CN104008433A - Method for predicting medium-and-long-term power loads on basis of Bayes dynamic model - Google Patents

Abstract

The invention discloses a method for predicting medium-and-long-term power loads on basis of a Bayes dynamic model. According to the internal change rule of medium-and-long-term power load data, the Bayes dynamic model with polynomial regression and index structures is built; according to the Bayes theory, the power load value of the next year is obtained in real time in a statistical interference mode according to prior information and test samples, and dynamic recursion prediction of the medium-and-long-term power loads in small sample capacity is achieved. The number of sample data required by the method is small, and the change rule of current power loads can be tracked in real time by means of model monitoring and subjective interference, so that a power load prediction result is more reliable, and an effective way is provided for improvement on medium-and-long-term power load prediction precision.

Description

Long-medium term power load forecasting method based on Bayesian Dynamic

Technical field

The present invention relates to Methods of electric load forecasting field, be specially a kind of long-medium term power load forecasting method based on Bayesian Dynamic.

Background technology

Electric Power Network Planning is directly connected to socio-economic development and energy resource supply, long-medium term power load forecasting is the basis of formulating Electric Power Network Planning, and its accuracy directly has influence on rationality, energy resources supply and balance, the power industry economy of Electricity Investment, network topology.

Mid-long term load forecasting is to take historical data as basis, sets up effective forecast model, seeks its variation tendency and the rule of development, predict future payload.Electric load development and change rule complexity is various, affected by the uncertain factor of Various Complex, load prediction is very difficult very accurately, its theoretical research is just taken much count of always, load forecasting method is varied, common are method of elasticity modulus, regression analysis, trend extrapolation, Grey Theory Forecast method etc.Affected by the multiple uncertain factors such as socio-economic development trend, climate change, the difficulty of prediction and the complicacy of model have been increased, for concrete network load data, to select appropriate forecast model, adopt if desired several mathematical model modelings simultaneously, comparative analysis, so that selection appropriate model, obtain and approach historical law, model that precision of prediction is high most.

Fast-developing Bayesian forecasting theory is to infer that according to prior imformation and sample information obtaining posterior information carries out modeling, has following distinguishing feature in recent years:

Bayesian model is a kind of dynamic model, and it regards conditional probability distribution as prediction distribution, and dopester can obtain prediction distribution according to prior imformation, and uses Bayes' theorem to try to achieve posteriority distribution, and constantly prior imformation is revised;

Bayes method utilizes prior imformation and sample information to forecast.Therefore, while obtaining fresh information in forecasting process (standard volume of inserting as interval etc.), as long as regard the forecast information in this moment as obtain priori, and and sample information combination now, just can revise original forecasting model, not only convenient but also quick, and then the time-varying characteristics of tracking sequence in time;

Bayesian Dynamic has various structures, as seaconal model, regression model, multinomial model, noise model etc., and these models can also be combined and be represented a complex sequence by superposition principle, so Bayes Modeling and Prediction does not have stationarity restriction to data, be applicable to very much the error random series Modeling and Prediction of kinetic measurement non-stationary;

Because Bayesian schools regards probability as the trusting degree of people to something or other as, rather than the stability of frequency, so the approval of some subjectivities also can be described as form of probability, and these are all the subjective prior imformations of relevant research object.During Bayesian forecasting, can not only utilize objective data message, can also utilize the subjective prior imformation of the platform reason that dopester itself can provide to intervene and correction model.In this case, dopester just can process the abnormal conditions that some can expect, for unforeseeable emergency case, can process by the sequential method for supervising of Bei Shisi.This will make Bayesian forecasting result more reliable, improve forecast precision.

