An options model for electric power markets

An options model for electric power markets

Electrical Power & Energy .~vsterns, Vol. 19, No. 2, pp. 75--85, 1997 Copyright ~';, 1996 Elsevier Science Lid Printed in Great Britain. All rights re...

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Electrical Power & Energy .~vsterns, Vol. 19, No. 2, pp. 75--85, 1997 Copyright ~';, 1996 Elsevier Science Lid Printed in Great Britain. All rights reserved ELSEVIER

PII: S0142-0615(96)00019-1

0142-0615,'97/$17.00+0.00

An options model for electric power markets Kanchan Ghosh anti V C Ramesh Department of Electrical and Computer Engineering, Illinois Institute of Technology. Chicago, IL 6061 6, USA

by the U.S. congress heralded a major structural change in the way the electric power business had hitherto been conducted. This research paper includes consideration of two major elements brought forward as a consequence of the changing situation in the power industry. They are:

The international electric utility industry is undergoing a radical transformation from an essentially regulated and monopolistic industry to an industry made uncertain with impending deregulation and the advent of competitive forces. This paper investigates the development of an options market for bulk power trading in a market setup while considering power system planning and operational constraints and/or requirer,nents. In so doing it considers the different market based fnancial derivative instruments which can be used to trade electrical power in bulk and examines how established tools such as Optimal Power Flow (OPF) may be appl,:ed in helping to develop a price for bulk power transactions under a market based setup. Copyright ~ 1996 Elsevier Science Ltd

(i) With increasing efforts towards making the electric power system respond to free market pressures the understanding of how to analyse and price power system performance under extreme events becomes even more critical. In this research, tools are examined which help us do so under a market based mechanism. (ii) Innovative approaches using optimization techniques are suggested. These approaches are based on proven free market and power system security analysis concepts.

Keywords." modelling of energ)' systems, optimal power .[tows, assessment

The basic assumptions that are made in the course of the research outlined in this paper are: (i) The existence of an electric bulk power market under an organized exchange setup. The New York Mercantile Exchange (NYMEX) is slated to begin the U.S.'s first futures market in bulk power transactions in March 1996. This research considers a more advanced and financially sophisticated market--an options market. (ii) The existence of a brokerage agency/firm where brokers set up bulk power transactions and purchase/ sell options (under an exchange setup) on such transactions. The brokerage house is also responsible for pricing such options. Many well known brokerage firms have indeed shown an interest in handling such market based bulk power transactions.

I. I n t r o d u c t i o n The international electric utility industry is undergoing a radical transformation from an essentially regulated and monopolistic industry to an industry made uncertain with impending deregulation and the advent of competitive forces. The United States, Great Britain and some developing countries such as India are already seeing the effects of deregulation. Great Britain is probably ahead of the other nations in this respect. This paper deals with the unification of free market forces with electric power system security concepts. One of the dynamic catalysts in the growth of American industrial power has been the triumphant spread of electric power technology. Though the United States has for some time now been considered a mature industrial power, legislative moves to make the U.S. electric utility industry less regulated and more amenable to free market competition have led to the revisiting of the electric power industry infrastructure question. The enactment of the National Energy Policy Act in 1992

This paper investigates the development of an options market for bulk power trading in a market setup while considering power system planning and operational constraints and/or requirements. In so doing it considers the different market based financial derivative instruments which can be used to trade electrical power in bulk and examines how established tools such as Optimal Power Flow (OPF) may be applied in helping to develop a price for bulk power transactions under a market based setup. In considering the different derivative instruments used in

Received 30 March 1995; revised 8 February 1996; accepted 18 April 1996

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Options model for electric power markets: K. Ghosh and V. C. Ramesh

a market setup for exchanging of commodities, the research outlined in this paper explains why an options market may be the best option for a bulk power market. In order to link the pricing of bulk power to electrical concerns such as parallel flows, voltage security and thermal overloads, OPF techniques are applied. The results obtained from the OPF runs are in turn linked to the pricing formulae adopted in the market structure. This research thus, in effect, develops an options market concept and an interface between the market and the power engineering principles involved in transmitting electric power. This will be a first step in developing a highly evolved bulk power market in an exchange setup while at the same time taking into consideration multiarea power system planning and operational issues. The enactment of the National Energy Policy Act (NEPA) in November 1992 by the U.S. Congress, heralded a major structural change in the way the electric power business has been conducted. The electric utility industry has long been classified as a natural monopoly (for a discussion of natural monopolies see References 1 and 2). However, it now seems that major changes to the industry are likely in the near future. This includes the possibility of considerably more freedom of access to utility transmission systems by Independent Power Producers (IPP) and Non Utility Generators (NUG). The Federal Energy Regulatory Commission (FERC) has indicated its willingness to mandate open access to utility owned transmission systems for IPPs and NUGs. The FERC's implementation of NEPA has begun the process of further opening up utility owned transmission systems to both buyers and sellers of electric power. A good example of regulatory activism is the California Public Utility Commission (CPUC) which has proposed extending complete access, ultimately to the smallest unit of customers. For the electricity market to achieve the extent of deregulation that the California PUC proposes, a market structure for large volume electricity transactions at market set prices is probably an inevitable consequence. The effectiveness of any tool is at least partially dependent on the details of the future structure of the utility industry with evolving deregulation. While the exact details of this structure are still rather unclear, there are some known options available to the electric utility industry. What is clear though, is the fact that in order to derive the maximum advantages from deregulation, FERC and regional and state regulatory bodies will have to outline a clearer vision of the U.S. electric utility industry in the coming years. A brief discussion [3] of the available options follows: In a 1989 FERC report [4] three options were presented as a future framework of the electric utility industry. They were, (i) The contract model (it) The joint planning model (iii) The U.K. model or the British model. As mentioned in Reference 4 the third option implies the vertical disintegration of the electric utility industry. That is probably an unfeasible option for the U.S. electric utility industry. The contract option would not require any major changes on the part of the utility industry i.e. the same utilities would own. operate and plan transmission capacity. On the other hand, to address questions such

