Processes, the ramping#
One of the 24 fragments of examples/pypsa.yaml: PyPSA's Process, the ramping rows. It reads Process_active, Process_big_m, Process_committable, Process_p, Process_p_nom_committed, Process_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
process:
description: generalized multi-port converters, each with an internal power that every port draws or delivers at its own rate
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:
Process_ramp_limit_up:
description: most a process may raise its internal power 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, process]
Process_ramp_limit_down:
description: most a process may lower its internal power 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, process]
Process_ramp_limit_start_up:
description: most internal power in the snapshot a process starts, per unit of nominal power
dims: [scenario, process]
Process_ramp_limit_shut_down:
description: most internal power in the snapshot before a process stops, per unit of nominal power
dims: [scenario, process]
Process_p_init:
description: >-
the internal power a process brought into the horizon — PyPSA's `p_init`, read
only where the process came in running; no value means it is unknown, so
the process carries no ramp row at the first snapshot
dims: [scenario, process]
given:
parameters:
Process_p_nom_extendable: { dims: [process], dtype: bool }
Process_committable: { dims: [process], dtype: bool }
Process_status_initial: { dims: [scenario, process], dtype: int }
Process_p_nom_mod: { dims: [process] }
Process_big_m: { dims: [scenario, process] }
Process_active: { dims: [snapshot, process], dtype: bool }
variables:
Process_p: { dims: [scenario, snapshot, process] }
Process_status: { dims: [scenario, snapshot, process], domain: integer }
Process_start_up: { dims: [scenario, snapshot, process], domain: integer }
Process_shut_down: { dims: [scenario, snapshot, process], domain: integer }
Process_p_nom_ext: { dims: [process] }
expressions:
Process_p_nom_effective: { dims: [scenario, process] }
Process_previous_status: { dims: [scenario, snapshot, process] }
Process_p_nom_committed: { dims: [scenario, process] }
expressions:
Process_previous_p:
description: >-
the internal power a process 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, process]
cases:
opening: { when: "position(snapshot) == 0", expression: Process_status_initial * Process_p_init }
otherwise: shift(Process_p, along=snapshot, offset=1)
Process_ramp_up_rate:
description: >-
the ramp limit a process'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, process]
cases:
given: { when: Process_ramp_limit_up, expression: Process_ramp_limit_up }
otherwise: 1
Process_ramp_down_rate:
description: >-
the ramp limit a process'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, process]
cases:
given: { when: Process_ramp_limit_down, expression: Process_ramp_limit_down }
otherwise: 1
Process_start_up_rate:
description: >-
the start-up ramp a process's up row reads — PyPSA's `ramp_limit_start_up`,
or the full build where it has none
dims: [scenario, process]
cases:
given: { when: Process_ramp_limit_start_up, expression: Process_ramp_limit_start_up }
otherwise: 1
Process_shut_down_rate:
description: >-
the shut-down ramp a process's down row reads — PyPSA's
`ramp_limit_shut_down`, or the full build where it has none
dims: [scenario, process]
cases:
given: { when: Process_ramp_limit_shut_down, expression: Process_ramp_limit_shut_down }
