Hydrothermal Scheduling Using Benders Decomposition- Accelerating Techniques

IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 22, NO. 3, AUGUST 2007 1351

Hydrothermal Scheduling Using BendersDecomposition: Accelerating TechniquesWilfredo S. Sifuentes, Student Member, IEEE, and Alberto Vargas, Senior Member, IEEE

AbstractA new decomposition method is presented that in-cludes the network through ac modeling within the hydrothermalscheduling optimization process including the losses. In short-termhydrothermal scheduling, the transmission network is typicallymodeled with dc power flow techniques. Such modeling, how-ever, can lead to impractical solutions when it is verified withac power flow. Another proposal considers in thermal systemsthe ac network modeling but not the optimization of losses. Theapproach presented here addresses issues such as congestionmanagement and control of service quality that often arise inlarge and weakly meshed networksthe typical pattern of powersystems in Latin America. Generalized Benders decompositionand traditional, well-known optimization techniques are usedto solve this problem. The master problem stage defines thegeneration levels by regarding the inter-temporal constraints,whereas the subproblem stage determines both the active and thereactive economical dispatches for each time interval of the loadcurve. It meets the electrical constraints through a modified acoptimal power flow (OPF). Another important contribution is theinclusion of accelerating techniques aimed at reducing the numberof iterations and CPU time. The methodology was proven in a realsystem and test systems. Results are discussed in this paper.

Index TermsBenders decomposition, hydrothermal sched-uling, optimal power flow, unit commitment.

I. INTRODUCTION

SHORT-TERM hydrothermal scheduling (STHS) is knownas one of the most challenging optimization tasks in powersystems [1]. Its purpose is to minimize the total generationcost over a time period (a day or a week) involving a mix ofhydro and thermal generation. This problem has been addressedwith several techniques, e.g., dynamic programming [1], [2],Lagrangean decomposition [3], [4], and generalized Bendersdecomposition [5]. In these works, however, the transmissionnetwork was not considered or was greatly simplified. Thesesimplifications could lead to inoperable dispatch or to the needfor adjustments at the level of unit commitment when verifyingoperational limits with ac power flow. This problem is notonly typical in Latin American countries that feature extensiveand weakly-meshed networks, highly-loaded power lines, andgeneration plants located far from the load [7], [8] but is alsopresent in other countries, like Spain [9].

The restructuring of electrical markets in the 1980s and 1990sfor better competitiveness also turned the STHS problem into

Manuscript received January 27, 2006; revised November 20, 2006. Thiswork was supported by the German Exchange Service (Deutscher Akademis-cher Austauschdienst/DAAD). Paper no. TPWRS-00044-2006.

The authors are with the Instituto de Energa Elctrica, Universidad Nacionalde San Juan, San Juan, Argentina (e-mail: [email protected]; [email protected]).

Digital Object Identifier 10.1109/TPWRS.2007.901751

a more complex one. Thus, when evaluating economically theoperation of a power system in this stringent competitive frame-work, the problem incorporates additional factors, e.g., the lackof electric power to supply the demand, as measured by thenon-supplied energy (NSE) factor. NSE is economically quan-tified by the cost of the NSE (CNSE) that must be evaluated foreach period and each system bus. In spite of this deregulationtrend, most countries in Latin America are still computing theshort-term operating planning under some kind of centralizeddispatch criteria.

The most successful optimization technique applied so farto solve STHS has been Lagrangean relaxation (LR). In earlyapplications, the optimization problem considered only thermalunits [1], [3], [10]. The methodology was soon extended to con-sider the constraints from hydroelectric power stations [4], [12]and, subsequently, to include the transmission network mod-eled by dc power flow [11], [13], [14]. The success of the LRapproach lies in the fact that it allows relaxing the hard con-straints by defining a dual problem, which turns the originalproblem into a separable one. However, on account of its ownnature, the solution attained for the dual problem is almost al-ways non-practical. It requires further complex adjustments tomeet every constraint of the primary problem, especially forinter-temporal constrains, such as the multi-period water bal-ance of hydropower schemes.

Other approaches of recent works [5], [6] have shownpromising results using generalized Benders decomposition(GBD) [15]. With this tool and with an appropriate selection ofvariables, it is possible to split the problem into several smallones. This transformation allows processing the inter-temporalconstraints separately from the static constraints, i.e., con-straints that cover a single period.

The issue of considering the transmission network within theSTHS problem implies adding a very large set of nonlinear con-straints that are equivalent to an OPF for each period. Since theOPF is regarded as a very complex problem [16], in STHS, thenetwork is modeled with little detail or, else, it is greatly sim-plified. At present, there are just a few works that include an en-tire network modeling and the losses included in optimization[6], [17]. Nonetheless, they consider only purely thermal gener-ation. Others authors include an ac network modeling and wereapplied in both small-scale and large-scale systems, includingsecurity and contingency constraints, but the approach is purelythermal feasibility oriented [18], [19] instead of hydrothermaloptimality oriented as is presented in this paper.