as risk bearing for wholesale wheeling and the priority ol" firm transmission service vis-fi-vis third party requirements there will be major structural changes involved in the planning option. The contract option also allows utilities a significant advantage in market power. This attains added significance in the event that FERC allows utilities to collect opportunity costs. Opportunity costs [5] are costs that a utility has incurred i.e. the profits it has foregone in providing the wheeling of transmission access service required. The potential profits would have stemmed from sales to native load or to other third party customers. Shirmohammadi et al. [6] outline a method for calculating operating costs. Basically the method involves calculating the change in the operating costs and and existing costs due to a transmission transaction. These changes are then expressed in terms of opportunity costs. Opportunity cost may be negative in a situation where a potential transaction relieves transmission congestion instead of increasing or worsening it. As seen from the discussions in References 5 and 6, opportunity costs are probably the most contentious issue in transmission pricing. One of the criticisms of allowing utilities to collect opportunity costs is that it could leave a utility with market power with no incentive to expand its transmission capacity. This, in turn increases transmission congestion. In an extreme situation this could potentially lead to accruing of monopoly profits. To prevent such a situation from occurring it is possible that utilities could arrive at a voluntary agreement among themselves to promote the expansion of transmission capacity whenever it is beneficial to do so. This would also be in the utility's long term interests. The planning option however, would ensure equal priority for all firm transmission contracts and thus ascribe no market power advantages to the utilities. Even though the planning option entails major structural changes to the existing utility structure, it is far less drastic than the British model. A good example of this would be the issue of retail wheeling of power. Both the contract and planning options disallow retail wheeling. This is in contrast to the British model which allows any customer to shop for the least expensive available power. A compromise structure will possibly involve a combination of a contract and joint planning model with more of the latter's features included to stimulate true competition in the deregulated industry. FERC and regional and state regulatory bodies are working on the details of the future structure.

II. The problem The possibility of limited or large scale deregulation in the electric power industry has given rise to a number of issues which may be broadly classified as: (a) The consideration of a future market structure for bulk power transactions. Such a structure would involve the application of different financial derivative instruments. (b) The effect of power system security issues on bulk power pricing under such a market structure which may be further classified as: • Inter utility power transaction levels considering system security issues like voltage and system stability.

Ol~tions model for electric power markets: K. Ghosh and V. C. Ramesh • Inadvertent parallel path flow problems through utility systems not it party to the power interchange transaction. • The use of certain analytical methods like Optimal Power Flow (OPF) analysis techniques to assign costs to such flows. I1.1 Terminology Financial derivatives are instruments whose values depend on a more basic underlying financial variable like a stock or bond. Examples of financial derivatives are forward contracts, futures, options, swaps etc. These are also known as derivative securities. Forward contracts are one of the simpler forms of financial derivatives. A forward contract is an agreement between financial institutions to buy or sell an asset at a future date at a fixed price. Forward contracts are not traded in an exchange setup. Futures contracts are a kind of forward contract. It is an agreement reached at one point in time calling for the delivery of some commodity at a specified later date at a price established at the time of contracting. In effect it is a forward contract traded on an organized exchange with contract terms clearly specified by the rules of exchange. At the close of business every day a futures contract is rewritten with a new cor, tract price equal to the futures price. This in turn means that a futures contract is a series of forward contracts over a period of time. Examples would be currencies, debt instruments and financial indices. Options contracts are the right to buy or sell for a limited time a particular commodity at a specified price. There are two varieties of optio:as, call options and put options. Depending on limitation.,+ of exercise time options can be classified as European options or American options. Option contracts are discussed in more detail in Sections III and IV. Swaps A swap is an agreement between two or more parties to exchange sets of cash flows over a period in the future. Two examples of swaps are interest rate swaps and currency swaps. Embedded costs primarily relate to costs of 'already built' or existing transmission equipment owned by an electric utility. There are four kinds of embedded costs: (i) (ii) (iii) (iv)

Postage stamp or rolled-in costs Contract path costs MW-mile costs Boundary flow cost~,..

