Generators#
One of the 24 fragments of examples/pypsa.yaml: PyPSA's Generator. It adds a term to primary_energy, operational_limit, tech_capacity_expansion, scenario_opex, Carrier_additions, Bus_injection. It reads CVaR_omega, Generator_committable, Generator_maintenance, Generator_maintenance_capacity, Generator_maintenance_pu, GlobalConstraint_energy_weight and 4 more under given.
dimensions:
scenario:
description: the futures dispatch is chosen in, each with a weight
snapshot:
description: dispatch periods
dtype: datetime
bus:
description: network nodes
generator:
description: generating units, each on one bus
global_constraint:
description: PyPSA's `GlobalConstraint` rows, one label per declared limit
period:
description: investment periods — PyPSA's `investment_periods`
dtype: int
carrier:
description: energy carriers, what a growth limit is set per
relations:
snapshot_period:
description: the investment period a snapshot falls in
key: snapshot
values: period
Generator_carrier:
description: the carrier a generator converts from
key: generator
values: carrier
Generator_bus:
description: the bus a generator sits on
key: generator
values: bus
parameters:
Generator_p_nom:
description: nominal power
dims: [scenario, generator]
Generator_p_nom_extendable:
description: whether the nominal power is a decision
dims: [generator]
dtype: bool
Generator_p_min_pu:
description: least output, per unit of nominal power
dims: [scenario, snapshot, generator]
Generator_p_max_pu:
description: most output, per unit of nominal power — an availability profile
dims: [scenario, snapshot, generator]
Generator_marginal_cost:
description: cost of one unit of output
dims: [scenario, snapshot, generator]
Generator_marginal_cost_quadratic:
description: cost of the square of one unit of output
dims: [scenario, snapshot, generator]
Generator_sign:
description: >-
the sign output enters its bus's balance with — PyPSA's `sign`, `1`
unless given, `-1` for a unit that draws power. PyPSA refuses one that
differs by scenario (`consistency.py:1187`)
dims: [generator]
Generator_p_nom_mod:
description: the module size a build comes in whole numbers of; no value means the build is continuous
dims: [generator]
Generator_modules_installed:
description: >-
how many whole modules a committable build has in place: `Generator_p_nom
/ Generator_p_nom_mod` where a fixed build is modular, one where it is
not, data prep. PyPSA refuses a fixed modular build whose nominal power
is not a whole number of modules
dims: [scenario, generator]
Generator_p_min_pu_nonneg:
description: >-
true where none of the generator's own minimums-per-unit is negative —
PyPSA's per-unit `(p_min_pu >= 0).all()` over every snapshot and scenario, data prep
dims: [generator]
dtype: bool
Generator_active:
description: whether a generator stands in a snapshot's period — PyPSA's `active`, from build year and lifetime, data prep
dims: [snapshot, generator]
dtype: bool
Generator_capital_weight:
description: the sum of period weights a generator stands in — PyPSA's `active * period_weighting`, summed, data prep
dims: [generator]
Generator_first_active:
description: >-
one in the first period a generator stands in, zero elsewhere, data prep.
