Uptime and downtime limits#

An extension of Calliope in fragments. Calliope's example uptime_downtime_limits.yaml: capacity factors over the whole time, forced downtime, and a cap on the time steps a unit-bought technology runs in.

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

parameters:
  capacity_factor_min:
    description: "`capacity_factor_min` — the least capacity factor a technology reaches over the whole time"
    dims: [nodes, techs]
  capacity_factor_max:
    description: "`capacity_factor_max` — the most capacity factor a technology reaches over the whole time; given only where set"
    dims: [nodes, techs]
  uptime_limit:
    description: "`uptime_limit` — the most time steps a technology runs in, weighted; given only where set"
    dims: [nodes, techs]
  downtime_periods:
    description: "`downtime_periods` — whether a technology is down for maintenance in a time step"
    dims: [nodes, techs, timesteps]
    dtype: bool

expressions:
  total_time:
    description: "`$total_time` — the hours the modelled time steps stand for"
    expression: sum(timestep_resolution * timestep_weights, over=timesteps)

given:
  parameters:
    carrier_out: { dims: [nodes, techs, carriers], dtype: bool }
    timestep_resolution: { dims: [timesteps] }
    timestep_weights: { dims: [timesteps] }
  variables:
    flow_out: { dims: [nodes, techs, carriers, timesteps] }
    flow_cap: { dims: [nodes, techs, carriers] }
    operating_units: { dims: [nodes, techs, timesteps] }

constraints:
  annual_capacity_factor_min:
    description: "`annual_capacity_factor_min` — a technology's outflow over the whole time is at least its least capacity factor"
    dims: [nodes, techs, carriers]
    where: carrier_out AND capacity_factor_min
    expression: sum(flow_out * timestep_weights, over=timesteps) >= flow_cap * capacity_factor_min * total_time
  annual_capacity_factor_max:
    description: "`annual_capacity_factor_max` — a technology's outflow over the whole time is at most its most capacity factor"
    dims: [nodes, techs, carriers]
    where: carrier_out AND capacity_factor_max
    expression: sum(flow_out * timestep_weights, over=timesteps) <= flow_cap * capacity_factor_max * total_time
  downtime_period:
    description: "`downtime_period` — a technology puts out nothing in a time step it is down"
    dims: [nodes, techs, timesteps]
    where: downtime_periods
    expression: sum(flow_out, over=carriers) == 0
  downtime_period_decision:
    description: >-
      `downtime_period_decision` — a unit-bought technology runs in at most
      its limit of time steps. Calliope's `where: operating_units` over a
      technology reads as the technology running in whole units at all
    dims: [nodes, techs]
    where: count(operating_units, over=timesteps) >= 1 AND uptime_limit
    expression: sum(operating_units * timestep_weights, over=timesteps) <= uptime_limit

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

Parameters#

Symbol	Meaning
$\mathrm{capacity\_factor\_min}$	`capacity_factor_min` over $\mathcal{N} \times \mathcal{I}$ — `capacity_factor_min` — the least capacity factor a technology reaches over the whole time
$\mathrm{capacity\_factor\_max}$	`capacity_factor_max` over $\mathcal{N} \times \mathcal{I}$ — `capacity_factor_max` — the most capacity factor a technology reaches over the whole time; given only where set
$\mathrm{uptime\_limit}$	`uptime_limit` over $\mathcal{N} \times \mathcal{I}$ — `uptime_limit` — the most time steps a technology runs in, weighted; given only where set
$\mathrm{downtime\_periods}$	`downtime_periods` over $\mathcal{N} \times \mathcal{I} \times \mathcal{T}$ — `downtime_periods` — whether a technology is down for maintenance 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
$\mathrm{timestep\_resolution}$	`timestep_resolution` over $\mathcal{T}$, data another file declares
$\mathrm{timestep\_weights}$	`timestep_weights` over $\mathcal{T}$, data another file declares
$\mathit{flow\_out}$	`flow_out` over $\mathcal{N} \times \mathcal{I} \times \mathcal{C} \times \mathcal{T}$
$\mathit{flow\_cap}$	`flow_cap` over $\mathcal{N} \times \mathcal{I} \times \mathcal{C}$
$\mathit{operating\_units}$	`operating_units` over $\mathcal{N} \times \mathcal{I} \times \mathcal{T}$

