dispatch

Least-cost generation against a load profile — the smallest model that is still a model.

The problem

Pick an output \(p_{s,g}\) for every generator in every snapshot, so that the fleet meets the load exactly and costs as little as possible:

\[\min \sum_{s,g} c_g \thinspace p_{s,g} \quad\text{s.t.}\quad \sum_g p_{s,g} = \ell_s ,\quad 0 \le p_{s,g} \le \bar p_g \quad\text{where}\quad \bar p_g > 0\]

The model

The same model, as math

Sets

Symbol	Meaning
\(\mathcal{S}\)	index \(s\) --- snapshot --- dispatch periods
\(\mathcal{G}\)	index \(g\) --- generator --- generating units

Parameters

Symbol	Meaning
\(\bar p\)	p_max over \(\mathcal{G}\) --- installed capacity
\(\ell\)	load over \(\mathcal{S}\) --- demand to be met
\(c\)	cost over \(\mathcal{G}\) --- marginal cost

Variables

Symbol	Meaning
\(p\)	p over \(\mathcal{S} \times \mathcal{G}\) --- output of generator \(g\) in snapshot \(s\)

Objective

total_cost

\[\min \sum_{s \in \mathcal{S},\enspace g \in \mathcal{G}} p_{s,g} \cdot c_{g}\]

Subject to

power_balance

\[\sum_{g \in \mathcal{G}} p_{s,g} = \ell_{s} \qquad \forall\thinspace s \in \mathcal{S}\]

Variable domains

p

\[0 \le p_{s,g} \le \bar p_{g} \qquad \forall\thinspace s \in \mathcal{S},\enspace g \in \mathcal{G} \thinspace:\thinspace \bar p_{g} > 0\]

dimensions:
  snapshot:
    dtype: int
  generator:
    values: [wind, solar, gas]

parameters:
  p_max:
    dims: [generator]
  load:
    dims: [snapshot]
  cost:
    dims: [generator]

variables:
  p:
    foreach: [snapshot, generator]
    where: "p_max > 0"
    bounds:
      lower: 0
      upper: p_max

constraints:
  power_balance:
    foreach: [snapshot]
    expression: sum(p, over=generator) == load

objectives:
  total_cost:
    sense: minimize
    expression: p * cost

What it exercises

where: "p_max > 0" is the one line worth pausing on. A generator with no capacity gets no columns at all — not a column pinned to zero — so a retired unit costs nothing to carry in the data. That is row absence, and it is how sparsity is spelled throughout: see where in the language reference.

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examples/dispatch.yaml · back to all models