PyPSA `1.3.0` takes `active.cumsum() == 1`, which also counts a generator
that has retired in every later period (`global_constraints.py:276`,
PyPSA/PyPSA#1938)
dims: [period, generator]
Generator_p_set:
description: a given output schedule; a generator without one has no row here
dims: [scenario, snapshot, generator]
Generator_p_nom_min:
description: least nominal power an extendable generator may be built at
dims: [scenario, generator]
Generator_p_nom_max:
description: most nominal power an extendable generator may be built at
dims: [scenario, generator]
Generator_capital_cost:
description: cost of one unit of nominal power — PyPSA's `capital_cost`, periodized as an annuity in data prep
dims: [scenario, generator]
Generator_p_nom_set:
description: a given nominal power for an extendable generator; one without a value has no row here
dims: [scenario, generator]
Generator_e_sum_min:
description: least energy over the horizon; minus infinity where no floor is meant
dims: [scenario, generator]
Generator_e_sum_max:
description: most energy over the horizon — a fuel or emission budget in energy terms; infinity where no cap is meant
dims: [scenario, generator]
Generator_primary_energy_weight:
description: >-
the constrained attribute per unit of energy at the bus — the carrier's
`co2_emissions` over the generator's efficiency at the snapshot, data prep; a generator
of an unweighted carrier has no row
dims: [scenario, global_constraint, snapshot, generator]
Generator_operational_limit_weight:
description: one where the generator is in the row's set — data prep; one outside it has no row
dims: [scenario, global_constraint, generator]
Generator_tech_capacity_weight:
description: >-
one where the generator is in the row's carrier-and-bus set — data
prep; one outside it, or one that does not stand in the row's
`investment_period`, has no row
dims: [global_constraint, generator]
variables:
Generator_p:
description: "`Generator-p` — output of a generator in a snapshot"
dims: [scenario, snapshot, generator]
where: Generator_active
Generator_n_mod:
description: "`Generator-n_mod` — how many modules of an extendable modular build"
dims: [generator]
where: Generator_p_nom_extendable AND Generator_p_nom_mod > 0
domain: integer
bounds:
lower: 0
Generator_p_nom_ext:
description: >-
`Generator-p_nom` — nominal power where it is a decision; the parameter
of the same PyPSA name carries the fixed regime
dims: [generator]
where: Generator_p_nom_extendable
given:
parameters:
snapshot_weightings_objective: { dims: [snapshot] }
Generator_committable: { dims: [generator], dtype: bool }
Generator_maintenance_pu: { dims: [scenario, generator] }
scenario_weight: { dims: [scenario] }
CVaR_omega: { dims: [] }
period_weight_objective: { dims: [period] }
snapshot_weightings_generators: { dims: [snapshot] }
variables:
Generator_maintenance: { dims: [scenario, snapshot, generator] }
Generator_maintenance_capacity: { dims: [scenario, snapshot, generator] }
expressions:
GlobalConstraint_energy_weight: { dims: [scenario, global_constraint, snapshot] }
primary_energy: { dims: [scenario, global_constraint] }
operational_limit: { dims: [scenario, global_constraint] }
tech_capacity_expansion: { dims: [global_constraint] }
scenario_opex: { dims: [scenario] }
Carrier_additions: { dims: [period, carrier] }
Bus_injection: { dims: [scenario, snapshot, bus] }
