Improvements to Benders' decomposition: systematic classification and performance comparison in a Transmission Expansion Planning problem

Transcription

1 Improvements to Benders' decomposition: systematic classification and performance comparison in a Transmission Expansion Planning problem Sara Lumbreras & Andrés Ramos July 2013

2 Agenda Motivation improvement improvement Transmission Expansion Planning Results Conclusions 2

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4 Motivation It was first proposed in 1962 In its 50 years of history it has been applied to very diverse fields: Scheduling Routing (e.g. traveling salesman) & vehicle assignment Computer network design Capacity allocation in telecommunications networks Manufacturing system design Portfolio optimization Specially, in Power systems Generation, Transmission & Distribution Expansion Planning [Pereira et al, A decomposition approach to automated generation/ transmission expansion planning, 1985] Hydrothermal co-ordination Unit Commitment Many improvements to the basic strategy have been proposed in an uncoordinated way, so they are not easily accessible or related to the cases where they can be useful Benders, Partitioning procedures for solving mixed-variables programming problems,

5 Motivation (II) This work aims at filling this gap: Classifies the improvements to Benders decomposition that appear in the literature Proposes new improvements to add to the ones in the literature Links these methodologies to the cases where they can be useful Compares their performance in a particularly relevant case study based on Transmission Expansion Planning Given its characteristics this problem has been extensively solved with Benders decomposition 5

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7 Benders decomposition is applied to a two-stage stochastic (linear) problem First-stage decisions (also known as investment decisions) are taken before an uncertain event occurs Second-stage decisions (also known as operation decisions or recourse decisions) are adjusted after the uncertainty has been revealed Second-stage scenarios can be solved independently Both stages are coupled through the tender constraints First-stage decisions complicate the resolution of the problem 1st stage 2nd stage Stochastic scenarios Constraints of the 1st stage Constraints that link both stages (tender constraints) Constraints of the 2nd stage 7

8 (cont ed) Benders decomposition divides the two-stage stochastic linear programming problem in two parts (master problem & subproblems) that are solved iteratively until convergence The process builds an increasingly accurate piecewise linear approximation of the recourse problem (Benders cuts) Complete problem lower bound iterations current iteration (solved independently for each scenario) upper bound 8

9 Feasibility cuts (cont ed) When a certain 1 st -stage solution is infeasible in the subproblem, a feasibility cut is generated with the aim of eliminating solutions that do not abide the constraints Usually, the sum of infeasibilities is minimized infeasibility The resulting feasibility cut has the form: For uniform notation l δ = l δ = 1 0 optimality cut feasibility cut 9

10 Scope of Benders decomposition This method is likely to yield time savings in the following cases: First-stage variables complicate the resolution of the problem Decoupling the resolution of 2nd stage scenarios is specially important Computational time taken by this method is related to the number of complicating variables Therefore, it is likely to yield time savings if the number of complicating variables is small The master problem and the subproblem have different natures (and hence it would be convenient to solve them with different methods, e.g. NLP / MIP vs LP) Both conditions apply to 10

11 Classification of improvement Improvement respond to the need to reduce either: solution time (solution time can be long because of size, integral variables or a large number of cuts), or solution time (solution time can be long because of a large number of scenarios or 2 nd stage conditions) Guides for their suitability will be provided However, the practical benefit achieved will have to be assessed on a case-by-case basis Most of them involve trade-offs that must be assessed individually A case study based on Transmission Expansion Planning will be presented 11

12 improvements

13 Improvements that modify the solution technique (I) relaxations Binary variables complicate the resolution, so solving the LP relaxation is much quicker It yields valid Benders cuts Linear-first approaches solve first the relaxed problem Recently, [Lumbreras & Ramos, Optimal design of an offshore wind farm applying decomposition strategies, 2012] proposed a progressive discretization of variables to improve convergence (semi-relaxed cuts) Can be useful if the linearized problem can be solved much quicker and the integrality gap is small upper bound variable discretization convergence lower bound 13

14 Improvements that modify the solution technique (II) Sub-optimal master problem solutions Early terminations of the master problem might improve convergence gap Using any feasible solution [Fortz& Poss, An improved Benders decomposition applied to a multi-layer network design problem, 2008] Rounding linearized solutions time [Costa et al, Accelerating Benders decomposition with heuristic master solutions, 2012] (feasibility must be checked) Alternative strategies to find master proposals Non-classical optimization (e.g., metaheuristics) can be applied to find near-optimal solutions in affordable times [Poojari & Beasley, Improving Benders decomposition using a Genetic Algorithm, 2009] Can be useful if the suboptimal solution is not far from the optimal one and is obtained in a reduced time 14

