Links, the ramping#
One of the 24 fragments of examples/pypsa.yaml: PyPSA's Link, the ramping rows. It reads Link_active, Link_big_m, Link_committable, Link_p, Link_p_nom_committed, Link_p_nom_effective and 8 more under given.
dimensions:
scenario:
description: the futures dispatch is chosen in, each with a weight
snapshot:
description: dispatch periods
dtype: datetime
link:
description: controllable connections, each from one bus to the buses it delivers to
period:
description: investment periods — PyPSA's `investment_periods`
dtype: int
relations:
snapshot_period:
description: the investment period a snapshot falls in
key: snapshot
values: period
parameters:
Link_ramp_limit_up:
description: most a link may raise its flow between snapshots, per unit of nominal power; no value means no limit — read at the later of the two snapshots, so the limit may change over time
dims: [scenario, snapshot, link]
Link_ramp_limit_down:
description: most a link may lower its flow between snapshots, per unit of nominal power; no value means no limit — read at the later of the two snapshots, so the limit may change over time
dims: [scenario, snapshot, link]
Link_ramp_limit_start_up:
description: most flow in the snapshot a link starts, per unit of nominal power
dims: [scenario, link]
Link_ramp_limit_shut_down:
description: most flow in the snapshot before a link stops, per unit of nominal power
dims: [scenario, link]
Link_p_init:
description: >-
the flow a link brought into the horizon — PyPSA's `p_init`, read
only where the link came in running; no value means it is unknown, so
the link carries no ramp row at the first snapshot
dims: [scenario, link]
given:
parameters:
Link_p_nom_extendable: { dims: [link], dtype: bool }
Link_committable: { dims: [link], dtype: bool }
Link_status_initial: { dims: [scenario, link], dtype: int }
Link_p_nom_mod: { dims: [link] }
Link_big_m: { dims: [scenario, link] }
Link_active: { dims: [snapshot, link], dtype: bool }
variables:
Link_p: { dims: [scenario, snapshot, link] }
Link_status: { dims: [scenario, snapshot, link], domain: integer }
Link_start_up: { dims: [scenario, snapshot, link], domain: integer }
Link_shut_down: { dims: [scenario, snapshot, link], domain: integer }
Link_p_nom_ext: { dims: [link] }
expressions:
Link_p_nom_effective: { dims: [scenario, link] }
Link_previous_status: { dims: [scenario, snapshot, link] }
Link_p_nom_committed: { dims: [scenario, link] }
expressions:
Link_previous_p:
description: >-
the flow a link carries into a snapshot — at the first, the
`p_init` it brought in where it came in running and nothing where it
came in off; the previous snapshot's after that
dims: [scenario, snapshot, link]
cases:
opening: { when: "position(snapshot) == 0", expression: Link_status_initial * Link_p_init }
otherwise: shift(Link_p, along=snapshot, offset=1)
Link_ramp_up_rate:
description: >-
the ramp limit a link's up row reads — PyPSA's `ramp_limit_up`, or the
full build where it has none, since a start-up ramp alone builds the row
dims: [scenario, snapshot, link]
cases:
given: { when: Link_ramp_limit_up, expression: Link_ramp_limit_up }
otherwise: 1
Link_ramp_down_rate:
description: >-
the ramp limit a link's down row reads — PyPSA's `ramp_limit_down`, or
the full build where it has none, since a shut-down ramp alone builds the row
dims: [scenario, snapshot, link]
cases:
given: { when: Link_ramp_limit_down, expression: Link_ramp_limit_down }
otherwise: 1
Link_start_up_rate:
description: >-
the start-up ramp a link's up row reads — PyPSA's `ramp_limit_start_up`,
or the full build where it has none
dims: [scenario, link]
cases:
given: { when: Link_ramp_limit_start_up, expression: Link_ramp_limit_start_up }
otherwise: 1
Link_shut_down_rate:
description: >-
the shut-down ramp a link's down row reads — PyPSA's
`ramp_limit_shut_down`, or the full build where it has none
