ByN Venkata Srinath,MS Power Systems.

Statistical approach to wind forecastingMaking use of past data future wind is forecasted.The simplest statistical prediction is an continuous forecast.The last measured value is assumed to persist into the

future without any change. Ŷk =Yk-1

Where Ŷk is the Predicted valueYk-1is the measured value at step k-1

This is the simplistic persistence method.As a forecasting technology, this method is not impressive,

but it is nearly costless, and performs surprisingly well.

The more sophisticated prediction will be some linear combination of the last n measured values, i.e.,

This is known as an nth order autoregressive model, or AR(n).

We can now define the prediction error at step k by

and then use the recent prediction errors to improve the prediction:

This is known as an nth order autoregressive, mth order moving average model, or ARMA(n, m).

The model parameters ai, bj can be estimated in various ways. A useful technique is the method of recursive least squares.

Assumed dataInstant Speed

1 332 33.53 33.84 345 34.26 34.67 34.28 349 34.5

10 3511 35.412 35.8

Instant Speed

13 3614 36.215 36.416 35.917 36.218 36.619 3720 37.221 37.422 3823 38.524 3925 40

Forecasted using persistence modelInstant Measured speed Forecasted speed Error

133 33 0

233.5 33 -0.5

333.8 33.5 -0.299999

434 33.8 -0.200001

534.2 34 -0.200001

634.6 34.2 -0.399998

734.2 34.6 0.399998

834 34.2 0.200001

934.5 34 -0.5

1035 34.5 -0.5

1135.4 35 -0.400002

1235.8 35.4 -0.399998

Instant Measured speed Forecasted speed Error

13 36 35.8 -0.200001

14 36.2 36 -0.200001

15 36.4 36.2 -0.200001

16 35.9 36.4 0.5

17 36.2 35.9 -0.299999

18 36.6 36.2 -0.399998

19 37 36.6 -0.400002

20 37.2 37 -0.200001

21 37.4 37.2 -0.200001

22 38 37.4 -0.599998

23 38.5 38 -0.5

24 39 38.5 -0.5

25 40 39 -1

Forecasting using a linear combination - AR(n)Here, the measured data of 1-8 instance is used to train the

model.4-9 speed’s are expressed as a linear equation’s.The considered order is n=3.

34=33a1+33.5a2+33.8a334.2=33.5a1+33.8a2+34a334.6=33.8a1+34a2+34.2a334.2=34a1+34.2a2+34.6a334=34.2a1+34.6a2+34.2a334.5=34.6a1+34.2a2+34a3

Calculated using recursive least square (X’X)-1X’Y. a1=0.5288; a2=-0.5725; a3=1.0501

Instant Speed Forecasted Error

1 332 33.53 33.84 345 34.26 34.67 34.28 349 34.5

10 35 34.5 -0.511 35.4 34.982 -0.41812 35.8 35.379 -0.421

Instant Speed Forecasted Error

13 36 35.835 -0.16514 36.2 36.0276 -0.172415 36.4 36.334 -0.06616 35.9 36.5359 0.635917 36.2 36.00215 -0.1978518 36.6 36.7091 0.109119 37 36.693 -0.30720 37.2 37.0427 -0.157321 37.4 37.235 -0.16522 38 37.5423 -0.457723 38.5 38.1636 -0.336424 39 38.4509 -0.549125 40 39.007 -0.99326 40.035

