572 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 16, NO. 4, NOVEMBER 2001

Application of the Optimal Power Flow Model inPower System Education

Viktor A. Levi, Member IEEE and Dusko P. Nedic

AbstractApplication of the optimal power flow based softwarepackage to aid instruction in classroom is given in this paper. Thepackage is used in courses Computer Methods in Power Engi-neering and Planning, Operation and Control of Power Systems.The essential idea is to utilize all calculation modules from thepreviously developed optimal power flow model. The software isapplied in four projects, that is, Constrained Optimization, Trans-mission Expansion Planning, Active and Reactive Dispatching forOperations Planning, and Corrective Rescheduling within RealTime Control.

Index TermsEducation, optimal power flow, optimization,power systems, software package, student projects.

I. INTRODUCTION

I N THE Electrical and Computer Department of theSchool of Engineering Sciences a new power engineeringprogram, based on recommendations from [1], was introducedin 1997/1998. The goal is to make the old program more attrac-tive, and emphasis is put on laboratory experience, applicationof software packages and individual elaboration of projects.

In recent years software packages have become irreplaceableclassroom teaching tools, especially in the field of powersystems where complexity and dimensionality of problems arethe main obstacles [2][15]. These packages can be divided intotwo global categories: 1) Integrated packages encompassingdifferent calculation (simulation) modules [2][5], [7][10];and 2) Specialized packages dedicated to a specific field [6],[11][15]. We have chosen the latter approach and apply thespecialized packages in different power engineering courses.

In this paper, application of the general purpose optimalpower flow (OPF) model for education of power engineersis presented. The OPF model was originally developed forthe voltage/reactive studies [16], [17]. Since it contains manydifferent modules, it became clear that this model could be avery efficient teaching tool. This idea is intensified by the factthat the OPF model is rarely used in educational packages,and even in cases of its application, it is an external blackbox [7], [13]. On the other hand, deeper understanding of theOPF model gives much insight into the solution techniquesand various power system problems. Thus, the developedOPF model is applied in the courses Computer Methods inPower Engineering and Planning, Operation and Control ofPower Systems, by devising four projects: 1) Constrained

Manuscript received October 8, 1999; revised February 12, 2001.V. A. Levi is with the School of Engineering Sciences, University of Novi

Sad, Novi Sad, Yugoslavia.D. P. Nedic is with the Power Distribution Company, Elektrovojvodina,

Novi Sad, Yugoslavia.Publisher Item Identifier S 0885-8950(01)08875-7.

Optimization; 2) Transmission Expansion Planning; 3) Ac-tive and Reactive Dispatching for Operations Planning; and4) Corrective Rescheduling for Security Enhancement. Theyare obligatory tasks for all senior students.

II. POWER CURRICULUM

The Electrical and Computer Department of the Schoolof Engineering Sciences offers a five year (ten semester)undergraduate program for the Bachelor of Science inElectrical Engineering (B.Sc.E.E.) diploma. During the firsttwo years, all students take basic subjects, such as, DiscreteMathematics, Mathematical Analysis, Physics, Fundamentalsof Electrical Engineering, Fundamentals of Computer En-gineering (first year), Electrical Circuits, Control Systems,Electronics, Telecommunications, Fundamentals of SoftwareEngineering, Electrical Measurements, and Fundamentals ofPower Engineering (second year). The power curriculum startsin the third year and it covers four subfields: 1) Electricalmachines; 2) Power electronics; 3) Design of power plants; and4) Modeling and analysis of power systems. The first subfieldcovers the courses Electrical Machines, Analytical Theory ofElectrical Machines, and Control of Electrical Drives, whilePower Electronics, Microprocessors, and Control of PowerConverters are offered in the second subfield. The courses fromthe third subfield are Design of High Voltage Power Plants,Design of Low Voltage Installations and Drives, and AdvancedDesign Techniques, while the forth subfield covers ComputerMethods in Power Engineering, Power System Analysis, Dis-tribution Systems, Planning Operation and Control of PowerSystems, and Power System Modeling. In the first and thesecond subfields the main attention is focused on laboratoryexperiments, the courses from the third subfield need to giveexperience in designing and completing project documentation,while the last group of courses is directed toward applyingthe ready-made software packages and writing own (small)programs. All lectures are finished in the ninth semester, andelaboration of the diploma thesis should be done in the tenthsemester.