dims: [scenario, link]
cases:
given: { when: Link_ramp_limit_shut_down, expression: Link_ramp_limit_shut_down }
otherwise: 1
Link_ramp_up_allowance:
description: >-
how far a link may raise flow between two snapshots — its ramp
limit of the build while it stays on, plus its start-up ramp in the
snapshot it turns on
dims: [scenario, snapshot, link]
cases:
committed:
when: Link_committable
expression: >-
Link_ramp_up_rate * Link_p_nom_committed * Link_previous_status
+ Link_start_up_rate * Link_p_nom_committed
* (Link_status - Link_previous_status)
otherwise: Link_ramp_up_rate * Link_p_nom_effective
Link_ramp_down_allowance:
description: >-
how far a link may lower flow between two snapshots — its ramp
limit of the build while it stays on, plus its shut-down ramp in the
snapshot it turns off
dims: [scenario, snapshot, link]
cases:
committed:
when: Link_committable
expression: >-
Link_ramp_down_rate * Link_p_nom_committed * Link_status
+ Link_shut_down_rate * Link_p_nom_committed
* (Link_previous_status - Link_status)
otherwise: Link_ramp_down_rate * Link_p_nom_effective
constraints:
Link_p_ramp_limit_up_run_big_m:
description: >-
`Link-p-ramp_limit_up-run-bigM` — a committed extendable link
raises flow no faster than its limit of the chosen build; the big M
releases the row in the snapshot it turns on
dims: [scenario, snapshot, link]
where: >-
Link_committable AND Link_p_nom_extendable AND NOT (Link_p_nom_mod > 0)
AND (Link_ramp_limit_up OR Link_ramp_limit_start_up)
AND (position(snapshot, by=snapshot_period, within=period) > 0 OR (position(snapshot) == 0 AND (Link_status_initial == 0 OR Link_p_init)))
AND Link_active
expression: >-
Link_p - Link_previous_p <=
Link_ramp_up_rate * Link_p_nom_ext
+ Link_big_m - Link_big_m * Link_previous_status
Link_p_ramp_limit_up_start_big_m:
description: >-
`Link-p-ramp_limit_up-start-bigM` — in the snapshot it turns on, a
committed extendable link ramps no further than its start-up ramp of
the chosen build; the big M releases the row everywhere else
dims: [scenario, snapshot, link]
where: >-
Link_committable AND Link_p_nom_extendable AND NOT (Link_p_nom_mod > 0)
AND (Link_ramp_limit_up OR Link_ramp_limit_start_up)
AND (position(snapshot, by=snapshot_period, within=period) > 0 OR (position(snapshot) == 0 AND (Link_status_initial == 0 OR Link_p_init)))
AND Link_active
expression: >-
Link_p - Link_previous_p <=
Link_start_up_rate * Link_p_nom_ext
+ Link_big_m - Link_big_m * Link_start_up
Link_p_ramp_limit_down_run_big_m:
description: >-
`Link-p-ramp_limit_down-run-bigM` — a committed extendable link
lowers flow no faster than its limit of the chosen build; the big M
releases the row in the snapshot it turns off
dims: [scenario, snapshot, link]
where: >-
Link_committable AND Link_p_nom_extendable AND NOT (Link_p_nom_mod > 0)
AND (Link_ramp_limit_down OR Link_ramp_limit_shut_down)
AND (position(snapshot, by=snapshot_period, within=period) > 0 OR (position(snapshot) == 0 AND (Link_status_initial == 0 OR Link_p_init)))
AND Link_active
expression: >-
Link_previous_p - Link_p <=
Link_ramp_down_rate * Link_p_nom_ext
+ Link_big_m - Link_big_m * Link_status
Link_p_ramp_limit_down_shut_big_m:
description: >-
`Link-p-ramp_limit_down-shut-bigM` — in the snapshot it turns off,
a committed extendable link ramps no further than its shut-down ramp of
the chosen build; the big M releases the row everywhere else
dims: [scenario, snapshot, link]
where: >-
Link_committable AND Link_p_nom_extendable AND NOT (Link_p_nom_mod > 0)
AND (Link_ramp_limit_down OR Link_ramp_limit_shut_down)
AND (position(snapshot, by=snapshot_period, within=period) > 0 OR (position(snapshot) == 0 AND (Link_status_initial == 0 OR Link_p_init)))
AND Link_active
expression: >-
Link_previous_p - Link_p <=
Link_shut_down_rate * Link_p_nom_ext
