Valuing Power Plants Under Emission Reduction Regulations and Investing in New Technologies: an Exchange Option on Real Options

University of Oxford

Kellogg College

Master of Science in Mathematical Finance

Valuing power plants under emission reduction regulations and investing in new technologies: An exchange option on real options

d-ﬁne GmbH1

Supervisor: Professor Sam Howison2

April 2010

1d-ﬁne GmbH, Opernplatz 2, 60313 Frankfurt, Germany (info@d-ﬁne.de) 2University of Oxford, Mathematical Institute, [email protected] Abstract

In this dissertation we model the value of a power generation asset through a real option approach. With electricity, fuel and emission allowances we express every essential uncertainty on the energy market by an own stochastic process and derive an optimal clean spark spread. Typical operational constraints of a power plant are taken into account. Beside analysing the behaviour of the generation asset under diﬀerent constraints, we want to evaluate the option to invest in new technologies to improve these constraints. In this dissertation, we do not set up the standard American option with strike equal to the investment as usual, but set up an exchange option on two real options with diﬀerent constraints. We show that this approach handles an option on new technology much more sensitive to the individual price uncertainties and considers all possible employments. If the intrinsic value of the exchange option exceeds the realization costs, it is time to invest. We also state an explicit Monte Carlo algorithm and present numerical results for the option to install a Carbon Capture and Storage unit. 1

Contents

1 Introduction 2

2 Modelling power plants 5 2.1 The clean spark spread ...... 5 2.2 Energy markets ...... 6 2.2.1 Electricity prices ...... 6 2.2.2 Fuel prices ...... 8 2.2.3 Emission allowance prices ...... 11 2.2.4 Correlations ...... 14 2.3 Operational constraints ...... 14 2.3.1 Heat rate, emission rate and optimal generation level . . . 15 2.3.2 Ramp rates and start up costs ...... 16 2.3.3 Minimum up time, minimum down time and cold time . . 17 2.3.4 Variable operational and maintenance costs ...... 18 2.3.5 Forced and scheduled outages ...... 18

3 A Real option approach for valuing power plants 20 3.1 The real option approach ...... 20 3.2 Real option valuation using dynamic programming ...... 20 3.2.1 Backward induction ...... 20 3.2.2 Least squares Monte Carlo ...... 22

4 Implementation aspects 26 4.1 Monte Carlo simulation ...... 26 4.2 Parameter calibration ...... 27 4.2.1 Electricity prices ...... 27 4.2.2 Fuel prices ...... 27 4.2.3 Emission allowance prices ...... 29 4.2.4 Correlations ...... 30 4.3 The algorithm ...... 32 4.4 Example valuations ...... 33

5 Investing in new technologies 37 5.1 An American style exchange option on real options ...... 37 5.2 Analytical valuation of American options ...... 38 5.3 Numeric valuation ...... 39 5.4 Examples ...... 40

6 Conclusions 44 2

1 Introduction

The world demand for energy is constantly rising. Nowadays, especially for new emerging economies like India, China and Brazil, energy is the basis for economic growth and wealth. In contrast, it is well known that the fossil energy sources on earth are limited and the accessible coal stocks, oil and gas sources will last for few more decades only. Additionally, pollution and global climate change becomes a big issue around the world with the power industry as the biggest pol- luter. Do these aspects put fuel-ﬁred power plants to the stack of old technologies?

A first argument against this hypothesis is, that by 2015 a generation of nuclear power plants in European nations will have to be shut down because they are becoming old and unsafe3 and no new ones could be build well-timed for years after that4. Secondly, alternative resources like Uranium will last for 83 years at the current rate of consumption5. This is shorter than other fossil energy resource like coal with its current reserves-to-production ratio of 137 years according to IEO20096. Thirdly, renewables are not reliable and even the latest technology improvements are far away from generating the needed huge amount of electricity. And last but not least, there are several technology improvements in efficiency, flexibility and emission reduction for coal and gas power plants. Thus, the major part of electricity produced worldwide currently comes and will come from fuel- driven power plants for the next decades, which makes them a desirable object for investors. For example, in the IEO2009 reference case, world coal consumption increases by 49% from 2006 to 2030.

For an investment in existing or new physical assets it is necessary to evaluate the proper value of the object. In this thesis we want to estimate the value of fuel driven power plants as electricity generation assets which consuming fuel and producing emissions. We do this by simulating electricity, fuel and emission allowances as uncertainties on the energy market, each through a selected stochastic process and derive the so called clean spark spread, the margin between these three commodities.

For valuation we use a real option approach. An alternative and common method is called Discounted Cash Flows (DCF) which sums the expected, discounted future cash ﬂows to estimate the present value. There are three mainly disadvan-

3ARD, www.tagesschau.de/inland/meldung1516.html Standorte und Laufzeiten deutscher Atomkraftwerke, 2004 4The Economist, How long till the lights go out?, Aug 6th 2009 5International Atomic Energy Agency, Nuclear Technology Report 2009, 2009 6Energy Information Administration, International Energy Outlook 2009, May 2009 3 tages of the DCF method. Firstly, the estimated value of the cash flows may be difficult to assess for distant years. Secondly, it is difficult to assess right discount factors including the risk aversion of investors and thirdly, it does not include technical properties of the facility or operational irregularity. For more details on DCF and its merits and limits see for example [Geman05]. Differently to DCF, the real option approach can captures typical operational constraints of a power plant. Also the operational, irreversible decisions a plant operator will have made are considered by setting the asset to different states. One more advantage of a real option approach is, that it captures the impact of price volatility to the value of a power plant more realistically. Keeping in mind that a modern peaker - a very flexible generation unit - has very high ramp rates, the operator has the option to adjust production over very short time periods to face price movements in a volatile market. Surely, a real option approach also has its difficulties. An appropriate stochastic process can be measured only if there are liquid markets. We will come back to this issue in section 4.2.

One special goal in this dissertation is to introduce the additional uncertainty emission allowance prices into the clean spark spread by a separate stochastic process and observe its affect to power plant values. Our motivation is that climate change due to industry emission has become a severe political issue because it affects the environment of the whole planet and is thought to be responsible for natural disasters. There are new regulations from advisors to meet their promises to reduce emission made in the Kyoto Protocol. The Kyoto protocol takes care of six different greenhouse gases: carbon dioxide CO2, methane CH4, nitrous ox- ide N2O, sulphur hexafluoride SF6, hydrofluorocarbons H − F KW/HF Cs and perfluorocarbons F KW/P F Cs. On the one hand, emissions of most of these gases are allowed under restrictions and taxation and the costs per emitted unit is rather deterministic. Thus we will capture these costs in the deterministic operational and maintaining costs. One the other hand, CO2 is part of the Kyoto emissions trading flexible mechanism and thus emission allowance prices depend on supply and demand. In the European Union, a very important instrument is the European Union Emission Trading Scheme (EU-ETS) where CO2 emission allowances can be traded between emission producers and emission reducers or other counterparties. Here we focus on this market. A short introduction to the EU-ETS is given in the Appendix.

Furthermore, we want to evaluate the option to invest into emission reduction technologies like a Carbon Capture and Storage (CCS) unit. Small pilot plants have been built with these new technologies but nobody knows whether or when it will be needed to install CCS in big projects. To handle this issue we do not 4 follow former articles on this topic which go from the costs point of view and try to identify building costs and average savings and model an American option. We come from the proﬁt point of view and use our real option model to value the actual returns including the new technology. This approach handles a new technology much more sensitive and consider all its possible employments than standard American options. In fact, a technology investment is originally nothing else than an exchange of operational parameters and thus can be modelled as an exchange option on two real options with diﬀerent constraints.

The thesis is organised as follows: we start by setting up the model to value a general fuel-driven generation asset. The clean spark spread is defined and for all uncertainties proper stochastic processes are identified. Then, in section 2.3 we introduce the treatment of a number of general constraints on generation assets, which leads us to a complex multi-state problem. In section 3 we explain the real option approach to model this multi-state problem. To solve it the dynamic programming technique is introduced in section 3.2 Then, we use Least Square Monte Carlo methods firstly introduced by Longstaff and Schwartz to find a solution. Numerical and implementation aspects are showed in section 4 and some analyses on assets with typical constraints are illustrated together. Going one step further, section 5 describes the valuation of an investment option on technological improvements or upgrades that influence model parameters. According to this American style option, price and volatility boundaries are identified to catch the optimal exercise time. We close the discussion by giving an outlook of further analysis and investigations.7

7Special thanks to Tilman Huhne and Yuri Ivanov for their useful comments. 5

2 Modelling power plants

In the following section we will set up a model to value a general fuel ﬁred power plant including its operational constraints.

2.1 The clean spark spread A fuel-fired power plant converts a particular fuel like coal, gas or biomass into electricity. This conversion involves at least two marketed commodities, fuel and electricity. Thus, the value of a fuel fired power plant V strongly depends on the so called spark spread Π, defined as the difference between the marked price of electricity P E (US$/MWh) and the price of gas, coal or biomass P F (US$/mmBtu) used for the generation of electricity. It also depends on the two prices individ- ually because a power plant needs starting fuel or consumes electricity by itself. We will come back to these issues in section 2.3. In this thesis, the amount of natural gas is measured in million British Thermal Units (mmBtu). The standard unit for coal or biomass is tonne but can be converted easily to mmBtu. For example, one tonne of anthracite coal is equivalent to 25.09 mBtu8. Sometimes, in literature the spark spread for a coal fired power plant is called dark spark spread.