otherwise: 1
Process_ramp_up_allowance:
description: >-
how far a process may raise internal power 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, process]
cases:
committed:
when: Process_committable
expression: >-
Process_ramp_up_rate * Process_p_nom_committed * Process_previous_status
+ Process_start_up_rate * Process_p_nom_committed
* (Process_status - Process_previous_status)
otherwise: Process_ramp_up_rate * Process_p_nom_effective
Process_ramp_down_allowance:
description: >-
how far a process may lower internal power 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, process]
cases:
committed:
when: Process_committable
expression: >-
Process_ramp_down_rate * Process_p_nom_committed * Process_status
+ Process_shut_down_rate * Process_p_nom_committed
* (Process_previous_status - Process_status)
otherwise: Process_ramp_down_rate * Process_p_nom_effective
constraints:
Process_p_ramp_limit_up_run_big_m:
description: >-
`Process-p-ramp_limit_up-run-bigM` — a committed extendable process
raises internal power no faster than its limit of the chosen build; the big M
releases the row in the snapshot it turns on
dims: [scenario, snapshot, process]
where: >-
Process_committable AND Process_p_nom_extendable AND NOT (Process_p_nom_mod > 0)
AND (Process_ramp_limit_up OR Process_ramp_limit_start_up)
AND (position(snapshot, by=snapshot_period, within=period) > 0 OR (position(snapshot) == 0 AND (Process_status_initial == 0 OR Process_p_init)))
AND Process_active
expression: >-
Process_p - Process_previous_p <=
Process_ramp_up_rate * Process_p_nom_ext
+ Process_big_m - Process_big_m * Process_previous_status
Process_p_ramp_limit_up_start_big_m:
description: >-
`Process-p-ramp_limit_up-start-bigM` — in the snapshot it turns on, a
committed extendable process ramps no further than its start-up ramp of
the chosen build; the big M releases the row everywhere else
dims: [scenario, snapshot, process]
where: >-
Process_committable AND Process_p_nom_extendable AND NOT (Process_p_nom_mod > 0)
AND (Process_ramp_limit_up OR Process_ramp_limit_start_up)
AND (position(snapshot, by=snapshot_period, within=period) > 0 OR (position(snapshot) == 0 AND (Process_status_initial == 0 OR Process_p_init)))
AND Process_active
expression: >-
Process_p - Process_previous_p <=
Process_start_up_rate * Process_p_nom_ext
+ Process_big_m - Process_big_m * Process_start_up
Process_p_ramp_limit_down_run_big_m:
description: >-
`Process-p-ramp_limit_down-run-bigM` — a committed extendable process
lowers internal power no faster than its limit of the chosen build; the big M
releases the row in the snapshot it turns off
dims: [scenario, snapshot, process]
where: >-
Process_committable AND Process_p_nom_extendable AND NOT (Process_p_nom_mod > 0)
AND (Process_ramp_limit_down OR Process_ramp_limit_shut_down)
AND (position(snapshot, by=snapshot_period, within=period) > 0 OR (position(snapshot) == 0 AND (Process_status_initial == 0 OR Process_p_init)))
AND Process_active
expression: >-
Process_previous_p - Process_p <=
Process_ramp_down_rate * Process_p_nom_ext
+ Process_big_m - Process_big_m * Process_status
Process_p_ramp_limit_down_shut_big_m:
description: >-
`Process-p-ramp_limit_down-shut-bigM` — in the snapshot it turns off,
a committed extendable process ramps no further than its shut-down ramp of
the chosen build; the big M releases the row everywhere else
dims: [scenario, snapshot, process]
where: >-
