Demand share as a decision#

An extension of Calliope in fragments. Calliope's example demand_share_per_timestep_decision.yaml: a technology meets a share of a demand that the model decides, the same in every time step.

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:
  decide_demand_share:
    description: >-
      `decide_demand_share` — the demand technology whose inflow a technology
      meets a share of. Calliope reads it with `select_from_lookup_arrays`,
      which is a read through the relation
    key: techs
    values: { demand: techs }
  demand_share_carrier:
    description: >-
      `demand_share_carrier` — the carrier a share of demand is counted in.
      Calliope slices `flow_out` by it, which is a test of the pair
    key: [techs, carriers]

parameters:
  demand_share_relaxation:
    description: "`demand_share_relaxation` — how far the share may stray from the one decided, as a fraction"
    dims: [nodes, techs]
  demand_share_limit:
    description: "`demand_share_limit` — the share of demand the technologies meet together; given only where set"
    dims: [nodes]

variables:
  demand_share_per_timestep_decision:
    description: "`demand_share_per_timestep_decision` — the share of demand a technology meets, the same in every time step"
    dims: [nodes, techs]
    where: decide_demand_share
    bounds: { lower: 0 }
    absence: zero

expressions:
  demand_share_flow_out:
    description: "`flow_out[carriers=$carrier]` — a technology's outflow of the carrier its share is counted in"
    dims: [nodes, techs, carriers, timesteps]
    cases:
      share_carrier:
        when: demand_share_carrier
        expression: flow_out
    otherwise: 0
  demand_share_sink:
    description: "`select_from_lookup_arrays(sink_use_equals, techs=decide_demand_share)` — the demand a technology meets a share of"
    expression: at(sink_use_equals, by=decide_demand_share, over=demand, into=techs)

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

constraints:
  demand_share_per_timestep_decision_main_min:
    description: "`demand_share_per_timestep_decision_main_min` — a technology puts out at least its decided share of demand, less the relaxation"
    dims: [nodes, techs, timesteps]
    where: demand_share_per_timestep_decision
    expression: >-
      sum(demand_share_flow_out, over=carriers)
      >= (1 - demand_share_relaxation) * demand_share_sink * demand_share_per_timestep_decision
  demand_share_per_timestep_decision_main_max:
    description: "`demand_share_per_timestep_decision_main_max` — a technology puts out at most its decided share of demand, plus the relaxation"
    dims: [nodes, techs, timesteps]
    where: demand_share_per_timestep_decision
    expression: >-
      sum(demand_share_flow_out, over=carriers)
      <= (1 + demand_share_relaxation) * demand_share_sink * demand_share_per_timestep_decision
  demand_share_per_timestep_decision_sum:
    description: >-
      `demand_share_per_timestep_decision_sum` — the decided shares at a node
      add up to the limit. Calliope's `where: demand_share_per_timestep_decision`
      over a node reads as any technology there deciding a share. Calliope
      builds the row in every time step, and it is the same in each; a row
      repeated along a dimension it does not read is refused, so it is one
      row per node
    dims: [nodes]
    where: count(demand_share_per_timestep_decision, over=techs) >= 1 AND demand_share_limit
    expression: sum(demand_share_per_timestep_decision, over=techs) == demand_share_limit

assumptions:
  demand_share_is_fraction:
    description: Calliope's `demand_share_is_fraction` — the demand share limit is a fraction
    holds: demand_share_limit >= 0 AND demand_share_limit <= 1
    where: demand_share_limit

Sets#

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

Parameters#

Symbol	Meaning
$\mathrm{demand\_share\_relaxation}$	`demand_share_relaxation` over $\mathcal{N} \times \mathcal{I}$ — `demand_share_relaxation` — how far the share may stray from the one decided, as a fraction
$\mathrm{demand\_share\_limit}$	`demand_share_limit` over $\mathcal{N}$ — `demand_share_limit` — the share of demand the technologies meet together; given only where set

Variables#

Symbol	Meaning
$\mathit{demand\_share\_per\_timestep\_decision}$	`demand_share_per_timestep_decision` over $\mathcal{N} \times \mathcal{I}$ — `demand_share_per_timestep_decision` — the share of demand a technology meets, the same in every time step

Given#

Symbol	Meaning
$\mathrm{sink\_use\_equals}$	`sink_use_equals` over $\mathcal{N} \times \mathcal{I} \times \mathcal{T}$, data another file declares
$\mathit{flow\_out}$	`flow_out` over $\mathcal{N} \times \mathcal{I} \times \mathcal{C} \times \mathcal{T}$

Definitions#

Symbol	Meaning
$\mathit{demand\_share\_flow\_out}$	`demand_share_flow_out` over $\mathcal{N} \times \mathcal{I} \times \mathcal{C} \times \mathcal{T}$ — `flow_out[carriers=$carrier]` — a technology's outflow of the carrier its share is counted in
$\mathrm{demand\_share\_sink}$	`demand_share_sink` over $\mathcal{N} \times \mathcal{I} \times \mathcal{T}$ — `select_from_lookup_arrays(sink_use_equals, techs=decide_demand_share)` — the demand a technology meets a share of

Upright is what the data supplies — a parameter such as $\mathrm{demand\_share\_relaxation}$, a coordinate map, a label — and italic is what the solver chooses, such as $\mathit{demand\_share\_per\_timestep\_decision}$. An index is italic too, being what a quantifier chooses, and a set is script.

