Accelerating the Convergence of Stochastic Unit Commitment Problems by Using Tight and Compact MIP Formulations

Transcription

1 Accelerating the Convergence of Stochastic Unit Commitment Problems by Using Tight and Compact MIP Formulations Germán Morales-España, and Andrés Ramos Delft University of Technology, Delft, The Netherlands Universidad Pontificia Comillas, Madrid, Spain PES General Meeting Denver-USA, July / 49

2 Outline 1 Introduction 2 Good and Ideal MIP formulations 3 Tight & Compact (TC) UC Formulations 4 Case Studies Deterministic Selft-UC Stochastic UCs: Different Solvers Stochastic UCs: IEEE-118 Bus System 5 Conclusions 2 / 49

3 Introduction UC and MIP Unit Commitment (UC): essential tool for day(week)-ahead planning in the electricity sector Decide on units physical operation (e.g., on-off) at minimum cost UC is an integer computationally demanding problem 3 / 49

4 Introduction UC and MIP Unit Commitment (UC): essential tool for day(week)-ahead planning in the electricity sector Decide on units physical operation (e.g., on-off) at minimum cost UC is an integer computationally demanding problem Significant breakthroughs in Mixed-Integer Programming (MIP) Solving MIP 100 million times faster than 20 years ago 1 The time to solve UC is still a critical limitation 1 T. Koch, T. Achterberg, E. Andersen, O. Bastert, T. Berthold, R. E. Bixby, E. Danna, G. Gamrath, A. M. Gleixner, S. Heinz, A. Lodi, H. Mittelmann, T. Ralphs, D. Salvagnin, D. E. Steffy, and K. Wolter, MIPLIB 2010, Mathematical Programming Computation, vol. 3, no. 2, pp , Jun / 49

5 Introduction UC and MIP Unit Commitment (UC): essential tool for day(week)-ahead planning in the electricity sector Decide on units physical operation (e.g., on-off) at minimum cost UC is an integer computationally demanding problem Significant breakthroughs in Mixed-Integer Programming (MIP) Solving MIP 100 million times faster than 20 years ago 1 The time to solve UC is still a critical limitation How to reduce solving times? Computer power (e.g., clusters) Solving algorithms (e.g., solvers, decomposition techniques) 1 T. Koch, T. Achterberg, E. Andersen, O. Bastert, T. Berthold, R. E. Bixby, E. Danna, G. Gamrath, A. M. Gleixner, S. Heinz, A. Lodi, H. Mittelmann, T. Ralphs, D. Salvagnin, D. E. Steffy, and K. Wolter, MIPLIB 2010, Mathematical Programming Computation, vol. 3, no. 2, pp , Jun / 49

6 Introduction UC and MIP Unit Commitment (UC): essential tool for day(week)-ahead planning in the electricity sector Decide on units physical operation (e.g., on-off) at minimum cost UC is an integer computationally demanding problem Significant breakthroughs in Mixed-Integer Programming (MIP) Solving MIP 100 million times faster than 20 years ago 1 The time to solve UC is still a critical limitation How to reduce solving times? Computer power (e.g., clusters) Solving algorithms (e.g., solvers, decomposition techniques) Improving the MIP-Based UC formulation solving times 1 T. Koch, T. Achterberg, E. Andersen, O. Bastert, T. Berthold, R. E. Bixby, E. Danna, G. Gamrath, A. M. Gleixner, S. Heinz, A. Lodi, H. Mittelmann, T. Ralphs, D. Salvagnin, D. E. Steffy, and K. Wolter, MIPLIB 2010, Mathematical Programming Computation, vol. 3, no. 2, pp , Jun / 49

7 Good & Ideal MIP Outline 1 Introduction 2 Good and Ideal MIP formulations 3 Tight & Compact (TC) UC Formulations 4 Case Studies Deterministic Selft-UC Stochastic UCs: Different Solvers Stochastic UCs: IEEE-118 Bus System 5 Conclusions 4 / 49

8 Good & Ideal MIP An MIP Has Infinite LP Formulations 5 / 49

9 Good & Ideal MIP An MIP Has Infinite LP Formulations LP1 and LP2 represent the same MIP problem 5 / 49

10 Good & Ideal MIP An MIP Has Infinite LP Formulations LP1, LP2 and CH represent the same MIP problem which one to choose? 5 / 49

11 Good & Ideal MIP Solving MIP Through The Powerful LP Shaping the linear feasible region to arrive from vertex Z LP to Z MIP To prove optimality Z MIP must become a vertex by: Branch and bound (divide and conquer) 6 / 49

12 Good & Ideal MIP Solving MIP Through The Powerful LP Shaping the linear feasible region to arrive from vertex Z LP to Z MIP To prove optimality Z MIP must become a vertex by: Branch and bound (divide and conquer) 6 / 49

13 Good & Ideal MIP Solving MIP Through The Powerful LP Shaping the linear feasible region to arrive from vertex Z LP to Z MIP To prove optimality Z MIP must become a vertex by: Branch and bound (divide and conquer) 6 / 49

14 Good & Ideal MIP Solving MIP Through The Powerful LP Shaping the linear feasible region to arrive from vertex Z LP to Z MIP To prove optimality Z MIP must become a vertex by: Branch and bound (divide and conquer) 6 / 49

15 Good & Ideal MIP Solving MIP Through The Powerful LP Shaping the linear feasible region to arrive from vertex Z LP to Z MIP To prove optimality Z MIP must become a vertex by: Branch and bound (divide and conquer) and/or by adding cuts 6 / 49

