Units with ports#

Units with ports is a specification of an optimisation model (a spec) in which each unit attaches to buses through numbered ports. Each port number matches one of the bus0, bus1, … columns of an open-source power system modelling framework (PyPSA). The relation connection lists the bus and the role of each port. Its key is the unit and the port, not the bus, so two ports of one unit may attach to the same bus.

efficiency is declared over every unit, port and snapshot. Its where: keeps only the output ports, and missing: 1 applies only there. Where connection has no row, injection takes its otherwise: and efficiency is absent, so injection is absent too. The sum in balance therefore has a term only for the ports that attach to each bus.

description: Units that draw power from one bus and deliver it to others, through numbered ports.

dimensions:
  snapshot: { dtype: int, description: dispatch periods }
  bus: { description: nodes of the network }
  unit: { description: "converters, such as an electrolyser or a heat pump" }
  port: { dtype: int, description: "an attachment of a unit, numbered as PyPSA's bus0, bus1, ..." }
  role: { description: input or output }

relations:
  connection:
    description: the bus each port of a unit attaches to, and whether the port draws or delivers
    key: [unit, port]
    values: [bus, role]
    missing: absent

parameters:
  capacity: { dims: [unit], description: the most a unit can draw at its input }
  efficiency:
    description: share of the drawn power that an output port delivers
    dims: [unit, port, snapshot]
    where: "connection[role] == 'output'"
    missing: 1
  load: { dims: [bus, snapshot], description: demand to be met }
  cost: { dims: [bus], description: price of power bought at a bus }

variables:
  flow:
    description: power a unit draws at its input
    dims: [unit, snapshot]
    bounds: { lower: 0, upper: capacity }
  supply:
    description: power bought at a bus
    dims: [bus, snapshot]
    bounds: { lower: 0 }

expressions:
  injection:
    description: what a port puts into its bus
    dims: [unit, port, snapshot]
    cases:
      draws: { when: "connection[role] == 'input'", expression: -flow }
    otherwise: efficiency * flow

constraints:
  balance:
    dims: [bus, snapshot]
    expression: supply + sum(injection, over=[unit, port], by=connection[bus]) == load

objective:
  sense: minimize
  expression: sum(cost * supply)

Units that draw power from one bus and deliver it to others, through numbered ports.

Sets#

Symbol	Meaning
\(\mathcal{B}\)	index \(b\) — `bus` with \(\mathrm{connection}: \mathcal{P} \times \mathcal{U} \to \mathcal{B} \times \mathcal{R}\), `connection` is `absent` where the data has no row — nodes of the network
\(\mathcal{P}\)	index \(p\) — `port` with \(\mathrm{connection}: \mathcal{P} \times \mathcal{U} \to \mathcal{B} \times \mathcal{R}\), `connection` is `absent` where the data has no row — an attachment of a unit, numbered as PyPSA's bus0, bus1, ...
\(\mathcal{R}\)	index \(r\) — `role` with \(\mathrm{connection}: \mathcal{P} \times \mathcal{U} \to \mathcal{B} \times \mathcal{R}\), `connection` is `absent` where the data has no row — input or output
\(\mathcal{T}\)	index \(t\) — `snapshot` — dispatch periods
\(\mathcal{U}\)	index \(u\) — `unit` with \(\mathrm{connection}: \mathcal{P} \times \mathcal{U} \to \mathcal{B} \times \mathcal{R}\), `connection` is `absent` where the data has no row — converters, such as an electrolyser or a heat pump

Parameters#

Symbol	Meaning
\(\mathrm{capacity}\)	`capacity` over \(\mathcal{U}\) — the most a unit can draw at its input
\(\mathrm{cost}\)	`cost` over \(\mathcal{B}\) — price of power bought at a bus
\(\mathrm{efficiency}\)	`efficiency` over \(\mathcal{P} \times \mathcal{T} \times \mathcal{U}\), only where \(\mathrm{connection.role}(p,\ u) = \text{'}\mathrm{output}\text{'}\), \(1\) where the data has no row — share of the drawn power that an output port delivers
\(\mathrm{load}\)	`load` over \(\mathcal{B} \times \mathcal{T}\) — demand to be met

Variables#

Symbol	Meaning
\(\mathit{flow}\)	`flow` over \(\mathcal{T} \times \mathcal{U}\) — power a unit draws at its input
\(\mathit{supply}\)	`supply` over \(\mathcal{B} \times \mathcal{T}\) — power bought at a bus

Definitions#

Symbol	Meaning
\(\mathit{injection}\)	`injection` over \(\mathcal{P} \times \mathcal{T} \times \mathcal{U}\) — what a port puts into its bus

Variable domains#

flow

\[ 0 \le \mathit{flow}_{t,u} \le \mathrm{capacity}_{u} \qquad \forall\, t \in \mathcal{T},\ u \in \mathcal{U} \]

supply

\[ \mathit{supply}_{b,t} \ge 0 \qquad \forall\, b \in \mathcal{B},\ t \in \mathcal{T} \]

Definitions#

injection

\[ \mathit{injection}_{p,t,u} = \begin{cases} -\mathit{flow}_{t,u} & \text{if } \mathrm{connection.role}(p,\ u) = \text{'}\mathrm{input}\text{'} \\ \mathrm{efficiency}_{p,t,u} \cdot \mathit{flow}_{t,u} & \text{otherwise} \end{cases} \qquad \forall\, p \in \mathcal{P},\ t \in \mathcal{T},\ u \in \mathcal{U} \]

Objective#

\[ \min \sum_{b \in \mathcal{B},\ t \in \mathcal{T}} \mathrm{cost}_{b} \cdot \mathit{supply}_{b,t} \]

Subject to#

balance

\[ \mathit{supply}_{b,t} + \sum_{p \in \mathcal{P},\ u \in \mathcal{U} \,:\, \mathrm{connection.bus}(p,\ u) = b} \mathit{injection}_{p,t,u} = \mathrm{load}_{b,t} \qquad \forall\, b \in \mathcal{B},\ t \in \mathcal{T} \]