Fast-developing Bayesian forecasting theory is applied in long-medium term power load forecasting in recent years.In order to overcome traditional combination forecasting method, do not take explicitly into account the uncertainty of model, document [open big. the prediction of Bayesian model average combined and applied research [J] based on EM. modern commerce and trade industry, 2010], in, application Bayesian model average combined Forecasting Methodology is accurately estimated the weight of Single model.Use support vector machine to carry out Combined Method of Mid-long Term Load Forecast, but support vector machine is to nuclear parameter, the selected possibility that exists of regularization parameter, document [Niu Dongxiao, Lv Haitao, Zhang Yun's cloud. the Middle-long Electric Power Load Forecast of least square method supporting vector machine [J] under Bayesian frame. North China Electric Power University's journal, 2008,35 (6): 62-66] the long-medium term power load forecasting model of application based on LS-SVM under Bayesian frame, the parameter that has provided support vector machine algorithm for estimating is selected and method of adjustment.

Whether Spatial Load Forecasting model validity depends on the division of grade under the piece of space region to be predicted correct to a great extent, [inscription on pottery is refined for document, open particle, Pan Hong etc. the Spatial Load Forecasting based on bilateral Bayes's classification [J]. Proceedings of the CSEE, 2007] adopt the Double-level Bayesian Classification model based on sample data to carry out Spatial Load Forecasting, improved the correctness to sample classification.Document [Hu Yunsheng, Zheng Jiming. the load forecast based on principal component analysis and genetic neural network [J]. control theory and application, 2008,27 (8): 1-3] for long-medium term power load forecasting, be subject to the factors such as economy, population, weather, policy, it is all taken into account to the input as forecast model, adopt the method for principal component analysis that the input of network is simplified, select Bayes's normalization method to carry out training network, the training speed that has improved network has also improved the precision of prediction.

When utilizing additive method prediction, introduce bayes method and carry out auxiliary parameter estimation, cluster or do the prediction of some parts, the bayes method that utilizes yet there are no in complete meaning carries out the correlative study of long-medium term power load forecasting.

Summary of the invention

The object of this invention is to provide a kind of long-medium term power load forecasting method based on Bayesian Dynamic, to realize the real-time estimate of the Mid-long Term Load under small sample capacity.

In order to achieve the above object, the technical solution adopted in the present invention is:

Long-medium term power load forecasting method based on Bayesian Dynamic, is characterized in that: comprise the following steps:

(1) set up Bayes's exponential polynomials regression model of Power system load data:

Most long Electric Power Load data have the feature that approaches exponential relationship, can consider that Bayes's exponential polynomials regression model of setting up Power system load data realizes Mid-long term load forecasting.Electric load model based on Bayesian Dynamic Prediction is generally comprised of observation equation and state equation, and the polynomial expression that is generally no more than second order just can provide the matching of good localized variation trend, and binomial Exponential Regression Model can be expressed as:

Observation equation: (logE) _i=F _i ^tθ _i+ v _iv _i～N (0, V _i),

State equation: θ _i=G θ _i-1+ ω _iω _i～N (0, W _i),

In formula, E is electric load value, (logE) _ielectric load time series, θ _i=(a _i, b _i, c _i) ^tfor i state parameter vector constantly, F _i=(1, t, t ²) ^ti dynamic regression matrix constantly, $G=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$ For state-transition matrix, ν _iand ω _ibe respectively mutually independently observational error and state error variable, and ω _n, ω _land ν _n, ν _l(n ≠ l) is separate;

(2) determine the prior imformation of associated arguments:

While adopting Bayesian Dynamic prediction recursion, need the prior imformation of known associated arguments, and generally, the prior imformation of parameter is difficult to obtain, and therefore adopts asemantic reference analysis method to determine the prior imformation of associated arguments;

In analyzing without information reference, suppose observational error ν _i(0, V), V is unknown parameter to Normal Distribution N, state equation error ω _imeeting average is 0, and variance is W _it distribute, in above-mentioned electric load model, have 3 state parameter a, b, c and 1 observational variance V, totally 4 unknown parameters, therefore can determine θ according to 4 of initial acquisition Power system load datas _iwith the initial information of V, due to the observation data adopting when determining initial information very little, can not estimate or detect any variation of parameter, so can establish W _i=0 (i=1,2,3,4);

