output value for daily maximum load;
step 9.3, according to the Epanechnikov kernel function K1(η) and the optimum window width h*And obtaining the probability density prediction result of the daily maximum load.
The method for predicting the probability density of the power load according to the present invention is also characterized in that the fuzzification processing 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 data of day jAverage temperature w'jThe fuzzy set to which it belongs; the fuzzy average temperature data W ═ W (W) is obtained1,w2,…,wj,…,wN)；
The low temperature membership function is:
the membership function of the intermediate temperature is as follows:
the membership function for high temperature is:
in the formula (6), the formula (7) and the formula (8), e is more than g, more than f, more than m, and more than n.
Compared with the prior art, the invention has the beneficial effects that:
1. the invention combines the traditional support vector regression method and quantile regression method and introduces fuzzy membership function to obtain fuzzy support vector quantile regression method, the nonlinear problem can be solved by adopting the support vector regression method, the quantile regression method estimates the condition quantile of the response variable by using the information provided by the regression variable to obtain the influence of the explanation variable on the response variable at different quantiles, and further the influence of the explanation variable on the change range and the condition distribution shape of the response variable can be accurately described, thus combining the two methods to obtain the complete probability distribution of future load and facilitating the scientific decision of a decision maker.
2. The method mainly considers the influence of the average temperature on the daily maximum load prediction, because the average temperature factor is fuzzy, a fuzzy theory is introduced, the membership function is adopted to fuzzify the average temperature to improve the prediction precision, probability density prediction results under different quantiles are given, and the effectiveness and the accuracy of the method are proved.
3. The invention not only effectively reduces the prediction error and improves the prediction precision of the power load, but also can accurately depict the volatility of the power load and provide a probability density prediction curve chart. Therefore, a reliable basis is provided for the power system dispatching department to adjust the power utilization plan, optimize the output of the generator set and the like.
Drawings
FIG. 1 is an overall flow chart of the method of the present invention;
FIG. 2 is a detailed flow chart of the method of the present invention;
FIG. 3 is a probability density chart of the method of the present invention.
Detailed Description
In the implementation process, the power load probability density prediction method based on fuzzy support vector quantile regression mainly considers the influence of average temperature on power load prediction. The flow chart is shown in figure 1 and is carried out according to the following steps:
step 1, factors influencing power load prediction are more, and the influence of average temperature factors on power load prediction results is obtained through research and analysis;
step 1.1, the invention selects the data of the global load forecasting competition of the EUNITE network organization for testing, which comprises load data of a 48-hour time interval (one load point every half hour) every day in 1997-1998 and average temperature data every day in 1997-1998. The data belongs to complete data. And predicting the daily maximum load data for 31 days in month 1 of 1999;
step 1.2, 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;
and 2, normalizing daily maximum load data in order to avoid the calculation saturation phenomenon in the calculation process.
Normalizing the daily maximum load data L' to obtain normalized daily maximum load data L ═ L1,l2,…,lj,…,lN)；ljThe normalized daily maximum load of the j day is determined, wherein the average temperature factor is determined according to weather forecast of a meteorological department, and daily maximum load data is determined according to data provided by an electric power company;
and 3, because the average temperature factor is uncertain or fuzzy in actual life, fuzzifying the average temperature factor by establishing a membership function by adopting a fuzzy theory.
Fuzzifying the average temperature data W' by adopting a fuzzy theory to obtain fuzzy average temperature data W ═ W1,w2,…,wj,…,wN)，wjMean temperature after blur for day j; wherein, the fuzzification treatment is carried out according to the following processes:
the average temperature w 'on day j is determined from the membership function for low temperature represented by formula (6), the membership function for medium temperature represented by formula (7) and the membership function for high temperature represented by 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,…,wj,…,wN)；
The low temperature membership function is:
the membership function for the mesophilic temperature is:
the membership function for high temperature is:
the formula (6), the formula (7) and the formula (8) satisfy the condition that e is more than g and less than f and more than m and more than n and p, and the values of the variables are determined according to specific conditions. Wherein e ∈ 10, -2, g ∈ [ -3, 3], f ∈ [5, 12], m ∈ [10, 16] n ∈ [17, 25], p ∈ [30, 40 ]. According to the data selected in the text, the values of the variables are respectively as follows: e-5, g-0, f-10, m-15, n-20, p-35.