Doa [7] considers a complementary heuristic procedure todetect and correct reactive power deficit problems within anoptimization process for a large-scale hydrothermal system,and Serrano [8] proposes an improvement of voltage levels and

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1352 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 22, NO. 3, AUGUST 2007

losses reduction process by considering the smallest deviationof the previously computed economic dispatch. A heuristicdecomposition is presented in [9] applied to the Spanish marketto correct both voltage levels and congestion issues.

As a conclusion on the state of the art, at present, there is notany strong mathematical formulation available applied to STHSwith an ac modeling. The present work offers a contributionon this sense. It combines the approach of Alguacil [5] to treatthe inter-temporary constraints and the approach of Ma [20] forthe electrical constraints, though with an extension to considera complete power flow modeling. This compounded approachalso allows representing cascaded hydro power plants, an issuethat is hard to address with other approaches.

Two drawbacks (always present in Benders decomposition)are its slow final convergence, also called tailing-off effect, andthe high computational burden for the master problem [21]. Theformer is minimized by introducing accelerating techniques ina specific stage of the procedure and the latter by using a spe-cialized optimization tool.

The solution of hydrothermal systems with hydro predomi-nance is significantly more complex than the thermal system.This problem is extremely difficult to deal with using LR ap-proaches but relatively straightforward using the MIP technique,as is recognized in [22].

This paper is organized as follows. The generic hydrothermalcoordination is formulated in Section II. Section III describesthe proposed decomposition and the addition of heuristics toreduce the number of iterations. Section IV gives the numer-ical results and discusses the application to an actual system.Section V gives concluding remarks.

II. PROBLEM FORMULATION

The following notation is introduced.Sets:

Time horizon of scheduling.Number of thermal units.Number of hydro-power plants.Number of buses in the system.Number of lines in the system.

Constants:

Free, linear, and quadratic terms of the costscurve of thermal unit .Startup cost of thermal unit .Active and reactive power load for periodon bus .Spinning reserve required for period .

Minimum active and reactive power outputof thermal unit .Maximum active and reactive power outputof thermal unit .Ramp rate of thermal unit .Minimum up time of thermal unit .Maximum active and reactive power outputsof hydro-power plant .Minimum active and reactive power outputsof hydro-power plant .Minimum and maximum limits active powerof the line or branch l.Module and angle of the admittance matrixelements.Minimum and maximum voltage limits ofbus k.Penalty costs due to active and reactivedeficits.

Rx Penalty costs due to reactive excess.Minimum and maximum water volume of thedam associated to the hydro-power plant .

Main Variables:

Active and reactive power output of thermal unitfor period .

Active and reactive power output of hydro-powerplant for period .Operation state of thermal unit for period

.

Operation state of hydro-power plant forperiod .

Startup of thermal unit for periodStartup.

Water volume and flow rate of the damassociated to hydro-power plant for period .Active and reactive power deficits for period

on bus .Excess values of active and reactive power forperiod on bus .Voltage value for period on bus .Voltage angle for period on bus .

The STHS can be formulated as shown in (1) at the bottomof the page.

(1)

SIFUENTES AND VARGAS: HYDROTHERMAL SCHEDULING USING BENDERS DECOMPOSITION 1353

The first group represents the production and startup costs,and the second group represents the total amount of penaltycosts. The production costs are those incurred by the com-mitted thermal units. They can be modeled either linearly or inquadratic form, in relation to the generated power output .

The introduction of variables , and meet two impor-tant objectives in the formulation: First, they prevent the occur-rence of non-feasibilities in the optimization approach, whichallows closing the nodal balance (active and/or reactive balance)for any condition. Second, the penalty magnitudes correspondto the costs resulting from being unable to supply active or re-active power to the system (an extension of the concept of faultmachine). These variables will not be zero when the proposedgenerating schedule and the transmission network cannot sat-isfy the nodal balance.

Problem (1) is subject to the following constrains:1) Auxiliary constraint to detect start-up.

(2)

2) Minimum up-time operation.

(3)

3) Ramping.

(4)

4) Minimum and maximum active and reactive power outputof hydraulic unit generation.

(5)(6)

5) Hydraulic balance considering continuity and time-delayof inflow.The constraints from hydroelectric power plants are fre-quently modeled linearly. Nevertheless, there exists someexception when a nonlinear modeling must be considered[23]a case not regarded in this work. A particular case isthe consideration of time delay in cascaded hydroelectricpower plants; these constraints are easily handled througha linear representation.

