Residual Demand Modeling and Application to Electricity Pricing.

AuthorWagner, Andreas


Electricity price models, which use demand as a state variable are known as structural, hybrid or supply/demand models. In this class of models, the supply side (electricity production) and the demand side (electricity consumption) are described separately. (1) The market price of electricity is determined by the marginal production unit in the merit order. All models in the literature use demand as a state variable, which is the only one in the early work by Barlow (2002). More recent approaches further include capacity (cf. Cartea and Villaplana (2008)), fuel prices (cf. Pirrong and Jermakyan (2008), Coulon and Howison (2009), D'Aertrycke and Smeers (2010), Carmona, Coulon, and Schwarz (2011)), (2) or both (cf. Aid et al. (2009), Aid, Campi, and Langrene (2012)). Lyle and Elliott (2009) and Burger et al. (2004) also have a load dependent component in their model. Demand therefore plays a key role in those models and should be modeled carefully. A survey on this class of models including its benefits and obstacles as well as some extensions is given in Carmona and Coulon (2012).

In recent years there has been a rapid growth in installed capacity of renewable energy sources (hydro, wind, solar, (3) biomass, geothermal), which is expected to continue for decades. Half of the worldwide newly added capacity in 2010 has been of renewable technologies (including hydro). In the European Union, renewables account for more than 40% of yearly capacity additions since 2005. (4) Due to strong political support, especially Germany already has a considerable high share of wind and solar power plants in its electricity system. By September 2012, Germany had 29 GW installed capacity of wind power plants and 30 GW installed capacity of solar power plants, which together accounts for about 35% of total installed capacity. (5) In 2011, 20% of German electricity consumption has been produced by renewable sources. (6) Scenarios (7) for the year 2022 see renewable installed capacity in Germany between 93 GW and 150 GW, which is (much) more than the German yearly peak demand (in 2010 at about 83 GW). In all those scenarios, conventional installed capacity is predicted to decrease to between 82 GW and 92 GW.

This considerable amount of renewables sources in the system and the regulatory circumstances (see below) heavily influence the electricity price for Germany, which is traded at the EEX and EPEX. In Figure 1 an hourly production stack and the corresponding market prices are displayed. Supply is split into production from wind, production from solar and residual production, which is covered by technologies other than wind and solar, i.e. mainly conventional generation units like nuclear, lignite, coal, and gas. The prices follow the profile of residual production (which is determined by subtracting renewable infeed from demand), so we conclude that the intra-day price shape heavily depends on the amount of renewable infeed

In this paper we suggest a model which extends existing models in this direction, as this is not accounted for in existing supply/demand models in the literature. Moreover, the model components can be used in economic models for the generation of wind and solar production time-series. (8)

The effect of renewable infeed is also seen in time-series of hourly spot prices as plotted in Figure 2 from 2005 to 2011. There are less positive spikes since 2009 and they also reduced in magnitude. On the other hand, negative prices started to occur and especially in 2009 negative spikes are much more present than positive ones. In 2010 and 2011, the size of negative spikes reduced, indicating that producers are learning and trying to avoid negative prices (e.g. by assembling a more flexible power plant portfolio). The time series for 2011 in Figure 3 reveals that there are still some moderate spikes in the market, both positive and negative. The typical situation in case of negative spikes is a very high renewable infeed (usually for a few hours only, caused by fluctuating wind infeed) on days with generally low demand (public holiday, weekend). The electricity oversupply during those hours is caused by baseload plants, which cannot be economically switched off for a few hours only (there are also technical restrictions). They are prepared to accept negative prices for a short time period in order to be able to continue with their production. Positive spikes occur usually in times of high conventional demand. Germany has enough conventional capacity to meet peak demand even in times of zero renewable infeed (Bundesnetzagentur (2011)), but the conventional capacity is near its limit in those situations. Further causes of positive spikes are unexpected outages. The increasing share of solar infeed is reducing the (positive) spike risk. Solar produces mainly during hours of peak demand, so it reduces conventional peakload. This is the peak-shaving effect, which is also visible since 2010 in Figure 1. A third characteristic of spotprices seen in the figures is the seasonality on different time-scales. There is intra-day seasonality (Figure 1) weekly seasonality (Figure 3) and a yearly seasonality (Figure 2 and Figure 3).

