Export#

One of the base fragments of Calliope in fragments. Export out of the system: a technology may export a carrier it produces, and the export leaves the balance and has a cost.

dimensions:
  nodes:
    description: Calliope's `nodes` — the places technologies stand at
  techs:
    description: Calliope's `techs` — technologies
  carriers:
    description: Calliope's `carriers` — energy and commodity carriers
  timesteps:
    description: Calliope's `timesteps` — time steps, in order
    dtype: datetime
  costs:
    description: Calliope's `costs` — cost classes, such as monetary and CO2

parameters:
  carrier_export:
    description: "`carrier_export` — whether a technology may export a carrier it produces out of the system"
    dims: [nodes, techs, carriers]
    dtype: bool
  export_min:
    description: "`export_min` — least export. Calliope's default is 0, and data prep fills it"
    dims: [nodes, techs, carriers]
  export_max:
    description: "`export_max` — most export. Calliope's default is `.inf`, and data prep fills it"
    dims: [nodes, techs, carriers]
  cost_export:
    description: "`cost_export` — the cost of one unit of export, usually negative"
    dims: [nodes, techs, costs, timesteps]

variables:
  flow_export:
    description: "`flow_export` — what a technology exports out of the system in a time step"
    dims: [nodes, techs, carriers, timesteps]
    where: carrier_export
    bounds: { lower: export_min, upper: export_max }
    absence: zero

expressions:
  export_carrier_flow: -sum(flow_export, over=techs)
  export_cost_operation_variable: timestep_weights * sum(cost_export * flow_export, over=carriers)

given:
  parameters:
    carrier_out: { dims: [nodes, techs, carriers], dtype: bool }
    timestep_weights: { dims: [timesteps] }
  variables:
    flow_out: { dims: [nodes, techs, carriers, timesteps] }
  expressions:
    carrier_flow: { dims: [nodes, carriers, timesteps], term: export_carrier_flow }
    cost_operation_variable: { dims: [nodes, techs, costs, timesteps], term: export_cost_operation_variable }

constraints:
  export_balance:
    description: "`export_balance` — a technology exports at most what it puts out"
    dims: [nodes, techs, carriers, timesteps]
    where: flow_export
    expression: flow_out >= flow_export

assumptions:
  export_only_for_outflows:
    description: Calliope's `export_only_for_outflows` — an exported carrier is one the technology produces
    holds: NOT carrier_export OR count(carrier_out, over=nodes) >= 1

Sets#

Symbol	Meaning
\(\mathcal{N}\)	index \(n\) — `nodes` — Calliope's `nodes` — the places technologies stand at
\(\mathcal{I}\)	index \(i\) — `techs` — Calliope's `techs` — technologies
\(\mathcal{C}\)	index \(c\) — `carriers` — Calliope's `carriers` — energy and commodity carriers
\(\mathcal{T}\)	index \(t\) — `timesteps` — Calliope's `timesteps` — time steps, in order
\(\mathcal{K}\)	index \(k\) — `costs` — Calliope's `costs` — cost classes, such as monetary and CO2

Parameters#

Symbol	Meaning
\(\mathrm{carrier\_export}\)	`carrier_export` over \(\mathcal{N} \times \mathcal{I} \times \mathcal{C}\) — `carrier_export` — whether a technology may export a carrier it produces out of the system
\(\mathrm{export\_min}\)	`export_min` over \(\mathcal{N} \times \mathcal{I} \times \mathcal{C}\) — `export_min` — least export. Calliope's default is 0, and data prep fills it
\(\mathrm{export\_max}\)	`export_max` over \(\mathcal{N} \times \mathcal{I} \times \mathcal{C}\) — `export_max` — most export. Calliope's default is `.inf`, and data prep fills it
\(\mathrm{cost\_export}\)	`cost_export` over \(\mathcal{N} \times \mathcal{I} \times \mathcal{K} \times \mathcal{T}\) — `cost_export` — the cost of one unit of export, usually negative