16 Good & Ideal MIP Convex Hull: The Tightest Formulation Convex Hull (CH) Smallest convex feasible region containing all feasible integer points 2 2 L. Wolsey, Integer Programming. Wiley-Interscience, H. P. Williams, Model Building in Mathematical Programming, 5th Edition. John Wiley & Sons Inc, Feb / 49

17 Good & Ideal MIP Convex Hull: The Tightest Formulation Convex Hull (CH) Smallest convex feasible region containing all feasible integer points 2 The convex hull problem solves an MIP as an LP Each vertex satisfies the integrality constraints So an LP optimum is also an MIP optimum Unfortunately, 2 L. Wolsey, Integer Programming. Wiley-Interscience, H. P. Williams, Model Building in Mathematical Programming, 5th Edition. John Wiley & Sons Inc, Feb / 49

18 Good & Ideal MIP Convex Hull: The Tightest Formulation Convex Hull (CH) Smallest convex feasible region containing all feasible integer points 2 The convex hull problem solves an MIP as an LP Each vertex satisfies the integrality constraints So an LP optimum is also an MIP optimum Unfortunately, the convex hull is typically too difficult to obtain 2,3 To solve an MIP is usually easier than trying to find its convex hull 2 L. Wolsey, Integer Programming. Wiley-Interscience, H. P. Williams, Model Building in Mathematical Programming, 5th Edition. John Wiley & Sons Inc, Feb / 49

19 Good & Ideal MIP Choosing The Best Formulation Measuring The Tightness Integrality Gap (IGap) Relative distance between MIP and LP optima IGap LP1 = Z MIP Z LP1 Z MIP > IGap LP2 = Z MIP Z LP2 Z MIP As an MIP problem: LP2 is expected to be solved faster than LP1 8 / 49

20 Good & Ideal MIP Choosing The Best Formulation Measuring The Tightness Integrality Gap (IGap) Relative distance between MIP and LP optima IGap LP1 = Z MIP Z LP1 Z MIP > IGap LP2 = Z MIP Z LP2 Z MIP > IGap CH = Z MIP Z CH Z MIP =0 As an MIP problem: LP2 is expected to be solved faster than LP1 CH will be solved way faster than LP2 8 / 49

21 Good & Ideal MIP Concepts: Tightness and Compactness Tightness: defines the search space (relaxed feasible region) that the solver needs to explore to find the solution Compactness (problem size): defines the searching speed (data to process) that the solver takes to find the solution 9 / 49

22 Good & Ideal MIP Concepts: Tightness and Compactness Tightness: defines the search space (relaxed feasible region) that the solver needs to explore to find the solution Compactness (problem size): defines the searching speed (data to process) that the solver takes to find the solution Convex hull: The tightest formulation MIP solved as LP 9 / 49

23 Good & Ideal MIP Tightening an MIP Formulation The most common strategy is adding cuts In fact, this is the most effective strategy of current MIP solvers 4 4 R. Bixby and E. Rothberg, Progress in computational mixed integer programming a look back from the other side of the tipping point, Annals of Operations Research, vol. 149, no. 1, pp , Jan / 49

24 Good & Ideal MIP Tightening an MIP Formulation The most common strategy is adding cuts In fact, this is the most effective strategy of current MIP solvers 4 They should be added as cuts in the B&B 5,6 Time and not directly to the model, huge number of inequalities Time 4 R. Bixby and E. Rothberg, Progress in computational mixed integer programming a look back from the other side of the tipping point, Annals of Operations Research, vol. 149, no. 1, pp , Jan P. Damci-Kurt, S. Kucukyavuz, D. Rajan, and A. Atamturk, A Polyhedral Study of Ramping in Unit Committment, University of California-Berkeley, Research Report BCOL IEOR, Oct J. Ostrowski, M. F Anjos, and A. Vannelli, Tight mixed integer linear programming formulations for the unit commitment problem, IEEE Transactions on Power Systems, vol. 27, no. 1, pp , Feb / 49

25 Good & Ideal MIP Tightening an MIP Formulation The most common strategy is adding cuts In fact, this is the most effective strategy of current MIP solvers 4 They should be added as cuts in the B&B 5,6 Time and not directly to the model, huge number of inequalities Time Trade-off: Tightness vs. Compactness 4 R. Bixby and E. Rothberg, Progress in computational mixed integer programming a look back from the other side of the tipping point, Annals of Operations Research, vol. 149, no. 1, pp , Jan P. Damci-Kurt, S. Kucukyavuz, D. Rajan, and A. Atamturk, A Polyhedral Study of Ramping in Unit Committment, University of California-Berkeley, Research Report BCOL IEOR, Oct J. Ostrowski, M. F Anjos, and A. Vannelli, Tight mixed integer linear programming formulations for the unit commitment problem, IEEE Transactions on Power Systems, vol. 27, no. 1, pp , Feb / 49

26 Good & Ideal MIP Tightening an MIP Formulation The most common strategy is adding cuts In fact, this is the most effective strategy of current MIP solvers 4 They should be added as cuts in the B&B 5,6 Time and not directly to the model, huge number of inequalities Time Trade-off: Tightness vs. Compactness Improving the MIP formulation Provide the convex hull for some set of constraints If available, use the convex hull for some set of constraints 4 R. Bixby and E. Rothberg, Progress in computational mixed integer programming a look back from the other side of the tipping point, Annals of Operations Research, vol. 149, no. 1, pp , Jan P. Damci-Kurt, S. Kucukyavuz, D. Rajan, and A. Atamturk, A Polyhedral Study of Ramping in Unit Committment, University of California-Berkeley, Research Report BCOL IEOR, Oct J. Ostrowski, M. F Anjos, and A. Vannelli, Tight mixed integer linear programming formulations for the unit commitment problem, IEEE Transactions on Power Systems, vol. 27, no. 1, pp , Feb / 49

