Weather inputs are an important factor in load forecasting. The arrival of a weather front or introduction of weather

forecast errors lead to a change in the significance of weather as input. A data fusion of individual models is introduced, which uses a priori information regarding the significance of weather inputs in these situations, to produce a superior forecast to any individual model

Condon, Marissa

Fay, Damien

Kelly, Michael

Ringwood, John

English

Maynooth University ePrints and eTheses Archive

Data Fusion Model for Irish Electricity Load Forecasting

Maynooth EPrints and ETheses Archive

NUI Maynooth Eprint Archive

MURAL - Maynooth University Research Archive Library

A Data Fusion Model for Irish Electricity Load ForecastingDamien Fay† John V. Ringwood††  Marissa Condon† Michael Kelly†††† Dublin City University †† NUI Maynooth ††† Electricity Supply Board AbstractWeather inputs are an important factor in load forecasting. The arrival of a weather front or introduction of weatherforecast errors lead to a change in the significance of weather as input. A data fusion of individual models is introduced,which uses  a priori information regarding the significance of weather inputs in these situations, to produce a superiorforecast to any individual model. 1. INTRODUCTIONShort term forecasting is required by electricity utilities for unit commitment and allocation of spinning reserve. Errors inshort term forecasts cost the Irish Electricity Supply Board a significant amount of money per annum. In accordance withthe Electricity Regulation Act of 1999, a deregulated market structure was set up which should lead to increased impetusto reducing forecast error and the associated costs.Load forecasting models are usually trained using actual past weather readings [1-2] as opposed to past weather forecasts.This is based on the assumption that to use the latter essentially adds forecast noise to the training data. Often weatherforecasts are unavailable for the entire training period and/or can be subject to increasing accuracy of meteorologicalmodels, as mathematical weather models are constantly improved. Therefore, training load models with actual weathercan be justified. However, weather forecast errors not present in the training set can have a disproportionate influence onthe load in models as illustrated in [1-2]. Changing the load model parameters to account for this can be impossible inmany conventional models once training is completed. Douglas et al. [4] approached this problem by use of a Bayesianframework, but restricted analysis to the use of dynamic linear models. As pointed out by [3], the literature is sparse withrespect to load forecasting in the presence of weather forecast error. An incoming weather front will lead to a shift in temperature and a subsequent shift in the load. In this situation, theinfluence of temperature on load is increased relative to other factors. Park et al [5] found that by considering theseeffects, improved results from their model could be obtained.The three load forecasting methods in [3-5] have the ability to change the parameters of an individual load model given apriori information regarding the accuracy of the inputs. However, combining the outputs of several different models, ordata fusion, is an alternative technique that has been applied to many fields [6,7,8] to solve similar problems to thosefaced in [3-5]. References [7,8] compare several data fusion techniques for rainfall run-off models and find that a fuzzylogic approach is the superior technique in that situation. The focus of this paper is to present a fusion technique for dealing with a priori knowledge regarding weather forecasterror and shifts in temperature. The data fusion algorithm of McCabe [9] was chosen as the fusion parameters can beeasily changed to cater for perceived shifts  in the relative significance of the various data streams used to form thecomposite output.2. DATA SET DETAILSA database of electricity demand, actual temperature, wind speed and cloud cover from 1987 to 2000 on an hourly basisis available. Data between Tuesday and Thursday in the months January to March is selected so as to avoid the exceptionsassociated with weekend, Christmas and changes due to the daylight saving hour. Weather forecast data for temperature wind speed and cloud cover is also available for February 2000 also on an hourlybasis. Four sets of data are used to train and test the models (Table 1).  