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IEEE Transactions on Power Systems, Vo1.6, No.1, February 1991 23

REAL-TIME PRICING OF REACTIVE POWER: THEORY AND CASE STUDY RESULTSbyMartin L. BaughmanSenior Member Student MemberProfessor Graduate Research AssistantDepartment of Electrical and Computer EngineeringThe University of Texas at AustinAustin, Texas 78712

ShamsN. iddiqi

ABSTRACTAn analysis is made of real-time pricing policies of reactivepower using a modification of the OPF model. The theory of real-time pricing of reactive power is presented, followed by a case studyillustrating the magnitudes and ranges that real-time prices ofreactive power might take on under different circumstances. Theefficiency implications of real-time pricing of reactive power arecompared with traditional power factor penalties. It is concludedthat real-time pricing of reactive power should developsimultaneously with that of active power for maximum economicefficiency and smooth operation of the electricity marketplace.KEY WORDS;Reactive Power, Real-Time Pricing, Optimal PowerFlowI NTRODUCTIONReal-time pricing of electrical energy is an area of intenseresearch at present. Real-time pricing of reactive power is closelyrelated to that of active (or eal) power.The development of spot pricing theory and the analysis ofthe practice of spot pricing for real power have been carried out byCaramanis, Bohn and Schweppe [1],[2], among others. Schweppe,et. al. [3] see spot pricing becoming more attractive n a competitiveelectricity market, especially at the generation stage whereindependent power producers would adjust their generation levelsaccording to the spot prices paid to the producers for their powerproduction.An extensive discussion on the implementation andfunctioning of a spot price based energy marketplace is made in [4].It includes topics ranging from spot price based rates and revenuereconciliation to optimal investment conditions. a summary ofexisting rates that are related to real-time prices and loadmanagement schemes that reflect real-time pricing policies isprovided in [5]. A utility perspective on spot pricing is given in [6].Most discussions on spot pricing of real power are equally relevantto real-time pricing of reactive power.

Unfortunately, the pricing of reactive power has receivedvery little at tention. A reason for this negligence is the inherentdifficulty in understanding the concept, especially by economists.Berg, et. al. [7] point out the inconsistency and inadequacy of thepresent pricing policies based on power factor penalties. Theysuggest that, given the present high cost of additional investmentsby electric utilities, prices should be derived from economicprinciples, which support a pricing approach that has price equalmarginal costs, that would also reflect today's technologicalconstraints.As power system margins are reduced because of emphasison the greater use of generation and transmission, power systemsdispatchers must operate their systems much closer to their technicallimits. Real-time prices for reactive power provide information toboth users and dispatchers of electricity about the cost and value ofreactive power usage, flows, and sources.

90 SM 466-3 PWRS A paper recommended and approvedby the IEEE Power System Engineering Committee of theIEEE Power Engineering Society for present a t ion a t t heIEEE/PES 1990 Summer Meeting, Minn eapoli s, Minn esota,July 15-19, 1990.1990; made available f o r pr i n t i ng Apr i l 24 , 1990.Manuscript submitted February 1,

Ray and Alvarado [8] use a modification of the OptimalPower Flow (OPF) model which allows for the priceresponsiveness of real power demand to analyze the effects of spotpricing policies. A similar model, but one which also allows for theprice responsiveness of reactive power demand, will be used here toanalyze the impact of real-time pricing of realIn this paper, an analysis is made of real-time pricing ofreactive power using a modification of the OPF model. In the nextsection the theory of real-time pricing of reactive power is presented.In Section 111, a case study illustrating he magnitudes and rangesthat real-time prices of reactive power might take on under differentcircumstances is presented. The next section then discusses theefficiency implications of real-time pricing of reactive power whencompared with traditional power factor penalties. The last sectionconcludes that real-time pricing of reactive power should developsimultaneously with that of active power for maximum economicefficiency and smooth operation of the electricity marketplace.II.THEORY OF REAL-TIME PRIC NG OFREAC'I?VE POWERA theory of real-time pricing of reactive power thataccurately reflects the underlying physical and engineeringproperties of electricity is presented below.The following notationwillbe used in the derivation:C = total system operating costCi(.)= cost function of generating plant at Bus iPgi= active power generation at Bus iQgi= reactive power generation at Bus iPdi = active power demand at Bus iQdi= reactive power demand at Bus iVi = voltage at Bus i6i = voltage angle at Bus iYBUS= Yij]= admittance matrix of the transmission network.8ij = phase angle of YijPi, = active power flow from Bus i to Bus jN = set of all buses in the systemG = set of all buses having generating capacityMCpi= Lagrange multiplier on the active power equation at Bus iM Q = Lagrange multiplier on the reactive power equation at Bus i&, i n =Lagrange multiplier on the minimum active powerhi,m= =Lagrange multiplier on the maximum active powerpi,min= Lagrange multiplier on the minimum reactive powerpi,"=Lagrange multiplier on the maximum reactive powerqi, =Lagrange multiplier on the active power flow limit from Bus i

