Download pdf - Pricing of electricity tariffs in competitive markets

Ž .Energy Economics 21 1999 213]223

Pricing of electricity tariffs in competitivemarkets

Jussi KeppoU, Mika Rasanen1¨ ¨Helsinki Energy, Kampinkuja 2, PL 469, 00101 Helsinki, Finland

Abstract

In many countries electricity supply business has been opened for competition. In this paperwe analyze the problem of pricing of electricity tariffs in these open markets, when both thecustomers’ electricity consumption and the market price are stochastic processes. Specifi-cally, we focus on regular tariff contracts which do not have explicit amounts of consumptionunits defined in the contracts. Therefore the valuation process of these contracts differsfrom the valuation of electricity futures and options. The results show that the more there isuncertainty about the customer’s consumption, the higher the fixed charge of the tariffcontract should be. Finally, we analyze the indication of our results to the different methodsfor estimating the customer’s consumption in the competitive markets. Since the consump-tion uncertainties enter into the tariff prices, the analysis indicates that the deterministicstandard load curves do not provide efficient methods for evaluating the customers’consumption in competitive markets. Q 1999 Elsevier Science B.V. All rights reserved.

JEL classifications: D4; L94

Keywords: Electricity pricing; Stochastic demand; Competitive markets; Tariff design

1. Introduction

The Scandinavian countries provide the first multinational electricity markets,where traders can buy and sell electricity between nations. In this market area each

U Corresponding author. Present address: Department of Statistics, Columbia University, New York, NY10027, USA. Tel.: q1 212 854 3652; fax: q1 212 663 2454; e-mail: [email protected] address: Cap Gemini, Management Consulting, Nuttymaentie 9, FIN-02200, Espoo, Finland.¨ Ê-mail: [email protected]

0140-9883r99r$ - see front matter Q 1999 Elsevier Science B.V. All rights reserved.Ž .PII: S 0 1 4 0 - 9 8 8 3 9 9 0 0 0 0 5 - 5

( )J. Keppo, M. Rasanen r Energy Economics 21 1999 213]223¨ ¨214

individual customer can buy electricity from any company providing electricitysupply services. In Norway the markets were opened to all customers in 1993.Sweden and Finland followed this trend in 1995 when markets were opened forlarge and medium scale customers. In 1997 all customers were allowed to enterinto the free markets. In addition to the Scandinavian countries, the UK and NewZealand have already opened their supply business for full competition. Thecompetition in the supply business requires that distribution networks and nationalgrids must have equal pricing principles for each operator in the market. There-fore, the governments regulate the distribution business in all of the abovecountries at the moment.

In this paper we consider the pricing of the electricity supply tariffs in thecompetitive markets. Specifically, we focus on regular tariff contracts which do nothave the explicit amount of consumption units defined in the contracts. The pricingof distribution services is not considered, since in the pure competition it shouldnot affect customers’ electricity supply contract decision. In the valuation of tariffs

Žwe assume that the additional services, i.e. billing services supply and transmis-.sion , reporting services, etc., are constants. Thus, they are added into the fixed

charge of the tariff. In our analysis, the price of the supply service is derived fromthe customer’s hourly electricity consumption and the hourly energy price processes.The price process is observed from the electricity exchange places. Both, the priceand customer’s consumption processes are stochastic processes by their nature. For

Ž .a different analysis of consumption processes see, e.g. Brown and Johnson 1969 ,Ž . Ž .Chao 1983 and Rasanen et al. 1995 .¨ ¨

The measurement of customer’s consumption is one of the main issues in thecompetitive markets, since each supplier must have a balance between his salesand supply contracts. In each of the countries having free markets, the regulateddistribution companies are responsible for the customers’ energy measurements. InScandinavia, the energy measurement interval is 1 h. In UK and New Zealand, thetime span is 30 min. It would be simple for free competition, if all customers in themarket had measurement devices that collected these hourly or half-hourly energymeasurements. Unfortunately, the energy measurement devices are quite expen-sive. For example, in Finland the annual hourly measurement cost per customerwas approximately US$500 at the end of 1996. If a customer had 10% lower bills inthe free market than with his local supplier, his annual electricity bill should be atleast US$5000 to make it profitable for her to enter into the competitive markets.This is beyond the electricity bill of an average household without electricity space