+ Link_big_m - Link_big_m * Link_shut_down
Link_p_ramp_limit_up:
description: >-
`Link-p-ramp_limit_up` — a link raises flow no faster than
its ramp limit of the build, and a committed one no further than its
start-up ramp in the snapshot it turns on. A link that came into the
horizon running carries a row at the first snapshot only where its
`p_init` gives the flow it brought in, and no link carries one at
the start of a later investment period — nor does any link a big M releases instead
dims: [scenario, snapshot, link]
where: >-
(Link_ramp_limit_up OR Link_ramp_limit_start_up)
AND NOT (Link_committable AND Link_p_nom_extendable AND NOT (Link_p_nom_mod > 0))
AND (position(snapshot, by=snapshot_period, within=period) > 0 OR (position(snapshot) == 0 AND (Link_status_initial == 0 OR Link_p_init)))
AND Link_active
expression: Link_p - Link_previous_p <= Link_ramp_up_allowance
Link_p_ramp_limit_down:
description: >-
`Link-p-ramp_limit_down` — a link lowers flow no faster than
its ramp limit of the build, and a committed one no further than its
shut-down ramp in the snapshot it turns off. A link that came into the
horizon running carries a row at the first snapshot only where its
`p_init` gives the flow it brought in, and no link carries one at
the start of a later investment period — nor does any link a big M releases instead
dims: [scenario, snapshot, link]
where: >-
(Link_ramp_limit_down OR Link_ramp_limit_shut_down)
AND NOT (Link_committable AND Link_p_nom_extendable AND NOT (Link_p_nom_mod > 0))
AND (position(snapshot, by=snapshot_period, within=period) > 0 OR (position(snapshot) == 0 AND (Link_status_initial == 0 OR Link_p_init)))
AND Link_active
expression: Link_previous_p - Link_p <= Link_ramp_down_allowance
assumptions:
Link_came_in_running_unless_committable:
holds: "Link_status_initial == 1"
where: "NOT Link_committable AND (Link_ramp_limit_up OR Link_ramp_limit_down)"
description: >-
PyPSA reads `up_time_before` of a link that is not committable in its
ramp rows. Where it is zero, PyPSA builds a row at the first snapshot
with nothing carried in, and caps the link there at zero, or at its
start-up ramp where another link of the component is committable with a
fixed build (`constraints.py:1091-1094`, `1110-1112`). PyPSA documents
the attribute as read only for a committable link and does not check
it. PyPSA has not decided which row is intended (PyPSA/PyPSA#1943). The
spec does not state that row, so it refuses the data
Sets#
| Symbol | Meaning |
|---|---|
| \(\Xi\) | index \(\xi\) — scenario — the futures dispatch is chosen in, each with a weight |
| \(\mathcal{T}\) | index \(t\) — snapshot with \(\mathrm{snapshot\_period}: \mathcal{T} \to \mathcal{Y}\) — dispatch periods |
| \(\mathcal{L}\) | index \(l\) — link — controllable connections, each from one bus to the buses it delivers to |
| \(\mathcal{Y}\) | index \(y\) — period with \(\mathrm{snapshot\_period}: \mathcal{T} \to \mathcal{Y}\) — investment periods — PyPSA's investment_periods |
Parameters#
| Symbol | Meaning |
|---|---|
| \(\mathrm{ru}^{f}\) | Link_ramp_limit_up over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — most a link may raise its flow between snapshots, per unit of nominal power; no value means no limit — read at the later of the two snapshots, so the limit may change over time |
| \(\mathrm{rd}^{f}\) | Link_ramp_limit_down over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — most a link may lower its flow between snapshots, per unit of nominal power; no value means no limit — read at the later of the two snapshots, so the limit may change over time |
| \(\mathrm{ru}^{f,\mathrm{up}}\) | Link_ramp_limit_start_up over \(\Xi \times \mathcal{L}\) — most flow in the snapshot a link starts, per unit of nominal power |