The ﬁrst proposals for this valuation approach came from Hsu [Hsu98] and Deng, Johnson and Sogomonian [Deng98], who modelled the payoﬀ of a generator as the spark spread:

E F Π = P − HRP (1) where HR (mmBtu/MWh) denotes the heat rate, the amount of fuel burned to produce 1 unit of electricity. In their approach, the value of a power plant V is the sum of European style options with this spark spread as payoﬀ for every 1 MWh electricity generation. An extension to this model includes the costs for emission produced by burning fuel according to the EU-ETS. It is called the clean spark spread and deﬁned as:

c E F A Π = P − HRP − ERP (2) A where P (US$/tonne CO2) stands for the allowance price to emit one tonne of carbon dioxide and ER (tonnes CO2/MWh) is the number of tonnes of carbon dioxide emitted by production of 1 MWh of electricity. If we deﬁne q (MW) as the generation level of the power plant - power output or energy generated per

8A energy units conversion table can be found at: http://www.energystar.gov/ia/business /tools resources/target ﬁnder/help/Energy Units Conversion Table.htm 6

time - and redeﬁne HR = HR(q) (mmBtu/h) and ER = ER(q) (tonnes CO2/h) accordingly, the clean spark spread Πc(q) of the power plant is

c E F A Π (q) = qP − HR(q)P − ER(q)P . (3) For the purpose of maximizing proﬁt, a power plant operator will switch generation on whenever it is proﬁtable to do so, i.e. when at time t the clean spark c spread Πt (qt) exceeds the operational costs OMt (US$/h). Assuming a very short switching time, fast notice of price movements and deterministic interest rates, we can state that the value of the generation asset at time t0 is

tN X c Vt0 = D(t0, t)Et0 [∆t max (Πt (qt) − OMt, 0) |Ft0 ] (4) t=t0 where the number N depends on the granularity ∆t = ti+1 − ti for all i = 0...N and the plant’s lifetime T = tN . For practical applications, the granularity should comply with the operational precision of the generator. Here D(t0, t) stands for the discount factor from t0 to t and Et0 stands for the expectation at time t0 under an appropriate filtration Ft0 . A filtration is an increasing sequence of σ-algebras { } on a measurable space Ω. Here Ω is the probability space for price move- Ft t≥t0 ments with the risk neutral measure Q. In (4) we used the principle of efficient and complete markets without arbitrage where a Q-Martingale exists under the risk neutral measure Q. For more details on probability spaces, filtrations and risk neutral measures in arbitrage free financial markets see for example [Hull02]. We notice that (4) is equal to the value of a strip of standard European style spread options.

In literature there exist many models to describe price uncertainties with stochastic processes that are trying to catch observable market behaviour. Before we will extend our model to real options by considering operational constraints, we want to discuss the behaviour of the three price uncertainties.

2.2 Energy markets 2.2.1 Electricity prices Electricity is one of the most exotic commodity due to its lack of efficient stora- bility. We have to keep in mind that power itself is not a tradable asset, i.e. spot power can be sold during a price spike but it cannot be borrowed, shorted, and then bought back and returned several days later. Thus, intertemporal concepts like efficient markets for asset pricing do not apply to power spot dynamics. Nev- ertheless, efficient markets still can be applied to the pricing of derivatives on 7 power and we will use observable market values for spot hourly auction prices in order to calibrate implied model parameters. For more details on this topic see [Seppi02].

€/MWh MWh

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0 0 12-13 12-13 12-13 12-13 12-13 12-13 12-13 Price Volume 15.09. 00-01 16.09. 00-01 17.09. 00-01 18.09. 00-01 Time 19.09. 00-01 20.09. 00-01 21.09. 00-01

Figure 1: Spot Hourly Auction prices at the European Energy Exchange, Phelix, from Tue. 15th Sep. 2009 to Mon. 21st Sep. 2009

Regarding liquid markets like the European Energy Exchange EEX in Fig. 19, we can observe some special behaviour of electricity spot prices. First, there is a hourly structure in the price level that reflects the average schedule of private and industrial consumers. Market participants distinguish these hourly periods into the following so called Blocks: Offpeak, Off Peak I, Off Peak II, Night, Morning, High Noon, Afternoon, Rush Hour, Evening and Business Hours. Second there is a difference between working days and weekends or bank holidays on total demand and price levels consequently. Third, on the intraday spot market we can observe price spikes emerging when demand exceeds the current supply, i.e. when a base load plant in the power grid has an unscheduled forced outage.

For our purpose to value a power plant as an option to produce electricity only if it is proﬁtable, according to the constraints given in section 2.3, we explicitly take

9source: www.eex.com 8 care of the hourly intraday variations. We do this by incorporating a periodic E E hourly mean Lt into our model, to which Pt is reverting to. In section 4.2.1 the model parameters are calibrated to working days as well as non-working days to capture both kinds of behaviours in right proportion. Thus, the electricity spot E price Pt follows the mean-reverting process

E E E E E E E E dPt = κ Lt − ln(Pt ) Pt dt + σ Pt dWt (5) E E E where Wt is a Wiener process and κ , σ are positive constants. As mentioned E above, Lt is a periodic step-function of the hourly mean deﬁned according to Table 4. Parameter κE captures the mean reverting speed when prices deviate E from Lt and parameter σ is the price volatility. This price model is an extension of the Schwartz one-factor model [Schwartz97] and equivalent to that in [Barz99]. For valuation in (4) we need the risk neutral price process under the risk neutral measure Q:

E E E E E E E E E dPt = κ Lt − λ − ln(Pt ) Pt dt + σ Pt dQt (6) E E where λ > 0 is the market price of energy risk and Qt is a Wiener process E under Q. To see that Pt follows an exponential Ornstein Ulenbeck process set E E Xt := ln(Pt ) and use Ito’s lemma to get:

E E ˜E E E E dXt = κ (Lt − Xt )dt + σ dQt (7) ˜E E E σE2 E where Lt = Lt − λ − 2κE . A proof that Pt is log-normally distributed can be found in [Uhlenbeck30].

One possible extension would be to model the price spikes explicitly by setting up multi price states, but we do not do this here. One argument is that power producers sell some parts of their output on one-day ahead auctions and forward delivery contracts, where spikes become less relevant. A second argument is, that outages are handled only implicitly in this thesis. The reason why a common additional Poisson jump term together with a mean-reverting model is not appropriate for electricity is that spikes ends as abruptly as they start. For more analyses on this topic see [Baron01]. Spikes would become necessary if we focus on pure peak power plant.

2.2.2 Fuel prices To model uncertainties in energy commodity markets, Schwartz and Smith sepa- rated the price movements into long-term dynamics ξt and short term deviations χt, see [Schwartz00]. Short term deviations could capture the actual delivery and availability risk at a specific location. The long term dynamics models the global 9 long term demand and supply risk of exhausting resources. A third aspect is the weather. Besides the usage as fuel gas in power plants, natural gas is needed in cold winters to heat homes, offices and facilities. The additional demand increases prices during these periods. If we consider the pure spark spread without any constraints and look at period of years we could average these effects out. But later we will introduce constraints and we also take a look at timescales of a few month. Thus we model the additional demand with a seasonal factor s(t). All three aspects are observable in the current market Futures illustrated in Fig. 210.

$/mmBtu

3 Jul. 09 Jul. 10 Jul. 11 Jul. 12 Jul. 13 Jul. 14 Jul. 15 Jul. 16 Jul. 17

Time

Figure 2: Natural Gas Future prices with monthly delivery for diﬀerent maturities at New York Mercantile Exchange, Henry Hub, on 14th Sep. 2009

Putting everything together, we assume the following 2-factor Schwartz-Smith F process with seasonality for fuel spot price Pt at time t:

F χt+ξt Pt = e s(t) (8) where 10source: Reuters 10

χ χ dχt = −κχtdt + σ dWt ξ ξ dξt = µdt + σ dWt (9) χ ξ dWt dWt = ρχξdt and

12 X t − ti s(t) := θ (t) γ + (γ − γ ) . (10) i i i+1 i t − t i=1 i+1 i χ ξ Here Wt and Wt are two Wiener processes with correlation ρχξ. κ > 0 is the χ ξ reverting speed of χt to zero, µ > 0 is the drift of ξt and σ , σ > 0 measuring both risks accordingly. In the seasonality function s(t), γi > 0 are the monthly seasonal 1 P12 factors normalized to 1 by the constraint 12 i=1 γi = 1 and θi(t) represents the characteristic indicator function:

1 if t ∈ [t , t ] θ (t) := i i+1 (11) i 0 otherwise. Thus s(t) is a periodic function deﬁned through linear interpolation between the 12 seasonal factors γi, see Figure 3.

1.1

1.05

1 seasonalfactor

0.95

0.9 0 1 2 3 time (years)

Figure 3: A normalized seasonality curve s(t) with γi estimated from market data January 2010. 11

Because Natural Gas is traded in monthly contract, 12 is the maximum number of seasonal factors one can estimate out of market data at once. In literature it is quite common to model the seasonality by an additive term likes ˜(t) = PS E 2πit E 2πit i=1 bi sin( P ) + ci cos( P ). In one of our last module hand in [Ludwig09] we showed that constant seasonal factors are much more applicable and easier to calibrate to market data than additive trigonometric functions. The reason is that the market amplitudes of seasonal oscillations are proportional to the future prices, see Figure 2. Especially they are not constant as an additive term would suggest. This is the reason why we multiply prices with the seasonal factor s(t) here. As in the previous section, for option valuation we deﬁne the risk-neutral process

χ χ χ dχt = (−λ − κχt)dt + σ dQt ξ ξ ξ (12) dξt = (µ − λ )dt + σ dQt χ ξ χ χ where λ , λ > 0 and Qt ,Qt are Wiener processes under the risk-neutral measure F F Q. For the log-price Xt = ln(Pt ) we get the following evolution equation:

F χ χ χ ξ ξ dXt = (˜µ − λ − κχt)dt + σ dQt + σ dQt + ln(s(t)) (13) whereµ ˜ = µ − λξ.