Ž .heating see, e.g. Rasanen et al., 1997 . To solve this measurement problem and to¨ ¨give all customers a possibility to benefit from the free markets, an alternative tothe hourly or half-hourly measurements has been discussed. In Norway for exam-ple, the customers with small consumption are associated to a single standard loadprofile, which is scaled according to the customers past total annual or monthlybilling measurements. For different statistical approaches for building the standard

Ž . Ž .load profiles see, e.g. Taylor and Lester 1975 , Bunn and Farmer 1985 , Bartels etŽ . Ž .al. 1992 and Rasanen et al. 1996 . The use of this standard load profile based¨ ¨

( )J. Keppo, M. Rasanen r Energy Economics 21 1999 213]223¨ ¨ 215

method is also under parliamentarian discussion in Finland and in the UK.Therefore, we present a general analysis, which covers both standard load profilesand hourlyrhalf-hourly measurement. Methods to improve the standard loadprofiles are also discussed.

In our pricing model, a single customer is interested only in the amount ofmoney that she will spend in her electricity consumption. This money amount is astochastic variable that depends on the electricity price and the amount ofconsumption at each moment of time. Because different customers have differentconsumption behaviors, the dynamics of the money amounts are different. Weassume that a single customer will consume electricity in the future according to agiven stochastic consumption model and we price the energy options on thatamount of money. Most of the electricity contracts in an electricity supplier’scontract portfolio are these kinds of tariff-contracts.

Energy is bought for consumption and it can not be considered as a tradableasset. Therefore, the market price of risk is liable to enter into the pricing of thetariff. We show how to evaluate this price of risk by using the futures prices on themoney amount that the customer will spend on consumption of electricity. Insteadof assuming that the variable underlying in the tariff design process is the moneyamount, we use the future price. In Scandinavia, electricity future prices can beobtained from the electricity exchange market places, e.g. NordPool, ELEX, etc.,where the electricity future contracts are traded. However, there are no standardinstruments for the money amount.

The main contribution of this paper is that it develops a tariff pricing methodthat takes into account both the price and the consumption uncertainties. Theproposed methods have already been implemented as a part of the contractportfolio management system in Helsinki energy.

The paper is divided as follows: Section 2 introduces the price and consumptionmodels used in the paper. The stochastic processes for the money amount that thecustomer spends on consumption of electricity as well the dynamics for futureprices are derived. These processes are then applied in the pricing problem. Thepricing rules are summarized in Section 3. A numerical example shows how themodel is applied in practice in Section 4. The main results of this paper aresummarized in Section 5.

2. Consumption and market price models

2.1. Price and consumption processes

We consider an electricity market where energy instruments are traded continu-w xously within a time horizon 0,t . This kind of market exists in Scandinavian

countries, where electricity producers and suppliers trade electricity 24 h each dayin a year. The market consists of a set of customers, M, and the number of

( )elements in M is n . Each customer m g M has consumption q t at timem m


w x Ž .t g 0,t . The total consumption at time t is q t . In describing theÝ mmgM

probabilistic structure of the markets, we will refer to an underlying probabilityŽ .space V,F,P . Here V is a set, F is a s-algebra of subsets of V, and P is a

probability measure on F. The following assumptions characterize our electricitymarkets.

Assumption A1: The stochastic äriables of the market follow an Ito stochasticˆdifferential equation:

Ž . Ž . Ž . Ž . Ž .d x t s a x ,t d t q e x ,t dz t 1

w x w x nmq1where a : R = 0,t ª R and e:R = 0,t ª R are gi en functions that satisfyŽ .Lipschitz and growth conditions on x and z t is a standard Brownian motion on the

Ž . � w x4probability space V,F,P , along with the standard filtration F : t g 0,t .t

Assumption A1 guarantees the existence and uniqueness of the solution to Eq.Ž .1 and it says that there are n q 1 independent Brownian motions in them

Ž .electricity markets. For electricity price Eq. 1 becomes

Ž . Ž . Ž . Ž . Ž . Ž . Ž .dW t s W t a t d t q W t e t dz t 2w w

Ž .where W is the energy price, and for the consumption of m g M Eq. 1 is

Ž . Ž . Ž . Ž . Ž . Ž . Ž .dq t s q t a t d t q q t e t dz t 3m m q m qm m

Ž . Ž .Eqs. 2 and 3 mean that uncertainty in electricity price and in a singlecustomer’s consumption is generated from the n q 1 Brownian motion processes.mThat is, the price process and the consumption processes can be correlated,uncorrelated or partially correlated. The first general analysis where the electricityconsumption and price processes are divided into correlated and non-correlated

Ž .processes can be found from Chao 1983 .

