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uc: Add quadratic terms

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Alinson S. Xavier
2025-09-23 11:39:39 -05:00
parent 4eeb1c1ab3
commit ca05429203
2 changed files with 72 additions and 49 deletions
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docs/guide/problems.ipynb
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@@ -1221,7 +1221,6 @@
"id": "7048d771",
"metadata": {},
"source": [
"\n",
"<div class=\"alert alert-info\">\n",
"Note\n",
"\n",
@@ -1230,7 +1229,7 @@
"\n",
"### Formulation\n",
"\n",
"Let $T$ be the number of time steps, $G$ be the number of generation units, and let $D_t$ be the power demand (in MW) at time $t$. For each generating unit $g$, let $P^\\max_g$ and $P^\\min_g$ be the maximum and minimum amount of power the unit is able to produce when switched on; let $L_g$ and $l_g$ be the minimum up- and down-time for unit $g$; let $C^\\text{fixed}$ be the cost to keep unit $g$ on for one time step, regardless of its power output level; let $C^\\text{start}$ be the cost to switch unit $g$ on; and let $C^\\text{var}$ be the cost for generator $g$ to produce 1 MW of power. In this formulation, we assume linear production costs. For each generator $g$ and time $t$, let $x_{gt}$ be a binary variable which equals one if unit $g$ is on at time $t$, let $w_{gt}$ be a binary variable which equals one if unit $g$ switches from being off at time $t-1$ to being on at time $t$, and let $p_{gt}$ be a continuous variable which indicates the amount of power generated. The formulation is given by:"
"Let $T$ be the number of time steps, $G$ be the number of generation units, and let $D_t$ be the power demand (in MW) at time $t$. For each generating unit $g$, let $P^\\max_g$ and $P^\\min_g$ be the maximum and minimum amount of power the unit is able to produce when switched on; let $L_g$ and $l_g$ be the minimum up- and down-time for unit $g$; let $C^\\text{fixed}$ be the cost to keep unit $g$ on for one time step, regardless of its power output level; let $C^\\text{start}$ be the cost to switch unit $g$ on; let $C^\\text{prod-lin}$ be the linear cost coefficient for generator $g$ to produce 1 MW of power; and let $C^\\text{prod-quad}$ be the quadratic cost coefficient for unit $g$. For each generator $g$ and time $t$, let $x_{gt}$ be a binary variable which equals one if unit $g$ is on at time $t$, let $w_{gt}$ be a binary variable which equals one if unit $g$ switches from being off at time $t-1$ to being on at time $t$, and let $p_{gt}$ be a continuous variable which indicates the amount of power generated. The formulation is given by:"
]
},
{
@@ -1238,14 +1237,14 @@
"id": "bec5ee1c",
"metadata": {},
"source": [
"\n",
"$$\n",
"\\begin{align*}\n",
"\\text{minimize} \\;\\;\\;\n",
" & \\sum_{t=1}^T \\sum_{g=1}^G \\left(\n",
" x_{gt} C^\\text{fixed}_g\n",
" + w_{gt} C^\\text{start}_g\n",
" + p_{gt} C^\\text{var}_g\n",
" + p_{gt} C^\\text{prod-lin}_g\n",
" + p_{gt}^2 C^\\text{prod-quad}_g\n",
" \\right)\n",
" \\\\\n",
"\\text{such that} \\;\\;\\;\n",
@@ -1287,12 +1286,11 @@
"id": "01bed9fc",
"metadata": {},
"source": [
"\n",
"### Random instance generator\n",
"\n",
