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764 IEEE Transactions on Power Systems, Vol. 12, No. 2, May 1997HYDRO UNIT COMMITMENT IN HYDRO-TChao-an L i, E ric Hsu, AlvaJ . Svoboda (M ember, IEEE), Chung-l i T seng, Raymond B. J ohnson (M ember, IEEE)

Pacific Gas andElectric Company, SanFranciscoAbstract- In this paper we develop a model and technique forsolving the combined hydro and thermal unit commitment problem,taking into full account the hydro unit dynamic constraints inachieving overall economy of power system operation. The combinedhydrothermal unit commitment problem is solved by a decompositionand coordination approach. Thermal unit commitment is solved usinga conventional L agrangian relaxation technique. The hydro system isdivided into watersheds, which are further broken down intoreservoirs. The watersheds are optimized by Network FlowProgramming (NFP). Priority-list-based Dynamic Programming isused to solve the Hydro Unit Commitment (HUC) problem at thereservoir level. A successive approximation method i s used forupdating the marginal water values (Lagrange multipliers) to improvethe hydro unit commitment convergence, due to the large size andmultiple couplings of water conservation constraints. The integrationof the hydro unit commitment into the existing Hydro-ThermalOptimization (HTO) package greatly improves the quality of itssolution in the PG&E power system.K eywords: Large scale hydro-thermal optimization, Hydro networkflow, Hydro unit commitment, Dynamic programming

1. INTRODUCTIONUntil now almost all papers have addressed the hydro-thermaloptimization problem without consideration of the dynamicconstraints of hydro units (e.g. minimum up-time and down-time) andhydro plant ramp rate constraints. As a result, the solution maycontain some unsatisfactory behavior, such as frequent switching ofhydro units. Frequent cycling of hydro units in daily operations isusually not allowed because of the resulting mechanical stress.Minimizing hydro unit cycling with minimum up-time and minimumdown-time and plant ramp rate constraints may also help to decreasewear and tear costs and other start-up costs of hydro units which candepend on the frequency of the cycling constraint violationslRecently we have developed a model and solution technique forsolving the hydro unit commitment problem with dynamicconstraints, and integrated it into PG&Es existing HTO package,which was built using Lagrangian Relaxation for the thermal UC andNetwork Flow Programming (NFP) for the hydro generationscheduling so as to improve the quality of its applications. Thegeneral solution of the new HTO is divided into the following steps:96 SM 497-8 PWRS A paper recommended and approved by theIEEE Power System Engineering Committee of the IEEE PowerEngineering Society for presentation at the 1996 IEEUPES SummerMeeting, July 28 - August 1, 1996, in Denver, Colorado. Manuscriptsubmitted J anuary 2, 1996; made available for printing June27, 1996.

1. The combined hydro and thermal unit commitment problem isdecomposeded into thermal and hydro subproblems. The thermalunit generation schedules are optimized by DynamicProgramming.The hydro system is divided into watersheds. Each watershed isoptimized by Network Flow Programming, ignoring the unitminimum up- and minimum down-time constraints, and start-upand shut-down costs. A ll available units in hydro plants arecombined into a single equivalent unit with an aggregatedinput/output curve. The network flow solution serves as thestarting point for the hydro unit commitment.Each watershed is further divided into reservoirs. Each reservoirsupplies one or more hydro plants. T he hydro unit commitmenis performed to determine an optimal combination of units ineach hour in each reservoir wi th constraints of minimum up- andminimum down-time, and start-up and shut-down costs. Thiscommitment is more complicated when the units in the plant arenot identical. To decrease the number of combinations, all unitsat a reservoir are optimized by a priority-list-based DynamicProgramming.

