Flow share over the whole time#

An extension of Calliope in fragments. Calliope's example share_all_timesteps.yaml: a technology meets a share of a demand, or of a node's outflow of a carrier, over the whole time.

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

relations:
  demand_share_tech:
    description: >-
      `demand_share_tech` — the demand technology whose inflow a technology
      meets a share of. Calliope slices `flow_in` by it; a read through the
      relation is that slice
    key: techs
    values: { demand: techs }
  supply_share_carrier:
    description: >-
      `supply_share_carrier` — the carrier a technology's share of outflow is
      counted in. Calliope slices `flow_out` by it, which is a test of the
      pair
    key: [techs, carriers]

parameters:
  demand_share_equals:
    description: "`demand_share_equals` — the share of a demand technology's inflow a technology meets; given only where set"
    dims: [nodes, techs]
  supply_share_equals:
    description: "`supply_share_equals` — the share of a node's outflow of a carrier a technology puts out; given only where set"
    dims: [nodes, techs]

expressions:
  supply_share_flow_out:
    description: "`flow_out[carriers=$carrier]` — a technology's outflow of its share carrier"
    dims: [nodes, techs, carriers, timesteps]
    cases:
      share_carrier:
        when: supply_share_carrier
        expression: flow_out
    otherwise: 0
  supply_share_all_flow_out:
    description: "`sum(flow_out[carriers=$carrier], over=techs)` — every technology's outflow of a technology's share carrier"
    dims: [nodes, techs, carriers, timesteps]
    cases:
      share_carrier:
        when: supply_share_carrier
        expression: sum(flow_out, over=techs)
    otherwise: 0

given:
  variables:
    flow_out: { dims: [nodes, techs, carriers, timesteps] }
    flow_in: { dims: [nodes, techs, carriers, timesteps] }

constraints:
  demand_share_equals_per_tech:
    description: "`demand_share_equals_per_tech` — a technology puts out its share of a demand technology's inflow over the whole time"
    dims: [nodes, techs]
    where: demand_share_equals
    expression: >-
      sum(flow_out, over=[timesteps, carriers])
      == sum(at(flow_in, by=demand_share_tech, over=demand, into=techs), over=[timesteps, carriers])
      * demand_share_equals
  supply_share_equals_per_tech:
    description: "`supply_share_equals_per_tech` — a technology puts out its share of a node's outflow of a carrier over the whole time"
    dims: [nodes, techs]
    where: supply_share_equals
    expression: >-
      sum(supply_share_flow_out, over=[carriers, timesteps])
      == sum(supply_share_all_flow_out, over=[carriers, timesteps]) * supply_share_equals

Sets#

Symbol	Meaning
$\mathcal{N}$	index $n$ — `nodes` — Calliope's `nodes` — the places technologies stand at
$\mathcal{I}$	index $i$ — `techs` with $\mathrm{demand\_share\_tech}: \mathcal{I} \to \mathcal{I},\ \mathrm{supply\_share\_carrier} \subseteq \mathcal{I} \times \mathcal{C}$ — Calliope's `techs` — technologies
$\mathcal{C}$	index $c$ — `carriers` with $\mathrm{supply\_share\_carrier} \subseteq \mathcal{I} \times \mathcal{C}$ — Calliope's `carriers` — energy and commodity carriers
$\mathcal{T}$	index $t$ — `timesteps` — Calliope's `timesteps` — time steps, in order

Parameters#

Symbol	Meaning
$\mathrm{demand\_share\_equals}$	`demand_share_equals` over $\mathcal{N} \times \mathcal{I}$ — `demand_share_equals` — the share of a demand technology's inflow a technology meets; given only where set
$\mathrm{supply\_share\_equals}$	`supply_share_equals` over $\mathcal{N} \times \mathcal{I}$ — `supply_share_equals` — the share of a node's outflow of a carrier a technology puts out; given only where set

Given#

Symbol	Meaning
$\mathit{flow\_out}$	`flow_out` over $\mathcal{N} \times \mathcal{I} \times \mathcal{C} \times \mathcal{T}$
$\mathit{flow\_in}$	`flow_in` over $\mathcal{N} \times \mathcal{I} \times \mathcal{C} \times \mathcal{T}$

Definitions#

Symbol	Meaning
$\mathit{supply\_share\_flow\_out}$	`supply_share_flow_out` over $\mathcal{N} \times \mathcal{I} \times \mathcal{C} \times \mathcal{T}$ — `flow_out[carriers=$carrier]` — a technology's outflow of its share carrier
$\mathit{supply\_share\_all\_flow\_out}$	`supply_share_all_flow_out` over $\mathcal{N} \times \mathcal{I} \times \mathcal{C} \times \mathcal{T}$ — `sum(flow_out[carriers=$carrier], over=techs)` — every technology's outflow of a technology's share carrier

