Generators, the ramping#
One of the 24 fragments of examples/pypsa.yaml: PyPSA's Generator, the ramping rows. It reads Generator_active, Generator_big_m, Generator_committable, Generator_p, Generator_p_nom_committed, Generator_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
generator:
description: generating units, each on one bus
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:
Generator_ramp_limit_up:
description: most a generator may raise its output 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, generator]
Generator_ramp_limit_down:
description: most a generator may lower its output 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, generator]
Generator_ramp_limit_start_up:
description: most output in the snapshot a unit starts, per unit of nominal power
dims: [scenario, generator]
Generator_ramp_limit_shut_down:
description: most output in the snapshot before a unit stops, per unit of nominal power
dims: [scenario, generator]
Generator_p_init:
description: >-
the output a unit brought into the horizon — PyPSA's `p_init`, read
only where the unit came in running; no value means it is unknown, so
the unit carries no ramp row at the first snapshot
dims: [scenario, generator]
given:
parameters:
Generator_p_nom_extendable: { dims: [generator], dtype: bool }
Generator_committable: { dims: [generator], dtype: bool }
Generator_status_initial: { dims: [scenario, generator], dtype: int }
Generator_p_nom_mod: { dims: [generator] }
Generator_big_m: { dims: [scenario, generator] }
Generator_active: { dims: [snapshot, generator], dtype: bool }
variables:
Generator_p: { dims: [scenario, snapshot, generator] }
Generator_status: { dims: [scenario, snapshot, generator], domain: integer }
Generator_start_up: { dims: [scenario, snapshot, generator], domain: integer }
Generator_shut_down: { dims: [scenario, snapshot, generator], domain: integer }
Generator_p_nom_ext: { dims: [generator] }
expressions:
Generator_previous_status: { dims: [scenario, snapshot, generator] }
Generator_p_nom_effective: { dims: [scenario, generator] }
Generator_p_nom_committed: { dims: [scenario, generator] }
expressions:
Generator_previous_p:
description: >-
the output a generator 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, generator]
cases:
opening: { when: "position(snapshot) == 0", expression: Generator_status_initial * Generator_p_init }
otherwise: shift(Generator_p, along=snapshot, offset=1)
Generator_ramp_up_rate:
description: >-
the ramp limit a unit'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, generator]
cases:
given: { when: Generator_ramp_limit_up, expression: Generator_ramp_limit_up }
otherwise: 1
Generator_ramp_down_rate:
description: >-
the ramp limit a unit'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, generator]
cases:
given: { when: Generator_ramp_limit_down, expression: Generator_ramp_limit_down }
otherwise: 1
Generator_start_up_rate:
description: >-
the start-up ramp a unit's up row reads — PyPSA's `ramp_limit_start_up`,
or the full build where it has none
dims: [scenario, generator]
cases:
given: { when: Generator_ramp_limit_start_up, expression: Generator_ramp_limit_start_up }
otherwise: 1
Generator_shut_down_rate:
description: >-
the shut-down ramp a unit's down row reads — PyPSA's
`ramp_limit_shut_down`, or the full build where it has none
dims: [scenario, generator]
cases:
given: { when: Generator_ramp_limit_shut_down, expression: Generator_ramp_limit_shut_down }
otherwise: 1
Generator_ramp_up_allowance:
description: >-
how far a generator may raise output 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, generator]
cases:
committed:
when: Generator_committable
expression: >-
Generator_ramp_up_rate * Generator_p_nom_committed * Generator_previous_status
+ Generator_start_up_rate * Generator_p_nom_committed
* (Generator_status - Generator_previous_status)
otherwise: Generator_ramp_up_rate * Generator_p_nom_effective
Generator_ramp_down_allowance:
description: >-
how far a generator may lower output 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, generator]
cases:
committed:
when: Generator_committable
expression: >-
Generator_ramp_down_rate * Generator_p_nom_committed * Generator_status
+ Generator_shut_down_rate * Generator_p_nom_committed
* (Generator_previous_status - Generator_status)
otherwise: Generator_ramp_down_rate * Generator_p_nom_effective
constraints:
Generator_p_ramp_limit_up_run_big_m:
description: >-
`Generator-p-ramp_limit_up-run-bigM` — a committed extendable unit
raises output no faster than its limit of the chosen build; the big M
releases the row in the snapshot it turns on
dims: [scenario, snapshot, generator]
where: >-
Generator_committable AND Generator_p_nom_extendable AND NOT (Generator_p_nom_mod > 0)
AND (Generator_ramp_limit_up OR Generator_ramp_limit_start_up)
AND (position(snapshot, by=snapshot_period, within=period) > 0 OR (position(snapshot) == 0 AND (Generator_status_initial == 0 OR Generator_p_init)))
AND Generator_active
expression: >-
Generator_p - Generator_previous_p <=
Generator_ramp_up_rate * Generator_p_nom_ext
+ Generator_big_m - Generator_big_m * Generator_previous_status
Generator_p_ramp_limit_up_start_big_m:
description: >-
`Generator-p-ramp_limit_up-start-bigM` — in the snapshot it turns on, a
committed extendable unit ramps no further than its start-up ramp of
the chosen build; the big M releases the row everywhere else
dims: [scenario, snapshot, generator]
where: >-
Generator_committable AND Generator_p_nom_extendable AND NOT (Generator_p_nom_mod > 0)
AND (Generator_ramp_limit_up OR Generator_ramp_limit_start_up)
AND (position(snapshot, by=snapshot_period, within=period) > 0 OR (position(snapshot) == 0 AND (Generator_status_initial == 0 OR Generator_p_init)))
AND Generator_active
expression: >-
Generator_p - Generator_previous_p <=
Generator_start_up_rate * Generator_p_nom_ext
+ Generator_big_m - Generator_big_m * Generator_start_up
Generator_p_ramp_limit_down_run_big_m:
description: >-
`Generator-p-ramp_limit_down-run-bigM` — a committed extendable unit
lowers output no faster than its limit of the chosen build; the big M
releases the row in the snapshot it turns off
dims: [scenario, snapshot, generator]
where: >-
Generator_committable AND Generator_p_nom_extendable AND NOT (Generator_p_nom_mod > 0)
AND (Generator_ramp_limit_down OR Generator_ramp_limit_shut_down)
AND (position(snapshot, by=snapshot_period, within=period) > 0 OR (position(snapshot) == 0 AND (Generator_status_initial == 0 OR Generator_p_init)))
AND Generator_active
expression: >-
Generator_previous_p - Generator_p <=
Generator_ramp_down_rate * Generator_p_nom_ext
+ Generator_big_m - Generator_big_m * Generator_status
Generator_p_ramp_limit_down_shut_big_m:
description: >-
`Generator-p-ramp_limit_down-shut-bigM` — in the snapshot it turns off,