(7)

6) Minimum and maximum volume of each reservoir.

(8)

7) Spinning reserve requirement (SRR). In this work, SRR isconsidered, and it has a fixed value previously computedthrough studies comparing reliability and operation cost.

(9)

Main electrical operation constrains are the following ones:

8) Nodal balance on bus (active and reactive) for every pe-riod .

(10)

(11)

9) Capacity limits of transmission lines or branches for everyperiod . Due to regulation in some Latin American coun-tries, the active power limit is shown. However, constraint(12) can be replaced for the apparent power limit or currentthermal limits for the methodology presented here.

(12)

10) Power output limits (active and reactive) of thermal units.

(13)(14)

11) Bus voltage limits.

(15)

Constraints (6) and (10)(15) correspond to the formulationof various ac OPFs, though with a difference in that they containvariables that link these constraints through several timeintervals, whichin turnprevents their individual treatment.

Constraints (2)(5) and (7)(9) correspond to a mixed integerproblem, whereas constraints (6) and (10)(15) correspond toa nonlinear problem with continuous variables (provided somevariables are fixed equal to a given value).

III. PROPOSED DECOMPOSITION

The original GBD [15] technique was developed with thepurpose of manipulating continuous and discrete variablesseparately. The master problema mixed integer optimizationproblemdeals with the discrete variables and with only onecontinuous variable, . The continuous variable togetherwith the Benders cuts represent the subproblem into the masterproblem. The subproblem is a nonlinear optimization problem.The Benders cuts are linear approximations to the subproblemabout the solution point found in the master problem. There-fore, the master problem must be solved successively (byaccumulating the Benders cuts), and the subproblem as well,until reaching the convergence criterion.

The proposed decomposition is slightly different from theoriginal GBD approach. In the present work, there are morecontinuous variables in the master problem, which allow


transforming the subproblem into a set of independent opti-mization problems as well as greatly reducing its computationalburden.

For sake of simplicity, an intuitive decomposition is presentedfirst.

Subproblem: The subproblem is defined as

(16)

Objective function (16) is subject to constraints (6) and(10)(15). This problem is nonlinear and separable. This way,the subproblem becomes a set of independent optimal powerflows. Then, (16) will actually be as follows:

(17)

The original variables, computed in the master problem asinput data to the OPF, must be included as a constraint explicitly,because it is necessary to know their dual value once the solutionof the subproblem has been attained. Each dual value is usedto construct the Benders cuts, which are added to the masterproblem in each iteration to improve the proposed dispatch.

Considering the case for variable for a specific period anda thermal unit , the additional constraint that should be addedto the OPF would be

(18)

where is the previous value obtained in the master problem,and is the Lagrange multiplier of the constraint; it givesa linear approximation of the rate-of-change of the subproblemcosts caused by a unitary change of .

The observed sensitivity arises from the fact that controlsthe limits of constrains (13) and (14). These constrains havevariables that participate in the objective function of the sub-problem.

As in the previous case, it is necessary to add specific con-strains of and variables into the subproblem.

The strict application of GBD requires that the subproblembe convex; else, the linearized Benders cut may cut off feasiblesolutions. The non-convexities in the subproblem result fromthe presence of sine and cosine functions in the nodal balance(active and reactive) equations. However, some conditions canprevent or reduce much of these occurrences.

1) It is necessary to reach large bus angle-differences in orderto change the convexity of these functions. Luckily, thissituation is seldom found in practice.

2) The penalized variables altogether with the magnitude ofthe penalization significantly reduce the non-convex zonesand prevent the subproblem from becoming non-feasible.Moreover, the first tool used in order to optimize the sub-problem is sequential quadratic programming; then, the re-gion is approximated by quadratic functions.

3) It is highly advisable to fix the bounds of variables. Thesolution gets reduced to an operative and feasible region(e.g., voltage angles bounds: to degrees).

It is very important to remark that, with this formulation, thesolution for the subproblem solution will always be feasible.Hence, the nonlinear solver must guarantee this is reached inorder to attain the Lagrangean multipliers. The main advantageof this formulation is that it prevents the need of building feasi-bility cuts, because in nonlinear programming, it is not alwayspossible to know when the optimization problem is not feasibleor that the solver cannot reach at the right solution.

Master Problem: The master problem includes the minimiza-tion of the fixed costs (free term of the cost curve) and the startupcosts of the thermal units subject to the operative inter-temporalconstrains (minimum operation times and minimum out-of-ser-vice time). It also contains the complete set of constraints of thehydroelectric power plants

(19)

The objective function (19) is subject to constraints (2)(5),(7), and (8). In Latin America, a common practice is to assign thespinning reserve only to hydraulic plants. Therefore, the masterproblem can also process constraint (9). It is then necessary toadd the Benders cuts when computing each iteration. The addedBenders cuts represent the subproblem in a linear way. The for-mulation of the master problem is of mixed integer lineal pro-gramming (MIP) type.