2.2.3 Emission allowance prices The European Union Emission Trading Scheme (EU-ETS) is a system whereby CO2 emission allowances are traded among the market participants and thus vary with the level of demand. For the power plant valuation, we want to find an appropriate price process to model this uncertainty. At the present time, one problem to find a stochastic process with stable parameters is the frequent changes of the EU-ETS by regulators. As mentioned above, the Member States set up their own National Allocation Plan (NAP) and choose the industry sectors that have to par- ticipate nationwide. Great Britain for example, add a big amount of allocations to their NAP after the trial period (2005-2007) already started which caused a big price drop throughout the European Union. Later, at the beginning of the second period (2008-2012), there were more changes like the convertibility to the Kyoto Certified Emission Reduction units (CER) and for the beginning of the third period, 2013, more changes are constituted already, see section 6. A second problem is, that at least 95% of allowances were free of charge in the fist trial period. This fact prevented a strong exchange and thus the development of a very liquid market. Bigger nations like Germany even allocated 100% of allowances for 12 this period free of charge11. A change away from these free allocations towards more auctions will continue in 2013. Thirdly, emitters have to deliver the needed amount of allowances at the end of each year and not at the time of emission. Furthermore, holders can transfer unused allowances to the next year within the same period, but not across the border of different periods. Both facts can support the occurrence of strong price movements or even jumps.

In literature, we found different approaches to model this uncertainty emission allowance prices. For example, Abadie and Chamorro [Chamorro08] observed the market behaviour of European emission allowance future contracts over a period of one year. With 1,325 daily observations for 5 futures contracts maturing from Dec-08 to Dec-12 they figured out that the behaviour of prices is hardly con- sistent with a mean reversion process. Trück [Trueck08] analysed GARCH and Regime-switching models and conclude that: ’the best example fit to the data is provided by a regime-switching model with an autoregressive process in the base regime and a normal distribution for the spike regime. Results for the GARCH and normal mixture regime-switching models are only slightly worse’. Daskalakis [Daskal05] suggests that CO2 emission allowance price levels are non-stationary and exhibit abrupt discontinuous shifts. For logarithmic returns they find that the distribution is clearly non-normal and characterized by heavy tails. They further find that the best model fits for allowance prices in terms of likelihood is obtained by a geometric Brownian motion with an additional jump-diffusion component. ’This model is also able to produce the discontinuous shifts in the underlying diffusion that are observed in the CO2 emission allowances prices’. An analysis at a big European bank - name is confidential - give us the insight that ’the jumps’ - of a Merton jump-diffusion model - ’seem to be a key ingredient in the CO2 modelling’. Some compared GARCH models fit quite good also, but require more parameters to estimate and thus loose robustness.

Considering the market behaviour - the structure of demand and supply and the arrival of important new information as described above - the presence of jumps is reasonable. Regarding Fig. 4, it is also reasonable to expect that beside the times when such information arrives, there will be ”quiet” times as Merton described in his paper [Merton75]. In addition, the historical returns in the bank analysis show fat tails and some negative skewness that could corresponds to the presence of jumps with negative jump mean. Thus, in this thesis we will apply the jump- diﬀusion model proposed by Merton to model the price of the allowance to emit 1 tonne of CO2 at time t: 11Federal Ministry for the Environment, Nature Conservation and Nuclear Safety, National Allocation Plan for the Federal Republic of Germany, March 2004 13

€ / t t CO 2

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- 0 Settle 30.11.09

Jul. 09 Jul. Settle 30.11.10 Apr. 09 Apr. Jun. 09 Jun. Mrz.09 Feb. 09 Feb. Sep. 09 Sep. Time Volume

Figure 4: Second Period European Carbon Future settlement prices for two different maturities. The third line displays the 30.11.09 contract volume traded at the European Energy Exchange

A A A A A A A dPt = (µ − λk)Pt dt + σ Pt dWt + P (J − 1)dNt (14) A where Wt is a Wiener processes and Nt is a Poisson process where arrivals of jumps are independently and identically distributed and the mean number of arrivals per unit time is λ. µA is the drift and σA is the volatility of the diﬀusion, A conditional on the Poisson event not occurring. dWt and dNt are assumed to be independent. Regarding the formula, when a jump occurs at time t the relation of the limits from both sides is Pt+ = JPt− . The magnitude J of the Poisson process itself is assumed to be log-normally distributed, ln(J) ∼ N(µJ , σJ2) and k A is deﬁned as k := E[J −1]. With these assumptions Pt is log-normally distributed with the variance parameter a Poisson-distributed random variable; for a proof see [Merton75]. Applying the martingale representation theorem, the risk neutral process under the measure Q is:

A A A A A A A A dPt = (µ − λ − λk)Pt dt + σ Pt dQt + P (J − 1)dNt (15) A A where λ > 0 is the market price of risk for emission allowances and Qt is a Wiener A A process under Q. Adopting the transformation Xt = ln(Pt ) and applying It´o’s 14

Lemma12 yields:

Nt+dt A A A A X dXt =µ ˜ dt + σ dQt + ln(Ji) (16) i=Nt A A A =µ ˜ dt + σ dQt + ln(J)dNt. (17)

A A A σA2 whereµ ˜ = µ − λ − λk − 2 .

2.2.4 Correlations From the nature of things it is clear that electricity prices, fuel prices and emission allowance prices are not independent and their processes that are responsible for price development are correlated. The correlations between the four stochastic processes are:

E χ dW dW = ρEχ, E ξ dW dW = ρEξ, E A dW dW = ρEA, χ ξ (18) dW dW = ρχξ, χ A dW dW = ρχA, ξ A dW dW = ρξA.

2.3 Operational constraints Thus far we defined the Clean Spark Spread for a fuel fired power plant as a strip of European style spread options the operator holds. But a power plant can’t be switched on and off immediately whenever it is profitable to do so. There are several constraints on the operating states transforming the operator’s decision possibilities into real options. Les Clewlow and Chris Strickland [Clewlow09], Chung-Li Tseng [Tseng09] or Graydon Barz [Barz02], for example, introduced methods to model complex operational constraints. The following list summarizes operational constraints that can be found in literature applied on spark spreads and power plant valuation approaches. The text in parenthesis indicates if the constraint is handled in this dissertation or not:

• variable generation level (yes) • non-constant heat rate (yes) • non-constant emission rate (yes) • maximum capacity (yes)

12Using the generalised It´oformula for semi-martingales; proof see e.g. [Duevel01] 15

• minimum stable generation (yes) • ramp-up rates (yes) • ramp-down rate (yes) • minimum up time (yes) • minimum down time (yes) • cold time (yes) • start-up costs (yes) • shut-down costs (yes) • variable operational and maintenance cost (yes) • self used electricity (yes) • forced outages (no) • scheduled maintenance (no) • over ﬁreing (no) • outside temperature (no) • fuel transportation costs (no) • power transmission costs (no)

Before we discuss every constraint in detail, we want to recall the time discreti- sation into N time intervals with ti ∈ [t0,T ] for all i = 1...N where tN = T is the remaining lifetime of the power plant and ∆t = ti+1 − ti is constant.

2.3.1 Heat rate, emission rate and optimal generation level

We already introduced the power plant generation level qt in section 2.1. The capacity qmax is the maximum generation level of the plant, qmin the level of minimum stable generation and qt ∈ [qmin, qmax]. Once the generator is producing electricity at time t, the operator can switch qt to any level within [qmin, qmax]. That means qt is a real time decision process and does not restrict any further decisions.

As is standard, the heat rate HR is modelled as a quadratic function of the generation level qt. For arguments on this assumption see [Wood84]. We deﬁne:

2 HR(qt) = a0 + a1qt + a2qt (19) 13 where a0, a1, a2 > 0 are positive constants. A chemical engineer at Didcot provides us with the information that the emission rate has a constant proportion to the fuel burned, no matter which temperature or pressure exists in the reactor. So we deﬁne the emission rate ER at time t to be: 13The RWE npower site at Didcot UK is host to two power stations - 2000MW dual-ﬁred coal station and the 1360 MW combined cycle gas turbine station. 16

ER(qt) = qt · E (20) where E > 0 (tonnes CO2/MWh) is the constant emission quotient specifying how many tonnes of CO2 are emitted by producing one MWh of electricity.

The dispatch problem of the power unit is to determine the optimal generation ∗ level qt at time t. Since the clean spark spread (3) is a convex quadratic function ∗ of market prices observed at time t, qt can be calculated by maximizing:

c E F A Π (qt) = qt · Pt − HR(qt)Pt + ER(qt)Pt E A 2 F (21) = qt(Pt − E · Pt ) − (a0 + a1qt + a2qt )Pt . 0 E A F 0 We know that Π (q) = (P − E · P ) − (a1 + 2a2q)P , set Π (q) = 0 and note 00 P E −E·P A 1 that Π (q) < 0 always, to get q = F − a1 . So the optimal generation P 2a2 ∗ level qt at time t is:

E A ∗ Pt − E · Pt 1 qt = min(qmax, max(qmin, F − a1 )). (22) Pt 2a2

2.3.2 Ramp rates and start up costs Most power plants cannot change their generation level instantaneously. A more realistic dispatch algorithm has to consider ramp rates (MW/h) defined as the rate at which the unit’s generation level q can increase or decrease in time. Nor- mally, ramp up rates can be classified as cold rc, warm rw or hot rh depending on how long the unit has been off-line. The ramp down rate is equal to rh.