expressions:
Generator_p_nom_effective:
description: the build a generator's limits are taken against — the chosen one where it is extendable, the given one otherwise
dims: [scenario, generator]
cases:
extendable: { when: Generator_p_nom_extendable, expression: Generator_p_nom_ext }
otherwise: Generator_p_nom
Generator_p_nom_committed:
description: >-
the build a committed unit's ramp rows are taken against — one module
where the build is extendable and modular, the given build otherwise
dims: [scenario, generator]
cases:
modular_build: { when: Generator_p_nom_extendable AND Generator_p_nom_mod > 0, expression: Generator_p_nom_mod }
otherwise: Generator_p_nom
Generator_primary_energy:
expression: >-
sum(sum((Generator_p * GlobalConstraint_energy_weight) * Generator_primary_energy_weight, over=snapshot), over=generator)
adds_to: primary_energy
Generator_operational_limit:
expression: >-
sum(sum((Generator_p * GlobalConstraint_energy_weight) * Generator_operational_limit_weight, over=snapshot), over=generator)
adds_to: operational_limit
Generator_tech_capacity_expansion:
expression: sum(Generator_p_nom_ext * Generator_tech_capacity_weight, over=generator)
adds_to: tech_capacity_expansion
Generator_opex:
expression: >-
sum(sum(((Generator_p * Generator_marginal_cost) * snapshot_weightings_objective) * at(period_weight_objective, by=snapshot_period, over=period, into=snapshot), over=generator), over=snapshot)
+ sum(sum((((Generator_p * Generator_p) * Generator_marginal_cost_quadratic) * snapshot_weightings_objective) * at(period_weight_objective, by=snapshot_period, over=period, into=snapshot), over=generator), over=snapshot)
adds_to: scenario_opex
Generator_additions:
expression: >-
sum(Generator_p_nom_ext * Generator_first_active, by=Generator_carrier, over=generator, into=carrier)
adds_to: Carrier_additions
Generator_injection:
expression: sum(Generator_sign * Generator_p, by=Generator_bus, over=generator, into=bus)
adds_to: Bus_injection
constraints:
Generator_fix_p_lower:
description: "`Generator-fix-p-lower` — a fixed generator outputs at least its minimum"
dims: [scenario, snapshot, generator]
where: not Generator_p_nom_extendable AND not Generator_committable AND Generator_active
expression: Generator_p >= Generator_p_min_pu * Generator_p_nom * (1 - Generator_maintenance_pu * Generator_maintenance)
Generator_fix_p_upper:
description: "`Generator-fix-p-upper` — a fixed generator outputs at most what is available"
dims: [scenario, snapshot, generator]
where: not Generator_p_nom_extendable AND not Generator_committable AND Generator_active
expression: Generator_p <= Generator_p_max_pu * Generator_p_nom * (1 - Generator_maintenance_pu * Generator_maintenance)
Generator_ext_p_lower:
description: "`Generator-ext-p-lower` — an extendable generator outputs at least its minimum of the chosen build"
dims: [scenario, snapshot, generator]
where: Generator_p_nom_extendable AND not Generator_committable AND Generator_active
expression: Generator_p >= Generator_p_min_pu * (Generator_p_nom_ext - Generator_maintenance_pu * Generator_maintenance_capacity)
Generator_ext_p_upper:
description: "`Generator-ext-p-upper` — an extendable generator outputs at most what is available of the chosen build"
dims: [scenario, snapshot, generator]
where: Generator_p_nom_extendable AND not Generator_committable AND Generator_active