If D _irepresent the i moment and the set of all effective informations constantly in the past thereof, D _i(i=0) be the set of i=0 initial information, by initial apriority, θ ₁with V without D under information condition ₀condition joint probability distribution be proportional to the V reciprocal of variance ^-1:

P(θ ₁,V|D ₀)∝V ^-1 V>0

According to Bayesian formula and Power system load data point y ₁, y ₂, y ₃, y ₄can recursion obtain posteriority joint probability distribution P (θ ₄, V|D ₄), and then obtain θ ₄| D ₄and V ^-1| D ₄condition marginal distribution P (θ ₄| D ₄) and P (V ^-1| D ₄), during recursion, need first define following each amount:

H _i＝(G ^-1) ^TK _i-1G ^-1

h _i＝(G ^-1) ^Tk _i-1

K _i＝H _i+F _iF _i ^T

k _i＝h _i+F _iy _i

Wherein, r _i=r _i-1+ 1

λ _i＝δ _i-1

δ _i＝λ _i+y _i ²

Here, H _i, h _i, λ _i, r _i, δ _i, K _i, k _ibe all the intermediate variable for initial information recursion, recursion initial value is H ₁=0, h ₁=0, λ ₁=0, r ₀=0.

Through deriving, obtain θ _i, associating priori, the posteriority of V (i=1,2,3,4) are respectively:

$P({\mathrm{\θ}}_i,V|D_{i-1})\∝V^{-(1+r_{i-1}/2)}\exp \{-0.5V^{-1}({\mathrm{\θ}}_i^T{H}_i{\mathrm{\θ}}_i-{2\mathrm{\θ}}_i^T{h}_i{+\mathrm{\λ}}_i)\}$

$P({\mathrm{\θ}}_i,V|D_i)\∝V^{-(1+r_i/2)}\exp \{-0.5V^{-1}({\mathrm{\θ}}_i^T{K}_i{\mathrm{\θ}}_i-{2\mathrm{\θ}}_i^T{k}_i+{\mathrm{\δ}}_i)\}$

According to above-mentioned derivation, obtain P (θ ₄, V|D ₄) after, just can obtain θ ₄| D ₄and V ^-1| D ₄posteriority distribute:

(θ _i|D _i)～T[M _i,C _i]

(V ^-1|D _i)～Γ[n _i/2,d _i/2]

State variable θ _iposteriority conditional probability to obey average be M _i, variance is C _it distribute, V ^-1it is n that posteriority conditional probability is obeyed average _i/ 2, variance is d _i/ 2 Γ distributes.Wherein, $M_i={K_i}^{-1}{k}_i,C_i=S_i{K}_i^{-1},S_i=d_i/n_i,n_i=r_i-3,d_i={\mathrm{\δ}}_i-k_i^T{M}_i,$ Try to achieve θ ₄| D ₄and V ^-1| D ₄condition marginal distribution after, just using this as initial information, model is revised;

(3) recursion correction and the prediction of Power system load data:

If initial information is:

ω _i～T[0,W _i]

(θ _i-1|D _i-1)～T[M _i-1,C _i-1]

(θ _i|D _i-1)～T[A _i,R _i],A _i＝GM _i-1,R _i＝GC _i-1G ^T+W _i

(V ^-1|D _i-1)～Γ(n _i-1/2,d _i-1/2),S _i-1＝d _i-1/n _i-1

In formula, A _i, R _istate variable θ _ithe average of prior distribution and variance; S _iit is the point estimation of V.