Given the unit is hot at time t already, assume that qmax−qmin < ∆t. The ramp rh |q∗−q∗ | rates are captured in the calculation by deﬁning the changing time τ = t t−1 rh and adding the following penalty term to the operational costs:

q∗ + q∗ S (t) = τ Πc(q∗) − Πc( t t−1 ) . (23) h t t t 2

We noticed that Sh(t) ≥ 0 always and Sh(t) = 0 if the plant operates on the optimal level already. This is a quite good approximation neglecting the small quadratic term of HR for the short time τ only. Moreover, the turbine channel of a generator can only be opened when a minimum level of pressure holds in the connected steam-pipes. Otherwise the turbine would start to toggle. This implies, that from a cold or a warm start the ﬁrst fuel is burned only to heat the steam-cycle and build up pressure in the pipes. 17

To capture this fact we deﬁne the time needed to reach the minimum stable generation level t = qmin and assume that the time to reach the optimal generation c,w rc,w level tc,w + τ < ton. Here ton is the minimum up time deﬁned in section 2.3.3. The cost for the fuel burned and the CO2 emitted are: ˜ R tc,w F R tc,w A St = 0 HR(rc,wt)dt Pt + 0 ER(rc,wt)dt Pt R tc,w 2 2 F R tc,w A = 0 a0 + a1rc,wt + a2rc,wt dt Pt + 0 Erc,wtdt Pt (24) a1 a2 2 F a1 A = a0tc,w + 2 qmintc,w + 3 qmintc,w Pt + E 2 qmintc,w Pt a1 a2 2 F a1 A = a0 + 2 qmin + 3 qmin Pt + E 2 qmin Pt tc,w. ˜ The complete start up costs Sc,w(t) include these fuel and emission costs St, the c ∗ optimal clean spark spread Πt (qt ) lost during this heat up time and the penalty term Sh(t) to reach the optimal generation level:

˜ c ∗ min Sc,w(t) = St + (Πt (qt ) − Sh(t)) tc,w + Sh (t + tc,w) − Sh(t + tc,w) . (25)

c ∗ Here, (Πt (qt ) − Sh(t)) tc,w stands for a sum over discrete time intervals until tc,w ∗ is reached. Smin indicates that τ min = qt −qmin is used instead of the original τ. h rh min This is the reason why we add the difference Sh (t + tc,w) − Sh(t + tc,w) here. Sometimes in literature these values are called switching cash flows, whish take care of the fractional part of ∆t, i.e. 60 minutes, after the switch is completed. As an example, if the switch of qt takes 15 minutes, the switching cash flow is the spark spread produced minus the optimal spark spread for these 15 minutes. After that, the plant is operating in the new optimal mode for the residuary 45 minutes.

2.3.3 Minimum up time, minimum down time and cold time The minimum time a unit must remain on after start up is called minimum up time ton > 0. The minimum time the unit must remain off after shut down is called minimum down time toff > 0. During these periods no operating decisions can be made. Furthermore, leaving the power plant off-line for a longer period, the reactor starts to cool down continuously. The time from shut down to the time when the reactor reaches the surrounding temperature is called cold time tcold > toff > 0. We define the state variable:

xi ∈ X = {−tcold, ..., −toﬀ, ..., −1, 1, ..., ton} (26) which indicates the reactor state at time ti. After start up the state is increasing from 1 to ton, after shut down the state is decreasing from −1 to −toﬀ or −tcold 18

probably. We notice that 0 is not a state. The future state xi+1 depends on the former state xi and on the decision ui at time ti by the plants operator: ui = 0 for turning down and ui = 1 for turning up. Using these definitions, the constraints for minimum up time, minimum down time and cold time are: min(ton, max(xt, 0) + 1) if ut = 1 xt+1(ut, xt) := (27) max(−toff, min(xt, 0) − 1) if ut = 0, with the following possible decisions ui ∈ {0, 1} at time t:  0 ∨ 1 if xt ∈ [−tcold, −toff]   0 if − toff < xt ≤ −1 ut(xt) := (28) 1 if 1 ≤ xt < tup   0 ∨ 1 if xt = tup.

For a start-up from xi = −tcold the ramp rate is rc and for a start up from −tcold < xi ≤ −toﬀ the ramp rate is rw. For later purpose, we deﬁne the set of all possible decision paths Ux0 for ui from t0 to T starting with x0 at time t0; so

(ut)t=t0...T ∈ Ux0 .

2.3.4 Variable operational and maintenance costs The operational and maintenance costs OM > 0 typically include costs such as workforce, administration, facility management and quick maintenance during production. Often OM is measured on a yearly basis. The variable operational costs VOM additionally include costs that occur according to the ﬂexible generation like start up costs discussed in the sections above:

  Sh(t) + Sc(t) + Su if ut = 1 ∧ xt = −tcold   Sh(t) + Sw(t) + Su if ut = 1 ∧ −tcold < xt ≤ −toﬀ VOMt(xt, ut) = OM + Sh(t) if ut = 1 ∧ xt > 0   Sd if ut = 0 ∧ xt > 0  0 otherwise (29) where Sd,Su > 0 are the constant shut down/start-up costs respectively.

2.3.5 Forced and scheduled outages The scheduled outages represent the planned downtime for maintenance of the plant. Typically, this downtime is of the order of 2-4 weeks per annum [Geman05]. Scheduled outages from time t1 to time t2 can be simulated in the Monte Carlo algorithm introduced below by setting ut = 0 for all t ∈ [t1, t2] and for all simulations, as soon as ut is free to be chosen. Forced outages represent unplanned 19 downtime caused by a technical failure of the unit. Power producers are very concerned by the Equivalent Forced Outage Rate (EFOR) which is deﬁned as the number of outage hours in a given period divided by the number of generation hours in the same period. Outages rates depend on the type of unit and can vary according to the season from 3% to 20%, see [Geman05]. In this thesis outages are c included indirectly by multiplying the clean spark spread Πt with (1 − EFOR) at every time when the power unit is at state xt = ton. 20

3 A Real option approach for valuing power plants

3.1 The real option approach By simply including all path-dependent constraints from section 2.3 into the option valuation in (4), the value of the power asset at time t0 would be:

" t # XN V = E max D(t , t)∆t (u Π (q∗) − VOM (x , u )) . (30) t0 t0 0 t t t t t t Ft0 u∈Ux0 t=t0 Braz called this a ‘single-stage’ problem because the commitment decisions for all time periods i= 1...T are determined after the optimisation problem is solved. This approach is ﬂawed because of the following two reasons: First, prices are not known before the commitment decisions are made. However, since the commitment is optimised in every scenario - if we think in terms of Monte Carlo simulation to solve the problem - V0 obtained in Equation (30) is an upper bound of the true power plant value. This is analogous to all path-dependent options like Bermudan-style or American-style options. Second, the commitment decision obtained in each scenario iteration is independent, i.e., no useful information about the commitment decisions can be extracted from the simulation. This problem is well known from pricing path dependent American style options.

A better way is to incorporate uncertainty into the dynamic commitment decision- making by a ‘multi-stage’ stochastic model. Thus, the true value of the real option is:

" tN # X ∗ Vt0 (x0) = max Et0 D(t0, t)∆t (utΠt(qt ) − VOMt(xt, ut)) |Ft0 (31) u∈Ux0 t=t0 where the optimal exercise strategy is fundamentally determined by the conditional expectation of the payoﬀ. The boundary conditions at time T are:

∗ VT (xT ) = max (∆t(uT ΠT (qT ) − VOMT (xT , uT ))) (32) uT for all states xT ∈ X.

3.2 Real option valuation using dynamic programming 3.2.1 Backward induction

Regarding (31), the problem now is to ﬁnd the optimal exercise strategy u ∈ Ux0 conditional to the information given at time t0. First, we reformulate (31) by using 21

Bellman’s principle of optimality, see [Bellman57]. We decompose the problem into two components: the actual payoﬀ structure for a small time period ∆t and the value of the continuing option Vt1 according to (31). Assuming that it is appropriate to use the methods of dynamic programming, Bellman’s principle of optimality states that the expected value of the sum of both components is equal to Vt0 (x0) as ut0 is chosen optimal:

∗ Vt0 (x0) = max Et0 [∆t (u0Π0(q0) − VOM0(x0, u0)) + D(t0, t1)Vt1 (x1(x0, u0))|Ft0 ] . u0(x0) (33)

The speciality of this recursive formulation is, that if we know Vt1 (x1) for all x1, we only have to optimize the current decision u0 at time point t0. Applying the same principle to Vt1 (x1), Vt2 (x2) and so on, we get a recursive formulation of the optimization problem with the boundary given in (32). We want to repeat this recursive formulation in more detail:

In order to value a power plant with flexible operating characteristics X, one must determine the optimal operating policy u ∈ U of the plant. Thus, the operator must consider not only the cash flow (Πi − VOMi) for the next time interval [ti, ti+1], but also the ’residual value’ Vi+1 of the state xi+1 that the decision ui gives the plant after ∆t. This residual value Vi+1 is the value of the cash flow for the remainder of the plant’s lifetime T , given the decision ui. Each possible decision ui(xi) at time ti implies the ’switching cash flow’:

∗ sc(xi, ui,Pi) = ∆t (uiΠti (qi ) − VOMti (xi, ui)) (34) where P := (P E,P F ,P A) should indicate the dependency of the ’switching cash i ti ti ti ﬂow’ on the current market prices. If the operator changes the operating mode from xi at time ti to xi+1(ui, xi) at time ti+1 - by making a possible decision ui(xi) according to (27) - the expected value of the power plant at time ti is: νti (xi, ui,Pi) = Eti sci(xi, ui,Pi) + D(ti, ti+1)Vti+1 (xi+1(xi, ui))|Fti . (35) where Eti is the expected value operator conditional on information available at time ti. As mentioned above, we call this a ’multiple-stage’ problem because every decision could change the state of the unit. The value of the power plant is given by choosing the optimal decision:

Vti (xi) = max νti (xi, ui,Pi). (36) ui(xi) The solution method uses backward induction; we start at the ﬁnal time T with the boundary given in (32) and determine the ’switching cash ﬂows’ going from 22

state xT −1 to all other possible states xT ∈ X through decision uT −1. The value function in is then simply the optimum given by (36). We do this for all states xT −1 ∈ X. Note that if xT −1 = xT , the plant stays at the current operating state but even though, there is a ’switching cash ﬂow’. The same thing is done for time T − 2, T − 3,... until we reach t0.

The only remaining item for valuation is to determine the conditional expectations of the relevant values. Due to the complicated nature of the problem as well as the stochastic behaviour of the clean spark spread, an analytical or closed form solution is not feasible. Thus, we turn to the power of using Least Squares Monte Carlo (LSM) next.