| \(\mathrm{rd}^{f,\mathrm{dn}}\) | Link_ramp_limit_shut_down over \(\Xi \times \mathcal{L}\) — most flow in the snapshot before a link stops, per unit of nominal power |
| \(\mathrm{f}^{0}\) | Link_p_init over \(\Xi \times \mathcal{L}\) — the flow a link brought into the horizon — PyPSA's p_init, read only where the link came in running; no value means it is unknown, so the link carries no ramp row at the first snapshot |
Given#
| Symbol | Meaning |
|---|---|
| \(\mathrm{ext}^{f}\) | Link_p_nom_extendable over \(\mathcal{L}\), data another file declares |
| \(\mathrm{com}^{f}\) | Link_committable over \(\mathcal{L}\), data another file declares |
| \(\mathrm{u}^{f,0}\) | Link_status_initial over \(\Xi \times \mathcal{L}\), data another file declares |
| \(\mathrm{f}^{\mathrm{mod}}\) | Link_p_nom_mod over \(\mathcal{L}\), data another file declares |
| \(\mathrm{M}^{f}\) | Link_big_m over \(\Xi \times \mathcal{L}\), data another file declares |
| \(\mathrm{on}^{f}\) | Link_active over \(\mathcal{T} \times \mathcal{L}\), data another file declares |
| \(f\) | Link_p over \(\Xi \times \mathcal{T} \times \mathcal{L}\) |
| \(u^{f}\) | Link_status over \(\Xi \times \mathcal{T} \times \mathcal{L}\) |
| \(\mathit{up}^{f}\) | Link_start_up over \(\Xi \times \mathcal{T} \times \mathcal{L}\) |
| \(\mathit{dn}^{f}\) | Link_shut_down over \(\Xi \times \mathcal{T} \times \mathcal{L}\) |
| \(F\) | Link_p_nom_ext over \(\mathcal{L}\) |
| \(\widetilde{\mathrm{f}}^{\mathrm{nom}}\) | Link_p_nom_effective over \(\Xi \times \mathcal{L}\), an expression another file defines |
| \(\overleftarrow{u}^{f}\) | Link_previous_status over \(\Xi \times \mathcal{T} \times \mathcal{L}\), an expression another file defines |
| \(\widehat{\mathrm{f}}^{\mathrm{nom}}\) | Link_p_nom_committed over \(\Xi \times \mathcal{L}\), an expression another file defines |
Definitions#
| Symbol | Meaning |
|---|---|
| \(\overleftarrow{f}\) | Link_previous_p over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — the flow a link carries into a snapshot — at the first, the p_init it brought in where it came in running and nothing where it came in off; the previous snapshot's after that |
| \(\widetilde{\mathrm{ru}}^{f}\) | Link_ramp_up_rate over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — the ramp limit a link's up row reads — PyPSA's ramp_limit_up, or the full build where it has none, since a start-up ramp alone builds the row |
| \(\widetilde{\mathrm{rd}}^{f}\) | Link_ramp_down_rate over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — the ramp limit a link's down row reads — PyPSA's ramp_limit_down, or the full build where it has none, since a shut-down ramp alone builds the row |
| \(\widetilde{\mathrm{ru}}^{f,\mathrm{up}}\) | Link_start_up_rate over \(\Xi \times \mathcal{L}\) — the start-up ramp a link's up row reads — PyPSA's ramp_limit_start_up, or the full build where it has none |
| \(\widetilde{\mathrm{rd}}^{f,\mathrm{dn}}\) | Link_shut_down_rate over \(\Xi \times \mathcal{L}\) — the shut-down ramp a link's down row reads — PyPSA's ramp_limit_shut_down, or the full build where it has none |
| \(\Delta^{f,+}\) | Link_ramp_up_allowance over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — how far a link may raise flow between two snapshots — its ramp limit of the build while it stays on, plus its start-up ramp in the snapshot it turns on |
| \(\Delta^{f,-}\) | Link_ramp_down_allowance over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — how far a link may lower flow between two snapshots — its ramp limit of the build while it stays on, plus its shut-down ramp in the snapshot it turns off |
\(\mathrm{pos}(t)\) denotes where index \(t\) sits along its dimension's own order — the order shift steps along, not the order labels sort in — counted from \(0\). The index itself stays the coordinate, so \(t\) compares against labels and \(\mathrm{pos}(t)\) against positions.