Process_committable AND Process_p_nom_extendable AND NOT (Process_p_nom_mod > 0)
AND (Process_ramp_limit_down OR Process_ramp_limit_shut_down)
AND (position(snapshot, by=snapshot_period, within=period) > 0 OR (position(snapshot) == 0 AND (Process_status_initial == 0 OR Process_p_init)))
AND Process_active
expression: >-
Process_previous_p - Process_p <=
Process_shut_down_rate * Process_p_nom_ext
+ Process_big_m - Process_big_m * Process_shut_down
Process_p_ramp_limit_up:
description: >-
`Process-p-ramp_limit_up` — a process raises internal power 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 process that came into the
horizon running carries a row at the first snapshot only where its
`p_init` gives the internal power it brought in, and no process carries one at
the start of a later investment period — nor does any process a big M releases instead
dims: [scenario, snapshot, process]
where: >-
(Process_ramp_limit_up OR Process_ramp_limit_start_up)
AND NOT (Process_committable AND Process_p_nom_extendable AND NOT (Process_p_nom_mod > 0))
AND (position(snapshot, by=snapshot_period, within=period) > 0 OR (position(snapshot) == 0 AND (Process_status_initial == 0 OR Process_p_init)))
AND Process_active
expression: Process_p - Process_previous_p <= Process_ramp_up_allowance
Process_p_ramp_limit_down:
description: >-
`Process-p-ramp_limit_down` — a process lowers internal power 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 process that came into the
horizon running carries a row at the first snapshot only where its
`p_init` gives the internal power it brought in, and no process carries one at
the start of a later investment period — nor does any process a big M releases instead
dims: [scenario, snapshot, process]
where: >-
(Process_ramp_limit_down OR Process_ramp_limit_shut_down)
AND NOT (Process_committable AND Process_p_nom_extendable AND NOT (Process_p_nom_mod > 0))
AND (position(snapshot, by=snapshot_period, within=period) > 0 OR (position(snapshot) == 0 AND (Process_status_initial == 0 OR Process_p_init)))
AND Process_active
expression: Process_previous_p - Process_p <= Process_ramp_down_allowance
assumptions:
Process_came_in_running_unless_committable:
holds: "Process_status_initial == 1"
where: "NOT Process_committable AND (Process_ramp_limit_up OR Process_ramp_limit_down)"
description: >-
PyPSA reads `up_time_before` of a process 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 process there at zero, or at its
start-up ramp where another process 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 process 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 |
| \(\mathcal{J}\) | index \(j\) — process — generalized multi-port converters, each with an internal power that every port draws or delivers at its own rate |
| \(\mathcal{Y}\) | index \(y\) — period with \(\mathrm{snapshot\_period}: \mathcal{T} \to \mathcal{Y}\) — investment periods — PyPSA's investment_periods |
Parameters#
| Symbol | Meaning |
|---|---|
| \(\mathrm{ru}^{z}\) | Process_ramp_limit_up over \(\Xi \times \mathcal{T} \times \mathcal{J}\) — most a process may raise its internal power 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 |
| \(\mathrm{rd}^{z}\) | Process_ramp_limit_down over \(\Xi \times \mathcal{T} \times \mathcal{J}\) — most a process may lower its internal power 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 |
| \(\mathrm{ru}^{z,\mathrm{up}}\) | Process_ramp_limit_start_up over \(\Xi \times \mathcal{J}\) — most internal power in the snapshot a process starts, per unit of nominal power |