"The class `UnitCommitmentGenerator` can be used to generate random instances of this problem.\n",
"\n",
"First, the user-provided probability distributions `n_units` and `n_periods` are sampled to determine the number of generating units and the number of time steps, respectively. Then, for each unit, the probabilities `max_power` and `min_power` are sampled to determine the unit's maximum and minimum power output. To make it easier to generate valid ranges, `min_power` is not specified as the absolute power level in MW, but rather as a multiplier of `max_power`; for example, if `max_power` samples to 100 and `min_power` samples to 0.5, then the unit's power range is set to `[50,100]`. Then, the distributions `cost_startup`, `cost_prod` and `cost_fixed` are sampled to determine the unit's startup, variable and fixed costs, while the distributions `min_uptime` and `min_downtime` are sampled to determine its minimum up/down-time.\n",
"First, the user-provided probability distributions `n_units` and `n_periods` are sampled to determine the number of generating units and the number of time steps, respectively. Then, for each unit, the probabilities `max_power` and `min_power` are sampled to determine the unit's maximum and minimum power output. To make it easier to generate valid ranges, `min_power` is not specified as the absolute power level in MW, but rather as a multiplier of `max_power`; for example, if `max_power` samples to 100 and `min_power` samples to 0.5, then the unit's power range is set to `[50,100]`. Then, the distributions `cost_startup`, `cost_prod`, `cost_prod_quad` and `cost_fixed` are sampled to determine the unit's startup, linear variable, quadratic variable, and fixed costs, while the distributions `min_uptime` and `min_downtime` are sampled to determine its minimum up/down-time.\n",
"\n",
"After parameters for the units have been generated, the class then generates a periodic demand curve, with a peak every 12 time steps, in the range $(0.4C, 0.8C)$, where $C$ is the sum of all units' maximum power output. Finally, all costs and demand values are perturbed by random scaling factors independently sampled from the distributions `cost_jitter` and `demand_jitter`, respectively.\n",
"\n",
@@ -1309,7 +1307,7 @@
},
{
"cell_type": "code",
"execution_count": 8,
"execution_count": 2,
"id": "6217da7c",
"metadata": {
"ExecuteTime": {
@@ -1333,78 +1331,94 @@
"min_power[0] [117.79 245.85 271.85 207.7 81.38]\n",
"cost_startup[0] [3042.42 5247.56 4319.45 2912.29 6118.53]\n",
"cost_prod[0] [ 6.97 14.61 18.32 22.8 39.26]\n",
"cost_fixed[0] [199.67 514.23 592.41 46.45 607.54]\n",
"cost_prod_quad[0] [0.02 0.0514 0.0592 0.0046 0.0608]\n",
"cost_fixed[0] [170.52 65.05 948.89 965.63 808.4 ]\n",
"demand[0]\n",
" [ 905.06 915.41 1166.52 1212.29 1127.81 953.52 905.06 796.21 783.78\n",
" 866.23 768.62 899.59 905.06 946.23 1087.61 1004.24 1048.36 992.03\n",
" 905.06 750.82 691.48 606.15 658.5 809.95]\n",
" [ 869.31 897.58 1212.29 1124.08 930.48 1012.62 869.31 606.15]\n",
"\n",
"min_power[1] [117.79 245.85 271.85 207.7 81.38]\n",
"max_power[1] [218.54 477.82 379.4 319.4 120.21]\n",
"min_uptime[1] [7 6 3 5 7]\n",
"min_downtime[1] [7 3 5 6 2]\n",
