Modeling Electricity Prices with Regime Switching Models

We address the issue of modeling spot electricity prices with regime switching models. After reviewing the stylized facts about power markets we propose and fit various models to spot prices from the Nordic power exchange. Afterwards we assess their performance by comparing simulated and market prices. 1 Electricity Spot Prices: Markets and Models The deregulation of the power industry has given way to a global trend toward the commoditization of electric energy. Electricity has transformed from a primarily technical business, to one in which the product is treated in much the same way as any other commodity, with trading and risk management as key tools to run a successful business [2,12,15]. However, we have to bear in mind that electricity is a very unique commodity. It cannot be economically stored, demand of end users is largely weather dependent, and the reliability of the transmission grid is far from being perfect. This calls for adequate models of price dynamics capturing the main characteristics of spot electricity prices. The spot electricity market is actually a day-ahead market. A classical spot market would not be possible, since the system operator needs advanced notice to verify that the schedule is feasible and lies within transmission constraints. The spot is an hourly (in some markets – a daily) contract with physical delivery. In our analysis we use spot prices from the Nordic power exchange (Nord Pool) covering the period January 1, 1997 – April 25, 2000. The system price is calculated as the equilibrium point for the aggregated supply and demand curves and for each of the 24 hours [14]. Due to limited space, in this paper we restrict the analysis to average daily prices. The averaged time series, however, retains the typical characteristics of electricity prices, including seasonality (on the annual and weekly level), mean reversion and jumps [20,21]. The seasonal character of electricity spot prices is a direct consequence of the fluctuations in demand. These mostly arise due to changing climate conditions, M. Bubak et al. (Eds.): ICCS 2004, LNCS 3039, pp. 859–867, 2004. c © Springer-Verlag Berlin Heidelberg 2004 860 M. Bierbrauer, S. Trück, and R. Weron like temperature and the number of daylight hours. In the analyzed period the annual cycle can be quite well approximated by a sinusoid with a linear trend [20,21]. The weekly periodicity is not sinusoidal, though, with peaks during the weekdays and troughs over the weekends. Spot electricity prices are also regarded as mean reverting – for time intervals ranging from a day to almost four years the Hurst exponent is significantly lower than 0.5 [18,19]. In addition to seasonality and mean reversion, spot electricity prices exhibit infrequent, but large jumps caused by extreme load fluctuations (due to severe weather conditions, generation outages, transmission failures, etc.). The spot price can increase tenfold during a single hour but the spikes are normally quite short-lived [2,12, 15,21]. Now, that we have discussed the properties of spot electricity prices we can turn to modeling issues. The starting point is the analysis of seasonal components. On the annual level this can be done through approximation by sinusoidal functions [15,21], fitting a piecewise constant function of a one year period [1, 13] or wavelet decomposition [18]. On the weekly (or daily) time scale, the seasonality is usually removed by subtracting an average week (or day) from the data. Once the seasonal components are removed we are left with the stochastic part of the process. In what follows we will analyze the logarithm dt of the deseasonalized average daily spot prices Dt, see the bottom panel in Figure 1. For details on obtaining dt from raw data see [20,21]. The stochastic part dt can be modeled by a diffusion-type stochastic differential equation (SDE) of the form: dXt = μ(X, t)dt+ σ(X, t)dBt, which is the standard model for price processes of stochastic nature. Mean reversion is typically induced into the model by having a drift term μ(X, t) that is negative if the spot price is higher than the mean reversion level and positive if it is lower, like in the arithmetic Ornstein-Uhlenbeck process: dXt = (α− βXt)dt+ σdBt = β(L−Xt)dt+ σdBt, (1) where μ(X, t) = (α − βXt) is the drift, σ(X, t) = σ is the volatility and dBt are the increments of a standard Brownian motion. This is a one-factor model that reverts to the mean L = αβ with β being the magnitude of the speed of adjustment. The equilibrium level L can be also made time dependent to reflect the fact that electricity prices tend to revert to different levels over the year. The second main feature of electricity spot prices, the ”jumpy” character, calls for spot price modeling which is not continuous. One approach is to introduce to eqn. (1) a jump component Jtdqt, where Jt is a random jump size and qt is a Poisson variate [2,10]. After a spike the price is forced back to its normal level by the mean reversion mechanism or mean reversion coupled with downward jumps. Alternatively, a positive jump may be always followed by a negative jump of the same size to capture the rapid decline – especially on the daily level – of electricity prices after a spike [20,21]. Since spot prices after a jump tend to remain high for several time periods (hours, sometimes even days) there is also need for models that are able to capture this behavior. The so-called regime switching models offer such a possibility and be discussed in the next section. Modeling Electricity Prices with Regime Switching Models 861 0 200 400 600 80