Improved Grey predictor rolling models for windpower prediction

T.H.M. El-Fouly, E.F. El-Saadany and M.M.A. Salama

Abstract: A new technique for one step ahead average hourly wind speed forecasting and windturbines’ output power prediction based on using the Grey predictor models is presented. Therequired mathematical formulation for developing the Grey predictor models is also presented.The obtained results from the proposed models are compared with the corresponding resultsobtained when using the persistent model. Utilising the traditional Grey model, GM(1,1) wasfirst investigated and showed good improvement over the persistent model. However, the generatedresults demonstrate the presence of intervals with overshoots in the predicted values. To reducesuch overshoots, a modified version for the Grey predictor model referred to as the adaptivealpha GM(1,1) model is investigated and two new models are proposed, hereafter, referred to asthe improved Grey model and the averaged Grey model. The presented results demonstrate theeffectiveness, the accuracy and the superiority of the proposed averaged Grey model for windspeed and wind power prediction.

1 Introduction

Wind power resources are expected to supply about 12% ofthe total world electricity demands by 2020 [1]. TheCanadian Wind Energy Association is expecting to supplyat least 5% of the Canadian electricity from wind resourcesby 2010. To achieve such goal, the association is targeting atotal wind generation of more than 10 GW by 2010 [2].Wind generation is highly dependent on the variable inci-dent wind speed at the wind turbine sites that results inhigh variability and uncertainty in the generated windpower. This has motivated researches to develop accurateand reliable wind speed forecasting and wind power predic-tion tools. Wind power prediction is very important forseveral reasons, such as wind farm’s unit maintenance,energy reserves planning and scheduling, power systemgenerators scheduling, optimal power flow between conven-tional units and wind farms and electricity market bidding.Table 1 presents the various prediction horizons, their fieldsof application and the developed techniques for each predic-tion horizon.

There are two main approaches for predicting wind power.The first approach is based on forecasting wind speed at windturbines locations and then uses wind turbine power curvesto predict wind power production. The second approach isbased on predicting the wind power directly from previouslyrecorded historical wind production data. During the lastdecades, several techniques have been utilised and devel-oped for wind speed forecasting and wind power prediction.Among the developed models, the time series based modelsinclude: auto regressive models, auto regressive movingaverage models and auto regression integrated moving

# The Institution of Engineering and Technology 2007

doi:10.1049/iet-gtd:20060564

Paper first received 26th December 2006 and in revised form 30th April 2007

The authors are with the Department of Electrical and Computer Engineering,University of Waterloo, 200 University Avenue West, Waterloo, Ontario,Canada N2L 3G1

E-mail: tarek_el_fouly@sympatico.ca

928

average models [3–7]. In addition, time series modelsbased on ANN have also been used for wind speed forecast-ing. The ANN-based methods are Elman recurrent network,adaptive network-based fuzzy inference system, radial basisfunction network and neural logic networks [3–10].However, these models are characterised by the need forlarge sets of recorded historical data for model training andparameters estimation purposes.

Techniques, based on utilising the wind speed data fromthe neighbouring sites have been proposed in [11–14].These techniques require the usage of large sets of datafrom more than one site to achieve reasonable accuracy ofprediction. The most advanced developed techniques arethe numerical weather prediction-based techniques, referredto as the physical models, which uses weather data withsophisticated meteorological for wind power predictions[15–17]. Yet, these models are very complicated andexpensive. In addition, these models have been reportedto be inefficient for short-term prediction (few hoursahead, up to 6 h). This increase the necessity for developingmore accurate short-term prediction tools to be used inassociation with such models.

This paper presents new models for one hour ahead pre-diction of wind speed and wind power based on utilising theGrey predictor models GM(1, 1). The results obtained fromthe proposed model are compared with the correspondingresults obtained from the persistent model (assuming thatthe next hour predicted value is equal to the current obser-vation). The presented paper is organised such that Section2 introduces the preparation and the analysis process carriedout for the data samples under investigation. Section 3 pre-sents the procedure for developing the traditional GM(1, 1)rolling model, and the generated prediction results whenusing such model. The proposed modification for theGM(1, 1) model to reduce the prediction error are intro-duced in Section 4. In this section, wind speed predictionresults when using these modified models are presentedand compared with those generated when using the tra-ditional model and the persistent model. Wind power gener-ation prediction results, analyses and accuracy evaluation

IET Gener. Transm. Distrib., 2007, 1, (6), pp. 928–937

IET Gener

Table 1: Different horizons for wind power prediction

Purpose Horizon Approaches

Unit maintenance 0 (Nowcasting) Time series/statistical approaches

Control purposes Few seconds or minutes Time series/statistical approaches

Small power systems operation such as one

hour electricity market bidding and optimal

power flow analysis.