27 Tight & Compact UCs Outline 1 Introduction 2 Good and Ideal MIP formulations 3 Tight & Compact (TC) UC Formulations 4 Case Studies Deterministic Selft-UC Stochastic UCs: Different Solvers Stochastic UCs: IEEE-118 Bus System 5 Conclusions 11 / 49

28 Tight & Compact UCs Tight and Compact (TC) Formulation Let s focus on the core of UC formulations: Min/max outputs SU & SD ramps Minimum up/down (T U/T D) times 12 / 49

29 Tight & Compact UCs Tight and Compact (TC) Formulation Let s focus on the core of UC formulations: Min/max outputs SU & SD ramps Minimum up/down (T U/T D) times, convex hull already available 7 The whole formulation can be found in the paper TC-UC 8 and the convex hull proof in gentile et al. 9 7 D. Rajan and S. Takriti, Minimum up/down polytopes of the unit commitment problem with start-up costs, IBM, Research Report RC23628, Jun G. Morales-Espana, J. M. Latorre, and A. Ramos, Tight and compact MILP formulation for the thermal unit commitment problem, IEEE Transactions on Power Systems, vol. 28, no. 4, pp , Nov C. Gentile, G. Morales-Espana, and A. Ramos, A tight MIP formulation of the unit commitment problem with start-up and shut-down constraints, Institute for Research in Technology (IIT), Technical Report IIT A, / 49

30 Tight & Compact UCs Formulation for a generating unit (I) Generation limits taking into account: ( ) ( ) p t P P u t P SD w t+1 max (SD SU, 0) v t ( ) ( ) p t P P u t P SU v t max (SU SD, 0) w t+1 t (1) t (2) Total generation = P u t + p t. Variables Parameters p t Energy production above P P Minimum power output u t Commitment status P Maximum power output v t Startup status SU Startup ramp w t Shutdown status SD Shutdown ramp 13 / 49

31 Tight & Compact UCs Formulation for a generating unit (II) Logical relationship: u t u t 1 = v t w t t (3) v t u t t (4) w t 1 u t t (5) where (4) and (5) avoid the simultaneous startup and shutdown. Variable bounds p t 0 t (6) 0 u t, v t, w t 1 t (7) 14 / 49

32 Tight & Compact UCs Tightness of the Formulation Let s study the polytope (1)-(7) using PORTA 10 : PORTA enumerates all vertices of a convex feasible region 10 T. Christof and A. Löbel, PORTA: POlyhedron representation transformation algorithm, version 1.4.1, Konrad-Zuse-Zentrum für Informationstechnik Berlin, Germany, / 49

33 Tight & Compact UCs Tightness of the Formulation Let s study the polytope (1)-(7) using PORTA 10 : PORTA enumerates all vertices of a convex feasible region Example: 3 periods and P = 200, P = SU = SD = 100 for: Case 1: T U = T D = 1 Case 2: T U = T D = 2 10 T. Christof and A. Löbel, PORTA: POlyhedron representation transformation algorithm, version 1.4.1, Konrad-Zuse-Zentrum für Informationstechnik Berlin, Germany, / 49

34 Tight & Compact UCs Case 1: Providing The Convex Hull Formulation: PORTA results for (T U =T D =1) p t ( P P ) u t ( P SD ) w t+1 max (SD SU, 0) v t (1) p t ( P P ) u t ( P SU ) v t max (SU SD, 0) w t+1 (2) u t u t 1 = v t w t (3) v t u t (4) w t 1 u t (5) 16 / 49

35 Tight & Compact UCs Case 1: Providing The Convex Hull Formulation: p t ( P P ) u t ( P SD ) w t+1 max (SD SU, 0) v t (1) p t ( P P ) u t ( P SU ) v t max (SU SD, 0) w t+1 (2) u t u t 1 = v t w t (3) v t u t (4) w t 1 u t (5) PORTA results for (T U =T D =1) u 1, u 2, u 3, v 2, v 3, w 2, w 3, p 1, p 2, p 3 : DIM = 10 CONV_SECTION ( 1) ( 2) ( 3) ( 4) ( 5) ( 6) ( 7) ( 8) ( 9) ( 10) ( 11) ( 12) ( 13) ( 14) ( 15) ( 16) ( 17) END 16 / 49

36 Tight & Compact UCs Case 1: Providing The Convex Hull Formulation: p t ( P P ) u t ( P SD ) w t+1 max (SD SU, 0) v t (1) p t ( P P ) u t ( P SU ) v t max (SU SD, 0) w t+1 (2) u t u t 1 = v t w t (3) v t u t (4) w t 1 u t (5) All vertices are integer Convex Hull PORTA results for (T U =T D =1) u 1, u 2, u 3, v 2, v 3, w 2, w 3, p 1, p 2, p 3 : DIM = 10 CONV_SECTION ( 1) ( 2) ( 3) ( 4) ( 5) ( 6) ( 7) ( 8) ( 9) ( 10) ( 11) ( 12) ( 13) ( 14) ( 15) ( 16) ( 17) END 16 / 49