The first two categories have been extensively used by FERC and the utilities to calculate bulk power wheeling costs. The applicability of both these categories is being increasingly questioned with the advent of open access. The last two categories use actual power flow data and hence may attain increasing relevance in an open access environment. Opportunity costs Opportunity costs are costs of the opportunities that a util:ty forgoes to provide wheeling service. These lost costs may include service to native load or other foregone wheeling opportunities. For a suboptireally operated power system, opportunity costs may theoretically be negative. Spot pricing The spot price of electric power is based on two components: (i) Cost of generation of electrical energy

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(ii) Cost of transmission used as common carrier to wheel electrical energy. Part (i) has been dealt with in the literature extensively [7,8]. Part (ii) which concerns transmission line usage pricing is an issue which remains to be resolved. FERC and several utility groups are debating various related issues such as parallel path flows. The Inter-regional Transmission Coordination Forum (ITCF) is examining parallel path pricing through its General Agreement on Parallel Paths (GAPP) committee. At present there is no industry consensus on parallel path pricing. Major exchange houses across the country have been examining for some time now the possibility of setting up a large volume electricity market based on the model of existing commodities markets. Two such major markets are the New York Mercantile Exchange (NYMEX) and the Chicago Board of Trade (CBOT). N Y M E X is expected to have a futures market in electricity in operation by March 1996. While CBOT is not involved in such a market at the present time, studies concerning such a market have been undertaken in the recent past and there is a possibility of CBOT involvement in the future. Other non-traditional entities such as financial houses have expressed an interest in participating in the electricity market as brokers. Since bulk electricity being traded in these markets will be treated as a commodity with title rights to the purchaser, the activities of such firms will be under the jurisdiction of FERC. The issue of inter utility transaction levels is of critical importance in an open access environment. This is because mandated open access transmission may have an effect on firm sales of electricity to a utility's existing customers. This is even more so when the utility concerned has+ limited transmission assets i.e. inadequate capacity for serving both existing and third party wheeling customers. In circumstances like this, the use of opportunity costs could act as a measure of prioritizing between competing transactions. In the past FERC has been unwilling to consider opportunity costs in its rulings. However, in an open access environment, this may no longer be the case. The issue of inadvertent parallel flows also attains added significance in an open access structure. Parallel flows may occur when wheeled power between two utilities flows through the transmission system of a third utility not a party to the transaction. Unfortunately, from a financial point of view there is no easy way to account for the cost of this inadvertent flow of power through the facilities of a third party. Various utility groups have launched initiatives to better understand this problem. One of these groups is the Interregional Transmission Coordination Forum (ITCF) which has formed a committee to explore the issue of parallel path flows in an open access environment. This committee [9,10] is the General Agreement on Parallel Paths (GAPP). Analytical tools like OPF are one of many tools which can be applied by the utility industry as a way to analyse power flows on the open access transmission systems of the future including the issue of inadvertent parallel flows. Examples of other attempts to better understand and analyse parallel path flows can be found in References 9 and 10. Optimal Power Flow (OPF) is a tool which may be used to some advantage [11] to resolve this question. This would help quantify the problems of inadvertent parallel

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Options model for electric power markets. K. Ghosh and V. C. Ramesh

path flows by assigning a cost factor to such flows. The concerned areas (utilities) could then negotiate a general form of agreement over such occurrences. Initially, this method would be more of a planning than an operating tool. In all the above cases, thermal overloads, voltage security and contingency constraints will form part of the OPF analysis. In the next section a solution will be proposed which examines the different choices for a bulk power market. Subsequently, the suggested method will include an approach for linking the pricing mechanism used for an options market to power system security issues concerning thermal overloads, voltage security and contingency constraints by the use of OPF.

III. E v o l u t i o n of bulk p o w e r m a r k e t s The major types of markets considered by exchanges and financial organizations when treating electricity as a commodity in a bulk power trading setup are: • • • • •

Cash markets. Forward markets. Futures markets. Options markets. Swap markets.

II1.1 Cash markets Cash trading in the bulk power market exists as a relatively crude form of bulk power transactions between utilities. By some estimates about 50% of the electricity produced in a given year in the United States is traded in the cash market. Examples of bulk power trading mechanisms which can be compared to a cash market are the Florida Power Pool system and the electronic bulletin board based systems. Examples of electronic bulletin board systems are the IPEX System with members in the Mid West and the East Coast, the Brokering system with membership in California and neighboring states and a third system called the Continental Power Exchange with an interactive power trading network. Use of these systems is primarily confined among utilities and they have none of the sophistication of the financial derivatives markets which allow parties to hedge against price risk and volatility. 111.2 Price risk In a market based scenario where electricity is treated like any other commodity, considerations of risk and volatility in bulk power prices, especially for long term contracts become very important. This is especially true since wholesale electricity prices have historically been very volatile. Until now most of this price risk has been passed on to consumers. However, progress towards increasing deregulation and market based electricity rates will shift part of the price risk onto electricity producers. Some entities which are either purely producers or consumers of electricity and do not own transmission facilities will bear a greater part of the risk. Hence, the relevance of financial derivatives as applied to electric power markets to control price risk and volatility. 111.3 Forward markets A forward contract between two entities for trading electric power involves an agreement to do so at a