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The combined problem is solved by L agrangian relaxation. The finalsolution is obtained by solving iteratively the combined thermal UC,watershed NFP and HUC problems. This paper uses a successiveapproximation method for updating the marginal water values(Lagrange multipliers on hydro conservation constraints) to improvethe hydro unit commitment convergence, due to the large size andmultiple coupling of the hydro system. To decrease thecomputationalburden of the hydro solution, special modeling for hydro units andhydro plants ispresented.The paper consists of the following sections. The combined hydro andthermal unit commitment problem is formulated in the next section.The hydro modeling is described in Section 3. The dynamicprogramming model is presented in Section 4. Section 5 describesthe general solution algorithms. Section 6 demonstrates some resultsof the implementation of the proposed approach ona test system.

2. FORMXJLATIONF PRNotationst , , ,w indexes of hour, unit, reservoi r and watershed1 number of thermal units of the systemJ setof hydrounit indexesT number of hours of the study periodW number of watersheds o f the systemR number of reservoirs of the systemR ( w) umber of reservoirs in watershed w

0885-8950/ 97/$10.00 0 1996 IEEE

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R (r ) et of reservoirs immediately upstream with respect toJ (r )J ' (r ) set of units immediately upstream with respect to res r

reservoir rnumber of hydro units at reservoir roperating cost of unit i at hour t including startup costgeneration of unit i at hour thydro unit startup coststate variable indicating hours when unit is on /off-linedecision variable of unit i at hour t1 -- unit on-line, 0 --unit off-linesystem load at hour tspinning capacity of thermal unit i at hour tspinning capacity of hydro unit j at hour trequired system spinning reserveminimum down time of hydro uni tjminimum up time of hydo unit j

(contentof reservoir r at hour twater releaseof hydro unit j at hour twater release of reservoir r at hour tspillage of reservoir r at hour tnatural inflow to reservoir r at hour ttime delay between reservoirs m and r

ObjectiveThis paper concentrates its discussion on the hydro unit commitment.The thermal unit commitment in PG&E's existing HTO has beendescribed in detail in [l ]. To simplify the description only hydro unitstartup (costs are considered in the formulation of the problem. Thethermal operating cost cltakes into consideration startup and shutdown costs. Assuming that the reservoir targets are not fixed at theend of the study period, the optimal short-term hydrothermal resourcescheduling problem is defined as the following optimization problem:

where the fi rst term represents the thermal operating cost includingfuel, stairt-up and shut-down costs; the second term represents thestartup costs of hydro units, the third term represents the future valueof water in the reservoirs of the power system.ConstraintsTotal hydro and thermal generation meets the system demand:gpt =22PIt +c lt -Dt =0 (2)

I C: j dSystem spinning reserve must be satisfied:

g s , = C R i t + C R j t - R ~ q 1 0i d j d

Water conservation for each reservoir must be observed:P r t =V r ,t+l - vr t +Qrt +@rt- C ( Qm,t-r, +sPl mt- r, , 1=0md?+(r)Release balance in the reservoir is:Qrt = C 4;tj t .J ( r )Maximum and minimum unit release limits are:-1tReservoir maximum and minimum content limits are:

-4 . qj t '4 j t

E r t 2 vr t I vr t-Reservoir target condition is:-! r ~ v,.~ r TWater spillage constraints:spl, 2 0Hydro unit cycling condition:

Dual problemThedual problem is constructed by incorporating constraints (2), ( 3 )and (4) into objective function (1) with multipliers ht , pt andmarginal water valuesyrtrespectively.

"(A, P ,Y>=f inPit 9 Uzt >U i , t - l ) +tsT i d2 U I t . 1- Ujt-1 ) -s tc; ) -At 'gPt - P t .gst +

CYrt ' p r t > - C Y r T * vr T >

j e J

(11)reR r6RSubstituting gpt,gstwth (2)and (3), the dual (11) s rearranged as:The dual function (12) is divided into three independent parts. Thefirst part of (12) is related to the thermal unit indices only, and isdefined as the thermal unit commitment problem. The correspondin,gthermal dual function is as follows:

w,,Y)=d W , )+dWA,P ,r )+ d W, ) (12)dlt(n, I =dn { C (Cit( X i , t - l , Pit 9 U jt > U i , t - l )t c T i d

-At ' P i t - PtRit 11 (13)The second part of (12) is related to the hydro indices r and j only,and is defined as the hydro optimization problem. The correspondinghydro dual function is as follows:dh(A,p,y)=mint C{C(ujt( 1 - ~ , ~ - ~ ) . s t c ~teT j e J

-At * P jt - tRjt)+cY t * p r f 1-2Y T "rT 1rGR rERsubject toconstraints(5)-(10).