For each single period , the Benders cut is

(20)

The upper line means a fixed value for the variable computedin the previous iteration in the master problem andthe remaining values in the subproblem .Hence, represents the subproblem within the master problem.

Master Problem versus Subproblem: As a clarifying ex-ample, let us consider the case of a given OPF, with its activepower totally supplied (as ensured by unit commitment com-puted in the master problem), though incapable of reaching theimposed voltage level for a given bus. This condition will becompensated for by activating a reactive power deficit machineon this bus. In this scenario, the dual value of variable willdepend on the electrical proximity of the generator to thereactive-deficit bus. If the generator is located on a bus that hasreactive power reserve electrically far from the bus showing thevoltage-problem, the dual value of will be zero or very small,because an increase of reactive power injection in that buswill contribute very little or nothing to raise the voltage level.Therefore, the generators (thermal and hydro units) nearest tothe reactive-deficit bus will have greater values associated tothe dual value of the variable that controls its operation state

. With all this information, the master problem determines a


Fig. 1. Proposed algorithm structure.

new generation schedule. A similar analysis can be done whencongestion appears. Fig. 1 shows the algorithm structure.

Decomposition Strategy: In the actual decomposition, thelinear part of the cost-curve of thermal units was moved to themaster problem (including its active power limits constraint).This consideration allows processing inter-temporary con-straintse.g., (9) or (4)without making simplifications, andit obtains a stronger lower bound, though the computationalburden is increased as well. The master problem is redefined as

(21)

The authors analyzed several scenarios and got as a conclu-sion that it is necessary to implement a particular decompo-sition strategy appropriate for the specific system, e.g., in hy-drothermal or thermal systems, the quadratic term can also bemoved to the master problem when the presented decomposi-tion presents a slow convergence but the master problem needsto be solved using a QMIP solver.

We adopt to include the quadratic term into the subproblemand then testing this decomposition provided that actualLatin-Americans hydrothermal systems present both hydro andthermal units very different characteristics and the quadraticterms are not significant (in fact, in some countries, only alinear cost curve representation is used).

Then in our case, the subproblem is redefined as

(22)

It is necessary to add the constraint to con-sider the influence of the quadratic part of the cost curve in themaster problem. Under this decomposition framework, the state

variables in the subproblem are the penalized variables, angles,and voltages. For each period , the Benders cut are defined as

(23)

Accelerating Techniques (ACCT): The inclusion of ACCT isaimed at reducing the computing time for the entire problem. Itis described briefly below.

1) When Benders decomposition is used in linear program-ming, it is only necessary to keep the active Benders cutsto the next iterations. In mixed integer programming, thisapproach cannot be used in the same way. The problemarises from the fact that a Benders cut cannot be active butthat cut is actually forcing to an integer solution of somevariable. The distance of the cut can be evaluated as thedifference between zero and the constraint evaluated at thesolution point. The selected distance in order to delete acut is chosen by trial and error, just to prevent the masterproblem giving the same solutions in the next iterations.

2) The first iterations are performed with relaxed mixedinteger programming (RMIP). In the first iterations, themaster problem has little information, and the proposedschedule does not have to be exactly computed. Therefore,the next better point (schedule) is calculated with RMIP.

3) The upper and lower bounds of the solution for hydraulicpower plants in the last iterations are tightened. After acareful observation to understand why GBD shows a longtail (slow final convergence), it was realized that, in specificperiods, the hydraulic generation undergoes a hydro gener-ation complementary effect. This means, for example, thatthe total amount of active generation of two power plantschange slightly between two iterations but change muchmore between them with a very little improvement of thelower and upper bounds. This problem is brought up bythe linear representation of the entire problem in the masterproblem. Controlling this behavior will lead to a faster con-vergence, with slightly worsening of the solution.

4) The stopping criteria adopted with GBD, and GBD withACCT, is that the difference between the upper and lowerbounds should be smaller than the adopted tolerance.

Results Validation: It is not possible to show the contrastwith other methodology applied to this problem (STHS) withac modeling. However, in order to guarantee the optimality ofthe solution, the following scheme is proposed.

1) Solve the original problem with the proposed methodology.2) By fixing all binary variables and using the remaining re-

sults found in step one as initial variable values, solve againthe original problem without considering any decomposi-tion (totally coupled problem).

