Tight and Compact MILP Formulation for the Thermal Unit Commitment Problem

Transcription

1 Online Companion for Tight and Compact MILP Formulation for the Thermal Unit Commitment Problem Germán Morales-España, Jesus M. Latorre, and Andres Ramos Universidad Pontificia Comillas, Spain Institute for Research in Technology (IIT This document is an online companion 1 for [1]. The Unit Commitment (UC formulations that were tested in [1] as well as some extra tests are here presented. Contents 1 Formulations Nomenclature One-binary UC formulation [2] Three-binary UC formulation [3] Tight & Compact UC Formulation [1] Implemented Formulations Numerical Results Comparing 1bin with 3bin Convergence evolution A Results Data with Startup Costs 10 B Results Data without Startup Costs 10 1 Formulations This section present all the four Unit Commitment (UC formulations tested in [1]. It is important to note that all the formulations are characterizing the same optimization problem. The difference between them is how the constraints are formulated. In other words, for a given case study, all the formulations obtain the same optimal results, e.g. commitments, generating outputs and operation costs. 1 Research Report No. IIT , last update

2 1.1 Nomenclature The main definitions and notation used are presented in this section for quick reference. Upper-case letters are used for denoting parameters and sets; and lower-case letters for variables and indexes Indexes and Sets g G Generating units, running from 1 to N G. l L g t T Start-up intervals, running from 1 (hottest to N Lg (coldest. Hourly periods in the time horizon, running from 1 to N T hours Constants System Related C NSE D t R t Non-served energy cost [$/MWh]. Load demand in period t [MW]. Spinning reserve requirements for the system in period t [MW]. Unit Related C LV g C NL g Linear variable cost of unit g [$/MWh]. No-load cost of unit g [$/h]. C SD g Shut-down cost of unit g [$]. C SU g,l Cost of the interval l of the stairwise startup cost function of unit g [$]. P g P g RD g RU g SD g SU g T D g T U g T SU g,l Maximum power output of unit g [MW]. Minimum power output of unit g [MW]. Down ramp limit of unit g [MW/h]. Up ramp limit of unit g [MW/h]. Shutdown ramp capability of unit g [MW]. Startup ramp capability of unit g [MW]. Minimum down time of unit g [h]. Minimum up time of unit g [h]. Times defining the interval l limits, which are [ Tl SU, Tl+1 SU, of the stairwise startup cost function of unit g [h]. 2

3 1.1.3 Variables Positive and Continuous Variables c SD g,t Shutdown cost of unit g in period t [$]. c SU g,t Startup cost of unit g in period t [$]. nse t Non-served energy in period t [MW]. p g,t p g,t p g,t r g,t Power output at period t of unit g, production above the minimum output [MW]. Total power output at period t of unit g [MW]. Maximum available power output at time t of unit g [MW]. Spinning reserve contribution of unit g in period t [MW]. Binary Variables u g,t Commitment status of the unit g for period t, which is equal to 1 if the unit is online and 0 otherwise. v g,t Start-up status of unit g, which takes the value of 1 if the unit starts up in period t and 0 otherwise. w g,t δ g,l,t Shut-down status of unit g, which takes the value of 1 if the unit shuts down in period t and 0 otherwise. Start-up-type l of unit g, which takes the value of 1 in the period where the unit starts up and the previous shutdown was within [ Tl SU, Tl+1 SU hours. 1.2 One-binary UC formulation [2] The UC formulation proposed in [2] is presented in this section Objective function min [ ] Cg NL u g,t +Cg LV p g,t + c SU g,t +c SD g,t +Ct NSE nse t g G t T ( Startup Costs The following MILP formulation for the stairwise startup cost was proposed in [4]. c SU g,t C SU g,k ( u g,t k u g,t i i=1 g, t, k [ ] 1, TN SU Lg (2 where C SU g,k as as: is the cost of turning on the unit g after being offline for k time periods. CSU g,k C SU g,k = C SU g,1 if k < [ T SU if k C SU g,l C SU g,n Lg g,2 g T SU g,l, T SU g,l+1 if k = T SU g,n Lg g, l (1, N Lg g is defined (2a 3