The training set is used to calculate model parameters. Inthe case of neural network models the validation set is used for early stopping and structure determination. For the linearmodels the validation set is used to determine the number of inputs to the model.  Since the validation set has someinfluence on model determination a novelty set is used to evaluate model performance using actual weather readings.Finally, the forecast set is used to evaluate model performance using weather forecasts.Table 1 Segmentation of data set.Set Training Validation Novelty ForecastRange 20th Jan 198720th Mar 199621st Mar 199620th Mar 199827th Mar 199830th Mar 20003rd Feb 200026th Feb 2000Size (Days) 320 40 60 143. FUSION MODEL3.1 Preliminary AR linear modelIt was previously found by these authors [10] that decomposing load data into 24 parallel series , one for each hour of theday, is advantageous as the parallel series have a degree of independence. The parallel series for hour j on day k, y(j,k),has a low frequency trend d(j,k), which is first removed using a Basic Structural Model (BSM) leaving a residual x(j,k)(Figure 1) which is composed of weather, non-linear AR and white noise components [10]. Figure 1. Preliminary AR linear model overview3.2 Sub-ModelsThree models were chosen which have different types of inputs. These are chosen so that forecast errors can be attributedto particular inputs. A fourth model is included using all the available inputs to capture any non-linear relationshipsbetween the inputs and the residual. The fusion technique combines the forecasts of the sub-models x1…4(j,k) to give afused forecast xf(j,k) of the residual for series j on day k (Figure2). Figure 2. Data fusion model overviewThe Temperature Model (TM) input, t(j,k), is a vector of the previous 24 hours of temperature from hour j on day k. Thisvector is transformed using Principal Component Analysis (PCA), to produce a vector of principal components as in [10].Temperature  Model (TM) other Weather  Model (WM) Non-linear Auto-Regressive residual model (NLAR) Non-Linear Model (NLM) Residual forecast 1  Residual forecast 2 Residual forecast 4 Residual forecast 3  Data fusion algorithm Temperature t(j,k)  Wind speed  ws(j,k) Cloud cover c(j,k)  Temperature t(j,k)  Wind speed ws(j,k)  Cloud cover c(j,k)  AR residuals x(j,k-1), x(j,k-2)... x2(j,k)  x1(j,k)  x f(j,k)  x3(j,k)  x4(j,k)  BSM Actual load y(j,k) Trend d(j,k) residual x(j,k) + _ The number of components retained, N, is determined by the model performance over the validation set [10]. The residualx(j,k) is then modeled using a simple linear model of the form where  ai  is  the  linear  coefficient  applied  to  PCA component wi and  (j,k)  is  a  white noise  error  term.  The linearcoefficients are calculated by means of least squares as in [10]. The other Weather Model (WM) inputs, ws(j,k) and c(j,k), are vectors containing the previous 24 hours of the wind speedand cloud cover  from hour j on day k, respectively. These inputs are transformed via PCA as in the TM and the residualis modeled using a linear model similar to (1). The Non-Linear AutoRegressive model (NLAR) uses a feed forward neural network trained using the back-propagationalgorithm with  early  stopping  using  the  validation  set  as  in  [10].  The  network topology  is  determined  from theperformance of networks with one to five nodes in both hidden layers over the validation set (refer to [10] for moredetails). The optimum topology was found to be three nodes in the first hidden layer with two nodes in the second.  The Non-Linear Model  (NLM) using all inputs employs a neural  network trained the same way as  the NLAR. Thedimensionality of the weather inputs is reduced using PCA as in the TM and WM (refer to [10] for more details). Theoptimum topology for the NLM was found to be two nodes in the first hidden and three in the second. 3.3 Fusion algorithmThe data fusion algorithm described in [9] seeks to minimize the variance of the fused forecast based on the covariancematrix of the sub-model forecasts. The cross-covariance of the forecasts is considered and the distribution of the forecasterror noise is not restricted to Gaussian but merely required to be unbiased.  