predetermined and mutually agreeable future date and at a predetermined mutually agreeable price. There are no costs involved in entering into a forward contract. In a forward contract, at maturity or at the agreed upon delivery time, one party delivers electric power to another in exchange for a cash amount equal to the predetermined price. The party delivering the bulk power holds a short position in the forward contract as opposed to the party paying the case price who holds the long position. If the spot price of the bulk power at time of delivery is S ( t ) per MW where t is the current time, and the predetermined delivery price of the bulk power is P per MW, then the payoff from the short position is defined as ( P - S ( t ) ) per MW and the payoff from the long position is defined as ( S ( t ) - P) per MW. From the definition in Section I, we note that a forward contract occurs outside an exchange setup. At the same time, the risk and volatility of electric prices are reflected in the payoffs. Thus a forward contract combines features of both cash trading and the derivatives markets. However due to reasons of risk credibility i.e. guarantees whether a contract will be honored, this kind of contract may be possible only between parties who are already acquainted with each other. To make the bulk power market work in an unrestricted manner between parties not necessarily known to each other or otherwise, an exchange setup with its guarantees of risk worthiness will be more appropriate. This is where the concept of a futures market becomes important [12-14]. 111.4 Futures markets Futures contracts are very similar to forward contracts. As in forward contracts, it does not cost any money to enter into a futures contract. However futures contracts are entered into in an exchange setup. The rules of the exchange ensure that though the parties in a contract do not necessarily know each other, the contract will be honored. This feature of the futures market makes trading flexibility higher for futures contracts relative to forward contracts. Examples of exchanges where futures are traded are the Chicago Board of Trade (CBOT), the Chicago Mercantile Exchange (CME) and the New York Mercantile Exchange (NYMEX). Unlike a forward contract however, a futures contract does not have an exact predetermined date but has a specified delivery month instead and the exchange concerned sets an exact date during the delivery month. Extending a concept outlined in Reference 12, futures markets provide a way for bulk power traders to diversify risk. However, an even more important advantage to be derived from the existence of a futures market in bulk power is that utilities, IPPs, QFs and transmission facility owners can make decisions on power production, reserve generation and building or upgrading of transmission facilities by observing the pattern of bulk power futures markets. The efciencies inherent in the futures markets can save millions of dollars by the development of such a market. A market like this would not necessarily require the complete deregulation of the utility industry. A lot of the efficiencies may be obtained from partial deregulation. Such a partially regulated structure accompanied by an efficient market would transform millions of dollars of economic benefits into societal benefits as well [15,16]. As mentioned earlier in this paper N Y M E X is expected to have a futures market in electricity in operation by March 1996. NYMEX, which is the world's largest

Options model for electric power markets: K. Ghosh and V. C. Ramesh energy exchange, is exp.oring a 14 state region in the Western U.S. for this futures exchange. The reasons N Y M E X looked to the Western U.S. are: (i) As mentioned earlier in this paper, California has been considering thz issue of open access very seriously. Relative to the rest of the country, the regulatory climate for the growth of IPPs has been very conducive in this region. Specifically, the power pools in the Northeastern region of the country are tightly interconnected and will probably be slower in developing a free market. (ii) A large part of the geographical region which forms part o f the Western Systems Power Pool (WSPP) specifically the Callfornia-Oregon corridor, has a substantial and successful operating history in a cash market for bulk power transactions. This, as we have seen from the discussion in this section, is a precursor of evolution into a futures market. (iii) NYMEX's expertise as an energy exchange as well as its experience in managing price risk and volatility make it well suited to be involved in an evolving electric power market. The typical duration for a N Y M E X electricity futures contract will be about 1'3 months. CBOT, though not involved in such a scheme at present, has considered a cash market in the past and has plans to be involved in the future. The CME is also not involved at the present, though given the fact that there are enormous opportunities in the bulk power market that situation might be expected to change. One issue which is likely to influence the decision of exchanges to be involved in such markets is the question of transmission interconnections. The interconnection facilities between power systems across the country vary according to geographical region. The Midwest is a relatively high load density area and heavily interconnected. This is true for the Northeastern part of the country as well. However, in the Western United States, (especially in the Rockies) transmission lines stretch over long distances and are relatively few in number. In this context, large bulk power transactio;~s raise issues of thermal and stability limits as well as transmission losses on transmission lines. This limits the extent of possible bulk power trading in an organized market and exchange structure. It is still not clear whether futures markets in bulk power will follow across areas with limited transmission interconnection facilities. As the players in the market begin to realize the economic efficiencies to be made, transmission interconnections will most likely increase [17-19].

111.5 Options markets One limitation of a forward or futures market from a market point of view is that the parties to a forward or futures contract are under a contractual obligation to honor the contract reg~,rdless of market behavior at the time of the honoring of the contract. An options contract on the other hand binds the parties to no similar obligations to exercise the opt:.on before the party is ready to do so. In Section I, we introduced the terms call options and put options. Expressed in terms of an electricity market, a call option gives the holder the right to buy a certain amount of power by a o,~rtain date and for a certain price. A put option on the otl~.er hand gives the owner the right to sell electric power by a certain date for a certain price. The date is known as the strike date and the price is

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known as the strike price. If the option can only be exercised on the exercise date it is known as a European option. If it can be exercised at any time up to the exercise date, it is an American option. The underlying asset which in this case is electric power can also be an electricity futures contract. A detailed discussion on options markets and options futures markets follows in Section V.

