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 E-mail: german.morales@iit.upcomillas.es 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 1 1.1 Nomenclature.......................................... 2 1.2 One-binary UC formulation [2]................................ 3 1.3 Three-binary UC formulation [3]............................... 4 1.4 Tight & Compact UC Formulation [1]............................ 5 1.5 Implemented Formulations................................... 6 2 Numerical Results 6 2.1 Comparing 1bin with 3bin................................... 6 2.2 Convergence evolution..................................... 7 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-12-072, last update 2012-10-10 1
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. 1.1.1 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. 1.1.2 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 [$]. P g P g RD g RU g SD g SU g T D g T U g T SU g,l Maximum power output of unit g [MW]. Minimum power output of unit g [MW]. Down ramp limit of unit g [MW/h]. Up ramp limit of unit g [MW/h]. Shutdown ramp capability of unit g [MW]. Startup ramp capability of unit g [MW]. Minimum down time of unit g [h]. Minimum up time of unit g [h]. Times defining the interval l limits, which are [ Tl SU, Tl+1 SU, of the stairwise startup cost function of unit g [h]. 2
1.1.3 Variables Positive and Continuous Variables c SD g,t Shutdown cost of unit g in period t [$]. 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. 1.2.1 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 (1 1.2.2 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
1.2.3 Shutdown Costs c SU g,t C SD g (u g,t 1 u g,t g, t (3 1.2.4 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 1.2.5 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 1.2.6 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 (12 1.2.7 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 (15 1.3 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 1.2. 4
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 (16 1.3.2 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 (17 1.3.3 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 ] (19 1.3.4 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 (20 1.3.5 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 (22 1.4 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 1.3.3. 1.4.1 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 (23 1.4.2 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
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 1.4.4 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 1.4.5 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 (30 1.5 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 1.4.2 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 1.4.2. 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
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 11-20. 2.2 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 01-10 36,4 70,2 116,7 91,7 54,8 Cases 11-20 19,3 71,1 1,8 135,9 40,5 Cases 01-20 26,5 70,6 14,4 111,6 47,1 Cases 01-10 37,5 56,6 69,7 348,3 65,7 Cases 11-20 17,8 68,2 192,5 1612,4 51,6 Cases 01-20 25,9 62,1 115,8 749,4 58,2 7
