CN101556664A - Cooperative load forecasting method based on maximum informational entropy - Google Patents

Abstract

The invention belongs to the field of medium-term and long-term load forecasting in power distribution system planning and short-term load forecasting in power distribution system planning, relating to a cooperative load forecasting method based on maximum informational entropy. The method comprises the following steps of: calculating the statistical characteristics of an original forecasting scheme; analyzing the confidence level of the original forecasting scheme; obtaining a cooperative probability distribution function; simultaneously taking the statistical characteristics of an upper level and a lower level as constraint information, and obtaining the cooperative probability distribution function based on the maximum informational entropy principle; and obtaining a cooperative forecasting principle: based on the cooperative probability distribution function, calculating mathematical expectation and maximum probability, and finally determining the high scheme, medium scheme and low scheme of cooperative load forecasting. The method applies the maximum informational entropy principle in the theoretical research of load forecasting under power grid cooperative planning mode, and can realize the information comprehension of multi-section, multi-path and multi-scheme, thereby effectively solving the problem of data collision of an upper level network and a lower level network, realizing the cooperative forecasting of the upper network and the lower network and providing the reference basis for the planning and operation of the power distribution system planning.

Description

Cooperative load forecasting method based on maximum informational entropy

Technical field

The invention belongs to distribution system and plan medium-term and long-term load prediction and distribution system operation short-term load forecasting field, relate to a kind of cooperative load forecasting method.

Background technology

Load prediction is the element task of urban power network planning, and traditional prediction mode is replaced by the working method of multidisciplinary division of labor asynchronous cooperation prediction just gradually.Multistage information can be fully resolved in collaborative prediction, reduces the influence of randomness error to predicted value, realizes that load prediction becomes more meticulous.The working method of multidisciplinary division of labor asynchronous cooperation prediction need be finished by city high pressure electric network synthetic planning information platform, and step is as follows:

(1) each subordinate unit predicts respectively and uploads and predicts the outcome;

(2) superior unit is responsible for the total amount prediction and uploads predicting the outcome simultaneously;

(3) superior unit gathers predicting the outcome of each department of subordinate, balance higher level department total amount predict the outcome with gather after department of subordinate predict the outcome, and finally determine unique predicting the outcome;

(4) last, grass-roots unit predicts the outcome according to total amount and adjusts separately result.

As can be seen, cooperative load forecasting has multidisciplinary collaborative, multipath collaborative, multi-scheme is collaborative characteristics:

Multidisciplinary collaborative: as to comprise higher level department and department of a plurality of subordinate in the forecasting process and participate in;

Multipath is collaborative: need carry out balance at predicting the outcome of higher level and subordinate's two paths in the forecasting process;

Multi-scheme is collaborative: in the forecasting process, according to the requirement of urban power network planning and design guide rule, predicting the outcome that departments at different levels submit to comprises 2-3 prediction scheme (high, medium and low scheme) usually.

The change of prediction mode has brought brand-new problem, in forecasting process, because the difference that each link such as the participation of multistage department, raw data quality, algorithm screening, manual intervention exists, higher level department total amount occurs and predicted the outcome and the inconsistent situation of the aggregation of forecasts result of department of subordinate, now this situation has been defined as the superior and the subordinate's electrical network synergistic data conflict.Whole Electric Power Network Planning work all is based on unique expansion that predicts the outcome, how solve the data collision problem of department of the superior and the subordinate by multidisciplinary, multipath and multivariant informix, determine unique rationally, objectively predict the outcome, be the key that solves cooperative load forecasting.All there is the similar data collision problem in numerous areas in Electric Power Network Planning, and how effectively to address these problems is the key that realizes that further multidisciplinary collaborative planning, planning become more meticulous.

The target of cooperative load forecasting is that the basis of design is drawn in selected unique predicting the outcome as the city network planning.Therefore, in collaborative forecasting process, needing fully, parsing higher level, subordinate predict the outcome, the statistical nature that predicts the outcome that various approach are obtained is as constraint information, by the reduction of prediction realization data reliability, integrality, accuracy again, make the most objective deduction to predicting the outcome, thereby realize multidisciplinary, multipath, multivariant informix.

Present planning theory, method and aid decision making instrument are based on all that traditional independent planning mode sets up, and present load forecasting method can improve precision of prediction to a certain extent, but can't realize the informix of the superior and the subordinate's prediction scheme.In the cooperative load forecasting process, the superior and the subordinate's electrical network predict the outcome (all comprising high, medium and low scheme usually separately) often inconsistent, be called the data collision problem, the main planning personnel's of dependence experience manual intervention at present solves.This thinking is theoretically unsound, and influence planning precision can't satisfy the demand that planning becomes more meticulous.

The information theory development is rapid, and wherein Maximum Entropy Principle Method provides good idea for addressing the above problem.Maximum Entropy Principle Method is meant in all compatible distributions, selects satisfying to make information entropy reach the distribution of the distribution of maximum value as system under some constraint condition.This principle is widely used, and is used for researchs such as communication system fragility venture analysis in the communications field; Be used for researchs such as parking lot addressing, public transport demand forecast at field of traffic; Be used for researchs such as seimic frequency-earthquake magnitude relation, wave Wave Height Distribution at meteorological field.At present, the aspect such as will assess in this principle of field of power temporarily at medium-term and long-term load prediction, short-term load forecasting, voltage and obtain application.Zhang Qingbao for example, the paper of Cheng Haozhong etc. " based on the research of the medium-term and long-term load prediction unified model of principle of maximum entropy " (source: relay, 2006,34 (3): 24-27) and Zhu Chengqi, the paper of Sun Hongbin etc. " based on the short-term load forecasting unified model of Maximum Entropy Principle Method " (source: Proceedings of the CSEE, 2005,25 (19): 1-6).The thinking of these two pieces of documents for will various single forecast models predict the outcome and the distribution of historical predicated error as constraint information, the distribution that utilizes Maximum Entropy Principle Method to obtain predicting the outcome.Above-mentioned document is applied to traditional electrical network independence planning field with Maximum Entropy Principle Method, the unified model that proposes is applicable to the independence load prediction of single department, can handle the uncertain problem of load variations, can effectively improve the precision that predicts the outcome, but the working method that can't adapt to the superior and the subordinate's cooperative load forecasting in the electrical network collaborative planning field can't effectively solve the data collision problem in the multidisciplinary collaborative forecasting process.