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766The third part of (12) is related to the system load and spinningreserve requirement:dls(A,p)=min{c(/2,.D,+pf Rr4) (15)tETWith known A, p, andy, the third part is a constant term and willbe ignored when optimizing the thermal and hydro unit commitment.The thermal unit commitment and hydro optimization problems areoptimized independently. The remainder of the paper will address thehydro optimization, especially the hydro unit commitment problem.Hydro SubproblemThe water conservation equations (4) are highly sensitive to theLagrangian multipliers y. Clearly, the choice of the initial value andthe proper subsequent updating of y (see section 5 ) is crucial to thefinal solution of the hydro subproblem. The next sections introduce anew model and solution technique for solving the hydro subproblem,First we formulate the hydro network flow problem by dividing thehydro system into individual watersheds, ignoring the hydro unitcycling constraints (10). The hydro network flow and economicdispatch provide reservoir release schedules and marginal watervalues as good approximations for input to the HUC. We thenformulate the hydro unit commitment problem for each reservoir byfurther dividing the watershed into individual reservoirs, taking intoaccount hydro constraints (5)-(10).Watershed Network Flow ProblemRelaxing the hydro unit cycling constraints (10) for the moment, wereformulate the hydro dual problem (14) as a non-l inear convexproblem, considering the hydro unit generation a function of therelease and water head:dlh(A ,~,y)=mi n{C CC -At ' P j t ( q j t 2 V r t )t c T j t J

- p t .R jt) - C Y r T " r T> (16)r tRsubject to constraints (4)-(9).Considering the independence of each watershed in the system, thehydro system can be divided into individual watersheds. Regrouping(16) according to the watershed index ,we formulate the optimizationproblem for each watershed as the following convex problem:~~w ( A ,P , Y ) =n{ C CC -At . P j t ( q j t , v r t ) -pt * R j t )t cT ] cJ (w)

- C Y r T ' vr T} w=1,2,..,w (17)rcR(w)subject to constraints (4)-(9).For hydro units, the water conservation constraints (4) arecomplicated by the network interdependencies resulting from thelocations of hydro units in a watershed containing reservoirs andconnected by river segments. Each watershed as a whole is treated asa resource, and optimized using a Network F low algorithm asdescribed in [1,5-81. The network flow model generates water releaseschedules and unit commitment schedules for each reservoir. It isobvious that if these schedules respect the minimum up- andminimum down-time of all units in the watersheds, the solution isfinal and optimal. Unfortunately, the network flow solution oftencontains infeasible schedules in terms of unit minimum up-time andminimum downtime constraints. The objective of the hydro unitcommitment is to eliminate the violations of such constraints.

Hydro Unit Commitment Pr oblemThe general formulation of the hydro unit commitment problem habeen represented in (14). Suppose good approximations of threservoir releases and marginal water values y have already beedetermined from the hydro network flow and economic dispatcmodel and fed into the hydro unit commitment. Substitutingwrtwith (4), regrouping hydro units according to the reservoindex, the hydro dual function (14) can be rewri tten as:whered W 4 P , Y ) =W 4 P , Y , V ) + d W , V ) (18)dlr(A,pu,~,v)=in{C CCult ' ( l - u j t - 1 ) . s t c jtc Tr cR j d ( r )