Observed reading y _ia step forward prediction distribution obey average f _i, variance Q _it distribute:

(y _i|D _i-1)～T[f _i,Q _i],f _i＝F _i ^TA _i,Q _i＝F _i ^TR _iF _i+S _i-1

Recursion correction relation:

(θ _i|D _i)～T[M _i,C _i],(V ^-1|D _i)～Γ[n _i/2,d _i/2],

M _i＝A _i+B _ie _i

C _i＝(S _i/S _i-1)[R _i-B _iF _i ^TQ _i]

$n_i=n_{i-1}+1,d_i=d_{i-1}+S_i{e}_i^2/Q_i,S_i=d_i/n_i$

Wherein, e _i=y _i-f _ifor predicated error, B _i=R _if _i/ Q _ifor correction factor matrix.

K walks prediction distribution forward: to k>0,

(θ _i+k|D _i)～T[A _i(k),R _i(k)]

(y _i+k|D _i)～T[f _i(k),Q _i(k)]

A _i(k)＝GA _i(k-1)

R _i(k)＝GR _i(k-1)G ^T+W _i

Wherein, $f_i\left(k\right)=F_{i++k}^T{A}_i(k-1)$

$Q_i\left(k\right)=F_{i+k}^T{R}_i\left(k\right)F_{i+k}+S_i$

Initial value is A _i(0)=M _i, R _i(0)=C _i,

The predicted value E of electric load _ifor y _iprediction average f _iexponential function:

To Comprehensive analysis load, prediction has practical value in the present invention, for load prediction in Electric Power Network Planning provides effective analytical calculation instrument, further promotes the accuracy of predicting under small sample capacity, has good development prospect.

The present invention is directed to one group of electric load measured data, set up dynamic linear built-up pattern, according to bayesian theory, by prior imformation with measure sample, the electric load value of statistical inference next year in real time, realizes the dynamic stepwise predict of long Electric Power Load.The method, according to the inherent Changing Pattern of long Electric Power Load data, has been set up the Bayesian Dynamic with polynomial regression and construction of indexes, has solved the shortcoming of the single model prediction of long Electric Power Load.The sample data that the method requires is less, and can be by the form of Model Monitoring and Subjective Intervention, follow the tracks of in real time the Changing Pattern of current electric load, this will make load forecast result more reliable, for improving long-medium term power load forecasting precision, provide an effective way.

Accompanying drawing explanation

Fig. 1 is Bayesian forecasting recursive algorithm figure in step of the present invention (4).

Embodiment

(1) set up Bayes's exponential polynomials regression model of Power system load data:

Electric load model based on Bayesian Dynamic Prediction is generally comprised of observation equation and state equation, and its binomial Exponential Regression Model can be expressed as:

Observation equation: (logE) _i=F _i ^tθ _i+ v _iv _i～N (0, V _i),

State equation: θ _i=G θ _i-1+ ω _iω _i～N (0, W _i),

In formula, E is electric load value, (logE) _ielectric load time series, θ _i=(a _i, b _i, c _i) ^tfor i state parameter vector constantly, F _i=(1, t, t ²) ^ti dynamic regression matrix constantly, $G=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$ For state-transition matrix, ν _iand ω _ibe respectively mutually independently observational error and state error variable, and ω _n, ω _land ν _n, ν _l(n ≠ l) is separate;

(2) determine the prior imformation of associated arguments:

While adopting Bayesian Dynamic prediction recursion, need the prior imformation of known associated arguments, and generally, the prior imformation of parameter is difficult to obtain, and therefore adopts asemantic reference analysis method to determine the prior imformation of associated arguments;

In analyzing without information reference, suppose observational error ν _i(0, V), V is unknown parameter to Normal Distribution N, state equation error ω _imeeting average is 0, and variance is W _it distribute, in above-mentioned electric load model, have 3 state parameter a, b, c and 1 observational variance V, totally 4 unknown parameters, therefore can determine θ according to 4 of initial acquisition Power system load datas _iwith the initial information of V, due to the observation data adopting when determining initial information very little, can not estimate or detect any variation of parameter, so can establish W _i=0 (i=1,2,3,4);