4 1.2.3 Shutdown Costs c SU g,t C SD g (u g,t 1 u g,t g, t ( Power System Requirements p g,t = D t nse t t (4 g G p g,t D t + R t nse t t (5 g G Minimum Up and Down The minimum up time is set by: t+n T t+t U g 1 i=t u g,i T U g (u g,t u g,t 1 g, t ( T U 0 g, N T T U g + 1 ] (6 [u g,i (u g,t u g,t 1 ] 0 g, t (N T T U g + 1, N T ] (7 i=t The minimum down time is set by t+n T t+t D g 1 i=t (1 u g,i T D g (u g,t 1 u g,t g, t ( T D 0 g, N T T D g + 1 ] (8 [1 u g,i (u g,t 1 u g,t ] 0 g, t (N T T D g + 1, N T ] (9 i=t Unit Capacity Limits P g u g,t p g,t g, t (10 p g,t p g,t g, t (11 p g,t P g u g,t g, t ( Unit Ramping Limits p g,t p g,t 1 RU g u g,t 1 + SU g (u g,t u g,t 1 + P g (1 u g,t g, t (13 p g,t P g u g,t+1 + SD g (u g,t u g,t+1 g, t [1, N T (14 p g,t 1 p g,t RD g u g,t + SD g (u g,t 1 u g,t + P g (1 u g,t 1 g, t ( Three-binary UC formulation [3] The t formulation described in [3] is presented in this section. Note that this formulation is the equivalent three-binary formulation in [2], see Subsection 1.2. Actually, the power-system and unitcapacity constraints for the three-binary formulation are the same as for the one-binary formulation (see Subsection

5 1.3.1 Objective Function min [ Cg NL u g,t +Cg LV p g,t + c SU g G t T g,t +Cg SD ] w g,t +Ct NSE ens t ( Logic constraint between commitment, startup and shutdown This formulation can be found in models published approximately fifty years ago [5]. u g,t u g,t 1 = v g,t w g,t t ( Minimum up and down Constraints The minimum up and down times are ensured with [6]: p v g,i u g,t g, t [T U g, N T ] (18 i=t T U g+1 p i=t T D g+1 w g,i 1 u g,t g, t [T D g, N T ] ( Unit Capacity Limits Apart from (10 and (11, the following constraint is also needed [7]: p g,t P g (u g,t w g,t+1 + SD g w g,t+1 g, t ( Unit Ramping Limits p g,t p g,t 1 RU g u g,t 1 + SU g v g,t g, t (21 p g,t 1 p g,t RD g u g,t + SD g w g,t g, t ( Tight & Compact UC Formulation [1] The formulation proposed in [1] is presented in this section. The minimum up and down time constraints in [6] are used in [1], see Subsection Objective Function min [ Cg NL u g,t +C LV ( g P g u g,t +p g,t + g G t T l L g C SU ] g,l δ g,l,t +Cg SD w g,t +Ct NSE ens t ( Startup Cost δ g,l,t T SU g,l+1 1 i=t SU g,l w g,t i g, t [ T SU g,l+1, N T ], l [1, NL (24 l L g δ g,l,t = v g,t g, t (25 5