1–6 h Time series/statistical approaches

Interconnected power systems operation

such as generator scheduling and dispatch,

unit commitments, and day ahead

electricity market bidding, and reserve

1–72 h Time series/statistical approaches

Physical models

Maintenance and planning 1–7 days Time series/statistical approaches

Physical models

are presented in Section 5. Finally, in Section 6 conclusionsare presented.

2 Data preparation and analysis

Four actual wind speed data sets, recorded at Madisonweather station [18], are used in this analysis, as presentedin Fig. 1. Samples 1 and 2 were chosen to represent the mostcommon pattern for the incident wind speed (consequentlymost common wind power generation pattern) [19], wherethe peak wind power (or wind speed) intervals occur over-night. Moreover, Sample 1 has higher average wind speedthan Sample 2. Sample 3 represents a different pattern(almost an opposite wind speed trend to that observed bySample 1 and Sample 2), where the wind speed timeseries has an increasing trend at the beginning of the dayand then a decreasing trend. The above mentioned threesamples have approximately smooth wind speed variationswith time; however, the last tested sample, Sample 4, rep-resents a wind speed time series with higher wind speed ran-domness. Finally, Sample 2 and Sample 3 are alsocharacterised by having time intervals with wind speedvalues below the cut in wind speed of the wind turbinepower curve used in this analysis, as will be discussed

. Transm. Distrib., Vol. 1, No. 6, November 2007

later. Each wind speed sample consists of 24 h data setrecorded over 15 min periods. The average hourly windspeed time series is obtained by applying an averagingprocess, before the prediction analysis, as follows

Xaverage( j) ¼1

4

X4

i¼1

Xrecorded(i) 8j ¼ 1, 2, . . . , 24 (1)

where Xaverage( j) is the hourly averaged data point duringthe hour j, and Xrecorded(i) is the 15 min recorded datapoints during the hour j.

After predicting the wind speed, the predicted wind speedtime series is used as an input to the manufacturer powercurve of the VESTAS V66-1.65 MW wind turbine, givenin Fig. 2, to predict the output wind power production.The power curve under investigation is characterised by acut in wind speed of 4 m/s, rated wind speed of 16 m/sand a cut out wind speed of 25 m/s.

3 Traditional Grey rolling model GM(1,1)

The Grey systems have been first introduced by ProfessorJulong Deng in 1982 [20, 21]. These systems refer to anysystem with partially unknown information about its

Fig. 1 Actual wind speed time series samples under investigation

929

parameters, structure and/or characteristics. Grey predictormodels are characterised by [22]:

† Their need for only fewer historical data requirement todevelop the prediction model (as few as four historicaldata points);† Being highly adaptive to the dynamic behaviour of thedata; and† Their little computation effort and processing time.

Grey predictor models have been involved in many pre-diction applications such as:

† Forecasting non-periodic time series such as stock pricesindices [23];† Predicting objects position and targets tracking in whichthe grey system predicts the future trend of an object ortarget based on few historical measurements [24];† Power system’s yearly peak load forecasting [22];† Predicting the changes in the inertia and damping coeffi-cients of the mechanical parts of induction servo motors andits control based on indirect field orientation [25].

There are several Grey predictor models currently in use.This section investigates the application of the most com-monly used model; the traditional GM(1,1) model [26].The required procedure to develop this model and theassociated prediction process is illustrated in Fig. 3. Themathematical formulations describing each stage can besummarised as follows:

1. Generate the first order accumulated generating oper-ation (AGO) series. This resulted in generating a new accu-mulated data series (X(1)) that is characterised by moresmoothed regular pattern (less randomness) than the orig-inal data series (X(0)) under analysis. This operation canbe expressed mathematically as follows

X (1)(k) ¼Xk

i¼1X (0)(i) 8k ¼ 1, . . . , n; (2)

where X(1) represents the first order AGO data series, X(0)

represents the original data series, n represents the sampledata (n ¼ 4 for all investigated models) and k and i rep-resent the step for the AGO and the original data series,respectively.2. Formulate the model’s differential equation that relatesits dependent variables with the independent ones. Thisdifferential equation is known as the Grey dynamicmodel. For the traditional GM(1,1) model, this differentialequation is represented by one independent variable andno dependent variables and can be expressed as follows

dX (1)

dtþ aX (1)