140 120 1bin 3bin 140 120 1bin 3bin 3 35 10 35 7 6 5 20 6 3 0h23 1h35 2h0 1h56 1h12 100 7 32 1 47 1 29 3 27 2 19 0h59 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 3 100 6.8e 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 [%] 80 60 Proportion [%] 80 60 40 40 20 20 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Case [#] 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Case [#] (a x7 days CPU time (b x7 days Integrality Gap 140 120 1bin 3bin 140 120 1bin 3bin 4 5 3 36 7 27 4 28 4 27 1h10 0h48 1h6 1h47 1h41 100 6 13 1 52 2 3 4 14 3 4 0h57 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 3 100 0.5e 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 [%] 80 60 Proportion [%] 80 60 40 40 20 20 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Case [#] 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 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 1-10 2,70 Cases 11-20 5,39 cases 1-20 3,82 8
Objective function [M $] 5.685 5.68 5.675 5.67 Prop2: Upper bound 3bin: Upper bound 1bin: Upper Bound Prop2: Lower bound 3bin: Lower bound 1bin: Lower Bound Optimality Gap [p.u.] 10 0 10 1 10 2 10 3 Prop2 3bin 1bin 5.665 10 0 10 1 10 2 10 3 CPU Time [s] 10 4 10 0 10 1 10 2 10 3 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. 2.2.1 10-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. 2.2.2 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
Objective function [M $] 1.1425 1.142 1.1415 1.141 1.1405 1.14 Prop2: Upper bound 3bin: Upper bound 1bin: Upper Bound Prop2: Lower bound 3bin: Lower bound 1bin: Lower Bound Optimality Gap [p.u.] 10 0 10 1 10 2 10 3 Prop2 3bin 1bin 1.1395 10 0 10 1 10 2 10 3 CPU Time [s] 10 4 10 0 10 1 10 2 10 3 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-00779-2012.R1, online preprint. [Online]. Available: http://www.iit.upcomillas.es/~aramos/papers/v3.3_tight_compact_uc.pdf [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. 1371 1378, 2006. [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. 39 46, Feb. 2012. [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. 251 272 272, Dec. 2000. [Online]. Available: http://dx.doi.org/10.1023/a:1019248506301 10
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 Prop2 01 103965 103617 51368 37284 739025 746855 245418 217138 18984 14280 9576 9576 4704 14112 23507 23507 02 130164 129718 64111 46674 931838 941896 307898 273094 23688 17808 11928 11928 5880 17640 29372 29372 03 155015 154527 80630 59002 1057263 1070901 373737 330905 29736 22344 14952 14952 7392 22176 36909 36909 04 152326 151876 82541 60410 1001914 1016098 372666 328830 30408 22848 15288 15288 7560 22680 37754 37754 05 172654 172114 89762 65283 1163474 1177978 415714 366854 33096 24864 16632 16632 8232 24696 41085 41085 06 162061 161621 91774 67464 1013243 1029927 403089 355241 33768 25368 16968 16968 8400 25200 41915 41915 07 176342 175810 93456 68475 1186086 1202210 427509 378153 34440 25872 17304 17304 8568 25704 42801 42801 08 169784 169296 93533 69056 1106658 1124018 418483 370639 34440 25872 17304 17304 8568 25704 42778 42778 09 179195 178653 95284 69968 1197099 1213891 435492 385636 35112 26376 17640 17640 8736 26208 43607 43607 10 177343 176853 99059 72905 1128725 1146833 438660 387300 36456 27384 18312 18312 9072 27216 45284 45284 11 453941 452537 241324 177616 3088889 3132199 1106534 981902 88872 66696 44520 44520 22176 66528 110799 110799 12 533219 531627 285238 208954 3551575 3601401 1299035 1148795 105000 78792 52584 52584 26208 78624 130889 130889 13 530359 528771 285255 209307 3523151 3573671 1296368 1147136 105000 78792 52584 52584 26208 78624 130868 130868 14 599762 597778 301187 218360 4173710 4220836 1428769 1263445 111048 83328 55608 55608 27720 83160 138370 138370 15 570002 568310 305348 223363 