A brief description of the two courses, for which the projectsare developed, is given in the Appendix.

III. OPTIMAL POWER FLOW MODEL

Simplified structure of the developed OPF model is given inFig. 1. When data input is finished, a user chooses one of thefollowing problems to be solved: 1) Production cost minimiza-tion; 2) Transmission losses minimization; 3) Maximization ofvoltage/reactive security; 4) Solution of the secondary voltage

08858950/01$10.00 2001 IEEE

LEVI AND NEDIC: APPLICATION OF THE OPTIMAL POWER FLOW MODEL IN POWER SYSTEM EDUCATION 573

Fig. 1. Overall structure of the OPF model.

control problem; 5) Minimization of control variable deviations;6) Planning of new reactive sources; 7) Calculation of systemreliability; and 8) Determination of the maximum loadabilitypoint. Although all these problems are very similar, actual solu-tion processes can be very different. Therefore, much attentionis paid to each problem initialization, resulting in various aux-iliary modules that are used to obtain a good starting point(Fig. 1). Initial setting of generator active productions is doneby using either the uniform relative production principle, orthe economic dispatching (ED) of the one-point system. Thesevalues can be further improved by running the transmission con-strained economic dispatching (TCED) model [20]. In case ofreliability analysis, initialization of curtailed loads is performedby using the minimum load curtailment (MLC) model [21]. Theinitialization stage is finished by running the fast decoupled loadflow (FDLF) model and a linear programming model that im-proves the voltage-reactive variables.

The OPF model is solved with the aid of the exterior/interiorpoint based Newton approach. Since the solution processes ofdifferent problems can be completely diverse, several optionsare incorporated into the solution modules. The main contri-bution is introduction of the barrier functions, special logic forinclusion/extraction of barrier terms, and the adaptive stepsizelength. Testing of the OPF model has shown that it is more ad-vantageous to use Lagrange multipliers in case of branch flowconstraints, while the rest of inequality constraints can be mod-eled via the penalty and/or inverse and logarithmic barrier func-tions. Sensitivity analysis of the optimal solution is the last stageof the OPF model.

IV. PROJECTS

Four projects are introduced to help instructors in teaching thecourses Computer Methods in Power Engineering and Power

System Planning, Operation and Control. All projects are re-alized in the eighth and the ninth semesters when students arefamiliar with power systems. A brief description of the projectsfollows.

A. Constrained Optimization

The aim is to renew students knowledge in linear and non-linear optimization methods. Teaching objectives are:

Dual simplex algorithm is a natural solution method forpower system problems.

Direct solution of the KuhnTucker optimality conditionscan be a cumbersome task even for simple problems.

Primal methods are robust, but the convergence can beslow.

Lagrange methods (Newton approach) are fast and reliablewhen a good starting point is available.

Penalty and barrier function methods give only an approx-imate solution.

A nonlinear programming method gives usually a localoptimum, which is obvious in case of multiple optima.

The project consists of following stages:1) The MLC model (TCED model) is solved with the aid

of the dual simplex method. Initial setting of load curtail-ments (MLC model) and incremental active power gener-ations (TCED model) to zero values gives the optimalitycriteria. Since branch flow constraints are not satisfied,the primal feasibility does not hold. Students are invitedto follow the dual feasibility criteria ( ), primal fea-sibility (slack variables for the branch flow constraints)and increase of the objective function.

2) The ED of the one-point system is solved with theaid of two methods. The first is direct solution of theKuhnTucker optimality conditions by using the devel-oped algorithm. Students are invited to change the costcoefficients and to follow identification of the bindingset of inequality constraints. Next, the same problem issolved via the reduced gradient approach [20]. Incre-mental costs and changes of generations are observedthrough the iterative process.

