Evaluation of Two Lagrangian Dual Optimization Algorithms for Large-Scale Unit Commitment Problems

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1 Journal of Electrcal Engneerng & Technology Vol. 7, No., pp. 7~22, Evaluaton of Two Lagrangan Dual Optmzaton Algorthms for Large-Scale Unt Commtment Problems Wen Fan*, Yuan Lao*, Jong-Beom Lee and Yong-Kab Km** Abstract Lagrangan relaxaton s the most wdely adopted method for solvng unt commtment (UC) problems. It conssts of two steps: dual optmzaton and prmal feasble soluton constructon. The dual optmzaton step s crucal n determnng the overall performance of the soluton. Ths paper ntends to evaluate two dual optmzaton methods one based on subgradent (SG) and the other based on the cuttng plane. Large-scale UC problems wth hundreds of thousands of varables and constrants have been generated for evaluaton purposes. It s found that the evaluated SG method yelds very promsng results. Keywords: Dual optmzaton, Lagrangan relaxaton, Resource schedulng, Unt commtment. Introducton Unt commtment (UC) s a specal optmzaton problem for determnng the startup and shutdown schedules of generaton unts n a power system consderng unt characterstcs, power networ operaton constrants, and fuel costs, among others [, 2]. Among the dfferent study objectves related to UC, one common am s to mnmze the total operatng cost over a study horzon whle satsfyng the demands. UC problems are normally formulated as mxed nteger programmng problems consstng of nteger varables (startup and shutdown of generators) and contnuous varables (dspatch level of turned-on generators). The soluton resoluton s usually one hour, and the study horzon could be one day, one wee, or one month. Solvng ths type of problem could be very dffcult due to the huge sze of the problem. Whle there are a great deal of efforts to tacle ths problem utlzng dverse technques such as genetc algorthm and benders decomposton, the most promsng one remans to be the Lagrangan relaxaton (LR) method, whch manly comprses two steps [3, 4]. The frst step s called dual optmzaton, whle the second step s called prmal feasble soluton generaton [5-7]. The dual optmzaton step s crucal n determnng the overall performance of an LR method n terms of speed and optmalty [8, 9]. Whle varous algorthms for dual optmzaton have been proposed n the past, there are only a few studes utlzng large-scale UC problems for Correspondng Author: Department of Electrcal Engneerng, Wonwang Unversty, Korea (power@wonwang.ac.r) * Department of Electrcal and Computer Engneerng, Unversty of Kentucy, Lexngton, KY, USA (ylao@engr.uy.edu) ** Department of Informaton and Communcaton Engneerng (ym@wonwang.ac.r) Receved: January 5, 20; Accepted: September 9, 20 evaluatng dual optmzaton algorthms. Meanwhle, benchmarng studes are helpful for the better understandng of the performance and characterstcs of the algorthms. Ths paper ntends to provde certan evaluaton results for large-scale UC problems. Two common approaches for updatng Lagrangan multplers, namely the subgradent (SG) based approach and the cuttng-plane (CP) based approach, have been evaluated [2, 3]. A lbrary of large-scale UC cases s generated for ths purpose. Due to the dverse varants of algorthms publshed n the lterature, ths paper does not ntend to cover the evaluaton of all the multpler update approaches. Instead, we focus on two selected dual optmzaton methods. The LR-based optmzaton method s frst revewed. The SG-based and CP-based multpler update approaches are then presented. The development of a lbrary of largescale UC cases s also descrbed, and the szes of the varous cases are reported. Evaluaton studes on the multpler update methods usng the generated cases are then presented, followed by the concluson. 2. LR and Multpler Update The followng generc formulaton for resource optmzaton problems, such as UC problems [2, 3], s consdered: Mnmze f ( x ) = f ( x ) () x Subject to: s( x) 0 g ( x ) G h( x ) = H