expression: Generator_p <= Generator_p_max_pu * (Generator_p_nom_ext - Generator_maintenance_pu * Generator_maintenance_capacity)
Generator_ext_p_nom_lower:
description: "`Generator-ext-p_nom-lower` — the chosen build is at least its floor in every scenario"
dims: [scenario, generator]
where: Generator_p_nom_extendable
expression: Generator_p_nom_ext >= Generator_p_nom_min
Generator_ext_p_nom_upper:
description: "`Generator-ext-p_nom-upper` — the chosen build is at most its cap in every scenario; a cap of infinity is no row"
dims: [scenario, generator]
where: Generator_p_nom_extendable AND Generator_p_nom_max
expression: Generator_p_nom_ext <= Generator_p_nom_max
Generator_p_nom_set:
description: "`Generator-p_nom_set` — the chosen build pinned, wherever a value is given"
dims: [scenario, generator]
where: Generator_p_nom_extendable AND Generator_p_nom_set
expression: Generator_p_nom_ext == Generator_p_nom_set
Generator_e_sum_min:
description: "`Generator-e_sum_min` — energy over the horizon is at least its floor; a floor of minus infinity is no row"
dims: [scenario, generator]
where: Generator_e_sum_min
expression: sum(Generator_p * snapshot_weightings_generators, over=snapshot) >= Generator_e_sum_min
Generator_e_sum_max:
description: "`Generator-e_sum_max` — energy over the horizon is at most its budget; a budget of infinity is no row"
dims: [scenario, generator]
where: Generator_e_sum_max
expression: sum(Generator_p * snapshot_weightings_generators, over=snapshot) <= Generator_e_sum_max
Generator_p_nom_modularity:
description: "`Generator-p_nom_modularity` — the chosen build is a whole number of modules"
dims: [generator]
where: Generator_p_nom_extendable AND Generator_p_nom_mod > 0
expression: Generator_p_nom_ext == Generator_p_nom_mod * Generator_n_mod
Generator_p_set:
description: "`Generator-p_set` — output pinned to the given schedule, wherever one is given"
dims: [scenario, snapshot, generator]
where: Generator_p_set AND Generator_active
expression: Generator_p == Generator_p_set
assumptions:
Generator_marginal_cost_quadratic_without_risk_preference:
holds: "Generator_marginal_cost_quadratic == 0"
where: "CVaR_omega > 0"
description: >-
a quadratic cost puts a square into every `CVaR-excess` row, and PyPSA
refuses quadratic costs under any risk preference
(`optimize.py:467-474`). The spec cannot tell no risk preference from
one with `omega = 0`, so it refuses only where `omega` is positive
objective:
sense: minimize
expression: >-
sum(((scenario_weight * Generator_p_nom_ext) * Generator_capital_cost) * Generator_capital_weight)
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 |
| \(\mathcal{N}\) | index \(n\) — bus with \(\mathrm{Generator\_bus}: \mathcal{G} \to \mathcal{N}\) — network nodes |
| \(\mathcal{G}\) | index \(g\) — generator with \(\mathrm{Generator\_carrier}: \mathcal{G} \to \mathcal{I},\ \mathrm{Generator\_bus}: \mathcal{G} \to \mathcal{N}\) — generating units, each on one bus |
| \(\mathcal{L}\) | index \(l\) — global_constraint — PyPSA's GlobalConstraint rows, one label per declared limit |
| \(\mathcal{Y}\) | index \(y\) — period with \(\mathrm{snapshot\_period}: \mathcal{T} \to \mathcal{Y}\) — investment periods — PyPSA's investment_periods |
| \(\mathcal{I}\) | index \(i\) — carrier with \(\mathrm{Generator\_carrier}: \mathcal{G} \to \mathcal{I}\) — energy carriers, what a growth limit is set per |
