#### A quick primer

A series of recent articles and papers have highlighted issues around valuation of physical generation assets and linked in with some work I was doing on Real and financial options. I thought it would be good to combine these into a blog post but, as I starting writing, it became clear that the material would be better served spread over a couple of posts.

Therefore, I have split the topic in two; the first half covers the background to valuation of assets, consideration of real and financial options and sets the ground work for the second half. (not yet uploaded) This covers the implementation and issues surrounding asset valuation whilst presenting some examples from theory and practice. The two posts draw heavily upon the work of Clewlow, Meador, Sobey and Stricklany (Energy Risk, June 2009) and Tseng, Barz (Operations Research, 2002) and I include links to the source material and supporting documentation at the bottom of the post.

Please also note that spark spread in US terms refers to any fuel, whilst European operators use named spreads designating the fuel type; spark-spread for gas, dark-spread for coal etc. The term spark spread in this post can be used in lieu of any thermal generation asset.

#### Background and overview

With deregulation of the electricity industry a global trend, utilities and power generators must adjust to the new risks of volatile spot prices in the competitive marketplace. Decisions to run the plant based solely upon satisfying demand have been replaced by an optimisation problem which is at least price-based and takes into account price stochastics. Solving such a problem not only yields the optimal commitment decision, but, also reveals the power plant’s value over the operating period.

#### Options – from the classroom to the coal face

One of the benefits of treating an asset as a real option is that we can make use of the many techniques that have been developed for the valuation of financial options. It is worth understanding the difference between financial and real options so that we understand the limitations these techniques impose on us when they are used to value generation assets.

Firstly, typically financial options are paid for ‘up front’ and there is no significant cost to exercising the option. As we have seen, there usually is a start cost associated with the generation asset. Since the start charge is accounted per start and not per hour run, it is more complicated to implement start costs in a closed-form solution than in a Monte Carlo solution.

The second major difference between financial options and generation assets is that once the financial option matures, we can immediately exercise it. Generation assets, on the other hand, have a ramp rate, which implies that we can’t instantly go from having the unit off to running at maximum capacity. In other words, we need to decide to exercise the real option of the generation asset before its ‘expiry’.

Thirdly, most financial options can have the payoff described in a single payoff function that can be easily written down. This is even true for some path dependent options like Asian options. The operational constraints of a generation asset such as start costs, ramp rates, and minimum up /down times, require us to keep track of prior states of the unit. This requirement makes it difficult to write out a simple payoff function for a generation asset with all the operational constraints. In order to make use of many of the standard techniques from financial options, many of the constraints of the generation asset are typically ignored or modelled in a less than ideal way.

Fourthly, direct application of financial options theory ignores the crucial dynamics inherent in physical assets making a number of incorrect assumptions. Notably, a zero start-up time ie the unit can be started immediately when favourable prices are observed implying no commitment-decision lead time,  no minimum up/downtime constraints ie a unit can be turned down when prices are unfavourable and finally, the unit heat rate is constant at all levels of power generation.

Ignoring these factors overestimates a power plants value since these approaches always results in a non-negative payoff. Incorrectly suggesting that the operator face no risk of loss. Examination of any assets operators’ financial statements would suggest that this is an optimistic view of reality.

Generation assets are beastly, complex and brutish pieces of machinery built to extract energy from a fuel source and transform it into electricity.  Their function is built around a conversion process; a power plant, with its associated heat rate, converts a particular fuel into electricity. Because this conversion involves two commodities with different market prices, owning a power plant can be regarded as holding call options of spark spreads, defined as the electricity price less the product of the heat rate associated with the fuel and its price.

When electricity prices are high, but the fuel price is low the power plant should run to capitalise on the profitable spread. When the spread is negative, the power plant should not run. Because the power plant profits increase as the spread increases, and because its losses are bounded, the power plant resembles a call option on the price spread.