111.6 Swap markets Swap contracts are cash exchange agreements between two entities over a period of a few years. As applied to an electricity market, swaps are very similar to an electric market for forward contracts. Most existing swap contracts are done outside of the exchanges and hence are not subject to exchange regulations. At present there are very few swap transactions in the electric industry. The same concerns regarding credibility of the parties involved and risk worthiness in a forward contract also apply to swap contracts. This would mean that swap markets for bulk power are not likely in the near future. 111.7 Summary From the discussion above, it is apparent that a futures market is a potentially effective market structure for bulk power trading. However an options market in electricity futures would be a further step forward in making the bulk power market highly sophisticated. This would take into account considerations such as price risk and volatility along with the added flexibility that an options market in futures offers over a pure futures market. Though N Y M E X is expected to have only a futures market in operation by March 1996, it is reasonable to expect an options market in electricity futures as the next logical step in the evolution of electricity markets. The following section presents a model of the pricing mechanism of an options market in electricity futures contracts.

IV. The proposed solution IV.1 Option pricing in electricity futures prices To develop a pricing mechanism in a futures market, two scenarios have to be considered. One scenario is where the asset concerned is held for investment purposes and the other is where the asset is being held for consumption purposes. In the second case, i.e. where the asset is being held for consumption purposes, it is not possible to express the futures price as a function of the spot price. Expressed in terms of electricity markets, this translates into the idea that we cannot express a futures price in electricity in terms of its spot price if it is being held for consumption purposes. Entities are more likely to try to maximize a profit from an asset held for consumption based on short term shortages rather than purchase futures contracts. A measure of such profits is given by a financial term called 'convenience yields'. A good example of assets being held for consumption are generation reserves during short term generation shortages during a peak load period which consequently could imply that futures contracts are not applicable to short term generation reserves. However, in this proposal we shall consider electrical power as an asset held for investment purposes. Under this scenario, assuming storage costs for electrical power (e.g. spinning reserves) to be zero, the futures price F at

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Options model for electric power markets: K. Ghosh and V. C. Ramesh

Modelling of the

electrical power system in a power flow

Determination of "base optimal cost"

Determination of "optimal cost"

software

i

3~ I Difference between the f 'base optimal cost" and "optimal cost" to calculate "actual

optimal cost"

J

T .......

1

Use 'actual optimal i cost" to calculate ] 'Modified spot price"

r

I

J Use "Modified spot "7 price" to determine I price volatility

ad level reached

loo%~ (85% ......... loo%)

I J~ ......

Use volatility numbers in Black-Scholes formula to determine options on futures prices

STOP

)

Figure 1. Flow chart illustrating method for a specific transaction time t is given by, F = Se r(T-')

(1)

where, t = current time in years S -- current spot price of electricity r = risk free interest rate per year at time t T = time in years when futures contract matures. The electric power being considered here as an investment asset is any power deemed as excess by a utility and available for sale. This power may be traded in an exchange setup through brokers or brokerage firms to be used in case of future need by another entity which would be the purchaser of the power. Before we continue with the application of options to the futures contract discussed above, it will be helpful to discuss the development of the model on which the options future pricing equation depends. Financial pricing mechanisms are based on what is known as a Markov stochastic process. One specific

example of such a process is known as the Wiener process. A Wiener process for a variable x may be represented as: dx = adt + bdz

(2)

where a and b are constants and z is a variable which follows the Wiener process. A generalized form of this process can be extended to express a futures price F asl3: dF -- = #dt + adz F

(3)

where, # and cr are, respectively, the expected rate of return and the volatility of the spot price. The pricing of any derivative security is a stochastic function of the asset (i.e. price of electric power) underlying it and that of time. Using a fundamental result in mathematics known as lto's Lemma, the futures price F can be expressed as: dr =

~---~+ -~- + 0.5 0 - ~ tr-S-

d t + -O- F ~ciSdz

where, dz = measure of uncertainty in the price of electricity. All other quantities are as defined before. It can be

Options model [or electric power markets: K. Ghosh and V. C. Ramesh shown [1] that with the choice of an appropriate portfolio of shares and a derivative security the Wiener process can be eliminated to allow F to be expressed in the form of the following equation (4) where all quantities are as defined previously: OF

OF

--~- + r S ~

2 , 02F

+ 0.5a S ' ~

= rF

(4)

Equation (4) is known ;is the Black-Scholes [12] differential equation. The Biack-Scholes equation may be applied to different kinds of financial derivatives, depending on the boundary conditions prevailing at the time of exercise of the contract. As applied to options contracts (call and put) on electricity futures prices, the BlackScholes expression for the options price is: c = e - r ( r - t ) [ F U ( d l ) --XN(d2)]

(5)

p = e - r ( r - , ) [ X U ( _ d 2 } _ V N ( _ d l )]

(6)

where, c and p are the call and put options prices respectively and N ( . ) are the cumulative normal probability values of dl and d2. The values of dl and d 2 are given by: In d~ =