\(\mathrm{pos}_{\mathrm{relation}(t)}(t)\) counts within the group a relation puts \(t\) in: the subscript names the map, \(\mathcal{T}_{\mathrm{relation}(t)}\) is the group it lands in, and that group has a first position of its own.
Subject to#
Link_p_ramp_limit_up_run_big_m
\[
f_{\xi,t,l} - \overleftarrow{f}_{\xi,t,l} \le \widetilde{\mathrm{ru}}^{f}_{\xi,t,l} \cdot F_{l} + \mathrm{M}^{f}_{\xi,l} - \mathrm{M}^{f}_{\xi,l} \cdot \overleftarrow{u}^{f}_{\xi,t,l} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{com}^{f}_{l} \wedge \mathrm{ext}^{f}_{l} \wedge \neg \left( \mathrm{f}^{\mathrm{mod}}_{l} > 0 \right) \wedge \left( \mathrm{ru}^{f}_{\xi,t,l} \text{ is defined} \vee \mathrm{ru}^{f,\mathrm{up}}_{\xi,l} \text{ is defined} \right) \wedge \left( \mathrm{pos}_{\mathrm{snapshot\_period}(t)}(t) > 0 \vee \mathrm{pos}(t) = 0 \wedge \left( \mathrm{u}^{f,0}_{\xi,l} = 0 \vee \mathrm{f}^{0}_{\xi,l} \text{ is defined} \right) \right) \wedge \mathrm{on}^{f}_{t,l}
\]
Link_p_ramp_limit_up_start_big_m
\[
f_{\xi,t,l} - \overleftarrow{f}_{\xi,t,l} \le \widetilde{\mathrm{ru}}^{f,\mathrm{up}}_{\xi,l} \cdot F_{l} + \mathrm{M}^{f}_{\xi,l} - \mathrm{M}^{f}_{\xi,l} \cdot \mathit{up}^{f}_{\xi,t,l} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{com}^{f}_{l} \wedge \mathrm{ext}^{f}_{l} \wedge \neg \left( \mathrm{f}^{\mathrm{mod}}_{l} > 0 \right) \wedge \left( \mathrm{ru}^{f}_{\xi,t,l} \text{ is defined} \vee \mathrm{ru}^{f,\mathrm{up}}_{\xi,l} \text{ is defined} \right) \wedge \left( \mathrm{pos}_{\mathrm{snapshot\_period}(t)}(t) > 0 \vee \mathrm{pos}(t) = 0 \wedge \left( \mathrm{u}^{f,0}_{\xi,l} = 0 \vee \mathrm{f}^{0}_{\xi,l} \text{ is defined} \right) \right) \wedge \mathrm{on}^{f}_{t,l}
\]
Link_p_ramp_limit_down_run_big_m
\[
\overleftarrow{f}_{\xi,t,l} - f_{\xi,t,l} \le \widetilde{\mathrm{rd}}^{f}_{\xi,t,l} \cdot F_{l} + \mathrm{M}^{f}_{\xi,l} - \mathrm{M}^{f}_{\xi,l} \cdot u^{f}_{\xi,t,l} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{com}^{f}_{l} \wedge \mathrm{ext}^{f}_{l} \wedge \neg \left( \mathrm{f}^{\mathrm{mod}}_{l} > 0 \right) \wedge \left( \mathrm{rd}^{f}_{\xi,t,l} \text{ is defined} \vee \mathrm{rd}^{f,\mathrm{dn}}_{\xi,l} \text{ is defined} \right) \wedge \left( \mathrm{pos}_{\mathrm{snapshot\_period}(t)}(t) > 0 \vee \mathrm{pos}(t) = 0 \wedge \left( \mathrm{u}^{f,0}_{\xi,l} = 0 \vee \mathrm{f}^{0}_{\xi,l} \text{ is defined} \right) \right) \wedge \mathrm{on}^{f}_{t,l}