Summary of the invention

The objective of the invention is to overcome the above-mentioned deficiency of prior art, Maximum Entropy Principle Method is applied to the theoretical research of cooperative load forecasting in the electrical network collaborative planning field, provide a kind of and can obtain the confidence level cooperative load forecasting method of rational prediction scheme automatically, the method that the present invention proposes, can realize multidisciplinary, multipath, multivariant informix, effectively solve the superior and the subordinate's electric network data collision problem, realize upper and lower level electrical network cooperative load forecasting, for distribution system planning provides reference frame with operation.

For this reason, the present invention adopts following technical scheme:

Department of subordinate prediction scheme after the first step is at first gathered higher level department prediction scheme and gathered, definition $g_u\left(x\right)={[(x-{\hat{l}}_{\mathrm{tu}})/{\hat{l}}_{\mathrm{tu}}]}^2,$ $E\left[g_u\left(x\right)\right]=m_{\mathrm{tu}2}/{\left({\hat{l}}_{\mathrm{tu}}\right)}^2,$ $g_d\left(x\right)={[(x-{\hat{l}}_{\mathrm{td}})/{\hat{l}}_{\mathrm{td}}]}^2,$ $E\left[g_d\left(x\right)\right]=m_{\mathrm{td}2}/{\left({\hat{l}}_{\mathrm{td}}\right)}^2,$ In the formula,

Be the mean value of t higher level's prediction scheme, m _Tu2It is the second-order moment around mean of t higher level's prediction scheme;

Be the t subordinate prediction scheme mean value of (gathering the back), m _Td2It is the t subordinate prediction scheme second-order moment around mean of (gathering the back); Next calculates the statistical nature of the superior and the subordinate's prediction scheme: mean value

Second-order moment around mean (m _Tu2, m _Td2), and foundation

m _Tu2, m _Td2Determine g _u(x), g _d(x) expression formula and E[g _u(x)], E[g _d(x)];

Second step separately with the statistical nature of upper and lower level prediction scheme as constraint information, obtain the probability distribution function of upper and lower level prediction scheme correspondence based on following load prediction formula, and then obtain original prediction scheme confidence level:

max?h(X)＝-∫p _i(x)ln?p _i(x)dx (1)

st?∫p _i(x)g _i(x)dx＝E[g _i(x)] i＝u，d (2)

∫p _i(x)dx＝1 i＝u，d (3)

Formula (1) is an objective function, and wherein h (X) is the entropy of stochastic variable X, and p (x) is the probability density of x for the X value; In the formula (2),, i=u should satisfy the constraint of the statistical nature correspondence of higher level department prediction when representing probability distribution function to be asked; Represent that when i=d probability distribution function to be asked should satisfy the constraint of the statistical nature correspondence of department of the subordinate prediction scheme after gathering; Formula (3) is the constraint of the probability distribution function self of the superior and the subordinate's prediction scheme correspondence;

The 3rd step simultaneously with the statistical nature of upper and lower level prediction scheme as constraint information, the cooperative load forecasting formula based on following maximum informational entropy obtains cooperative probability distribution function:

max?h(X)＝-∫p(x)ln?p(x)dx (4)

st?∫p(x)g _u(x)dx＝E[g _u(x)] (5)

∫p(x)g _d(x)dx＝E[g _d(x)] (6)

∫p(x)dx＝1 (7)

Formula (4) is an objective function, and wherein h (X) is the entropy of stochastic variable X, and p (X) is the probability density of x for the X value; Formula (5) expression probability distribution function to be asked should satisfy the constraint of the statistical nature correspondence of higher level department prediction scheme; Formula (6) expression probability distribution function to be asked should satisfy the constraint of the statistical nature correspondence of the subordinate's prediction scheme after gathering; Formula (7) is the constraint of probability distribution function self;

The 4th step was calculated its mathematical expectation and maximum probability based on the cooperative probability distribution function that obtains in the 3rd step, based on the interval correlation theory of estimating of theory of probability, finally determined the high, medium and low scheme of cooperative load forecasting.

As preferred implementation, the cooperative load forecasting method based on maximum informational entropy of the present invention makes F=h (X)-(λ ₀+ 1) (∫ p (x) dx-1)-λ _i(∫ p _i(x) g _i(x) dx-E[g _i(x)]), i=u, d; And order $\∂F/\∂p\left(x\right)=0,$ Can get upper and lower level probability distribution function p _u(x), p _d(x):

p _u(x)＝exp(-λ ₀-λ _ug _u(x))

p _d(x)＝exp(-λ ₀-λ _dg _d(x))

p _u(x) and p _d(x) be respectively the most possible probability distribution function that higher level and subordinate's prediction scheme satisfy;

Make F=h (X)-(λ ₀+ 1) (∫ p (x) dx-1)-λ _u(∫ p _u(x) g _u(x) dx-E[g _u(x)])-λ _d(∫ p _d(x) g _d(x) dx-E[g _d(x)], and the order $\∂F/\∂p\left(x\right)=0,$ Get p (x)=exp (λ ₀-λ _ug _u(x)-λ _dg _d(x))=exp (λ ₁x ²+ λ ₂X+ λ ₃), the g that calculates in again the first step and second being gone on foot _u(x), g _d(x), E[g _u(x)], E[g _dAnd the following system of equations of substitution as a result of following formula (x)]:

$\underset{-\∞}{\overset{+\∞}{\∫}}\exp ({\mathrm{\λ}}_1{x}^2+{\mathrm{\λ}}_{2}x+{\mathrm{\λ}}_3)\mathrm{dx}=1$

$\underset{-\∞}{\overset{+\∞}{\∫}}\exp ({\mathrm{\λ}}_1{x}^2+{\mathrm{\λ}}_{2}x+{\mathrm{\λ}}_3){\left(\frac{x-{\hat{l}}_{\mathrm{tu}}}{{\hat{l}}_{\mathrm{tu}}}\right)}^2\mathrm{dx}=E\left[g_u\left(x\right)\right]$

$\underset{-\∞}{\overset{+\∞}{\∫}}\exp ({\mathrm{\λ}}_1{x}^2+{\mathrm{\λ}}_{2}x+{\mathrm{\λ}}_3){\left(\frac{x-{\hat{l}}_{\mathrm{td}}}{{\hat{l}}_{\mathrm{td}}}\right)}^2\mathrm{dx}=E\left[g_d\left(x\right)\right]$

Obtain parameter lambda ₁, λ ₂, λ ₃, and then obtain cooperative probability distribution function p (x)=exp (λ ₁x ²+ λ ₂X+ λ ₃);

Make confidence level 1-α=k * p _Max, k is that confidence level is adjusted coefficient, k＜1 meets the characteristic of normal distribution according to cooperative probability distribution function, finds the confidence lower limit X of X ^-, confidence upper limit X ⁺Make p{X ^-≤ X≤X ⁺} 〉=1-α gets the confidence upper limit X of X ⁺, mathematical expectation E (X), confidence lower limit X ^-High, medium and low scheme as final collaborative prediction.

Substantive distinguishing features of the present invention is: Maximum Entropy Principle Method is applied to the theoretical research of cooperative load forecasting in the electrical network collaborative planning field, with the cooperative load forecasting process as typically multidisciplinary, multipath, multivariant informix process, cooperative load forecasting method based on maximum informational entropy has been proposed, with the statistical nature of the superior and the subordinate's prediction scheme as constraint information, utilize Maximum Entropy Principle Method to obtain the satisfied cooperative probability distribution function that predicts the outcome, and the utilization theory of probability obtains the most rational collaborative prediction height of confidence level automatically, in, low scheme, thereby the data collision problem in the solution cooperative load forecasting realizes cooperative load forecasting between the superior and the subordinate.

Comparing tradition relies on planning personnel's experience to carry out data collision problem in the manual intervention solution cooperative load forecasting process, the present invention can obtain probability distribution function based on Maximum Entropy Principle Method in the information theory, obtain final high, medium and low scheme based on the theory of probability relevant knowledge, its theoretical foundation is abundant, and range of application is more extensive.The collaborative probability distribution curve that the present invention obtains is partial to the high original prediction scheme of confidence level automatically, and the collaborative prediction scheme confidence level that obtains significantly improves, and efficiently solves the data collision problem in the cooperative load forecasting process.

Description of drawings

Fig. 1: the whole implementation process flow diagram of the cooperative load forecasting method based on maximum informational entropy of the present invention;

Fig. 2: the higher level of the invention process example correspondence, subordinate and adjusted collaborative probability distribution curve.

Embodiment

The present invention proposes a kind of cooperative load forecasting method based on maximum informational entropy, the method with a plurality of departments of the superior and the subordinate, two paths, overlap the statistical nature of prediction scheme as constraint information more, information entropy is maximized as objective function, find the solution the satisfied cooperative probability distribution function that predicts the outcome, and the relevant knowledge of applied probability, obtain the most reliable unique prediction scheme automatically.Below cooperative load forecasting method of the present invention is elaborated.

One, expression formula

The mathematic(al) representation based on the cooperative load forecasting method of maximum informational entropy that the present invention makes up is:

max?h(X)＝-∫p(x)ln?p(x)dx (1)

st?∫p(x)g _u(x)dx＝E[g _u(x)] (2)

∫p(x)g _d(x)dx＝E[g _d(x)] (3)

∫p(x)dx＝1 (4)

Formula (1) is the objective function of the method, and wherein h (X) is the entropy of stochastic variable X, and p (X) is the probability density of x for the X value.

Formula (2) represents that probability distribution function to be asked should satisfy the constraint of the statistical nature correspondence of the high, medium and low scheme of higher level department prediction (being called for short higher level's prediction scheme), wherein $g_u\left(x\right)={[(x-{\hat{l}}_{\mathrm{tu}})/{\hat{l}}_{\mathrm{tu}}]}^2$ Be the special function of the present invention's definition, $E\left[g_u\left(x\right)\right]=m_{\mathrm{tu}2}/{\left({\hat{l}}_{\mathrm{tu}}\right)}^2$ Be its mathematical expectation.In the following formula

Be the mean value of t higher level's prediction scheme, m _Tu2It is the second-order moment around mean (sample variance) of t higher level's prediction scheme.

Formula (3) represents that probability distribution function to be asked should satisfy the constraint of the statistical nature correspondence of the high, medium and low scheme of department of the subordinate prediction after gathering (abbreviation subordinate prediction scheme), wherein $g_d\left(x\right)={[(x-{\hat{l}}_{\mathrm{td}})/{\hat{l}}_{\mathrm{td}}]}^2$ Be the special function of the present invention's definition, $E\left[g_d\left(x\right)\right]=m_{\mathrm{td}2}/{\left({\hat{l}}_{\mathrm{td}}\right)}^2$ Be its mathematical expectation.In the following formula

Be the mean value of t subordinate prediction scheme, m _Td2It is the second-order moment around mean (sample variance) of t subordinate prediction scheme.