+ ~- ( T -- t) d2 = d~ - o v ~ T

- t)

where X is the strike/delivery price of the option. All other terms are as defined before. Of the terms in the equations shown above, F, X, T and t are directly observable from the market. However, a which is the price volatility is not directly observable. This is probably the most critical issue in pricing electricity options contracts. Specifically, a has to be considered in a number of different contexts: Scenario I: Depending on the transaction, pricing of power system voltage stability and thermal loads on transmission lines for different interconnected systems have to be considered [20,21]. Scenario I1: Control and/or pricing of parallel path flows for certain transactions over transmission lines of systems not involved in the transaction have to be considered [22,23]. Scenario II1: Bulk power transactions in a contingency constrained system [24]. Note (l) An options contract can take the form of a call or a put option. Consequently, there is a call price and a put price. This will be illustrated in the next section. (2) An important point lo note here is that the spot price S of the bulk power is set by the market, i.e. what the market can bear. Our objective in this research is to make a determination of the price of the call/put option on the bulk power transaction. For this purpose we will use OPF as a tool. OPF will be used extensively to calculate costs for bulk power transactions betwcen systems for all three scenarios mentioned above. OPF will be used primarily as a tool to analyze power system functioning on an optimum basis. This would enable different options on electricity

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futures pricing to be compared on the same basis i.e. 'optimal system operation'. This might be of importance also, in case these options markets come under the jurisdiction of FERC; in which case the agency may require different entities to compare prices based on a standardized methodology. Hence for these reasons, OPF will be used to link options prices to electrical issues by the following steps (see Figure 1): (i) Modelling of an IEEE test system in power flow software. (ii) An optimal solution of the base test case to calculate the 'base optimal cost'. (iii) An optimal solution with each of the three scenarios modelled above to find the 'optimal cost'. This will have to be done exhaustively for all multi area combinations for each transaction. (iv) The difference between the 'base optimal cost' and 'optimal cost' is the 'actual optimal cost' due to the occurrence of a certain scenario for a particular multi area transaction. (v) These 'actual cost' numbers from multiple OPF runs will be related back to a, or price volatility, in the Black-Scholes pricing formula for options. The entire process will have to be repeated for different load levels. Depending on the values of the other parameters in the Black-Scholes formula, this will enable the calculation of the price of a call (put) option for a specific multi area bulk power transaction. Step (v) is important because it enables us to express the volatility of bulk power pricing in power system security issue terms [25]. A large database of OPF calculated prices will allow a reasonable derivation of volatility, or.

V. Test results In this section we illustrate a procedure to calculate the option price of a bulk power transaction using the concepts outlined in the previous section. For the purposes of this paper we will consider the modified IEEE 30 bus test system with three control areas as shown in Figure 2. The control areas are labeled 1, 2 and 3. A commercially available OPF program is employed for the test. The problem is formulated as follows: Given: Area 1 approaches a broker to sell power in blocks of 20, 50 or 70 MW at a certain time T in the future. Area 3 approaches the same broker to buy power in the same quantities (blocks) at the same time T. We will assume that a contract path exists between areas 1 and 3 through area 2. The power transaction may occur for load levels ranging from 85 to 100% of peak load level. However, both areas 1 and 3 are not certain about the exact amount of power they will sell/buy at what load level. Also, areas 1 and 3 want to include any power system security issues that this transaction might entail. Hence, they plan to hedge their risks by buying an options contract from the broker on the futures price of the transaction, which will take into account the power system security impacts of all three blocks of power. All transactions are assumed to take place over 1 hour. Find." The call and the put prices of the options contracts that the broker signs with area 1 and area 3. Since any power transaction from area 1 to area 3 has to go through area 2, the options price must also reflect the security impact of all the possible transactions on area 2. For the purposes of this paper assumed costs were used

82

Options model for electric power markets." K. Ghosh and V. C. Ramesh

ii (~ I

From the above calculations, the volatility, c~, is obtained as 0.07. This is comparable to typical volatility numbers obtained for options pricing of other commodities as seen in Reference 13.

V.1 Calculation of futures price

,~,ea3

For the purposes of this paper we will assume the following:

@ Area _~



Spot price of electricity, S = $10/MW; This price, as mentioned in Section IV.l, is set by the market. Risk free interest rate, r = 5%, = 0.05 and (T - t) = 5 yrs. With the above assumptions, we can calculate a value for the futures price of the bulk power transaction as:

F = Se r(T t ) = 10eO05isi = 12.84.

V.2 Calculation of options price from the BlackScholes formula for area 1

AreaI

We will assume that the strike price for the option on futures contract between area 1 and the broker is X =$11/MWh. Therefore, F=

Figure 2. Modified IEEE 30 bus test system for production costs for generators. The procedure for the modelling of the IEEE 30 bus test system and the application of the O P F program to calculate costs were according to steps (i)-(v) as outlined in Section IV. 1. To reiterate these steps in the context of the example problem: (i) The modified 30 bus IEEE test system is modelled in a commercially available power flow software. (ii) An optimal power flow solution of the base 30 bus IEEE system is run to calculate the 'base optimal cost' for control areas 1 and 2. (iii) An optimal solution is performed with three transactions (20, 50 and 70 MW) modelled in the case separately. All the transactions flow from area 1 to area 3 through area 2. The costs for these transactions are defined as the 'optimal costs'. This will be done for each load level, i.e. 85, 90, 95 and 100% of peak load level. (iv) The difference between the 'base optimal cost' and "optimal cost' is the 'actual optimal cost' (expressed as 'Transaction Price' in Table 1) due to the occurrence of each multi area transaction. (v) From the 'actual optimal cost', a $/MWh cost for each transaction at each load level is determined. To the $/MWh costs for each transaction is added the bulk power spot price to get the "modified spot price of bulk power' in $/MWh. From the $/MWh costs volatility a is determined by statistical methods. For the purposes of this paper we will assume a market set spot price of bulk power as $10/MWh. Table 1 shows the results of these runs. From the data in Table 1 we obtain the mean cost and the volatility of the transactions (by the standard statistical methods) for both areas 1 and 2 as:

Eci 272.66 c . n. . . 24

11.36

/ • ( c i - c) a = g -~ ~-1) - 0.07.

