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The coupling surface#

The coupling surface is the one file of the library (a fragment) that every other fragment is written against. The surface lists total_cost under given, and its objective minimises total_cost.

An open-source power system framework (PyPSA) gives each class of component, such as Generator, a column of bus labels, and sums the classes into Bus-nodal_balance. Here Bus_nodal_balance is the same however many fragments merge joins.

description: >-
  The coupling surface every component in this library is written against: one
  flow per port, one balance per bus, and one cost to minimise. A component is
  wired to a port, the port to a bus, and the balance contains no component class.
  A flow is positive where the port injects into its bus. A component that
  costs something adds its cost to `total_cost`.
given:
  expressions:
    total_cost: { dims: [], description: what running the system costs }
dimensions:
  snapshot: { dtype: datetime, description: dispatch periods }
  bus: { dtype: str, description: network nodes }
  port: { dtype: str, description: "the connections components make, one label per connection" }
relations:
  Port_bus: { key: port, values: bus }
variables:
  Port_p:
    dims: [snapshot, port]
    description: what a port puts into its bus in a snapshot, negative for a withdrawal
constraints:
  Bus_nodal_balance:
    description: "`Bus-nodal_balance` — what the ports on a bus put in sums to zero"
    dims: [snapshot, bus]
    expression: sum(Port_p, over=port, by=Port_bus[bus]) == 0
objective:
  sense: minimize
  expression: total_cost

The coupling surface every component in this library is written against: one flow per port, one balance per bus, and one cost to minimise. A component is wired to a port, the port to a bus, and the balance contains no component class. A flow is positive where the port injects into its bus. A component that costs something adds its cost to total_cost.

Given#

Symbol Meaning
\(\mathit{total\_cost}\) total_cost (scalar), an expression another file defines — what running the system costs

Sets#

Symbol Meaning
\(\mathcal{N}\) index \(n\) — bus with \(\mathrm{Port\_bus}: \mathcal{J} \to \mathcal{N}\) — network nodes
\(\mathcal{J}\) index \(j\) — port with \(\mathrm{Port\_bus}: \mathcal{J} \to \mathcal{N}\) — the connections components make, one label per connection
\(\mathcal{T}\) index \(t\) — snapshot — dispatch periods

Variables#

Symbol Meaning
\(f\) Port_p over \(\mathcal{J} \times \mathcal{T}\) — what a port puts into its bus in a snapshot, negative for a withdrawal

Variable domains#

Port_p

\[ f_{j,t} \in \mathbb{R} \qquad \forall\, j \in \mathcal{J},\ t \in \mathcal{T} \]

Objective#

\[ \min \mathit{total\_cost} \]

Subject to#

Bus_nodal_balance

\[ \sum_{j \in \mathcal{J} \,:\, \mathrm{Port\_bus}(j) = n} f_{j,t} = 0 \qquad \forall\, n \in \mathcal{N},\ t \in \mathcal{T} \]

See also