6 1.4.3 Power System Requirements [ ] P g u g,t + p g,t = Dt nse t t (26 g G r g,t R t t (27 g G Unit Capacity Limits p g,t + r g,t ( P g P g ug,t ( P g SU g vg,t ( P g SD g wg,t+1 g / G1, t (28 where the subset G1 is defined as the units in G with T U g = 1. Constraint (28 is infeasible for units with T U g = 1, then the less compact and less tight formulation is used for these units: p g,t + r g,t ( P g P g ug,t ( P g SU g vg,t p g,t + r g,t ( P g P g ug,t ( P g SD g wg,t+1 g G1, t (28a g G1, t (28b Unit Ramping Limits (p g,t + r g,t p g,t 1 RU g g, t (29 p g,t + p g,t 1 RD g g, t ( Implemented Formulations Four formulations were tested in [1]. 1 the formulation presented in Subsection 1.2 labeled as 1bin 2. 2 the formulation presented in Subsection 1.3 labeled as 3bin. 3 The formulation labeled as Prop1 is the same as the 3bin, however, the exponential startup-cost constraints presented in Subsection were used instead. 4 The formulation labeled as Prop2 is the complete formulation proposed in [1], see Subsection 1.4. The formulations 1bin, 3bin and Prop2 were detailed in Subsection 1.2, Subsection 1.3 and Subsection 1.4. Prop1 is the 3bin formulation with the startup-cost constraints presented in Subsection The objective function for Prop1 is then: min g G [ t T Cg NL u g,t +Cg LV p g,t + C SU l L g g,l δ g,l,t +Cg SD ] w g,t +Ct NSE ens t For the sake of clarity, Table 1 shows the constraints that belong to each formulation. 2 Numerical Results 2.1 Comparing 1bin with 3bin As discussed in [1], although the 3bin is tighter than 1bin (always presented a smaller integrability gap, 3bin did not show a clear computational-performance dominance over 1bin. In order to observe a more clear impact of the minimum up and down constraints used in 3bin, all the 40 instances, detailed in [1], were run where the exponential startup-cost constraints were deactivated. Fig. 1 shows the CPU times and intergrality gaps for 3bin in comparison with 1bin for all the instances (using ratios, where 1bin always represents the 100%. The CPU time and the integrability 2 Note that the formulations labeled in this document as 1bin and 3bin are labeled in [1] as [14] and [15] respectively. (31 6

7 Table 1: Set of Constraints for each formulation 1bin 3bin Prop1 Prop2 Objective Function (1 (16 (31 (23 SU & SD Costs (2-(3 (2 (24-(25 (24-(25 Power System Requiriments (4-(5 (4-(5 (4-(5 (26-(27 Logic Constraint (17 (17 (17 Min Up/Down Times (6-(9 (18-(19 (18-(19 (18-(19 Capacity Limits (10-(12 (10, (11 and (20 (10, (11 and (20 (28 Ramping Limits (13-(15 (21-(22 (21-(22 (29-(30 Table 2: Problem size summary, 1bin in relation with 3bin (% Case # of constraints # of nonzero elements # of real variables # of binary variables Mean 87,5 95,2 50,2 300,0 min 87,4 95,2 50,0 300,0 max 87,4 95,2 50,1 300,0 gaps of 1bin are shown within the squares to give an idea of the different problem magnitudes. The summaries for the problem size and computational performance comparison are presented in Table 2 and Table 3 respectively. In short, the is a clear computational-performance dominance of 3bin over 1bin. Table 4 shows the speed ups of 3bin over 1bin, where 3bin shows in average to be almost four time faster presenting a better performance for the large cases (Cases Convergence evolution In practice, the main goal o solving a MILP problem is often to find good feasible solutions as quickly as possible rather than the optimal solution [8, 9]. The quality of a feasible solution, under the branch-and-bound framework, is measured with the optimality tolerance, which is basically the difference between the upper and lower bounds. The upper bounds are actually the feasible integral solutions, and the lower bounds are the optimal objective value for the LP relaxation (among all current branch-and-cut nodes [10, 11]. Even if the optimal integer solution has been found by the by the upper bound, it can only be proven to be optimal if the lower bound is equal to the upper bound. The branch-and-cut algorithm improves both, (i the upper bound, by heuristics and node presolve; and (ii the lower bound, by cuts and node presolve [8]. The evolution of the upper bound depends on the evolution of he lower bound, because heuristics and node presolve are mainly applied using the relaxed LP solution of he current active branch-and-cut node [8, 10]. That is, heuristics, for example explore the neighborhood of the current active LP relaxation to finding potentially better integer Table 3: Computational Performance (% 7-days 10-gen CPU Time (s Integrality Gap Opt. Tolerance Nodes Iterations Cases ,4 70,2 116,7 91,7 54,8 Cases ,3 71,1 1,8 135,9 40,5 Cases ,5 70,6 14,4 111,6 47,1 Cases ,5 56,6 69,7 348,3 65,7 Cases ,8 68,2 192,5 1612,4 51,6 Cases ,9 62,1 115,8 749,4 58,2 7