Forecasted using ARMA(n , m)The order is n=3, m=1 i.e., ARMA(3,1).Expressing the data from the instant 3-10 as a linear

combination

33.8a1+34a2+34.2a3+2b1=34.634a1+34.2a2+34.6a3+3b1=34.2 34.2a1+34.6a2+34.2a3+3b1=3434.6 a1+34.2a2+34a3+4b1=34.5 34.2a1+34a2+34.5a3+3b1=35

a1=2.0338; a2=-1.3898; a3=0.4246; b1= -0.6826

Instant Speed Forecasted Error

1 332 33.53 33.84 345 34.2 26 34.6 37 34.2 38 34 49 34.5 3

10 35 33.7477 -1.252311 35.4 35.7677 0.367712 35.8 35.1368 -0.6632

Instant Speed Forecasted Error

13 36 36.4544 0.454414 36.2 36.02 -0.1815 36.4 37.06058 0.6605816 35.9 36.6937 0.793717 36.2 36.5126 0.312618 36.6 38.0633 1.463319 37 36.0307 -0.969320 37.2 37.9051 0.705121 37.4 37.09121 -0.3087922 38 38.39025 0.3902523 38.5 38.2898 -0.210224 39 38.4781 -0.521925 40 39.40834 -0.5916626 40.1856

Comparisons

Instant

Measured Persistence model

Error AR Model

Error ARMA Error

1 332 33.5 33 -0.5

3 33.8 33.5 -0.299999

4 34 33.8 -0.200001

5 34.2 34 -0.200001 26 34.6 34.2 -0.399998 37 34.2 34.6 0.399998 38 34 34.2 0.200001 49 34.5 34 -0.5 3

10 35 34.5 -0.5 34.5 -0.5 33.7477 -1.252311 35.4 35 -0.400002 34.982 -0.418 35.7677 0.367712 35.8 35.4 -0.399998 35.379 -0.421 35.1368 -0.6632

Instant Measured

Persistence

model

Error AR Model

Error ARMA Error

13 36 35.8 -0.200001 35.835 -0.165 36.4544 0.454414 36.2 36 -0.200001 36.0276 -0.1724 36.02 -0.1815 36.4 36.2 -0.200001 36.334 -0.066 37.06058 0.6605816 35.9 36.4 0.5 36.5359 0.6359 36.6937 0.793717 36.2 35.9 -0.299999 36.00215 -0.19785 36.5126 0.312618 36.6 36.2 -0.399998 36.7091 0.1091 38.0633 1.463319 37 36.6 -0.400002 36.693 -0.307 36.0307 -0.969320 37.2 37 -0.200001 37.0427 -0.1573 37.9051 0.705121 37.4 37.2 -0.200001 37.235 -0.165 37.09121 -0.3087922 38 37.4 -0.599998 37.5423 -0.4577 38.39025 0.3902523 38.5 38 -0.5 38.1636 -0.3364 38.2898 -0.210224 39 38.5 -0.5 38.4509 -0.5491 38.4781 -0.521925 40 39 -1 39.007 -0.993 39.40834 -0.5916626

4040.035 40.1856

Graph Showing all the Measured and Forecasted speeds

Series 1. Measured 2. Persistence model3. AR Model 4. ARMA Model

Series 1. Persistence model 2. AR Model 3. ARMA Model

Illustrative Example

Statistical model for wind forecasting, for wind farm located in US [1]

Here, ARMA model is considered for wind forecasting.

Lake Benton 2 kw forecasts: Jan/Feb 2001.

Lake Benton 2 kw 1-hour forecasts vs. Actual: Jan/Feb 2001. ARMA(1,24).

Lake Benton 2 kw 2-hour forecasts vs. Actual: Jan/Feb 2001

ConclusionsThere is a clear difference in the ability of ARMA forecast

models applied to different time periods. In some cases, the model that does the best job forecasting

1-2 hours. In several cases, we found many alternative ARMA models

that did a good job forecasting over the testing time frame. It is apparent that a one-size-fits-all approach will not work.

Reference

M. Milligan, M. Schwartz, Y. Wan ‘Statistical Wind Power Forecasting Models: Results forU.S. Wind Farms’ WINDPOWER 2003 Austin, Texas May 18-21, 2003

Tony Burton, David Sharpe ‘Wind Energy Hand Book’Dr. Matthias Lange, Dr. Ulrich Focken ‘Physical Approach to

Short-Term Wind Power Prediction’