a committed extendable unit ramps no further than its shut-down ramp of
the chosen build; the big M releases the row everywhere else
dims: [scenario, snapshot, generator]
where: >-
Generator_committable AND Generator_p_nom_extendable AND NOT (Generator_p_nom_mod > 0)
AND (Generator_ramp_limit_down OR Generator_ramp_limit_shut_down)
AND (position(snapshot, by=snapshot_period, within=period) > 0 OR (position(snapshot) == 0 AND (Generator_status_initial == 0 OR Generator_p_init)))
AND Generator_active
expression: >-
Generator_previous_p - Generator_p <=
Generator_shut_down_rate * Generator_p_nom_ext
+ Generator_big_m - Generator_big_m * Generator_shut_down
Generator_p_ramp_limit_up:
description: >-
`Generator-p-ramp_limit_up` — a generator raises output 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 unit that came into the
horizon running carries a row at the first snapshot only where its
`p_init` gives the output it brought in, and no unit carries one at
the start of a later investment period — nor does any unit a big M releases instead
dims: [scenario, snapshot, generator]
where: >-
(Generator_ramp_limit_up OR Generator_ramp_limit_start_up)
AND NOT (Generator_committable AND Generator_p_nom_extendable AND NOT (Generator_p_nom_mod > 0))
AND (position(snapshot, by=snapshot_period, within=period) > 0 OR (position(snapshot) == 0 AND (Generator_status_initial == 0 OR Generator_p_init)))
AND Generator_active
expression: Generator_p - Generator_previous_p <= Generator_ramp_up_allowance
Generator_p_ramp_limit_down:
description: >-
`Generator-p-ramp_limit_down` — a generator lowers output 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 unit that came into the
horizon running carries a row at the first snapshot only where its
`p_init` gives the output it brought in, and no unit carries one at
the start of a later investment period — nor does any unit a big M releases instead
dims: [scenario, snapshot, generator]
where: >-
(Generator_ramp_limit_down OR Generator_ramp_limit_shut_down)
AND NOT (Generator_committable AND Generator_p_nom_extendable AND NOT (Generator_p_nom_mod > 0))
AND (position(snapshot, by=snapshot_period, within=period) > 0 OR (position(snapshot) == 0 AND (Generator_status_initial == 0 OR Generator_p_init)))
AND Generator_active
expression: Generator_previous_p - Generator_p <= Generator_ramp_down_allowance
assumptions:
Generator_came_in_running_unless_committable:
holds: "Generator_status_initial == 1"
where: "NOT Generator_committable AND (Generator_ramp_limit_up OR Generator_ramp_limit_down)"
description: >-
PyPSA reads `up_time_before` of a unit 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 unit there at zero, or at its
start-up ramp where another unit 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 unit 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 |
| \(\mathcal{G}\) | index \(g\) — generator — generating units, each on one bus |
| \(\mathcal{Y}\) | index \(y\) — period with \(\mathrm{snapshot\_period}: \mathcal{T} \to \mathcal{Y}\) — investment periods — PyPSA's investment_periods |
Parameters#
| Symbol | Meaning |
|---|---|
| \(\mathrm{ru}\) | Generator_ramp_limit_up over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — most a generator may raise its output 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 |
