The use of piecewise linear models to predict hydroelectric load for Manitoba Hydro

Techniques for forecasting hourly load and monthly peak load have been developed and used at Manitoba Hydro for many years.One part of the methodology involves temperature-adjusting the historical load data.This process involves fitting piecewise iinear models to the relationship between temperature and load.In this practicum we investigate models with one or two knots, or turning points, with the locations of the knots being either predetermined or estimated from the data.If the knots are predetermined, then the standard theory for linear models can be utilized.However, if the knots are estimated from the data, then this involves fitting nonlinear models.In this latter case, iterative procedures such as "Dud" may be used, and approxi- mate confidence intervals and regions for unknown parameters may be constructed, based on linear approximations to the nonlinear problem.In the problems that are investigated it is shown that the approximate intervals and regions provide good ap- proximations to the likelihood intervals based on the profile sum of squares functions.This is due to the fact that there is a small degree of nonlinearity in the model-data set combinations.For the data sets considered, the two-knot model with the knots estimated from the data, appears to provide the best fit.It is also shown that the addition of variables such as day, year, and month increase the predictive power of the model' Fbture research in this area will focus on the utilization of these additional variables in random coefficient, piecewise linear models, in order to provide a better understanding of the influence of noise factors, such as temperature' on load forecasts' 67 67 68 68 69 ltl 4.3 4.4 4.5 4.6 4.2.3Linear vs. Nonlinear Models 7L The Profiie Likelihood Intervals Based on the Sum of Squares F\:nction 73 4.3.1The Piecewise Linear Model with one unknown Knot 73 4.3.2The Nonlinear Model With Two Knots 75 Advantages and Disadvantages of Linear Least Squares 78 4.4.L The Advantages .78 4.4.2The Disadvantages 78 Advantages and Disadvantages of Nonlinear Least Squares 80 4.5J The Advantages .80 4.5.2The Disadvantages 82 Conclusion 86 Modifying The Model 4.4 Sum of squares contours for two knot nonlinear model.74 77 81 VI 4.5 Proc NLIN output for BOD data.4.6 Sum of squares contours for the BOD data 4.7 Sum of squares function for BOD data for 91' 5.1 Plots of 01, 02, 0z and 0a each versus hour of day.5.2 Plot of residuals vs. temperature for piecewise linear fit with two un- known knots.5.3 Plot of residuals vs. day of the week for original piecewise linear model with two unknown knots.5.4 PROC NLIN output for Hydro data using nonlinear model with added day cornponent.5.5 Plot of residuals vs. day of the week after adjusting nonlinear model for day variable.5.6 Plot of residuals vs. month for our original piecewise linear modei with two unknown knots.5.7 PROC NLIN output for Hydro data using nonlinear model with added season component.5.8 Plot of residuals vs. month after adjusting nonlinear model for seasonal variable.98 5.9 Plot of residuals vs. year for our original nonlinear piecewise model with two unknown knots.99 5.10 PROC NLIN output for Hydro data using nonlinear model with added year component.100 5.11 Plot of residuals vs. year after adjusting nonlinear model for year variable.102vll S,ist of T'abtres 4.1 The extra sum of squares analysis for nested models' 4.2 The extra sum of squares analysis for the 4 and 5 parameter piecewise model fitted to the Manitoba Hydro data.4.J The extra sum of squares analysis for the 3 and 4 parameter piecewise model fitted to the Manitoba Hydro data. 4.4The extra sum of squares analysis for the 3 and 5 parameter piecewise model fitted to the Manitoba Hydro data.4.5 BOD versus time.5.1 0 versus hour of day.'5.2 The extra sum of squares analysis for the 5 and 8 parameter piecewise model fitted to the Manitoba Hydro data. 5.3The extra sum of Squares analysis for the 5 and 8 parameter piecewise model fitted to the Manitoba Hydro data. 5.4The extra sum of squares analysis for the 10 and 18 parameter piecewise model fitted to the Manitoba Hydro data.106 70 70 7L 97 4.1 Partiai data set collected by Manitoba Hydro vrlI Clnapten l- lntroductior-i.and tverview 1.1 FrearnbleFor electrical companies, such as Manitoba Hydro, it is very important to be able to accurateiy forecast the amount of power that they will need to produce at a particular point in time.Capacity in excess of that required by domestic customers is becoming increasingly more valuable, as excess capacity can be sold to other utilities.The value of excess capacity is dependent on when it is sold, and accurate prediction of anticipated demand is critical in maximizing the value of any excess.Effective cost-benefit analysis requires reliable assessments of the uncertainties associated with predictions of excess capacity, and achieving a good understarding of the influence of temperature and other noise factors on predictions is particularly crucial.Estimates of future load requirements affect many management decisions.Areas that are affected include capital investment in generation, transmission and distribution, purchasing decisions on fuel, and tariffs and revenue calculations [f0].Techniques for forecasting hourly load and monthly peak loads have been devel- oped and used at Manitoba Hydro for many years.During this time the models have undergone a number of changes, and the process has evolved.In the past the tradi-

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