What formula (4) embodied is the constraint of probability distribution function self.

Two, find the solution the summary step

The solution procedure based on the cooperative load forecasting method of maximum informational entropy that the present invention makes up is: the first step is calculated original prediction scheme statistical nature: department of the subordinate prediction scheme after gathering higher level department prediction scheme and gathering, and calculate its statistical nature: mean value

Second-order moment around mean (m _Tu2, m _Td2).

Second step was resolved original prediction scheme confidence level: separately with the statistical nature of upper and lower level prediction scheme as constraint information, obtain the probability distribution function of the superior and the subordinate's correspondence based on Maximum Entropy Principle Method, based on the theory of probability relevant knowledge, resolve original prediction scheme confidence level.

The 3rd step obtained cooperative probability distribution function: simultaneously with the statistical nature of upper and lower level as constraint information, obtain cooperative probability distribution function based on Maximum Entropy Principle Method, the collaborative probability distribution curve that obtains can be partial to the high prediction scheme of confidence level in second step automatically.

The 4th step obtained collaborative prediction scheme: based on the cooperative probability distribution function that obtains in the 3rd step, calculate its mathematical expectation and maximum probability, based on the interval correlation theory of estimating of theory of probability, finally determine the high, medium and low scheme of cooperative load forecasting.

Three, find the solution detailed step

Concrete solution procedure can be referring to Fig. 1.

1, calculates original prediction scheme statistical nature

In the cooperative load forecasting process, for raw data provided by the invention comprises: t higher level's prediction scheme l _Tu ^-, l _Tu, l _Tu ⁺The prediction scheme l of t subordinate _Td ^-, l _Td, l _Td ⁺

The statistical nature of the prediction scheme that cooperative load forecasting method obtains various approach is as constraint information, and constraint information comprises:

T higher level's prediction scheme mean value

${\hat{l}}_{\mathrm{tu}}=\frac{1}{3}(l_{\mathrm{tu}}^{-}+l_{\mathrm{tu}}+l_{\mathrm{tu}}^{+})$

The second-order moment around mean m of t higher level's prediction scheme _Tu2:

$m_{\mathrm{tu}2}=\frac{1}{3}[{(l_{\mathrm{tu}}^{-}-{\hat{l}}_{\mathrm{tu}})}^2+{(l_{\mathrm{tu}}-{\hat{l}}_{\mathrm{tu}})}^2+{(l_{\mathrm{tu}}^{+}-{\hat{l}}_{\mathrm{tu}})}^2]$

T subordinate prediction scheme mean value

${\hat{l}}_{\mathrm{td}}=\frac{1}{3}(l_{\mathrm{td}}^{-}+l_{\mathrm{td}}+l_{\mathrm{td}}^{+})$

The second-order moment around mean m of t subordinate prediction scheme _Td2:

$m_{\mathrm{td}2}=\frac{1}{3}[{(l_{\mathrm{tf}}^{-}-{\hat{l}}_{\mathrm{td}})}^2+{(l_{\mathrm{td}}-{\hat{l}}_{\mathrm{td}})}^2+{(l_{\mathrm{td}}^{+}-{\hat{l}}_{\mathrm{td}})}^2]$

By

m _Tu2, m _Td2Above-mentioned four formulas can be determined g in the collaborative Forecasting Methodology _u(x), g _d(x) expression formula and E[g _u(x)], E[g _d(x)] result.

2, resolve original prediction scheme confidence level

Separately with the statistical nature of higher level or subordinate's prediction scheme as constraint information, the cooperative load forecasting method mathematic(al) representation can be reduced to

max?h(X)＝-∫p _i(x)ln?p _i(x)dx

st?∫p _i(x)g _i(x)dx＝E[g _i(x)]

∫p _i(x)dx＝1 i＝u，d

Utilize method of Lagrange multipliers, order

F＝h(X)-(λ ₀+1)(∫p(x)dx-1)-λ _i(∫p _i(x)g _i(x)dx-E[g _i(x)]) i＝u，d

And order $\∂F/\∂p\left(x\right)=0,$ Can get p _u(x), p _d(x):

p _u(x)＝exp(-λ ₀-λ _ug _u(x))

p _d(x)＝exp(-λ ₀-λ _dg _d(x))

The probability distribution function p that obtains based on Maximum Entropy Principle Method _u(x) and p _d(x) be exactly the most possible probability distribution function that higher level or subordinate's prediction scheme satisfy.

Make X equal l _Tu ^-, l _Tu ⁺, l _Td ^-, l _Td ⁺, can obtain the Probability p of the high and low scheme correspondence of upper and lower level _u(l _Tu ^-), p _u(l _Tu ⁺), p _d(l _Td ^-), p _d(l _Td ⁺).To any L ∈ (0-∞), have

$p\{l_{\mathrm{tu}}^{-}\≤L\≤l_{\mathrm{tu}}^{+}\}\≥\min \{p_u\left(l_{\mathrm{tu}}^{-}\right),p_u\left(l_{\mathrm{tu}}^{+}\right)\}$

$p\{l_{\mathrm{td}}^{-}\≤L\≤l_{\mathrm{td}}^{+}\}\≥\min \{p_d\left(l_{\mathrm{td}}^{-}\right),p_d\left(l_{\mathrm{td}}^{+}\right)\}$

According to the interval estimation of theory of probability key concept, the confidence level of upper and lower grade of department's prediction scheme is respectively min{p _u(l _Tu ^-), p _u(l _Tu ⁺) and min{p _d(l _Td ^-), p _d(l _Td ⁺).