12.84, X = l l , ( T - t ) = 5

From d(2)) as for d(l) N(d(2))

andcr=0.07.

the formulae for dl and d2 (herewith d(l) and given in Section IV. 1, we can calculate the values and d(2) and obtain the values of N(d(1)) and from the probability distribution tables as:

d ( l ) = 1.07, N ( d ( l ) ) = 0.9525, N ( - d ( 1 ) ) = 0.0475 d(2) = 0.91, N(d(2)) = 0.9345, N ( - d ( 2 ) ) = 0.0655. From equations (5) and (6) in Section IV. 1, we obtain the values of the call and put options as: c = e-°5i5}i12.84(0.9525 ) - 10(0.9345)] = 2.25 p = e -° 5(5)[10(0.0655) - 12.84(0.0475)] = 0.04. Here, area 1 has a put option on the bulk power transaction while the broker has a call option on the transaction.

V.3 Calculation of options price from the BlackScholes formula for area 3 We will assume that the strike price for the option on futures contract between area 3 and the broker is X = $ 9/MWh. Therefore,

F = 12.84. X = 9 , ( T - t ) =

5

and

a=0.07

From the formulae for dl and d2 (herewith d(1) and d(2)) as given in Section IV.I, we can calculate the values for d(1) and d(2) and obtain the values of N ( d ( l ) ) and N(d(2)) from the probability distribution tables as: d(1) = 2.35, U ( d ( l ) ) = 0.9525, N ( - d ( l ) ) = 0.0475 d(2) = 2.19. U(d(2)) = 0.9345, U ( - d ( 2 ) ) = 0.0655. From equations (5) and (6) in Section IV. 1. we obtain the values of the call and put options as: c = e °°5(5~[12.84(0.9525) - 10(0.9345)] = 2.25 p = e-°°515/[10(0.0655) - 12.84(0.0475)] = 0.04. Here, area 3 has a call option on the bulk power transaction while the broker has a put option on the transaction.

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Options model for electric power markets: K. Ghosh and V. C. Ramesh

The example problem illustrated above may be moditied to suit different needs for different transactions. The broker can thus apply the method above to come up with an options price for a transaction which also includes consideration of power system security issues. Thus, with the above test example we have linked together power system security issues, which are important for bulk power transaction purposes, with options pricing issues in free market environments. Two apparently completely different concepts have been linked together through basic principles. As mentioned before, call and put option prices as derived above can be obtained for various permutations of the problem outlined at the beginning of Section V. The example given in Section V however does not include consideration of Scenario ill in Section IV.I. Examples including this scenario will be published in forthcoming research reports. However, qualitatively the fundamental concepts remain the same [26-28].

VI. Conclusion The research in this paper has considered two widely differing fields of work and linked them together. The issues of bulk power transaction security and market infrastructure have been studied in isolation for some time. However, until the present no attempt has been made to unify these two areas of work, perhaps because it was not needed. It should be recognized that unifying these two aspects of the bulk power market infrastructure problem is only a first step towards making this unification more comprehensive. An example of this would be the possibility of introducing emission constraints in the power system infrastructure formulation [29,30]. Research on price volatility as it relates to the unbundling of electricity services is the interface between the market realities and technical realities of electricity commodity pricing. The technical realities that have been and will be considered in further research include [31,32]: (i) Consideration of parallel path flows including consideration of PSTs on transmission lines. (ii) Thermal overloads on transmission lines. (iii) Voltage stability considerations on transmission systems. (iv) Contingency constraints on the transmission system, Large scale implementation of the research outlined in this paper will lead to its application by large brokerage firms and trading houses which at present may not have the infrastructure to deal with security problems in large scale bulk power transfers. The proposed interface model will enable brokers and similar agencies to price the value of a call/put option of a certain bulk power transaction with relative ease. The issues addressed in this research are of increasing concern to utilities around the world [33]. The free market, or relatively open bulk power market, structure will probably be given its final form in the next two years. Crucial to its successful implementation will be the full participation of major financial exchanges and electric utilities. Massive, fundamental change in the electric power supply industry is just around the corner.

VII. References I Stigler, G. J., The Theory of Price (3rd Edn), 1966.

2 Gould, J. P. and Ferguson, C. E., Microeconomic Theory (5th Edn). 1980. 3 Ghosh, K. and Behera, A. K., 'The future of the U.S. electric utility industry technical and economic issues and possible solutions' Proceedings of the North American Power Symposium, Manhattan, KS, Sept. 26-27, 1994. pp. 1-97.