3762018 3814950 1388889 1227101 112392 84336 56280 56280 28056 84168 140045 140045 16 590987 589213 314416 230756 3969699 4025323 1436796 1272508 115752 86856 57960 57960 28896 86688 144339 144339 17 625763 623853 332653 243123 4174389 4231341 1522714 1345866 122472 91896 61320 61320 30576 91728 152634 152634 18 628285 626385 332650 242952 4190253 4246939 1524177 1346825 122472 91896 61320 61320 30576 91728 152633 152633 19 632650 630726 334455 244422 4240030 4297192 1534460 1356608 123144 92400 61656 61656 30744 92232 153502 153502 20 638696 636784 341861 249648 4223514 4281874 1556543 1374171 125832 94416 63000 63000 31416 94248 156834 156834 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 Prop2 01 416,9 422,5 112,7 43,4 8,50E-05 7,91E-05 6,56E-05 5,11E-05 499 790 620 570 580323 259595 107627 81375 02 641,8 1150,6 246,6 67,6 7,49E-05 6,55E-05 5,16E-05 3,96E-05 495 508 1048 493 770176 439386 294440 97012 03 749,0 1095,6 303,6 85,2 6,47E-05 5,17E-05 4,10E-05 3,05E-05 521 490 650 950 720958 306995 302165 118903 04 1393,7 604,3 146,1 33,6 5,38E-05 3,95E-05 3,22E-05 2,51E-05 2142 487 499 1 1187817 200172 158784 51056 05 805,2 1380,8 299,0 82,9 6,48E-05 4,71E-05 3,74E-05 3,02E-05 536 482 620 650 565851 322222 199697 118349 06 320,8 670,0 201,5 183,2 8,08E-05 3,25E-05 2,79E-05 2,53E-05 550 510 560 1983 364345 220930 194765 151903 07 1053,6 1674,0 244,8 78,5 4,21E-05 3,55E-05 2,93E-05 1,98E-05 570 620 710 452 894056 512861 207498 105286 08 778,6 650,7 376,8 72,4 6,37E-05 4,69E-05 3,85E-05 2,74E-05 475 534 526 550 685877 235348 273674 86749 09 1304,3 1102,9 416,9 87,7 6,48E-05 4,89E-05 4,01E-05 3,04E-05 501 475 740 530 724609 259676 320328 107435 10 609,2 877,0 314,9 55,9 5,83E-05 3,98E-05 3,27E-05 2,44E-05 530 486 496 530 464102 220025 230454 79371 11 5145,9 8232,2 1431,9 518,7 1,43E-05 1,23E-05 9,76E-06 5,84E-06 518 517 463 456 1604092 912925 369498 263175 12 24212,6 18376,7 2564,6 534,2 1,46E-05 1,20E-05 9,49E-06 5,79E-06 3863 517 740 527 6136091 1464725 585503 244173 13 11833,6 27794,4 2559,7 498,6 1,64E-05 1,28E-05 1,01E-05 6,24E-06 487 7160 520 508 1819344 4116774 578681 264027 14 34503,0 31220,8 2837,2 790,8 1,66E-05 1,25E-05 9,24E-06 7,06E-06 3395 2005 485 519 5624960 3320459 552201 392813 15 19619,6 18611,9 3955,0 457,0 1,69E-05 1,18E-05 9,18E-06 6,54E-06 498 1899 488 476 3073186 2024014 726095 234892 16 12195,6 32819,7 2400,6 631,9 1,22E-05 1,03E-05 8,18E-06 4,62E-06 485 3898 560 493 2039868 3855190 550904 242635 17 14861,3 32065,7 1840,3 2654,5 1,47E-05 1,04E-05 8,10E-06 5,58E-06 488 692 590 3327 2704214 2111573 1010762 664465 18 36002,4 29123,1 2866,7 2710,5 1,40E-05 9,70E-06 7,35E-06 5,12E-06 800 2330 530 4379 4607583 2602570 631353 809198 19 15483,1 25001,5 3005,1 797,5 1,31E-05 9,50E-06 7,13E-06 4,90E-06 477 500 467 518 2619471 1305841 957259 352670 20 15495,2 25240,7 2934,7 681,1 1,32E-05 8,94E-06 6,76E-06 4,73E-06 511 661 550 460 2247429 1788217 561277 306349 11
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 Prop2 01 147849 144369 66839 46959 923993 921653 302163 262939 26904 20184 13464 13464 6720 20160 33470 33470 02 185199 180739 83389 58779 1164443 1161343 378323 330019 33624 25224 16824 16824 8400 25200 41720 41720 03 220589 215709 106019 75499 1331493 1332513 466473 407009 42264 31704 21144 21144 10560 31680 52290 52290 04 216739 212239 109289 78059 1267603 1271203 470163 409299 43224 32424 21624 21624 10800 32400 53540 53540 05 245779 240379 118139 83589 1470563 1471603 520483 452579 47064 35304 