8 bin 3bin bin 3bin h23 1h35 2h0 1h56 1h h59 1h48 1h11 1h16 0h44 8.8e 3 6.3e 3 5.7e 3 3.8e 3 5.7e 3 1.7e 3 1.4e 3 1.6e 3 1.4e 3 1.2e e 3 4.9e 3 7.8e 3 5.9e 3 5.4e 3 1.5e 3 1.4e 3 1.1e 3 1.3e 3 1.1e 3 Proportion [%] Proportion [%] Case [#] Case [#] (a x7 days CPU time (b x7 days Integrality Gap bin 3bin bin 3bin h10 0h48 1h6 1h47 1h h57 2h0 1h16 1h32 1h14 0.5e 3 0.5e 3 0.4e 3 0.3e 3 0.4e 3 0.1e 3 0.1e 3 0.1e 3 0.1e 3 0.1e e 3 0.3e 3 1.2e 3 0.5e 3 0.4e 3 0.1e 3 0.2e 3 0.1e 3 0.1e 3 0.1e 3 Proportion [%] Proportion [%] Case [#] Case [#] (c x10 gen CPU Time (d x10 gen Integrality Gap Fig. 1: Improvements in relation with [2] (%. White areas correspond with small cases and gray to big ones. Table 4: Speed ups of 3bin over 1bin Cases ,70 Cases ,39 cases ,82 8

9 Objective function [M $] Prop2: Upper bound 3bin: Upper bound 1bin: Upper Bound Prop2: Lower bound 3bin: Lower bound 1bin: Lower Bound Optimality Gap [p.u.] Prop2 3bin 1bin CPU Time [s] CPU Time [s] Fig. 2: Convergence evolution for a 100 units power system. The figure at the left shows the evolution of the upper and lower bounds, and the figure at the right presents the evolution of the optimality tolerance. solutions. This section shows the convergence evolution for two different power systems for the 1bin, 3bin and Prop2 formulations described in Section 1. All tests were carried out using CPLEX 12.4 under GAMS [12] on an Intel-i7 2.4-GHz personal computer with 4 GB of RAM memory. The case studies are solved with a CPU time limit of 1200 seconds. Apart from this, CPLEX defaults were used for all the experiments unit The 10-unit system in [2] was replicated 10 times and the convergence evolution for the 1bin, 3bin and Pro2 formulations are shown in Fig. 2. The curves in Fig. 2 correspond to the results since the first integer solution was found till the time limit is achieved IEEE 118bar -54 gen The modified IEEE 118-bus system with 54 thermal units presented in [13] is implemented. This power system was implemented with a single node. The converge evolution curves in Fig. 3 correspond to the results since the first integer solution was found till the time limit is achieved for the 1bin, 3bin and Prop2 formulations. Note in Fig. 2 and Fig. 3 that the increment evolution of the lower bounds depends on the formulation tightness. On the other hand, the evolution speed is highly dependent on the compactness of the formulation. 9

10 Objective function [M $] Prop2: Upper bound 3bin: Upper bound 1bin: Upper Bound Prop2: Lower bound 3bin: Lower bound 1bin: Lower Bound Optimality Gap [p.u.] Prop2 3bin 1bin CPU Time [s] CPU Time [s] Fig. 3: Convergence evolution for a 54 units power system. The figure at the left shows the evolution of the upper and lower bounds, and the figure at the right presents the evolution of the optimality tolerance. A Results Data with Startup Costs The extended set of results which were summarized in [1] are shown from Table 5 to Table 8 B Results Data without Startup Costs The extended set of results which were summarized in Subsection 2.1 are shown from Table 9 to Table 12. References [1] G. Morales-España, J. M. Latorre, and A. Ramos, Tight and compact MILP formulation for the thermal unit commitment problem, IEEE Transactions on Power Systems, 2012, paper under Review (Manuscrit ID: TPWRS R1, online preprint. [Online]. Available: [2] M. Carrion and J. Arroyo, A computationally efficient mixed-integer linear formulation for the thermal unit commitment problem, Power Systems, IEEE Transactions on, vol. 21, no. 3, pp , [3] 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 [4] M. P. Nowak and W. Römisch, Stochastic lagrangian relaxation applied to power scheduling in a hydro-thermal system under uncertainty, Annals of Operations Research, vol. 100, no. 1, pp , Dec [Online]. Available: 10