2 8 Evaluaton of Two Lagrangan Dual Optmzaton Algorthms for Large-Scale Unt Commtment Problems Where x = ( x ; ) ; =, 2,..., N ; the objectve functon f s addtvely separable nto N functons f ; x s the varable set assocated wth f ; s s the local constrant set nvolvng only varable set x ; and g and h are the components for couplng constrants and nvolve only the varable set x. In UC problems, N typcally represents the number of generaton unts, whch wll be referred to as the prmal problem (P). In practcal problems, local constrants can nclude the mnmum uptme, mnmum downtme, rampng constrant, and the avalable capacty lmts of each generatng unt. The equalty couplng constrants nclude energy balance requrements, whle the nequalty couplng constrants nclude the reserve requrements. The presence of such couplng constrants maes t dffcult to fnd the soluton of the problem P. The LR of P s derved by dualzng the couplng constrants, resultng n the relaxed problem (RP), as shown below [2]: Mnmze Lxλ (,, λ ) (2) x Subject to s ( x ) 0, =, 2,... N Where Lxλ (,, λ ) = Lxλ (, ) = f( x ) + e e g T λ T λ ( H h( x )) + ( G g ( x )) e g λe λ = λ s the Lagrangan multpler vector; λ e and g λ g are the multpler vectors correspondng to the equalty and nequalty constrants, respectvely; λg 0 ; and λ e s unrestrcted n sgn. All vectors n ths study are column vectors. Once the couplng constrants are removed, the resultng problem, RP, can be decomposed nto N ndependent problems, each of whch corresponds to a sngle generaton unt. A sngle problem s usually of much smaller sze and thus s much easer to solve. LR soluton method s based on the theorems of dualty, whch establsh certan propertes governng the relatonshp between solutons to the Lagrangan dual problem, DP, and solutons to P. The Lagrangan dual problem, DP, s defned as follows: Maxmze λ L( λ ) (3) Subject to λg 0 Where L( λ ) = Mnmze Lxλ (, ) x The dual optmzaton ams to fnd a set of multplers that maxmze the Lagrangan dual functon. There are two major varatons of LR method dependng on how the Lagrangan multplers are updated n the teratve process: the SG-based method and CP-based method. g The SG for the Lagrangan dual s calculated as follow [2, 3]: G g( x) ξ = H h( x) The soluton of the dual problem usually taes an teratve procedure, the Lagrangan multplers are updated teratvely dependng on the calculated SG. The method of updatng the multplers s pvotal n determnng the speed and qualty of the soluton. The two commonly used multpler update methods are presented n the followng sectons; both are evaluated by applyng the developed case lbrary. 2. SG-based multpler update method Assumng that for teraton the SG vector for the ( ) ( ) Lagrangan dual s ξ and the multpler vector s λ, then the multpler vector s updated as follows [2]: λ = λ + α ξ + ( + ) ( ) ( ) Approprate values for α, the scale factors, are selected accordng to [2]. 2.2 CP-based multpler update method The updated multpler s obtaned by solvng the followng lnear programmng problem [3]: Maxmze z (6) z, λ () () () Subject to z φ + [ ξ ] T ( λ λ ), =, L, λ 0 g Where φ s the value of the Lagrangan functon, and T s the vector transpose operator. 3. Generaton of a UC Case Lbrary Ths secton descrbes the generated UC case lbrary based on typcal, f not comprehensve, models [-5]. The UC problem s formulated as a mxed-nteger lnear programmng (MILP) problem wth a tme resoluton of one hour, whch mnmzes the total producton cost subject to specfc constrants. The nputs of problem formulaton nclude the startup/shutdown cost, the energy cost, the maxmum/mnmum power output, the mnmum uptme/downtme, and the maxmum ramp up/down (4) (5)