¼ b (3)

where X represents the independent variable for the

Fig. 2 Power curve for the VESTAS V66-1.65 MW wind turbine

930

traditional GM(1,1), and a and b are the model coefficients(parameters) determined using the least square method.3. The parameters a and b of the GM(1,1) can be deter-mined using the least square method as follows

A ¼a

b

� �¼ bTb

� ��1bT

Y (4)

where

b ¼

�Z(1)(2) 1

�Z(1)(3) 1

..

. ...

�Z(1)(n) 1

266664

377775 (5)

Y ¼

X (0)(2)

X(0)(3)

..

.

X (0)(n)

266664

377775 (6)

and

Z(1)(i) ¼

X(1)(i� 1) þ X

(1)(i)

2(7)

4. This step is dedicated to calculate the forecasted orpredicted values of the AGO series (X (1)). The equationused in this process can be written as follows

X (1)(iþ 1) ¼ X (0)(1) �b

a

� �e�ai

þb

a(8)

where i represents the step, and X(0) represents the first datain the original time series.5. Transform the forecasted AGO series of data back to itsoriginal form (series) using the inverse accumulated gener-ating operation (IAGO). This operation is considered theinverse of the AGO and can be represented mathematically

Fig. 3 Prediction process using the traditional GM(1,1) rollingmodel

IET Gener. Transm. Distrib., Vol. 1, No. 6, November 2007

Fig. 4 Wind speed prediction using the traditional GM(1,1) rolling model and persistent model

by

X (0)(1) ¼ X (1)(1) (9)

X(0)(iþ 1) ¼ X

(1)(iþ 1) � X(1)(i) 8i ¼ 1, 2, 3, . . . (10)

6. Use the provided manufacturer power curves to calculatethe predicted power generation of the turbine.7. After the predicted interval becomes an observation,update the used input data using the rolling modellingmechanism [27]. This is carried out by eliminating theoldest historical data and adding the recent observation topredict the next future interval.

Actual and predicted wind speed time series for the fourtested samples are presented in Fig. 4. This figure presentsthe predicted time series when using both the traditionalGM(1,1) rolling and the persistent models. The figure demon-strates the effectiveness of the traditional GM(1,1) rollingmodel in tracking the actual time series better than the persist-ent model. Using the traditional GM(1,1) rolling modelresulted in mean absolute error (MAE) values of 0.76, 0.94,1.12 and 1.1 m/s for Samples 1–4, respectively, that corre-sponds to 0.79, 0.97, 1.19 and 1.13 m/s, respectively, whenusing the persistent model. Moreover, for the studiedsamples, the traditional GM(1,1) rolling model forecastswind speed time series with an improvement, in the MAE,over the persistent model of 3.08, 3.09, 5.88 and 2.65%.

4 Modified Grey rolling models

Using the traditional GM(1,1) model in predicting continu-ously variable time series, normally resulted in the occur-rence of overshoots [27] that reduce the predictionaccuracy. This section is devoted to investigate a previouslyproposed technique to reduce these overshoots [27], here-after, referred to as the adaptive alpha-based GM(1,1)model. In addition, two new models, hereafter, referred toas the improved and the averaged Grey models are proposed.

4.1 Adaptive alpha-based GM(1,1) model

The adaptive alpha-based GM(1,1) model is developed fol-lowing the same procedure (stages) described earlier for the

IET Gener. Transm. Distrib., Vol. 1, No. 6, November 2007

traditional GM(1,1) model except for the formula used incalculating the Z(1)(i) terms shown in Fig. 5. In 1999,Chang et al. [28] proposed an adaptive a(i) for the Z(1)(i)terms so that the formula used in calculating these termscan be expressed by

Z(1)(i) ¼ [1 � a(i)]X (1)(i� 1) þ a(i)X (1)(i) (11)

where a(i) is a weighting factor within the range0 � a(i) � 1. For the traditional GM(1,1) model, thisfactor is set equal to 0.5.