3) Minimization of the production cost is done with the aidof the OPF model. All inequality constraints are relaxed,and the Newton approach is applied to the Lagrangian. In-cremental changes of the problem variables and Lagrangemultipliers are followed until the convergence criteria aresatisfied.

4) The complete OPF model is used to minimize the produc-tion cost, and inequality constraints are modeled throughthe penalty and barrier functions. The main points arechanges of the current set of active inequality constraints,and the solution increments. Students are invited tochange the penalty and barrier function coefficients inorder to find out the convergence range of the analyzedOPF problem, as well as influence of these coefficientson the optimum of the objective function.

574 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 16, NO. 4, NOVEMBER 2001

B. Transmission Expansion Planning

The aim is to determine the network reinforcements andnew VAR sources that satisfy the (n-1) security principle. Thisproject complies with the real-life practice in Yugoslav powerindustry. Thus, the network reinforcements are calculated fromthe sensitivity analysis of the optimal solution of the MLCmodel [21], while the voltage/reactive subproblem is studiedby using the OPF model. Teaching objectives of the project are:

Rather complex transmission expansion planning shouldbe broken into smaller subproblems.

The essential principle of the transmission expansion plan-ning is the (n-1) security criterion.

Methods based on the DC load flow are simpler but notsufficiently accurate, so the AC load flow based modelsshould be applied.

Expansion planning should be interactive, that is, aplanning engineer must take place in the decision makingprocess.

The project consists of following stages:1) The MLC model is run for the base case regime and all

single outages. If the optimal solution is not zero, a sensi-tivity based planning criterion that shows reduction of theoverall curtailed load with respect to the unit increase of abranch susceptance is applied for network reinforcements[21]. One branch is added to the network, and the wholeprocedure is repeated until the (n-1) security criterion ismet.

2) The previously obtained transmission expansion plan isselected. The OPF model is used to minimize the costof reactive sources in the base case peak regime. Singleoutage states are analyzed next, and the optimal valuesof new reactive injections are cumulatively added. If theQV contingency ranking procedure is used, the new ca-pacitive sources are necessary only within the first mostsevere outage states.

3) One complete expansion plan satisfying the (n-1) securitycriterion is chosen. Multiple contingency cases are ana-lyzed with the aid of the OPF model (the objective func-tion is a sum of curtailed loads). Students should realizethat cascade outages can bring the system to a blackout.

C. Active and Reactive Dispatching for Operations PlanningThe goal is to show variability of the production cost and

the voltage profile when using different objective functions andassociated constraints. Teaching objectives are:

Active power dispatching of the one-point system is abasic, approximate method, while the OPF solution isexact.

The transmission network increases the overall productioncost.

Reactive power dispatching is necessary to improve thevoltage/reactive power security.

Combined activereactive power dispatching gives atrade-off between the production cost and the voltage/reactive security.

The project encompasses following stages:1) The active power dispatching of the one-point system

is performed. Next, three OPF runs are executed. Inthe first case all inequality constraints are relaxed, thenlower/upper limits on the active and reactive powergenerations are taken into account, while the rest of in-equality constraints are added in the third case. Studentsare invited to change the voltage magnitude and branchflow limits and to compare the results.

2) The reactive power dispatching is done with the activepower generations set to the economic values. Trans-mission losses are minimized first, and it is followedby maximization of the voltage/reactive security, inwhich squared relative reactive power generations areminimized. Combination of these two objectives is thenmade and computation is repeated. Students need tocompare the production costs, reactive generations andvoltage profiles in the considered regimes.

3) A trade-off between the active and reactive powerdispatching is sought for. The objective function is aweighted sum of the production cost and the squaredrelative reactive generations. Students are invited to usea range of weighting coefficients.