Production from renewables introduces new uncertainty ( = risk) to the market due to their volatile production profile. Therefore in order to apply supply/demand models to the German power market (or any other market with a high share of renewables in its system), the infeed from wind and solar power plants should be considered, especially as the seasonalities in electricity prices are mainly generated by the seasonalities in residual production. As supply and demand must coincide at every time we use residual demand in the following, as we include the renewables on the demand side of the model.

The main contributions of this work are explicit models for wind and solar power infeed, which are used to refine existing supply/demand models for electricity. Moreover, within a simple structural framework, the residual demand model is applied to the German market. Our empirical work is also based on the German power market.

The regulatory circumstances in Germany are based on the EEG. (9) It implies that all production from renewables must be fed into the electricity grid. Conventional generation units (nuclear, lignite, hard coal, gas, oil, pumped storage) are to cover the residual demand only. As an incentive to investors, a guaranteed tariff for the produced electricity is paid for 20 years after the installation of the plant (feed-in tariff), so the renewable producer is not exposed at all to spot price risk. We will model wind and solar only, as they have a particularly uncertain and fluctuating infeed profile (unlike hydro or biomass). As most of the wind and solar power plants have been built in recent years they are eligible for the feed-in tariff and infeed priority and bid as must-run in the day-ahead auction (basically they bid at the lowest price allowed). Therefore we can consider the production from renewables as a demand reduction, i.e.

residual demand = (stochastic)total demand-(stochastic)infeed from renewables.

Residual (or conventional) demand is the electricity to be generated by conventional technologies. This is the demand which should be used in the merit order of conventional generation units to determine the market price. A common approach in demand modeling in the literature (i.e. the supply/demand models cited above) is to choose a deterministic seasonal component plus a stochastic process modeling random deviations from the seasonal level. They occur mainly due to weather conditions, i.e. an unusual cold spell in spring will cause a rise in demand for a few days due to electric heating. As those deviations are only temporary the stochastic process is meanreverting to level zero. Any trend or seasonality is contained in the deterministic component. As an example we introduce a model for total system load with the desired properties using an OrnsteinUhlenbeck process, which is used in Aid et al. (2009), Coulon and Howison (2009), and Lyle and Elliott (2009). We measure time continuously in years and denote it by t[member of] [0,T], where T>0 is some finite time horizon. All processes are defined on a probability space ([OMEGA],P,[??]) supporting Brownian motion with the filtration [??] generated by all the Brownian motions [W.sub.t] used in this paper.

Model 1.1 (Model for total system load)

Denote total system load at time t[member of] [0,T] by Lt and assume

[L.sub.t] = [[psi].sub.t] +[l.sub.t], (1)

where [[psi].sub.t] is a time-dependent deterministic load forecast, and [l.sub.t] is an Ornstein-Uhlenbeck process with mean-reversion speed [[theta].sup.load] >0 and volatility [[sigma].sup.load] >0, i.e.

[mathematical expression not reproducible] (2)

Model 1.1 is formulated in a rather general matter. It can also be applied to log-system load (10) to ensure that total system load is always positive (Coulon and Howison (2009), D'Aertrycke and Smeers (2010)). However, as the size of the seasonal component usually overweighs the stochastic fluctuations by far, the probability of negative values is negligible. An extension with time-dependent parameters is possible (e.g. seasonal volatility as in Cartea and Villaplana (2008)). It is important to note that demand is assumed to be price inelastic, i.e. [L.sub.t] depends not on the price for electricity. This assumption is common in the literature on supply/demand models. This is obviously a simplification, as in particular market coupling and the trade over interconnectors leads to part of demand to be price sensitive. However, this amount is so far considered negligible, but we expect it to become more important in the future. Nevertheless market coupling is not in the scope of this work, so we assume price-inelastic demand in the following.

The intra-day load pattern is very strong and possible deviations from...

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