Fig. 5 Prediction process using the adaptive alpha-basedGM(1,1) rolling model

931

In this model, the weighting factor a(i) is determinedusing the ‘average system slope’ technique [27] that is sum-marised as follows:

1. Calculate the average slope coefficient (savg) as follows

savg ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiX (0)(n)

X (0)(1)

n�1

s(12)

where X(0)(n) and X(0)(1) represent the last and the first datapoints in the original data series, respectively, and n rep-resents the total number of data points used in developingthe GM(1,1) model.2. Determine the relative positions (kj) for the remainingdata points (excluding the first and the last points that areforced to match the data values of the real system endpoints) to force the data of the theoretical system modelto equate with the data of the real system model. This canbe mathematically formulated as follows (for four datapoints model)

X(0)(1)sk2

avg ¼X(0)(2)) k2 ¼ log X

(0)(2)=X (0)(1)�

=log(savg)

(13)

X (0)(1)sk3avg ¼X (0)(3)) k3 ¼ log X (0)(3)=X (0)(1)

� =log(savg)

(14)

X(0)(1)s3

avg ¼X(0)(4) (15)

3. Calculate the adaptive alpha set, a(i), using the follow-ing ‘if’ rules:

a: IF k2 � 0 then a(2) ¼ 0 else IF k2 [ (0,1)

then a(2) ¼ k2 else IF k2 � 0 then a(2) ¼ 1 (16)

b: IF k3 � 1 then a(3) ¼ 0 else IF k3 [ (1,2)

then a(3) ¼ k3 � 1 else IF k3 � 2 then a(3) ¼ 1 (17)

c: a(4) ¼ 1 (18)

932

4. Proceed with step 3 as illustrated in the traditionalGM(1,1) model while using the adaptive Z(1)(i) expressionpresented in (11).

The corresponding actual and predicted wind speed timeseries, for the four tested samples, when using the adaptivealpha-based model are presented in Fig. 6. Using thismodel resulted in MAE values of 0.75, 0.87, 1.09 and1.1 m/s for Samples 1–4, respectively, that correspondsto an improvement in the MAE over the persistent modelof 5.06, 10.31, 8.4 and 2.65%. This discloses that moreimprovements in the MAE, over the persistent model,have been achieved by this modified model compared tothose produced by the traditional model for all samplesexcept the fourth sample that reveals the same improve-ment level. Comparing the results presented in Fig. 6with their corresponding results presented in Fig. 4reveals that the adaptive alpha-based model reduces theovershoots in the predicted time series; however, for theintervals not experiencing overshoots, tracking the actualtime series is better when using the traditional model.This motivated the authors to investigate and developother Grey-based models, which are discussed in the fol-lowing subsections.

4.2 Improved Grey model

In an attempt to increase the prediction accuracy, animproved Grey model is proposed for wind speed andwind power prediction. This model is based on generatingtwo shifted prediction models from the traditionalGM(1,1) model, discussed in Section 3; then hybrid thesenew models to develop the final prediction model. Fig. 7presents a flow chart representing the different stages fordeveloping the proposed improved Grey model. The math-ematical formulations describing each stage can be sum-marised as follows:

1. Follow steps 1–4 discussed in Section 3 for developingthe traditional GM(1,1) model. Step 4 ends by developingthe traditional GM(1,1) prediction model to predict valuesfor the AGO series (X

(1)GM) that is represented

Fig. 6 Wind speed prediction using the adaptive alpha-based GM(1,1) rolling model

IET Gener. Transm. Distrib., Vol. 1, No. 6, November 2007

mathematically by

X(1)GM(iþ 1) ¼ X (0)(1) �

b

a

� �e�ai

þb

a(19)

2. Generate two shifted prediction models (X(1)1 and X

(1)2 ).

The aim of this stage is to generate an envelope-like timeseries that contain the actual AGO series. This can be

Fig. 7 Prediction process using the improved GM(1,1) rollingmodel

IET Gener. Transm. Distrib., Vol. 1, No. 6, November 2007

formulated as follows

X(1)1 (iþ 1) ¼ X

(1)GM(iþ 1) þ shift (20)

X(1)2 (iþ 1) ¼ X

(1)GM(iþ 1) � shift (21)

where ‘shift’ represents the value by which the traditionalGM(1,1) model is shifted and it was chosen to be equal tothe last data point in the first AGO series; that is X(1)(n)where n¼ 4 (four data points are used to build the GMmodels), to ensure that the generated envelope containsthe actual AGO series.3. Hybrid the new prediction models to generate the finalimproved Grey model (X