Parameters#
| Symbol | Meaning |
|---|---|
| \(\mathrm{p}^{\mathrm{nom}}\) | Generator_p_nom over \(\Xi \times \mathcal{G}\) — nominal power |
| \(\mathrm{ext}\) | Generator_p_nom_extendable over \(\mathcal{G}\) — whether the nominal power is a decision |
| \(\underline{\mathrm{p}}\) | Generator_p_min_pu over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — least output, per unit of nominal power |
| \(\overline{\mathrm{p}}\) | Generator_p_max_pu over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — most output, per unit of nominal power — an availability profile |
| \(\mathrm{c}\) | Generator_marginal_cost over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — cost of one unit of output |
| \(\mathrm{c}^{(2)}\) | Generator_marginal_cost_quadratic over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — cost of the square of one unit of output |
| \(\mathrm{sgn}\) | Generator_sign over \(\mathcal{G}\) — the sign output enters its bus's balance with — PyPSA's sign, 1 unless given, -1 for a unit that draws power. PyPSA refuses one that differs by scenario (consistency.py:1187) |
| \(\mathrm{p}^{\mathrm{mod}}\) | Generator_p_nom_mod over \(\mathcal{G}\) — the module size a build comes in whole numbers of; no value means the build is continuous |
| \(\mathrm{N}^{\mathrm{fix}}\) | Generator_modules_installed over \(\Xi \times \mathcal{G}\) — how many whole modules a committable build has in place: Generator_p_nom / Generator_p_nom_mod where a fixed build is modular, one where it is not, data prep. PyPSA refuses a fixed modular build whose nominal power is not a whole number of modules |
| \(\mathrm{nonneg}\) | Generator_p_min_pu_nonneg over \(\mathcal{G}\) — true where none of the generator's own minimums-per-unit is negative — PyPSA's per-unit (p_min_pu >= 0).all() over every snapshot and scenario, data prep |
| \(\mathrm{on}\) | Generator_active over \(\mathcal{T} \times \mathcal{G}\) — whether a generator stands in a snapshot's period — PyPSA's active, from build year and lifetime, data prep |
| \(\mathrm{W}\) | Generator_capital_weight over \(\mathcal{G}\) — the sum of period weights a generator stands in — PyPSA's active * period_weighting, summed, data prep |
| \(\mathrm{new}\) | Generator_first_active over \(\mathcal{Y} \times \mathcal{G}\) — one in the first period a generator stands in, zero elsewhere, data prep. PyPSA 1.3.0 takes active.cumsum() == 1, which also counts a generator that has retired in every later period (global_constraints.py:276, PyPSA/PyPSA#1938) |
| \(\mathrm{p}^{\mathrm{set}}\) | Generator_p_set over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — a given output schedule; a generator without one has no row here |
| \(\underline{\mathrm{p}}^{\mathrm{nom}}\) | Generator_p_nom_min over \(\Xi \times \mathcal{G}\) — least nominal power an extendable generator may be built at |
| \(\overline{\mathrm{p}}^{\mathrm{nom}}\) | Generator_p_nom_max over \(\Xi \times \mathcal{G}\) — most nominal power an extendable generator may be built at |
| \(\mathrm{c}^{\mathrm{cap}}\) | Generator_capital_cost over \(\Xi \times \mathcal{G}\) — cost of one unit of nominal power — PyPSA's capital_cost, periodized as an annuity in data prep |
| \(\mathrm{p}^{\mathrm{nom,set}}\) | Generator_p_nom_set over \(\Xi \times \mathcal{G}\) — a given nominal power for an extendable generator; one without a value has no row here |