37 Tight & Compact UCs Case 2: Providing and Using Convex Hulls (I) Formulation + T U/T D Convex hull: p t ( P P ) u t ( P SD ) PORTA results for (T U =T D =2) w t+1 max (SD SU, 0) v t (1) p t ( P P ) u t ( P SU ) v t max (SU SD, 0) w t+1 (2) u t u t 1 = v t w t (3) t v i u t (4) i=t T U+1 t i=t T D+1 w i 1 u t (5) 17 / 49

38 Tight & Compact UCs Case 2: Providing and Using Convex Hulls (I) Formulation + T U/T D Convex hull: p t ( P P ) u t ( P SD ) w t+1 max (SD SU, 0) v t (1) p t ( P P ) u t ( P SU ) v t max (SU SD, 0) w t+1 (2) u t u t 1 = v t w t (3) t v i u t (4) i=t T U+1 t i=t T D+1 w i 1 u t (5) How to remove the fractional vertices? PORTA results for (T U =T D =2) u 1, u 2, u 3, v 2, v 3, w 2, w 3, p 1, p 2, p 3 : DIM = 10 CONV_SECTION ( 1) ( 2) 1/2 1 1/2 1/ / ( 3) 1/2 1 1/2 1/ / ( 4) 1/2 1 1/2 1/ / ( 5) 1/2 1 1/2 1/ / ( 6) ( 7) ( 8) ( 9) ( 10) ( 11) ( 12) ( 13) ( 14) ( 15) ( 16) ( 17) ( 18) ( 19) END 17 / 49

39 Tight & Compact UCs Case 2: Providing and Using Convex Hulls (II) Reformulating (1) and (2) for T U 2: p t ( P P ) u t ( P SD ) w t+1 max (SD SU, 0) v t (1) p t ( P P ) u t ( P SU ) v t max (SU SD, 0) w t+1 (2) p t ( P P ) u t ( P SU ) v t ( P SD ) w t+1 (8) u t u t 1 = v t w t (3) t v i u t (4) i=t T U+1 t i=t T D+1 w i 1 u t (5) PORTA results for (T U =T D =2) u 1, u 2, u 3, v 2, v 3, w 2, w 3, p 1, p 2, p 3 : DIM = 10 CONV_SECTION ( 1) ( 2) ( 3) ( 4) ( 5) ( 6) ( 7) ( 8) ( 9) ( 10) ( 11) ( 12) ( 13) ( 14) ( 15) END 18 / 49

40 Tight & Compact UCs Case 2: Providing and Using Convex Hulls (II) Reformulating (1) and (2) for T U 2: p t ( P P ) u t ( P SD ) w t+1 max (SD SU, 0) v t (1) p t ( P P ) u t ( P SU ) v t max (SU SD, 0) w t+1 (2) p t ( P P ) u t ( P SU ) v t ( P SD ) w t+1 (8) u t u t 1 = v t w t (3) t v i u t (4) i=t T U+1 t i=t T D+1 Convex Hull w i 1 u t (5) PORTA results for (T U =T D =2) u 1, u 2, u 3, v 2, v 3, w 2, w 3, p 1, p 2, p 3 : DIM = 10 CONV_SECTION ( 1) ( 2) ( 3) ( 4) ( 5) ( 6) ( 7) ( 8) ( 9) ( 10) ( 11) ( 12) ( 13) ( 14) ( 15) END 18 / 49

41 Case Studies Deterministic Selft-UC Outline 1 Introduction 2 Good and Ideal MIP formulations 3 Tight & Compact (TC) UC Formulations 4 Case Studies Deterministic Selft-UC Stochastic UCs: Different Solvers Stochastic UCs: IEEE-118 Bus System 5 Conclusions 19 / 49

42 Case Studies Deterministic Selft-UC Self-UC case Study Case Study: Self-UC for 10-units, for days time span Basic constraints: max/min, SU/SD and TU/TD All results are expressed as percentages of 1bin results 20 / 49

43 Case Studies Deterministic Selft-UC Self-UC case Study Case Study: Self-UC for 10-units, for days time span Basic constraints: max/min, SU/SD and TU/TD Formulations tested modeling the same MIP problem: TC 11 : Proposed Tight & Compact 1bin 12 : 1-binary variable (u) 3binTUTD 13 : 3-binary variable version (u,v,w) + T U/T D convex hull All results are expressed as percentages of 1bin results 11 G. Morales-Espana, J. M. Latorre, and A. Ramos, Tight and compact MILP formulation for the thermal unit commitment problem, IEEE Transactions on Power Systems, vol. 28, no. 4, pp , Nov M. Carrion and J. Arroyo, A computationally efficient mixed-integer linear formulation for the thermal unit commitment problem, IEEE Transactions on Power Systems, vol. 21, no. 3, pp , J. Ostrowski, M. F Anjos, and A. Vannelli, Tight mixed integer linear programming formulations for the unit commitment problem, IEEE Transactions on Power Systems, vol. 27, no. 1, pp , Feb / 49

44 Case Studies Deterministic Selft-UC Case Study: Self-UC (I) Results presented as percentages of 1bin: 3binTUTD (%) TC (%) Constraints <78 <48 Nonzeros Real Vars Bin Vars =300 =300 TC is more Compact 21 / 49

45 Case Studies Deterministic Selft-UC Case Study: Self-UC (I) Results presented as percentages of 1bin: 3binTUTD (%) TC (%) Constraints <78 <48 Nonzeros Real Vars Bin Vars =300 =300 Integrality Gap 34 =0 TC is Tighter and Simultaneously more Compact 21 / 49

