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Regression-based Approachfor Calculating CBL

Dr. Sunil MaheshwariDominion Virginia Power

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Benefits of Regression Approach

• This approach can be used to calculate CBL for both weather-sensitive and non-weather-sensitive loads.

• The science of regression theory is well developed.

• Most statistical packages such as SAS, STATA, SPSS, etc. can perform regression analysis.

• Regression equations can be easily updated on a periodic basis (perhaps annually).

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Description of Regression Approach

• The idea is to treat load as a function of explanatory factors such as weather, time of day, day of the week, etc.

• Estimate the relationship between load and explanatory variables using a variety of functional forms.

• Pick the functional form that gives the highest R-sq adjusted or the lowest Root Mean Squared Error (RMSE)

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Functional Forms for Weather-Sensitive Loads

Form 1: Load = a + b*CDD + c*HDD

Form 2: Load = a + Σ (bi * CDDi) + Σ (cj * HDDj);

Temperature breakpoints to be established based on Regression Analysis

Form 3: Load = a + Σ (bi * CDDi) + Σ (cj * HDDj) + Σ (dk * hourk);

In addition to weather, each hour impacts the load as well.

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Functional Forms for Non-Weather Sensitive Loads

• The following forms may do a good job of estimating CBL for Industrial loads:

Form 4: Load = a + Σ (bk * hourk) + Σ (cj * monthj);

Form 5: Load = a + b * TimeTrend + Σ (ck * hourk) + Σ (dj * monthj);

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Applying Theory into Practice…

• For one of our DSR participants (a Building Complex), we estimated the relationship between 2006 hourly Load and Weather using Functional Form 3:

Load = a + Σ (bi * CDDi) + Σ (cj * HDDj) + Σ (dk * hourk);

• Following temperature breakpoints were used:

– For Heating Degree Days (HDD) – 65, 55, 40, 25

– For Cooling Degree Days (CDD) – 65, 80, 90, 100

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• We further sliced the data by:– Day type

• Weekdays

• Weekends and Holidays

– Season

• Winter – December - March

• Summer – June - September

• Shoulder – April, May, October, November

Applying Theory into Practice…

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Partial Regression Output (Summer, Weekday)

regress load cdd_65to80 cdd_80to90 cdd_90to100 cdd_over100 hdd hddsq hour2-hour24 if year==2006 & weekdayflag==1 & holiday==0 & season=="Summer"

Source | SS df MS Number of obs = 2040-------------+------------------------------ F( 28, 2011) = 518.86 Model | 339127510 28 12111696.8 Prob > F = 0.0000 Residual | 46942715.2 2011 23342.9713 R-squared = 0.8784-------------+------------------------------ Adj R-squared = 0.8767 Total | 386070225 2039 189342.925 Root MSE = 152.78------------------------------------------------------------------------------ load | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- cdd_65to80 | 33.08646 .9634143 34.34 0.000 31.19707 34.97586 cdd_80to90 | 31.6024 .5931229 53.28 0.000 30.4392 32.7656 cdd_90to100 | 32.58845 .68489 47.58 0.000 31.24528 33.93162 cdd_over100 | (dropped) hdd | -104.137 6.194134 -16.81 0.000 -116.2846 -91.98941 hddsq | 4.866126 .6342737 7.67 0.000 3.622224 6.110028 hour2 | -33.26744 23.44299 -1.42 0.156 -79.24253 12.70765 hour3 | 19.0973 23.452 0.81 0.416 -26.89547 65.09006 hour4 | 25.50128 23.46632 1.09 0.277 -20.51955 71.52211

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Predicted Load (CBL) based on 2006 data applied to 2007 data

• Using regression parameters from previous slide, predict the load for 2007.

• Compare predicted load (CBL) to actual load.

• Absolute average % deviation between Actual and Predicted Load was less than 5%.

• Regression Equations will be re-estimated every year

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Actual vs Predicted (CBL) - 2007

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1000

2000

3000

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Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

kW

Time

January - November 2007

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Actual Load vs Predicted (CBL) - Summer, Weekday(Absolute Average % Deviation = 3%)

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500

1000

1500

2000

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62 67 71 72 73 74 76 77 78 78 79 80 80 82 83 83 84

Temperature

KW

H

Hourly Predicted Load (CBL) Averaged by Day

Hourly Actual Load Averaged by Day

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Actual Load vs Predicted (CBL) - Winter, Weekday(Absolute Average % Deviation = 3.6%)

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500

1,000

1,500

2,000

2,500

3,000

3,500

4,000

4,500

5,000

17 23 27 30 32 35 37 41 44 45 52 56 63

Temperature

KW

H

Hourly Predicted Load (CBL) Averaged by Day

Hourly Actual Load Averaged by Day

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Actual Load vs Predicted (CBL) - Shoulder, Weekday(Absolute Average % Deviation = 3.7%)

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500

1000

1500

2000

2500

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41 44 46 50 51 53 56 58 60 62 66 67 69 71 72 74 76

Temperature

KW

H

Hourly Predicted Load (CBL) Averaged by Day

Hourly Actual Load Averaged by Day

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Actual Load vs Predicted (CBL) - Weekend/Holiday(Absolute Average % Deviation = 4.7%)

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500

1000

1500

2000

2500

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3500

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4500

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27 29 34 38 45 47 50 54 55 60 62 65 67 71 74 75 76 77 78 80 82

Temperature

KW

H

Hourly Predicted Load (CBL) Averaged by Day

Hourly Actual Load Averaged by Day

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Conclusions

• 4 Equations with single variable hourly temperature

– Summer, Weekday

– Winter, Weekday

– Shoulder, Weekday

– Weekends / Holidays

• Good fit (R-sq adjusted > 78% in all cases).

• Simplified calculations, and the regression equations can be easily updated.