| \(\mathrm{rd}^{z,\mathrm{dn}}\) | Process_ramp_limit_shut_down over \(\Xi \times \mathcal{J}\) — most internal power in the snapshot before a process stops, per unit of nominal power |
| \(\mathrm{z}^{0}\) | Process_p_init over \(\Xi \times \mathcal{J}\) — the internal power a process brought into the horizon — PyPSA's p_init, read only where the process came in running; no value means it is unknown, so the process carries no ramp row at the first snapshot |
Given#
| Symbol | Meaning |
|---|---|
| \(\mathrm{ext}^{z}\) | Process_p_nom_extendable over \(\mathcal{J}\), data another file declares |
| \(\mathrm{com}^{z}\) | Process_committable over \(\mathcal{J}\), data another file declares |
| \(\mathrm{u}^{z,0}\) | Process_status_initial over \(\Xi \times \mathcal{J}\), data another file declares |
| \(\mathrm{z}^{\mathrm{mod}}\) | Process_p_nom_mod over \(\mathcal{J}\), data another file declares |
| \(\mathrm{M}^{z}\) | Process_big_m over \(\Xi \times \mathcal{J}\), data another file declares |
| \(\mathrm{on}^{z}\) | Process_active over \(\mathcal{T} \times \mathcal{J}\), data another file declares |
| \(z\) | Process_p over \(\Xi \times \mathcal{T} \times \mathcal{J}\) |
| \(u^{z}\) | Process_status over \(\Xi \times \mathcal{T} \times \mathcal{J}\) |
| \(\mathit{up}^{z}\) | Process_start_up over \(\Xi \times \mathcal{T} \times \mathcal{J}\) |
| \(\mathit{dn}^{z}\) | Process_shut_down over \(\Xi \times \mathcal{T} \times \mathcal{J}\) |
| \(Z\) | Process_p_nom_ext over \(\mathcal{J}\) |
| \(\widetilde{\mathrm{z}}^{\mathrm{nom}}\) | Process_p_nom_effective over \(\Xi \times \mathcal{J}\), an expression another file defines |
| \(\overleftarrow{u}^{z}\) | Process_previous_status over \(\Xi \times \mathcal{T} \times \mathcal{J}\), an expression another file defines |
| \(\widehat{\mathrm{z}}^{\mathrm{nom}}\) | Process_p_nom_committed over \(\Xi \times \mathcal{J}\), an expression another file defines |
Definitions#
| Symbol | Meaning |
|---|---|
| \(\overleftarrow{z}\) | Process_previous_p over \(\Xi \times \mathcal{T} \times \mathcal{J}\) — the internal power a process 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 |
| \(\widetilde{\mathrm{ru}}^{z}\) | Process_ramp_up_rate over \(\Xi \times \mathcal{T} \times \mathcal{J}\) — the ramp limit a process'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 |
| \(\widetilde{\mathrm{rd}}^{z}\) | Process_ramp_down_rate over \(\Xi \times \mathcal{T} \times \mathcal{J}\) — the ramp limit a process'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 |
| \(\widetilde{\mathrm{ru}}^{z,\mathrm{up}}\) | Process_start_up_rate over \(\Xi \times \mathcal{J}\) — the start-up ramp a process's up row reads — PyPSA's ramp_limit_start_up, or the full build where it has none |
| \(\widetilde{\mathrm{rd}}^{z,\mathrm{dn}}\) | Process_shut_down_rate over \(\Xi \times \mathcal{J}\) — the shut-down ramp a process's down row reads — PyPSA's ramp_limit_shut_down, or the full build where it has none |
| \(\Delta^{z,+}\) | Process_ramp_up_allowance over \(\Xi \times \mathcal{T} \times \mathcal{J}\) — how far a process may raise internal power between two snapshots — its ramp limit of the build while it stays on, plus its start-up ramp in the snapshot it turns on |
| \(\Delta^{z,-}\) | Process_ramp_down_allowance over \(\Xi \times \mathcal{T} \times \mathcal{J}\) — how far a process may lower internal power 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#