reactive power.

generation limit at Bus igeneration limit at Bus igeneration at Bus igeneration at Bus ito Bus j

Vimh= Lagrange multiplier on the minimum voltage level at Bus iVim== Lagrange multiplieron themaximumvoltage level at Bus iEpi = price elasticity of demand of active power at Bus iQ = price elasticity of demand of reactive power at Bus iEpqi = cross-price elasticity of demand for active power with

respect to a change in price of reactive power at Bus iEspi = cross-price elasticity of demand for reactive power withrespect to a change in price of active power at Bus iDpi= demand for active power at Bus i for active power andreactive power prices equal to unityDq j = demand for reactive power at Bus i for active powerand reactive power prices equal to unity.0885-8950/91/0200-0023$01.000 1991 EEE

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24It is assumed that a single welfare-maximizingpublic utilityowns and operates the generating plants and transmission networkof the electric power system under consideration and it sells activeand reactive power to independent customers. The utility is assumedto be able to set and communicate prices instantly, and can set adifferent price for each customer location at each moment. Thecustomer responses to these prices are assumed to be known and aregiven by the demand functions. It is also assumed that demands areindependent of past and future prices and, correspondingly, forgenerating costs. Therefore, the model can be solved as a singleperiod model.

Objective:The objective of the pricing policy is to maximize socialwelfare, that is, to maximize consumers' plus producers' surplus,subject to the operational constraints. This is accomplished bysetting prices of real and reactive power at each bus at a particulartime equal to the marginal costs of supplying real and reactivepower, respectively, at that bus and that time, where the marginalcosts are determined on the basis of an optimal load dispatch, that is,real and reactive power dispatch so as to minimize total operatingcosts of the utility, subject to the operational constraints. The patternofoptimal load dispatch is dependent on the real and reactive powerdemands, and, on the other hand, real and reactive power demandsare dependent on the prices, that is, marginal costs determined fromthe optimal load dispatch. Thus, this is a bi-level problem consistingof an upper and a lower part. The upper level problem is that ofsatisfying the demand functions, where the prices are implicitlydetermined by the lower level problem, which is that of minimizingtotal operating costs subject to he operational constraints.Demand Functions:The demand functions are assumed to be of log-linear formexhibiting constant price elasticity and cross-price elasticity.Thedemand functions may be mathematically expressed in the followingequation forms:

Pd;= Dpi ( MCp;)Epi MCqi)Ep9'wi= %i (MQi)E4ph(MCqi)'1 (2)

(1)which give the demand of real and reactive power at Bus i as afunction of prices of real and reactive power at that bus, for all i E N.The parameters of the equations, which are Dpi, Dqi, Epi,Eqi,. Epqi, and Ew i, vary with customer classes, time underconsideration, and other exogenous factors. For example, they maydiffer for residential and commercial customers, the time of day, theseason, weather conditions, etc.The prices of real and reactive power, which are equal tomarginal costs MCpi and MCqi, respectively, when load isdispatched optimally, are implicitly defined by the following lowerlevel problem.Lower-level Objective Function:The lower level problem is basically the optimal power flowproblem which has the objective of minimizing the total cost ofoperating the spatially separated generating units subject to he set ofequations that characterize the flow of power throughout the systemand all operational constraints. Since the operating cost of producingreactive power is much less than that of real power where thecapacity exists, the operating cost of generating reactive power isassumed to be negligible compared to that of real power in theanalysis to follow. Thus, the objective function may be expressed inthe following form:

Minimize C = Ci ( Pgi) (3 )i e Gwhere Ci ( Pgi ) is the operating cost of producing Pgi units of realpower at the generating plant at Bu s i.There are several constraints to the lower level problem.Load Flow Equations:The set of equations, determined by Kirchoffs laws, thatcharacterize the flow of power throughout the system are givenbelow:

(4)gi - Pdi - c Vi I IVj I 1YijICOS( iJ+6j-6i ) = 0j E N

Qg; - Qdi + c Vi I IV,j e Nwhich are the active and

respectively, for all i E N.

lYijeaci

Sin( Bij+6,-6i ) =0 ( 5 )ve power flow equations,

The Lagrange multipliers of the above constraints, M Qi andMCqi respectively, give the marginal costs of supplying real andreactive power at Bus i and thus determine the prices of real andreactive power, respectively, at that bus.Generation Limits:

The generating plants of the utility have a maximumgenerating capacity, above which it is not feasible to generate due totechnical or economic reasons. Generation limits are important indetermining the operating points and marginal costs of generation.Generating limits are usually expressed as maximum and minimumactive power and reactive power outputs,Pgi,min I Pgi I Pgi,max (6)Qgi,max 5 Qgi I Qgi,max (7)where Pgi,min and Pgi,max are the minimum and maximum activepower output and Qgi,min and Qgi,max are the minimum and

maximum reactive power output, respectively, of the generatingplant at Bus i, for all i EG.If the plant at Bus i has only reactive power generatingcapability, for example, capacitor banks or synchronouscondensers, then Pgi min = 0, Pgi,max = 0, Qgi,min = 0, nd the setof constraints effectivkly reduce to0 2 Qgi I Qgi,max . (8)Transmission Limits:Transmission limits refer to the maximum power or currentthat a given transmission line is capable of transmitting under givenconditions. These limits can be based on thermal considerationsoron stability considerations. Thermal limits usually dominate forshorter lines. Dynamic stabilitylimitsdominate longer line behavior.These limits affect the marginal costs of operation. Transmissionlimits are expressed here in terms of the maximum active powerflow through the lines, (9)where Pij = IVi I IVj I IYi, I COS(8ij+6,-6i ) - IVi l2 IYij ICoseij,assuming that shunt admittance is negligible, where Pij,min andPij,max are the minimum and maximum active power flow,respectlvely, through the line connecting Bus i to Bus j , or i # j,and all i, j EN .Voltage Limits:Voltage limits refer to the requirement for the system busvoltages to remain within a narrow range of levels. Since voltagesare affected primarily by reactive power flows, the marginal cost ofsupplying reactive power to a bus is directly dependent in thevoltage level requirement at that bus. The voltage limits can beexpressed by the following constraints, (10)where Vi,min and Vi," are the minimum and maximum voltagelevels, respectively, hat are acceptable atBus i, for all i E N.Model Solution:A method of solution for the bi-level nonlinear programmingproblem is presented below.The Lagrangian to be minimized over all active powergeneration levels Pg, reactive power generation levelsQg, voltagelevels V, and voltage angles 6, is:

Pij,min 5 Pij 5 Pij,max

Vi,min I IViI I ViDa

L ( Pg, Qg, V, 6 ) =c Ci(Pgi)[ operating costs IE G- ( M Q i ) [P gi - Pdi - N~INjITY~~ICos(8i$6J-6i)]

i E N ' j e N[ active power flow equations ]

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26Results:The model formulated in the previous section was solved onthe CDC Dual Cyber 170/750 using GRG2 [9], a program whichsolves nonlinear programming problems using generalized reducedgradient methods. The results obtained from the analysis arepresented below.Behavior of Real-time Prices:The variations of real- time prices of active and reactivepower at Bus 3 for a 24 hour period are shown in Figs. 2 and 3 forcustomer demands under flat-rate pricing as well as customerdemands which are price responsive under real-time pricing. Real-time price trajectories are closely related to demand, except that thepercentage change in prices is greater than the correspondingpercentage change in demand.Effect of Operational Constraints:The effect of operational constraints on the utility and itscustomers under real-time pricing is studied by observing the impactof varying a single constraint while keeping the rest of theconstraints fixed. The observed effects at the time of system peakaredescribed below.(a) Generation limits:In general, the most economically efficient generating unitsare used to supply the base load, generating units of lesser efficiencyto supply the intermediate load, and the most inefficientunits o meetpeak load. Thus, with the increase of load, s more inefficient unitsare operated to supply the demand, the marginal cost and hence thereal-time price of real power increases due to the introduction ofinefficient units as well as due to the increase in losses. Taking thepartial derivative of equation (1 1) with respect to Pgi and Qgi, weget,

