A Threshold Autoregressive Model for Wholesale Electricity Prices

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Temperature (deg F) 80 60 40 20 0 Wi99 Su99 Fa99 Wi00 Sp00 Su00 Fa00 Wi01 Sp01 Su01 Fa01 Season of Year Figure 4: Temperature data corresponding to price data over the period Jan 1999 - Dec 2001. Log of Daily Max Prices ($/MwH) 3 4 5 6 7 • • •• • • • • • • • • • • • • • • • • • • • • • •• •• • • •• •• • •• •• • •• •• • • • • • • • • • •• • • •• •• •• • • • • • • • • • • •• • • • • • •• •• • • •• • • • • •• • • • • • • • •• • • • • • • •• • •• •• •• •• • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • •• • • •• • • •• • • • • • • • •• • • • • • • • • • • • • 0 20 40 60 80 Temperature (deg F) Figure 5: Log maximum prices vs. corresponding temperature. 6
¥ £ ¡ £ ¥ I 0 The temperature time series exhibits a clear periodic component, with higher temperatures associated with summer months. In order to illustrate the relationship between temperatures and log-electricity prices, we give a scatter-plot of log-price versus temperature in Figure 5. The figure shows a parabolic relationship between the log of the maximum prices and temperature. We use the loess function in S-Plus (see [18]) in order to obtain a smooth fit to the plot (the smoothing parameter was chosen to be 2/3). This technique fits polynomials locally so that nearby data points are most influential in the fit (see Cleveland et al. [5] for details on LOESS). Using this method of smoothing, we find that the minimizing temperature is about 57.0 degrees Fahrenheit. In light of this, we assume that the log-daily maximum price ¢¡¤£¦¥ includes a component of the §©¨£ form , for some positive § constant , £ where denotes the temperature on the -th day, and represents the minimizing temperature. ¨¡¤£¡£ Hence should include a difference of this component, that is, §©¨%©£&' (¨£!'©£ *) (1) £$# §©¨£! "§©¨£ 3 The Proposed Model The sample PACF plot in Figure 3 suggests that a low-order autoregressive model might be appropriate for the data. We use the discrete-time version of the model, that is an AR(1) model, but we generalize it in several ways. We introduce a time-varying mean, along with a time-varying threshold. The mean-reversion parameter is then allowed to take two different values - one when the process is above the threshold, and another when it is below. In a difference equation for logdaily price, we also include the term (1) to model the effect of temperature, as well as a term which mixes spikes into the process. 3.1 The Model Our discrete-time model for the daily log-electricity price series ,+£¦¥ is given by / 01¨%¡£ 324¡£657§©¨%©£! 98:£;5=£;5@?;A£B (2) ¡£.- E£65"¡£FB (3) +C£D- where (4) represents the volatility, §RP 01¨%GH- 0J*BKGMLONB BKGMPQN©B is a constant which determines the magnitude of the ?.P S temperature-effect, is the temperature at which price is typically £ minimal, is the temperature that corresponds to the maximum price recorded on the -th 8T£U-V£W©£ day, XA£¦¥ , is a sequence of independent and identically distributed Gaussian random variables with zero mean and unit variance, YE£¦¥ and represents the time-varying mean of the process, which we introduced 7
Page 1 and 2: A Threshold Autoregressive Model fo
Page 3 and 4: 1200 Dollars per MwH 1000 800 600 4
Page 5: Partial ACF 0.0 0.2 0.4 0 5 10 15 2
Page 9 and 10: 0 ¡ ¥ ) ) ) ) That is, we assi
Page 11 and 12: ¥ ¥ B B ¡ ¡ able forecasts are
Page 13 and 14: 8 8 7 7 Log Dollars per MwH 6 5 Log

A Threshold Autoregressive Model for Wholesale Electricity Prices

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