Variables#

Symbol	Meaning
\(\mathit{flow\_export}\)	`flow_export` over \(\mathcal{N} \times \mathcal{I} \times \mathcal{C} \times \mathcal{T}\) — `flow_export` — what a technology exports out of the system in a time step

Given#

Symbol	Meaning
\(\mathrm{carrier\_out}\)	`carrier_out` over \(\mathcal{N} \times \mathcal{I} \times \mathcal{C}\), data another file declares
\(\mathrm{timestep\_weights}\)	`timestep_weights` over \(\mathcal{T}\), data another file declares
\(\mathit{flow\_out}\)	`flow_out` over \(\mathcal{N} \times \mathcal{I} \times \mathcal{C} \times \mathcal{T}\)
\(\mathit{carrier\_flow}\)	`carrier_flow` over \(\mathcal{N} \times \mathcal{C} \times \mathcal{T}\), an expression this file adds `export_carrier_flow` to
\(\mathit{cost\_operation\_variable}\)	`cost_operation_variable` over \(\mathcal{N} \times \mathcal{I} \times \mathcal{K} \times \mathcal{T}\), an expression this file adds `export_cost_operation_variable` to

Definitions#

Symbol	Meaning
\(\mathit{export\_carrier\_flow}\)	`export_carrier_flow` over \(\mathcal{N} \times \mathcal{C} \times \mathcal{T}\)
\(\mathit{export\_cost\_operation\_variable}\)	`export_cost_operation_variable` over \(\mathcal{N} \times \mathcal{I} \times \mathcal{T} \times \mathcal{K}\)

Upright is what the data supplies — a parameter such as \(\mathrm{carrier\_export}\), a coordinate map, a label — and italic is what the solver chooses, such as \(\mathit{flow\_export}\). An index is italic too, being what a quantifier chooses, and a set is script.

Subject to#

export_balance

\[ \mathit{flow\_out}_{n,i,c,t} \ge \mathit{flow\_export}_{n,i,c,t} \qquad \forall\, n \in \mathcal{N},\ i \in \mathcal{I},\ c \in \mathcal{C},\ t \in \mathcal{T} \,:\, \mathit{flow\_export}_{n,i,c,t} \text{ exists} \]

Definitions#

export_carrier_flow

\[ \mathit{export\_carrier\_flow}_{n,c,t} = -\left( \sum_{i \in \mathcal{I}} \mathit{flow\_export}_{n,i,c,t} \right) \qquad \forall\, n \in \mathcal{N},\ c \in \mathcal{C},\ t \in \mathcal{T} \]

export_cost_operation_variable

\[ \mathit{export\_cost\_operation\_variable}_{n,i,t,k} = \mathrm{timestep\_weights}_{t} \cdot \left( \sum_{c \in \mathcal{C}} \mathrm{cost\_export}_{n,i,k,t} \cdot \mathit{flow\_export}_{n,i,c,t} \right) \qquad \forall\, n \in \mathcal{N},\ i \in \mathcal{I},\ t \in \mathcal{T},\ k \in \mathcal{K} \]

Variable domains#

flow_export

\[ \mathrm{export\_min}_{n,i,c} \le \mathit{flow\_export}_{n,i,c,t} \le \mathrm{export\_max}_{n,i,c} \qquad \forall\, n \in \mathcal{N},\ i \in \mathcal{I},\ c \in \mathcal{C},\ t \in \mathcal{T} \,:\, \mathrm{carrier\_export}_{n,i,c} \]

Assumptions#

export_only_for_outflows

\[ \neg \mathrm{carrier\_export}_{n,i,c} \vee \lvert \{ n' \in \mathcal{N} \,:\, \mathrm{carrier\_out}_{n',i,c} \} \rvert \ge 1 \qquad \forall\, n \in \mathcal{N},\ i \in \mathcal{I},\ c \in \mathcal{C} \]