= L [ C i Pgi)] - MCp;- hi,m;l+ hisnax (14)spgi spgi6Qgi = - MCg - pi,- + pisnax (15)Since the function L is minimized with respect to Pgi and

MCqi = - Pi,min + Pi,max4 Flat-rates361 + Real-time rates

0 6 12 18 24Hours

4 Flat-rates

0 6 12 18 24Hours

Fig.3 Variations of Real-time Price of Reactive Power at Bus 3 fora 24 Hour PeriodThus, at a generating bus, the price of active power is equalto the marginal cost of production and the price of reactive power iszem until the generating capacity limitsarereached. Figures 4 and 5

show the effects of decreasing the generating capacity of thegenerating unit at Bus 1under real-time pricing of active and reactivepower..

4 Bus1w,i "1 + Bus2

(18)0 5 0 100 150 200

Power Generation t Bus 119)Fig.4 Real-time Price of Active Power vs.Power Generation at Bus1

h

0 Bus3+ Bus4

0 5 0 100 150 200PowerGenerationatBUB (MW)

Fig.5 Real-time Price of Reactive Power vs. Power Generation atBus 1

A reduction of reactive power generating capacity at Bus 3has a much greater affect on real-time prices of reactive power,especially at Bus 3, than on the real-time prices of active power. Theeffects of tightening the constraint on reactive power generationcapacityt B~~ 3 aTe iig.2 Variations of Real-time Price of Active Power at Bus 3 for a24 Hour Period in figures through 1

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27Figures 6 and 7 show the effects of decreasing the reactivegenerating capacity of the generating unit at Bus 3on the real-timeprices of active and reactive power. Figures 8.9, and 10 show theeffects on active power de,mand. reactive power demand, and therevenue, costs and profits of the utility, respectively. Figure 11shows that the revenue, and hence profit, of the reactive powergenerating plant alone initially increases due to increasing prices butthen falls as the quantity produced is reduced more than thecorresponding rise in prices.(b) Transmission Limits:

Transmission limits affect real-time prices of reactive powermuch in the same way as it does real power prices. When the powerflow constrainton a line ismade increasingly tight, the prices of realand reactive power on the receiving end of the line also increase.This is because the line flow constraint forces the use of a higherloss path to satisfy the demand requirements of the bus at thereceiving end of the line, and may also require the reallocation ofgeneration which would increasetotaloperating costs.(c) Voltage limit:The greatest impact on the real-time prices of reactive poweras well as the generation and consumption pattem of reactive powerby the utility and customers is due to voltage constraints. This isbecause voltages are affected mainly by react ive power flows andvoltage constraints are usually relieved by adding sources of reactivepower.+b Busl+ Bus29 Bus3+ Bus4

cz 321++----3 1 : . . I . I . 1

0 10 20 30 40ReactivePowerGenerationatBus 3

Fig.6 Real-time Price of Active Power vs.Reactive PowerGeneration at Bus 31

0fz

0.8

0.6

0.4

P, 0 Bus3+ Bus40 Bus3+ Bus4

0.2+0 1 0 2 0 30 4 0

ReactivePower Oeneration t Bus 3 MVAR)Fig.7 Real-time Price of Reactive Power vs. Reactive PowerGeneration at Bus 3

s -0.0-0.2

-0.4

-0.6 44 Busl+ Bus24 Bus3+ Bus4

-0.80 1 0 20 30 4 0

ReactivePowerGenerationatBus 3 (MVAR)Fig.8 Change in Active Power Demand vs. Reactive PowerGenerationat Bus 3

0d 4-'t 0 Busl+- Bus29 Bus3+ Bus4-5I

10 20 30 4 0ReactivePowerGenerationatBus 3 (MVAR)

Fig9 Change in Reactive Power Demand vs. Reactive PowerGeneration at Bus 3

+ Profit+ cost4 Rev.