A combined forecast  xf(j,k) of the load iscreated using a weighted average of the individual forecasts x1…4(j,k)  [9]:where Ai(j,k) is the weight applied to forecast i and is derived from the error covariance matrices of  x1…4(j)  as: where with where E[] denotes the expectation operator and M is the number of samples. The  P’i,n(j) are calculated from:The final weight A4 is determined using the constraint that xf(j) is unbiased:  Finally the fused load yf(i,j) is estimated by reintroducing the trend:x j k a j w j k a j w j k a j w j k j kN N( , ) ( ) ( , ) ( ) ( , ) ..... ( ) ( , )  1 1 2 2  + ( , )                                               (1)x j k A j x j kf i ii( , ) ( ) ( , )14                                                                         (2)   A j A j A j P j P j P jP j P j P jP j P j P jP j P j P j1 2 3 4 1 4 2 4 31 1 1 2 1 32 1 2 2 2 33 1 3 2 3 313( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ),',',',',',',',',',',','                                 P j P j P j ii i4 4 4 4 4,', ,( ) ( )  ( )                                                                                              (4))6(                                            4,4    )()()()()( 4,44,,4,',  nijPjPjPjPjP inniniA j A jii4131( ) ( )                                                                            (7)P j x j x j x j x jMx j k x j k x j k x j ki n i n i nkM,=( ) = E[( ( ) - ( ))( ( ) - ( ))] = ( ( ) - ( ))( ( ) - ( ))                                 1 51, , , , ( )y j k d j k x j kf f( , ) ( , )  ( , )                                                                        (8)4. COVARIANCE ESTIMATION4.1 Calculating the covariance matrixThe sample covariance matrix was calculated on a combination of the training and validation sets using (5). It is foundthat the covariance matrix doesn't change considerably over time and so is not tracked. A normalised sample covariancematrix* is shown (for the 11pm load series as an example) in Table 2. Table 2 Normalised sample covariance matrix for 11pm load series.Model TM WM NLAR NLMTM 0.6763    0.6240    0.6201    0.6605WM 0.6240    0.9548    0.9015    0.7711NLAR 0.6201    0.9015    1.0000    0.8398NLM 0.6605    0.7711    0.8398    0.9397As can be seen, the off-diagonal elements of the matrix are significant showing strong cross-correlation between sub-models. 4.2 Covariance matrix for the forecast setWhen forecast data is applied to the inputs of the sub-models the covariance matrix will change as weather forecast errorsaffect the estimated load. The sample covariance matrix is calculated using (5) from the forecast set. A separate set forcalculating the covariance matrix would be superior, but due to the small size of the forecast set this is not possible. 4.3 Covariance matrix for a shift in temperatureA shift  in  temperature  indicates  a  corresponding  reaction  in  the  load.  Since  the  NLAR and  WM  model  have notemperature input their  forecasts will be  less accurate than before.  Thus their  expected errors  will be  larger,  or  theexpected errors of the TM and NLM will be relatively smaller.  To investigate if this could be taken advantage of asimple formula was applied. The shift is identified by a large difference between the TM forecast and the NLAR forecast.If the difference was above a certain threshold  then the entries in the covariance matrix relating to the TM are reducedby a factor :5. RESULTSThe value for is tuned over the training set and the results show that as alpha is increased from 1 (i.e. no effect) to 5 theMAPE in both the training and forecast sets falls until  reaches 2.4.  Adjusting the covariance matrix for a shift intemperature is shown to be consistent over both sets (Figure 3). Figure 3. MAPE as a function of The performance of the sub-models and the fusion model (with and without adjusting the covariance matrix for a shift intemperature) are shown in Table 3. The results are compared using the standard load forecasting measure, the Mean* Normalised for confidentiality reasons.IF   THEN                                                                                                                           (9 )x j k x j kP j P jP j P j nn n1 31 1 1 121 1 2 3 4( , ) ( , )( ) ( ) /( ) ( ) / , ,, ,, ,  1 1.5 2 2.5 3 3.5 4 4.51.9922.012.022.032.042.05 TrainForecastMAPEAbsolute Percentage Error (MAPE) and the Mean Squared Error (MSE) over the training, novelty and weather forecastdata sets. Table 3 MAPE & normalised MSE*.Model TM WM NLAR NLM Fused/ Fused /Training MAPE 1.93 2.61 2.43 2.06 1.99 1.97Novelty MAPE 1.65 2.21 2.19 1.90 1.74 1.68Forecast MAPE 2.10 6.28 2.03 2.27 2.10 1.98Training MSE .98 1.82 1.67 1.15 1 .96Novelty MSE 1.29 1.92 2.10 1.70 1.34 1.27Forecast MSE 2.41 30.75 2.19 2.46 2.39 1.94It is interesting to note that the TM is superior to the other sub-models except in the weather forecast set. When weatherforecasts are fed into the sub-models all degrade in performance