D. Corrective Rescheduling for Security EnhancementSecurity assessment of a power system consists of three

steps: 1) Contingency selection; 2) Contingency evaluation; and3) Preventive/corrective control actions. In the first step, contin-gencies are ranked in an approximate order of severity by usingthe branch flow performance index (branch overloads)and the voltage performance index (voltage/reactiveproblems) [8]. A full load flow solution is then found for theworst contingencies from the top of the ranking list. Finally, apreventive control strategy is developed for contingencies withthe most severe consequences, while the corrective control isdefined in cases when fast remedial actions are sufficient. Inthis project, the main attention is paid to the corrective controls.Teaching objectives are:

Severity of a contingency can be well predicted by usingthe performance indices, which should be separatelydefined for the branch overload and the voltage/reactiveproblems.

The violated operation constraints can be efficiently elim-inated by the corrective controls.

Since various controls are involved in different contin-gency cases, it is difficult to adjust them and cover sev-eral contingencies.

A large number of involved corrective controls should bereduced in order to enable the operator to realize the task.

The project consists of following stages:1) A base case regime is selected and minimization of the

production cost is done with the aid of the OPF model.A single outage is defined and the FDLF is run, givingthe indices and after the first iteration. Thecontingency evaluation is repeated until a sufficient

LEVI AND NEDIC: APPLICATION OF THE OPTIMAL POWER FLOW MODEL IN POWER SYSTEM EDUCATION 575

Fig. 2. Environment structure of the developed software package.

number of outages is analyzed. Values of the indicesand are compared with the true indices obtainedafter the complete load flow solutions.

2) A severe contingency is selected and the OPF model isused to minimize the production cost. The control vari-able settings and the total production cost are comparedwith the base case values. This procedure is repeated withthe modified objective function representing a sum of theproduction cost and the cost of control variable devia-tions. Students are invited to change the weighting coef-ficients in the objective function and to follow variationof the production cost and the control variables.

3) Analysis of the previously selected contingency is donewith the reduced number of control variables. The vari-ables closest to the base case values are fixed and the OPFmodel is rerun. Students need to choose different typesand number of the fixed variables. Dependence betweenthe production cost and the types and number of the con-trol variables can be made.

V. DESCRIPTION OF THE SOFTWARE PACKAGE

The previously developed OPF software has modular struc-ture and it is coded in the FORTRAN POWER STATION

programming language. The following models can be solvedwithin the initialization stage: ED, DC load flow, TCED, MLC,FDLF and four linear programming models in incremental formused to improve the voltage-reactive variables. Two special-ized algorithms (modules) based on either the direct solutionof the KuhnTucker optimality conditions, or the reduced gra-dient method are applied for the ED model. The sparse linearequation solvers from the IMSL library (routines LFTXGand LFSXG) are used in case of the DC load flow and theFDLF models. The MLC model is solved with the aid of thespecialized reduced-basis sparse dual simplex linear program-ming routine [21], while the dense primal simplex linear pro-gramming IMSL routine (DLPRS) is applied for the TCED(node angles eliminated) and the voltage/reactive power incre-mental models. Solution of the OPF model is obtained by devel-oping the modules in which the Lagrange method (Newton ap-proach) is combined with the penalty/barrier function methods.Construction of the problem gradient and Hessian, identifica-tion of the lower/upper limit binding constraints and the activebranch current constraints, and the solution of the system ofsparse linear equations with the same IMSL routines are themajor steps here.

A simple software package, intended to be used as a teachingtool in the classroom, is then developed. This package is based

576 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 16, NO. 4, NOVEMBER 2001

Fig. 3. Five-node test system.

on the described OPF software, and the essential idea is to useits individual modules either separately, or together for OPFcalculations. The complete graphical interface is programmedin the VISUAL BASIC and the package is run under theWINDOWS 95 operating system. Input data are organizedas ASCII files, and results of calculations can be displayed onthe single-line diagrams of the analyzed systems. Environmentstructure of the software package is in the form of a tree, andit is given in Fig. 2.

The main menu contains five options: 1) Input data; 2) Systemanalysis; 3) Graphical display; 4) Tabular display; and 5) Exit.All input data are divided into several groups, and they are givenin [p.u.]. It is possible to analyze the 5-node test system and the39-node IEEE power system. Within the system analysis, theone-point ED can be performed, and the MLC, TCED, FDLFand OPF models can be run. Only basic options of the OPFmodel objectives are given in Fig. 2. The results of calculationsare graphically displayed either on the one-point system (one-point ED), or on the entire network (the rest of models). Finally,relatively detailed reports on the iterative solution processes aregiven in tabular form.