3 Wen Fan, Yuan Lao, Jong-Beom Lee and Yong-Kab Km 9 dspatch ncrement of each unt, as well as the demand and reserve requrement n each hour. There are four sets of varables for each hour and for each unt: dspatch, commtment status, startup acton, and shutdown acton. The constrants consdered nclude energy balance and reserve, unt capacty, unt ramp up/ramp down, and unt mnmum uptme/downtme constrants. The energy balance constrants and the reserve requrement constrants are the two types of global constrants, or complcatng constrants. The objectve functon s to mnmze the total producton cost, ncludng the energy, startup, and shutdown costs. The mathematcal equatons for the constrants and objectve functon are presented as follows based on references [, 0, ]. 3. Constrants The constrants of the formulated optmzaton problem are presented as follows, referrng to [, 0], especally [0]. Unt capacty constrants: Pmn u p 0, SK, SI (7a) p Pmax u 0, SK, SI (7b) Where, P max : Maxmum capacty of unt P mn : Mnmum capacty of unt p : Dspatch of unt n hour, whch s a contnuous varable u : Commtment status of unt n hour, whch s a bnary varable S I : Set {,..., I } S K : Set {,..., K } I : The number of unts K : The number of hours Startup and shutdown relatonshp: s d u + u( ) = 0, SK, SI (8) s : Startup acton of unt n hour, whch s a bnary varable d : Shutdown acton of unt n hour, whch s a bnary varable Intal unt status and output: u p = U, SI (9a) = P, SI (9b) U 0 : The unt status precedng hour, ( for onlne and 0 for off lne) P : The ntal dspatch of unt precedng hour 0 Energy balance constrants: I ( p ) = D, SK (0) = D s the demand requrement n hour. Ramp up constrants: L unt. p( + ) p Lu, = 0,..., K, SI () u s the maxmum ramp up dspatch ncrement of Ramp down constrants: p p( + ) Ld, = 0,..., K, SI (2) Ld s the maxmum ramp down dspatch decrement of unt. Reserve requrements: I ( Pmax u) D + R, SK = (3) R s the reserve requrement n hour. Mnmum uptme constrants: K Mut + um Muts, =,..., K + Mut, SI (4a) um ( K + ) s, = K + 2 M ut,..., K, SI M ut s the mnmum uptme of unt. Mnmum downtme constrants: Mdt + dt m dt (4b) M u M d, =,..., K + Mdt, SI K ( K + ) um ( K + ) d, (5a) = K + 2 Mdt,..., K, SI (5b) M dt s the mnmum downtme of unt.

4 20 Evaluaton of Two Lagrangan Dual Optmzaton Algorthms for Large-Scale Unt Commtment Problems The abovementoned constrants are not meant to be allnclusve, and other constrants such as transmsson, contngency, emsson, and fuel constrants are not consdered n the present study [2, 3]. 3.2 Objectve functon The objectve functon s to mnmze the total producton cost of all unts durng the entre optmzaton horzon, ncludng the energy, startup, and shutdown costs. Mnmze K I ( Cp + sw + dv ) (6) = = W : Startup cost of unt V : Shutdown cost of unt C : Energy rate of unt Based on the formulaton descrbed above, a case lbrary of large-scale UC problem s generated, whch s used for testng the dual optmzaton problem. 4. Evaluaton Studes for Dual Optmzaton Ths secton presents the evaluaton studes for dual optmzaton. Varous cases are generated, wth dfferent number of unts and of hours n the optmzaton horzon. The scale and complexty of the cases are comparable to or exceed those of real-world power system UC problems. The szes of the generated cases are shown n Table, the number of contnuous varables, bnary varables, constrants, non-zero coeffcents n the constrants, couplng equalty constrants (CECs), and couplng nequalty constrants (CIECs) are shown. Some of the cases are the same as those reported n [], and the software development wor utlzed to