ARMA Model for Wind Forecasting

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ARMA Model for Wind Forecasting

• By
• N Venkata Srinath,
• MS Power Systems.

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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.
• Yk Yk-1
• Where
• Yk is the Predicted value
• Yk-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.

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

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Assumed data
Instant Speed
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
Instant Speed
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
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Forecasted using persistence model
Instant Measured speed Forecasted speed Error
1 33 33 0
2 33.5 33 -0.5
3 33.8 33.5 -0.299999
4 34 33.8 -0.200001
5 34.2 34 -0.200001
6 34.6 34.2 -0.399998
7 34.2 34.6 0.399998
8 34 34.2 0.200001
9 34.5 34 -0.5
10 35 34.5 -0.5
11 35.4 35 -0.400002
12 35.8 35.4 -0.399998
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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
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Forecasting using a linear combination - AR(n)

• Here, the measured data of 1-8 instance is used
  to train the model.
• 4-9 speeds are expressed as a linear equations.
• The considered order is n3.
• 3433a133.5a233.8a3
• 34.233.5a133.8a234a3
• 34.633.8a134a234.2a3
• 34.234a134.2a234.6a3
• 3434.2a134.6a234.2a3
• 34.534.6a134.2a234a3
• Calculated using recursive least square
  (XX)-1XY.
• a10.5288 a2-0.5725 a31.0501

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Instant Speed Forecasted Error
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 34.5 -0.5
11 35.4 34.982 -0.418
12 35.8 35.379 -0.421
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Instant Speed Forecasted Error
13 36 35.835 -0.165
14 36.2 36.0276 -0.1724
15 36.4 36.334 -0.066
16 35.9 36.5359 0.6359
17 36.2 36.00215 -0.19785
18 36.6 36.7091 0.1091
19 37 36.693 -0.307
20 37.2 37.0427 -0.1573
21 37.4 37.235 -0.165
22 38 37.5423 -0.4577
23 38.5 38.1636 -0.3364
24 39 38.4509 -0.5491
25 40 39.007 -0.993
26 40.035
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Forecasted using ARMA(n , m)

• The order is n3, m1 i.e., ARMA(3,1).
• Expressing the data from the instant 3-10 as a
  linear combination
• 33.8a134a234.2a32b134.6
• 34a134.2a234.6a33b134.2
• 34.2a134.6a234.2a33b134
• 34.6 a134.2a234a34b134.5
• 34.2a134a234.5a33b135
• a12.0338 a2-1.3898 a30.4246 b1 -0.6826

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Instant Speed Forecasted Error
1 33
2 33.5
3 33.8
4 34
5 34.2 2
6 34.6 3
7 34.2 3
8 34 4
9 34.5 3
10 35 33.7477 -1.2523
11 35.4 35.7677 0.3677
12 35.8 35.1368 -0.6632
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Instant Speed Forecasted Error
13 36 36.4544 0.4544
14 36.2 36.02 -0.18
15 36.4 37.06058 0.66058
16 35.9 36.6937 0.7937
17 36.2 36.5126 0.3126
18 36.6 38.0633 1.4633
19 37 36.0307 -0.9693
20 37.2 37.9051 0.7051
21 37.4 37.09121 -0.30879
22 38 38.39025 0.39025
23 38.5 38.2898 -0.2102
24 39 38.4781 -0.5219
25 40 39.40834 -0.59166
26 40.1856
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Comparisons
Instant Measured Persistence model Error AR Model Error ARMA Error
1 33
2 33.5 33 -0.5
3 33.8 33.5 -0.299999
4 34 33.8 -0.200001
5 34.2 34 -0.200001 2
6 34.6 34.2 -0.399998 3
7 34.2 34.6 0.399998 3
8 34 34.2 0.200001 4
9 34.5 34 -0.5 3
10 35 34.5 -0.5 34.5 -0.5 33.7477 -1.2523
11 35.4 35 -0.400002 34.982 -0.418 35.7677 0.3677
12 35.8 35.4 -0.399998 35.379 -0.421 35.1368 -0.6632
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Instant Measured Persistence model Error AR Model Error ARMA Error
13 36 35.8 -0.200001 35.835 -0.165 36.4544 0.4544
14 36.2 36 -0.200001 36.0276 -0.1724 36.02 -0.18
15 36.4 36.2 -0.200001 36.334 -0.066 37.06058 0.66058
16 35.9 36.4 0.5 36.5359 0.6359 36.6937 0.7937
17 36.2 35.9 -0.299999 36.00215 -0.19785 36.5126 0.3126
18 36.6 36.2 -0.399998 36.7091 0.1091 38.0633 1.4633
19 37 36.6 -0.400002 36.693 -0.307 36.0307 -0.9693
20 37.2 37 -0.200001 37.0427 -0.1573 37.9051 0.7051
21 37.4 37.2 -0.200001 37.235 -0.165 37.09121 -0.30879
22 38 37.4 -0.599998 37.5423 -0.4577 38.39025 0.39025
23 38.5 38 -0.5 38.1636 -0.3364 38.2898 -0.2102
24 39 38.5 -0.5 38.4509 -0.5491 38.4781 -0.5219
25 40 39 -1 39.007 -0.993 39.40834 -0.59166
26 40 40.035 40.1856
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Graph Showing all the Measured and Forecasted
speeds
Series 1. Measured 2. Persistence model 3.
AR Model 4. ARMA Model
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Series 1. Persistence model 2. AR Model 3. ARMA
Model
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Illustrative Example

• Statistical model for wind forecasting, for wind
  farm located in US 1
• Here, ARMA model is considered for wind
  forecasting.

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

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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