\]
Link_p_ramp_limit_down_shut_big_m
\[
\overleftarrow{f}_{\xi,t,l} - f_{\xi,t,l} \le \widetilde{\mathrm{rd}}^{f,\mathrm{dn}}_{\xi,l} \cdot F_{l} + \mathrm{M}^{f}_{\xi,l} - \mathrm{M}^{f}_{\xi,l} \cdot \mathit{dn}^{f}_{\xi,t,l} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{com}^{f}_{l} \wedge \mathrm{ext}^{f}_{l} \wedge \neg \left( \mathrm{f}^{\mathrm{mod}}_{l} > 0 \right) \wedge \left( \mathrm{rd}^{f}_{\xi,t,l} \text{ is defined} \vee \mathrm{rd}^{f,\mathrm{dn}}_{\xi,l} \text{ is defined} \right) \wedge \left( \mathrm{pos}_{\mathrm{snapshot\_period}(t)}(t) > 0 \vee \mathrm{pos}(t) = 0 \wedge \left( \mathrm{u}^{f,0}_{\xi,l} = 0 \vee \mathrm{f}^{0}_{\xi,l} \text{ is defined} \right) \right) \wedge \mathrm{on}^{f}_{t,l}
\]
Link_p_ramp_limit_up
\[
f_{\xi,t,l} - \overleftarrow{f}_{\xi,t,l} \le \Delta^{f,+}_{\xi,t,l} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \left( \mathrm{ru}^{f}_{\xi,t,l} \text{ is defined} \vee \mathrm{ru}^{f,\mathrm{up}}_{\xi,l} \text{ is defined} \right) \wedge \neg \left( \mathrm{com}^{f}_{l} \wedge \mathrm{ext}^{f}_{l} \wedge \neg \left( \mathrm{f}^{\mathrm{mod}}_{l} > 0 \right) \right) \wedge \left( \mathrm{pos}_{\mathrm{snapshot\_period}(t)}(t) > 0 \vee \mathrm{pos}(t) = 0 \wedge \left( \mathrm{u}^{f,0}_{\xi,l} = 0 \vee \mathrm{f}^{0}_{\xi,l} \text{ is defined} \right) \right) \wedge \mathrm{on}^{f}_{t,l}
\]
Link_p_ramp_limit_down
\[
\overleftarrow{f}_{\xi,t,l} - f_{\xi,t,l} \le \Delta^{f,-}_{\xi,t,l} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \left( \mathrm{rd}^{f}_{\xi,t,l} \text{ is defined} \vee \mathrm{rd}^{f,\mathrm{dn}}_{\xi,l} \text{ is defined} \right) \wedge \neg \left( \mathrm{com}^{f}_{l} \wedge \mathrm{ext}^{f}_{l} \wedge \neg \left( \mathrm{f}^{\mathrm{mod}}_{l} > 0 \right) \right) \wedge \left( \mathrm{pos}_{\mathrm{snapshot\_period}(t)}(t) > 0 \vee \mathrm{pos}(t) = 0 \wedge \left( \mathrm{u}^{f,0}_{\xi,l} = 0 \vee \mathrm{f}^{0}_{\xi,l} \text{ is defined} \right) \right) \wedge \mathrm{on}^{f}_{t,l}
\]
Definitions#
Link_previous_p
\[
\overleftarrow{f}_{\xi,t,l} = \begin{cases} \mathrm{u}^{f,0}_{\xi,l} \cdot \mathrm{f}^{0}_{\xi,l} & \text{if } \mathrm{pos}(t) = 0 \\ f_{\xi,t - 1,l} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L}
\]
Link_ramp_up_rate
