"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",