3, obtain cooperative probability distribution function

Cooperative load forecasting method mathematic(al) representation at the present invention proposes utilizes method of Lagrange multipliers equally, order

F＝h(X)-(λ ₀+1)(∫p(x)dx-1)-λ _u(∫p _u(x)g _u(x)dx-E[g _u(x)])-λ _d(∫p _d(x)g _d(x)dx-E[g _d(x)]

And order $\∂F/\∂p\left(x\right)=0,$ Can get

p(x)＝exp(-λ ₀-λ _ug _u(x)-λ _dg _d(x))＝exp(λ ₁x ²+λ ₂x+λ ₃) (5)

λ in the formula ₁, λ ₂, λ ₃Be unknown number, can be by λ ₀, λ _u, λ _dExpression.

With the g that obtains in 1 _u(x), g _d(x), E[g _u(x)], E[g _d(x)] and the result of formula (5) be updated to formula (2) (3) (4) and can obtain following system of equations:

$\underset{-\∞}{\overset{+\∞}{\∫}}\exp ({\mathrm{\λ}}_1{x}^2+{\mathrm{\λ}}_{2}x+{\mathrm{\λ}}_3)\mathrm{dx}=1$

$\underset{-\∞}{\overset{+\∞}{\∫}}\exp ({\mathrm{\λ}}_1{x}^2+{\mathrm{\λ}}_{2}x+{\mathrm{\λ}}_3){\left(\frac{x-{\hat{l}}_{\mathrm{tu}}}{{\hat{l}}_{\mathrm{tu}}}\right)}^2\mathrm{dx}=E\left[g_u\left(x\right)\right]$

$\underset{-\∞}{\overset{+\∞}{\∫}}\exp ({\mathrm{\λ}}_1{x}^2+{\mathrm{\λ}}_{2}x+{\mathrm{\λ}}_3){\left(\frac{x-{\hat{l}}_{\mathrm{td}}}{{\hat{l}}_{\mathrm{td}}}\right)}^2\mathrm{dx}=E\left[g_d\left(x\right)\right]$

Separate this system of equations, obtain parameter lambda ₁, λ ₂, λ ₃, and then obtain cooperative probability distribution function p (x)=exp (λ ₁x ²+ λ ₂X+ λ ₃).

4, obtain collaborative prediction scheme

Can be in the hope of the expectation of X according to cooperative probability distribution function p (x):

$E\left(X\right)={\∫}_0^{\∞}x*\exp ({\mathrm{\λ}}_1{x}^2+{\mathrm{\λ}}_{2}x+{\mathrm{\λ}}_3)\mathrm{dx}$

The present invention adopts the second-order moment around mean of the superior and the subordinate's prediction scheme as constraint information, and probability distribution function meets normal distribution, so the expectation of X is exactly the variate-value of maximum probability correspondence.Can utilize mathematical expectation to try to achieve maximum probability p _Max=p[E (X)].

Make 1-α=k * p _Max(definition k is that confidence level is adjusted coefficient, k＜1, this paper gets k=0.7), can find such X ^-, X ⁺Make

p{X ^-≤X≤X ⁺}≥1-α

According to the interval estimation of theory of probability key concept, interval [X ^-, X ⁺] be the 1-α (0.7*p of X _Max) fiducial interval, X ^-, X ⁺Be respectively confidence lower limit and the confidence upper limit of X, 1-α is called confidence level (degree of confidence), L=X ⁺-X ^-Length for fiducial interval.Wherein, confidence level 1-α has reflected the credibility of Estimating Confidence Interval parameter X, and the length of fiducial interval L has then reflected the levels of precision of Estimating Confidence Interval parameter X.

The present invention gets the confidence upper limit X of X ⁺, mathematical expectation E (X), confidence lower limit X ^-High, medium and low scheme as final collaborative prediction.

Comparing tradition relies on planning personnel's experience to carry out data collision problem in the manual intervention solution cooperative load forecasting process, the present invention can obtain probability distribution function based on Maximum Entropy Principle Method in the information theory, obtain final high, medium and low scheme based on the theory of probability relevant knowledge, its theoretical foundation is abundant, and range of application is more extensive.The collaborative probability distribution curve that the present invention obtains is partial to the high original prediction scheme of confidence level automatically, and the collaborative prediction scheme confidence level that obtains significantly improves, and efficiently solves the data collision problem in the cooperative load forecasting process.

To sum up, the present invention is applied to Maximum Entropy Principle Method the theoretical research of cooperative load forecasting in the electrical network collaborative planning field,, multipath multidisciplinary by realizing, multivariant informix effectively solve the superior and the subordinate's electric network data collision problem that the distribution system planning field exists.

The present invention proposes a kind of cooperative load forecasting method, in the process that obtains final prediction scheme, the statistical nature of the superior and the subordinate's prediction scheme is taken into account as constraint information, effectively improved the confidence level of prediction scheme based on maximum informational entropy.Below in conjunction with the collaborative prediction of somewhere the superior and the subordinate the present invention is elaborated.

1, calculates original prediction scheme statistical nature

Higher level's prediction scheme, the subordinate's prediction scheme after gathering are as shown in table 1.

Table 1 the superior and the subordinate prediction scheme

T higher level's prediction scheme mean value

${\hat{l}}_{\mathrm{tu}}=\frac{1}{3}(l_{\mathrm{tu}}^{-}+l_{\mathrm{tu}}+l_{\mathrm{tu}}^{+})=\frac{1}{3}(62+60+58)=60;$

The second-order moment around mean m of t higher level's prediction scheme _Tu2:

$m_{\mathrm{tu}2}=\frac{1}{3}[{(l_{\mathrm{tu}}^{-}-{\hat{l}}_{\mathrm{tu}})}^2+{(l_{\mathrm{tu}}-{\hat{l}}_{\mathrm{tu}})}^2+{(l_{\mathrm{tu}}^{+}-{\hat{l}}_{\mathrm{tu}})}^2]=\frac{1}{3}[{(62-60)}^2+{(60-60)}^2+{(58-60)}^2]=2.6667;$