Process_p_ramp_limit_up_run_big_m
\[
z_{\xi,t,j} - \overleftarrow{z}_{\xi,t,j} \le \widetilde{\mathrm{ru}}^{z}_{\xi,t,j} \cdot Z_{j} + \mathrm{M}^{z}_{\xi,j} - \mathrm{M}^{z}_{\xi,j} \cdot \overleftarrow{u}^{z}_{\xi,t,j} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{com}^{z}_{j} \wedge \mathrm{ext}^{z}_{j} \wedge \neg \left( \mathrm{z}^{\mathrm{mod}}_{j} > 0 \right) \wedge \left( \mathrm{ru}^{z}_{\xi,t,j} \text{ is defined} \vee \mathrm{ru}^{z,\mathrm{up}}_{\xi,j} \text{ is defined} \right) \wedge \left( \mathrm{pos}_{\mathrm{snapshot\_period}(t)}(t) > 0 \vee \mathrm{pos}(t) = 0 \wedge \left( \mathrm{u}^{z,0}_{\xi,j} = 0 \vee \mathrm{z}^{0}_{\xi,j} \text{ is defined} \right) \right) \wedge \mathrm{on}^{z}_{t,j}
\]
Process_p_ramp_limit_up_start_big_m
\[
z_{\xi,t,j} - \overleftarrow{z}_{\xi,t,j} \le \widetilde{\mathrm{ru}}^{z,\mathrm{up}}_{\xi,j} \cdot Z_{j} + \mathrm{M}^{z}_{\xi,j} - \mathrm{M}^{z}_{\xi,j} \cdot \mathit{up}^{z}_{\xi,t,j} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{com}^{z}_{j} \wedge \mathrm{ext}^{z}_{j} \wedge \neg \left( \mathrm{z}^{\mathrm{mod}}_{j} > 0 \right) \wedge \left( \mathrm{ru}^{z}_{\xi,t,j} \text{ is defined} \vee \mathrm{ru}^{z,\mathrm{up}}_{\xi,j} \text{ is defined} \right) \wedge \left( \mathrm{pos}_{\mathrm{snapshot\_period}(t)}(t) > 0 \vee \mathrm{pos}(t) = 0 \wedge \left( \mathrm{u}^{z,0}_{\xi,j} = 0 \vee \mathrm{z}^{0}_{\xi,j} \text{ is defined} \right) \right) \wedge \mathrm{on}^{z}_{t,j}
\]
Process_p_ramp_limit_down_run_big_m
\[
\overleftarrow{z}_{\xi,t,j} - z_{\xi,t,j} \le \widetilde{\mathrm{rd}}^{z}_{\xi,t,j} \cdot Z_{j} + \mathrm{M}^{z}_{\xi,j} - \mathrm{M}^{z}_{\xi,j} \cdot u^{z}_{\xi,t,j} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{com}^{z}_{j} \wedge \mathrm{ext}^{z}_{j} \wedge \neg \left( \mathrm{z}^{\mathrm{mod}}_{j} > 0 \right) \wedge \left( \mathrm{rd}^{z}_{\xi,t,j} \text{ is defined} \vee \mathrm{rd}^{z,\mathrm{dn}}_{\xi,j} \text{ is defined} \right) \wedge \left( \mathrm{pos}_{\mathrm{snapshot\_period}(t)}(t) > 0 \vee \mathrm{pos}(t) = 0 \wedge \left( \mathrm{u}^{z,0}_{\xi,j} = 0 \vee \mathrm{z}^{0}_{\xi,j} \text{ is defined} \right) \right) \wedge \mathrm{on}^{z}_{t,j}
\]
Process_p_ramp_limit_down_shut_big_m
\[
\overleftarrow{z}_{\xi,t,j} - z_{\xi,t,j} \le \widetilde{\mathrm{rd}}^{z,\mathrm{dn}}_{\xi,j} \cdot Z_{j} + \mathrm{M}^{z}_{\xi,j} - \mathrm{M}^{z}_{\xi,j} \cdot \mathit{dn}^{z}_{\xi,t,j} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \mathrm{com}^{z}_{j} \wedge \mathrm{ext}^{z}_{j} \wedge \neg \left( \mathrm{z}^{\mathrm{mod}}_{j} > 0 \right) \wedge \left( \mathrm{rd}^{z}_{\xi,t,j} \text{ is defined} \vee \mathrm{rd}^{z,\mathrm{dn}}_{\xi,j} \text{ is defined} \right) \wedge \left( \mathrm{pos}_{\mathrm{snapshot\_period}(t)}(t) > 0 \vee \mathrm{pos}(t) = 0 \wedge \left( \mathrm{u}^{z,0}_{\xi,j} = 0 \vee \mathrm{z}^{0}_{\xi,j} \text{ is defined} \right) \right) \wedge \mathrm{on}^{z}_{t,j}
\]
Process_p_ramp_limit_up
\[
z_{\xi,t,j} - \overleftarrow{z}_{\xi,t,j} \le \Delta^{z,+}_{\xi,t,j} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \left( \mathrm{ru}^{z}_{\xi,t,j} \text{ is defined} \vee \mathrm{ru}^{z,\mathrm{up}}_{\xi,j} \text{ is defined} \right) \wedge \neg \left( \mathrm{com}^{z}_{j} \wedge \mathrm{ext}^{z}_{j} \wedge \neg \left( \mathrm{z}^{\mathrm{mod}}_{j} > 0 \right) \right) \wedge \left( \mathrm{pos}_{\mathrm{snapshot\_period}(t)}(t) > 0 \vee \mathrm{pos}(t) = 0 \wedge \left( \mathrm{u}^{z,0}_{\xi,j} = 0 \vee \mathrm{z}^{0}_{\xi,j} \text{ is defined} \right) \right) \wedge \mathrm{on}^{z}_{t,j}