Definitions#

Symbol	Meaning
$\mathrm{total\_time}$	`total_time` (scalar) — `$total_time` — the hours the modelled time steps stand for

Subject to#

annual_capacity_factor_min

\[ \sum_{t \in \mathcal{T}} \mathit{flow\_out}_{n,i,c,t} \cdot \mathrm{timestep\_weights}_{t} \ge \mathit{flow\_cap}_{n,i,c} \cdot \mathrm{capacity\_factor\_min}_{n,i} \cdot \mathrm{total\_time} \qquad \forall\, n \in \mathcal{N},\ i \in \mathcal{I},\ c \in \mathcal{C} \,:\, \mathrm{carrier\_out}_{n,i,c} \wedge \mathrm{capacity\_factor\_min}_{n,i} \text{ is defined} \]

annual_capacity_factor_max

\[ \sum_{t \in \mathcal{T}} \mathit{flow\_out}_{n,i,c,t} \cdot \mathrm{timestep\_weights}_{t} \le \mathit{flow\_cap}_{n,i,c} \cdot \mathrm{capacity\_factor\_max}_{n,i} \cdot \mathrm{total\_time} \qquad \forall\, n \in \mathcal{N},\ i \in \mathcal{I},\ c \in \mathcal{C} \,:\, \mathrm{carrier\_out}_{n,i,c} \wedge \mathrm{capacity\_factor\_max}_{n,i} \text{ is defined} \]

downtime_period

\[ \sum_{c \in \mathcal{C}} \mathit{flow\_out}_{n,i,c,t} = 0 \qquad \forall\, n \in \mathcal{N},\ i \in \mathcal{I},\ t \in \mathcal{T} \,:\, \mathrm{downtime\_periods}_{n,i,t} \]

downtime_period_decision

\[ \sum_{t \in \mathcal{T}} \mathit{operating\_units}_{n,i,t} \cdot \mathrm{timestep\_weights}_{t} \le \mathrm{uptime\_limit}_{n,i} \qquad \forall\, n \in \mathcal{N},\ i \in \mathcal{I} \,:\, \lvert \{ t \in \mathcal{T} \,:\, \mathit{operating\_units}_{n,i,t} \text{ exists} \} \rvert \ge 1 \wedge \mathrm{uptime\_limit}_{n,i} \text{ is defined} \]

Definitions#

total_time

\[ \mathrm{total\_time} = \sum_{t \in \mathcal{T}} \mathrm{timestep\_resolution}_{t} \cdot \mathrm{timestep\_weights}_{t} \]

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

Symbol	Meaning
\(\mathrm{capacity\_factor\_min}\)	`capacity_factor_min` over \(\mathcal{N} \times \mathcal{I}\) — `capacity_factor_min` — the least capacity factor a technology reaches over the whole time
\(\mathrm{capacity\_factor\_max}\)	`capacity_factor_max` over \(\mathcal{N} \times \mathcal{I}\) — `capacity_factor_max` — the most capacity factor a technology reaches over the whole time; given only where set
\(\mathrm{uptime\_limit}\)	`uptime_limit` over \(\mathcal{N} \times \mathcal{I}\) — `uptime_limit` — the most time steps a technology runs in, weighted; given only where set
\(\mathrm{downtime\_periods}\)	`downtime_periods` over \(\mathcal{N} \times \mathcal{I} \times \mathcal{T}\) — `downtime_periods` — whether a technology is down for maintenance in a time step

Symbol	Meaning
\(\mathrm{carrier\_out}\)	`carrier_out` over \(\mathcal{N} \times \mathcal{I} \times \mathcal{C}\), data another file declares
\(\mathrm{timestep\_resolution}\)	`timestep_resolution` over \(\mathcal{T}\), data another file declares
\(\mathrm{timestep\_weights}\)	`timestep_weights` over \(\mathcal{T}\), data another file declares
\(\mathit{flow\_out}\)	`flow_out` over \(\mathcal{N} \times \mathcal{I} \times \mathcal{C} \times \mathcal{T}\)
\(\mathit{flow\_cap}\)	`flow_cap` over \(\mathcal{N} \times \mathcal{I} \times \mathcal{C}\)
\(\mathit{operating\_units}\)	`operating_units` over \(\mathcal{N} \times \mathcal{I} \times \mathcal{T}\)