transport¶

A network: generators sit on buses, lines connect buses, and power balances at every bus.

The problem¶

\[\sum_{g \thinspace:\thinspace \mathrm{bus}(g) = b} p_{s,g} \quad+\quad \sum_{\ell \thinspace:\thinspace \mathrm{to}(\ell) = b} f_{s,\ell} \quad-\quad \sum_{\ell \thinspace:\thinspace \mathrm{from}(\ell) = b} f_{s,\ell} \quad=\quad d_{s,b}\]

Each sum is over the lines or generators a coordinate map sends to bus \(b\) — \(\mathrm{bus}\), \(\mathrm{to}\) and \(\mathrm{from}\) are the coordinates the dimensions declare, not sets in their own right. Load is \(d\) here, because \(\ell\) is already the line index.

The model¶

The same model, as math

Sets¶

Symbol	Meaning
\(\mathcal{S}\)	index \(s\) --- `snapshot` --- dispatch periods
\(\mathcal{G}\)	index \(g\) --- `generator` with \(\mathrm{bus}: \mathcal{G} \to \mathcal{B}\) --- generating units
\(\mathcal{B}\)	index \(b\) --- `bus` --- network nodes
\(\mathcal{L}\)	index \(\ell\) --- `line` with \(\mathrm{from}: \mathcal{L} \to \mathcal{B},\enspace \mathrm{to}: \mathcal{L} \to \mathcal{B}\) --- transmission lines, each joining two buses

Parameters¶

Symbol	Meaning
\(\bar p\)	`p_max` over \(\mathcal{G}\) --- installed capacity
\(c\)	`cost` over \(\mathcal{G}\) --- marginal cost
\(\bar f\)	`cap` over \(\mathcal{L}\) --- forward transmission limit
\(\underline{f}\)	`neg_cap` over \(\mathcal{L}\) --- reverse transmission limit
\(d\)	`load` over \(\mathcal{S} \times \mathcal{B}\) --- demand at each bus

Variables¶

Symbol	Meaning
\(p\)	`p` over \(\mathcal{S} \times \mathcal{G}\) --- output of generator \(g\) in snapshot \(s\)
\(f\)	`f` over \(\mathcal{S} \times \mathcal{L}\) --- flow on line \(\ell\), signed towards its `to` bus

Objective¶

total_cost

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

Subject to¶

balance

\[\sum_{g \in \mathcal{G} \thinspace:\thinspace \mathrm{bus}(g) = b} p_{s,g} + \sum_{\ell \in \mathcal{L} \thinspace:\thinspace \mathrm{to}(\ell) = b} f_{s,\ell} - \left( \sum_{\ell \in \mathcal{L} \thinspace:\thinspace \mathrm{from}(\ell) = b} f_{s,\ell} \right) = d_{s,b} \qquad \forall\thinspace s \in \mathcal{S},\enspace b \in \mathcal{B}\]

Variable domains¶

p

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

f

\[\underline{f}_{\ell} \le f_{s,\ell} \le \bar f_{\ell} \qquad \forall\thinspace s \in \mathcal{S},\enspace \ell \in \mathcal{L}\]

dimensions:
  snapshot:
    dtype: int
  generator:
    dtype: str
    coords: [bus]  # every generator sits on a bus
  bus:
    dtype: str
  line:
    dtype: str
    coords: {from: bus, to: bus}  # both endpoints are buses

parameters:
  p_max:
    dims: [generator]
  cost:
    dims: [generator]
  cap:
    dims: [line]
  neg_cap:
    dims: [line]
  load:
    dims: [snapshot, bus]

variables:
  p:
    foreach: [snapshot, generator]
    bounds:
      lower: 0
      upper: p_max
  f:
    foreach: [snapshot, line]
    bounds:
      lower: neg_cap
      upper: cap

# Naming the two halves of the nodal balance. Pure substitution before either
# backend sees the model, so this costs nothing at build and nothing at solve —
# what it buys is a constraint that reads as the sentence it is.
expressions:
  gen_at_bus: group_sum(p, over=generator, by=bus)
  net_inflow: group_sum(f, over=line, by=to) - group_sum(f, over=line, by=from)

constraints:
  balance:
    foreach: [snapshot, bus]
    expression: gen_at_bus + net_inflow == load

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

What it exercises¶

Three group_sum calls, and they are what a network is in this language. A dimension can carry coordinates — generator carries bus, line carries from and to — and group_sum(f, over=line, by=to) sums along a line's to coordinate, landing the result on bus. The same f is summed twice through two different coordinates, once as an inflow and once as an outflow.

No adjacency matrix, and no join written by the modeller: the topology is data on the dimension.

examples/transport.yaml · back to all models