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ARMA Model for Wind Forecasting

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Title: ARMA Model for Wind Forecasting


1
ARMA Model for Wind Forecasting
  • 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.
  • 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.

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

5
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
6
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
7
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
8
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

9
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
10
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
11
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

12
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
13
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
14
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
15
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
16
Graph Showing all the Measured and Forecasted
speeds
Series 1. Measured 2. Persistence model 3.
AR Model 4. ARMA Model
17
Series 1. Persistence model 2. AR Model 3. 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
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