"min_power[1] [117.79 245.85 271.85 207.7 81.38]\n",
"cost_startup[1] [2458.08 6200.26 4585.74 2666.05 4783.34]\n",
"cost_prod[1] [ 6.31 13.33 20.42 24.37 46.86]\n",
"cost_fixed[1] [196.9 416.42 655.57 52.51 626.15]\n",
"cost_startup[1] [3710.99 6283.5 4530.89 3526.6 4859.62]\n",
"cost_prod[1] [ 5.91 11.29 16.72 21.53 34.77]\n",
"cost_prod_quad[1] [0.0233 0.0477 0.0527 0.0047 0.0499]\n",
"cost_fixed[1] [ 196.29 51.21 1179.89 1097.07 686.62]\n",
"demand[1]\n",
" [ 981.42 840.07 1095.59 1102.03 1088.41 932.29 863.67 848.56 761.33\n",
" 828.28 775.18 834.99 959.76 865.72 1193.52 1058.92 985.19 893.92\n",
" 962.16 781.88 723.15 639.04 602.4 787.02]\n",
" [ 827.37 926.76 1166.64 1128.59 939.17 948.8 950.95 639.5 ]\n",
"\n",
"Gurobi Optimizer version 12.0.3 build v12.0.3rc0 (linux64 - \"Ubuntu 24.04.3 LTS\")\n",
"\n",
"CPU model: AMD Ryzen 9 3950X 16-Core Processor, instruction set [SSE2|AVX|AVX2]\n",
"Thread count: 16 physical cores, 32 logical processors, using up to 32 threads\n",
"\n",
"Optimize a model with 578 rows, 360 columns and 2128 nonzeros\n",
"Model fingerprint: 0x4dc1c661\n",
"Variable types: 120 continuous, 240 integer (240 binary)\n",
"Optimize a model with 162 rows, 120 columns and 512 nonzeros\n",
"Model fingerprint: 0x1e3651da\n",
"Model has 40 quadratic objective terms\n",
"Variable types: 40 continuous, 80 integer (80 binary)\n",
"Coefficient statistics:\n",
" Matrix range [1e+00, 5e+02]\n",
" Objective range [7e+00, 6e+03]\n",
" QObjective range [9e-03, 1e-01]\n",
" Bounds range [1e+00, 1e+00]\n",
" RHS range [1e+00, 1e+03]\n",
"Presolve removed 341 rows and 133 columns\n",
"Presolve time: 0.01s\n",
"Presolved: 237 rows, 227 columns, 725 nonzeros\n",
"Variable types: 114 continuous, 113 integer (113 binary)\n",
"Found heuristic solution: objective 475243.89360\n",
"Found heuristic solution: objective 282371.35206\n",
"Presolve removed 61 rows and 40 columns\n",
"Presolve time: 0.00s\n",
"Presolved: 137 rows, 98 columns, 362 nonzeros\n",
"Presolved model has 40 quadratic objective terms\n",
"Variable types: 58 continuous, 40 integer (40 binary)\n",
"\n",
"Root relaxation: objective 3.361348e+05, 96 iterations, 0.00 seconds (0.00 work units)\n",
"Root relaxation: objective 1.995341e+05, 126 iterations, 0.00 seconds (0.00 work units)\n",
"\n",
" Nodes | Current Node | Objective Bounds | Work\n",
" Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time\n",
"\n",
" 0 0 336134.820 0 18 475243.894 336134.820 29.3% - 0s\n",
"H 0 0 471441.37480 336134.820 28.7% - 0s\n",
"H 0 0 410679.27820 336134.820 18.2% - 0s\n",
"H 0 0 391706.31610 336134.820 14.2% - 0s\n",
"H 0 0 374515.31390 336134.820 10.2% - 0s\n",
"H 0 0 369369.87450 336134.820 9.00% - 0s\n",
"H 0 0 368600.14450 344055.048 6.66% - 0s\n",
"H 0 0 368180.65796 364657.488 0.96% - 0s\n",
"H 0 0 364721.76610 364657.488 0.02% - 0s\n",
" 0 0 364721.766 0 6 364721.766 364721.766 0.00% - 0s\n",
" 0 0 199534.058 0 8 282371.352 199534.058 29.3% - 0s\n",
"H 0 0 212978.80635 199534.058 6.31% - 0s\n",
"H 0 0 208079.69920 199534.058 4.11% - 0s\n",
" 0 0 203097.772 0 16 208079.699 203097.772 2.39% - 0s\n",
" 0 0 203097.772 0 4 208079.699 203097.772 2.39% - 0s\n",
" 0 0 203097.772 0 4 208079.699 203097.772 2.39% - 0s\n",