(1)imp) as follows

X(1)imp(iþ 1) ¼ w1X

(1)1 (iþ 1) þ w2X

(1)2 (iþ 1) (22)

and

w1 þ w2 ¼ 1 (23)

where w1 and w2 represent the weights for the shiftedmodels. These weights are being updated using the leastmean square (LMS) technique, known as the Widro-Hoffdelta rule [29, 30], which can be expressed mathematicallyby

W (iþ 1) ¼ W (i) þ dX (1)(i)e(i)

X (1)(i)� �T

X (1)(i)(24)

where W(i) and W(iþ 1) represent the current and theupdated weights vector, d is the learning parameter,X(1)(i) and [X(1)(i)]T represent the shifted AGO seriesvector and its transposed vector, and e(i) is the errorvector given by

e(i) ¼ X (1)(i) � X(1)

imp(i) (25)

4. Proceed with step 5 as illustrated in the traditionalGM(1,1) model (Section 3).

Fig. 8 presents the corresponding actual and predictedwind speed time series when using the improved

Fig. 8 Wind speed prediction using the improved GM(1,1) rolling model

933

Fig. 9 Wind speed prediction using the averaged GM(1,1) rolling model

Grey model. This model achieves levels of the MAEof 0.71, 0.95, 1 and 1.03 m/s for Samples 1–4, respectively.This corresponds to an improvement over the persistentmodel of 10.13, 2.06, 15.97 and 8.85%, respectively.More improvements in the MAE are achieved forSamples 1, 3 and 4. However, Sample 2 experienceda drop in the percentage improvement of the MAE com-pared with the adaptive alpha-based model. Comparedwith the traditional model, Fig. 8 reveals that the improvedmodel managed to reduce the overshoots, however, the

934

tracking feature for the actual time series is still notimproved.

4.3 Averaged Grey model

This model attempts to combine, to some extent, the goodfeatures of both the traditional GM(1,1) model (representedby good tracking and lower prediction error for the intervalswithout prediction overshoots) and the improved Greymodel (represented by reducing the overshoots and hence

Table 2: MAE (m/s), RMSE (m/s) and improvements for wind speed prediction

Parameter Persistent GM(1,1) Adaptive Improved Averaged

Sample 1

MAE Value 0.79 0.76 0.75 0.71 0.69

% Improvement – 3.80 5.06 10.13 12.66

RMSE Value 1.09 1.04 1.06 0.99 0.95

% Improvement – 4.59 2.75 9.17 12.84

Sample 2

MAE Value 0.97 0.94 0.87 0.95 0.85

% Improvement – 3.09 10.31 2.06 12.37

RMSE Value 1.22 1.10 1.09 1.22 1.06

% Improvement – 9.84 10.66 0.00 13.11

Sample 3

MAE Value 1.19 1.12 1.09 1.00 0.87

% Improvement – 5.88 8.40 15.97 26.89

RMSE Value 1.58 1.42 1.48 1.47 1.26

% Improvement – 10.13 6.33 6.96 20.25

Sample 4

MAE Value 1.13 1.10 1.10 1.03 0.99

% Improvement – 2.65 2.30 8.85 12.39

RMSE Value 1.26 1.38 1.26 1.27 1.21

% Improvement – 29.52 0.00 20.79 3.97

IET Gener. Transm. Distrib., Vol. 1, No. 6, November 2007

Fig. 10 Wind power prediction using the averaged GM(1,1) rolling model

reducing the prediction errors at the intervals where over-shoots occur). This is carried out by averaging the resultantof both models as follows

X (1)avg(iþ 1) ¼

X(1)GM(iþ 1) þ X

(1)imp(iþ 1)

2(26)

where X (1)avg(iþ 1) represents the predicted future point using

the proposed averaged Grey model.The corresponding actual and predicted wind speed time

series, for the four tested samples, when using the averagedGrey model are presented in Fig. 9. Using this modelresulted in MAE values of 0.69, 0.85, 0.87 and 0.99 m/sfor Samples 1–4, respectively, that corresponds to animprovement over the persistent model of 12.66, 12.37,26.89 and 12.39%, respectively. This reveals the superiorityof the proposed averaged model in wind speed forecastingover all the presented models in this paper. Moreover,Fig. 9 discloses that the tracking feature has been enhancedwhen using the proposed averaged model and it is overallsuperior when compared with the persistent model and theother presented Grey models.

