# Quantile Regression for Peak Demand Forecasting

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1 Quantile Regression for Peak Demand Forecasting Charlie Gibbons Ahmad Faruqui July 1, 2014 Copyright 2013 The Brattle Group, Inc.

2 Outline Approaches to forecasting peak demand Our empirical approach OLS versus QR Results 1 brattle.com

3 Forecasting peak demand In this presentation, we discuss methodologies for forecasting peak demand. Any proposed methodology should be evaluated by its accuracy. An accurate model is just as likely to over-forecast as it is to under-forecast peak demand in a given year. 2 brattle.com

4 Motivation for the study Annual peak demand is anomalous and the forecasting methodology must take this into account. Current approaches largely rely on OLS, which is best suited to forecasting average consumption and not peak demand: OLS with one data point for each annual peak OLS on a subset of relatively extreme days We propose Quantile Regression (QR) as a superior solution. We compare QR and OLS methods of the same functional form using 32 utilities in a meta-study. 3 brattle.com

5 Data ( ) We model Daily maximum hourly demand using all summer (May-September) days Using Monthly energy Maximum daily temperature (contemporaneous and lag) Minimum daily temperature (contemporaneous and lag) Month fixed effects Day of week fixed effects Dummy variable for weekday holiday Quarterly index of economic activity* Daily humidity* Monthly CDD and HDD* * when available 4 brattle.com

6 Approach Daily maximum demand is predicted separately by utility and the same functional form is used for all utilities. We estimate QR and OLS using data from (the in sample or estimation period). We evaluate the methods using the data (the out of sample period). 5 brattle.com

7 Decomposing maximum demand We can decompose maximum demand into two parts: Predictable demand We can use weather, sales, and other variables to predict maximum demand. Unpredictable shock There are idiosyncratic components of demand that are not predictable. 6 brattle.com

8 OLS versus QR The day establishing peak demand likely has large values for both of these components. Many days in the summer exhibit severe weather; the one with the highest demand will be the one with the highest demand shock. OLS can be used to create forecasts for days with large predictable demand. OLS cannot be used to create forecasts for days with large shocks. QR can be used to create forecasts for days with both large predictable demand and large shocks. An accurate peak demand model must incorporate both components. 7 brattle.com

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10 The questions that OLS and QR answer OLS QR For a given level of the predictors, what is the average level of demand that is to be expected? This model incorporates a shock of size 0 (the average). QR requires specifying a percentile of interest; 90%, say. For a given level of the predictors, what is the 90th percentile of demand that is to be expected? This model incorporates a 90th percentile shock. 9 brattle.com

11 Quantiles The quantile is estimated to provide accurate in-sample predictions of peak demand. The further the estimated quantile is from 50, the greater the difference between the QR and OLS models. 10 brattle.com

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13 Evaluation For each year and each utility, we compare: Highest level of hourly demand predicted by the model Highest level of hourly demand actually witnessed (non-coincident) We provide errors for two sets of predictions: In sample prediction Oracle forecasts These out-of-sample forecasts use the true values of the predictors. 12 brattle.com

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18 Performance summary Across utilities, QR: Has a mean out-of-sample prediction error of 0.05% compared to -4.82% for OLS Has a mean out-of-sample absolute prediction error of 3.05% compared to 5.28% for OLS, a reduction of 42% By utility, QR: Has a negative MPE for 15 of 32 utilities, while OLS has a negative MPE for 30 of 32 utilities Has a smaller MAPE than OLS for 24 out of 32 utilities Has a median MAPE that is 25% smaller than that for OLS 17 brattle.com

19 Results Peak demand is a function of predictable demand and unpredictable shocks. OLS only considers the predictable component of demand. Shocks are larger on annual peak days compared to non-peak days. QR incorporates the size of the shock term in addition to the predictable component. We see that The OLS model generally under-predicts peak demand. The QR model generally provides an accurate estimate of peak demand. These results generally hold across utilities. 18 brattle.com

20 Forecast intervals We also develop a bootstrapping method to produce forecast intervals for peak demand. 95% interval coverage In sample 95.1 Oracle brattle.com

21 Conclusions Peak demand days are differentiated from other high demand days by the presence of extreme shocks to demand. OLS does not account for this distinguishing feature of these days and the approach s predictions are biased downward. Focusing on days with high levels of predictors (rather than high levels of shocks) does not counteract this bias. Quantile regression can incorporate extreme shocks to better predict peak demand. 20 brattle.com

22 Presenter information CHARLIE GIBBONS Associate San Francisco, CA Dr. Charles Gibbons is an associate with The Brattle Group who specializes in applying sophisticated econometric and statistical models to legal, regulatory, and policy issues. He has produced analysis for a variety of litigation matters, including models of mortgage delinquencies, forensic analysis of product defects, and local impacts of environmental damages. His work has been used for class certification, liability determination, and damage calculations. He has also developed models for forecasting electricity sales and peak demand for an international utility. Additionally, Dr. Gibbons is a lecturer at the University of California, Berkeley and last taught a graduate-level course in probability and statistics. He received a Ph.D. in economics and an M.A. in statistics from the University of California, Berkeley. His dissertation proposed new methods in applied econometrics and a theory of competition for online advertising platforms. The views expressed in this presentation are strictly those of the presenter(s) and do not necessarily state or reflect the views of The Brattle Group, Inc. 21 brattle.com

24 About Brattle The Brattle Group provides consulting and expert testimony in economics, finance, and regulation to corporations, law firms, and governments around the world. We aim for the highest level of client service and quality in our industry. We are distinguished by our credibility and the clarity of our insights, which arise from the stature of our experts, affiliations with leading international academics and industry specialists, and thoughtful, timely, and transparent work. Our clients value our commitment to providing clear, independent results that withstand critical review. 23 brattle.com

25 Our practices PRACTICES Antitrust/Competition Commercial Damages Environmental Litigation and Regulation Forensic Economics Intellectual Property International Arbitration International Trade Product Liability Regulatory Finance and Accounting Risk Management Securities Tax Utility Regulatory Policy and Ratemaking Valuation INDUSTRIES Electric Power Financial Institutions Natural Gas Petroleum Pharmaceuticals, Medical Devices, and Biotechnology Telecommunications and Media Transportation 24 brattle.com

26 Offices NORTH AMERICA Cambridge New York San Francisco Washington, DC EUROPE London Madrid Rome 25 brattle.com

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