-I t . t (4 t > V j t1- t'jt 1+( ~ r t~ d , t + . r , ~.Qrt (19)

dlc@,v)=min{ CE{ yr t .idrt yrT .vrT-tcTrcR

(Y l -Y ,t+l) 'vrt -t Y t -Y d,t+z,d ' (20)subject to constraints(5)-(10)where yd, t+. r l is thewater value of the downstream reservoir d time t+qdSuppose that the multipliers li. , p, and y are given. Also assume ththe reservoir contents v and water spills spl are determined a priofrom the network flow model. From (1 8-20), we see that the hyddual function consists of two terms. The first term dlr(A,p, ,v)dependent on the unit state variable x in stc, , the unit ordodecisionsu, and the unit water release or generation variables q orThe second term dlc(y,v) is constant. Because dlr(A,p,y,v)additive and separable in the reservoir index r , we are able decompose the hydro optimization problem into subproblems in threservoir index. Then the following dual function i s defined as thhydrounit commitment problem (HUC) for the reservoir:(HUC):dlr(d,p,y,v)=min{C{ C(ujt ( l - ~~~- ~tcj

t tT j d ( )-At . j t (4 t ,V j t 1 P t R j t )+( y t - d,t+z, 'Qrt 1r =1,2,..R(w),w=42 ..,W (21)subject to hydro constraints (5)-(10). If we ignore the impact of watheads on the hydro unit commitment, it can be shown that tmarginal water values are constant over the study time horizon. T hthe hydro unit commitment problem (HUC) of (21) is simplif ied as:

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761Priority l ist-based dynamic programming i s used for solving the hydrounit commitment problem to reduce the problem's dimensions (from2" -1 to n+l) . A ll available hydro units at the reservoir are sequencedin increasing order of average full-load water rate.

The size of multipliers y in (22) is greatly decreased in comparisonwith (21).The hydro unit commitment problem(HUC) can be solvedby Dynamic Programming. Unl ike the thermal unit commitment inwhich only one unit is involved in the DP solution, the hydro unitcommitment is to determine the optimal combination of unitsavailable in each hour in each reservoir. To decrease the DPcomputafional burden we use a unit priority list instead of a full-blown search of all combinations of units at the reservoir (Section4)Hydro Economic Dispatch problemWith the fixed unit schedules in the reservoir, the objective function(22) is separable and additive in index of time.Then we formulate thehydro eoonomic dispatch problem for each reservoir for each houras:h ed ~ (A : ,~ ,~ ,v ,u )fin{-At 'pjt(4jt vjt)+

( ~ r ~ d. Cq j t ) t=1,2,...,T (24)j c J ( r )

s.t. (5). 'The difference of the marginal water values of reservoir rand its downstream reservoir d represents the plant or unit marginalwater valueof the reservoir r as:Yrj =Y r - Y d ( 2 5 )

The hydao economic dispatch problem is solved by the equalincremental watrer rate principle.3. H Y D R O I /O CURVE MODELING

This section is confined to describing the creation of water rate curvesfor different combinations of identical units with consideration of thehead effects. The typical curves with 3 units for a specified waterhead are shown in Fig. 1. The cross points of two consecutive curvesrepresent. the switch points from one combination to another.1 unit switch points

Figure 1 Typical curves of unit combinationsThe water rate curves are modeled by quadratic functions given forthe miniimum and maximum water heads. The coefficients of thequadratic forms for intermediate heads are determined by linearinterpolation. With the quadratic model the commitment switchpoints can be determined analytically.