" 0 0 203097.772 0 3 208079.699 203097.772 2.39% - 0s\n",
" 0 0 203097.772 0 3 208079.699 203097.772 2.39% - 0s\n",
" 0 0 203097.772 0 3 208079.699 203097.772 2.39% - 0s\n",
" 0 0 205275.299 0 - 208079.699 205275.299 1.35% - 0s\n",
" 0 0 205777.846 0 2 208079.699 205777.846 1.11% - 0s\n",
" 0 0 205789.407 0 - 208079.699 205789.407 1.10% - 0s\n",
" 0 0 postponed 0 208079.699 205789.407 1.10% - 0s\n",
" 0 0 postponed 0 208079.699 205789.408 1.10% - 0s\n",
" 0 0 205789.408 0 4 208079.699 205789.408 1.10% - 0s\n",
" 0 0 205789.408 0 4 208079.699 205789.408 1.10% - 0s\n",
" 0 0 205789.408 0 1 208079.699 205789.408 1.10% - 0s\n",
" 0 0 205789.408 0 - 208079.699 205789.408 1.10% - 0s\n",
" 0 0 205789.408 0 - 208079.699 205789.408 1.10% - 0s\n",
" 0 0 205789.408 0 - 208079.699 205789.408 1.10% - 0s\n",
" 0 0 205789.408 0 - 208079.699 205789.408 1.10% - 0s\n",
" 0 0 205789.408 0 2 208079.699 205789.408 1.10% - 0s\n",
"H 0 0 207525.63560 205789.408 0.84% - 0s\n",
" 0 0 205789.408 0 2 207525.636 205789.408 0.84% - 0s\n",
" 0 2 205789.408 0 2 207525.636 205789.408 0.84% - 0s\n",
"* 9 0 5 205789.40812 205789.408 0.00% 0.0 0s\n",
"\n",
"Cutting planes:\n",
" Gomory: 2\n",
" Cover: 7\n",
" Implied bound: 1\n",
" Clique: 19\n",
" MIR: 3\n",
" RLT: 10\n",
" Relax-and-lift: 1\n",
" Cover: 1\n",
" Implied bound: 6\n",
" MIR: 2\n",
" Flow cover: 2\n",
" RLT: 2\n",
"\n",
"Explored 1 nodes (181 simplex iterations) in 0.03 seconds (0.01 work units)\n",
"Explored 11 nodes (621 simplex iterations) in 0.32 seconds (0.19 work units)\n",
"Thread count was 32 (of 32 available processors)\n",
"\n",
"Solution count 7: 364722 368600 374515 ... 475244\n",
"Solution count 5: 205789 207526 208080 ... 282371\n",
"No other solutions better than 205789\n",
"\n",
"Optimal solution found (tolerance 1.00e-04)\n",
"Best objective 3.647217661000e+05, best bound 3.647217661000e+05, gap 0.0000%\n",
"Best objective 2.057894080889e+05, best bound 2.057894081187e+05, gap 0.0000%\n",
"\n",
"User-callback calls 816, time in user-callback 0.00 sec\n"
"User-callback calls 647, time in user-callback 0.00 sec\n"
]
}
],
@@ -1418,14 +1432,15 @@
"random.seed(42)\n",
"np.random.seed(42)\n",
"\n",
"# Generate a random instance with 5 generators and 24 time steps\n",
"# Generate a random instance with 5 generators and 8 time steps\n",
"data = UnitCommitmentGenerator(\n",
" n_units=randint(low=5, high=6),\n",
" n_periods=randint(low=24, high=25),\n",
" n_periods=randint(low=8, high=9),\n",
" max_power=uniform(loc=50, scale=450),\n",
" min_power=uniform(loc=0.5, scale=0.25),\n",
" cost_startup=uniform(loc=0, scale=10_000),\n",
" cost_prod=uniform(loc=0, scale=50),\n",
" cost_prod_quad=uniform(loc=0, scale=0.1),\n",
" cost_fixed=uniform(loc=0, scale=1_000),\n",
" min_uptime=randint(low=2, high=8),\n",
" min_downtime=randint(low=2, high=8),\n",
@@ -1443,6 +1458,7 @@
" print(f\"min_power[{i}]\", data[i].min_power)\n",
" print(f\"cost_startup[{i}]\", data[i].cost_startup)\n",
" print(f\"cost_prod[{i}]\", data[i].cost_prod)\n",
" print(f\"cost_prod_quad[{i}]\", data[i].cost_prod_quad)\n",
" print(f\"cost_fixed[{i}]\", data[i].cost_fixed)\n",
" print(f\"demand[{i}]\\n\", data[i].demand)\n",
" print()\n",