Column-generation and interior point methods applied to the long-term electric power-planning problem PhD Student: Adela Pag`es Directed by: Prof. Narc´ıs Nabona Universitat Polit`ecnica de Catalunya 9th November 2006 In the preparation of this thesis Adela Pag`es was co-advised by Prof. J. Gondzio during two periods of study totalling 5 months at the School of Mathematics, Univ. of Edinburgh, Edinburgh, U.K., and by Prof. J. E. Mitchell during a one-month stay at Rensselaer Polytechnic Institute, Troy, NY, USA. Adela Pag`es received a Scholarship from the Programa Nacional de Formaci´on de Profesorado Universitario of Ministerio de Educaci´on y Ciencia (Spain), during the period January 2003 – December 2006. The work in this thesis was carried out in the Research Group GNOM (Group on Numerical Op- timization and Modeling) in the Department of Statistics and Operations Research, UPC, Barcelona, under Projects DPI2002-03330 (Coordinaci´on hidrot´ermica a corto y largo plazo de la generaci´on el´ectrica en un mercado competitivo) and DPI2005-09117-C02-01 (Planificaci´on de la generaci´on el´ectrica a corto y largo plazo en un mercado liberalizado con contratos bilaterales) of Ministerio de Educaci´on y Ciencia (Spain). Contents 1 Introduction 1 1.1 Motivation ...................................... .... 1 1.2 Objectives...................................... ..... 2 1.2.1 Objectivesinmodeling. ...... 2 1.2.2 Computationobjectives . ...... 3 1.3 Contributions................................... ...... 4 1.3.1 Mainmodelingcontributions . ....... 4 1.3.2 Main computational contributions . ......... 5 1.4 Contents........................................ .... 7 2 The long-term generation planning problem 11 2.1 The Spanish electricity market evolution . .............. 12 2.1.1 Electricitycycle.............................. ...... 12 2.1.2 Regulated and liberalized electricity systems in Spain .............. 12 2.1.3 TheMIBEL...................................... 13 2.1.4 The emission allowance trading system . .......... 13 2.2 Basicmodelingconcepts . ........ 14 2.2.1 Thelong-termhorizon . 14 2.2.2 Generationunits ............................... 14 2.2.3 Theload-durationcurve . ..... 15 2.2.4 Expected generation given a loading order . .......... 16 2.3 StateoftheArtinLTGP ............................. ..... 17 2.3.1 Other long-term generation planning approaches . ............ 18 2.3.2 Market-priceanalysis . ..... 20 2.4 The Bloom and Gallant formulation . ......... 21 2.4.1 Loadingorder .................................. 21 2.4.2 Loading order and load-matching constraints . ........... 22 2.5 The non-load-matching constraints . .......... 24 2.6 Thegenerationdurationcurve . ......... 25 2.6.1 Externalenergy................................ 26 2.7 The long-term market-price function . .......... 27 1 2 Contents 2.7.1 Market-price function with respect to load duration . ............. 27 2.7.2 Profit maximization versus other approaches . ........... 29 2.8 Change of the market-price function with hydro generation ............... 30 2.9 Emission allowance constraints and objective function .................. 32 2.10 The long-term power planning problem with maintenance . ............... 33 2.10.1 StateoftheArtinLTGMP . 33 2.10.2 Modelassumptions. ..... 34 2.10.3 Variables and maintenance constraints . ............ 34 2.11 Mathematicalmodels. ........ 36 2.11.1 The long-term generation planning problem . ........... 36 2.11.2 Model of joint long-term generation planning and unit maintenance . 37 2.12 Scopeofthisthesis.............................. ........ 38 3 Column-generation methods 41 3.1 Decomposition methods: Dantzig-Wolfe . ........... 41 3.2 Transformation of the LTGP problem . ......... 42 3.2.1 Definitionofavertex. ..... 42 3.2.2 Feasiblepoints ................................ 43 3.2.3 Problemtransformation . ...... 43 3.3 Murtagh and Saunders algorithm using column generation ................ 43 3.3.1 Variableclassification . ....... 44 3.3.2 Theprojectedgradient. ...... 45 3.3.3 Computation of the multipliers and modified costs . ............ 46 3.3.4 Findingadescentdirection . ....... 47 3.3.5 Computation of the step length . ....... 48 3.3.6 Changesinthevariablesets. ....... 49 3.3.7 Choosing a superbasic variable to enter the basis . ............. 49 3.3.8 Initialpoint.................................. 51 3.3.9 Management of the nonbasic set . ...... 52 3.4 ComputationalResults. ........ 53 3.4.1 Comparison of the two implementations of the Dantzig-Wolfe algorithm . 53 3.4.2 Performance of the Dantzig-Wolfe algorithm compared to the active set method 53 3.4.3 Convergence ................................... 55 4 Direct solution methods and heuristic procedure 57 4.1 Heuristictodeterminealoadingorder . ............ 