T subordinate prediction scheme mean value

${\hat{l}}_{\mathrm{td}}=\frac{1}{3}(l_{\mathrm{td}}^{-}+l_{\mathrm{td}}+l_{\mathrm{td}}^{+})=\frac{1}{3}(63.5+61+59)=61.1667;$

The second-order moment around mean m of t subordinate prediction scheme _Td2:

$m_{\mathrm{td}2}=\frac{1}{3}[{(l_{\mathrm{td}}^{-}-{\hat{l}}_{\mathrm{td}})}^2+{(l_{\mathrm{td}}-{\hat{l}}_{\mathrm{td}})}^2+{(l_{\mathrm{td}}^{+}-{\hat{l}}_{\mathrm{td}})}^2]=\frac{1}{3}[{(63.5-61.1667)}^2+{(61-61.1667)}^2+{(59-61.1667)}^2]=3.3889.$

By

m _Tu2, m _Td2Above-mentioned four formulas can be determined g _u(x), g _d(x) expression formula and E[g _u(x)], E[g _d(x)] result:

$g_u\left(x\right)={[(x-{\hat{l}}_{\mathrm{tu}})/{\hat{l}}_{\mathrm{tu}}]}^2={[(x-60)/60]}^2$

$g_d\left(x\right)={[(x-{\hat{l}}_{\mathrm{td}})/{\hat{l}}_{\mathrm{td}}]}^2={[(x-61.1667)/61.1667]}^2$

$E\left[g_u\left(x\right)\right]=m_{\mathrm{tu}2}/{\left({\hat{l}}_{\mathrm{tu}}\right)}^2=2.6667/{60}^2=0.000741$

$E\left[g_d\left(x\right)\right]=m_{\mathrm{td}2}/{\left({\hat{l}}_{\mathrm{td}}\right)}^2=3.389/{61.1667}^2=0.000906$

2, resolve original prediction scheme confidence level

Separately with the statistical nature of higher level's prediction scheme as constraint information, the collaborative Simplification on Forecasting Method mathematic(al) representation that obtains is:

max?h(X)＝-∫p _u(x)ln?p _u(x)dx

$\begin{array}{cc}\mathrm{st}& {\∫}_0^{\∞}{p}_u\left(x\right){[(x-60)/60]}^2\mathrm{dx}=0.000741\end{array}$

∫p _u(x)dx＝1

Utilize method of Lagrange multipliers, obtain p _u(x)=exp (λ ₀-λ _ug _u(x)), it is taken back above-mentioned mathematic(al) representation, obtain corresponding Lagrange multiplier λ ₀, λ _u:

λ ₀＝-1.410，λ _u＝674.764

Therefore, p _u(x)=exp (λ ₀-λ _ug _u(x))=exp[-1.410-674.764* (x/60-1) ²].

In like manner can obtain the corresponding Lagrange multiplier λ of subordinate's prediction scheme ₀, λ _uAnd probability distribution function p _d(x):

λ ₀＝-1.529，λ _d＝551.876

p _d(x)＝exp(-λ ₀-λ _dg _d(x))＝exp(-1.529-551.876*(x/61.1667-1) ²)

Make X equal respectively $l_{\mathrm{tu}}^{+}=62,$ $l_{\mathrm{tu}}^{-}=58,$ $l_{\mathrm{td}}^{+}=63.5,$ $l_{\mathrm{td}}^{-}=659,$ Can obtain the probability of the high and low scheme correspondence of upper and lower level $p_u\left(l_{\mathrm{tu}}^{+}\right)=0.115,$ $p_u\left(l_{\mathrm{tu}}^{-}\right)=0.115,$ $p_d\left(l_{\mathrm{td}}^{+}\right)=0.097,$ $p_d\left(l_{\mathrm{td}}^{-}\right)=0.108.$ To any L ∈ (0-∞), have

$p\{58\≤L\≤62\}\≥\min \{p_u\left(l_{\mathrm{tu}}^{-}\right),p_u\left(l_{\mathrm{tu}}^{+}\right)\}=\min \{0.115,0.115\}=0.115$

$p\{59\≤L\≤63.5\}\≥\min \{p_d\left(l_{\mathrm{td}}^{-}\right),p_d\left(l_{\mathrm{td}}^{+}\right)\}=\min \{0.108,0.097\}=0.097$

According to the interval estimation of theory of probability key concept, the confidence level of upper and lower grade of department's prediction scheme is respectively 0.115 and 0.097.

3, obtain cooperative probability distribution function

The mathematic(al) representation of the cooperative load forecasting method correspondence that proposes at the present invention utilizes method of Lagrange multipliers, order

F＝h(X)-(λ ₀+1)(∫p(x)dx-1)-λ _u(∫p _u(x)g _u(x)dx-E[g _u(x)])-λ _d(∫p _d(x)g _d(x)dx-E[g _d(x)]

And order $\∂F/\∂p\left(x\right)=0,$ Can get

p(x)＝exp(-λ ₀-λ _ug _u(x)-λ _dg _d(x))＝exp(λ ₁x ²+λ ₂x+λ ₃)

λ in the formula ₁, λ ₂, λ ₃Be unknown number, can be by λ ₀, λ _u, λ _dExpression.

With the g that obtains in 1 _u(x), g _d(x), E[g _u(x)], E[g _d(x)] and the result of following formula be updated to formula (2) (3) (4) and can obtain following system of equations:

$\underset{-\∞}{\overset{+\∞}{\∫}}\exp ({\mathrm{\λ}}_1{x}^2+{\mathrm{\λ}}_{2}x+{\mathrm{\λ}}_3)\mathrm{dx}=1$

$\underset{-\∞}{\overset{+\∞}{\∫}}\exp ({\mathrm{\λ}}_1{x}^2+{\mathrm{\λ}}_{2}x+{\mathrm{\λ}}_3){[(x-60)/60]}^2\mathrm{dx}=0.000741$

$\underset{-\∞}{\overset{+\∞}{\∫}}\exp ({\mathrm{\λ}}_1{x}^2+{\mathrm{\λ}}_{2}x+{\mathrm{\λ}}_3){[(x-61.1667)/61.1667]}^2\mathrm{dx}=0.000906$

Separate this system of equations, can obtain parameter lambda ₁=-0.193, λ ₂=23.257, λ ₃=-702.275.