generate the case lbrary s detaled n []. The name of each test system s composed of the number of generaton unts and the number of hours n the optmzaton horzon. For example, UK_500_68 means that the system conssts of 500 unts and the optmzaton horzon s 68 hours. The number of CECs and CIECs are the same and equal to the number of hours n the optmzaton horzon. The developed cases were appled to evaluate the two multpler update methods. The dual optmzaton procedure outlned n secton 2 s followed n the present study. The dual solutons obtaned by usng the SG and CP methods are shown n Table 2. The maxmum number of teraton s set at 300 n the smulaton studes. In the tables, the column Iteraton ndcates the least number of teratons requred to acheve a dual soluton gap no more than %; f such soluton gap cannot be acheved, the teraton number wll be equal to 300. The actual dual value and soluton gap acheved are shown n the column Dual and Gap, respectvely. The dual soluton gap s calculated as the dfference between the prmal feasble optmal value and the dual value, dvded by the prmal feasble optmal value. Columns 2 4 show the results for the SG method, and columns 5 7 show the results for the CP method. Table 2. Dual soluton based on the SG and CP approaches Table. Szes of the UC cases under study The evaluated SG method sgnfcantly outperformed the evaluated CP method. The SG method can usually acheve a hgh-qualty dual soluton wthn 20 teratons, whle the CP method ether requres hundreds of teratons for reachng the desred soluton gap or fals to do so wthn the specfed maxmum teraton lmt. To better understand the convergence behavor of the evaluated algorthms, the convergence locus for the SG method s depcted n Fg. for case UK200_68. Its zoom-n vew s shown n Fg. 2. The dual optmzaton convergence locus based on the CP method for case UK200_68 s plotted n Fg. 3. The dual optmzaton convergence loc based on the CP method for case UK00_24 are llustrated n Fgs. 4 and 5.

5 Wen Fan, Yuan Lao, Jong-Beom Lee and Yong-Kab Km 2 4 x 08 2 x Iteraton Fg.. Dual optmzaton convergence locus based on SG for UK200_68 overall vew Iteraton Fg. 4. Dual optmzaton convergence locus based on CP for case UK00_24 overall vew x Iteraton Fg. 2. Dual optmzaton convergence locus based on SG for UK200_68 zoom n vew 0.5 x x Iteraton Fg. 5. Dual optmzaton convergence locus based on CP for case UK00_24 zoom-n vew The Fgs. show that the tested SG method can qucly converge to the desred optmal dual soluton wth few or no oscllatons. Meanwhle, the CP method suffers consderable oscllatons and thus taes a sgnfcant number of teratons to get close to the dual optmal soluton Iteraton Fg. 3. Dual optmzaton convergence locus based on CP for UK200_68 5. Conclusons A lbrary of UC problems, ncludng large-scale problems wth hundreds of thousands of varables and constrants, was created and appled for the evaluaton of the performance of an SG-based and a CP-based multpler update methods for Lagrangan dual optmzaton. The evaluated SG algorthm can acheve hgh-qualty dual solutons, usually wthn 20 teratons for large-scale UC problems wthout any sgnfcant oscllatons. Meanwhle, the evaluated CP method suffers sgnfcant oscllatons and thus s very slow n reachng the optmal dual soluton. The evaluaton results can provde researchers wth useful

