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.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 customers 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 customers 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 customersconsumption 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: jkeppo@stat.columbia.edu1Present address: Cap Gemini, Management Consulting, Nuttymaentie 9, FIN-02200, Espoo, Finland. E-mail: Mika.Rasanen@capgemini.fi

0140-9883r99r$ - see front matter Q 1999 Elsevier Science B.V. All rights reserved. .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

we 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 customers hourly electricity consumption and the hourly energy price processes.The price process is observed from the electricity exchange places. Both, the priceand customers consumption processes are stochastic processes by their nature. For

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

The measurement of customers 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

.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

. .load profiles see, e.g. Taylor and Lester 1975 , Bunn and Farmer 1985 , Bartels et . .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 supplierscontract 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

( )J. Keppo, M. Rasanen r Energy Economics 21 1999 213]223 216

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

probabilistic structure of the markets, we will refer to an underlying probability .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 ariables of the market follow an Ito stochasticdifferential equation:

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

. w x4probability space V,F,P , along with the standard filtration F : t g 0,t .tAssumption A1 guarantees the existence and uniqueness of the solution to Eq.

.1 and it says that there are n q 1 independent Brownian motions in them .electricity markets. For electricity price Eq. 1 becomes

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

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

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

. .Eqs. 2 and 3 mean that uncertainty in electricity price and in a singlecustomers 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

.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

customers 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 customers

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

( )J. Keppo, M. Rasanen r Energy Economics 21 1999 213]223 217

means the amount of money that the customer spends on the consumption ofelectricity at time t. That is, the value of the customers consumption pattern

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

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

. . . . . .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 . . . . w x .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.

. . .Proof: Using Itos lemma, see, e.g. ksendal 1995 , and Eq. 4 we get Eq. 5 .Q.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 alue 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 customers consumptionpattern is given a follows:

w .xE g Tt m . w . 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 .xE q Tt m .6

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

. x w . .x .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

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

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

( )J. Keppo, M. Rasanen r Energy Economics 21 1999 213]223 218

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

T . . . . . .g T s g t exp a y q a y q e y e y 9Hm m w q w qm mt

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

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

and

T T1 . . . . . . . .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 q2 5m m m mt t

. . .Eqs. 6 , 8 and 9 give

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

.By using Itos lemma, we get Eq. 7 .Q.E.D.

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

.2 . . .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 customers 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 customers 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 customers consumption based on actual measure-ments are, in general, smaller than forecast errors based on standard load profiles .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

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

3. Pricing of tariffs

This section considers the valuation of tariffs for a single customers consump-tion pattern, which is based on standard pricing...