46 Case Studies Deterministic Selft-UC Case Study: Self-UC (I) Results presented as percentages of 1bin: 3binTUTD (%) TC (%) Constraints <78 <48 Nonzeros Real Vars Bin Vars =300 =300 Integrality Gap 34 =0 MIP runtime (speedup) 4.9 (20x) (995x) TC is Tighter and Simultaneously more Compact 21 / 49

47 Case Studies Deterministic Selft-UC Case Study: Self-UC (II) Results presented as percentages of 1bin: 3binTUTD (%) TC (%) Constraints <78 <48 Nonzeros Real Vars Bin Vars =300 =300 Integrality Gap 34 =0 MIP runtime LP runtime The TC formulation describe the convex hull then solving MIP (non-convex) as LP (convex) 22 / 49

48 Case Studies Stochastic UCs: Different Solvers Outline 1 Introduction 2 Good and Ideal MIP formulations 3 Tight & Compact (TC) UC Formulations 4 Case Studies Deterministic Selft-UC Stochastic UCs: Different Solvers Stochastic UCs: IEEE-118 Bus System 5 Conclusions 23 / 49

49 Case Studies Stochastic UCs: Different Solvers Stochastic UC: Case Study 10 generating units for a time span of 2 days 10 to 200 scenarios in demand 24 / 49

50 Case Studies Stochastic UCs: Different Solvers Stochastic UC: Case Study 10 generating units for a time span of 2 days 10 to 200 scenarios in demand Comparing TC, 1bin, 3binTUTD and Two additional formulations Sh 14 and 3bin T. Li and M. Shahidehpour, Price-based unit commitment: a case of lagrangian relaxation versus mixed integer programming, IEEE Transactions on Power Systems, vol. 20, no. 4, pp , Nov J. Arroyo and A. Conejo, Optimal response of a thermal unit to an electricity spot market, Power Systems, IEEE Transactions on, vol. 15, no. 3, pp , / 49

51 Case Studies Stochastic UCs: Different Solvers Stochastic UC: Case Study 10 generating units for a time span of 2 days 10 to 200 scenarios in demand Comparing TC, 1bin, 3binTUTD and Two additional formulations Sh 14 and 3bin 15 Different Solvers Cplex Gurobi XPRESS Stop criteria: Time limit: 5 hours or Optimality tolerance: 0.1 % 14 T. Li and M. Shahidehpour, Price-based unit commitment: a case of lagrangian relaxation versus mixed integer programming, IEEE Transactions on Power Systems, vol. 20, no. 4, pp , Nov J. Arroyo and A. Conejo, Optimal response of a thermal unit to an electricity spot market, Power Systems, IEEE Transactions on, vol. 15, no. 3, pp , / 49

52 Case Studies Stochastic UCs: Different Solvers Stochastic: Cplex TC 3binTUTD 3bin Sh 1bin Time [s] Demand Scenarios 25 / 49

53 Case Studies Stochastic UCs: Different Solvers Stochastic: Cplex TC 3binTUTD 3bin Sh 1bin Time [s] Demand Scenarios TC deals with 200 scenarios within the time that others deal with / 49

54 Case Studies Stochastic UCs: Different Solvers Stochastic: Gurobi 10 3 TC 3binTUTD 3bin Sh 1bin 10 2 Time [s] Demand Scenarios TC deals with 200 scenarios within the time that others deal with / 49

55 Case Studies Stochastic UCs: Different Solvers Stochastic: XPRESS TC 3binTUTD 3bin Sh 1bin 10 2 Time [s] Demand Scenarios TC deals with 200 scenarios within the time that others deal with / 49

56 Case Studies Stochastic UCs: IEEE-118 Bus System Outline 1 Introduction 2 Good and Ideal MIP formulations 3 Tight & Compact (TC) UC Formulations 4 Case Studies Deterministic Selft-UC Stochastic UCs: Different Solvers Stochastic UCs: IEEE-118 Bus System 5 Conclusions 28 / 49

57 Case Studies Stochastic UCs: IEEE-118 Bus System IEEE-118 Bus System 54 thermal units; 118 buses; 186 transmission lines; 91 loads 24 hours time span 3 wind farms, 20 wind power scenarios Stop Criteria in Cplex % opt. tolerance or 24h time limit 29 / 49

58 Case Studies Stochastic UCs: IEEE-118 Bus System UC performance comparisons (I) Traditional Energy-Block Scheduling 3binTUTD 16 TC o.f. [k$] opt.tol [%] IntGap [%] Compared with 3binTUTD, TC: lowered IntGap by 53.3% 16 FERC, RTO unit commitment test system, Federal Energy and Regulatory Commission, Washington DC, USA, Tech. Rep., Jul. 2012, p / 49

59 Case Studies Stochastic UCs: IEEE-118 Bus System UC performance comparisons (I) Traditional Energy-Block Scheduling 3binTUTD 16 TC o.f. [k$] opt.tol [%] IntGap [%] MIP runtime [s] Compared with 3binTUTD, TC: lowered IntGap by 53.3% is more than 420x faster 16 FERC, RTO unit commitment test system, Federal Energy and Regulatory Commission, Washington DC, USA, Tech. Rep., Jul. 2012, p / 49

60 Case Studies Stochastic UCs: IEEE-118 Bus System UC performance comparisons (II) Traditional Energy-Block Scheduling 3binTUTD 17 TC o.f. [k$] opt.tol [%] IntGap [%] MIP runtime [s] LP runtime [s] TC solved the MIP before 3binTUTD solved the LP within the required opt. tolerance (0.05%) 17 FERC, RTO unit commitment test system, Federal Energy and Regulatory Commission, Washington DC, USA, Tech. Rep., Jul. 2012, p / 49