If D _irepresent the i moment and the set of all effective informations constantly in the past thereof, D _i(i=0) be the set of i=0 initial information, by initial apriority, θ ₁with V without D under information condition ₀condition joint probability distribution be proportional to the V reciprocal of variance ^-1:

P(θ ₁,V|D ₀)∝V ^-1 V>0

According to Bayesian formula and Power system load data point y ₁, y ₂, y ₃, y ₄can recursion obtain posteriority joint probability distribution P (θ ₄, V|D ₄), and then obtain θ ₄| D ₄and V ^-1| D ₄condition marginal distribution P (θ ₄| D ₄) and P (V ^-1| D ₄), during recursion, need first define following each amount:

H _i＝(G ^-1) ^TK _i-1G ^-1

h _i＝(G ^-1) ^Tk _i-1

K _i＝H _i+F _iF _i ^T

k _i＝h _i+F _iy _i

Wherein, r _i=r _i-1+ 1

λ _i＝δ _i-1

δ _i＝λ _i+y _i ²

Here, H _i, h _i, λ _i, r _i, δ _i, K _i, k _ibe all the intermediate variable for initial information recursion, recursion initial value is H ₁=0, h ₁=0, λ ₁=0, r ₀=0.

Through deriving, obtain θ _i, associating priori, the posteriority of V (i=1,2,3,4) are respectively:

$P({\mathrm{\θ}}_i,V|D_{i-1})\∝V^{-(1+r_{i-1}/2)}\exp \{-0.5V^{-1}({\mathrm{\θ}}_i^T{H}_i{\mathrm{\θ}}_i-{2\mathrm{\θ}}_i^T{h}_i{+\mathrm{\λ}}_i)\}$

$P({\mathrm{\θ}}_i,V|D_i)\∝V^{-(1+r_i/2)}\exp \{-0.5V^{-1}({\mathrm{\θ}}_i^T{K}_i{\mathrm{\θ}}_i-{2\mathrm{\θ}}_i^T{k}_i+{\mathrm{\δ}}_i)\}$

According to above-mentioned derivation, obtain P (θ ₄, V|D ₄) after, just can obtain θ ₄| D ₄and V ^-1| D ₄posteriority distribute:

(θ _i|D _i)～T[M _i,C _i]

(V ^-1| D _i)～Γ [n _i/ 2, d _i/ 2] state variable θ _iposteriority conditional probability to obey average be M _i, variance is C _it distribute, V ^-1it is n that posteriority conditional probability is obeyed average _i/ 2, variance is d _i/ 2 Γ distributes.Wherein, $M_i={K_i}^{-1}{k}_i,C_i=S_i{K}_i^{-1},S_i=d_i/n_i,n_i=r_i-3,d_i={\mathrm{\δ}}_i-k_i^T{M}_i,$ Try to achieve θ ₄| D ₄and V ^-1| D ₄condition marginal distribution after, just using this as initial information, model is revised;

(3) recursion correction and the prediction of Power system load data:

The recursion correction of electric load and the basic ideas of prediction are as shown in Figure 1:

If initial information is:

ω _i～T[0,W _i]

(θ _i-1|D _i-1)～T[M _i-1,C _i-1]

(θ _i|D _i-1)～T[A _i,R _i],A _i＝GM _i-1,R _i＝GC _i-1G ^T+W _i

(V ^-1|D _i-1)～Γ(n _i-1/2,d _i-1/2),S _i-1＝d _i-1/n _i-1

In formula, A _i, R _istate variable θ _ithe average of prior distribution and variance; S _iit is the point estimation of V.