\[
\widetilde{\mathrm{ru}}^{f}_{\xi,t,l} = \begin{cases} \mathrm{ru}^{f}_{\xi,t,l} & \text{if } \mathrm{ru}^{f}_{\xi,t,l} \text{ is defined} \\ 1 & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L}
\]
Link_ramp_down_rate
\[
\widetilde{\mathrm{rd}}^{f}_{\xi,t,l} = \begin{cases} \mathrm{rd}^{f}_{\xi,t,l} & \text{if } \mathrm{rd}^{f}_{\xi,t,l} \text{ is defined} \\ 1 & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L}
\]
Link_start_up_rate
\[
\widetilde{\mathrm{ru}}^{f,\mathrm{up}}_{\xi,l} = \begin{cases} \mathrm{ru}^{f,\mathrm{up}}_{\xi,l} & \text{if } \mathrm{ru}^{f,\mathrm{up}}_{\xi,l} \text{ is defined} \\ 1 & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ l \in \mathcal{L}
\]
Link_shut_down_rate
\[
\widetilde{\mathrm{rd}}^{f,\mathrm{dn}}_{\xi,l} = \begin{cases} \mathrm{rd}^{f,\mathrm{dn}}_{\xi,l} & \text{if } \mathrm{rd}^{f,\mathrm{dn}}_{\xi,l} \text{ is defined} \\ 1 & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ l \in \mathcal{L}
\]
Link_ramp_up_allowance
\[
\Delta^{f,+}_{\xi,t,l} = \begin{cases} \widetilde{\mathrm{ru}}^{f}_{\xi,t,l} \cdot \widehat{\mathrm{f}}^{\mathrm{nom}}_{\xi,l} \cdot \overleftarrow{u}^{f}_{\xi,t,l} + \widetilde{\mathrm{ru}}^{f,\mathrm{up}}_{\xi,l} \cdot \widehat{\mathrm{f}}^{\mathrm{nom}}_{\xi,l} \cdot \left( u^{f}_{\xi,t,l} - \overleftarrow{u}^{f}_{\xi,t,l} \right) & \text{if } \mathrm{com}^{f}_{l} \\ \widetilde{\mathrm{ru}}^{f}_{\xi,t,l} \cdot \widetilde{\mathrm{f}}^{\mathrm{nom}}_{\xi,l} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L}
\]
Link_ramp_down_allowance
\[
\Delta^{f,-}_{\xi,t,l} = \begin{cases} \widetilde{\mathrm{rd}}^{f}_{\xi,t,l} \cdot \widehat{\mathrm{f}}^{\mathrm{nom}}_{\xi,l} \cdot u^{f}_{\xi,t,l} + \widetilde{\mathrm{rd}}^{f,\mathrm{dn}}_{\xi,l} \cdot \widehat{\mathrm{f}}^{\mathrm{nom}}_{\xi,l} \cdot \left( \overleftarrow{u}^{f}_{\xi,t,l} - u^{f}_{\xi,t,l} \right) & \text{if } \mathrm{com}^{f}_{l} \\ \widetilde{\mathrm{rd}}^{f}_{\xi,t,l} \cdot \widetilde{\mathrm{f}}^{\mathrm{nom}}_{\xi,l} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L}
\]
Assumptions#
Link_came_in_running_unless_committable
\[
\mathrm{u}^{f,0}_{\xi,l} = 1 \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \neg \mathrm{com}^{f}_{l} \wedge \left( \mathrm{ru}^{f}_{\xi,t,l} \text{ is defined} \vee \mathrm{rd}^{f}_{\xi,t,l} \text{ is defined} \right)
\]