| \(\mathrm{rd}\) | Generator_ramp_limit_down over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — most a generator may lower its output 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 |
| \(\mathrm{ru}^{\mathrm{up}}\) | Generator_ramp_limit_start_up over \(\Xi \times \mathcal{G}\) — most output in the snapshot a unit starts, per unit of nominal power |
| \(\mathrm{rd}^{\mathrm{dn}}\) | Generator_ramp_limit_shut_down over \(\Xi \times \mathcal{G}\) — most output in the snapshot before a unit stops, per unit of nominal power |
| \(\mathrm{p}^{0}\) | Generator_p_init over \(\Xi \times \mathcal{G}\) — the output a unit brought into the horizon — PyPSA's p_init, read only where the unit came in running; no value means it is unknown, so the unit carries no ramp row at the first snapshot |
Given#
| Symbol | Meaning |
|---|---|
| \(\mathrm{ext}\) | Generator_p_nom_extendable over \(\mathcal{G}\), data another file declares |
| \(\mathrm{com}\) | Generator_committable over \(\mathcal{G}\), data another file declares |
| \(\mathrm{u}^{0}\) | Generator_status_initial over \(\Xi \times \mathcal{G}\), data another file declares |
| \(\mathrm{p}^{\mathrm{mod}}\) | Generator_p_nom_mod over \(\mathcal{G}\), data another file declares |
| \(\mathrm{M}\) | Generator_big_m over \(\Xi \times \mathcal{G}\), data another file declares |
| \(\mathrm{on}\) | Generator_active over \(\mathcal{T} \times \mathcal{G}\), data another file declares |
| \(p\) | Generator_p over \(\Xi \times \mathcal{T} \times \mathcal{G}\) |
| \(u\) | Generator_status over \(\Xi \times \mathcal{T} \times \mathcal{G}\) |
| \(\mathit{up}\) | Generator_start_up over \(\Xi \times \mathcal{T} \times \mathcal{G}\) |
| \(\mathit{dn}\) | Generator_shut_down over \(\Xi \times \mathcal{T} \times \mathcal{G}\) |
| \(P\) | Generator_p_nom_ext over \(\mathcal{G}\) |
| \(\overleftarrow{u}\) | Generator_previous_status over \(\Xi \times \mathcal{T} \times \mathcal{G}\), an expression another file defines |
| \(\widetilde{\mathrm{p}}^{\mathrm{nom}}\) | Generator_p_nom_effective over \(\Xi \times \mathcal{G}\), an expression another file defines |
| \(\widehat{\mathrm{p}}^{\mathrm{nom}}\) | Generator_p_nom_committed over \(\Xi \times \mathcal{G}\), an expression another file defines |
Definitions#
| Symbol | Meaning |
|---|---|
| \(\overleftarrow{p}\) | Generator_previous_p over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — the output a generator 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 |
| \(\widetilde{\mathrm{ru}}\) | Generator_ramp_up_rate over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — the ramp limit a unit'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 |
| \(\widetilde{\mathrm{rd}}\) | Generator_ramp_down_rate over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — the ramp limit a unit'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 |
| \(\widetilde{\mathrm{ru}}^{\mathrm{up}}\) | Generator_start_up_rate over \(\Xi \times \mathcal{G}\) — the start-up ramp a unit's up row reads — PyPSA's ramp_limit_start_up, or the full build where it has none |
| \(\widetilde{\mathrm{rd}}^{\mathrm{dn}}\) | Generator_shut_down_rate over \(\Xi \times \mathcal{G}\) — the shut-down ramp a unit's down row reads — PyPSA's ramp_limit_shut_down, or the full build where it has none |
| \(\Delta^{+}\) | Generator_ramp_up_allowance over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — how far a generator may raise output between two snapshots — its ramp limit of the build while it stays on, plus its start-up ramp in the snapshot it turns on |