15 Box-step method ( of additional constraints): Ideally they are conditions that should be met at the optimum, so they only eliminate not useful zones of the feasible region From expert opinion Data mining If the additional constraint is active at the optimal solution the constraint should be relaxed by an specified amount (the step) and the problem resolved. Use of a more suitable solution technique E.g. Constraint programming and logic-based methods [Benoist et al, Constraint programming contribution to Benders decomposition, 2002] In many cases, complex problems include many logical constraints that make use of auxiliary binary variables This greatly complicates the problem There are that have been specially developed for these problems, where the logical constraints are included explicitly (e.g. LOGMIP) Improvements that modify the solution technique (III) Can be useful if these constraints are available Can be useful if there is a technique that is better suited to the specific problem 15

16 Extracting non-dominated cuts / Pareto-optimal cuts [Sherali & Fraticelli, On generating maximal nondominated Benders cuts, 2011] A cut or constraint dominates another if any evaluation of first stage decisions is larger than or equal to the previous one Generating covering cuts [Saharidis et al, Accelerating benders method using covering cut bundle generation, 2010] Generating cuts so that they carry the maximum amount of information possible They include the maximum possible number of 1 st -stage variables Removing inactive cuts Improvements that modify Benders cuts (I) Can be useful if there are too many cuts and most are dominated Can be useful if there are too many cuts Can be useful if there are too many cuts [Marin & Salmeron, Electricity capacity expansion under uncertain demand: decomposition approaches, 1998] Or dynamically defining the master problem so that only the cuts that are likely to be active constraints are taken into account 16

17 Minimal Infeasible Subsystems (MIS) can be used to modify the way feasibility cuts are calculated [Saharidis & Ierapitrou, Improving benders decomposition using maximum feasible subsistem (MFS) cut generation, 2009] Instead of minimizing the sum of infeasibilities the problem minimizes the number of equations that are infeasible This enables faster convergence in some cases Conversely, if most of the solutions are infeasible, it is possible to keep a maximum feasible set to derive optimality cuts to better guide the search Modifications to Benders cuts (II) Can be useful if most of the cuts are feasibility cuts 17

18 improvements

19 Scenario structure design (I) Subtree partition [Birge & Loveaux, A multicut algorithm for stochastic linear programs, 1988] The second stage scenarios can be arranged differently In general, the most efficient arrangement cannot be known beforehand (tradeoff between the accuracy of the cuts and solution time for the master problem) ws: wind scenario ss: system state 19

20 Scenario structure design (II) Scenario aggregation: the second stage corresponding to the scenarios with the highest impact on the final design are added to the master problem [Cerisola & Ramos, Node aggregation in stochastic nested benders decomposition applied to hydrothermal coordination,2000] The master problem proposes solutions that are closer to the optimal Convergence speed can be increased Can be useful if one of the scenarios has a much higher impact on the final solution 20

21 Solution technique modifications Bunching: if the 2 nd stage scenarios are similar we can solve only one scenario and calculate the others using the calculated sensitivities [Birge & Loveaux, Introduction to Stochastic Programming, 1988] Application of specific algorithms [Marin & Salmeron, Electricity Capacity Expansion Models Under Uncertain Demand. Decomposition Approaches, 1988] Can be useful if there are many similar scenarios being calculated Can be useful if there is an specific more efficient technique available Or even a series of increasingly accurate versions of the subproblem (as long as they have increasing values of the o.f.) so that time is not wasted in the first few iterations [Romero & Monticelli, A hierarchical decomposition approach for transmission network expansion planning, 1994] Sub-optimal subproblem solutions (Zakeri s cuts) [Zakeri et al, Inexact Cuts in Benders Decomposition, 1999] Any infeasible solution in the subproblem will give a valid cut (can use IPM) Can be useful if solving the subproblem until optimality, even for one scenario, is computationally very expensive 21

22 Transmission Expansion Planning

23 The power system in a nutshell Supply is composed by generators of different technologies with different operation costs Demand has to be served instantaneously (there is no possibility for storage) The transmission network enables (and constrains) the physical transactions (and the operation outcome) Flows follow Kirchhoff s laws, so bilateral transactions do not have a physical meaning 23

24 Transmission Expansion Planning The Transmission Expansion Planning () problem consists in deciding the optimal transmission investments (lines or otherwise) that should be added to the existing transmission network in order to minimize the total investment and operation costs (generation costs and reliability). 24