58 4.1.1 Notation...................................... 58 4.1.2 Stepsoftheheuristic. ...... 58 4.2 Checkingthefeasibility . ......... 60 4.2.1 Characteristics of the active load-matching constraints of a feasible point . 61 Contents 3 4.2.2 Feasibility with a perfect ordering entails complete feasibility . 62 4.2.3 Potentially violated constraints with respect to a partialordering . 63 4.3 Implementation of the GP heuristic for the LTGP problem . .............. 64 4.3.1 Using a modeling language and a general linearly constrained optimization solver 64 4.3.2 Using a special purpose code and a special purpose linearly constrained opti- mizationsolver .................................... 65 4.4 Computationalresults . ........ 65 4.5 Application of the heuristic to a small single-interval problem .............. 66 4.6 Solution to the LTGP with hydro generation influence . .............. 69 4.6.1 Estimationofthenewcoefficients. ........ 70 5 Application of interior point methods 73 5.1 Infeasible primal-dual interior point method . ............... 73 5.1.1 Thecentralpath ................................ 74 5.1.2 Newton’smethod................................ 75 5.1.3 Newtonstepdirection . ..... 75 5.1.4 Interiorpointsteps ............................ ..... 76 5.1.5 Mehrotra predictor-corrector direction . ............ 77 5.1.6 Multiplecentralitycorrections . .......... 78 5.2 SolutionoftheNewtonsystem . ........ 80 5.2.1 The Newton system applied to the LTGP problem . ........ 80 5.2.2 Solution of an indefinite system . ........ 82 5.2.3 Controlofsystemstability. ........ 82 5.3 Computationalresults . ........ 83 5.3.1 Solution of the test cases with the heuristic . ............ 83 5.3.2 Multiple-centralitycorrections . .......... 84 6 Warmstarting for interior point methods 87 6.1 Warm-startFramework .............................. ..... 88 6.2 Restoring Primal and Dual Feasibility . ............ 89 6.3 InitialPoint.................................... ...... 89 6.4 Recoveringstep.................................. ...... 91 6.4.1 WeightedNewtonStep. 91 6.5 Outlineofthewarm-startprocedure . .......... 93 6.6 Computationalresults . ........ 94 6.6.1 Implementationissues . ...... 94 6.6.2 Heuristicstatistics . ....... 95 6.6.3 Warm-startresults .............................. 95 4 Contents 7 Long- short- term coordination 103 7.1 Theshort-termplanningmodel . ........ 104 7.1.1 Generation units in short-term planning . .......... 105 7.1.2 Coordination with the long-term planning results . ............. 108 7.1.3 Thesupply-bidfunction . 108 7.1.4 Formulation of the unit commitment . ........ 112 7.2 Short-termmodels ................................. 113 7.2.1 Linearizedunitcommitmentmodel . ........ 113 7.2.2 Nonlinearschedulingmodel . ....... 115 7.3 Thethree-stageprocedure . ........ 116 7.4 Results......................................... 118 8 Approach to the solution of the long-term planning problem with maintenances 119 8.1 DirectsolutiontotheLTGMPproblem . ......... 120 8.1.1 SolutionwiththeCPLEXB&B. 121 8.1.2 Heuristic for finding a maintenance schedule: MS1 heuristic ........... 121 8.2 Branchandboundalgorithm . ....... 122 8.2.1 Boundingthesolution . 123 8.2.2 Heuristic MS2 to find an all-binary solution . .......... 123 8.2.3 Notation...................................... 124 8.2.4 StepsoftheMS2heuristic. 124 8.2.5 Branchingstrategy. 126 8.2.6 Searchstrategy................................ 128 8.2.7 Variableselectionrules. ........ 128 8.2.8 DiagramoftheB&B............................... 130 8.3 Computationalresults . ........ 130 8.3.1 Computational results of the self-developed B&B . ............ 130 8.3.2 Results using the MS2 heuristic in the self-developed B&Bcode ........ 131 9 Conclusions and further research 135 9.1 Publications and presentations generated by this thesis.................. 138 9.2 Topicsforfurtherresearch. .......... 138 9.2.1 Extensions to the long-term generation planning model ............. 138 9.2.2 Extensions to the solution procedures proposed for solving the models . 139 A Test cases 141 A.1 Cases for testing the LTGP problem . ......... 141 A.2 Cases for testing the LTGMP problem . ......... 142 Contents 5 B Full model and solution of case ltp 14 145 B.1 Basicmodelingconcepts . ........ 145 B.1.1 Thelong-termhorizon . 145 B.1.2 Generationunits ............................... 145 B.1.3 Theload-durationcurve . 146 B.1.4 Load-matching constraints . ....... 148 B.1.5 Non-load-matching constraints . ....... 148 B.1.6 Market-price function with respect to load duration . ............. 151 B.2 Solution ........................................ 153 B.2.1 LTGPproblemsolution . 153 B.2.2 LTGPH problemsolution .............................. 154 B.2.3 LTGMPproblemsolution . 155 B.2.4 Commentsontheresults