| \(\Delta^{-}\) | Generator_ramp_down_allowance over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — how far a generator may lower output 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#
Generator_p_ramp_limit_up_run_big_m
\[
p_{\xi,t,g} - \overleftarrow{p}_{\xi,t,g} \le \widetilde{\mathrm{ru}}_{\xi,t,g} \cdot P_{g} + \mathrm{M}_{\xi,g} - \mathrm{M}_{\xi,g} \cdot \overleftarrow{u}_{\xi,t,g} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{com}_{g} \wedge \mathrm{ext}_{g} \wedge \neg \left( \mathrm{p}^{\mathrm{mod}}_{g} > 0 \right) \wedge \left( \mathrm{ru}_{\xi,t,g} \text{ is defined} \vee \mathrm{ru}^{\mathrm{up}}_{\xi,g} \text{ is defined} \right) \wedge \left( \mathrm{pos}_{\mathrm{snapshot\_period}(t)}(t) > 0 \vee \mathrm{pos}(t) = 0 \wedge \left( \mathrm{u}^{0}_{\xi,g} = 0 \vee \mathrm{p}^{0}_{\xi,g} \text{ is defined} \right) \right) \wedge \mathrm{on}_{t,g}
\]
Generator_p_ramp_limit_up_start_big_m
\[
p_{\xi,t,g} - \overleftarrow{p}_{\xi,t,g} \le \widetilde{\mathrm{ru}}^{\mathrm{up}}_{\xi,g} \cdot P_{g} + \mathrm{M}_{\xi,g} - \mathrm{M}_{\xi,g} \cdot \mathit{up}_{\xi,t,g} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{com}_{g} \wedge \mathrm{ext}_{g} \wedge \neg \left( \mathrm{p}^{\mathrm{mod}}_{g} > 0 \right) \wedge \left( \mathrm{ru}_{\xi,t,g} \text{ is defined} \vee \mathrm{ru}^{\mathrm{up}}_{\xi,g} \text{ is defined} \right) \wedge \left( \mathrm{pos}_{\mathrm{snapshot\_period}(t)}(t) > 0 \vee \mathrm{pos}(t) = 0 \wedge \left( \mathrm{u}^{0}_{\xi,g} = 0 \vee \mathrm{p}^{0}_{\xi,g} \text{ is defined} \right) \right) \wedge \mathrm{on}_{t,g}
\]
Generator_p_ramp_limit_down_run_big_m
\[
\overleftarrow{p}_{\xi,t,g} - p_{\xi,t,g} \le \widetilde{\mathrm{rd}}_{\xi,t,g} \cdot P_{g} + \mathrm{M}_{\xi,g} - \mathrm{M}_{\xi,g} \cdot u_{\xi,t,g} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{com}_{g} \wedge \mathrm{ext}_{g} \wedge \neg \left( \mathrm{p}^{\mathrm{mod}}_{g} > 0 \right) \wedge \left( \mathrm{rd}_{\xi,t,g} \text{ is defined} \vee \mathrm{rd}^{\mathrm{dn}}_{\xi,g} \text{ is defined} \right) \wedge \left( \mathrm{pos}_{\mathrm{snapshot\_period}(t)}(t) > 0 \vee \mathrm{pos}(t) = 0 \wedge \left( \mathrm{u}^{0}_{\xi,g} = 0 \vee \mathrm{p}^{0}_{\xi,g} \text{ is defined} \right) \right) \wedge \mathrm{on}_{t,g}
\]
Generator_p_ramp_limit_down_shut_big_m
\[
\overleftarrow{p}_{\xi,t,g} - p_{\xi,t,g} \le \widetilde{\mathrm{rd}}^{\mathrm{dn}}_{\xi,g} \cdot P_{g} + \mathrm{M}_{\xi,g} - \mathrm{M}_{\xi,g} \cdot \mathit{dn}_{\xi,t,g} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{com}_{g} \wedge \mathrm{ext}_{g} \wedge \neg \left( \mathrm{p}^{\mathrm{mod}}_{g} > 0 \right) \wedge \left( \mathrm{rd}_{\xi,t,g} \text{ is defined} \vee \mathrm{rd}^{\mathrm{dn}}_{\xi,g} \text{ is defined} \right) \wedge \left( \mathrm{pos}_{\mathrm{snapshot\_period}(t)}(t) > 0 \vee \mathrm{pos}(t) = 0 \wedge \left( \mathrm{u}^{0}_{\xi,g} = 0 \vee \mathrm{p}^{0}_{\xi,g} \text{ is defined} \right) \right) \wedge \mathrm{on}_{t,g}
\]
Generator_p_ramp_limit_up
\[
p_{\xi,t,g} - \overleftarrow{p}_{\xi,t,g} \le \Delta^{+}_{\xi,t,g} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \left( \mathrm{ru}_{\xi,t,g} \text{ is defined} \vee \mathrm{ru}^{\mathrm{up}}_{\xi,g} \text{ is defined} \right) \wedge \neg \left( \mathrm{com}_{g} \wedge \mathrm{ext}_{g} \wedge \neg \left( \mathrm{p}^{\mathrm{mod}}_{g} > 0 \right) \right) \wedge \left( \mathrm{pos}_{\mathrm{snapshot\_period}(t)}(t) > 0 \vee \mathrm{pos}(t) = 0 \wedge \left( \mathrm{u}^{0}_{\xi,g} = 0 \vee \mathrm{p}^{0}_{\xi,g} \text{ is defined} \right) \right) \wedge \mathrm{on}_{t,g}