And then obtain cooperative probability distribution function p (x)=exp (λ ₁x ²+ λ ₂X+ λ ₃)=exp (0.193x ²+ 23.257x-702.275).

4, obtain collaborative prediction scheme

Can be in the hope of the expectation of X according to cooperative probability distribution function p (x):

$E\left(X\right)={\∫}_0^{\∞}x*\exp ({\mathrm{\λ}}_1{x}^2+{\mathrm{\λ}}_{2}x+{\mathrm{\λ}}_3)\mathrm{dx}={\∫}_0^{\∞}x*\exp (-{0.193x}^2+23.257x-702.275)\mathrm{dx}=60.274$

The present invention adopts the second-order moment around mean of the superior and the subordinate's prediction scheme as constraint information, and probability distribution function meets normal distribution, so the expectation of X is exactly the variate-value of maximum probability correspondence.Can utilize mathematical expectation to try to achieve maximum probability p _Max:

p _max＝p[E(X)]＝p[60.274]＝exp(-0.193*60.274 ²+23.257*60.274-702.275)＝0.248

Make 1-α=k * p _Max=0.7 * 0.248=0.173 (definition k is that confidence level is adjusted coefficient, k＜1, this paper gets k=0.7), make p (x)=exp (λ ₁x ²+ λ ₂X+ λ ₃)=0.173 is found the solution this quadratic equation with one unknown and can be obtained x ₁=61.634, x ₂=58.914.

To sum up, can find such X ^-=58.914, X ⁺=61.634 make

p{58.914≤X≤61.634}≥0.173

According to the interval estimation of theory of probability key concept, interval [58.914,61.634] are that the confidence level of X is 0.173 fiducial interval, X ^-=58.914, X ⁺=61.634 are respectively confidence lower limit and the confidence upper limit of X.

The present invention gets the confidence upper limit X of X ⁺=61.634, mathematical expectation E (X)=60.274, confidence lower limit X ^-=58.914 high, medium and low schemes as final collaborative prediction.

The collaborative prediction scheme of table 2 and the superior and the subordinate's prediction scheme comprehensively compare

Comprehensive more upper and lower level prediction scheme and collaborative prediction scheme, as shown in table 2.Can see that the corresponding maximum probability of the cooperative probability distribution function that application invention obtains is greater than the maximum probability of higher level, subordinate's probability distribution function, the lifting of maximum probability indicates the lifting of prediction scheme reliability, accuracy; The confidence level of the collaborative prediction scheme that obtains is apparently higher than the confidence level of higher level, subordinate's prediction scheme, and confidence level is the direct embodiment of data reliability, and the lifting of confidence level indicates that collaborative prediction scheme is more objective, reliable.

In addition, as can be seen from Figure 2, the probability distribution curve that application the present invention obtains is partial to the high probability distribution curve of confidence level automatically.Curve is followed successively by from left to right among the figure: f _u, f _U-d, f _dF wherein _U-dBe the probability distribution curve (collaborative probability distribution curve) that simultaneously statistical nature of higher level, subordinate's prediction scheme is obtained as constraint information; f _uBe the independent probability distribution curve (higher level's probability distribution curve) that the statistical nature of higher level's prediction scheme is obtained as constraint information; f _dBe the independent probability distribution curve (subordinate's probability distribution curve) that the statistical nature of subordinate's prediction scheme is obtained as constraint information.As can be seen, higher level's prediction scheme confidence level is higher than the confidence level of subordinate's prediction scheme; f _uCorresponding maximum probability is greater than f _dCorresponding maximum probability, probability distribution curve f _U-dAutomatically be partial to higher level's probability distribution curve f _uOne side.

The innovation of multidisciplinary division of labor asynchronous cooperation prediction work mode, the working method that makes the simple planning personnel of dependence experience carry out manual intervention can't satisfy the requirement that planning becomes more meticulous.Under new working method, the present invention has stronger adaptability.The present invention is applied to the theoretical research of cooperative load forecasting in the electrical network collaborative planning field with Maximum Entropy Principle Method, and, multipath multidisciplinary by realizing, multivariant informix can solve the data collision problem in the cooperative load forecasting process; The present invention can resolve the confidence level of upper and lower level prediction scheme, realizes the reduction of data reliability, integrality, accuracy; The present invention can improve the confidence level of prediction scheme, finally obtains unique reliable prediction scheme, satisfies the demand that collaborative prediction, prediction become more meticulous.

Claims (5)

1. the cooperative load forecasting method based on maximum informational entropy comprises the following steps:

Department of subordinate prediction scheme after the first step is at first gathered higher level department prediction scheme and gathered, definition $g_u\left(x\right)={[(x-{\hat{l}}_{\mathrm{tu}})/{\hat{l}}_{\mathrm{tu}}]}^2,$ $E\left[g_u\left(x\right)\right]=m_{\mathrm{tu}2}/{\left({\hat{l}}_{\mathrm{tu}}\right)}^2,$ $g_d\left(x\right)={[(x-{\hat{l}}_{\mathrm{td}})/{\hat{l}}_{\mathrm{td}}]}^2,$ $E\left[g_d\left(x\right)\right]=m_{\mathrm{td}2}/{\left({\hat{l}}_{\mathrm{td}}\right)}^2,$ In the formula,

