Monthly peak flow charge#

An extension of Calliope in fragments. Calliope's example monthly_peak_flow_charge.yaml: a cost on the peak outflow of each month. Calliope restates cost_operation_fixed to add it; here it is a term. The month of a time step is a relation, so the row is one per time step, not one per time step and month.

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
  costs:
    description: Calliope's `costs` — cost classes, such as monetary and CO2
  timesteps:
    description: Calliope's `timesteps` — time steps, in order
    dtype: datetime
  months:
    description: Calliope's `months` — the months of the year
    dtype: int

relations:
  lookup_month:
    description: >-
      `lookup_month` — the month a time step falls in. Calliope ships it as a
      boolean table over time step and month, and builds its row over both;
      as a relation the row is one per time step
    key: timesteps
    values: months

parameters:
  monthly_peak_mode:
    description: "`monthly_peak_mode` — whether a technology's peak outflow in a month is priced"
    dims: [nodes, techs, carriers]
    dtype: bool
  cost_month_peak:
    description: "`cost_month_peak` — the cost of one unit of peak outflow in a month"
    dims: [nodes, techs, costs]

variables:
  flow_peak_month:
    description: "`flow_peak_month` — a technology's peak outflow in a month"
    dims: [nodes, techs, carriers, months]
    where: carrier_out AND monthly_peak_mode
    bounds: { lower: 0, upper: flow_cap_max }
    absence: zero

expressions:
  cost_month_peak_charge:
    description: "`sum(cost_month_peak * flow_peak_month, over=[carriers, months])` — the term Calliope writes into `cost_operation_fixed`, by restating it whole"
    expression: sum(cost_month_peak * flow_peak_month, over=[carriers, months])

given:
  parameters:
    carrier_out: { dims: [nodes, techs, carriers], dtype: bool }
    flow_cap_max: { dims: [nodes, techs] }
  variables:
    flow_out: { dims: [nodes, techs, carriers, timesteps] }
  expressions:
    cost_operation_fixed: { dims: [nodes, techs, costs], term: cost_month_peak_charge }

constraints:
  set_peak_month_flow:
    description: "`set_peak_month_flow` — the peak outflow in a month is at least the outflow in each of its time steps"
    dims: [nodes, techs, carriers, timesteps]
    where: at(flow_peak_month, by=lookup_month, over=months, into=timesteps)
    expression: flow_out <= at(flow_peak_month, by=lookup_month, over=months, into=timesteps)

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
\(\mathcal{K}\)	index \(k\) — `costs` — Calliope's `costs` — cost classes, such as monetary and CO2
\(\mathcal{T}\)	index \(t\) — `timesteps` with \(\mathrm{lookup\_month}: \mathcal{T} \to \mathcal{M}\) — Calliope's `timesteps` — time steps, in order
\(\mathcal{M}\)	index \(m\) — `months` with \(\mathrm{lookup\_month}: \mathcal{T} \to \mathcal{M}\) — Calliope's `months` — the months of the year

Parameters#

Symbol	Meaning
\(\mathrm{monthly\_peak\_mode}\)	`monthly_peak_mode` over \(\mathcal{N} \times \mathcal{I} \times \mathcal{C}\) — `monthly_peak_mode` — whether a technology's peak outflow in a month is priced
\(\mathrm{cost\_month\_peak}\)	`cost_month_peak` over \(\mathcal{N} \times \mathcal{I} \times \mathcal{K}\) — `cost_month_peak` — the cost of one unit of peak outflow in a month

Variables#

Symbol	Meaning
\(\mathit{flow\_peak\_month}\)	`flow_peak_month` over \(\mathcal{N} \times \mathcal{I} \times \mathcal{C} \times \mathcal{M}\) — `flow_peak_month` — a technology's peak outflow in a month

Given#

Symbol	Meaning
\(\mathrm{carrier\_out}\)	`carrier_out` over \(\mathcal{N} \times \mathcal{I} \times \mathcal{C}\), data another file declares
\(\mathrm{flow\_cap\_max}\)	`flow_cap_max` over \(\mathcal{N} \times \mathcal{I}\), data another file declares
\(\mathit{flow\_out}\)	`flow_out` over \(\mathcal{N} \times \mathcal{I} \times \mathcal{C} \times \mathcal{T}\)
\(\mathit{cost\_operation\_fixed}\)	`cost_operation_fixed` over \(\mathcal{N} \times \mathcal{I} \times \mathcal{K}\), an expression this file adds `cost_month_peak_charge` to

Definitions#

Symbol	Meaning
\(\mathit{cost\_month\_peak\_charge}\)	`cost_month_peak_charge` over \(\mathcal{N} \times \mathcal{I} \times \mathcal{K}\) — `sum(cost_month_peak * flow_peak_month, over=[carriers, months])` — the term Calliope writes into `cost_operation_fixed`, by restating it whole

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

Subject to#

set_peak_month_flow

\[ \mathit{flow\_out}_{n,i,c,t} \le \mathit{flow\_peak\_month}_{n,i,c,\mathrm{lookup\_month}(t)} \qquad \forall\, n \in \mathcal{N},\ i \in \mathcal{I},\ c \in \mathcal{C},\ t \in \mathcal{T} \,:\, \mathit{flow\_peak\_month}_{n,i,c,\mathrm{lookup\_month}(t)} \text{ exists} \]

Definitions#

cost_month_peak_charge

\[ \mathit{cost\_month\_peak\_charge}_{n,i,k} = \sum_{c \in \mathcal{C},\ m \in \mathcal{M}} \mathrm{cost\_month\_peak}_{n,i,k} \cdot \mathit{flow\_peak\_month}_{n,i,c,m} \qquad \forall\, n \in \mathcal{N},\ i \in \mathcal{I},\ k \in \mathcal{K} \]

Variable domains#

flow_peak_month

\[ 0 \le \mathit{flow\_peak\_month}_{n,i,c,m} \le \mathrm{flow\_cap\_max}_{n,i} \qquad \forall\, n \in \mathcal{N},\ i \in \mathcal{I},\ c \in \mathcal{C},\ m \in \mathcal{M} \,:\, \mathrm{carrier\_out}_{n,i,c} \wedge \mathrm{monthly\_peak\_mode}_{n,i,c} \]