Observed reading y _ia step forward prediction distribution obey average f _i, variance Q _it distribute:

(y _i|D _i-1)～T[f _i,Q _i],f _i＝F _i ^TA _i,Q _i＝F _i ^TR _iF _i+S _i-1

Recursion correction relation:

(θ _i|D _i)～T[M _i,C _i],(V ^-1|Di)～Γ[n _i/2,d _i/2],

M _i＝A _i+B _ie _i

C _i＝(S _i/S _i-1)[R _i-B _iF _i ^TQ _i]

$n_i=n_{i-1}+1,d_i=d_{i-1}+S_i{e}_i^2/Q_i,S_i=d_i/n_i$

Wherein, e _i=y _i-f _ifor predicated error, B _i=R _if _i/ Q _ifor correction factor matrix.

K walks prediction distribution forward: to k>0,

(θ _i+k|D _i)～T[A _i(k),R _i(k)]

(y _i+k|D _i)～T[f _i(k),Q _i(k)]

A _i(k)＝GA _i(k-1)

R _i(k)＝GR _i(k-1)G ^T+W _i

Wherein, $f_i\left(k\right)=F_{i++k}^T{A}_i(k-1)$

$Q_i\left(k\right)=F_{i+k}^T{R}_i\left(k\right)F_{i+k}+S_i$

Initial value is A _i(0)=M _i, R _i(0)=C _i,

The predicted value E of electric load _ifor y _iprediction average f _iexponential function:

Specific embodiment:

In order to verify the validity of the Methods of electric load forecasting based on Bayesian Dynamic, the total electricity consumption of a certain area several years has been carried out to modeling and forecasting, and contrasted with at present conventional improved grey model model prediction result.Original Power system load data, Bayesian Dynamic and improved grey model model prediction result (unit: comparison hundred million kilowatt hours) is in Table 1.

Table 1 Bayesian Dynamic and improved grey model model prediction result (unit: hundred million kilowatt hours)

Time	Original Power system load data	Improved grey model model prediction result	Bayesian Dynamic predicts the outcome
				1	0.4754	0.4754	0.47544
2	0.6058	0.41692	0.58608
				3	0.6103	0.46933	0.63791
4	0.6063	0.53012	0.59241
				5	0.6098	0.60075	0.51313
6	0.4898	0.32414	0.53936
				7	0.5183	0.40558	0.39319
8	0.6152	0.50231	0.4271
				9	0.6963	0.61706	0.57183
10	0.9880	1.1722	0.72989
				11	1.1878	1.3498	1.1495
12	1.5672	1.5575	1.5646
				13	2.2420	1.8005	2.1743

By relatively finding, in this example, the Methods of electric load forecasting precision of prediction based on Bayesian Dynamic is high, effective, feasible effective.

Claims (1)

1. the long-medium term power load forecasting method based on Bayesian Dynamic, is characterized in that: comprise the following steps:

(1) set up Bayes's exponential polynomials regression model of Power system load data:

Most long Electric Power Load data have the feature that approaches exponential relationship, can consider that Bayes's exponential polynomials regression model of setting up Power system load data realizes Mid-long term load forecasting.Electric load model based on Bayesian Dynamic Prediction is generally comprised of observation equation and state equation, and the polynomial expression that is generally no more than second order just can provide the matching of good localized variation trend, and binomial Exponential Regression Model can be expressed as:

Observation equation: (logE) _i=F _i ^tθ _i+ v _iv _i～N (0, V _i),

State equation: θ _i=G θ _i-1+ ω _iω _i～N (0, W _i),

In formula, E is electric load value, (logE) _ielectric load time series, θ _i=(a _i, b _i, c _i) ^tfor i state parameter vector constantly, F _i=(1, t, t ²) ^ti dynamic regression matrix constantly, $G=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$ For state-transition matrix, ν _iand ω _ibe respectively mutually independently observational error and state error variable, and ω _n, ω _land ν _n, ν _l(n ≠ l) is separate;

(2) determine the prior imformation of associated arguments:

While adopting Bayesian Dynamic prediction recursion, need the prior imformation of known associated arguments, and generally, the prior imformation of parameter is difficult to obtain, and therefore adopts asemantic reference analysis method to determine the prior imformation of associated arguments;