\]
Generator_p_ramp_limit_down
\[
\overleftarrow{p}_{\xi,t,g} - p_{\xi,t,g} \le \Delta^{-}_{\xi,t,g} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \left( \mathrm{rd}_{\xi,t,g} \text{ is defined} \vee \mathrm{rd}^{\mathrm{dn}}_{\xi,g} \text{ is defined} \right) \wedge \neg \left( \mathrm{com}_{g} \wedge \mathrm{ext}_{g} \wedge \neg \left( \mathrm{p}^{\mathrm{mod}}_{g} > 0 \right) \right) \wedge \left( \mathrm{pos}_{\mathrm{snapshot\_period}(t)}(t) > 0 \vee \mathrm{pos}(t) = 0 \wedge \left( \mathrm{u}^{0}_{\xi,g} = 0 \vee \mathrm{p}^{0}_{\xi,g} \text{ is defined} \right) \right) \wedge \mathrm{on}_{t,g}
\]
Definitions#
Generator_previous_p
\[
\overleftarrow{p}_{\xi,t,g} = \begin{cases} \mathrm{u}^{0}_{\xi,g} \cdot \mathrm{p}^{0}_{\xi,g} & \text{if } \mathrm{pos}(t) = 0 \\ p_{\xi,t - 1,g} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G}
\]
Generator_ramp_up_rate
\[
\widetilde{\mathrm{ru}}_{\xi,t,g} = \begin{cases} \mathrm{ru}_{\xi,t,g} & \text{if } \mathrm{ru}_{\xi,t,g} \text{ is defined} \\ 1 & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G}
\]
Generator_ramp_down_rate
\[
\widetilde{\mathrm{rd}}_{\xi,t,g} = \begin{cases} \mathrm{rd}_{\xi,t,g} & \text{if } \mathrm{rd}_{\xi,t,g} \text{ is defined} \\ 1 & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G}
\]
Generator_start_up_rate
\[
\widetilde{\mathrm{ru}}^{\mathrm{up}}_{\xi,g} = \begin{cases} \mathrm{ru}^{\mathrm{up}}_{\xi,g} & \text{if } \mathrm{ru}^{\mathrm{up}}_{\xi,g} \text{ is defined} \\ 1 & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ g \in \mathcal{G}
\]
Generator_shut_down_rate
\[
\widetilde{\mathrm{rd}}^{\mathrm{dn}}_{\xi,g} = \begin{cases} \mathrm{rd}^{\mathrm{dn}}_{\xi,g} & \text{if } \mathrm{rd}^{\mathrm{dn}}_{\xi,g} \text{ is defined} \\ 1 & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ g \in \mathcal{G}
\]
Generator_ramp_up_allowance
\[
\Delta^{+}_{\xi,t,g} = \begin{cases} \widetilde{\mathrm{ru}}_{\xi,t,g} \cdot \widehat{\mathrm{p}}^{\mathrm{nom}}_{\xi,g} \cdot \overleftarrow{u}_{\xi,t,g} + \widetilde{\mathrm{ru}}^{\mathrm{up}}_{\xi,g} \cdot \widehat{\mathrm{p}}^{\mathrm{nom}}_{\xi,g} \cdot \left( u_{\xi,t,g} - \overleftarrow{u}_{\xi,t,g} \right) & \text{if } \mathrm{com}_{g} \\ \widetilde{\mathrm{ru}}_{\xi,t,g} \cdot \widetilde{\mathrm{p}}^{\mathrm{nom}}_{\xi,g} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G}
\]
Generator_ramp_down_allowance
\[
\Delta^{-}_{\xi,t,g} = \begin{cases} \widetilde{\mathrm{rd}}_{\xi,t,g} \cdot \widehat{\mathrm{p}}^{\mathrm{nom}}_{\xi,g} \cdot u_{\xi,t,g} + \widetilde{\mathrm{rd}}^{\mathrm{dn}}_{\xi,g} \cdot \widehat{\mathrm{p}}^{\mathrm{nom}}_{\xi,g} \cdot \left( \overleftarrow{u}_{\xi,t,g} - u_{\xi,t,g} \right) & \text{if } \mathrm{com}_{g} \\ \widetilde{\mathrm{rd}}_{\xi,t,g} \cdot \widetilde{\mathrm{p}}^{\mathrm{nom}}_{\xi,g} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G}
\]
Assumptions#
Generator_came_in_running_unless_committable
\[
\mathrm{u}^{0}_{\xi,g} = 1 \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \neg \mathrm{com}_{g} \wedge \left( \mathrm{ru}_{\xi,t,g} \text{ is defined} \vee \mathrm{rd}_{\xi,t,g} \text{ is defined} \right)
\]