11 Table 5: Problem Size: x7-day # of Equations # of Non-Zero Elements # of Real Variables # of Binary Variables 1bin 3bin Prop1 Prop2 1bin 3bin Prop1 Prop2 1bin 3bin Prop1 Prop2 1bin 3bin Prop1 Prop Table 6: Computational Performance: x7-day Time (s Gap (p.u. Nodes Iterations 1bin 3bin Prop1 Prop2 1bin 3bin Prop1 Prop2 1bin 3bin Prop1 Prop2 1bin 3bin Prop1 Prop ,9 422,5 112,7 43,4 8,50E-05 7,91E-05 6,56E-05 5,11E ,8 1150,6 246,6 67,6 7,49E-05 6,55E-05 5,16E-05 3,96E ,0 1095,6 303,6 85,2 6,47E-05 5,17E-05 4,10E-05 3,05E ,7 604,3 146,1 33,6 5,38E-05 3,95E-05 3,22E-05 2,51E ,2 1380,8 299,0 82,9 6,48E-05 4,71E-05 3,74E-05 3,02E ,8 670,0 201,5 183,2 8,08E-05 3,25E-05 2,79E-05 2,53E ,6 1674,0 244,8 78,5 4,21E-05 3,55E-05 2,93E-05 1,98E ,6 650,7 376,8 72,4 6,37E-05 4,69E-05 3,85E-05 2,74E ,3 1102,9 416,9 87,7 6,48E-05 4,89E-05 4,01E-05 3,04E ,2 877,0 314,9 55,9 5,83E-05 3,98E-05 3,27E-05 2,44E ,9 8232,2 1431,9 518,7 1,43E-05 1,23E-05 9,76E-06 5,84E , ,7 2564,6 534,2 1,46E-05 1,20E-05 9,49E-06 5,79E , ,4 2559,7 498,6 1,64E-05 1,28E-05 1,01E-05 6,24E , ,8 2837,2 790,8 1,66E-05 1,25E-05 9,24E-06 7,06E , ,9 3955,0 457,0 1,69E-05 1,18E-05 9,18E-06 6,54E , ,7 2400,6 631,9 1,22E-05 1,03E-05 8,18E-06 4,62E , ,7 1840,3 2654,5 1,47E-05 1,04E-05 8,10E-06 5,58E , ,1 2866,7 2710,5 1,40E-05 9,70E-06 7,35E-06 5,12E , ,5 3005,1 797,5 1,31E-05 9,50E-06 7,13E-06 4,90E , ,7 2934,7 681,1 1,32E-05 8,94E-06 6,76E-06 4,73E

