A curve as a special-ordered set#
The same restriction as the adjacency page, handed to
the solver instead of built. method: sos2 says that at most two weights may
be non-zero and they must be neighbours, which is the definition of a type-2
set.
Read the two pages together. The binaries are gone here, and so are the two rows that constrained them. What replaces them is a declaration rather than a row, because a special-ordered set is something a solver enforces directly. Whether a given solver does is that solver's business, not the file's.
description: >-
A piecewise-linear cost curve stated as a special-ordered set, so the solver
is handed the adjacency restriction rather than binaries that encode it.
dimensions:
snapshot:
description: dispatch periods
dtype: int
generator:
description: dispatchable units
dtype: str
bp:
description: breakpoints of the cost curve
dtype: int
parameters:
capacity:
description: maximum dispatch
dims: [generator]
load:
description: demand to be met
dims: [snapshot]
bp_x:
description: breakpoint dispatch levels, one curve per generator
dims: [generator, bp]
bp_y:
description: cost at each breakpoint, one curve per generator
dims: [generator, bp]
variables:
dispatch:
description: dispatched power
dims: [snapshot, generator]
bounds:
lower: 0
upper: capacity
op_cost:
description: operating cost, piecewise-linear in dispatch
dims: [snapshot, generator]
bounds:
lower: 0
piecewise:
cost_curve:
description: >-
cost read off the generator's curve, with at most two adjacent weights
non-zero — the restriction the default method builds out of binaries,
declared as a set instead
along: bp
dims: [snapshot, generator]
links:
dispatch: [dispatch, bp_x]
op_cost: [op_cost, bp_y]
method: sos2
constraints:
balance:
dims: [snapshot]
expression: sum(dispatch, over=generator) == load
objective:
sense: minimize
description: total operating cost, taken off the curves rather than from a marginal rate
expression: sum(op_cost)
A piecewise-linear cost curve stated as a special-ordered set, so the solver is handed the adjacency restriction rather than binaries that encode it.
Sets#
| Symbol | Meaning |
|---|---|
| \(\mathcal{T}\) | index \(t\) — snapshot — dispatch periods |
| \(\mathcal{G}\) | index \(g\) — generator — dispatchable units |
| \(\mathcal{B}\) | index \(b\) — bp — breakpoints of the cost curve |
Parameters#
| Symbol | Meaning |
|---|---|
| \(\mathrm{capacity}\) | capacity over \(\mathcal{G}\) — maximum dispatch |
| \(\mathrm{load}\) | load over \(\mathcal{T}\) — demand to be met |
| \(\mathrm{x}\) | bp_x over \(\mathcal{G} \times \mathcal{B}\) — breakpoint dispatch levels, one curve per generator |
| \(\mathrm{y}\) | bp_y over \(\mathcal{G} \times \mathcal{B}\) — cost at each breakpoint, one curve per generator |
Variables#
| Symbol | Meaning |
|---|---|
| \(\mathit{dispatch}\) | dispatch over \(\mathcal{T} \times \mathcal{G}\) — dispatched power |
| \(\mathit{op\_cost}\) | op_cost over \(\mathcal{T} \times \mathcal{G}\) — operating cost, piecewise-linear in dispatch |
Upright is what the data supplies — 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#
Subject to#
balance
cost_curve
Variable domains#
dispatch
op_cost
Assumptions#
cost_curve_complete