Step 4, constructing a training set: selecting the day maximum load value 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,…,Ntrain，NtrainThe total number of training samples;
step 5, constructing a test set: selecting the day maximum load value d days before the k dayAnd the mean temperature after blurringAs input vectors for the day k test 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,…,Ntest，NtestThe 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 respectivelyiAnd the ith linear component uiOne-dimensional actual output value of ith row in training setAs the ith actual output value yiEstablishing a fuzzy support vector quantile regression model shown as the formula (1) as follows:
in the formula (1), T is transposition; w is aiRepresents the average temperature factor after fuzzification on day i; tau isrDenotes the r-th quantile andr∈(0,1)，r＝1,2,…,Nτ；Nτrepresenting the number of quantiles;denotes the r-th quantilerA vector of parameters, C being a penalty parameter,is the r-th quantilerThe lower coefficient vector, phi (-) represents a non-linear mapping function,representing a checking function; and comprises the following components:wherein, denotes the r-th quantilerThe following thresholds, in combination:wherein alpha isiAndrepresents the ith optimal Lagrange multiplier; k is a kernel matrix in the input space and has:
K(xi,xv)＝φ(xi)Tφ(xv) (ii) a v is an element of I; i is the subscript set of the resulting support vectors
Step 7, solving parameter values: introducing a relaxation variable to construct a Lagrange function and solving a formula (1) according to a fuzzy support vector quantile regression model to obtain an r quantile point tau shown in a formula (2)rParameter vector ofThreshold valueSum coefficient vector
in the formula (2), α*For the optimal Lagrange multiplier vector, y ═ yiI belongs to I; design the matrix as
And 8, substituting the obtained parameter values into the model and solving the output value of the test target:
dividing the r-th quantilerParameter vector ofThreshold valueSum coefficient vectorSubstituting the fuzzy support vector quantile regression prediction model; and d-dimensional row vector input by the k line in the test setKth nonlinear component x as a test input variablekAnd the k-th linear component ukThereby obtaining the r-th quantile τ in the test set using equation (3)rOutput value of line k below
In the formula (3), KkA K-th row vector representing the kernel matrix K;
and 9, carrying out probability density prediction on the daily maximum load by using kernel density estimation: the flow chart is shown in FIG. 2;
step 9.1, solving the output values of the test targets under different quantiles: let the r-th quantile τ on day krThe 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, solving the function value of the probability density: obtaining w quantile tau at day k by using formula (4)wCorresponding probability density function value
In the formula (4), zk,wW quantile τ representing day kwThe prediction results of (1); number of quantile points w is 1,2, …, Nτ(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 9.2, solving the optimal window width: in the research of the kernel density estimation method, the window width selection is a very important problem in the local smoothness problem of the probability density prediction function. Epanechnikov kernel function K is calculated by using thumb principle shown in formula (5)1(η) optimum window width h*：
h*＝1.06sXNτ -1/5(5)
In the formula (5), sXThe standard deviation of the test output value of the daily maximum load is obtained, and then the standard deviation of the sample is 1.06 times, so that the optimal window width is obtained. The thumb principle is to directly obtain the optimal window width according to the standard deviation of the sample by a fixed standard.
Step 9.3, according to the Epanechnikov kernel function K1(η) and optimum window width h*And obtaining the probability density prediction result of the daily maximum load.
Step 9.4, the probability density graph is shown in fig. 3: FIG. 3 shows probability density distribution plots of predicted values at day 1, day 6, day 11, day 16, day 21, and day 26. As can be seen from the figure, the predicted values all appear with a greater probability at the mode of the probability density prediction curve.
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,…,lj,…,lN)；ljNormalized 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,…,wj,…,wN)，wjMean 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,…,Ntrain，NtrainThe 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,…,Ntest，NtestThe 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 respectivelyiAnd the ith linear component uiAnd one-dimensional actual output value of the ith row in the training setAs the ith actual output value yiEstablishing a fuzzy support vector quantile regression model shown as the formula (1) as follows:
<mrow> <munder> <mi>min</mi> <mrow> <msub> <mi>&omega;</mi> <msub> <mi>&tau;</mi> <mi>r</mi> </msub> </msub> <mo>,</mo> <msub> <mi>b</mi> <msub> <mi>&tau;</mi> <mi>r</mi> </msub> </msub> </mrow> </munder> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>|</mo> <mo>|</mo> <msub> <mi>&omega;</mi> <msub> <mi>&tau;</mi> <mi>r</mi> </msub> </msub> <mo>|</mo> <msup> <mo>|</mo> <mn>2</mn> </msup> <mo>+</mo> <mi>C</mi> <munderover> <mo>&Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <msup> <mi>N</mi> <mrow> <mi>t</mi> <mi>r</mi> <mi>a</mi> <mi>i</mi> <mi>n</mi> </mrow> </msup> </munderover> <msub> <mi>w</mi> <mi>i</mi> </msub> <msub> <mi>&rho;</mi> <msub> <mi>&tau;</mi> <mi>r</mi> </msub> </msub> <mrow> <mo>(</mo> <msub> <mi>y</mi> <mi>i</mi> </msub> <mo>-</mo> <msub> <mi>b</mi> <msub> <mi>&tau;</mi> <mi>r</mi> </msub> </msub> <mo>-</mo> <msubsup> <mi>&beta;</mi> <msub> <mi>&tau;</mi> <mi>r</mi> </msub> <mi>T</mi> </msubsup> <msub> <mi>u</mi> <mi>i</mi> </msub> <mo>-</mo> <msubsup> <mi>&omega;</mi> <msub> <mi>&tau;</mi> <mi>r</mi> </msub> <mi>T</mi> </msubsup> <mi>&phi;</mi> <mo>(</mo> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow>
in the formula (1), T is transposition; w is aiRepresents the average temperature factor after fuzzification on day i; tau isrDenotes the r-th quantile andr∈(0,1)，r＝1,2,…,Nτ；Nτrepresenting the number of quantiles;denotes the r-th quantilerParameter vector ofAnd C is a penalty parameter,is the r-th quantilerThe lower coefficient vector, phi (-) represents a non-linear mapping function,representing a checking function; and comprises the following components:wherein, denotes the r-th quantilerThe following thresholds, in combination:wherein alpha isiAndrepresents the ith optimal Lagrange multiplier; k is a kernel matrix in the input space and has:
K(xi,xv)＝φ(xi)Tφ(xv) (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
<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>&omega;</mi> <msub> <mi>&tau;</mi> <mi>r</mi> </msub> </msub> <mo>=</mo> <munderover> <mo>&Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <msup> <mi>N</mi> <mrow> <mi>t</mi> <mi>r</mi> <mi>a</mi> <mi>i</mi> <mi>n</mi> </mrow> </msup> </munderover> <mrow> <mo>(</mo> <msub> <mi>&alpha;</mi> <mi>i</mi> </msub> <mo>-</mo> <msubsup> <mi>&alpha;</mi> <mi>i</mi> <mo>*</mo> </msubsup> <mo>)</mo> </mrow> <mi>&phi;</mi> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msup> <mrow> <mo>(</mo> <msub> <mi>b</mi> <msub> <mi>&tau;</mi> <mi>r</mi> </msub> </msub> <mo>,</mo> <msub> <mi>&beta;</mi> <msub> <mi>&tau;</mi> <mi>r</mi> </msub> </msub> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>U</mi> <mi>T</mi> </msup> <mi>U</mi> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>U</mi> <mi>T</mi> </msup> <mrow> <mo>(</mo> <mi>y</mi> <mo>-</mo> <mi>K</mi> <mo>(</mo> <mrow> <mi>&alpha;</mi> <mo>-</mo> <msup> <mi>&alpha;</mi> <mo>*</mo> </msup> </mrow> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow>
in the formula (2), α*For the optimal Lagrange multiplier vector, y ═ yiI belongs to I; design the matrix as
Step 8, dividing the r th quantile point taurParameter 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 variablekAnd the k-th linear component ukThereby obtaining the r-th quantile τ in the test set using equation (3)rOutput value of line k below
<mrow> <msub> <mi>Q</mi> <msub> <mi>y</mi> <mi>k</mi> </msub> </msub> <mrow> <mo>(</mo> <msub> <mi>&tau;</mi> <mi>r</mi> </msub> <mo>|</mo> <msub> <mi>u</mi> <mi>k</mi> </msub> <mo>,</mo> <msub> <mi>x</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>b</mi> <msub> <mi>&tau;</mi> <mi>r</mi> </msub> </msub> <mo>+</mo> <msubsup> <mi>&beta;</mi> <msub> <mi>&tau;</mi> <mi>r</mi> </msub> <mi>T</mi> </msubsup> <msub> <mi>u</mi> <mi>k</mi> </msub> <mo>+</mo> <msub> <mi>K</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>&alpha;</mi> <mo>-</mo> <msup> <mi>&alpha;</mi> <mo>*</mo> </msup> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>
In the formula (3), KkA 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 krThe 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>&tau;</mi> </msub> <mi>h</mi> </mrow> </mfrac> <munderover> <mo>&Sigma;</mo> <mrow> <mi>r</mi> <mo>=</mo> <mn>1</mn> </mrow> <msub> <mi>N</mi> <mi>&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), zk,wW quantile τ representing day kwThe prediction results of (1); number of quantile points w is 1,2, …, Nτ(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*＝1.06sXNτ -1/5(5)
In the formula (5), sXStandard 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,…,wj,…,wN)；
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>&prime;</mo> </msubsup> <mo>></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>&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>&le;</mo> <msubsup> <mi>w</mi> <mi>j</mi> <mo>&prime;</mo> </msubsup> <mo><</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>&prime;</mo> </msubsup> <mo><</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>&prime;</mo> </msubsup> <mo><</mo> <mi>n</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mrow> <msubsup> <mi>w</mi> <mi>j</mi> <mo>&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>&le;</mo> <msubsup> <mi>w</mi> <mi>j</mi> <mo>&prime;</mo> </msubsup> <mo>&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>&prime;</mo> </msubsup> <mo>></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.
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