Table 2 provides a comparison between all introducedGrey models in this paper and the persistent model fromthe point of view of the samples’ MAE and the samples’root mean squares of errors (RMSE). Moreover, this tabledemonstrates the achieved percentage improvements, com-pared with the persistent model, for all the developedmodels. The table reveals that the best percentage improve-ments, for all the four tested samples, is achieved whenusing the proposed averaged GM(1,1) model. Moreover,the table also discloses that the overshoots occurrencecharacteristic of the traditional GM(1,1) model wouldresult in a negative percentage improvement on theRMSE scale especially for samples with high randomnessin wind speed variations such as Sample 4.

5 Wind power prediction

This section is dedicated to investigate the usage of the pro-posed averaged model in wind power prediction. The gener-ated results for the predicted wind speed time series usingthe proposed model is used as an input to the VESTASV66-1.65 MW wind turbine power curve, as previously

IET Gener. Transm. Distrib., Vol. 1, No. 6, November 2007

highlighted, to predict the wind power production. The gen-erated actual and predicted wind power time series, whileusing the averaged model, for the four tested samples arepresented in Fig. 10. The figure demonstrates the effective-ness of the proposed averaged GM(1,1) rolling model intracking the actual generated wind power time series.Table 3 presents the MAE, the RMSE and the average per-centage error referred to the rated power (% error, for thepower prediction) for the generated results using the aver-aged GM(1,1) rolling model. This table reveals lowerMAE, lower RMSE and lower average percentage errorvalues while using the averaged GM(1,1) rolling modelcompared to the persistence model. The table also disclosesthat the averaged model, for the studied samples, predictswind power with improvements over the persistent modelup to 36.31% for the MAE, 25.83% for the RMSE and36.34% for the average percentage error. This table also

Table 3: MAE (kW), RMSE (kW), average percentageerror (% error) and improvements for wind powerprediction

Parameter Persistent Averaged % Improvement

Sample 1

MAE 116.01 95.50 17.68

RMSE 184.81 150.97 18.31

% error 7.03 5.79 17.64

Sample 2

MAE 109.84 93.80 14.60

RMSE 179.70 150.96 15.99

% error 6.62 5.69 14.05

Sample 3

MAE 131.60 83.81 36.31

RMSE 194.90 144.55 25.83

% error 7.98 5.08 36.34

Sample 4

MAE 159.89 139.84 12.54

RMSE 194.74 194.16 0.30

% error 9.69 8.48 12.49

935

Fig. 11 Relationships between the actual and predicted wind power values when using the averaged GM(1,1) rolling model

demonstrates that for wind sample with high randomnesslevel, such as Sample 4, using the RMSE only in evaluatingthe accuracy of the prediction technique does not provide asuitable indication.

Fig. 11 presents the obtained scattering and the linearrelationships between the predicted and the actual valuesfor wind power prediction for the tested four data setswhen predicted using the averaged GM(1,1) model. Thislinear relationship is expressed by

yi ¼ mxi þ c (27)

where yi and xi are the predicted and the actual wind speedor power values at time interval i, m and c are the linearrelationship coefficients obtained using the least squaremethod where m is the scaling factor (slope of the linearrelationship) and c is the y-axis (prediction-axis) interceptof the linear relation. The best linear relationship betweenthe actual and the predicted values is achieved whenc ¼ 0 and m ¼ 1, which indicates that the predictor is notbiased to generate higher (or lower) predicted values thanthe actual values.

Fig. 11 reveals good scattering of the predicted values ofwind power. Table 4 presents the corresponding linearrelationships between the predicted and the actual values,and reveals the following:

† For low actual wind power values: The averaged Greymodel is not biased to generate higher (or lower) predictedvalues than their corresponding actual values. This is

Table 4: Actual and predicted values linear relationshipcoefficients when using the averaged GM(1,1) rollingmodel

Samples Linear relationship (MW)

Sample 1 20.015þ 1.032xi

Sample 2 0.037þ 0.809xi

Sample 3 20.016þ 1.015xi

Sample 4 0.037þ 0.836xi

936

revealed by the y-axis intercept coefficients (c) that arevery close to zero.† For high actual wind power values: The averaged Greymodel is sometimes biased to generate lower predictedvalues for wind power than their corresponding actualvalues. This is revealed by the scaling factors (m) that aresometimes less than unity (Samples 2 and 4).