23544 23544 11760 35280 58050 58050 06 230569 226169 122419 88119 1294493 1302933 515193 448769 48024 36024 24024 24024 12000 36000 59150 59150 07 250979 245659 123399 88149 1495083 1497923 536193 467649 48984 36744 24504 24504 12240 36720 60810 60810 08 241559 236679 124169 89639 1402083 1408643 529453 463069 48984 36744 24504 24504 12240 36720 60580 60580 09 255029 249609 125839 90119 1513053 1516813 546903 477679 49944 37464 24984 24984 12480 37440 61670 61670 10 252349 247449 131909 95009 1436993 1445273 558423 487119 51864 38904 25944 25944 12960 38880 64040 64040 11 646649 632609 319039 229159 3882473 3889333 1385003 1211699 126744 95064 63384 63384 31680 95040 157590 157590 12 759749 743829 378019 270379 4487413 4498953 1635293 1426229 149784 112344 74904 74904 37440 112320 185690 185690 13 755629 739749 378189 271029 4453733 4466453 1633103 1425479 149784 112344 74904 74904 37440 112320 185480 185480 14 854939 835099 394949 278039 5251163 5244343 1773433 1543129 158424 118824 79224 79224 39600 118800 195700 195700 15 812219 795299 404879 289189 4767363 4781163 1751913 1526729 160344 120264 80184 80184 40080 120240 198050 198050 16 842069 824329 416359 298319 5006733 5018653 1804743 1576159 165144 123864 82584 82584 41280 123840 204990 204990 17 891749 872649 440329 313989 5279553 5290353 1915123 1668939 174744 131064 87384 87384 43680 131040 215940 215940 18 895369 876369 440299 313719 5301393 5312413 1916793 1669889 174744 131064 87384 87384 43680 131040 215930 215930 19 901579 882339 442509 315459 5357083 5367103 1927463 1679879 175704 131784 87864 87864 43920 131760 217420 217420 20 910199 891079 453209 323079 5348883 5362243 1962533 1708629 179544 134664 89784 89784 44880 134640 221940 221940 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 Prop2 01 239,7 69,7 54,3 41,5 6,26E-06 5,69E-06 4,82E-06 3,52E-06 10 200 520 486 316956 72168 64370 87583 02 921,4 332,0 127,3 44,1 5,86E-06 4,81E-06 3,69E-06 2,86E-06 400 489 190 310 706343 292994 121537 52965 03 499,0 285,0 101,5 93,4 5,87E-06 4,54E-06 3,49E-06 2,73E-06 459 504 494 506 211488 136257 78378 98816 04 499,3 169,3 133,9 64,5 4,37E-06 2,96E-06 2,11E-06 1,74E-06 494 469 360 100 164890 106445 139080 96677 05 1032,3 380,2 209,1 117,5 5,72E-06 4,00E-06 2,68E-06 2,30E-06 500 514 510 570 248045 175455 159007 119753 06 346,1 186,7 208,8 67,5 1,26E-05 2,00E-06 1,56E-06 7,32E-07 581 469 590 510 198980 142294 211854 81462 07 988,1 277,2 141,2 127,3 3,63E-06 2,94E-06 2,51E-06 1,81E-06 377 510 210 526 517339 190861 149133 149592 08 443,0 270,6 165,9 85,2 5,62E-06 4,06E-06 2,99E-06 2,14E-06 524 620 494 430 175821 209557 142942 88263 09 2984,3 320,7 237,1 132,0 5,72E-06 4,14E-06 2,92E-06 2,28E-06 750 499 519 496 503778 159668 140322 131226 10 785,3 425,6 238,2 117,2 4,75E-06 2,97E-06 1,88E-06 1,35E-06 474 474 499 531 196718 223781 164785 155122 11 4483,3 3793,1 2122,7 435,8 1,39E-06 1,17E-06 1,01E-06 7,25E-07 100 529 300 516 492856 1590376 648622 178918 12 7292,4 6027,3 766,6 595,8 1,40E-06 1,11E-06 8,88E-07 6,29E-07 40 479 489 476 1083370 923382 234876 210086 13 6538,5 9844,1 936,1 507,2 1,61E-06 1,26E-06 9,87E-07 6,93E-07 479 1018 494 380 592088 1032660 262054 241089 14 26631,0 8993,6 1140,2 1115,5 1,00E+00 1,00E+00 9,04E-07 7,78E-07 539 550 1 511 2493346 773759 282934 399203 15 31509,1 4570,3 1399,9 725,2 1,61E-06 1,12E-06 7,63E-07 5,87E-07 479 459 519 486 1727931 646113 294070 311273 16 14532,7 7708,8 2967,6 513,4 1,22E-06 1,00E-06 8,42E-07 5,88E-07 110 509 1192 531 2011434 989274 428585 213496 17 26809,0 10377,4 1427,7 1027,2 1,46E-06 1,00E+00 7,25E-07 5,63E-07 504 504 529 220 4211399 937355 340078 373192 18 32271,6 4778,9 1223,2 790,0 1,00E+00 9,73E-07 6,47E-07 5,24E-07 474 514 504 536 2131791 588581 318800 299103 19 14584,7 16819,7 1055,9 595,8 1,00E+00 9,38E-07 6,70E-07 5,33E-07 519 470 489 501 1050736 1514756 270816 215751 20 36003,6 7428,0 1549,4 705,1 1,00E+00 1,00E+00 5,83E-07 4,69E-07 485 489 509 506 6638676 981330 327880 252598 12