3.2.2 Least squares Monte Carlo The issues that mandates the use of Monte Carlo techniques is the fact that the operational constraints, introduced in the previous section, make the problem highly path dependent. Although the problem exhibits a complex structure, Monte Carlo simulations are still relatively easy to implement and provide a high grade of ﬂexibility which we will take advantage of in later sections.√ A disadvan- tage is the slow convergence of the error variance, which is ∼ 1/ M, where M is the number of simulations. An alternative method would be a 3-dimensional lattice approach in which for every single state an individual tree is built. This approach often is called forests and should have better performance but less ﬂex- ibility. For an example of using a multi-dimensional forest method for real option valuation, we refer to [Barz02].

As we mentioned in section 3.1, we can’t optimize along a Monte Carlo path because this would imply the knowledge of the future price development and thus lead to perfect decisions - an overestimation. Differently, to solve the dynamic programming problem in (33), we have to compute the conditional expectations and make a decision conditional on the information given at the current time t. To achieve that, we use the Least Squares Monte Carlo method (LSM) introduced by Longstaff and Schwartz [Longstaff01] for American options. The idea is to approximate the conditional expectation by a finite set of analytic basis functions of the current market prices and regress the function parameters by minimizing the least squares error. We do not want to go into details on the LSM method here; for the complete theory see [Longstaff01]. Transferred to our ’multi-stage’ problem (33), we first fix the state xt ∈ X at time t. Then, for a fixed possible decision ut, we assume that ν(xt, ut,Pt) can be approximated by a second order polynomial: 23

t νt(xt, ut,Pt) ≈ ft,xt,ut (Pt) := bt,0 + bt,1Pt + Pt bt,2Pt (37) where for every t, bt,0 is a scalar, bt,1 a 3-dimensional vector and bt,2 a 3x3-Matrix j of coeﬃcients. For every Monte Carlo simulation Pt at time t, j = 1..M, we j calculate sc(xt, ut,Pt ) according to (34) and deﬁne

j j j νt (xt, ut) := sc(xt, ut,Pt ) + D(ti, ti+1)Vt+1(xt+1) . (38) j which is (34) for one simulation Pt . Notice that due to our backward induction we j j already know Vt+1(xt+1) - deﬁned in (41) - here. From these simulated {νt }J=1..M we can now regress the coeﬃcients {bt,k} by minimizing the least square approximation error:

M X j j 2 {bt,k} = arg min νt (xt, ut) − ft,xt,ut (Pt ) . (39) {b } t,k j=1

νt(,,Pt) and thereby the regression coefficients {bt,k} depend on seasonal price variables known at time t, including the information about the season of the year, the day of the week and the hour of the day. Thus, these regression coefficients {bt,k} are different for each time step ti ∈ [t0,T ]. For simplicity, we skip the index for x and u here. If we repeat this procedure for every possible decision ut, it gives us the estimates fxt,ut of the switching alternatives in (36). Thus, for every j j j simulation Pt , we choose the ut that maximizes fxt,ut (Pt ):

j j ut = arg max ft,xt,ut (Pt ). (40) ut j Note this function depends only on current information Pj at time t. Now we can j use ut to evaluate equation (36):

j j j Vt (xt) = νt (xt, ut ). (41) j We recall this optimization for every initial state xt ∈ X to ﬁnd all Vt (xt). After all, we can go one step backward in time to ti−1 and resume this method as long as we reach t0. Then,

M 1 X V (x ) = V j (x ). (42) t0 0 M t0 0 j=1 j We want to notice, that ft,xt,ut (Pt ) is only used to ﬁnd the appropriate decision j j j ut . For the real option value itself, we take the exact value νt (xt, ut ). Longstaﬀ and Schwartz stated, that otherwise this expectation function is biased. Since we 24 are using these approximations to arrive at an optimal decision, any error in the approximation leads to a sub-optimal decision. Hence this will undervalue the power plant.

E For the approximation function ft,xt,ut (Pt) we used the single order prices Pt , F A Pt , PT and every 2nd order combination of them as basis functions. Fig. 5 show the νj’s and the accordingly approximated f(P j) at the same price-nodes for free chosen time step t, state xt and decision ut. The values are taken form one of our example calculations in section 4.4.

Figure 5: Comparison of sample points νj and the accordingly approximated A function f(P ) for a ﬁxed P . The time step t, the state xt and the decision ut were chosen randomly.

To get an impression how appropriate a second order polynomial is to approximate the expectation function, Fig 6 illustrates the distribution of the relative differ- ences between f(P j) and νj as histogram. We can examine a light distortion. One explanation is that a polynomial function couldn’t match the big relative differ- ences of the power plant value between different price paths close to time T where absolute values are very small according to our backward induction. Since we do not want to focus on differnet types of basis functions for the LSM method in this thesis, we don’t start trying many different ones. Longstaff and Schwartz stated that ”extensive numerical tests indicate that the results from the LSM algorithm 25 are remarkably robust to the choice of basis functions” and that ”few basis functions are needed to closely approximate the conditional expectation function” in their tests. We found very little difference in 2nd and 3rd order approximations, hence the degree of under-valuation is unlikely to be significant. Thus, a second order polynomial should be adequate for our purposes. In their latest work on real options, Sick and Cassano [Sick09] also used 2nd and 3rd order polynomials.

25000

20000

15000

Frequency 10000

5000

Difference

j Figure 6: Histogram of the relative diﬀerences between sample points νt and j the accordingly approximated ft(Pt ) for 7*24 time steps. The state xt and the decision ut were chosen randomly.

In our example calculations we also measured the convergence of the asset√ value for diﬀerent numbers of simulations. As expected we observed a ∼ 1/ M rela- tionship. For example, for the later introduced base-generation unit we measured a standard deviation of 0.9% with 4000 simulations. 26

4 Implementation aspects

The following section provide key ﬁgures about the Monte Carlo simulation of the price paths, the calibration of price parameters to historical market data and the eﬃcient implementation of the model algorithm. Additionally, some results of example calculations are presented. Here the asset properties are chosen according to typical, real world power plants. Furthermore, two short analysis show the behaviour of the asset value when input parameters or asset constraints change.

4.1 Monte Carlo simulation To simulate random price paths for Monte Carlo valuation the transition equation for each price process is needed. We begin with the transition equation for electricity prices according to the mean reverting process (6):

" r −2κE ∆t # E E (1 − e ) P = P e−κ ∆t exp (1 − e−κ ∆t)L˜ + σE ΦE (43) t+∆t t t 2κE t with ΦE a collection of i.i.d. standard normal random variables. Here ∆t is the inﬁnitesimal time step. The transition equation for fuel prices according to the 2-factor Schwartz-Smith model (13) is

χ q χ χ = χ e−κχ∆t − λχ 1−e−κ ∆t + σχ 1−e−2κ ∆t Φχ, t+1 t √ κχ 2κχ t ξ ξ (44) ξt+1 = ξt +µ ˜∆t + σ ∆tΦ , F χt+1+ξt+1 Pt+1 = e s(t), with Φχ, Φξ collections of i.i.d. standard normal random variable. Determining the value of s(t), we have to be careful about the right season at time t0 when the process starts. For proofs of (43) and (44) see [Dixit94], p.76. Last but not least the transition equation for allowance prices according to (15) is: √ A A A N Pt+1 = Pt exp µ˜ ∆t + σ ∆tΦt + ln(Jt)Φt (45) where ΦN is a collection of i.i.d events of a standard Poisson process with λ mean arrivals per time. Above we deﬁned k := E[J − 1] = exp(µJ + σJ2/2) − 1. If we transform this equation to express µJ with k, we get the jump amplitude:

σJ2 ln(J ) = ln(k + 1) − + σJ ΦJ . (46) t 2 t Here ΦA, ΦJ are collections of i.i.d. standard normal random variables. So for the Poisson process we simulate the jump times from an exponential distribution ﬁrst and then simulate ΦJ only for these times. To include the correlations given in 27

(18) into the simulation, we generate four independent collections of i.i.d. standard normal random events and multiply them with the Cholesky decomposition matrix U of the correlation matrix   1 ρEχ ρEξ ρEA ρEχ 1 ρχξ ρχA  Corr :=   (47) ρEξ ρχξ 1 ρξA  ρEA ρχA ρξA 1

4.2 Parameter calibration Before we can simulate price paths according to (43), (44), (45), (46) and (47), we have to ﬁnd appropriate values for all relevant parameters. To obtain these, we calibrate each price process to current or historical market data of liquidly traded contracts. The following sections describe how we did this for our example calculations.

4.2.1 Electricity prices First, we observed end-of-day prices of hourly contracts from the intraday spot market of the European Energy Exchange (EEX) over a two-year period from 19.01.2007 to 19.01.200914. Then, for each of the 24 one-hour contracts, we took 15 ˜E the 60 day average of historic log-prices as an estimation of Lt . By applying a linear regression on the same data as an alternative and comparing the results, we ˜E found out that both estimates for Lt are very close. The results can be found in Table 4. Secondly, after bringing the contracts into the right chronological order, we used the following description of log-price movements, derived from (44):

x − x = (1 − eκ∆t)(L˜ − x ) + t+1 t t t t (48) = a + mxt + t, to regress κ by minimizing the mean square error to the market log-returns. The standard deviation from this linear regression multiplied by p−2κ/(e−2κ − 1) is an appropriate estimation of the volatility σE. Note that the time unit of these estimations is one hour.

4.2.2 Fuel prices The power plant model can be applied to oil, coal, gas and biomass as fuels. Since an observation and comparison of all diﬀerent fuel types would to go beyond the

14Source: Bloomberg 15with a lower bound c > e 10.00 28

electricityprice (€) 35

Mon. 00 12 Tue. 00 12 Wed. 00 12 Thu. 00 12 Fri. 00 12 time (hours)

Figure 7: Three example paths of electricity prices simulated by transition equation (43) using calibrated parameters from Table 1 & 4 over a ﬁve day period. The mean reversion to intraday levels can be obtained clearly. scope of this thesis we focus on one fuel type, natural gas, here. As market data, we choose Natural Gas monthly future contracts traded at the New York Merchan- dise Exchange (NYMEX). This exchange has a higher liquidity of short maturing contract and also much more and longer maturing futures than any other market, which is essential to observe appropriate seasonal factors. We used daily foreign exchange rates to convert currencies.