4.DYNAM IC PROGRAMM ING MODELUnit Combination is defined as a set of units for on-line operationin a reservoir. A plant with n identical units hasn combinations. Aplant with n nonidentical units has 2" -1 combinations. A ll units off-line is a special combination called the 0combination.Deckion is defined as the transition of one combination at hour t toanother combination at t+l. Any change in the unit combination isalways accompanied by a change of one or more additional units toon-lineor off-line status

State Transition DiagramLet 0 represent the combination state variable of all units off-line,and 1,2 ...n, -- unit 1, units 1,2, ..., and units 1,2 ..,n committed on-line respectively. The state transition diagram is depicted in Figure 2.To reduce further the number of combinations to consider, we mallalso account for all manual-schedule and must-run units as onecombination and give it a state 1after the state0. We will record thenumber of hours that each unit has been on or off in each state foirthe optimal path. To avoid frequent cycling of units, we will use therecord of hours on and off to determine if a transition between statesis feasible given the minimum up- and down-time constraints.

+Units 1,2,3,4,5 committed+Units 1,2,3,4 committed+Units 1,2,3 committed2 Units 1,2 committed0!+ +Allnitsnitscommittedff-linet t+ 1

Figure 2. State transition diagram

5. SOLUTIONLGORITHML agrangian multipl ier updatesThe update of 1 and p is described in detail in PG&E's existingHTO program [l]. The difficulty here is in updating the marginalwater value. Our experiences have shown that the conventionallmethod of choosing the initial marginal water value (e.g. usingaverage water value) and its subsequent updating (e.g. using Polyak;[ 9 ] or another updating formula) often results in non-convergence oroscillation of the hydro schedules. The large number of y multipliers;(e.g. there are more than 12000 in the one weekPG&E problem) andlthe multiple couplings of the river system both in space (reservoirs in1cascade) and in time (limited usage of water over the time horizon)almost exclude the use of the conventional method. In this paper a[successive approximation method is used for updating the 1multipliers. With the initial y values determined from NFP and hydrcleconomic dispatch we run the hydro unit commitment. If thereservoir release balance equations ( 5 ) are violated due tclrescheduling hydro units in HUC to meet the cycling constraints, WE:will reallocate the reservoir water flow using the following rules:increase water releases in hours when marginal water values are:large and decrease water releases in hours when marginal watervalues ar e small We then update the marginal water value andlrepeat the hydro unit commitment again. The water reallocation in1dif ferent hours and in different units at each reservoir continues until1the marginal water values in different hours are close to each other.This successive approximation method of updating marginal watervalues has several advantages over the conventional iterative method:1)The conventional iterative updates are very sensitive to the waterconservation equations (23) due to the near-flat hydro incremental.characteristics and the coupling feature of the hydro system, i.e. a.small changeof marginal water value often results in a big change in.

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768the imbalance of the use of theconventional iterative update often leads to the non-convergence i nthe hydro optimization. 2)T he network flow pr.ovides a good startingpoint for the hydro unit commitment, i.e. initial marginal watervalues in different hours calculated in NFP by hydro economicdispatch are usually close to each other. The marginal water valuesneed to be updated only when the unit minimum up and minimumdown-times are violated. These updates are usually small and can bedone much more easily by the successive approximation method thanby using the conventional updating formula.

equation (23). This is why

Flow chart of solutionThe flow chart of the algorithm for solving the combined hydro andthermal unit commitment is shown inFigure 3.

unit commitment

I I

+ VPSRun system economic dispatch

stopFig. 3 Flow chart of the algorithm

Computation procedureThe computational procedure is broken into the steps:1.2.3.4.

5.6.

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Initialize the system lambdash andp at the master coordinator.Run Thermal Unit Commitment to give the unit commitment andgeneration schedules of all thermal units.Run the hydro network flow programming for watersheds togive the water release schedules for all reservoirs.Initialize the marginal water values by running the hydroeconomic dispatch program with the reservoir water releaseschedules determined from hydro network flow.Run HUC to give the unit commitment and generation schedulesof all hydro units in the reservoir.Check if the reservoir inflow and outflow are balanced. Alsocheck if the absolute value of the difference of marginal watervalues between two different hours is less than a prespecifiedtolerance. If yes, go to step 7. Otherwise, reallocate waterreleases, and update y.Check the optimality of the hydrothermal unit commitment. Theoptimization phase stops, if the number of iterations of thisphase exceeds a specified minimum number, and the difference

of norms of the system lambdas h and p in consecutiviterations are small enough. Otherwise, update h and p, anrepeat step2to step 7.If the system reserve requirements are observed, go to step Otherwise, repeat step 2 to step 8.Run the system economic dispatch prgrani to schedule the powegenerationof the committed units and stop computation.