except the NLAR which becomes the best sub-model.The NLAR model does not degrade, since it is not dependent on weather information. The fusion model achieves acomparable  performance to  the best  sub-model in all  data  sets,  but  is  seen to  achieve better  consistency across thedifferent data sets. When the value of  is set to 2.4, the fusion model performance is superior in all data sets except the in the novelty setwhere the TM MAPE is superior. The MSE for the fusion model is, however, superior and this is due to the fact that thefusion model is trained to minimize the MSE and not the MAPE. As can be seen in Figure 4, the performance of thefusion model for certain hours of the day is preferable to the performance of the other models and thus it may not bedesirable to apply the fusion model for all parallel series. Figure 4. MAPE as a function of hour of day over the forecast set&Figure 5 shows a sample forecast made using weather forecast data with a shift in temperature occurring at 17:00 and18:00. Figure 5. Sample forecast*& The WM and NLM are excluded for clarity.* Normalised for confidentiality.00:00 05:00 10:00 15:00 20:00 25:000.550.60.650.70.750.80.850.90.951ActualTMNLARFusion with int.TimeLOAD0 5 10 15 20 2511.522.533.5BSMTMNLARFusedHour of dayMAPE5. CONCLUSIONA fusion model has been introduced which can adapt to  a priori information via the covariance matrix giving superiorresults to  any individual model.   The  type of sub-model used is  not constrained in any way. The  majority of  loadforecasting models use weather forecasts online. The danger of training models with inputs, some of which will later beforecast, as in the NLM has been shown. The NLM was trained in the absence of weather forecast noise. Its inability toreject this noise is shown in the forecast set. By using the TM and NLAR, the weather forecast error could be de-coupledfrom the AR error. Thus, the fusion technique can reduce the relative importance of the temperature inputs and maintainperformance. Although the algorithm applied for a shift in temperature was possibly too simple and ignored the relativeimprovement of the NLM, it was found to give considerable improvement and warrants further research.  REFERENCES1.  H.  Yoo,  R.L.  Pimmel,  "Short  term  load  forecasting  using  a  self-supervised  adaptive  neural  network",  IEEETransactions on Power Systems, Vol. 14, No. 2, May 1999, pp 779-784.2. P.K.Dash, A.C. Liew, S.Rahman, "A real-time short-term load forecasting system using functional link network", IEEETransactions on Power Systems, Vol. 12, No. 2, May 1997, pp 675-680. 3. T.Miyake, J.Murata, K.Hirasawa, "One day through seven day ahead electrical load forecasting in consideration ofuncertainties of weather information", Elec. Eng. Jpn (USA), Vol. 115, no. 8, 15th December 1995, pp 22-32.4. A.P. Douglas, A.M. Breipol, F.N.Lee, R. Adapa, "The impacts of temperature forecast uncertainty on Bayesian loadforecasting", IEEE Transactions on Power Systems, Vol. 14, No. 4, November 1998, pp 1507-1513.5.  D.Park,  O.Mohammed, R.Merchant,  T.Dinh,  C.Tong,  A.Azeem, J.  Farah,  C.Drake,  "Forecasting  abnormal  loadconditions with neural networks", Proc. of the Second International Forum on Applications of Neural Networks to PowerSystems, Yokohama, Japan, 19-22 April 1993, pp 73-78. 6. M.P. Perrone, L.N. Cooper, "When networks disagree: Ensemble methods for hybrid neural networks", in ArtificialNeural Networks for Speech and Vision, R.J. Mammone (eds),Chapman & Hall, 1993.7.  A.Y.  Shamseldin, K.M.  O'Connor,  G.C.  Liang,  "Methods  for  combining the  outputs  of  different  rainfall-runoffmodels", Journal of Hydrology, 197, 1997, pp 203-229.8.  L.Xiong, A.Y. Shamseldin, K.M. O'Connor, "A non-linear combination of the forecasts of rainfall-runoff models bythe first order Takagi-Sugeno fuzzy system", Journal of Hydrology, 245, 2001, pp 196-217.9. H. McCabe, “Minimum trace fusion of N sensors with arbitrary correlated sensor to sensor errors”,  Proceedings ofIFAC Conference on Distributed Intelligent Systems, Virginia, USA, August 1991, pp 229-234.10. D.Fay, J.V.Ringwood, M.Condon, M.Kelly, "Comparison of linear and neural parallel time series models for shortterm load forecasting in the Republic of Ireland", 35rd Universities Power Engineering Conference, Vol 1Belfast, UK., 6-8th September 2000,  (Not paginated on CD-ROM).

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Data Fusion Model for Irish Electricity Load Forecasting

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