On special thing of the chosen two-factor Schwartz-Smith model is the analytic formulation of future prices, which is at least partly responsible for its high usage:

−κ(T −t) F (T, t) = exp e χ0 + ξ0 + A(T, t) s(t) −κ(T −t) λχ A(T, t) =µ ˜(T − t) − (1 − e ) κ (49) 2 1 −2κ(T −t) σχ 2 −κ(T −t) ρχξσχσξ + 2 (1 − e ) 2κ + σξ (T − t) + 2(1 − e ) κ . Our calibration of the model parameters is based on an optimization that try to minimize the mean square error between the analytic future prices and the market future prices for all traded maturities. During calibration, delivery periods are taken into account by integrating the analytic future values over the whole 29 delivery period. Due to time we can’t go into details here. Another recommended estimation method would be a ﬁltering technique like [Kalman60]. We had the opportunity to re-calibrate the implied parameters every day over a six-month period and observed very little variation in values, which is a good indicator that two-factor Schwartz-Smith is an appropriate model.

5.5

4.5

4 Natural Gas (€/mmBTU)

3.5

3 Jan. 01 Jan 02 Jan 03 time (month)

Figure 8: Three example paths of Natural Gas prices simulated by transition equation (44) using calibrated parameters from Table 1 & 2 over a two year period. The seasonal shape can be observed.

4.2.3 Emission allowance prices According to European regulatory requirements for carbon emissions, which are brieﬂy described in the Appendix, a CO2 allowance can be delivered for a speciﬁed amount of emission that was produced at an arbitrary time point of the year. If we think in terms of commodities, the ’delivery period’ of a carbon future, which holds this allowance to emit CO2, is the whole year. Thus, the front-year carbon future price can be interpreted as a kind of spot price for CO2 emissions.

Furthermore, as we see in Figure 4, carbon futures on different maturities have very similar price movements. In fact, one can measure a correlation of 99% between two certificates in sequence. Hence, for the parameter calibration of the 30 assumed spot process (15), we only considered the front-year future history. We took observed market prices over a two year period from 19.01.2007 to 19.01.2009 and apply a Maximum Likelihood Estimation (MLE). The estimator is the (log-) probability density function of the Merton jump-diffusion model (15), which can be found in closed form using Fourier Transformation techniques on the Charac- teristic function (proof see e.g. [Labahn G.]).

 2  − (˜µAT +nµJ −x) −λT ∞ n 2(T σ2 +nσ2 ) e X (λT ) e A J pdf(x) = √   (50)  n! p 2 2  2π n=0 T σA + nσJ

4.2.4 Correlations Firstly, we measured the 24-hour average of electricity price for each observed day and calculated daily returns. Additionally, we took the daily returns of the front- year future for emission allowances, the daily returns of the generic front-month Natural Gas future for χ and the daily returns of the last generic Natural Gas future for ξ. With these four histories of daily returns, each over a two-year period, all correlations between PE, χ, ξ and PA can be estimated. The results can be found in Table 3. All correlations found out to be tiny except for ρχξ. Especially the zero correlation between electricity and Natural Gas and the negative correlation between electricity and emission allowances are conspicuous. An argument for the former could be that electricity traders are focusing on actual, local power demand - standard market instruments are regional one-day ahead auctions - and Natural Gas traders are looking for longer periods - standard market instruments are monthly Future contracts with a one-month delivery period. An argument for the later could be that higher power prices make regenerative power assets more valuable and emission would decrease in the long run. But these are just hypotheses. 31

Electricity Natural Gas Emissions E A P0 37.7 e /MWh χ0 0.10 e /mmBTU P0 13.6 e /t κE 0.19 hourly κ 1.30 yearly µ˜A 0.47 yearly σE 17.0% hourly λχ 0.00 yearly σA 44.0% yearly ˜ χ Lt Table 4 σ 112% yearly k -6.00% J ξ0 1.20 e /mmBTU σ 0.01% yearly µ˜ 0.02 yearly λ 11.8 yearly σξ 1.40% yearly

Table 1: Estimated market parameters, calibrated from historical end of day spot and future prices at EEX and Nymex from Jan 2007 till Jan 2009.

Seasonality Jan 1.084 Feb 1.075 Mrz 1.038 Apr 0.945 Correlations Mai 0.947 ρEχ 0.013 Jun 0.954 ρEξ 0.015 Jul 0.962 ρEA -0.009 Aug 0.968 ρEA -0.009 Sep 0.969 ρχξ 0.400 Okt 0.982 ρχA 0.063 Nov 1.019 ρξA -0.072 Dez 1.058 Table 3: Estimated correlations, cal- Table 2: Estimated seasonal factors, ibrated from historical end of day calibrated from Natural Gas future spot and future prices at EEX and prices at Nymex from Jan 2009. Nymex from Jan 2007 till Jan 2009.

hour 1 2 3 4 5 6 7 8 9 10 11 12 ˜E Lt 3.48 3.34 3.22 3.06 3.01 3.07 3.12 3.41 3.53 3.65 3.70 3.80 hour 13 14 15 16 17 18 19 20 21 22 23 24 ˜E Lt 3.79 3.69 3.60 3.59 3.70 3.96 3.95 3.84 3.69 3.64 3.71 3.55

˜E Table 4: Hourly Lt values, estimated from one-hour period contracts traded at the intraday spot market at the EEX from Dec 2008 till Jan 2009. 32

4.3 The algorithm Before we present some results from example valuations of power assets with typical constraints, we want to give a short overview of the numerical algorithm. Once more, a real option valuation is more complex than other path-dependent instruments. Beside the high path-dependency, a valuation algorithm must consider several states simultaneously. Furthermore, at every valuation point on the simulated paths, at every node on the extended trees (if using forests) or on every grid point (if using a three-dimensional mesh), the possibility of switching to other states must be considered and valued.

As described in section 3.2.1, for every state and every Monte Carlo path we must step backwards in time from the asset’s maturity T to the actual valuation date t0 and ﬁnd the optimal state to switch to at every time point. The decision is made by estimating the expected value (35) for every possible decision via least square regression, which is described in section 3.2.2.

A first possible optimization of the algorithm is to recognize, that in our power plant model there are tcold + ton states in total but only for 2 + tcold − toff states, x ≤ toff or x = ton, can the operational decisions affect the future state. In all other states, toff < x < ton, the operational decision can optimize the clean spark spread but the future state is deterministic. This should be considered when al- locating cash.

A second simplification is the pre-generation of the complete, optimal clean spark spread paths. In section 2.3 we defined the variable operational and maintenance costs VOMt(xt, ut) for a fixed price path on every time point t, for every state xt and for every decision ut. The special trick was to include all predictable in- stant and future losses like losses from changing operational level to optimum or start up fuel costs. With this technique VOMt(xt, ut) looks a little bit more complex than defining only actual losses when they appear but gives us the oppor- c,j ∗ tunity to simulate the discrete Monte Carlo price paths and calculate Π (qt ) and j VOMt (xt, ut) in advance. The following pseudo-code shows the full algorithm to valuate the real option:

1. simulate M Monte Carlo price path {Pj}j∈M according to (43), (44), (45), (46) and (47) using parameters from Table 4, 1, 2 & 3

c,j ∗ 2. calculate Π (qt ) ∀j ∈ [1,M], t ∈ [t0,T ] according to (3) and (22) j 3. calculate VOMt (xt, ut), ∀j ∈ [1,M], t ∈ [t0,T ], xt ∈ X and ut ∈ {0, 1} according to (29) 33

j 4. calculate VT (xT ), ∀j ∈ [1,M], xT ∈ X according to (32)

5. for t = T − 1 until t0 do (t = t − 1):

(a) for all xt ∈ X do: j j i. calculate switching cash ﬂow sc(xt, 0,Pt ) and sc(xt, 1,Pt ) accord- j j j j ing to (34) and ν (xt, 0,Pt ) and ν (xt, 1,Pt ) according to (38) ∀j ∈ [1,M]

ii. calculate the coeﬃcients {bk}t through regression according to (61) j iii. make decision ut (xt) according to (40) ∀j ∈ [1,M] j iv. calculate Vt (xt) according to (41) ∀j ∈ [1,M]

P j 6. calculate V0(x0) = 1/M j V0 (x0).

4.4 Example valuations In this section we analyse how changes in some of the path-dependent constraints affect the value of a generation asset. We do this by comparing assets with various constraints to a base-generation unit with typical constraints. The properties of our base-generation unit, introduced in equations (19), (20), (26) & (29) can be found in Table 5. If not stated otherwise, for all calculations we used the calibrated market parameters from section 4.2. In our first example we will examine different ramp rates rh. The ramp rate mainly defines the time needed for the adjustment to the optimal operational level. Figure 9 should give an impression how the optimized level q∗ changes the value of the clean spark spread Πc without shutting down the power plant. Now we changed rh and revalued the asset leaving all other market parameters and constraints untouched. Of course, we also kept the same random numbers for all calculations. Figure 10 shows the results: a higher rh has only little affect on the average asset value - about 10.000 e per month.

As a second example we regarding start-up Su and shut-down costs Sd. As we saw in Figure 9 an immediate shut-down and start-up would be preferable at beginning and end of negative clean spark spread periods. These times appear often but persists only for a short while. Beside off- and on-times, shut-down and start-up costs are the essential criteria for the decision to shut down or not. In our analysis we changed Su to various levels and revalued the power plant. We could set Su = 0 without noticeable influence on the decision finding. Figure 11 and Figure 12 illustrate the results. 34

CCGT qmax 500 MWh qmin 300 MWh a0 725 a1 5.0 a2 0.00128 E 350 g/kWh rh 3.4 MW/min rw 3.0 MW/min rc 2.0 MW/min ton 4.0 hours toﬀ 3.0 hours tcold 6.0 hours Su 2500 e Sd 900 e OM 600 e /h

Table 5: Typical constraints for a Combined Cycle Gas Turbine (CCGT). The information come for the engineer at Didcot, the references given in the introduction and some google search on power plants.