Be the mean value of t higher level's prediction scheme, m _Tu2It is the second-order moment around mean of t higher level's prediction scheme;

Be the mean value of t subordinate prediction scheme, m _Td2It is the second-order moment around mean of t subordinate prediction scheme; Next calculates the statistical nature of the superior and the subordinate's prediction scheme: mean value

Second-order moment around mean (m _Tu2, m _Td2), and foundation m _Tu2, m _Td2Determine g _u(x), g _d(x) expression formula and E[g _u(x), E[g _d(x)];

Second step separately with the statistical nature of upper and lower level prediction scheme as constraint information, obtain the probability distribution function of upper and lower level prediction scheme correspondence based on following load prediction formula, and then obtain original prediction scheme confidence level:

max?h(X)＝-∫p _i(x)lnp _i(x)dx (1)

st∫p _i(x)g _i(x)dx＝E[g _i(x)]i＝u，d (2)

∫p _i(x)dx＝1i＝u，d (3)

Formula (1) is an objective function, and wherein h (X) is the entropy of stochastic variable X, and p (x) is the probability density of x for the X value; In the formula (2),, i=u should satisfy the constraint of the statistical nature correspondence of higher level department prediction when representing probability distribution function to be asked; Represent that when i=d probability distribution function to be asked should satisfy the constraint of the statistical nature correspondence of department of the subordinate prediction scheme after gathering; Formula (3) is the constraint of the probability distribution function self of the superior and the subordinate's prediction scheme correspondence;

The 3rd step simultaneously with the statistical nature of upper and lower level prediction scheme as constraint information, the cooperative load forecasting formula based on following maximum informational entropy obtains cooperative probability distribution function:

max?h(X)＝-∫p(x)lnp(x)dx (4)

st∫p(x)g _u(x)dx＝E[g _u(x)] (5)

∫p(x)g _d(x)dx＝E[g _d(x)] (6)

∫p(x)dx＝1 (7)

Formula (4) is an objective function, and wherein h (X) is the entropy of stochastic variable X, and p (x) is the probability density of x for the X value; Formula (5) expression probability distribution function to be asked should satisfy the constraint of the statistical nature correspondence of higher level department prediction scheme; Formula (6) expression probability distribution function to be asked should satisfy the constraint of the statistical nature correspondence of the subordinate's prediction scheme after gathering; Formula (7) is the constraint of probability distribution function self;

The 4th step was calculated its mathematical expectation and maximum probability based on the cooperative probability distribution function that obtains in the 3rd step, based on the interval correlation theory of estimating of theory of probability, finally determined the high, medium and low scheme of cooperative load forecasting.

2. the cooperative load forecasting method based on maximum informational entropy according to claim 1 is characterized in that, makes F=h (X)-(λ ₀+ 1) (∫ p (x) dx-1)-λ _i(∫ p _i(x) g _i(x) dx-E[g _i(x)]), i=u, d; And order $\∂F/\∂p\left(x\right)=0$ , can get upper and lower level probability distribution function p _u(x), p _d(x):

p _u(x)＝exp(-λ ₀-λ _ug _u(x))

p _d(x)＝exp(-λ ₀-λ _dg _d(x))

p _u(x) and p _d(x) be respectively the most possible probability distribution function that higher level and subordinate's prediction scheme satisfy.

3. cooperative load forecasting method according to claim 1 is characterized in that, makes F=h (X)-(λ ₀+ 1) (∫ p (x) dx-1)-λ _u(∫ p _u(x) g _u(x) dx-E[g _u(x)])-λ _d(∫ p _d(x) g _d(x) dx-E[g _d(x)], and the order $\∂F/\∂p\left(x\right)=0,$ Get p (x)=exp (λ ₀-λ _ug _u(x)-λ _dg _d(x))=exp (λ ₁x ²+ λ ₂X+ λ ₃), the g that calculates in again the first step and second being gone on foot _u(x), g _d(x), E[g _u(x)], E[g _dAnd the following system of equations of substitution as a result of following formula (x)]:

$\underset{-\∞}{\overset{+\∞}{\∫}}\exp ({\mathrm{\λ}}_1{x}^2+{\mathrm{\λ}}_{2}x+{\mathrm{\λ}}_3)\mathrm{dx}=1$

$\underset{-\∞}{\overset{+\∞}{\∫}}\exp ({\mathrm{\λ}}_1{x}^2+{\mathrm{\λ}}_{2}x+{\mathrm{\λ}}_3){\left(\frac{x-{\hat{l}}_{\mathrm{tu}}}{{\hat{l}}_{\mathrm{tu}}}\right)}^2\mathrm{dx}=E\left[g_u\left(x\right)\right]$

$\underset{-\∞}{\overset{+\∞}{\∫}}\exp ({\mathrm{\λ}}_1{x}^2+{\mathrm{\λ}}_{2}x+{\mathrm{\λ}}_3){\left(\frac{x-{\hat{l}}_{\mathrm{td}}}{{\hat{l}}_{\mathrm{td}}}\right)}^2\mathrm{dx}=E\left[g_d\left(x\right)\right]$

Obtain parameter lambda ₁, λ ₂, λ ₃, and then obtain cooperative probability distribution function p (x)=exp (λ ₁x ²+ λ ₂X+ λ ₃).

4. cooperative load forecasting method according to claim 1 is characterized in that, makes confidence level 1-α=k * p _Max, k is that confidence level is adjusted coefficient, k＜1 meets the characteristic of normal distribution according to cooperative probability distribution function, finds the confidence lower limit X of X ^-, confidence upper limit X ⁺Make p{X ^-≤ X≤X ⁺} 〉=1-α gets the confidence upper limit X of X ⁺, mathematical expectation E (X), confidence lower limit X ^-High, medium and low scheme as final collaborative prediction.

5. each described method of claim 1-4 is in distribution system planning or the operating application of distribution system.

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