Subject to#

demand_share_equals_per_tech

\[ \sum_{c \in \mathcal{C},\ t \in \mathcal{T}} \mathit{flow\_out}_{n,i,c,t} = \left( \sum_{c \in \mathcal{C},\ t \in \mathcal{T}} \mathit{flow\_in}_{n,\mathrm{demand\_share\_tech}(i),c,t} \right) \cdot \mathrm{demand\_share\_equals}_{n,i} \qquad \forall\, n \in \mathcal{N},\ i \in \mathcal{I} \,:\, \mathrm{demand\_share\_equals}_{n,i} \text{ is defined} \]

supply_share_equals_per_tech

\[ \sum_{c \in \mathcal{C},\ t \in \mathcal{T}} \mathit{supply\_share\_flow\_out}_{n,i,c,t} = \left( \sum_{c \in \mathcal{C},\ t \in \mathcal{T}} \mathit{supply\_share\_all\_flow\_out}_{n,i,c,t} \right) \cdot \mathrm{supply\_share\_equals}_{n,i} \qquad \forall\, n \in \mathcal{N},\ i \in \mathcal{I} \,:\, \mathrm{supply\_share\_equals}_{n,i} \text{ is defined} \]

Definitions#

supply_share_flow_out

\[ \mathit{supply\_share\_flow\_out}_{n,i,c,t} = \begin{cases} \mathit{flow\_out}_{n,i,c,t} & \text{if } \left( i,\ c \right) \in \mathrm{supply\_share\_carrier} \\ 0 & \text{otherwise} \end{cases} \qquad \forall\, n \in \mathcal{N},\ i \in \mathcal{I},\ c \in \mathcal{C},\ t \in \mathcal{T} \]

supply_share_all_flow_out

\[ \mathit{supply\_share\_all\_flow\_out}_{n,i,c,t} = \begin{cases} \sum_{i' \in \mathcal{I}} \mathit{flow\_out}_{n,i',c,t} & \text{if } \left( i,\ c \right) \in \mathrm{supply\_share\_carrier} \\ 0 & \text{otherwise} \end{cases} \qquad \forall\, n \in \mathcal{N},\ i \in \mathcal{I},\ c \in \mathcal{C},\ t \in \mathcal{T} \]

Symbol	Meaning
\(\mathcal{N}\)	index \(n\) — `nodes` — Calliope's `nodes` — the places technologies stand at
\(\mathcal{I}\)	index \(i\) — `techs` with \(\mathrm{demand\_share\_tech}: \mathcal{I} \to \mathcal{I},\ \mathrm{supply\_share\_carrier} \subseteq \mathcal{I} \times \mathcal{C}\) — Calliope's `techs` — technologies
\(\mathcal{C}\)	index \(c\) — `carriers` with \(\mathrm{supply\_share\_carrier} \subseteq \mathcal{I} \times \mathcal{C}\) — Calliope's `carriers` — energy and commodity carriers
\(\mathcal{T}\)	index \(t\) — `timesteps` — Calliope's `timesteps` — time steps, in order

Symbol	Meaning
\(\mathrm{demand\_share\_equals}\)	`demand_share_equals` over \(\mathcal{N} \times \mathcal{I}\) — `demand_share_equals` — the share of a demand technology's inflow a technology meets; given only where set
\(\mathrm{supply\_share\_equals}\)	`supply_share_equals` over \(\mathcal{N} \times \mathcal{I}\) — `supply_share_equals` — the share of a node's outflow of a carrier a technology puts out; given only where set

Symbol	Meaning
\(\mathit{flow\_out}\)	`flow_out` over \(\mathcal{N} \times \mathcal{I} \times \mathcal{C} \times \mathcal{T}\)
\(\mathit{flow\_in}\)	`flow_in` over \(\mathcal{N} \times \mathcal{I} \times \mathcal{C} \times \mathcal{T}\)

Symbol	Meaning
\(\mathit{supply\_share\_flow\_out}\)	`supply_share_flow_out` over \(\mathcal{N} \times \mathcal{I} \times \mathcal{C} \times \mathcal{T}\) — `flow_out[carriers=$carrier]` — a technology's outflow of its share carrier
\(\mathit{supply\_share\_all\_flow\_out}\)	`supply_share_all_flow_out` over \(\mathcal{N} \times \mathcal{I} \times \mathcal{C} \times \mathcal{T}\) — `sum(flow_out[carriers=$carrier], over=techs)` — every technology's outflow of a technology's share carrier