VI. EXAMPLES

Constrained optimization is studied on the 5-node test systemshown in Fig. 3. We present here some interesting results re-garding the comparison of the penalty and barrier function basedOPF solutions (Table I), as well as the multiple optima of theOPF model (Table II). In the first case, the greatest differencewas obtained within expansion planning of new reactive sources

, when generations were modeled either as implicitvariables (I), or with the aid of the barrier functions (II). Theformer approach requires less additional capacitive injections,

TABLE IPENALTY AND BARRIER FUNCTION BASED OPF SOLUTIONS

TABLE IIMULTIPLE OPTIMA IN CASE OF RELIABILITY ANALYSIS

since all reactive generations are limited by the highervoltage dependent boundaries (Table I). Minimization of equalproportions of the constant ( ) and voltage dependent loadcurtailments ( ) with active ( ) and reactive load exponents( ) equal to 2.00 was performed for the case when branch cur-rent was limited to 2.2 [p.u.] (Table II). In situation I,the current flow constraints were activated in the first OPF it-eration, while these constraints became active in the fourth it-eration in case II). The multiple optima are obtained, and themajor difference is between the active generations and ac-tive power flows (or, node angles ), since the voltage-reactivevariables (voltages , reactive productions and the tap po-sition ) do not differ significantly. Next, variation of the ac-tive and reactive load exponents was performed and the resultsare summarized in Table III. Case I) corresponds toand , while case II) is for and .The impact of the load exponents variation on the power flowsand voltage profile is obvious, but the convergence patterns arealso completely different.

Transmission expansion planning of the 39-node IEEEtest system, in which the branch flow limits are significantlytightened, is presented. In the first stage, after satisfactorybase-case run, single contingencies are studied with the aid ofthe MLC model. The most difficult contingencies, necessaryload curtailments and the expansion planning criterion are

LEVI AND NEDIC: APPLICATION OF THE OPTIMAL POWER FLOW MODEL IN POWER SYSTEM EDUCATION 577

TABLE IIIVARIATION OF ACTIVE AND REACTIVE LOAD EXPONENTS

shown in Table IV. It is obvious that the planning criteriongives much freedom in choosing the branches that should bereinforced [21]. One possible solution to this problem is thereinforcement of branches 1639, 1213, 723, 825, 2526and 130. In the next stage, it was supposed that the newreactive sources can be located in all consumption nodes,and the OPF model was applied to provide 0.951.05. The base-case of the reinforced network was successful,while the lined up contingencies, which required new reactivesources, are quoted in Table V. The new reactive injectionsare not cumulatively added in order to show that the voltage/reactive problem is of local nature, and that the OPF solutionsrequire too many locations for new reactive sources. Hence,a cumulative addition of the capacitive injections should beperformed when values exceed a certain threshold. Next, anew planning scenario can be defined by relaxing the flowlimits of branches that connect the generators to the grid(Table VI). In this case only the outage of generator 9 brings tothe same load curtailments in nodes no. 26 and 28 (the outagesof branches 219, 1819, 1213 and generators 2 and 4 arecompensated by additional generator delivery capabilities), sothat the reinforcement of only one branch (2526) gives thesolution which satisfies the (n-1) security criterion. Planning ofnew reactive sources is very similar to the results presented inTable V, which indicates clearly that the branch overload andthe voltage/reactive problems can be treated separately.