3) Contrast the results found in steps 1 and 2.This scheme is founded on the fact that GBD converges in one

iteration [24] if the subproblem is solved at point , where


means the entire set of binary variables, and are their op-timal values. If the status (0/1) of both thermal and hydro oper-ation units were correctly computed, then the original problemcan be transformed into a nonlinear optimization problem byfixing the binary variables, with the possibility of having somefurther cost reductions due to an improvement in water alloca-tion. The reason for this is that, in GBD, the nonlinear problemwas reconstructed by hyper-planes, and the minimum may beslightly different from the actual one.

This nonlinear problem, in spite of having fewer variablesand having as initial variable values the results obtained by theproposed decomposition, takes a long time to converge.

The methodology was implemented with Dash OptimizationSuite [25], using XpressMIP to solve the master problem,and Xpress-SLP for the subproblem. It was tested on an AthlonXP 2.0-GHz PC with 1 GBytes of RAM memory.

IV. RESULTSTest Example: The methodology was tested in the North-

Center Interconnected System of Peru (SICN), characterized byan annual energy demand of 11085 GWh, and a peak of 2200MW. The system was modeled considering 12 hydraulic powerplants associated to nine river basins, some of them in cascade,22 thermal plants, four SVCs. The electric network was modeledwith 61 buses and 103 lines. Active and reactive generationsof five noncontrollable hydro plants were directly discarded inthe nodal balance as a negative load. An entire survey day wassplit into 48 periods. The voltage limits were set to 1.1 and 0.95p.u. in all buses (excepting the lower limit in SICN-17 bus,which was set in 1.0 p.u. due to operational requirements). Thequadratic cost curve term for thermal units was neglected be-cause it was not available. However, the proposed algorithm hasnot presented any problem in various tests done with estimatedquadratic costs.

The penalty costs used were UM 1750 for active deficit andUM 1650 for reactive deficit.

During a typical day with a high hydropower share (89%),four simulations were made to observe the impact of ac mod-eling. In the first and second simulations, the reactive nodal bal-ance and other constraints containing reactive variables weredisabled, and the voltage was fixed in 1.0 p.u. in all buses inorder to reproduce a dc modeling, though maintaining a non-linear losses representation. Besides, in the first and the thirdsimulation, the line limit capacity constraints were not set. Fi-nally, the two last simulations were made with a full ac mod-eling. Due to the large number of results (voltages, angles, gen-eration levels, etc.), only a few of them are shown. All results arereferred to the proposed method, including the above-describedaccelerating techniques, unless stated otherwise.

Table I shows the total costs for the four simulations. It ispossible to see the impact over the cost of the activation of bothgroups of constraints: voltage limits and/or capacity limits.

The CPU time solution corresponds 55% to the masterproblem and 45% to the subproblem. These values are onlyrepresentative for this specific problem. For larger problems,the execution time increases quickly due to the MIP nature ofthe master problem.

Table II also shows the impact of not considering the voltageconstraints.

TABLE IIMPACT ON OPERATIVE COSTS DUE TO NETWORK MODELING CONSTRAINTS

TABLE IIBUS VOLTAGE (IN P.U.) COMPUTED FOR THE FIRST SIMULATION.

OUT-OF-LIMIT VOLTAGES IN BOLD

TABLE IIIBUS VOLTAGE (IN P.U.) COMPUTED TO THE FOURTH SIMULATION.

ALL VOLTAGES ARE AT THEIR OPERATIVE LIMITS

These voltages were obtained after fixing the commitmentstatus (0/1) of both thermal and hydro units and performing theproposed algorithm with the voltage constraints.1

The fourth simulation had all its constraints active. Table IIIshows the voltage levels for the fourth simulation.

Table IV shows the unit commitment status for dc (first sim-ulation) and ac results (fourth simulation). The voltage buseswith problems were resolved with a new unit commitment. Atpeak hour, the thermal unit U12 startup and stay online lastedone hour because of reactive requirements.

Fig. 2 shows the load diagram. The very flat thermal gener-ation dispatched suggests a good solution for the proposed de-composition.

Impact From the ACCT: Table V shows the impact fromACCT aimed at reducing the total CPU time. The CPU timeconsiderably decreases (five times), with no reduction in solu-tion quality. In various tests performed, the ACCT was able todiminish the solution quality, and it reached 0.4% of tolerance.The key reason for this reduction on CPU time lies on the lessiteration needed to achieve convergence. This fact is very im-portant because the master problem grows with each iteration.Then, its solution takes more CPU time.

Fig. 3 shows the convergence with the addition of ACCT (13iterations) and without ACCT (25 iterations).

Results Validation: As previously stated, once the discretevariables were computed, the original problem was turned intoa large-scale nonlinear (totally coupled) one.

