(This article was first published on

**Commodity Stat Arb**, and kindly contributed to R-bloggers)I can’t believe it has been nearly 6 months since I last posted. Given the sustained heat it seemed like a good idea to finish off this subject.

As hinted at in my last post, temperature is the missing variable to make sense of Residential electrical usage. Fortunately there are some reasonable freely available historical weather databases, most notably the one provided by NOAA. As covered in the last posting, we can download this data for a particular location, in this case Chicago O’Hare airport, KORD.

Next we have to find a suitable model for the usage pattern we observed in the last posting. Usage is lowest at around 70F, and rises slightly as temperatures fall but rises significantly as temperatures rise.

Operationally we can imagine that as temperatures rise beyond what is comfortable, more and more cooling devices turn on to combat the heat. However, eventually all the air conditioning equipment is on at full capacity and incremental demand drops off. Based on this, a logistic function [http://en.wikipedia.org/wiki/Logistic_function] seems appropriate.

Furthermore, depending on the time of day, the high temperature and the low temperature may both be influential predictors.

Continuing from the code presented in the last posting, we can fit a logistic function in temperature to the usage data for each hour of the day, taking into account whether it is a weekday, the time of the year and the temperature function:

allResults=NULL # create the variable for results for each hour

startList=list(intercept=1.5,weekday=-.02,Time=0,S1=.04,C1=-.03,S2=-.02,

C2=-.04,S3=.001,C3=.01,maxT=87,maxS=4,maxScalar=1,

maxSlope=0,minSlope=0)

# initial variable value

for (hourInd in 1:24){ # iterate through hours; create a model for each hour

test2<-nls(ComEd[,(hourInd+1)] ~ intercept

+ timcols%*%c(weekday,Time,S1,C1,S2,C2,S3,C3)

+ maxSlope*KORDtemps[dateInd,”MAX”]

+ minSlope*KORDtemps[dateInd,”MIN”]

+ maxScalar*plogis(KORDtemps[dateInd,”MAX”],

location = maxT, scale = maxS),

start=startList,

control=nls.control(maxiter = 500, tol = 1e-05,

minFactor = 1/10000,printEval = FALSE, warnOnly = TRUE))

# using nonlinear least squares to find the best result

allResults=rbind(allResults,coef(test2)) # combine all results

startList=list(intercept=coef(test2)[1],weekday=coef(test2)[2],

Time=coef(test2)[3],S1=coef(test2)[4],

C1=coef(test2)[5],S2=coef(test2)[6],C2=coef(test2)[7],

S3=coef(test2)[8],C3=coef(test2)[9],maxT=coef(test2)[10],

maxS=coef(test2)[11],maxScalar=coef(test2)[12],

maxSlope=coef(test2)[13],minSlope=coef(test2)[14])

# update starting value for next iteration to solution from previous hour

# although not always necessary this reduces computational time

}

Then we can examine the regression results to see the average response function in usage based on Max temperature:

maxResponse = mean(allResults[,’maxSlope’]) * KORDtemps[dateInd,”MAX”]

+ mean(allResults[,’maxScalar’])*plogis(KORDtemps[dateInd,”MAX”],

location = mean(allResults[,’maxT’]),

scale = mean(allResults[,’maxS’]), log = FALSE)

plot(KORDtemps[dateInd,”MAX”],maxResponse)

In terms of fit we can revisit the chart from the previous blog posting by plotting the observed data for a particular hour, in this case midday, overlaid with the fitted data for the same inputs.

We can see that the fitted data (in black) overlay the actual data (in red) well but there are still differences for very hot days.

So to summarize we can model residential power usage pretty accurately to first order based on knowledge of the high temperature for the day as well as the time of the year, the day of the week and the hour of the day. As expected, as temperatures rise, usage grows dramatically over a high temperature of 80F.

Stay cool!

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