\]
Process_p_ramp_limit_down
\[
\overleftarrow{z}_{\xi,t,j} - z_{\xi,t,j} \le \Delta^{z,-}_{\xi,t,j} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \left( \mathrm{rd}^{z}_{\xi,t,j} \text{ is defined} \vee \mathrm{rd}^{z,\mathrm{dn}}_{\xi,j} \text{ is defined} \right) \wedge \neg \left( \mathrm{com}^{z}_{j} \wedge \mathrm{ext}^{z}_{j} \wedge \neg \left( \mathrm{z}^{\mathrm{mod}}_{j} > 0 \right) \right) \wedge \left( \mathrm{pos}_{\mathrm{snapshot\_period}(t)}(t) > 0 \vee \mathrm{pos}(t) = 0 \wedge \left( \mathrm{u}^{z,0}_{\xi,j} = 0 \vee \mathrm{z}^{0}_{\xi,j} \text{ is defined} \right) \right) \wedge \mathrm{on}^{z}_{t,j}
\]
Definitions#
Process_previous_p
\[
\overleftarrow{z}_{\xi,t,j} = \begin{cases} \mathrm{u}^{z,0}_{\xi,j} \cdot \mathrm{z}^{0}_{\xi,j} & \text{if } \mathrm{pos}(t) = 0 \\ z_{\xi,t - 1,j} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J}
\]
Process_ramp_up_rate
\[
\widetilde{\mathrm{ru}}^{z}_{\xi,t,j} = \begin{cases} \mathrm{ru}^{z}_{\xi,t,j} & \text{if } \mathrm{ru}^{z}_{\xi,t,j} \text{ is defined} \\ 1 & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J}
\]
Process_ramp_down_rate
\[
\widetilde{\mathrm{rd}}^{z}_{\xi,t,j} = \begin{cases} \mathrm{rd}^{z}_{\xi,t,j} & \text{if } \mathrm{rd}^{z}_{\xi,t,j} \text{ is defined} \\ 1 & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J}
\]
Process_start_up_rate
\[
\widetilde{\mathrm{ru}}^{z,\mathrm{up}}_{\xi,j} = \begin{cases} \mathrm{ru}^{z,\mathrm{up}}_{\xi,j} & \text{if } \mathrm{ru}^{z,\mathrm{up}}_{\xi,j} \text{ is defined} \\ 1 & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ j \in \mathcal{J}
\]
Process_shut_down_rate
\[
\widetilde{\mathrm{rd}}^{z,\mathrm{dn}}_{\xi,j} = \begin{cases} \mathrm{rd}^{z,\mathrm{dn}}_{\xi,j} & \text{if } \mathrm{rd}^{z,\mathrm{dn}}_{\xi,j} \text{ is defined} \\ 1 & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ j \in \mathcal{J}
\]
Process_ramp_up_allowance
\[
\Delta^{z,+}_{\xi,t,j} = \begin{cases} \widetilde{\mathrm{ru}}^{z}_{\xi,t,j} \cdot \widehat{\mathrm{z}}^{\mathrm{nom}}_{\xi,j} \cdot \overleftarrow{u}^{z}_{\xi,t,j} + \widetilde{\mathrm{ru}}^{z,\mathrm{up}}_{\xi,j} \cdot \widehat{\mathrm{z}}^{\mathrm{nom}}_{\xi,j} \cdot \left( u^{z}_{\xi,t,j} - \overleftarrow{u}^{z}_{\xi,t,j} \right) & \text{if } \mathrm{com}^{z}_{j} \\ \widetilde{\mathrm{ru}}^{z}_{\xi,t,j} \cdot \widetilde{\mathrm{z}}^{\mathrm{nom}}_{\xi,j} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J}
\]
Process_ramp_down_allowance
\[
\Delta^{z,-}_{\xi,t,j} = \begin{cases} \widetilde{\mathrm{rd}}^{z}_{\xi,t,j} \cdot \widehat{\mathrm{z}}^{\mathrm{nom}}_{\xi,j} \cdot u^{z}_{\xi,t,j} + \widetilde{\mathrm{rd}}^{z,\mathrm{dn}}_{\xi,j} \cdot \widehat{\mathrm{z}}^{\mathrm{nom}}_{\xi,j} \cdot \left( \overleftarrow{u}^{z}_{\xi,t,j} - u^{z}_{\xi,t,j} \right) & \text{if } \mathrm{com}^{z}_{j} \\ \widetilde{\mathrm{rd}}^{z}_{\xi,t,j} \cdot \widetilde{\mathrm{z}}^{\mathrm{nom}}_{\xi,j} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J}
\]
Assumptions#
Process_came_in_running_unless_committable
\[
\mathrm{u}^{z,0}_{\xi,j} = 1 \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ j \in \mathcal{J} \,:\, \neg \mathrm{com}^{z}_{j} \wedge \left( \mathrm{ru}^{z}_{\xi,t,j} \text{ is defined} \vee \mathrm{rd}^{z}_{\xi,t,j} \text{ is defined} \right)
\]