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 01 47349 42297 203115 196861 18984 9576 4704 14112 02 59100 52774 255567 248020 23688 11928 5880 17640 03 74207 66327 309289 300795 29736 14952 7392 22176 04 75886 67876 308269 299818 30408 15288 7560 22680 05 82606 73834 343883 333740 33096 16632 8232 24696 06 84277 75437 332699 324233 33768 16968 8400 25200 07 85958 76858 352873 343344 34440 17304 8568 25704 08 85952 76896 345683 337390 34440 17304 8568 25704 09 87635 78357 359757 350393 35112 17640 8736 26208 10 90991 81429 361667 352613 36456 18312 9072 27216 11 221933 198353 916803 893717 88872 44520 22176 66528 12 262235 234435 1072625 1043983 105000 52584 26208 78624 13 262231 234435 1071983 1044035 105000 52584 26208 78624 14 277370 247666 1183837 1147968 111048 55608 27720 83160 15 280706 250958 1146377 1115308 112392 56280 28056 84168 16 289091 258421 1185557 1154665 115752 57960 28896 86688 17 305891 273405 1260079 1225485 122472 61320 30576 91728 18 305893 273417 1258453 1223593 122472 61320 30576 91728 19 307570 274902 1267511 1232624 123144 61656 30744 92232 20 314288 280960 1286435 1250734 125832 63000 31416 94248 Table 10: Computational Performance: x7-day Time (s Optimaliy Gap (p.u. Nodes Iterations 1bin 3bin 1bin 3bin 1bin 3bin 1bin 3bin 01 214,907 51,2 8,79E-05 8,10E-05 971 570 209721 84599 02 452,263 77,657 6,81E-05 5,69E-05 1162 680 349892 100941 03 635,096 82,353 6,27E-05 4,84E-05 1145 472 407881 119523 04 107,469 66,113 4,92E-05 3,45E-05 590 590 140880 96435 05 425,68 130,776 5,74E-05 3,88E-05 461 700 292751 132481 06 88,671 90,387 7,81E-05 3,04E-05 550 550 109585 121305 07 320,16 85,473 3,78E-05 3,09E-05 476 760 223801 124655 08 206,904 132,85 5,95E-05 4,13E-05 531 560 231279 176555 09 363,436 127,609 5,72E-05 4,04E-05 525 570 267005 160324 10 139,215 93,257 5,43E-05 3,57E-05 478 513 135731 111148 11 1387,519 350,878 1,67E-05 1,44E-05 503 487 428882 207764 12 3534,249 509,436 1,54E-05 1,25E-05 512 500 690915 302591 13 5724,222 593,755 1,44E-05 1,05E-05 511 536 1067243 314400 14 6467,334 949,157 1,43E-05 9,85E-06 501 475 1011811 285976 15 7201,131 2120,475 1,58E-05 1,05E-05 1008 4178 1827581 665357 16 4247,798 614,363 1,13E-05 9,13E-06 484 488 709605 284423 17 6984,991 3382,102 1,39E-05 9,49E-06 483 3175 1161753 772616 18 4558,224 788,257 1,27E-05 8,09E-06 505 490 872746 340794 19 4322,258 630,26 1,15E-05 7,65E-06 529 464 698044 295562 20 2646,276 665,594 1,14E-05 6,95E-06 496 484 783289 324417 13
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 01 66969 56769 275133 252913 26904 13464 6720 20160 02 83679 70819 346053 318103 33624 16824 8400 25200 03 105149 89709 421033 390813 42264 21144 10560 31680 04 107539 92239 420913 392563 43224 21624 10800 32400 05 117139 99979 468893 435143 47064 23544 11760 35280 06 119449 103049 456413 429353 48024 24024 12000 36000 07 121859 104299 481033 447663 48984 24504 12240 36720 08 121799 104679 472493 442843 48984 24504 12240 36720 09 124229 106329 490833 457673 49944 24984 12480 37440 10 