12 Table 7: Problem Size: x10-gen # of Equations # of Non-Zero Elements # of Real Variables # of Binary Variables 1bin 3bin Prop1 Prop2 1bin 3bin Prop1 Prop2 1bin 3bin Prop1 Prop2 1bin 3bin Prop1 Prop Table 8: Computational Performance: x10-gen Time (s Gap (p.u. Nodes Iterations 1bin 3bin Prop1 Prop2 1bin 3bin Prop1 Prop2 1bin 3bin Prop1 Prop2 1bin 3bin Prop1 Prop ,7 69,7 54,3 41,5 6,26E-06 5,69E-06 4,82E-06 3,52E ,4 332,0 127,3 44,1 5,86E-06 4,81E-06 3,69E-06 2,86E ,0 285,0 101,5 93,4 5,87E-06 4,54E-06 3,49E-06 2,73E ,3 169,3 133,9 64,5 4,37E-06 2,96E-06 2,11E-06 1,74E ,3 380,2 209,1 117,5 5,72E-06 4,00E-06 2,68E-06 2,30E ,1 186,7 208,8 67,5 1,26E-05 2,00E-06 1,56E-06 7,32E ,1 277,2 141,2 127,3 3,63E-06 2,94E-06 2,51E-06 1,81E ,0 270,6 165,9 85,2 5,62E-06 4,06E-06 2,99E-06 2,14E ,3 320,7 237,1 132,0 5,72E-06 4,14E-06 2,92E-06 2,28E ,3 425,6 238,2 117,2 4,75E-06 2,97E-06 1,88E-06 1,35E ,3 3793,1 2122,7 435,8 1,39E-06 1,17E-06 1,01E-06 7,25E ,4 6027,3 766,6 595,8 1,40E-06 1,11E-06 8,88E-07 6,29E ,5 9844,1 936,1 507,2 1,61E-06 1,26E-06 9,87E-07 6,93E ,0 8993,6 1140,2 1115,5 1,00E+00 1,00E+00 9,04E-07 7,78E ,1 4570,3 1399,9 725,2 1,61E-06 1,12E-06 7,63E-07 5,87E ,7 7708,8 2967,6 513,4 1,22E-06 1,00E-06 8,42E-07 5,88E , ,4 1427,7 1027,2 1,46E-06 1,00E+00 7,25E-07 5,63E ,6 4778,9 1223,2 790,0 1,00E+00 9,73E-07 6,47E-07 5,24E , ,7 1055,9 595,8 1,00E+00 9,38E-07 6,70E-07 5,33E ,6 7428,0 1549,4 705,1 1,00E+00 1,00E+00 5,83E-07 4,69E

13 Table 9: Problem Size: x7-day # of Equations # of Non-Zero Elements # of Real Variables # of Binary Variables 1bin 3bin 1bin 3bin 1bin 3bin 1bin 3bin Table 10: Computational Performance: x7-day Time (s Optimaliy Gap (p.u. Nodes Iterations 1bin 3bin 1bin 3bin 1bin 3bin 1bin 3bin ,907 51,2 8,79E-05 8,10E ,263 77,657 6,81E-05 5,69E ,096 82,353 6,27E-05 4,84E ,469 66,113 4,92E-05 3,45E ,68 130,776 5,74E-05 3,88E ,671 90,387 7,81E-05 3,04E ,16 85,473 3,78E-05 3,09E , ,85 5,95E-05 4,13E , ,609 5,72E-05 4,04E ,215 93,257 5,43E-05 3,57E , ,878 1,67E-05 1,44E , ,436 1,54E-05 1,25E , ,755 1,44E-05 1,05E , ,157 1,43E-05 9,85E , ,475 1,58E-05 1,05E , ,363 1,13E-05 9,13E , ,102 1,39E-05 9,49E , ,257 1,27E-05 8,09E , ,26 1,15E-05 7,65E , ,594 1,14E-05 6,95E

14 Table 11: Problem Size: x10-gen # of Equations # of Non-Zero Elements # of Real Variables # of Binary Variables 1bin 3bin 1bin 3bin 1bin 3bin 1bin 3bin Table 12: Computational Performance: x10-gen Time (s Optimaliy Gap (p.u. Nodes Iterations 1bin 3bin 1bin 3bin 1bin 3bin 1bin 3bin ,296 43,415 5,49E-06 5,01E , ,967 4,80E-06 3,72E ,952 58,921 4,80E-06 3,43E ,79 58,5 3,46E-06 2,10E , ,683 4,39E-06 2,60E , ,932 1,19E-05 1,43E ,338 97,423 3,12E-06 2,48E , ,429 4,73E-06 3,02E , ,382 4,44E-06 2,76E , ,561 3,74E-06 1,85E , ,657 1,22E-06 1,01E , ,409 1,17E-06 8,69E , ,867 1,33E-06 9,58E , ,182 2,16E-06 1,71E , ,202 1,25E-06 7,37E , ,609 1,05E-06 8,28E , ,813 1,14E-06 7,07E , ,479 1,09E-06 6,31E , ,611 1,03E-06 6,59E , ,628 9,96E-07 5,69E

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