| \(\underline{\mathrm{E}}\) | Generator_e_sum_min over \(\Xi \times \mathcal{G}\) — least energy over the horizon; minus infinity where no floor is meant |
| \(\overline{\mathrm{E}}\) | Generator_e_sum_max over \(\Xi \times \mathcal{G}\) — most energy over the horizon — a fuel or emission budget in energy terms; infinity where no cap is meant |
| \(\mathrm{a}\) | Generator_primary_energy_weight over \(\Xi \times \mathcal{L} \times \mathcal{T} \times \mathcal{G}\) — the constrained attribute per unit of energy at the bus — the carrier's co2_emissions over the generator's efficiency at the snapshot, data prep; a generator of an unweighted carrier has no row |
| \(\mathrm{b}\) | Generator_operational_limit_weight over \(\Xi \times \mathcal{L} \times \mathcal{G}\) — one where the generator is in the row's set — data prep; one outside it has no row |
| \(\mathrm{m}\) | Generator_tech_capacity_weight over \(\mathcal{L} \times \mathcal{G}\) — one where the generator is in the row's carrier-and-bus set — data prep; one outside it, or one that does not stand in the row's investment_period, has no row |
Variables#
| Symbol | Meaning |
|---|---|
| \(p\) | Generator_p over \(\Xi \times \mathcal{T} \times \mathcal{G}\) — Generator-p — output of a generator in a snapshot |
| \(N\) | Generator_n_mod over \(\mathcal{G}\) — Generator-n_mod — how many modules of an extendable modular build |
| \(P\) | Generator_p_nom_ext over \(\mathcal{G}\) — Generator-p_nom — nominal power where it is a decision; the parameter of the same PyPSA name carries the fixed regime |
Given#
| Symbol | Meaning |
|---|---|
| \(\mathrm{w}\) | snapshot_weightings_objective over \(\mathcal{T}\), data another file declares |
| \(\mathrm{com}\) | Generator_committable over \(\mathcal{G}\), data another file declares |
| \(\gamma\) | Generator_maintenance_pu over \(\Xi \times \mathcal{G}\), data another file declares |
| \(\pi\) | scenario_weight over \(\Xi\), data another file declares |
| \(\omega\) | CVaR_omega (scalar), data another file declares |
| \(\mathrm{w}^{y}\) | period_weight_objective over \(\mathcal{Y}\), data another file declares |
| \(\mathrm{w}^{\mathrm{gen}}\) | snapshot_weightings_generators over \(\mathcal{T}\), data another file declares |
| \(\mu\) | Generator_maintenance over \(\Xi \times \mathcal{T} \times \mathcal{G}\) |
| \(\mu^{\mathrm{nom}}\) | Generator_maintenance_capacity over \(\Xi \times \mathcal{T} \times \mathcal{G}\) |
| \(\mathit{w}^{\mathrm{gc}}\) | GlobalConstraint_energy_weight over \(\Xi \times \mathcal{L} \times \mathcal{T}\), an expression another file defines |
| \(\mathit{primary\_energy}\) | primary_energy over \(\Xi \times \mathcal{L}\), an expression this file adds Generator_primary_energy to |
| \(\mathit{operational\_limit}\) | operational_limit over \(\Xi \times \mathcal{L}\), an expression this file adds Generator_operational_limit to |
| \(\mathit{tech\_capacity\_expansion}\) | tech_capacity_expansion over \(\mathcal{L}\), an expression this file adds Generator_tech_capacity_expansion to |
| \(\mathit{scenario\_opex}\) | scenario_opex over \(\Xi\), an expression this file adds Generator_opex to |
| \(\mathit{Carrier\_additions}\) | Carrier_additions over \(\mathcal{Y} \times \mathcal{I}\), an expression this file adds Generator_additions to |