Assumption A2: There is no arbitrage.

Assumption A2 means that there are no opportunities for risk-less profits in themarket. If there exists such risk-less trading opportunities, the greed traders and

Žcustomers will collect those instruments out from the market see, e.g. Brown and.Sibley, 1986 .

2.2. Valuation of consumption patterns and future price dynamics

Here, we derive the value process of the consumption pattern corresponding to asingle customer and the process of the future prices on the value of customer’s

w xconsumption pattern. The value of the consumption pattern at time t g 0,t


means the amount of money that the customer spends on the consumption ofelectricity at time t. That is, the value of the customer’s consumption pattern

Ž . Ž . Ž . w x Ž .g t s q t W t for all m g M , t g 0,t 4m m

Lemma 1: The älue process of the consumption pattern for the customer m g M is

Ž . Ž . Ž . Ž . Ž . Ž .d g t s g t a t q a t q e t e t 9 d tm m w q W qm m

Ž . Ž . Ž . Ž . w x Ž .qg t e t q e t dz t for all m g M , t g 0,t 5m w qm

where e9 means the transpose of e.

Ž . Ž . Ž .Proof: Using Ito’s lemma, see, e.g. Øksendal 1995 , and Eq. 4 we get Eq. 5 .ˆQ.E.D.

Before we derive the process for the future prices we must make sure that thereexists this kind of contracts in our electricity market.

Assumption A3: There are future contracts on the älue of consumption patterns.

In a competitive electricity market, there are huge amounts of future contractscontinuously traded in exchange places. Typical contracts in these electricityexchanges are futures or bundles of futures where the amount of energy and futureprice per energy unit is fixed. Normally, a supplier calls these bundles to fill hersupply contracts and a producer puts this type of bundles to keep her production asoptimal and stable as possible. With the above assumption, we avoid a pricingmodel where there is a market price for the risk parameter. If assumption A3 is notvalid, it can be replaced by the valuation of the future contracts.

In our market, the future price of the value of the customer’s consumptionpattern is given a follows:

w Ž .xE g Tt mŽ . w Ž . x w x w xW t ,T s exp y T y t r for all m g M , t g 0,T , T g 0,tm w Ž .xE q Tt m

Ž .6

where r is the constant instantaneous discount rate and E is the conditionaltw Ž . Ž .x w Žexpectation operator with respect to P, since E W t,T q T s exp y T yt m m

. x w Ž . Ž .x Ž .t r E q T W T and W t,T is F -measurable. The assumption of the constantt m m tdiscount rate is made to simplify the model.

Lemma 2: The process of the future price is gi en by

Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .dW t ,T s W t r t y e t e t 9 d t q W t e t dz tm w q wm

w x w x Ž .for all m g M , t g 0,T , T g 0,t 7

TŽ . Ž . Ž . Ž . Ž .where W t,T s W t exp a y q e y e y 9 y r d yHm w w q½ 5mt


Ž .Proof: From Eq. 3 and Lemma 1 we get

TŽ . Ž . Ž . Ž . Ž . Ž .g T s g t exp a y q a y q e y e y 9Hm m w q w qm m½t

1Ž . Ž . Ž . Ž .y e y q e y e y q e y d yw q w qm m2

T Ž . Ž . Ž . Ž .q e y q e y dz y 8H w qm 5t

and

T T1Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .q T s q t exp a y y e y e y 9 d y q e y dz y 9H Hm m q q q q2½ 5m m m mt t

Ž . Ž . Ž .Eqs. 6 , 8 and 9 give

TŽ . Ž . Ž . Ž . Ž . Ž .W t ,T s W t exp a y q e y e y 9 y r d y 10Hm w w q½ 5mt

Ž .By using Ito’s lemma, we get Eq. 7 .ˆQ.E.D.

Ž .Eq. 7 says that the stochastic process for the future price follows Ito process,ˆŽ .w Ž . Ž . Ž . xwhere the conditional expected change is W t r t y e t e t 9 andw qm

Ž .2 Ž . Ž . Ž .W t e t e t 9 is the conditional variance of W t,T . That is, the consumptionw w mpattern affects only to the expected drift of the future price and the volatility of thefuture price is the same as the volatility of the electricity price. This result is simpleto explain: the more there is uncertainty about the customer’s hourly consumptionpattern the higher the future price will be. In practice, this means the customershaving a predictable consumption will have lower prices than customers having anunpredictable consumption behavior.