4 The Transmission Task Force's Report of the Commission: Electrieiu" Transmission: Realities, Theory and Policy Alternatives FERC Report. 1989.

5 Landon. J. H., Pace, J. D. and Jaskow P. L., 'Opportunity costs as a legitimate component of the cost of transmission service' Public Utilities Fortnightly, 1989. 6 Shirmohammadi, D., Rajagopalan, C., Alward, E. R. and Thomas, C. I., 'Cost of transmission transactions: an introduction', IEEE Trans Power Syst, 1991,6(3). 7 Bohn, R. E., Caramanis, M. C. and Schweppe, F. C., 'Optimal pricing in electrical networks over space and time' Rand Journal of Economics, 1984, 15(3), 360 376. 8 Siddiqi, S. and Baughman, M. L., "Optimal pricing of nonutility generated electric power' IEEE Trans. Power Syst, 1994, 9(1), 397-403. 9 Happ, H. H., 'Cost of wheeling methodologies'. IEEE Trans Power Syst, 1994, 9( I ). 10 Kavicky, J. A. and Shahidehpour, S. M., 'Parallel path aspects of transmission modelling' IEEE Winter Power Meeting. 1995, Paper # 141-2-PWRS. 11 Mukerji, R., Neugebauer, W., Ludorf, R. P. and Catelli, A., 'Evaluation of wheeling and non-utility generation (NUG) options using optimal powerflows" IEEE Trans. Power Syst, 1992, 7(1). 12 Black F., 'The pricing of commodity contracts' Journal of Financial Economics 3 (1976) 3- 51. 13 Hull, J. C., Options, Futures and other Derivative Securities Prentice Hall, 1993. 14 Smith, C., 'Option pricing--a review' Journal of Financial Economics, 1976, 3, 3-51. 15 Gedra, T. W., "Optimal forward contracts for electric power markets, IEEE 1994 Winter Power Meeting New York, NY, Paper 94WM 159-4PWRS. 16 Gedra, T. W. and Varaiya, P., 'Markets and pricing for interruptible electric power' IEEE Trans. Power Syst, 1993, 8(1), 122-128. 17 Mukerji, R., Ellis, S., Garver, L. and Simons, N., 'Analytic tools for evaluating transmission access issues' Proceedings of the 56th Annual American Power Conference, Chicago, IL pp. 300-305 18 Landgren, G. L., Terhune, H. L. and Angel, R. K., "Transmission interchange capability--analysis by computer' IEEE Summer Meeting and International Symposium on High Power Testing Portland, OR July 1971.

19 Landgren, G. L. and Anderson, S. W., 'Simultaneous power interchange capability analysis' IEEE PES Winter Meeting New York, NY January-February 1973. 20 Blake, M., 'GAPP: A proposed solution to the parallel flow problem' Proceedings of the 55th American Power Conference, 55-11, Chicago 1993. 21 Blake, M., 'Examining the dark side of competition' Public Utilities Fortnightly (December 6, 1990). 22 Boxall, J. A., "Washington and the utilities--FERC transmission task force report released' Public Utilities Fortnightly (November 23, 1989).

Options model for electric power markets." K. Ghosh and V. C. Ramesh 23 Graves, F. C., Carpenter, P. R., Ilic, M. and Zobian A., "Pricing of electricity network services to preserve network security and quality of frequency under transmission access' Response to the Federal Energy Regulatory Commission's Request for Comments in its Notice of Technical Conference Docket No. RM93-19-000 (November 8, 1993). 24 Kirk, D. E., Optimat Control Theory--An Introduction Prentice Hall. 1970. 25 Nwankpa, C., 'Stochastic models for small signal stability and voltage collapse in power systems' Ph.D Thesis Report Department of Elec':rical and Computer Engineering, Illinois Institute of Tezhnology, 1990. 26 Ramesh, V. C., Lubkeman, D. and Matulic, I., 'Decision support applications for electric power distribution facility design and management' Proceedings o f the second symposium on expert .wstem application to power systems. Seattle, Washington, July 17-20, 1989. 27 Talukdar, S. N. and Ramesh, V. C., 'A-Teams for real time operations' Int. J. Elect. Power Energy Syst, 1992 14(2/3).

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28 Talukdar, S. N. and Ramesh, V. C., 'A multi-agent technique for contingency-constrained optimal power flows' IEEE Trans. Power Syst, 1994, 9(2), 855-861. 29 Kaye, R. J., Outhred, H. R. and Bannister, C. H., 'Forward contracts for the operation of an electric industry under spot pricing' IEEE Trans. Power Syst, 1992, 5(1), 46-52. 30 IEEE Committee Report 'Operating Problems with Parallel Flows' IEEE Trans. Power Syst, 1991, 6(3), 1017- 1023. 31 Doty, K. W. and McEntire, P. I., "An analysis of electric power brokerage systems' IEEE Trans Power Appar. Syst, 1982, PAS-101(2), 389-396. 32 Monticelli, A., Pereira, M. V. F. and Granville, S., 'Security constrained optimal power flow with post contingency corrective rescheduling' IEEE Trans. Power Syst, 1987,

PWRS-2(1). 33 Ghosh, K., 'Electrical power system ancillary services pricing by application of options contracts and optimal power flows' Ph.D Thesis, 1996.