128989 111129 495533 465473 51864 25944 12960 38880 11 315209 269489 1248813 1161953 126744 63384 31680 95040 12 372629 319269 1464953 1365733 149784 74904 37440 112320 13 372589 319269 1464293 1366253 149784 74904 37440 112320 14 394379 334939 1610833 1486863 158424 79224 39600 118800 15 398939 341939 1567433 1462663 160344 80184 40080 120240 16 410789 351769 1617953 1507753 165144 82584 41280 123840 17 434789 372009 1720933 1602513 174744 87384 43680 131040 18 434809 372129 1719073 1600873 174744 87384 43680 131040 19 437179 374019 1730453 1610543 175704 87864 43920 131760 20 446759 382759 1758413 1639003 179544 89784 44880 134640 Table 12: Computational Performance: x10-gen Time (s Optimaliy Gap (p.u. Nodes Iterations 1bin 3bin 1bin 3bin 1bin 3bin 1bin 3bin 01 245,296 43,415 5,49E-06 5,01E-06 347 380 166258 54603 02 372,904 108,967 4,80E-06 3,72E-06 50 494 185825 114479 03 215,952 58,921 4,80E-06 3,43E-06 70 240 112119 66534 04 111,79 58,5 3,46E-06 2,10E-06 80 504 103732 64379 05 446,912 100,683 4,39E-06 2,60E-06 519 530 210114 95127 06 122,773 110,932 1,19E-05 1,43E-06 494 580 119277 139571 07 268,338 97,423 3,12E-06 2,48E-06 50 499 132549 83188 08 254,032 103,429 4,73E-06 3,02E-06 400 489 122106 101265 09 266,996 103,382 4,44E-06 2,76E-06 20 440 118996 90109 10 184,284 118,561 3,74E-06 1,85E-06 140 489 121992 117130 11 4203,509 498,657 1,22E-06 1,01E-06 50 534 498824 226930 12 3413,832 654,409 1,17E-06 8,69E-07 10 479 465294 198208 13 2850,372 689,867 1,33E-06 9,58E-07 1 534 390823 229122 14 7202,347 1041,182 2,16E-06 1,71E-06 4 524 661902 259373 15 3971,77 1097,202 1,25E-06 7,37E-07 499 630 489377 369344 16 4568,567 969,609 1,05E-06 8,28E-07 1 590 515728 367067 17 6444,526 948,813 1,14E-06 7,07E-07 524 494 554695 262605 18 5515,946 952,479 1,09E-06 6,31E-07 484 474 531537 311187 19 6084,554 821,611 1,03E-06 6,59E-07 499 464 528564 223187 20 4453,673 898,628 9,96E-07 5,69E-07 10 519 526509 252712 14
[5] L. L. Garver, Power generation scheduling by integer programming-development of theory, Power Apparatus and Systems, Part III. Transactions of the American Institute of Electrical Engineers, vol. 81, no. 3, pp. 730 734, Apr. 1962. [6] D. Rajan and S. Takriti, Minimum Up/Down polytopes of the unit commitment problem with start-up costs, IBM, Research Report RC23628, Jun. 2005. [Online]. Available: http://domino.research.ibm.com/library/cyberdig.nsf/ 1e4115aea78b6e7c85256b360066f0d4/cdcb02a7c809d89e8525702300502ac0?OpenDocument [7] 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. 1098 1104, 2000. [8] R. Bixby, M. Fenelon, Z. Gu, E. Rothberg, and R. Wunderling, MIP: theory and practice closing the gap, in System Modelling and Optimization: Methods, Theory and Applications, M. J. D. Powell and S. Scholtes, Eds. Boston: Kluwer Academic Publishers, 2000, vol. 174, p. 19 49. [9] 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. 37 41, Jan. 2007. [10] J. Hooker, Logic-Based Methods for Optimization: Combining Optimization and Constraint Satisfaction, 1st ed. Wiley-Interscience, May 2000. [11] E. L. Johnson, G. L. Nemhauser, and M. W. Savelsbergh, Progress in linear programming-based algorithms for integer programming: An exposition, INFORMS Journal on Computing, vol. 12, no. 1, pp. 2 23, 2000. [12] The GAMS development corporation website, 2012, www.gams.com. [13] L. Wu, M. Shahidehpour, and T. Li, Stochastic security-constrained unit commitment, Power Systems, IEEE Transactions on, vol. 22, no. 2, pp. 800 811, May 2007. 15