| \(\mathit{Bus\_injection}\) | Bus_injection over \(\Xi \times \mathcal{T} \times \mathcal{N}\), an expression this file adds Generator_injection to |
Definitions#
| Symbol | Meaning |
|---|---|
| \(\widetilde{\mathrm{p}}^{\mathrm{nom}}\) | Generator_p_nom_effective over \(\Xi \times \mathcal{G}\) — the build a generator's limits are taken against — the chosen one where it is extendable, the given one otherwise |
| \(\widehat{\mathrm{p}}^{\mathrm{nom}}\) | Generator_p_nom_committed over \(\Xi \times \mathcal{G}\) — the build a committed unit's ramp rows are taken against — one module where the build is extendable and modular, the given build otherwise |
| \(\mathit{Generator\_primary\_energy}\) | Generator_primary_energy over \(\Xi \times \mathcal{L}\) |
| \(\mathit{Generator\_operational\_limit}\) | Generator_operational_limit over \(\Xi \times \mathcal{L}\) |
| \(\mathit{Generator\_tech\_capacity\_expansion}\) | Generator_tech_capacity_expansion over \(\mathcal{L}\) |
| \(\mathit{Generator\_opex}\) | Generator_opex over \(\Xi\) |
| \(\mathit{Generator\_additions}\) | Generator_additions over \(\mathcal{Y} \times \mathcal{I}\) |
| \(\mathit{Generator\_injection}\) | Generator_injection over \(\Xi \times \mathcal{T} \times \mathcal{N}\) |
Objective#
\[
\min \sum_{\xi \in \Xi,\ g \in \mathcal{G}} \pi_{\xi} \cdot P_{g} \cdot \mathrm{c}^{\mathrm{cap}}_{\xi,g} \cdot \mathrm{W}_{g}
\]
Subject to#
Generator_fix_p_lower
\[
p_{\xi,t,g} \ge \underline{\mathrm{p}}_{\xi,t,g} \cdot \mathrm{p}^{\mathrm{nom}}_{\xi,g} \cdot \left( 1 - \gamma_{\xi,g} \cdot \mu_{\xi,t,g} \right) \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \neg \mathrm{ext}_{g} \wedge \neg \mathrm{com}_{g} \wedge \mathrm{on}_{t,g}
\]
Generator_fix_p_upper
\[
p_{\xi,t,g} \le \overline{\mathrm{p}}_{\xi,t,g} \cdot \mathrm{p}^{\mathrm{nom}}_{\xi,g} \cdot \left( 1 - \gamma_{\xi,g} \cdot \mu_{\xi,t,g} \right) \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \neg \mathrm{ext}_{g} \wedge \neg \mathrm{com}_{g} \wedge \mathrm{on}_{t,g}
\]
Generator_ext_p_lower
\[
p_{\xi,t,g} \ge \underline{\mathrm{p}}_{\xi,t,g} \cdot \left( P_{g} - \gamma_{\xi,g} \cdot \mu^{\mathrm{nom}}_{\xi,t,g} \right) \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{ext}_{g} \wedge \neg \mathrm{com}_{g} \wedge \mathrm{on}_{t,g}
\]
Generator_ext_p_upper
\[
p_{\xi,t,g} \le \overline{\mathrm{p}}_{\xi,t,g} \cdot \left( P_{g} - \gamma_{\xi,g} \cdot \mu^{\mathrm{nom}}_{\xi,t,g} \right) \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{ext}_{g} \wedge \neg \mathrm{com}_{g} \wedge \mathrm{on}_{t,g}
\]
Generator_ext_p_nom_lower
\[
P_{g} \ge \underline{\mathrm{p}}^{\mathrm{nom}}_{\xi,g} \qquad \forall\, \xi \in \Xi,\ g \in \mathcal{G} \,:\, \mathrm{ext}_{g}
\]
Generator_ext_p_nom_upper
\[
P_{g} \le \overline{\mathrm{p}}^{\mathrm{nom}}_{\xi,g} \qquad \forall\, \xi \in \Xi,\ g \in \mathcal{G} \,:\, \mathrm{ext}_{g} \wedge \overline{\mathrm{p}}^{\mathrm{nom}}_{\xi,g} \text{ is defined}
\]
Generator_p_nom_set
\[
P_{g} = \mathrm{p}^{\mathrm{nom,set}}_{\xi,g} \qquad \forall\, \xi \in \Xi,\ g \in \mathcal{G} \,:\, \mathrm{ext}_{g} \wedge \mathrm{p}^{\mathrm{nom,set}}_{\xi,g} \text{ is defined}
\]
Generator_e_sum_min
\[
\sum_{t \in \mathcal{T}} p_{\xi,t,g} \cdot \mathrm{w}^{\mathrm{gen}}_{t} \ge \underline{\mathrm{E}}_{\xi,g} \qquad \forall\, \xi \in \Xi,\ g \in \mathcal{G} \,:\, \underline{\mathrm{E}}_{\xi,g} \text{ is defined}