6 Conclusions

This paper presents a new tool for one hour ahead averagehourly wind speed and wind power prediction using theGrey prediction technique {GM(1,1)}. The studiedsamples have wind speed ranges close to and/or aroundthe cut-off range of wind power, which usually representa significant part of the working range of large windfarms. The presented results reveal very good wind powerprediction improvements while using the Grey modelsover the persistent model. Based on the studied samples,using the traditional GM(1,1) model reveals the effective-ness of the proposed model in tracking the actual windspeed time series with an improvement over the persistentmodel up to 5.88% for the MAE and 10.13% for theRMSE. However, such models are characterised by theoccurrence of some overshoots in the predicted timeseries. Such overshoots might result in predicting windspeed time series worse than the persistent model.

To overcome the problem of overshoots occurrence, thispaper investigated the application of the adaptive alpha-based grey model. This model achieved higher levels forthe percentage improvement over the persistent modelthan those achieved by the traditional model. However,the results show that this model lacks the good trackingcharacteristic for the actual wind speed time series achievedby the traditional model. For further reduction of the over-shoots and the prediction error, the paper proposed theimproved Grey model. However, using the improvedmodel did not achieve higher improvement levels for alltested wind speed data samples.

Finally, the paper proposed the averaged Grey model.This model has an overall superiority in wind speed fore-casting over the persistent model and the other presented

IET Gener. Transm. Distrib., Vol. 1, No. 6, November 2007

Grey models. Moreover, it also has a very good trackingfeature and a reduction in the overshoot occurrence.Based on the studied samples, using the averaged Greyrolling model discloses an improvement in the predictionaccuracy, compared with the persistent model, of windspeed up to 26.89% for the MAE, 20.25% for the RMSEand for wind power prediction up to 36.31% for theMAE, 25.83% for the RMSE and 36.34% for the averagepercentage error. The results also demonstrate that the gen-erated predicted results using the proposed averaged modelfor wind power prediction have a very good linear relation-ship with their corresponding actual values.

7 References

1 ‘Wind Force 12’. Report by the European Wind Energy Association(EWEA), October 2002, available at: http://www.ewea.org/doc/WindForce12.pdf

2 ‘Wind Vision for Canada’. Recommendations for achieving Canada’swind energy potential, Report by the Canadian Wind EnergyAssociation (CanWEA), June 2001, available at:http:\\www.canwea.ca\pdfs\CanWEA-WindVision.pdf

3 EL-Fouly, T.H.M., EL-Saadany, E.F., and Salama, M.M.A.: ‘A studyof wind farms output power prediction techniques’. Proc. NorthAmerican Power Symp., NAPS, August 2004, pp. 249–254

4 Huang, Z., and Chalabi, Z.S.: ‘Use of time-series analysis to modeland forecast wind speed’, J. Wind Eng. Ind. Aerodyn., 1995, 56,pp. 311–322

5 Kamal, L., and Jafri, Y.Z.: ‘Time series models to simulate andforecast hourly averaged wind speed in Quetta, Pakistan’, SolarEnergy, 1997, 61, pp. 23–32

6 Sfetsos, A.: ‘A comparison of various forecasting techniques appliedto mean hourly wind speed time series’, Renew. Energy, 2000, 21,pp. 23–35

7 More, A., and Deo, M.C.: ‘Forecasting wind with neural networks’,Mar. Struct., 2003, 16, (1), pp. 35–49

8 Costa, M., and Pasero, E.: ‘Artificial neural systems for Verglassforecast’. Proc. Int. Joint Conf. on Neural Networks, IJCNN, 2001,vol. 1, pp. 258–262

9 Alexiadis, M.C., Dokopoulos, P.S., Sahsamanoglou, H.S., andManousaridis, I.M.: ‘Short-term forecasting of wind speed andrelated electric power’, Solar Energy, 1998, 63, (1), pp. 61–68

10 Li, S., Wunsch, D.C., O’Hair, E., and Giesselmann, M.G.: ‘Neuralnetwork for wind power generation with compressing function’. Int.Conf. on Neural Networks, 9–12 June 1997, vol. 1, pp. 115–120

11 Damoisis, I.G., Alexiadis, M.C., Theocharis, J.B., and Dokopoulos,P.S.: ‘A fuzzy model for wind speed prediction and powergeneration in wind parks using spatial correlation’, IEEE Trans.Energy Convers., 2004, 19, (2), pp. 352–361