- l o ! . I . I * 1 - i0 10 20 30 4 0

ReactivePower Generation t Bus 3 MVAR)Fig.10Change in Revenue, Cost, and Profit vs. Reactive PowerGeneration at Bus 3

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28The effect of tightening the voltage constraint at Bus 4 underreal-time pricing of active and reactive power are shown in Figures12and 13.Figure 12 shows that the real-time price of active powerrises at Bu s 4 and changes very little at other buses with theincreasingly tight constraint on Bus 4 voltage. On the other hand,the real-time price of reactive power at Bus 4 rises very rapidly withthe tightening of the constraint, as shown in Figure 13. Withoutreal-time pricing of reactive power and the corresponding priceresponsiveness of react ive power demand, the tightening of thevoltage constraint at Bu s 4 could lead to skyrocketing costs of

supplying reactive power to Bus 4 and consequent load intemptionin order to maintain the desiredvoltage level.IV. COMPARISON WITH POWER F A O R PENALTIESReactive power pricing based on power factor penalties isunable to provide accurateprice signals to customers under voltageconstraints. In the case in question, the marginal cost of reactivepower at Bus 4 is $693.52 per MVARHr when the voltage limit is0.966 PA., but the price of reactive power would be zero underpower factor penalties since the power factor of the customer at Bus4 is greater than 85 percent. Under real-time pricing of reactivepower, the price of reactive power at Bus 4 would be $693.52 perMVARHr, equal to the marginal cost. Such a high price wouldprovide a strong incentive for thecustomer to reduce eactive powerdemand.Thus, power factor penalties are unable to give accurateprice signals to customers, while real-time prices provide suchsignals.Consider the case of two customers connected to Bus 4, onelarge customer having a demand of 200 MW and 113 MVAR andanother smaller customer having a demand of 20 MW and 19MVAR at the time of system peak, when the marginal cost ofreactive power is$0.56 per MVARHr t Bus 4. In order to calculatethe bill for reactive power consumption under power factor penalty,the following billing algorithm is used for customers having powerfactors below 85 percent: The totalKWh for the month is multipliedby 85 percent and divided by the average power factor for thatmonth for adjustment purpose. Under this penalty policy, the largercustomer who demands 113MVAR of reactive power will not bepenalized, that is. his monthly bill for real power consumption-willnot increase due to his reactive power consumption since thecugtomer's power factor is greater than 85 percent, namely 87percent. On the other hand, the smaller customer would be penalizedfor his demand of 19MVAR and, if a load factor of one is assumed,the customer's monthly bill would be 1.1724 times his bill for realpower consumption alone. This represents an increase in themonthly bill of 17.24percent for the smaller customer, while thelarger customer's bill does not increase. The smaller customereffectively pays $7.26perMVARHrwhile the larger customer pays$0.00 per MVARHr, even though the larger customer is the majorconsumer of reactive power. Thus, the cost burden is inequitablyshared by the customers under reactive power pricing based onpower factor penalties.Under real-time pricing of reactive power, both customerswould pay the same real-time price of $0.56 per MVARHr. Thus,each customer would pay in the exact proportion as the amount ofreactive power consumed by each, which results in an equitablesharing of the cost burden. The cost imposed on a utility due to thereactive power demand at a bus depends on the amount of reactivepower consumed and not on the power factors of the individualcustomers. Hence, reactive power pricing based on power factorpenalties does not result in equitable sharing of the cost burdenswhile real-time pricing of reactive power based on marginal costsdoes result in equitable sharing of cost burdens.V. CONCLUSIONThe importance of an efficient reactive power pricing policyis beginning to be recognized by the utility industry. Accurate pricesignals are essential for proper investment planning by the utility andthe customerssoas to maximize overallsocialwelfare.Real-time pricing of active and reactive power are necessaryingredients for a successful marketplace of electricity. Such a marketwould treat VARs like other market "m cd it ie s, thus providing amarket mechanism for buving and selling VARs. This wouldfacilitate marketplace transactions of reactive power includingbuying and selling of VARs to neighboring utilities, large industriesand independent power producers as well as to cletermine wheelingcharges for VARs.

