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dispatch

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

✔ Agrees with hand-written linopy 0.9.0 — objective 10500, matched to rtol=1e-09.

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

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

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 a generator in a snapshot

Objective

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

Subject to

power_balance

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

Variable domains

p

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

The tabs start from the instance’s tables — one frame per parameter.

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

dimensions:
  snapshot:
    description: dispatch periods
    dtype: int
  generator:
    description: generating units
    values: [wind, solar, gas]

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

variables:
  p:
    description: output of a generator in a snapshot
    foreach: [snapshot, generator]
    where: "p_max > 0"
    bounds:
      lower: 0
      upper: p_max

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

objective:
  sense: minimize
  description: total cost of generation over the horizon
  expression: p * cost

# sources: parameter name -> frame or parquet path
with lps.solve('examples/dispatch.yaml', sources) as solution:
    solution.objective  # 10500.0
    solution.dual('power_balance')

The model-building half of examples/ports/references/linopy/dispatch.py:

def build(tables: dict[str, pd.DataFrame]) -> linopy.Model:
    """The instance's tables as a linopy model, row for row.

    ``tables`` is the same mapping the lpspec call binds as ``sources``.
    """
    p_max: pd.Series = tables['p_max'].set_index('generator')['value']
    cost: pd.Series = tables['cost'].set_index('generator')['value']
    load: pd.Series = tables['load'].set_index('snapshot')['value']

    m = linopy.Model()
    p = m.add_variables(lower=0, upper=p_max, coords=[load.index, p_max.index], name='p')
    m.add_constraints(p.sum('generator') == load, name='power_balance')
    m.add_objective((p * cost).sum())
    return m

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.

examples/dispatch.yaml · back to all models