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6. COMPUTATI ON RESUL TSThe hydro unit commitment model proposed in this paper hrecently been built and integrated into PG&E's existing LagrangianRelaxation-based HTO program. The enhanced hydro and thermunit commitment has been implemented and tested on the PG&power system with a total of 243 units. 115 hydro units and 5thermal units participate in the combined hydro and thermal uncommitment program. The hydro system consists of 65 reservoirs cascade located on 14watersheds in Northern and Central Califomincluding a pumped storage facility with 3 pumping units. Thsmallest watershed contains 2 reservoirs with 2 plants and 5 unitthe largest watershed 11 reservoirs with 9 plants and 19 units. Tsystem parameters used to drive the test results can be found in oprevious paper [l]. The hydro and thermal unit incremental cocurves are modeled by piecewise linear functions. Hydro unit staup costs are set to zero in the study case.The computer program is coded in theFORTRAN77andruns on thHp9000/735 computer. Some test results are illustrated here:Table 1 shows the improvement in aunit's schedule by thehydro uncommitment in comparison with the schedule produced by NFP. Thminimum up and minimum down-time of this unit are 3 hours.

Table 1 Improvement of daily unit schedulesAfterNFP 0 0 1 0 1 1 1 0 1 1 1 0 1 0 1 0 1 0 1 1 1 1 1 1AfterHUC 0 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 1 1 1 1 1 1 1 1As indicated in the problem formulation, the marginal water valuare constant over the hours units are on-line, when ignoring twater head variation. The use of the successive approximatimethod to update these Lagrange multipliers takes advantages ovthe useof the conventional iterative updates. Fig. 4 shows the

I 6 0 0 T

ITop: AAer hydro network f low Bottom: After hydro unit commitmeFig.4 Marginal water values by hourmarginal water values over time. The higher values in the gracorrespond to the hours when units at the reservoir are all shut dowThe lower values correspond to the hours in which at least one unit

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on-line. As shown in Fig. 4, the hydro network flow provides goodinitial marginal water values for input to the hydro unit commitment.The marginal water values over the on-line hours are close to eachother. The rescheduling of units in the hydro unit commitment tomeet the unit cycling constraints will cause big changes of marginalwater values only in the hours when a unit switches on or off. Thereservoir water imbalance due to rescheduling in the unitcommitment will be reallocated to all on-line hours in proportion tothe hourly releases of the reservoir. Such reallocation has only aminor effect on the marginal water values in the on-line hours.Table 2 lists some summary results of Hydro-thermal Optimizationwith and without hydro unit commitment function for a one weekstudy case.

Table 2 Comparison of HTO with and without HUCComparison items HTO without HUC HTOwith HUCNo. ofiterations 21 21CPU tiime(sec) 253.92 269.05Total thermal cost($1000) 10247.078 '0247.282..........................................................................................................No. of 'Cycling constraint violations >60 0This table shows that the preferrebstart-up behavior of hydro units(see Table 1) from HUC can be obtained with only a small increase inCPU time and total system cost.

7. CONCLUSIONA combined hydro and thermal unit commitment taking into fullaccount hydro unit dynamic constraints, is developed by the authorsof the paper. The hydro system is divided into reservoir subsystemsthat cannot be broken down further due to the hydro networkstructure. All units at reservoirs are committed or decommitted byusing priority list-based Dynamic Programming. In order to improvethe convergence of the algorithm, a successive approximationapproach is used for updating the marginal water values instead ofusing the conventional iterative updates. The enhancement of theexisting HTO with hydro unit commitment improves its value in thePG&E power system.