If standard load profiles are used for approximating customer’s electricityconsumption, i.e. the consumption pattern is fixed, the customers having anassociated standard load profiles must obtain lower prices than similar customerswith hourlyrhalf-hourly measurements. This is a contradiction, since with theabsolute forecast error of the customer’s consumption based on actual measure-ments are, in general, smaller than forecast errors based on standard load profilesŽ .see Rasanen et al., 1996 . A possible solution to above problem is to set standard¨ ¨profiles proportional to some stochastic external factors. In Scandinavian countries

Ž .this factor is typically outdoor temperature Rasanen et al., 1995 .¨ ¨

3. Pricing of tariffs

This section considers the valuation of tariffs for a single customer’s consump-tion pattern, which is based on standard pricing of contingent claims. Therefore,


the presentation will be brief, since detailed analysis of contingent claims valuationŽ . Ž .can be found, e.g. Harrison and Kreps 1979 , Harrison and Pliska 1981 and

Ž .Heath et al. 1992 .Ž .Given A2 there exists equivalent martingale probabilities for W t,T . Usingm

Ž .Girsanov’s Theorem see, e.g. Øksendal, 1995 we get

Ž . Ž . Ž .dz t s dz t q n for all m g M 11˜m m

˜ ˜Ž .where z is a Brownian motion on V,F,P , P is an equivalent martingale˜m m mŽ . Ž . Ž . w xmeasure, and r y e t e t 9 s e t n for all m g M and t g 0,t . Becausew q w mm

there may be many different equivalent martingale measures, the market may be˜incomplete. However, there is no arbitrage because P is used only with m g M,m

and the customers can not take advantage of the diverging pricing functions in theeconomy.

Lemma 3: The process of the future price under the equi alent martingale measure isgi en by

Ž . Ž . Ž . Ž . w x w x Ž .dW t ,T s W t e t dz t for all m g M , t g 0,T , T g 0,t 12˜m w m

Ž . Ž .Proof: Lemma 2 and Eq. 11 give Eq. 12 .

Q.E.D.

The value of an electricity call option can now be calculated as the price ofŽ .regular call options see, e.g. Hull, 1993 where the energy charge is considered as

a strike price, fixed charge as the premium of the option, and the future price as anunderlying asset. We can state the following theorem.

Theorem 1: The time t T-maturity arbitrage-free price of a electricity call option Cmfor customer m g M is

m˜Ž . Ž . w Ž .x w x w xC t ,T s exp t y T r E C T ,T for all m g M , t g 0,T , T g 0,tm f t m

Ž .13

where r is the instantaneous risk-free rate at time t and it is assumed to be constant,f˜mE is the conditional expectation with respect to the risk-neutral probability measuret˜ ˜mw Ž .xP , and E C T ,T is calculated by using Lemma 3.m t m

Ž .Proof: Given A2, if Eq. 13 does not hold there exists arbitrage opportunities.For the more detailed proof and the use of equivalent martingale measures see,

Ž .e.g. Duffie 1992 .

Q.E.D.

Theorem 1 is a pricing rule only for one energy call option, i.e. a pricing rule foran option to buy at time T one energy unit at a given energy charge corresponding


to customer m g M. Typically tariffs are understood as a portfolio of differentmaturity options each of which are priced according to Theorem 1. That is, from

Ž .Eq. 6 we get that the value of the tariff for customer’s consumption pattern is

TŽ . Ž . w Ž .x w x w x Ž .D t ,T s C t , y E q y d y for all m g M , t g 0,T , T g 0,t 14H m t mt

Ž .By using the model of Black 1976 we get

Ž . Ž . w Ž . Ž . Ž . Ž .x Ž .C t ,T s exp t y T r W t ,T N d y X T N d 15m f m 1 2

where

1 Tw Ž . Ž .x Ž . Ž .ln W t ,T rX T q e y e y 9d yHm w w2 td s1T Ž . Ž .e y e y 9d yH( w w

t

T Ž . Ž .d s d y e y e y 9d yH(2 1 w wt

Ž . Ž .where N 9 is cumulative normal distribution and X t is the strike price, i.e. theenergy price, at time t.

4. Numerical example

In this section we illustrate our tariff-pricing model with a hypothetical numeri-cal example. The pricing interval is from 1 January to 29 January 1998. The currentdate is 1 January 1998 and the time interval is divided into 1-h periods, i.e. we

Žanalyze the case of Scandinavian markets. In this model, r s r s 5% annual,f.continuous-time and the electricity price and the consumption of a representative

customer are assumed to be distributed according to lognormal distributions. Theexpected electricity price is shown in Fig. 1. The standard deviation is assumed tobe equal to 0.2 during the time interval. The uncertainties of price and consump-tion are perfectly correlated in this example.