Subject to#

demand_share_per_timestep_decision_main_min

\[ \sum_{c \in \mathcal{C}} \mathit{demand\_share\_flow\_out}_{n,i,c,t} \ge \left( 1 - \mathrm{demand\_share\_relaxation}_{n,i} \right) \cdot \mathrm{demand\_share\_sink}_{n,i,t} \cdot \mathit{demand\_share\_per\_timestep\_decision}_{n,i} \qquad \forall\, n \in \mathcal{N},\ i \in \mathcal{I},\ t \in \mathcal{T} \,:\, \mathit{demand\_share\_per\_timestep\_decision}_{n,i} \text{ exists} \]

demand_share_per_timestep_decision_main_max

\[ \sum_{c \in \mathcal{C}} \mathit{demand\_share\_flow\_out}_{n,i,c,t} \le \left( 1 + \mathrm{demand\_share\_relaxation}_{n,i} \right) \cdot \mathrm{demand\_share\_sink}_{n,i,t} \cdot \mathit{demand\_share\_per\_timestep\_decision}_{n,i} \qquad \forall\, n \in \mathcal{N},\ i \in \mathcal{I},\ t \in \mathcal{T} \,:\, \mathit{demand\_share\_per\_timestep\_decision}_{n,i} \text{ exists} \]

demand_share_per_timestep_decision_sum

\[ \sum_{i \in \mathcal{I}} \mathit{demand\_share\_per\_timestep\_decision}_{n,i} = \mathrm{demand\_share\_limit}_{n} \qquad \forall\, n \in \mathcal{N} \,:\, \lvert \{ i \in \mathcal{I} \,:\, \mathit{demand\_share\_per\_timestep\_decision}_{n,i} \text{ exists} \} \rvert \ge 1 \wedge \mathrm{demand\_share\_limit}_{n} \text{ is defined} \]

Definitions#

demand_share_flow_out

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

demand_share_sink

\[ \mathrm{demand\_share\_sink}_{n,i,t} = \mathrm{sink\_use\_equals}_{n,\mathrm{decide\_demand\_share}(i),t} \qquad \forall\, n \in \mathcal{N},\ i \in \mathcal{I},\ t \in \mathcal{T} \]

Variable domains#

demand_share_per_timestep_decision

\[ \mathit{demand\_share\_per\_timestep\_decision}_{n,i} \ge 0 \qquad \forall\, n \in \mathcal{N},\ i \in \mathcal{I} \,:\, \mathrm{decide\_demand\_share}(i) \text{ is defined} \]

Assumptions#

demand_share_is_fraction

\[ \mathrm{demand\_share\_limit}_{n} \ge 0 \wedge \mathrm{demand\_share\_limit}_{n} \le 1 \qquad \forall\, n \in \mathcal{N} \,:\, \mathrm{demand\_share\_limit}_{n} \text{ is defined} \]

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

Symbol	Meaning
\(\mathrm{demand\_share\_relaxation}\)	`demand_share_relaxation` over \(\mathcal{N} \times \mathcal{I}\) — `demand_share_relaxation` — how far the share may stray from the one decided, as a fraction
\(\mathrm{demand\_share\_limit}\)	`demand_share_limit` over \(\mathcal{N}\) — `demand_share_limit` — the share of demand the technologies meet together; given only where set

Symbol	Meaning
\(\mathrm{sink\_use\_equals}\)	`sink_use_equals` over \(\mathcal{N} \times \mathcal{I} \times \mathcal{T}\), data another file declares
\(\mathit{flow\_out}\)	`flow_out` over \(\mathcal{N} \times \mathcal{I} \times \mathcal{C} \times \mathcal{T}\)

Symbol	Meaning
\(\mathit{demand\_share\_flow\_out}\)	`demand_share_flow_out` over \(\mathcal{N} \times \mathcal{I} \times \mathcal{C} \times \mathcal{T}\) — `flow_out[carriers=$carrier]` — a technology's outflow of the carrier its share is counted in
\(\mathrm{demand\_share\_sink}\)	`demand_share_sink` over \(\mathcal{N} \times \mathcal{I} \times \mathcal{T}\) — `select_from_lookup_arrays(sink_use_equals, techs=decide_demand_share)` — the demand a technology meets a share of