In the literature generation assets are often grouped into three categories, Base load, Mid-merit and Peaking units. Base-load units, as the name implies, handle the base load of the grid. They have high start costs and have long minimum up and down times. Mid-merit units, on the other hand, tend to be less eﬃcient than base-load units but typically have lower start costs, shorter minimum up/down times, and less time to ramp up to maximum capacity. Peaking units tend to have high heat rates and are designed to be able to meet sudden peaks in demand and consequently have low start costs, short minimum up/down times, and can quickly ramp up to maximum generation, see [Clewlow09b]. Regarding these aspects Clewlow and Strickland state that Base load, Mid-merit and Peaking units can be regarded as in-the-money, at-the-money and out-of-money options having more intrinsic or more extrinsic value. Here we don’t want to strictly divide physical assets into categories but want to emphasise that crossings are smooth.

Now we could analyse a lot more sensitivities of constraints or market parameters and valuate diﬀerent real units, but here we want to make a further step ﬁrst. We want to value the possibility to changed a constraint, maybe through technological improvements, as an exchange-option on two real options. 35

4 x 10 1.5

full capacity optimal level 1 only positive spread

0.5

0 clean sprak spread (€)

-0.5

5 10 15 20 25 30 35 40 45 50 time (hours)

Figure 9: The clean spark spread for maximum (blue) and optimal (red) level for our base-generation unit. A very ﬂexible asset could switch oﬀ during times with negative spread (green).

6 x 10 6.66

6.65

6.64 asset value (€)

6.63

6.62 1 2 3 4 5 6 7 8 9 10 ramp rate (MWh/min)

Figure 10: Asset value for diﬀerent ramp rates on valuation date 19.01.2010 with maturity 19.04.2010. 36

x 10 6 6.75

6.7

6.65

6.6

6.55

6.5 assetvalue (€)

6.45

6.4

6.35

6.3 0 0.5 1 1.5 2 2.5 3 3.5 4 start-up costs (€) x 10 4

Figure 11: Asset value for diﬀerent Su on valuation date 19.01.2010 with maturity 19.04.2010. Sd = 0.

100

60 numberswitches of

0 0.5 1 1.5 2 2.5 3 3.5 4 start-up costs (€) x 10 4

Figure 12: Average number of operational switches during a 3 month period for diﬀerent Su. Sd = 0. 37

5 Investing in new technologies

As we already pointed out in the introduction, there are a lot of new technologies to improve eﬃciency of the generator or reduce the emission output of the plant. On the one hand, the actual energy market condition does not force an immediate installation for most of these innovations. On the other hand, by holding a physical generation asset, the investor not only holds the real option but additionally some American style options to install these new technologies. We will call them upgrading options. Thus, when an investor of a new power plant has to decide the speciﬁcations, he should only compare the expected values of all possible constraints but also consider the value of upgrading options - if there are any. An example is the investment in a new coal power plant and the choice is whether or not to install a Carbon Capture Unit to reduce emissions. The question is: what is the value of an upgrading option and when is the time to invest?

Luis M. Abadie and Chamorro [Chamorro08] as well as Stein-Erik Fleten and Erkka Näsäkkälä[Näsäkkälä05]analysed investment options in power plants. Ei- ther they used DCF methods or they left out operational constraints and used a one dimensional stochastic process for the whole clean spark spread only, in order to get analytical solutions. As mentioned before, the generation asset and especially an upgrading option do not only depend on the pure clean spark spread. If we think of an emission reduction installation without any side effects, the market conditions for emission allowances would be the essential criteria and not the spark spread itself. Here we want to go this different way. We want to use the introduced real options as underlyings of the upgrading option while keeping each price process PE, PF , PA as uncertainty.

5.1 An American style exchange option on real options To model an upgrading option we first define the set of all feasible values for all constraint parameters as Ω. One point ω ∈ Ω represents a specific power plant. ω We denote the value of this power plant by Vt (Pt). For the sake of simplicity, the actual state xt are left out in the following. Especially for longer running maturities T , the actual state xt is insignificant for the whole value Vt. Now, a technical improvement would change the power plant ω1 to a plant with different specifications ω2. Let C(ω1, ω2) denote the constant or deterministic costs for this technical improvement from ω1 to ω2. Then, the payoff value of the option is

ω1,ω2 ω2 ω1 It (Pt) = Vt (Pt) − Vt (Pt) − C(ω1, ω2) (51) 38

ω1,ω2 ω1,ω2 with IT = 0. Notice that the real payoﬀ It depends on future cash-ﬂows ω1 which are not know at time t and thus could be negative. Notice also that Vt , ω1 Vt are expected values. Thus, holding the generation asset ω1 the value of the option to invest in the upgrade to ω2 is given by

ω1,ω2 ω1,ω2 vt0 (Pt0 ) = sup Et0 [D(t0, t)It (Pt)|Ft] (52) t This is an American style exchange option on the exchange of two real options with diﬀerent constraints ω1, ω2. The strike is equal to the upgrading costs C(ω1, ω2).

5.2 Analytical valuation of American options To value the upgrading option we could use the complementary formulation of the problem:

L[v] ≤ 0, vt(Pt) ≥ πt(Pt) and

(vt(Pt) − πt(Pt)) · L[v] = 0 where

3 3 X X 1 [v] = ∂ v + y(P i)∂ v + ρ σ σ P iP j∂ ∂ v − rv. L t i 2 i,j i j i j i=1 i,j=1 y(P i) is the corresponding drift and r the risk free rate. From option theory we know that the best exercise time is when the payoﬀ is equal to the intrinsic value of the option and there is a hold region L[v] = 0 and an exercise region L[v] < 0. Suppose we are not in the exercise region so we can set L[v] = 0. One chance for getting a solution is to assume that the generation asset never expires T = ∞. The value of this perpetual American option satisﬁes

3 3 X X 1 y(P i)d v + ρ σ σ P iP jd d v − rv = 0. i 2 i,j i j i j i=1 i,j=1 and some smooth pasting condition. With these assumptions and some simplifica- tions on Vt(Pt) we could try to find a general solution for this problem, determining some boundary conditions and try to find some suggestions of the exercise boundary. For analytical boundary conditions and exercise regions of exotic American exchange options see [Broadie94] for example. In this thesis we want to include all introduced constraints and thus have to use numerical methods to get a solution. 39

5.3 Numeric valuation To value the American style upgrading option numerically we are following the same steps as for the valuing of our real option in section 3. Our problem is to solve

ω1,ω2 ω1,ω2 vt0 = sup Et0 [D(t0, t)It (Pt)|Ft0 ] (53) t E F A where Pt := (Pt ,Pt ,Pt ) and the boundary conditions at time T are given by

ω1,ω2 ω1,ω2 vT (PT ) = max(IT (PT ), 0). (54) Again we perform a backward induction and using Bellman’s principle of optimality to get:

ω1,ω2 ω1,ω2 ω1,ω2 vt = max Et[It (Pt)|Ft],Et[D(t, t + 1)vt+1 |Ft]. (55) To simplify notation we deﬁne

ex ω1,ω2 νt (Pt) := Et[It (Pt)|Ft]) (56) hold ω1,ω2 νt (Pt) := Et[D(t, t + 1)vt+1 |Ft]) (57) as the expected values and (55) becomes

ω1,ω2 ex hold vt = max νt (Pt), νt (Pt) . (58)

We also leave out the index ω1, ω2 if it is clear. In section 3.2.2 we already have j j,ωi simulated M price paths Pt and calculated M value paths Vt for j = 1..M, t = 1..T and i = {1, 2} accordingly; see (41). We can use these already simulated paths to value the exchange option by applying the least square Monte Carlo j,ωi method once more. In the following we take the average of Vt (xt) over xt to hold exclude the irrelevant state. We assume again that νt (Pt) can be approximated by a second order polynomial:

hold hold t νt (Pt) ≈ ft (Pt) := et,0 + et,1Pt + Pt et,2Pt (59) where for every time t, et,0 is a scalar, et,1 a 3-dimensional vector and et,2 a 3x3- j Matrix of coeﬃcients. For every Monte Carlo simulation Pt at time t, j = 1..M, we deﬁne

hold,j j νt := D(t, t + 1)vt+1. (60) j which is (57) for one simulation Pt . Notice that due to the backward induction j hold,j we already know vt+1 - deﬁned in (62) - here. From these simulated {νt }J=1..M 40

we can now regress the coeﬃcients {et,k} by minimizing the least square approximation error:

M 2 X hold,j hold j {et,k} = arg min νt − ft (Pt ) . (61) {e } t,k j=1

ex ω1 Then we do the same kind of regression for νt but here we have to split into νt ω2 ω1 ω1 ω1 and νt to approximate the expected value of Vt and Vt separately by ft and ω2 j ft accordingly. After all, for every simulation Pt , we set:

ω2,j ω1,j ω2 j ω1 j hold j j νt − νt − C if ft (Pt ) − ft (Pt ) − C > ft (Pt ) vt = hold,j (62) νt othewise

Repeating this we go the steps backward in time as long as we reach t0. Then,

M 1 X v = vj . (63) t0 M t0 j=1 We notice that we have the opportunity to let the approximation of the conditional expectation ft not depend on all uncertainties but only on the essential ones. As mentioned before, if we imagine of an CO2 ﬁlter without any additional costs or E side eﬀects Pt := (Pt ) would be the right choice.