61 Case Studies Stochastic UCs: IEEE-118 Bus System UC performance comparisons (III) Traditional Power-Based Energy-based UC UC 3binTUTD TC P-TC o.f. [k$] opt.tol [%] IntGap [%] MIP runtime [s] LP runtime [s] P-TC 18 has a more detailed and accurate UC representation 18 G. Morales-Espana, A. Ramos, and J. Garcia-Gonzalez, An MIP formulation for joint market-clearing of energy and reserves based on ramp scheduling, IEEE Transactions on Power Systems, vol. 29, no. 1, pp , / 49

62 Case Studies Stochastic UCs: IEEE-118 Bus System UC performance comparisons (III) Traditional Power-Based Energy-based UC UC 3binTUTD TC P-TC o.f. [k$] opt.tol [%] IntGap [%] MIP runtime [s] LP runtime [s] P-TC 18 has a more detailed and accurate UC representation it solved 100x faster than 3binTUTD its UC core is also a convex hull G. Morales-Espana, A. Ramos, and J. Garcia-Gonzalez, An MIP formulation for joint market-clearing of energy and reserves based on ramp scheduling, IEEE Transactions on Power Systems, vol. 29, no. 1, pp , G. Morales-España, C. Gentile, and A. Ramos, Tight MIP formulations of the power-based unit commitment problem, en, OR Spectrum, pp. 1 22, May / 49

63 Conclusions Outline 1 Introduction 2 Good and Ideal MIP formulations 3 Tight & Compact (TC) UC Formulations 4 Case Studies Deterministic Selft-UC Stochastic UCs: Different Solvers Stochastic UCs: IEEE-118 Bus System 5 Conclusions 33 / 49

64 Conclusions Conclusions (I) Beware of what matters in good MIP formulations Tightness & Compactness 34 / 49

65 Conclusions Conclusions (I) Beware of what matters in good MIP formulations Tightness & Compactness Binaries Solving time False myth 34 / 49

66 Conclusions Conclusions (I) Beware of what matters in good MIP formulations Tightness & Compactness Binaries Solving time False myth Use the convex hull of some set of constraints 34 / 49

67 Conclusions Conclusions (I) Beware of what matters in good MIP formulations Tightness & Compactness Binaries Solving time False myth Use the convex hull of some set of constraints Minimum up/down times 20 Unit operation in Energy-based UC 21 Unit operation in Power-based UC 22 solving time by simultaneously T&Cing the final UCs 23,24 20 D. Rajan and S. Takriti, Minimum up/down polytopes of the unit commitment problem with start-up costs, IBM, Research Report RC23628, Jun C. Gentile, G. Morales-Espana, and A. Ramos, A tight MIP formulation of the unit commitment problem with start-up and shut-down constraints, Institute for Research in Technology (IIT), Technical Report IIT A, G. Morales-España, C. Gentile, and A. Ramos, Tight MIP formulations of the power-based unit commitment problem, en, OR Spectrum, pp. 1 22, May G. Morales-Espana, J. M. Latorre, and A. Ramos, Tight and compact MILP formulation for the thermal unit commitment problem, IEEE Transactions on Power Systems, vol. 28, no. 4, pp , Nov G. Morales-Espana, J. M. Latorre, and A. Ramos, Tight and compact MILP formulation of start-up and shut-down ramping in unit commitment, IEEE Transactions on Power Systems, vol. 28, no. 2, pp , / 49

68 Conclusions Conclusions (II) Better UC core in stochastic UCs Critical solving time reductions even when the UC is modeled in more detail, i.e., P-UC 35 / 49

69 Conclusions Conclusions (II) Better UC core in stochastic UCs Critical solving time reductions even when the UC is modeled in more detail, i.e., P-UC If convex hulls are not available Create simultaneously tight and compact models by reformulating the problem, e.g., CCGTs 25 Key hint: start removing all big-m parameters 25 G. Morales-Espana, C. Correa-Posada, and A. Ramos, Tight and Compact MIP Formulation of Configuration-Based Combined-Cycle Units, IEEE Transactions on Power Systems, vol. PP, no. 99, pp. 1 10, / 49

70 Conclusions Conclusions (II) Better UC core in stochastic UCs Critical solving time reductions even when the UC is modeled in more detail, i.e., P-UC If convex hulls are not available Create simultaneously tight and compact models by reformulating the problem, e.g., CCGTs 25 Key hint: start removing all big-m parameters Create tight cuts Don t add them directly to the model Use them as cuts in the B&B algorithm 26,27 time 25 G. Morales-Espana, C. Correa-Posada, and A. Ramos, Tight and Compact MIP Formulation of Configuration-Based Combined-Cycle Units, IEEE Transactions on Power Systems, vol. PP, no. 99, pp. 1 10, P. Damci-Kurt, S. Kucukyavuz, D. Rajan, and A. Atamturk, A Polyhedral Study of Ramping in Unit Committment, University of California-Berkeley, Research Report BCOL IEOR, Oct J. Ostrowski, M. F Anjos, and A. Vannelli, Tight mixed integer linear programming formulations for the unit commitment problem, IEEE Transactions on Power Systems, vol. 27, no. 1, pp , Feb / 49