Thus, in the works of Hsu (1998) and Deng (1999) the authors give the example of a generator with a heat rate, H, generating 1 MWh electricity, requiring H MMBtu of fuel heat content. Because this conversion involves two marketed commodities, the payoff of a generator can be modeled as a linear system of their market prices. Assuming H is known, for every 1 MWh electricity generation:

$\textup{Payoff} =p^{E} - H \cdot p^{F}$

where $p^{E} \textup{ (fuel cost ie GBP/MWh)}$ and $p^{F} \textup{ (fuel cost ie £/MMBtu)}$ stand for electricity and fuel prices respectively. Given this situation, a rational plant operator will decide to run a unit only if $p^{E} > H \cdot p^{F}$. Therefor, over the period $\left [ 0,T\right ]$ they propose:

$\textup{Power Plant Value} = \sum_{t=1}^{T} E_{0}\left [ max\left ( p_{t}^{E}-Hp_{t}^{F},0 \right ) \right ]$

So that given the price processes of electricity and the fuel, a power plant’s value may be estimates by a series of (European) spark-spread call options (expiring at t).

#### Reality draws us towards Real Options

The power generation process of a steam unit begins with heating water in its boiler and, therefore, requires time to start up or shut down the generator. Additionally, a thermal-generation unit cannot switch between online mode and offline mode at an arbitrary frequency, due to both the nonzero response time of the unit and the damaging effects of fatigue. Consequently, once a thermal unit if shut down (or started up), it is required to stay offline (or online) for a minimum period before it can resume. The following list highlights some of the common properties that must be considered when attempting to value a generation asset and indicated just how many different properties there are.

1. Maximum capacity
2. Minimum stable generation
3. Heat rate
4. Variable operational and maintenance cost
5. Start cost
6. Ramp-up rate
7. Ramp-down rate
8. Minimum up time
9. Minimum down time
10. Emissions
11. Outages
12. Scheduled maintenance
13. Fuel transportation costs
14. Power transmission costs

The maximum capacity (MEL) is probably the most widely known as represents the maximum amount of power than can be produced in a given time frame. This can change from month to month with the fluctuation occurring as a function of the thermal gradient between the plant and the ambient air or water temperature.

Additionally, the heat measures the efficiency of the asset and can also vary with the thermal gradient and load. The unit is least efficient when generating at the minimal stable level (SEL). Typically, the unit will be modelled as being dispatched at a capacity between the MEL and SEL. This generation profile is unlikely to be linear and can be modelled as a series of step increments or as a continuous function. Start costs are charges associated with the starting of the unit. Some of these related to the purchase of fuel to be consumed getting the unit up to SEL (heavy fuel oil for example). Other costs to consider are wear-and-tear caused by starting and stopping a unit.

Every asset also has the possibility of suffering a forced outage – random outages that reduce operating capacity or take it off entirely for a period of time. In addition, there are scheduled outages for maintenance and repair.

#### The case of capacity unused capacity

So far, we have only considered a plant based upon its energy output. Other “products” associated with a power plant, such as ancillary services and emission allowances which can also add value.

For example, when a unit is online (spinning) and its generation level is neither MEL nor SEL, the residual capacity can be sold to a spinning-reserve market. If the unit is called to generate on contingency it gain additional value from this energy. On the other hand, the power could sell its emission allowances in exchange for less generation (thus less emission) if it believes doing so is profitable. With the presence of markets other than energy, the valuation problem becomes a complicated optimisation problem where one must consider allocation of resources to different markets simultaneously.

#### Quick consideration of the price process

I’ll be covering more about spot and future prices in commodity markets in a future post but as a comment. Commodity price models are characterised by mean reversion and log normally distributed seasonal prices. Because, to varying degrees, both electricity and fuel have associated storage costs, their prices are determined by the forces of producer supply and consumer demand. This interplay is manifested in the mean-reverting nature of their price processes. In some sense, the mean reverting parameter represents the storability of the commodity, For electricity, which is difficult to store, this parameter is large, implying little autocorrelation between today’s price and tomorrow’s price. Furthermore, this parameter in conjunction with the volatility captures short and long-term price fluctuations.

Additional consideration should be given to the periodic nature of long-term expected prices as the interplay between the cost of production and consumer demand changes.

#### References

CHUNG-LI, T. and BARZ, G., 2002. Short-term generation asset valuation: A real options approach. Operations research, 50(2), pp. 297-310.

CLEWLOW, L., MEADOR, D., SOBEY, R. and STRICKLAND, C., 2009. Valuing Generation Assets: Overview & Spark-Spread Option Valuation. Energy Risk, , pp. 76-81.

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