8.ACKNOWLEDGMENTThe authors would like to acknowledge the essential contributions ofClaudia Greif and PG&E's Energy Trading department to the HUCproblem's definition, modeling and solution methodology. This workhas been supported in part by PG&E'sR&D department.

9.REFERENCES1. L A F M Ferreira, T A nderson, C F Imparato, T E Miller, C KPiing, A Svoboda and A F V ojdani, Short-tern resource

scheduling in multi-area hydrothermal power system, ElectricPower&Energy Systems, Vol. 11, no. 3, 1989.0.Nilsson, D. Sjelvgren, Mixed-Integer Programming Appliedto Short-Term Planning of a Hydro-Thermal System, PICAConf., 1995.

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10.BIOGRAPHIES

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Chao-an Li graduated from Electric Power System Department ofMoscow Energetic Institute, Moscow, USSR. He has broad interestsin power system optimization including hydrothermal coordination,economic dispatch, unit commitment, load forecasting, automa.ticgeneration control, power flow, power system state estimation, etc..He is currently working as a contractor on PG&E's Hydro-ThemnalOptimization project.Eric Hsu received a B.S. and an M.S. in Operations Research fromthe University of Calirornia, Berkeley, in 1982 and 1983,respectively. He has worked as a systems engineer for Pacific Gasand Electric Company since 1983, developing computer applicationsfor fuel and resource planning, hydro scheduling and forec,astmanagement, and hydro-thermal optimizationAlva J . Svoboda received a B.A. in mathematics from U.C. SantaBarbara in 1980, and an M.S and Ph.D. in Operations Research fromU.C. Berkeley in 1984 and 1992. He has worked on contract asanoperations research analyst at Pacific Gas and Electric Co. since1986. His current interest is the extension of utility operationsplanning models to incorporate new operating constraints.Chung-L i T seng received a B.S. in Electrical Engineering fromNational Taiwan University in 1988 and a M.S. in Electrical andComputer Engineering from U.C. Davis in 1992. He is currently aPh.D. candidate in Industrial Engineering and Operations Research atU.C. Berkeley.Raymond B. J ohnson received his B.A . in 1976 in ElectricalSciences from Trinity College, Cambridge University, and a Ph.D. inElectrical Engineering from Imperial College, London University in1985. His professional experience includes positions as a povmsystem design engineer with Hawker Siddeley Power Engineeringfrom 1976 to 1980 and an EM S applications developer with FerranltiInternational Controls from 1987 to 1989. Since 1989, has been withPG&E where he is currently a Systems Engineering Team Leaderresponsible for resource scheduling and energy trading applications.

769John J . Shaw, R F. Gendron, D P. Bertsekas, "OptimalScheduling of L arge Hydrothermal Power Systems", IEEE Trans.onPA S-104, No.2, Feb. 1985H Habillollahzadeh, J A Bubenko, Application of decompositiontechniques to short-term operation planning of hydrothermalpower system,EEE Trans. power system, Feb. 1986H Brannlund, J A Bubenko, D Sjelvgren, N A ndemon,"Optimal Short-term Operation Planning of a LzrgeHydrothemal Power System based on a Nonlinear NetworkFlow Concept", IEEE Trans. onPower Systems, Nov. 1986Chao-an Li , Philip J . J ap, Dan L. Streiffert, Implementation ofNetwork Flow Programming to the Hydrothermal Coordinationin an Energy Management System, IEEE Transactions on powerSystems, August 1993.R E Rosenthel, A nonlinear network f low algorithm formaximization of benefits in a hydroelectric power system,Oper.Res. Vol. 29 No. 4J L K ennington, and R V Helgason, Algorithms for NetworkProgramming Wiley, New Y ork (1 980)B T Polyak, Minimization of unsmooth functionals, USSRComput. Math. M ath. Phys. 1969