71 Conclusions Questions Thank you for your attention Contact Information: 36 / 49

72 Conclusions For Further Reading For Further Reading J. Arroyo and A. Conejo, Optimal response of a thermal unit to an electricity spot market, Power Systems, IEEE Transactions on, vol. 15, no. 3, pp , R. Bixby and E. Rothberg, Progress in computational mixed integer programming a look back from the other side of the tipping point, Annals of Operations Research, vol. 149, no. 1, pp , Jan M. Carrion and J. Arroyo, A computationally efficient mixed-integer linear formulation for the thermal unit commitment problem, IEEE Transactions on Power Systems, vol. 21, no. 3, pp , T. Christof and A. Löbel, PORTA: POlyhedron representation transformation algorithm, version 1.4.1, Konrad-Zuse-Zentrum für Informationstechnik Berlin, Germany, P. Damci-Kurt, S. Kucukyavuz, D. Rajan, and A. Atamturk, A Polyhedral Study of Ramping in Unit Committment, University of California-Berkeley, Research Report BCOL IEOR, Oct FERC, RTO unit commitment test system, Federal Energy and Regulatory Commission, Washington DC, USA, Tech. Rep., Jul. 2012, p / 49

73 Conclusions For Further Reading For Further Reading (cont.) C. Gentile, G. Morales-Espana, and A. Ramos, A tight MIP formulation of the unit commitment problem with start-up and shut-down constraints, Institute for Research in Technology (IIT), Technical Report IIT A, X. Guan, F. Gao, and A. Svoboda, Energy delivery capacity and generation scheduling in the deregulated electric power market, IEEE Transactions on Power Systems, vol. 15, no. 4, pp , Nov T. Koch, T. Achterberg, E. Andersen, O. Bastert, T. Berthold, R. E. Bixby, E. Danna, G. Gamrath, A. M. Gleixner, S. Heinz, A. Lodi, H. Mittelmann, T. Ralphs, D. Salvagnin, D. E. Steffy, and K. Wolter, MIPLIB 2010, Mathematical Programming Computation, vol. 3, no. 2, pp , Jun T. Li and M. Shahidehpour, Price-based unit commitment: a case of lagrangian relaxation versus mixed integer programming, IEEE Transactions on Power Systems, vol. 20, no. 4, pp , Nov G. Morales-Espana, C. Correa-Posada, and A. Ramos, Tight and Compact MIP Formulation of Configuration-Based Combined-Cycle Units, IEEE Transactions on Power Systems, vol. PP, no. 99, pp. 1 10, / 49

74 Conclusions For Further Reading For Further Reading (cont.) G. Morales-Espana, C. Gentile, and A. Ramos, Tight MIP formulations of the power-based unit commitment problem, Institute for Research in Technology (IIT), Technical Report, G. Morales-Espana, J. M. Latorre, and A. Ramos, Tight and compact MILP formulation for the thermal unit commitment problem, IEEE Transactions on Power Systems, vol. 28, no. 4, pp , Nov G. Morales-Espana, A. Ramos, and J. Garcia-Gonzalez, An MIP formulation for joint market-clearing of energy and reserves based on ramp scheduling, IEEE Transactions on Power Systems, vol. 29, no. 1, pp , G. Morales-España, C. Gentile, and A. Ramos, Tight MIP formulations of the power-based unit commitment problem, en, OR Spectrum, pp. 1 22, May G. Morales-Espana, J. M. Latorre, and A. Ramos, Tight and compact MILP formulation of start-up and shut-down ramping in unit commitment, IEEE Transactions on Power Systems, vol. 28, no. 2, pp , J. Ostrowski, M. F Anjos, and A. Vannelli, Tight mixed integer linear programming formulations for the unit commitment problem, IEEE Transactions on Power Systems, vol. 27, no. 1, pp , Feb / 49

75 Conclusions For Further Reading For Further Reading (cont.) D. Rajan and S. Takriti, Minimum up/down polytopes of the unit commitment problem with start-up costs, IBM, Research Report RC23628, Jun H. P. Williams, Model Building in Mathematical Programming, 5th Edition. John Wiley & Sons Inc, Feb L. Wolsey, Integer Programming. Wiley-Interscience, / 49

76 Conclusions For Further Reading Power-Based UC Tow main features are included: Schedules Power instead of Energy (for feasibility). To avoid: Infeasible energy delivery 28 Overestimated ramp availability X. Guan, F. Gao, and A. Svoboda, Energy delivery capacity and generation scheduling in the deregulated electric power market, IEEE Transactions on Power Systems, vol. 15, no. 4, pp , Nov G. Morales-Espana, A. Ramos, and J. Garcia-Gonzalez, An MIP formulation for joint market-clearing of energy and reserves based on ramp scheduling, IEEE Transactions on Power Systems, vol. 29, no. 1, pp , / 49

77 Conclusions For Further Reading Power-Based UC Tow main features are included: Schedules Power instead of Energy (for feasibility). To avoid: Infeasible energy delivery 28 Overestimated ramp availability 29 Includes startup (SU) and shutdown (SD) power trajectories SU & SD ramps are deterministic events in day-ahead UCs Ignoring them change commitment decisions and increase costs 21,30 28 X. Guan, F. Gao, and A. Svoboda, Energy delivery capacity and generation scheduling in the deregulated electric power market, IEEE Transactions on Power Systems, vol. 15, no. 4, pp , Nov G. Morales-Espana, A. Ramos, and J. Garcia-Gonzalez, An MIP formulation for joint market-clearing of energy and reserves based on ramp scheduling, IEEE Transactions on Power Systems, vol. 29, no. 1, pp , G. Morales-Espana, J. M. Latorre, and A. Ramos, Tight and compact MILP formulation of start-up and shut-down ramping in unit commitment, IEEE Transactions on Power Systems, vol. 28, no. 2, pp , / 49