Dispatching of active generations is performed on the39-node IEEE test system, and it starts with the one-point ED.All linear and quadratic cost coefficients are set to, respectively,1.00 and 0.10 [p.u.], and the same generator outputs are ob-tained. The active generation limits and the cost coefficients arethen changed, in order to establish relations between the activegenerations and these quantities. Next, the production costminimization is done four times by using the following OPFmodels: I) Without inequality constraints; II) Only constraintson reactive generations are used; III) Active and reactivegenerations are limited; and IV) All inequality constraints areapplied (Table VII). In case I), difference between the activegenerations is the consequence of transmission losses, reactivegenerations are large, and the voltage profile is very low. Thereactive generation limits reduce significantly reactive outputs,node voltages are increased, while the active productions

change negligibly. The most important consequence of in-cluding the active generation limits is rescheduling of theactive productions (generators no. 4 and 10 are limited) andactive power flows, while the voltage profile is very similarto the case II). Constraints on voltage magnitudes furtherincrease the node voltages and, consequently, decrease thereactive generations. Summary of these four cases is given inTable VIII. The increase of the overall production cost, activegenerations and the node voltages, as well as the decrease ofthe reactive outputs are the main conclusions.

Three cases are studied within the reactive power dis-patching: I) Transmission loss minimization; II) Maximizationof voltage/reactive security; and III) Combination of these twoobjectives. Reactive productions are given in Table IX, whilevoltage profiles are shown in Fig. 4. In case I), the reactivegenerations are relatively high and the node voltages are low.Maximization of reactive reserves reduces the overall reactiveproduction for 1.2 [p.u.], increases the node voltages (thelowest voltage is 0.967 [p.u.]) and gives rise to the transmissionlosses equal to 0.037 [p.u.]. Combination of these equallyweighted objectives gives very similar reactive generationsand the voltage profile as in case II), while the transmissionlosses are slightly reduced (0.001 [p.u.]). This is due to thehigher influence of the reactive productions in the objectivefunction. The combined active and reactive power (securitymaximization) dispatching depends on the choice of theweighting coefficients. The overall production cost varies in therange of {104.1 105.7} [p.u./h], the total reactive generationbelongs to the interval {13.5 16.8} [p.u.], while the voltageprofiles differ significantly in the boundary optimization cases.

Finally, we include a sample description of one of the correc-tive rescheduling projects, because students are usually givenslightly different tasks. The steps are: 1) Find a base case min-imum cost OPF solution of the 39-node IEEE system; 2) Set thecontrol variables to the base case OPF solution values, run theFDLF model for the prespecified single branch outages and rankthese outages according to the index ; 3) Select a severecontingency case, find the OPF solution, and if the load curtail-ments are necessary repeat the step no. 3) for another outage;otherwise continue; 4) Find a trade-off between the productioncost and the deviation of the control variables from the base casevalues; and 5) Select another contingency case and go back tothe step no. 3). The expected results are: ranking of branch out-ages, production cost and values of control variables in differentcontingency regimes, as well as the variation of these quanti-ties when modifying the objective function with the weightedsquared control variable deviations.

VII. CONCLUSION

In this paper, application of the OPF model for educationalpurposes is presented. Four projects are designed: constrainedoptimization, transmission expansion planning, active andreactive dispatching for operations planning, and correctiverescheduling for security enhancement. The individual modulesof the OPF model and the OPF model itself represent the core ofthe software package developed to aid instruction in two powersystem courses. The benefits for the students are multifold:

578 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 16, NO. 4, NOVEMBER 2001

TABLE IVEXPANSION PLANNING OF NETWORK REINFORCEMENTS

TABLE VEXPANSION PLANNING OF REACTIVE SOURCES

TABLE VIMODIFIED BRANCH FLOW LIMITS

TABLE VIIDISPATCHING OF ACTIVE GENERATIONS

TABLE VIIISUMMARY OF ACTIVE DISPATCHING

a) Tabular display of the solution processes results enables aclear insight into different optimization techniques; b) Solutionof different power system optimization models gives the oppor-tunity to compare the individual results and to realize that the

LEVI AND NEDIC: APPLICATION OF THE OPTIMAL POWER FLOW MODEL IN POWER SYSTEM EDUCATION 579

TABLE IXDISPATCHING OF REACTIVE GENERATIONS

Fig. 4. Voltage profiles within reactive dispatching.

system states depend dominantly on control variables settingsand operation constraints; c) Analysis of different scenarios andparameter variation shows that a wide range of diverse regimesis highly possible; and d) The interaction with this softwarepackage enables students to understand relatively easily com-plex power system phenomena and the role of a dispatcher in thepower system planning and operation. All these conclusions be-came evident after the oral examinations during the past period.