25 Simplified formulation The problem can be formulated as MIP in a centralized, costminimization framework Minimize the sum of investment and operation costs mininvc OpC + InvC ic x ijc ijc ijc CL Subject to Kirchhoff s laws jic OpC f f + g = d pns : σ f ijc jic ijc i i i i ijc θ θ i j = ijc EL X ijc = = ( gcg + pnspnlt) i i Respecting generation and transmission limits i i ( ) i j + M 1 x : ρ ijc ijc ijc ijc f X ijc ( ) i j M 1 x : ρ ijc ijc ijc ijc f θ θ X ijc CL θ θ ijc f f x ijc ( CL EL) : π + ijc ijc ijc ijc f f x : π ijc ijc ijc ijc g g g,0 pns d i i i i i 25

26 Benders decomposition applied to The master problem takes the form: min l c x And the subproblem: Z jic + P Θ ω ω ijc ijc 2 ijc CL ω Ω ω ωl lt lt Z + ω ω f ( ) 2 2 π π ijc ijc ijc ijc + ωlt ωlt l M ρ ρ x x ijc ijc ijc ijc ijc Θ + + = 1,..., n ωn ω ω 2 ω ω i i i g, pns i i ωi ( )( ) = min ( gcg + pns pnlt) f f + g = d pns f ω ijc : σ ω ω ω ω ω ω jic ijc i i i i ijc ω ω θ θ i j = ijc EL X ω ijc ω θ θ ω i j + ω M ( 1 x ) : ρ ijc ijc ijc ijc f X ijc ω ω θ θ ω i j ω M ( 1 x ) : ρ ijc ijc ijc ijc f X ijc CL ijc f f x ijc ( CL EL) 26 : π ω ω + ω ijc ijc ijc ijc ω ω ω f f x : π ijc ijc ijc ijc g g g,0 pns d i i i i i

27 Selection of the improvement The ratio of time spent solving the master problem and the subproblem is the most important factor in the decision Master Decomposition Case Eq Var D var Time (s) Eq Var Time (s) It. Time Time per it. (s) Garver N N scenarios 87N In all cases, time spent in the subproblem is less than 12% modification are more appropriate. 27

28 Selection of the improvement (cont ed) Are relaxations potentially interesting? The largest case study, with 152 discrete variables in the master problem, is solved in 2.9s for MIP and 0.1s in a relaxed version (85% savings) relaxations should be explored - Linear first - Semi-relaxed cuts Are suboptimal master solutions interesting? We examine the tolerance-performance curve Solution time (s) Iterations Values around 2.5% seem interesting 0 0 0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% Relative optimality tolerance 28

29 Selection of the improvement (cont ed) Other master problem modification Box-step could be interesting in general in problems where an additional constraint can be easily imposed. In this case we can limit investment. Values of 85%, 100% and 115% of optimal investment were used ij C cx MaxInv ij ij Alternative solution (other than MIP) are not available for this problem Modifications to Benders cuts are not interesting in this case The number of generated cuts is not excessively large, so cover cuts or cut removal are not indicated The subproblem is feasible in all cases, so MIS is not applicable 29

30 Results

31 Results Only the largest case study is reported for the sake of clarity Semi-relaxed cuts and suboptimal master solutions offer consistent savings that can be above 50% of solution time Linear-first approaches and box-step methods are not able to provide consistent results 87N Time (s) % Savings Benders' decomposition Linear first % Semi relaxed cuts % Suboptimal master εmaster = 5% % εstep = 0.025% Suboptimal master εmaster = 2.5% % εstep = 0% Suboptimal master Highest tolerance % Boxstep Boxstep Boxstep Initial maximum investment = 85% optimal Initial maximum investment = 100% optimal Initial maximum investment = 115% optimal % % % 31

32 Conclusions

33 Conclusions (I) Benders decomposition is a key tool in stochastic optimization which divides the problem in: A master problem that optimizes the first stage and a piecewise linear approximation of the second stage costs A subproblem which optimizes the second stage and creates the piecewise approximation by means of primal and marginal information The method can bring substantial benefits when: First-stage variables complicate the resolution of the problem and subproblems have a different nature 33

34 Conclusions (II) A slow master problem can be accelerated: Modifying the solution method Relaxations Sub-optimal solutions Box-step Use of a more suitable technique Modifying the calculation of cuts: Nondominated cuts / Pareto optimal cuts Covering cuts Removing inactive cuts Minimal Infeasible Subsystems 34

35 Conclusions (III) A slow subproblem can be accelerated: Selecting the most suitable structure for the scenario tree Scenario aggregation Bunching Application of specific algorithms Sub-optimal solutions A case study based on Transmission Expansion Planning has demonstrated How to select the most promising improvement for a particular problem Results show that for this case the semi-relaxed cuts (proposed by the authors) and sub-optimal master resolution were able to consistently offer savings above 50% 35

36 Thank you

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