78 Conclusions For Further Reading Power-Based UC Tow main features are included: Schedules Power instead of Energy (for feasibility). To avoid: Infeasible energy delivery 28 Overestimated ramp availability 29 Includes startup (SU) and shutdown (SD) power trajectories SU & SD ramps are deterministic events in day-ahead UCs Ignoring them change commitment decisions and increase costs 21,30 The convex hull of the Power-Based UC for basic operating constraints is provided X. Guan, F. Gao, and A. Svoboda, Energy delivery capacity and generation scheduling in the deregulated electric power market, IEEE Transactions on Power Systems, vol. 15, no. 4, pp , Nov G. Morales-Espana, A. Ramos, and J. Garcia-Gonzalez, An MIP formulation for joint market-clearing of energy and reserves based on ramp scheduling, IEEE Transactions on Power Systems, vol. 29, no. 1, pp , G. Morales-Espana, J. M. Latorre, and A. Ramos, Tight and compact MILP formulation of start-up and shut-down ramping in unit commitment, IEEE Transactions on Power Systems, vol. 28, no. 2, pp , G. Morales-Espana, C. Gentile, and A. Ramos, Tight MIP formulations of the power-based unit commitment problem, Institute for Research in Technology (IIT), Technical Report, / 49

79 Conclusions For Further Reading Case Study: Self-UC (II) Performance of the Energy-Based formulations: 3binTUTD (%) TC (%) R-TC (%) Constraints <78 <48 <56 Nonzeros Real Vars Bin Vars =300 =300 =300 Integrality Gap 34 =0 =0 MIP runtime MIP run (best-worst) LP runtime / 49

80 Conclusions For Further Reading Case Study: Self-UC (II) Performance of the Energy-Based formulations: 3binTUTD (%) TC (%) R-TC (%) Constraints <78 <48 <56 Nonzeros Real Vars Bin Vars =300 =300 =300 Integrality Gap 34 =0 =0 MIP runtime MIP run (best-worst) LP runtime The TC formulations describe the convex hull then solving MIP (non-convex) as LP (convex) 42 / 49

81 Conclusions UC for 40 power system mixes Outline UC for 40 power system mixes 43 / 49

82 Conclusions UC for 40 power system mixes UC for 40 power system mixes Case Study B: UC for 40 power system mixes, from 28 to 1870 units 32 Including demand, ramps, reserves, variable SU costs 32 G. Morales-Espana, J. M. Latorre, and A. Ramos, Tight and compact MILP formulation for the thermal unit commitment problem, IEEE Transactions on Power Systems, vol. 28, no. 4, pp , Nov / 49

83 Conclusions UC for 40 power system mixes Integrality Gap: Small Cases units x 1 day, OptTol: bin 3binTUTD TC 0.6e 3 0.6e 3 0.6e 3 0.4e 3 0.6e e 3 0.4e 3 1.3e 3 0.6e 3 0.5e 3 Proportion [%] Case [#] Geometric Averages: 3bin 64%; TC 35% 45 / 49

84 Conclusions UC for 40 power system mixes CPU Time: Small Cases units x 1 day, OptTol: bin 3binTUTD TC Proportion [%] Case [#] Geometric Averages: 3bin 36%; TC 12% 46 / 49

85 Conclusions UC for 40 power system mixes Integrality Gap: Large Cases units x 1 day, OptTol: bin 3binTUTD TC 0.1e e 3 0.2e 3 0.2e 3 0.2e 3 0.1e 3 0.1e 3 0.1e 3 0.1e 3 0.1e 3 Proportion [%] Case [#] Geometric Averages: 3bin 75%; TC 43% 47 / 49

86 Conclusions UC for 40 power system mixes CPU Time: Large Cases units x 1 day, OptTol: bin 3binTUTD TC 1h h2 1h49 7h24 8h45 4h2 7h27 8h58 4h3 10h0 Proportion [%] Case [#] Geometric Averages: 3bin 45%; TC 5% 48 / 49

87 Conclusions UC for 40 power system mixes UC for 40 power system mixes Results presented as percentages of 1bin: 3binTUTD (%) TC (%) Constraints <99 <40 Nonzeros ~100 <35 Real Vars Bin Vars =300 <500 Integrality Gap / 49

88 Conclusions UC for 40 power system mixes UC for 40 power system mixes Results presented as percentages of 1bin: 3binTUTD (%) TC (%) Constraints <99 <40 Nonzeros ~100 <35 Real Vars Bin Vars =300 <500 Integrality Gap TC is Tighter and Simultaneously more Compact 49 / 49

89 Conclusions UC for 40 power system mixes UC for 40 power system mixes Results presented as percentages of 1bin: 3binTUTD (%) TC (%) Constraints <99 <40 Nonzeros ~100 <35 Real Vars Bin Vars =300 <500 Integrality Gap Total Average runtime 71 7 runtime (best-worst) TC is Tighter and Simultaneously more Compact 49 / 49

90 Conclusions UC for 40 power system mixes UC for 40 power system mixes Results presented as percentages of 1bin: 3binTUTD (%) TC (%) Constraints <99 <40 Nonzeros ~100 <35 Real Vars Bin Vars =300 <500 Integrality Gap Total Average runtime 71 7 runtime (best-worst) Runtime Small Cases Runtime Large Cases TC is Tighter and Simultaneously more Compact 49 / 49