1To check a dc dispatch, normally an OPF is performed in each period, andthe active power difference is taken by the thermal units. But in the proposedcase, a more generic checking is performed by allowing further moves of boththermal and hydro generation.


TABLE IVUNIT COMMITMENT STATUS A MEANS ONLY PRESENT IN AC DISPATCH,AD PRESENT IN BOTH AC AND DC DISPATCH AND D ONLY PRESENT

IN DC DISPATCH

Fig. 2. Dispatch for full ac modeling: fourth simulation. Noncontrollable hydrounits are not included.

TABLE VTOTAL COST CONSIDERING PURE GBD AND GBD PLUS ACCT:

FOURTH SIMULATION

Fig. 3. Number of iterations required to achieve convergencelower and upperbound evolution (fourth simulation).

TABLE VITOTAL COST CONSIDERING THE PROPOSED METHOD INCLUDING ACCT,

AND THE FULLY COUPLED NONLINEAR PROBLEM

The additional cost savings noted in Table VI are caused byan improved water allocation resulting from the nonlinear rep-resentation of the total problem. It may be argued that 1.41%(the largest difference) could be non-negligible, but it is impor-tant to say that this full operative solution was achieved in rea-sonable computing CPU time, while avoiding any post-heuristicschedule correction.

Discussion: In the proposed master problem of decomposi-tion, the active power balance constraint is not explicitly con-sidered; rather, it is actually present as signal cost. The maindrawback to include this constraint is the fact that the activepower balance is actually the entire set of active nodal balancemade up of completely nonlinear constraints. Even though ifthey were linearized and added to the master problem, it wouldbecome so large and would take extremely long CPU times toreach the solution. In fact, there is an optimization techniquefor nonlinear mixed integer problems called outer approxima-tion [21] that solves MINL problems by adding linearized con-straints to the master problem, converging in fewer iterationsthan with Benders decomposition. Nevertheless, it is applicableonly to small-scale problems because of the above-mentionedreasons.

Another possibility is to add feasibility cuts, as in [5]; though,actually, these are not mathematically deducible. In some cases,they could cause the master problem to converge to a wrongsolution. For instance, if it includes the next constraint into themaster problem (losses computed in the previous iteration)

losses (24)

Several tests showed that this constraint can reduce thenumber of iterations, with a much worse solution than with theproposed method (see Table VII).


TABLE VIITOTAL COST, CPU TIME, AND ITERATIONS CONSIDERING

THE ADDITIONAL CONSTRAINT (24)

TABLE VIIITOTAL MISMATCHES DEFICIT OR EXCESS [MWH OR MV ARH]:

FOURTH SIMULATION

The implementation of constraint (24) in our proposed de-composition without any other ACCT does not help to reducecomputation time. The same tailing-off effect has been noted,though with a difference: The gap between the upper and lowerbounds in the first iterations is smaller than with the proposedmethod. This fact tells us that the proposed dispatch is better,though the water movement between sub-periods is again thegreatest problem that prevents achieving a faster convergence.

In addition, it is strictly necessary to be sure that the genera-tion is always able to cover both the demand and the losses. Ifcongestion appears, and the only solution to meet the nodal bal-ance is to shed load [18], [19] (equivalent to dispatching a faultmachine in that bus), this heuristic cut will lead to an incorrectsolution.

For actual systems, the authors have not noted any numer-ical difficulty as regards unit status. Most 0/1 definitive status ofthermal units are defined in the first iterations. However, whenreactive problems are present, this status could change for cer-tain units (near to the voltage problem bus) until the reactiveproblem is overcome. The main observed reason for slow con-vergence lies in the variations of hydro generation between iter-ations in the last iterations.

Due to the linear representation of the subproblem into themaster problem, the hydro generation can change abruptly insome units between two successive iterations. This behaviorcauses power flow variations, with the consequent variations oflosses. In this condition, the proposed generation does not ex-actly match the active power balance in the subproblem, so asmaller amount of penalized imbalance will be present. This im-balance, in spite of being small or very small, contributed largelyto the total cost due to its penalization. From the tenth iteration,hydro generation is allowed to change only % of its pre-vious value. Fewer percentages lead to faster convergence butalso could lead to a non-feasible master problem.

The last iterations of the process are used to reduce the deficitvery close to zero. In this instance, the deficit is very small (seeTable VIII), but it has a high effect over the cost. This is possible

to observe in the computed lower bound. This criterion requiresmore CPU time than the nodal mismatch criteria adopted in [18]and [19], but the quality of solution is better and a near globaloptimum solution is assured.