5 1

-150 10 2 0 30 40ReactivePowaGenerationatBus3 (MVAR)

Fig.11 Change in Revenue from VAR Sales at Bus 3 vs. ReactivePower Generation at Bus 33oo 1

100

0 10.90 0.92 0.94 0.96 0.98

VoltageatBus4 @er unit)Fig.12 Real-time Price of Active.Power vs. Voltage at Bus 4

M O 1PF3Eda2

f-200

0.90 0.92 0.94 0.96 0.98

VoltageatBus 4 @er unit)Fig.13Real-time Price of Reactive Power atBus 4 vs. VoltageatBus 4

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29

APPENDIX: TRANSMISSION LINE IMPEDGNCEDATALine, Lengthbus to R X R X Chargingbus km mi R R perunit perunit MVAR1-2 64.4 40 8 32 0.042 0.168 4.11-4 48.3 30 6 24 0.031 0.126 3.12-3 48.3 30 6 24 0.031 0.126 3.12-4 128.7 80 16 64 0.084 0.336 8.23-4 80.5 50 10 40 0.053 0.210 5.1REFERENCES1 ) Caramanis, M. C., R. E. Bohn, and F. C. Schweppe, "SpotPricing of Electricity: Practice and Theory", IEEE Transactionson Power Apparatus and Systems, Vol. PAS-101, No. 9,September 1982,pp. 3234 - 3245.2) Bohn, R. E., M. C. Caramanis, and F. C. Schweppe, "OptimalPricing in Electrical Networks Over Space and Time",TheRandJournal of Economics, Vol. 15, No. 3 (Autumn 1984),pp. 3603) Schweppe,F., et al, "Homeostatic Control for Electric PowerUsage", IEEE Spectrum, July 1982,pp. 44 - 48.4) Schweppe, F. C., Caramanis, M. C., Tabors, R. D. , and Bohn,R. E., Spot Pricing of Electricity, Kluwer Academic Publishers,Boston, MA, 1988.5) Tabors, R. D., F. C. Schweppe, and M. C. Caramanis, "UtilityExperience with Real Time Rates", IEEE Summer Power

Meeting, 1988.6) Garcia, E. V., and J. E. Runnels, "The Utility Perspective ofSpot Pricing", IEEE 1984 Summer Power Meeting, Seattle,Washington, July 15 - 20, 1984. Paper 84-SM-:54-2.7) Berg, Sanford V., J. Adams and B. Niekum, Power Factorsand the Efficient Pricing and Production of Reactive Power",The Energy Journal, Vol. 4, Special Electricity Issue, pp. 93 -102.8) Ray, D., and F. Alvarado, "Use of an Engineering Model forEconomic Analyses in the Electric Utility Industry", Departmentof Electrical and Computer Engineering, University ofWisconsin-Madison, 1988, presented at the AdvancedWorkshop on Regulation and Public Utility Economics, RutgersUniversity, May 25-27, 1988.9) Lasdon, L.S. and Waren, A.D., "Generalized Reduced GradientSoftware for Linearly and Nonlinearly Constrained Problems",in Design and Implementation of Optimization Software, H.Greenberg, ed., Sijthoff and Noordhoff, pubs., 1979.

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Martin L. Baughman (S , 72) wasbom on February 18,1946inPaulding,Ohio. He received hisBSin Electrical Engineering from OhioNorthem University in 1968 and hisMSEE andPhD degrees in electricalengineering at MIT in 1970and 1972,respectively.Dr.Baughman was a ResearchAssociate at Massachusetts Institute ofTechnology from 1972 to 1975,atwhich time he joined the Universityof Texas at Austin as a SeniorResearch Associate. In 1976hejoined the faculty of the Department of Electrical and ComputerEngineering as an Assistant Professor. In 1979 he coauthored abook with Paul Joskow on electricity supply planning entitledElectricitv in the U n i w s : Models and Policv Analv&. From1984 to 1986 he chaired the National Research Council Committeeon Electricity in Economic Growth. He ha s served as a consultantto several agencies, including Edison Electric Institute, the MITEnergy Laboratory, the Economic Councilof Canada, and theMinistry of Planning in Saudi Arabia, and the Electric PowerResearch Institute.Dr. Baughman is a member of the Intemational Associationof Economists and registered ProfessionalEngineer in the state ofTexas.

N. Siddiai was bom inChittagong, Bangladesh onSeptember 1,1964. He received aB.Sc. (Engineering) from BangladeshUniversity of Engineering andTechnology in 1988. He received aMasterdegree n ElectricalEngineeringfrom he University ofTexas at Austin in 1989. He has beena Graduate Research Assistant withthe Center for Energy Studies at theUniversity of Texas at Austin since September,1988, where he ispursuing a PhD in the Departmentof Electrical and ComputerEngineering.

economics, electricity pricing, and optimal power dispatch.His esearch interests are in the areas of power systems