5.4 Examples As an example we analyse the installation of a long-lived Carbon Capture and Storage (CCS) unit. On the one hand, a CCS is a huge plant by its own and can save up to 90% CO2 output of a generation asset. On the other hand, it consumes a signiﬁcant amount of the plant’s electricity output - about 15%. Additionally, there are CO2 transportation and storage costs - about 7.3e /t - and usual operation and maintenance costs - about 550e /h. Since it haven’t been build before, the initial installation costs of a CCS unit could only be estimated to about 200 Mio.e 16.

Yet, we haven’t introduced CO2 transportation and storage costs as well as self- used energy in our model. To do this we deﬁne fE as the proportional factor for CO2 output and fP as the proportional factor for electricity output. With these factors the clean spark spread in (3) changes to

16The values were taken from [Chamorro08] were more information on CCS units can be found. 41

c E F S Π (q) = qfP P − HR(q)P − fEER(q)PA − (1 − fE)ER(q)P (64) S where P is the constant price for CO2 transportation and storage. Furthermore, the optimal operational level deﬁned in (22) becomes

E A S ∗ fP Pt − fEE · Pt − (1 − fE)EP 1 qt = min(qmax, max(qmin, F − a1 )). Pt 2a2 (65) ˜ We do not change the start-up fuel and emission costs St in (24) because a CCS unit can’t operate during the start-up period. With these adjustments we can now deﬁne the two generation assets ω1 and ω2 for our exchange option. Their properties are the same as for the base-generation asset in Table 5. The additional and adjusted properties for ω1 and ω2 can be found in Table 6:

ω1 ω2 fP 100% fP 85% fE 100% fE 10% P S 0.0 e /t P S 7.3 e /t OM 600 e /h OM 1050 e /h Cω1,ω2 200 Mio. e

Table 6: The diﬀerent properties of the held asset without CCS ω1 and the up- graded asset with CCS ω2.

Since the biggest effect of a CCS installation is the emission reduction, the option ω1,ω2 A value v strongly depends on the emission allowance price level Pt . If T is A fixed, the question of best exercise becomes: at which price level P0 the intrinsic option value is equal to the payoff? Thus, we analysed the behaviour of the up- ω1,ω2 ω1,ω2 A grade option v0 and the payoff I0 under different P0 . In the first run we didn’t change P E and P F . Figure 13 shows the results17.

We observed that the value of the CCS is virtually zero for quite a large range of A allowance prices. For higher P0 the option value and the payoﬀ seems to converge, but they do not - see Figure 14. The reason here is, that the electricity price level left unchanged and does not allow for the higher emission allowance costs. Thus, the clean spark spread Πt decrease under the OM level, even for ω2. At that point, the electricity generation becomes unproﬁtable and V ω2 , vω1,ω2 decrease.

17Due to runtime, we reduced T and changed C accordingly 42

6 x 10

value (€) -1

-2

15 20 25 30 35 40 emission allowance price (€)

Figure 13: The value of the CCS upgrade option (green) and its payoﬀ (blue) according to Table 6 with runtime T = 3 month and suitable investment costs C = 1.250.000 e .

5 x 10

value (€) 0

-5

20 30 40 50 60 70 80 90 100 emission allowance price (€)

Figure 14: The value of the CCS upgrade option (green) and its payoﬀ (blue) for A high P0 according to Table 6 with runtime T = 1 month and suitable investment costs C = 416.666 e . 43

In the second run we adjusted the electricity prices to allow for the higher emission allowance costs. An argument is that power suppliers are forced to increase elec- E A tricity prices to still being proﬁtable. We did this by keeping the part qP0 −ERP0 ˜ of the clean spark spread at time 0 at a constant ratio and adjust the levels Lt - where electricity prices are mean-reverting to - for all t accordingly. Thus we A E ˜ increased P0 by δA and at the same time increased P0 and Lt by ln (EδA). E is the constant emission quotient from equation (20). Other process variables were ω1,ω2 A left unchanged. Figure 15 shows that v > 0 e if P0 > 25 e . The option and the payoﬀ values converge and intersect at about 35 e .

6 x 10 2

value (€) 0

-1

10 15 20 25 30 35 40 45 50 emission allowance price (€)

Figure 15: The value of the CCS upgrade option (green) and its payoﬀ (blue) with a constant spread ratio between electricity and emission allowance prices. T = 3 month and C = 1.250.000 e . 44

6 Conclusions

In this thesis we introduced a model for valuing generation assets which consider an own stochastic process for each price uncertainty of the clean spark spread. We included many different but typical power plant properties into the model at the same time. The resulting real option problem was solved by backward induction using a least square Monte Carlo method. With this model one can estimate the value for different types of generation assets. Especially when a power supplier has to extend or rebuild his portfolio of power plants he could estimate the expected profits. Additionally, he could analyse the affect of the asset profits and operation moods when there are essential changes in the energy market and he could figure out if an asset with different properties would show a different behaviour.

Beside the asset value, operational states and switching decisions are important issues for a power plant operator. Our real option model support a concrete decision for the actual state and a decision path according to price movements which can be used for a running unit. Especially the operator can figure out if it is valuable for a specified unit to switch off during the low-price period at night or not. Surely, after some time the model parameters must be recalibrated to the changed market.

Generation asset runtimes may be up to 40 years. This is a known problem for stochastic valuation of generation asset because the maturities of market contracts comes up to at least 10 years. Furthermore long maturing contracts becomes very illiquid. Thus, it is highly questionable if the calibrated parameters of the stochastic price processes could describe the behaviour of prices for longer maturities. In our example calculation we only used shorter maturities to avoid this kind of problem.

In section 5 we consider that a generation asset always comes with options to upgrade the plant properties and its constraints. We showed a way to value these kinds of options on real options. Here we could make full use of our model with three individual described price processes. A model with a single process for the clean spark spread could not capture the origin of the diﬀerences. In our CCS example we could ﬁnd boundaries for emission allowance prices where it is optimal to exercise. An emission allowance prices above this point would bring no use for emission abatement - in our model world - but increase power prices.

There are a lot of open issues and some possible extension of our model which leaves space for further analyses. At the end, we want to list a few here: 45

• A more complex but adequate model for electricity prices would be a regime switching process that consists of a spike regime and a base regime - see for examples [Ronn02]. This model would take care of observable price spikes in the electricity market that are occurring through supply lacks due to unscheduled or forced outages.

• More power plant properties and operational constraints could be considered which leads us to a wider range of operational states and possible decisions. A high sophisticated approach could even consider the opportunity to resell stored fuel of the fuel stack.

• To improve the least square Monte Carlo approach, one could compare different sets of base-function and provide additional details on adequacy and convergence. On the other side, the same regression function could be used for more than one time point to improve the algorithm speed. For example, one approximation function is regressed for every hour of the day but is used for several days of the month, to keep respect of the hourly electricity price levels.

• We could analyse hedging strategies with standard market future or option contracts.

• Last but not least we are sorry that there wasn’t time to look at special aspects of coal, bio fuel or multi-fuel power plants. 46

The European Union Emission Trading Scheme

The EU-ETS is the largest multi-national, emissions trading scheme in the world and is a major pillar of European Union climate policy18. The EU-ETS currently covers more than 10,000 installations which are collectively responsible for close 19 to half of the EU’s emission of CO2 . Since 1 January 2005, companies that are part of the EU-ETS have needed allowances to emit CO2. Sanctions are severe if companies decide not to comply with the regulation. If a ﬁrm does not deliver a suﬃcient amount of allowances at the end of each year, it will have to pay a penalty and will have to deliver the missing EU allowances the following year.

There was a trial period (2005-2007) mainly intended for large industrial emitters. Each Member State set up its own National Allocation Plan (NAP) which must indicate how many emission allowances the Member State intends to issue altogether. The Member must allocate at least 95% of allowances for this period free of charge20. This approach has been criticized as giving rise to windfall profits, being less efficient than auctioning, and providing too little incentive for innovative new competition to provide clean, renewable energy. To address these problems, the European Commission proposed various changes in a January 2008 package - start of the first Kyoto commitment period (2008-2012) - including the abolishment of NAP’s from 2013 and auctioning a far greater share (60%) of emission permits.

The EU scheme used to be a system of climate change policy that was com- pletely independent of International Climate Change Policy such as the United Nations’ Framework Convention on Climate Change (UNFCCC, 1992) or the Ky- oto Protocol that was subsequently (1997) established under it. When the Kyoto Protocol came into force on 16 February 2005, the EU ETS had already become operational. Only later, the EU decided to accept Kyoto flexible mechanism certificates as compliance tools within the EU ETS. The Linking Directive allows operators to use a certain amount of Kyoto certificates from flexible mechanism projects in order to cover their emissions21. According to the Kyoto Protocol, the European Union must reduce its emissions by 8% below the 1990 levels during 2008 to 201222. 18Oxford Journals, Review of Environmental Economics and Policy, 2007 19http://en.wikipedia.org/wiki/European Union Emission Trading Scheme 20Federal Ministry for the Environment, Nature Conservation and Nuclear Safety, National Allocation Plan for the Federal Republic of Germany, March 2004 21http://en.wikipedia.org/wiki/Clean Development Mechanism 22United Nations, Kyoto Protocol to the United Nations Framework Convention On Climate Change,1998 47

Like the Kyoto trading scheme, the EU scheme allows a regulated operator to use carbon credits in the form of Emission Reduction Units (ERU) to comply with its obligations. A Kyoto Certified Emission Reduction unit (CER), produced by a carbon project that has been certified by the UNFCCC’s Clean Develop- ment Mechanism Executive Board, or Emission Reduction Unit (ERU) certified by the Joint Implementation project’s host country or by the Joint Implementa- tion Supervisory Committee, are accepted by the EU as equivalent. Thus one EU Allowance Unit of one tonne of CO2 (EUA) was designed to be identical with the equivalent Assigned Amount Unit (AAU) of CO2 defined under Kyoto. Hence, because of the EU’s decision to accept Kyoto-CERs as equivalent to EU-EUAs, it will be possible to trade EUAs and UNFCCC-validated CERs on a one-to-one basis within the same system. 48

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