In our proposal, the quadratic term is located in the sub-problem with the goal to alleviate the computational burden ofthe master problem and to obtain faster solutions, because asis observed, this step requires the most computational effort.Our premise was based on the fact and experience about powersystems where the operative characteristics of unit generatorsare significantly different. This fact causes that commonly ap-pears one or two marginal units (in pure thermal systems) orthe uniform operation of thermal units in hydrothermal systemswith hydro predominance. The master problem, after severaliterations, begins to delimitate the final solution region of theproblem and exist few alternatives (thermal dispatch) with sim-ilar cost. In this manner, the last iterations are necessary to adjustthe hydro generation units for reducing the network losses (dueto its linear representation reconstructed by the Benders cuts).

We also test a purely thermal system (IEEE 118 bus) withmany similar (some of them equal) cost curves and importantquadratic terms. The proposed method showed a slow conver-gence because the linear representation2 of the cost curves intothe master problem, a thermal unit will always try to absorb mostof the deficit. Then we proceed to consider the quadratic term inthe master problem transforming this in a type of mixed integerquadratic problem. Under these test conditions, the problem wassolved in few iterations.

As a conclusion, the type of modeling (the quadratic term inthe subproblem or in the master problem) is problem-dependentlike was showed. There is not one methodological solution thatfits all cases in unit commitment and short-term hydrothermalcoordination.

V. CONCLUSION

The proposed methodology is innovative from the viewpointthat it allows modeling the transmission network using an acOPF in the STHS optimization. It has fundamental importancefor largely expanded and weakly meshed systems, such as thoseof Latin American countries or systems with a lack of reactivesupport.

The presented decomposition scheme is simple, pure, and ro-bust, even for dominant hydraulic power systems. It is also easyto implement, because it uses well-known, optimized, fast tech-niques, such as MIP and ac OPF algorithms. Its main disadvan-tage is that it may require a large number of iterations. The largeiteration numbers required to achieve convergence are reduced(and CPU time) with the addition of simple rules.

Important constraints like transmission congestion andvoltage control are considered in the proposed methodology,without needing any post-schedule correction. This ensuresthe near optimal global solution, while the quality servicerequirements are satisfied.

The chosen decomposition of the problem allows consideringthe networks entire modeling, with little impact on computingCPU time.

2In spite of having the Benders cuts that transfer the impact of the quadraticcost term, the representation still have been linear.


Complex hydraulic chains or additional constrains can beeasily modeled and/or added.

In contrast to the Lagrangean relaxation approach, it is notnecessary to build a feasible solution by resorting to complexheuristic techniques that do not always guarantee the near globaloptimum.

Finally, parallel processing can be used in both the masterproblems and the subproblems to reduce the computing times.

REFERENCES[1] A. J. Wood and B. F. Wollenberg, Power Generation, Operation and

Control, 2nd ed. New York: Wiley, 1986.[2] J. H. Walter, G. H. S. Warner, and G. B. Shebl, An enhanced

dynamic programming approach for unit commitment, IEEE Trans.Power Syst., vol. 3, no. 3, pp. 12011205, Aug. 1988.

[3] G. Xiaohong, L. H. Yan, and J. A. Amalfi, Short-term scheduling ofthermal power systems, in Proc. 17th PICA Conf., Baltimore, MD,May 1991, pp. 120126.

[4] Y. Houzhong, P. B. Luh, X. Guan, and P. M. Rogan, Scheduling ofHydrothermal Power Systems, IEEE Trans. Power Syst., vol. 8, no. 3,pp. 13581365, Aug. 1993.

[5] N. Alguacil and A. J. Conejo, Multiperiod optimal power flow usingBenders decomposition, IEEE Trans. Power Syst., vol. 15, no. 1, pp.196201, Feb. 2000.

[6] E. Kuan, O. Ao, and A. Vargas, Unit commitment optimization con-sidering the complete network modeling, in Proc. IEEE Porto PowerTech, 2001, vol. 3, p. 5.

[7] V. M. Doa, A. Hoese, and A. Vargas, Predespacho de Potencia Re-activa en el Despacho Econmico de Sistemas Hidrotrmicos Multin-odales, IX ERLAC Foz do Iguaz, Brazil, May 2001, IX/FI-38.26.

[8] B. R. Serrano, R. E. Laciar, and A. Vargas, Coordination of Voltageand Reactive Power Control Actions Based On Economics Criteria, XERLAC Puerto Iguaz, Argentina, May 2003, X/PI-38.4.

[9] E. L. Miguelez, L. R. Rodriguez, T. G. S. Roman, F. M. E. Cerezo, M. I.N. Fernandez, R. C. Lafarga, and G. L. Camino, A practical approachto solve power system constraints with application to the Spanish elec-tricity market, IEEE Trans. Power Syst., vol. 19, no. 4, pp. 20292037,Nov. 2004.

