piecewise¶

Per-generator convex cost curves, expanded into a λ-formulation.

The problem¶

Each generator gets its own breakpoint list, so the curve varies per unit — something a flat breakpoint list cannot express:

\[p_g = \sum_k \lambda_{g,k}\, x_{g,k}, \quad \mathrm{cost}_g = \sum_k \lambda_{g,k}\, y_{g,k}, \quad \sum_k \lambda_{g,k} = 1, \quad \lambda \ge 0\]

The model¶

The same model, as math

Sets¶

Symbol	Meaning
\(\mathcal{T}\)	index \(t\) --- `snapshot` --- dispatch periods
\(\mathcal{G}\)	index \(g\) --- `generator` --- dispatchable units
\(\mathcal{K}\)	index \(k\) --- `bp` --- breakpoints of the cost curve

Parameters¶

Symbol	Meaning
\(p^{\mathrm{max}}\)	`p_max` over \(\mathcal{G}\) --- maximum dispatch
\(\mathit{load}\)	`load` over \(\mathcal{T}\) --- demand to be met
\(x\)	`bp_x` over \(\mathcal{G} \times \mathcal{K}\) --- breakpoint dispatch levels
\(y\)	`bp_y` over \(\mathcal{G} \times \mathcal{K}\) --- cost at each breakpoint

Variables¶

Symbol	Meaning
\(p\)	`p` over \(\mathcal{T} \times \mathcal{G}\) --- dispatched power
\(\mathrm{cost}\)	`op_cost` over \(\mathcal{T} \times \mathcal{G}\) --- operating cost, piecewise-linear in dispatch
\(\lambda\)	`cost_curve_lam` over \(\mathcal{T} \times \mathcal{G} \times \mathcal{K}\) --- convex-combination weight on breakpoint \(k\)

Objective¶

total_cost

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

Subject to¶

balance

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

cost_curve_convexity

\[\sum_{k \in \mathcal{K}} \lambda_{t,g,k} = 1 \qquad \forall\thinspace t \in \mathcal{T},\enspace g \in \mathcal{G}\]

cost_curve_link0

\[p_{t,g} = \sum_{k \in \mathcal{K}} \lambda_{t,g,k} \cdot x_{g,k} \qquad \forall\thinspace t \in \mathcal{T},\enspace g \in \mathcal{G}\]

cost_curve_link1

\[\mathrm{cost}_{t,g} = \sum_{k \in \mathcal{K}} \lambda_{t,g,k} \cdot y_{g,k} \qquad \forall\thinspace t \in \mathcal{T},\enspace g \in \mathcal{G}\]

Variable domains¶

p

\[0 \le p_{t,g} \le p^{\mathrm{max}}_{g} \qquad \forall\thinspace t \in \mathcal{T},\enspace g \in \mathcal{G}\]

op_cost

\[\mathrm{cost}_{t,g} \ge 0 \qquad \forall\thinspace t \in \mathcal{T},\enspace g \in \mathcal{G}\]

cost_curve_lam

\[0 \le \lambda_{t,g,k} \le 1 \qquad \forall\thinspace t \in \mathcal{T},\enspace g \in \mathcal{G},\enspace k \in \mathcal{K}\]

# Dispatch with piecewise-linear generation costs via the piecewise: block.
#
# Each generator has its own convex cost curve: the breakpoint parameters
# carry [generator, bp], so curves vary per generator — something flat
# breakpoint lists cannot express. With convex: true the expansion emits no
# binaries (pure-LP convex hull, exact for convex curves under minimisation)
# and the model stays relational-eligible as an LP. Drop convex: true for
# nonconvex curves — the expansion then adds segment binaries + adjacency,
# and the model becomes a (still relational-eligible) MILP.

dimensions:
  snapshot:
    dtype: int
  generator:
    dtype: str
  bp:
    dtype: int

parameters:
  p_max:
    dims: [generator]
  load:
    dims: [snapshot]
  bp_x:
    dims: [generator, bp]  # per-generator breakpoint positions
  bp_y:
    dims: [generator, bp]  # per-generator cost at each breakpoint

variables:
  p:
    foreach: [snapshot, generator]
    bounds:
      lower: 0
      upper: p_max
  op_cost:
    foreach: [snapshot, generator]
    bounds:
      lower: 0

piecewise:
  cost_curve:
    over: bp
    links:
      - [p, bp_x]
      - [op_cost, bp_y]
    convex: true

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

objectives:
  total_cost:
    sense: minimize
    expression: op_cost

What it exercises¶

piecewise: is a declaration, not a helper — it expands before lowering into the λ-formulation above. With convex: true the expansion emits no binaries at all: the convex hull is exact for a convex curve under minimisation, so the model stays a pure LP. Drop convex: true and the expansion adds segment binaries and adjacency constraints, and the model becomes a MILP that is still entirely inside the relational subset.

By the time the logical plan exists there is nothing left called piecewise — which is why the construct matrix reads it from the surface declaration.

examples/piecewise.yaml · back to all models