The expected consumption of the customer is shown in Fig. 2. The standarddeviation of the predicted consumption is assumed to be equal to 150 at each hour.

Fig. 3 illustrates the value of electricity call options with different maturitieswhen the energy charge is fixed at the level of US$1.8. Since the tariff contracts areportfolios of call options, the value of the tariff is high when the value of theelectricity price is high.

Ž . Ž .By using Eqs. 14 , 15 and Figs. 2 and 3 we can calculate the value of the tariff,


Fig. 1. Expected electricity price.

which is US$952. In a case of deterministic consumption process, the value of theŽ . Ž .tariff is US$732. As Eqs. 10 and 13 show the tariff with the consumption risks is

more expensive than the corresponding deterministic case. In this example thedifference is US$220. That is approximately 30% increase in the fixed charge of thetariff.

5. Summary

In this paper we have derived a pricing model for tariffs on the value ofcustomer’s electricity consumption pattern in the competitive electricity supply

Fig. 2. Expected consumption pattern of the customer.


Ž .Fig. 3. The price of electricity call options with different maturities energy charge US$1.8 .

markets. In this valuation model, we use the future price of the customer’selectricity consumption pattern value instead of using directly the consumptionpattern value. In order to obtain a risk neutral probability distribution for thefuture prices, we apply equivalent martingale measure. However, usually theredoes not exist such future contracts and these hypothetical securities have to bepriced before the valuation model can be used. A customer’s consumption affectsthe prices in the tariffs only through the future price. The main result of this studyshows that the more there is uncertainty in the customer’s electricity consumptionpattern, the higher the fixed charge of the tariff must be. The analysis also offers aframework for understanding the net risks corresponding to the tariffs portfoliosfor different customer groups.

The analysis indicates that deterministic standard load profiles as such cannot beapplied in competitive supply business, since they do not include consumptionrisks. Therefore the customers associated with standard load profiles will facelower prices than similar customers having hourly or half-hourly measurementdevice installed.

References

Bartels, R., Fiebig, D., Garben, M., Lumsdaine, R., 1992. An end-use electricity load simulation model.Utilities Policy January, 71]82.

Black, F., 1976. The pricing of commodity contracts. J. Fin. Econ. 3, 167]179.Brown, G., Johnson, M., 1969. Public utility pricing and output under risk. Am. Econ. Rev. 59, 119]128.Brown, S., Sibley, D., 1986. The Theory of Public Utility Pricing. Cambridge University Press, New

York.Bunn, D., Farmer, E., 1985. Comparative Models for Electric Load Forecasting. Wiley, Chichester.Chao, H., 1983. Peak load pricing and capacity planning with demand and supply uncertainty. Bell J.

Econ. 4, 179]190.Duffie, D., 1992. Dynamic Asset Pricing Theory. Princeton University Press, Princeton.


Harrison, L., Kreps, S., 1979. Martingales and arbitrage in multiperiod securities markets. J. Econ.Theory 20, 381]408.

Harrison, J., Pliska, S., 1981. Martingales and stochastic integrals in the theory of continuous trading.Stochastic Processes Applications 11, 215]260.

Heath, D., Jarrow, R., Morton, A., 1992. Bond pricing and the term structure of interest rates: A newmethodology for contingent claims valuation. Econometrica 60, 77]106.

Hull, L., 1993. Options, Futures, and Other Derivative Securities. Prentice-Hall, New Jersey.Øksendal, B., 1995. Stochastic Differential Equation. Springer-Verlag, Berlin.Rasanen, M., Ruusunen, L., Hamalainen, R., 1995. Customer level analysis of dynamic pricing¨ ¨ ¨ ¨ ¨

experiments using consumption pattern models. Energy 20, 897]906.Rasanen, M., Ruusunen, J., Hamalainen, R., 1996. Object-oriented software for electric load analysis¨ ¨ ¨ ¨ ¨

and simulation. Simulation 66, 275]288.Rasanen, M., Ruusunen, J., Hamalainen, R., 1997. Optimal tariff design under customer self-selection.¨ ¨ ¨ ¨ ¨

Energy Econ. 19, 151]167.Taylor, R., Lester, D., 1975. The demand for electricity: A survey. Bell J. Econ. 6, 75]110.