[10] F. Zhuang and F. D. Galiana, Towards a more rigorous and practicalunit commitment by Lagrangian relaxation, IEEE Trans. Power Syst.,vol. 3, no. 2, pp. 763773, May 1988.

[11] J. Batut and A. Renaud, Daily generation scheduling optimizationwith transmission constraints: A new class of algorithms, IEEE Trans.Power App. Syst., vol. PAS-7, no. 3, pp. 982989, Aug. 1992.

[12] G. Xiaohong, N. Ernan, L. Renhou, and P. B. Luh, An optimiza-tion-based algorithm for scheduling hydrothermal power systems withcascaded reservoirs and discrete hydro constraints, IEEE Trans. PowerSyst., vol. 12, no. 4, pp. 17751780, Nov. 1997.

[13] R. Baldick, The generalized unit commitment problem, IEEE Trans.Power Syst., vol. 10, no. 1, pp. 465475, Feb. 1995.

[14] S. Al Agtash, Hydrothermal scheduling by augmented Lagrangian:Consideration of transmission constraints and pumped-storage units,IEEE Trans. Power Syst., vol. 16, no. 4, pp. 750756, Nov. 2001.

[15] A. M. Geoffrion, Generalized Benders Decomposition, J. Optim.Theory Appl., vol. 10, pp. 237260, 1972.

[16] M. E. El Hawary, Optimal power flow: Solution techniques, require-ments, and challenges, IEEE Tutorial Course, 1996.

[17] C. Murillo-Sanchez and R. J. Thomas, Thermal unit commitment in-cluding optimal AC power flow constraints, in Proc 31st Hawaii Int.Conf. System Sciences, 1998, vol. 3, pp. 8188.

[18] Y. Fu, M. Shahidehpour, and Z. Li, Security-constrained Unit Com-mitment with AC constraints, IEEE Trans. Power Syst., vol. 20, no. 2,pp. 15381550, May. 2005.

[19] Y. Fu, M. Shahidehpour, and Z. Li, AC Contingency dispatch basedon security-constrained unit commitment, IEEE Trans. Power Syst.,vol. 21, no. 2, pp. 897908, May 2006.

[20] H. Ma and S. M. Shahidehpour, Unit commitment with transmissionsecurity and voltage constraints, IEEE Trans. Power Syst., vol. 14, no.2, pp. 757764, May 1999.

[21] Watkins, Jr, D. W. McKinney, and C. Daene, Decomposition methodsfor water resources optimization models with fixed costs, Adv. WaterRes., vol. 21, no. 4, pp. 283295, Apr. 10, 1998.

[22] S. Dan, R. Philbrick, and A. Ott, A mixed integer programming solu-tion for market clearing and reliability analysis, in Proc. IEEE PowerEng. Soc. General Meeting, Jun. 1216, 2005, vol. 3, pp. 27242731.

[23] J. Garcia-Gonzalez and G. A. Castro, Short-term hydro schedulingwith cascaded and head-dependent reservoirs based on mixed-integerlinear programming, in Proc. IEEE Porto Power Tech, Sep. 1013,2001, vol. 3, p. 6.

[24] N. V. Sahinidis and I. E. Grossmann, Convergence properties of gen-eralized benders decomposition, Comput. Chem. Eng., vol. 15, no. 7,pp. 481491, Jul. 1991.

[25] Dash optimization Dashs Xpress-MP suite, Optimization SoftwareProduct for Modeling and Optimization. [Online]. Available: http://www.dashoptimization.com.

Wilfredo S. Sifuentes (S92) was born in Peru in 1969. He received theMechanical-Electrical degree from Universidad Nacional de Ingenieria (UNI),Lima, Per, in 1992. In 2001, he was awarded a DAAD scholarship to carry outhis Ph.D. studies at the Instituto de Energa ElctricaUniversidad Nacional deSan Juan (UNSJ), San Juan, Argentina.

He worked at COES-SINAC as a dispatcher and real-time system operator ofthe Peruvian interconnected system. His research interests include optimizationtheory, planning, and economics of electric energy systems.

Alberto Vargas (M97SM02) was born in Argentina. He received the Electro-mechanical Engineer degree in 1975 from the Universidad Nacional de Cuyo,Mendoza, Argentina, and the Ph.D. degree in electrical engineering in 2001from the National University of San Juan (UNSJ), San Juan, Argentina.

At present, he is a Professor of postgraduate studies at the Institute of Elec-trical Energy (IEE)UNSJ. Since 1985, he has been a Chief Researcher of theteam of Electrical and Competitive Market, at IEE.

Hydrothermal Scheduling Using Benders Decomposition- Accelerating Techniques

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