# [MPD, Tu 9:00] Multivariate Non-Standard Modelling Approaches

## 1.Time-Series Modelling of Power Output for Large-Scale Wind Fleets (Sturt, Strbac)

The operating costs of power systems with high wind penetration are strongly dependent on the statistical properties of the wind output. The most important of these properties are: the asymptotic distribution, which describes the probability of each output level over the long term; the distribution of short-term output changes (particularly up to around four hours, which is a typical start-up time for a steam turbine unit); diurnal and seasonal variations; and the forecast accuracy. Many costing studies use historical wind time series directly within power system scheduling simulations, in order to account for these properties in a natural way. However, there are often advantages in fitting a stochastic time-series model to the data and using the model to drive a Monte Carlo simulation. Monte Carlo methods allow us to quantify standard errors and reduce them with variance reduction techniques. Furthermore, characterising the statistics of the wind output allows us to generate forecasts for scheduling or pricing algorithms, and we can also manipulate the model parameters to represent the smoothing effect of increased diversity.

Most wind power modelling studies create models of wind *speeds* at individual sites, and convert them to power outputs using turbine power curves. In order to model the combined output from several sites, such methods must use multivariate models or create a representative windspeed for the whole area. In this study, we modelled the wind *power output* directly. At an individual site, the output is prone to large, sudden changes due to turbine cutout, which can lead to difficulties in parameterising a wind power model. However, when output is combined from a diverse group of sites, the dynamics are smoother so that it becomes easier to model the aggregate output.

We used wind power datasets from several sites across New Zealand and the United Kingdom, to create univariate autoregressive models representing both single-site and nationwide wind outputs. Diurnal variations and the long-term distribution were accounted for by fitting a periodic additive term and a transformation function, while the model parameters were adjusted to fit the transitional properties. We aimed to use models of as low an order as possible, in order to facilitate the generation of scenario trees within a stochastic scheduler. We found that, for the New Zealand case, there was no significant trending (autocorrelation of increments) apparent in the data, and a first-order autoregressive model provided a good fit to both long-term and transitional statistics. For the United Kingdom case, there was a small amount of trending in the data. This means that a higher-order model would be preferable if we needed to capture the probability of large output changes over both thirty-minute and four-hour horizons using the same parameter set.

## 2.Dynamic Behavior of CO2 Spot Prices – Modeling Multiple Trading Periods (Hitzemann, Uhrig-Homburg)

For traders and CO_{2} emitting companies it becomes increasingly important to have a valid CO_{2} spot price model in order to value derivatives or assess production costs and support emissions-related investment decisions. Seifert et al. (2008) have developed a stochastic equilibrium model depending on expected cumulative emissions and maturity for a single trading period. We extend this model, according to the current setting of the EU ETS, to a multi-period model accounting for inter-period banking and later delivery of lacking certificates.

Then we compare the spot price functions for a setting of only one trading period and settings with more than one trading period. In a single trading period setting, the lower price bound is zero and an upper price bound in the amount of the penalty costs exists. For very low cumulative expected emissions, the price is close to the lower bound, for very high emissions it is close to the upper bound. In between, there is a transition between both bounds. When time approaches the end of the trading period, this transition area becomes smaller and smaller and ends up in a discontinuity at the period end, where only two different spot prices are possible.

In a setting with multiple trading periods, at first there exists an additional spread for every additional consecutive trading period. That means that the upper bounds for the different settings differ, but the lower bound is still the same because for very low emissions it is not likely to pay penalty costs for any of the trading periods. Similarly to the single trading period setting, the steepness of the transition area between these two bounds increases when time passes, but at the end of the trading period a broader range of prices is possible. This is because the spot price still contains a value coming from the following trading periods. If we consider volatility surfaces for the different settings, it can be seen that the volatility surface becomes more moderate when adding consecutive trading periods.

Altogether, the inter-period incentive for emissions abatement given by the allowance of banking affects the CO_{2} spot price in a way that its dynamics and volatility become somewhat more moderate and uniform over the whole trading period.

## 3.Modeling Energy American Options in the Framework of the Non-Markovian Approach (Kholodnyi)

We present and further develop the non-Markovian approach to modeling energy spot prices with spikes proposed earlier by the author. In contrast to other approaches, we model energy spot prices with spikes as a non-Markovian stochastic process that allows for modeling spikes directly as self-reversing jumps.

While the stochastic process we propose is non-Markovian, it can be represented as a product of two Markov processes, the spike process and the inter-spike process. The spike process is responsible for modelling spikes in the spot prices while the inter-spike process is responsible for modelling the spot prices between spikes. In this regard, although the stochastic process we propose is non-Markovian, it can be represented as a Markov process with a suitably extended state space that, in addition to the spot price, also includes the magnitude of spikes.

This allows for the analytical valuation of European options on energy spots with spikes as well as for the analytical valuation and dynamic hedging of European options on energy forwards and swaps for energy spots with spikes.

This also allows for a relatively simple and computationally efficient valuation of American options on energy spots, forwards and swaps. The valuation is based on the semilinear evolution equation for American options in the entire domain of the state variables introduced earlier by the author. The nonlinear term in this semilinear equation can be financially interpreted as a cash flow that should be received to compensate for the losses due to holding an American option unexercised in the exercise region. Additionally, the valuation is based on the multi-layered tree method also introduced earlier by the author. Different layers in this multi-layered tree corresponds to different levels of the magnitude of spikes while the tree structure at each layer corresponds to a suitable diffusion process for the inter-spike spot prices.

We consider a practically important example of the crude oil American options. We show that the extracted risk-neutral probability distributions, among other things, allowed to conclude as early as at the end of March 2008 that the crude oil prices were at an upward spike and, due to the detected possible downward spikes, were highly likely to fall to the levels below $40 dollars by the end of 2008, well before they even picked later that summer.