1. By N Venkata Srinath, MS Power Systems.

2. Statistical approach to wind forecasting Making 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. Ŷ k =Y k-1 Where Ŷ k is the Predicted value Y k-1 is 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.

3. 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:

4. This is known as an nth order autoregressive, mth order moving average model, or ARMA(n, m). The model parameters a i, b j can be estimated in various ways. A useful technique is the method of recursive least squares.

5. Assumed data InstantSpeed 1 33 2 33.5 3 33.8 4 34 5 34.2 6 34.6 7 34.2 8 34 9 34.5 10 35 11 35.4 12 35.8 InstantSpeed 13 36 14 36.2 15 36.4 16 35.9 17 36.2 18 36.6 19 37 20 37.2 21 37.4 22 38 23 38.5 24 39 25 40

6. Forecasted using persistence model InstantMeasured speedForecasted speedError 1 33 0 2 33.533-0.5 3 33.833.5-0.299999 4 3433.8-0.200001 5 34.234-0.200001 6 34.634.2-0.399998 7 34.234.60.399998 8 3434.20.200001 9 34.534-0.5 10 3534.5-0.5 11 35.435-0.400002 12 35.835.4-0.399998

7. InstantMeasured speedForecasted speedError 13 3635.8-0.200001 14 36.236-0.200001 15 36.436.2-0.200001 16 35.936.40.5 17 36.235.9-0.299999 18 36.636.2-0.399998 19 3736.6-0.400002 20 37.237-0.200001 21 37.437.2-0.200001 22 3837.4-0.599998 23 38.538-0.5 24 3938.5-0.5 25 4039

8. 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+33.8a3 34.2=33.5a1+33.8a2+34a3 34.6=33.8a1+34a2+34.2a3 34.2=34a1+34.2a2+34.6a3 34=34.2a1+34.6a2+34.2a3 34.5=34.6a1+34.2a2+34a3 Calculated using recursive least square (X’X) -1 X’Y. a1=0.5288; a2=-0.5725; a3=1.0501

9. InstantSpeedForecastedError 1 33 2 33.5 3 33.8 4 34 5 34.2 6 34.6 7 34.2 8 34 9 34.5 10 3534.5-0.5 11 35.434.982-0.418 12 35.835.379-0.421

10. InstantSpeedForecastedError 13 3635.835-0.165 14 36.236.0276-0.1724 15 36.436.334-0.066 16 35.936.53590.6359 17 36.236.00215-0.19785 18 36.636.70910.1091 19 3736.693-0.307 20 37.237.0427-0.1573 21 37.437.235-0.165 22 3837.5423-0.4577 23 38.538.1636-0.3364 24 3938.4509-0.5491 25 4039.007-0.993 26 40.035

11. 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=34.6 34a1+34.2a2+34.6a3+3b1=34.2 34.2a1+34.6a2+34.2a3+3b1=34 34.6 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

12. InstantSpeedForecastedError 1 33 2 33.5 3 33.8 4 34 5 34.22 6 34.63 7 34.23 8 344 9 34.53 10 3533.7477-1.2523 11 35.435.76770.3677 12 35.835.1368-0.6632

13. InstantSpeedForecastedError 13 3636.45440.4544 14 36.236.02-0.18 15 36.437.060580.66058 16 35.936.69370.7937 17 36.236.51260.3126 18 36.638.06331.4633 19 3736.0307-0.9693 20 37.237.90510.7051 21 37.437.09121-0.30879 22 3838.390250.39025 23 38.538.2898-0.2102 24 3938.4781-0.5219 25 4039.40834-0.59166 26 40.1856

14. Comparisons InstantMeasuredPersistence model ErrorAR ModelErrorARMAError 1 33 2 33.533 -0.5 3 33.833.5 -0.299999 4 3433.8 -0.200001 5 34.234 -0.200001 2 6 34.634.2 -0.399998 3 7 34.234.6 0.399998 3 8 3434.2 0.200001 4 9 34.534 -0.5 3 10 3534.5 -0.5 34.5-0.533.7477-1.2523 11 35.435 -0.400002 34.982-0.41835.76770.3677 12 35.835.4 -0.399998 35.379-0.42135.1368-0.6632

15. InstantMeasuredPersisten ce model ErrorAR ModelErrorARMAError 13 3635.8-0.200001 35.835-0.16536.45440.4544 14 36.236-0.200001 36.0276-0.172436.02-0.18 15 36.436.2-0.200001 36.334-0.06637.060580.66058 16 35.936.40.5 36.53590.635936.69370.7937 17 36.235.9-0.299999 36.00215-0.1978536.51260.3126 18 36.636.2-0.399998 36.70910.109138.06331.4633 19 3736.6-0.400002 36.693-0.30736.0307-0.9693 20 37.237-0.200001 37.0427-0.157337.90510.7051 21 37.437.2-0.200001 37.235-0.16537.09121-0.30879 22 3837.4-0.599998 37.5423-0.457738.390250.39025 23 38.538-0.5 38.1636-0.336438.2898-0.2102 24 3938.5-0.5 38.4509-0.549138.4781-0.5219 25 4039 39.007-0.99339.40834-0.59166 26 40 40.03540.1856

16. Graph Showing all the Measured and Forecasted speeds Series 1. Measured2. Persistence model 3. AR Model 4. ARMA Model

17. Series1. Persistence model2. AR Model3. ARMA Model

18. Illustrative Example Statistical model for wind forecasting, for wind farm located in US [1] Here, ARMA model is considered for wind forecasting.

19. Lake Benton 2 kw forecasts: Jan/Feb 2001.

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

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

22. Conclusions There 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.

23. 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’