12 Barbounis, T.G., Theocharis, J.B., Alexiadis, M.C., and Dokopoulos,P.S.: ‘Long term wind speed and power forecasting using localrecurrent neural network models’, IEEE Trans. Energy Convers.,2006, 21, (1), pp. 273–284

IET Gener. Transm. Distrib., Vol. 1, No. 6, November 2007

13 Damousis, I.G., and Dokopoulos, P.: ‘A fuzzy expert system for theforecasting of wind speed and power generation in wind farms’.PICA 2001 22nd IEEE Power Engineering Society Int. Conf. onPower Industry Computer Applications. Innovative computing forpower—electric energy meets the market, 20–24 May 2001,pp. 63–69

14 Alexiadis, M.C., Dokopoulos, P.S., and Sahsamanoglou, H.S.: ‘Windspeed and power forecasting based on spatial correlation models’,IEEE Trans. Energy Convers., 1999, 14, (3), pp. 836–842

15 Landberg, L.: ‘A mathematical look at a physical power predictionmodel’, Wind Energy, 1998, 1, (1), pp. 23–28

16 Landberg, L.: ‘Short-term prediction of the power production fromwind farms’, J. Wind Eng. Ind. Aerodyn., 1999, 80, (1–2),pp. 207–220

17 Ackermann, T.: ‘Wind power in power systems’ (John Wiley & Sons,Ltd, England, 2005)

18 Madison weather station available at: http:\\www.oardc.ohio-state.edu\centernet\stations\mahome.asp

19 Piwko, R., Osborn, D., Gramlich, R., Jordan, G., Hawkins, D., andPorter, K.: ‘Wind energy delivery issues’, IEEE Power EnergyMag., 2005, 3, (6), pp. 47–56

20 Deng, J.L.: ‘Control problems of Grey systems’, Syst. Control Lett.,1982, 1, (5), pp. 288–294

21 Wen, K.: ‘Grey systems: modeling and prediction’ (Yang’s ScientificPress, Arizona, USA, October 2004)

22 Yang, H.T., Liang, T.C., Shih, K.R., and Huang, C.L.: ‘Power systemyearly peak load forecasting: a Grey system modeling approach’.Proc. EMPD, Int. Conf. on Energy Management and PowerDelivery, 1995, vol. 1, pp. 261–266

23 Chang, B.R., and Tsai, S.F.: ‘A grey-cumulative LMS hybrid predictorwith neural network based weighting for forecasting non-periodicshort-term time series’. IEEE Int. Conf. on Systems, Man andCybernetics, 2002, vol. 6, p. 5

24 Luo, R.C., and Chen, T.M.: ‘Target tracking by grey prediction theoryand look-ahead fuzzy logic control’. Proc. 1999 IEEE Int. Conf. onRobotics and Automation, 1999, vol. 2, pp. 1176–1181

25 Wai, R.J., Duan, R.Y., and Chang, L.J.: ‘Grey feedback linearizationspeed control for induction servo motor drive, IECON ‘01’. The 27thAnnual Conf. of the IEEE Industrial Electronics Society, 2001, vol. 1,pp. 580–585

26 EL-Fouly, T.H.M., El-Saadany, E.F., and Salama, M.M.A.: ‘Greypredictor for wind energy conversion systems output powerprediction’, IEEE Trans. Power Syst. (Letter), 2006, 21, (3),pp. 1450–1452

27 Yao, A.W.L., Chi, S.C., and Chen, J.H.: ‘An improved Grey-basedapproach for electricity demand forecasting’, Electric Power Syst.Res., 2003, 67, pp. 217–224

28 Chang, S.C., Wu, J., and Lee, C.T.: ‘A study on the characteristics ofa(k) of Grey prediction’. Proc. 4th Conf. on Grey Theory andApplications, 1999, pp. 291–296

29 Marei, M.I., El-Saadany, E.F., and Salama, M.M.A.: ‘Envelopetracking techniques for flicker mitigation and voltage regulation’,IEEE Trans. Power Delivery, 2004, 19, (4), pp. 1854–1861

30 Dash, P.K., Swain, D.P., Routary, A., and Liew, A.C.: ‘Harmonicestimation in a power system using adaptive perceptrons’, IEEProc., Gener. Transm. Distrib., 1996, 143, (6), pp. 565–574

937