\]
Generator_e_sum_max
\[
\sum_{t \in \mathcal{T}} p_{\xi,t,g} \cdot \mathrm{w}^{\mathrm{gen}}_{t} \le \overline{\mathrm{E}}_{\xi,g} \qquad \forall\, \xi \in \Xi,\ g \in \mathcal{G} \,:\, \overline{\mathrm{E}}_{\xi,g} \text{ is defined}
\]
Generator_p_nom_modularity
\[
P_{g} = \mathrm{p}^{\mathrm{mod}}_{g} \cdot N_{g} \qquad \forall\, g \in \mathcal{G} \,:\, \mathrm{ext}_{g} \wedge \mathrm{p}^{\mathrm{mod}}_{g} > 0
\]
Generator_p_set
\[
p_{\xi,t,g} = \mathrm{p}^{\mathrm{set}}_{\xi,t,g} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{p}^{\mathrm{set}}_{\xi,t,g} \text{ is defined} \wedge \mathrm{on}_{t,g}
\]
Definitions#
Generator_p_nom_effective
\[
\widetilde{\mathrm{p}}^{\mathrm{nom}}_{\xi,g} = \begin{cases} P_{g} & \text{if } \mathrm{ext}_{g} \\ \mathrm{p}^{\mathrm{nom}}_{\xi,g} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ g \in \mathcal{G}
\]
Generator_p_nom_committed
\[
\widehat{\mathrm{p}}^{\mathrm{nom}}_{\xi,g} = \begin{cases} \mathrm{p}^{\mathrm{mod}}_{g} & \text{if } \mathrm{ext}_{g} \wedge \mathrm{p}^{\mathrm{mod}}_{g} > 0 \\ \mathrm{p}^{\mathrm{nom}}_{\xi,g} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ g \in \mathcal{G}
\]
Generator_primary_energy
\[
\mathit{Generator\_primary\_energy}_{\xi,l} = \sum_{g \in \mathcal{G}} \sum_{t \in \mathcal{T}} p_{\xi,t,g} \cdot \mathit{w}^{\mathrm{gc}}_{\xi,l,t} \cdot \mathrm{a}_{\xi,l,t,g} \qquad \forall\, \xi \in \Xi,\ l \in \mathcal{L}
\]
Generator_operational_limit
\[
\mathit{Generator\_operational\_limit}_{\xi,l} = \sum_{g \in \mathcal{G}} \sum_{t \in \mathcal{T}} p_{\xi,t,g} \cdot \mathit{w}^{\mathrm{gc}}_{\xi,l,t} \cdot \mathrm{b}_{\xi,l,g} \qquad \forall\, \xi \in \Xi,\ l \in \mathcal{L}
\]
Generator_tech_capacity_expansion
\[
\mathit{Generator\_tech\_capacity\_expansion}_{l} = \sum_{g \in \mathcal{G}} P_{g} \cdot \mathrm{m}_{l,g} \qquad \forall\, l \in \mathcal{L}
\]
Generator_opex
\[
\mathit{Generator\_opex}_{\xi} = \sum_{t \in \mathcal{T}} \sum_{g \in \mathcal{G}} p_{\xi,t,g} \cdot \mathrm{c}_{\xi,t,g} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{snapshot\_period}(t)} + \sum_{t \in \mathcal{T}} \sum_{g \in \mathcal{G}} p_{\xi,t,g} \cdot p_{\xi,t,g} \cdot \mathrm{c}^{(2)}_{\xi,t,g} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{snapshot\_period}(t)} \qquad \forall\, \xi \in \Xi
\]
Generator_additions
\[
\mathit{Generator\_additions}_{y,i} = \sum_{g \in \mathcal{G} \,:\, \mathrm{Generator\_carrier}(g) = i} P_{g} \cdot \mathrm{new}_{y,g} \qquad \forall\, y \in \mathcal{Y},\ i \in \mathcal{I}
\]
Generator_injection
\[
\mathit{Generator\_injection}_{\xi,t,n} = \sum_{g \in \mathcal{G} \,:\, \mathrm{Generator\_bus}(g) = n} \mathrm{sgn}_{g} \cdot p_{\xi,t,g} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ n \in \mathcal{N}
\]
Variable domains#
Generator_p
\[
p_{\xi,t,g} \in \mathbb{R} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \mathrm{on}_{t,g}
\]
Generator_n_mod
\[
N_{g} \ge 0, N_{g} \in \mathbb{Z} \qquad \forall\, g \in \mathcal{G} \,:\, \mathrm{ext}_{g} \wedge \mathrm{p}^{\mathrm{mod}}_{g} > 0
\]
Generator_p_nom_ext
\[
P_{g} \in \mathbb{R} \qquad \forall\, g \in \mathcal{G} \,:\, \mathrm{ext}_{g}
\]
Assumptions#
Generator_marginal_cost_quadratic_without_risk_preference
\[
\mathrm{c}^{(2)}_{\xi,t,g} = 0 \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ g \in \mathcal{G} \,:\, \omega > 0
\]