6 22 Evaluaton of Two Lagrangan Dual Optmzaton Algorthms for Large-Scale Unt Commtment Problems benchmarng nformaton for the study of LR-based algorthms to solve large-scale resource schedulng problems such as UC problems. References [] Benjamn F. Hobbs, et al, The Next Generaton of Electrc Power Unt Commtment Models, Kluwer s Internatonal Seres, 200. [2] X Feng and Yuan Lao, A new Lagrangan multpler update approach for Lagrangan relaxaton based unt commtment, Electrc Power Components & Systems, 34 (2006), pp [3] N. J. Redondo, A. J. Conejo, Short-term hydrothermal coordnaton by Lagrangan relaxaton: soluton of the dual problem, IEEE Transactons on Power Systems, 4 (999), pp [4] C. P. Cheng, C. W. Lu, C. C. Lu, Unt commtment by Lagrangan relaxaton and genetc algorthms, IEEE Transactons on Power Systems, 5(2000), pp [5] R. T. Rocafellar, Lagrange multplers and optmalty, SIAM Rev., 35 (993), pp [6] Mohtar S. Bazaraa, Hanf D. Sheral, C. M. Shetty, Nonlnear Programmng, Theory and Algorthms, John Wley & Sons, Inc [7] C. A. Kasavels, M. C. Caramans, Effcent Lagrangan relaxaton algorthm for ndustry sze jobshop schedulng problems, IIE Transactons, 30 (998), pp [8] W. Ongsaul, N. Petcharas, Unt commtment by enhanced adaptve Lagrangan relaxaton, IEEE Transactons on Power Systems, 9 (2004), pp [9] A. Borghett, A. Frangon, F. Lacalandra, C.A. Nucc, Lagrangan Heurstcs Based on Dsaggregated Bundle Methods for Hydrothermal Unt Commtment, IEEE Transactons on Power Systems, 8 (2003), pp [0] H.P. Wllams, Model Buldng n Mathematcal Programmng, John Wley & Sons Ltd.,West Sussex, England, 999. [] Yuan Lao, Development of a generaton resource schedulng case lbrary, 38 th Southeastern Symposum on System Theory, Tennessee Technologcal Unversty, Cooevlle, TN, USA, March 5-7, [2] Yuan Lao, X Feng and J Pan, Impact of emsson complance program on compettve power maret, nvted paper, The 2 nd Internatonal Conference on Electrc Utlty Deregulaton, Restructurng and Power Technologes, Hong Kong, Aprl 5-8, [3] Yuan Lao, X Feng and J Pan, Analyss of nteracton between ancllary servce marets and energy maret usng power maret smulator, nvted paper, The 2 nd Internatonal Conference on Electrc Utlty Deregulaton, Restructurng and Power Technologes, Hong Kong, Aprl 5-8, [4] K. Chandram, N. Subrahmanyam and M. Sydulu, Improved pre-prepared power demand table and muller s method to solve the proft based unt commtment problem, Journal of Electrcal Engneerng & Technology, 4 (2009), pp [5] Yun-Won Jeong, Woo-Nam Lee, Hyun-Houng Km, Jong-Bae Par and Joong-Rn Shn, Thermal unt commtment usng bnary dfferental evoluton, Journal of Electrcal Engneerng & Technology, 4 (2009), pp Wen Fan s currently studyng at the Department of Electrcal and Computer Engneerng at the Unversty of Kentucy. Hs research nterests nclude power system smulaton, analyss and optmzaton, and smart grd. Yuan Lao s currently an assocate professor at the Department of Electrcal and Computer Engneerng at the Unversty of Kentucy, Lexngton, KY, USA. Hs research nterests nclude protecton, power qualty analyss, large-scale resource scheduleng optmzaton, and networ management system/supervsory control and data acquston system desgn. Jong-Beom Lee receved hs B.S., M.S., and Ph.D degrees n Electrcal Engneerng from Hanyang Unversty, Korea, n 98, 983, and 986, respectvely. He wored at the Korea Electrotechnology Research Insttute from 987 to 990. He was a vstng scholar at the Techncal Unversty of Berln, Germany, n 993; the Cty Unversty, UK, n 995; Texas A&M Unversty n 997; and Swss Federal Insttute of Technology (ETH) n He s currently a professor n the Department of Electrcal Engneerng, Wonwang Unversty, Korea. Yong-Kab Km receved hs B.S. degree n Electroncs Engneerng from Ajou Unversty and hs M.S.E. degree n Electrcal Engneerng from the Unversty of Alabama n Huntsvlle. He receved hs Ph.D. degree n Electrcal and Computer Engneerng n North Carolna State Unversty. He s currently a professor at School of Electrcal Informaton Communcaton Engneerng, Wonwang Unversty, Korea. Hs research nterests nclude remote sensng for vsble communcaton, optcal fber sensng, and power lne communcaton.