Least-cost dispatch

The smallest file that is a whole model: generators with a capacity, an hourly load to meet, and a cost to minimise. It is the model on the home page.

The where: on dispatch deletes the rows where a generator has no capacity (absence). sum(dispatch, over=generator) names the dimension it reduces, so the constraint's dims is what remains.

description: Least-cost dispatch of a generator fleet against an hourly load.

dimensions:
  snapshot: { dtype: int, description: dispatch periods }
  generator: { description: generating units }

parameters:
  capacity: { dims: [generator], description: installed capacity }
  load: { dims: [snapshot], description: demand to be met }
  cost: { dims: [generator], description: marginal cost }

variables:
  dispatch:
    description: output of a generator in a snapshot
    dims: [snapshot, generator]
    where: "capacity > 0"
    bounds: { lower: 0, upper: capacity }

constraints:
  power_balance:
    dims: [snapshot]
    expression: sum(dispatch, over=generator) == load

objective:
  sense: minimize
  expression: sum(dispatch * cost)

Least-cost dispatch of a generator fleet against an hourly load.

Sets

Symbol	Meaning
\(\mathcal{T}\)	index \(t\) — snapshot — dispatch periods
\(\mathcal{G}\)	index \(g\) — generator — generating units

Parameters

Symbol	Meaning
\(\mathrm{capacity}\)	capacity over \(\mathcal{G}\) — installed capacity
\(\mathrm{load}\)	load over \(\mathcal{T}\) — demand to be met
\(\mathrm{cost}\)	cost over \(\mathcal{G}\) — marginal cost

Variables

Symbol	Meaning
\(\mathit{dispatch}\)	dispatch over \(\mathcal{T} \times \mathcal{G}\) — output of a generator in a snapshot

Upright is what the model is given — a parameter such as \(\mathrm{capacity}\), a coordinate map, a label — and italic is what the solver chooses, such as \(\mathit{dispatch}\). An index is italic too, being what a quantifier chooses, and a set is script.

Objective

\[ \min \sum_{t \in \mathcal{T},\ g \in \mathcal{G}} \mathit{dispatch}_{t,g} \cdot \mathrm{cost}_{g} \]

Subject to

power_balance

\[ \sum_{g \in \mathcal{G}} \mathit{dispatch}_{t,g} = \mathrm{load}_{t} \qquad \forall\, t \in \mathcal{T} \]

Variable domains

dispatch

\[ 0 \le \mathit{dispatch}_{t,g} \le \mathrm{capacity}_{g} \qquad \forall\, t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{capacity}_{g} > 0 \]

Regenerate with pixi run python -m tools.gallery.