APPENDIX

A. Computer Methods in Power EngineeringThis course is taught in the third year and its objective is to

apply different computer methods in order to solve smallproblems. Classroom instructions are covered by the textbook[18], which contains the following main areas: 1) Systems oflinear algebraic equations; 2) Eigenvectors and eigenvalues;3) Systems of nonlinear algebraic equations; 4) Systemsof linear differential equations; 5) Integration of ordinarydifferential equations; 6) Stability of linear and nonlinear sys-tems; 7) Linear and network programming; and 8) Non-linearunconstrained/constrained optimization. Lectures are supportedby worked examples of two-by-two problems, and studentsapply the software package MATLAB in laboratory. Allproblems are discussed in general form, since students are notfamiliar with power system problems.

B. Planning, Operation and Control of Power SystemsThis course is taught in the eighth and the ninth semesters

and its objective is to give the most important functions ofthe power systems real-life practice. Expansion planning istaught from the textbook [19], while the book [20] is used foroperation and control of power systems. The main areas are:1) Engineering economy; 2) Long- and medium-term loadforecasting; 3) Generation expansion planning and produc-tion costing; 4) Transmission and reactive power expansionplanning; 5) Economic dispatching; 6) Unit commitment;7) Hydrothermal coordination; 8) Fuel scheduling; 9) Stateestimation; 10) Steady-state security assessment; and 11) Pre-ventive and corrective rescheduling. Worked examples of smallsystems are given, and students apply specialized softwarepackages for more realistic power systems. Before taking theexam, each student has to write a (small) program, which solvesone of the numerous problems in this field.

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[14] L. Goel, C. M. Tao, and Z. B. Osman, Software modeling of electricsub-transmission systems for educational purposes, IEEE Trans. PowerSystems, vol. 11, no. 3, pp. 11391145, Aug. 1996.

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[17] V. A. Levi and D. S. Popovic, Integrated methodology for transmissionand reactive power planning, IEEE Trans. Power Systems, vol. 11, no.1, pp. 370376, Feb. 1996.

[18] V. A. Levi and D. D. Bekut, Application of Computer Methods inPower Engineering. Novi Sad, Yugoslavia: Stylos, 1997. ISBN86-80 249-22-X.

[19] V. A. Levi, Computer-Aided Power System Expansion Planning. NoviSad, Yugoslavia: Stylos, 1998. ISBN 86-7473-012-4.

[20] A. Wood and B. Wollenberg, Power Generation, Operation and Control,2nd ed. New York, NY, USA: J. Wiley & Sons, 1996.

[21] V. Levi and M. Calovic, A new decomposition based method for op-timal expansion planning of large transmission networks, IEEE Trans.Power Systems, vol. 6, no. 3, pp. 937943, Aug. 1991.

Viktor A. Levi (M89) was born in Zrenjanin, Yugoslavia, on June 21, 1958.He graduated in 1982 from the School of Engineering Sciences, University ofNovi Sad, and received the M.Sc. degree in 1986 and the Ph.D. degree in 1991from the Faculty of Electrical Engineering, University of Belgrade. In 1982, hejoined School of Engineering Sciences, Novi Sad, where he is employed as theAssociate Professor in Power Systems Department. He teaches power systemcourses, and in past several years he was active in developing software for powersystem optimization.

Dusko P. Nedic was born in Sabac, Yugoslavia, on October 27, 1968. He grad-uated in 1994 from the School of Engineering Sciences, University of NoviSad, and received the M.Sc. degree in 1999 from the Faculty of Electrical Engi-neering, University of Belgrade. In 1994, he joined the School of EngineeringSciences, Novi Sad, as a research assistant, and in 1997, he moved to the PowerDistribution Company Elektrovojvodina, Novi Sad. Currently, he is with theUMIST, Manchester, UK, and he is working toward his Ph.D. thesis.

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