In analyzing without information reference, suppose observational error ν _i(0, V), V is unknown parameter to Normal Distribution N, state equation error ω _imeeting average is 0, and variance is W _it distribute, in above-mentioned electric load model, have 3 state parameter a, b, c and 1 observational variance V, totally 4 unknown parameters, therefore can determine θ according to 4 of initial acquisition Power system load datas _iwith the initial information of V, due to the observation data adopting when determining initial information very little, can not estimate or detect any variation of parameter, so can establish W _i=0 (i=1,2,3,4);

If D _irepresent the i moment and the set of all effective informations constantly in the past thereof, D _i(i=0) be the set of i=0 initial information, by initial apriority, θ ₁with V without D under information condition ₀condition joint probability distribution be proportional to the V reciprocal of variance ^-1:

P(θ ₁,V|D ₀)∝V ^-1 V>0

According to Bayesian formula and Power system load data point y ₁, y ₂, y ₃, y ₄can recursion obtain posteriority joint probability distribution P (θ ₄, V|D ₄), and then obtain θ ₄| D ₄and V ^-1| D ₄condition marginal distribution P (θ ₄| D ₄) and P (V ^-1| D ₄), obtain P (θ ₄, V|D ₄) after, just can obtain θ ₄| D ₄and V ^-1| D ₄posteriority distribute:

(θ _i|D _i)～T[M _i,C _i]

(V ^-1|D _i)～Γ[n _i/2,d _i/2]

State variable θ _iposteriority conditional probability to obey average be M _i, variance is C _it distribute, V ^-1it is n that posteriority conditional probability is obeyed average _i/ 2, variance is d _i/ 2 Γ distributes.Try to achieve θ ₄| D ₄and V ^-1| D ₄condition marginal distribution after, just using this as initial information, model is revised;

(3) recursion correction and the prediction of Power system load data:

If initial information is:

ω _i～T[0,W _i]

(θ _i-1|D _i-1)～T[M _i-1,C _i-1]

(θ _i|D _i-1)～T[A _i,R _i],A _i＝GM _i-1,R _i＝GC _i-1G ^T+W _i

(V ^-1|D _i-1)～Γ(n _i-1/2,d _i-1/2),S _i-1＝d _i-1/n _i-1

In formula, A _i, R _istate variable θ _ithe average of prior distribution and variance; S _iit is the point estimation of V.

Observed reading y _ia step forward prediction distribution obey average f _i, variance Q _it distribute:

(y _i|D _i-1)～T[f _i,Q _i],f _i＝F _i ^TA _i,Q _i＝F _i ^TR _iF _i+S _i-1

Recursion correction relation:

(θ _i|D _i)～T[M _i,C _i],(V ^-1|D _i)～Γ[n _i/2,d _i/2],

M _i＝A _i+B _ie _i

C _i＝(S _i/S _i-1)[R _i-B _iF _i ^TQ _i]

$n_i=n_{i-1}+1,d_i=d_{i-1}+S_i{e}_i^2/Q_i,S_i=d_i/n_i$

Wherein, e _i=y _i-f _ifor predicated error, B _i=R _if _i/ Q _ifor correction factor matrix.

K walks prediction distribution forward: to k>0,

(θ _i+k|D _i)～T[A _i(k),R _i(k)]

(y _i+k|D _i)～T[f _i(k),Q _i(k)]

A _i(k)＝GA _i(k-1)

R _i(k)＝GR _i(k-1)G ^T+W _i

Wherein, $f_i\left(k\right)=F_{i++k}^T{A}_i(k-1)$

$Q_i\left(k\right)=F_{i+k}^T{R}_i\left(k\right)F_{i+k}+S_i$

Initial value is A _i(0)=M _i, R _i(0)=C _i,

The predicted value E of electric load _ifor y _iprediction average f _iexponential function: