Links#
One of the 24 fragments of examples/pypsa.yaml: PyPSA's Link. It adds a term to transmission_volume_expansion, transmission_expansion_cost, tech_capacity_expansion, scenario_opex, Carrier_additions, Bus_injection. It reads CVaR_omega, Link_committable, Link_maintenance, Link_maintenance_capacity, Link_maintenance_pu, period_weight_objective and 2 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
link:
description: controllable connections, each from one bus to the buses it delivers to
link_output:
description: >-
a link's output ports, one label per port a link declares — PyPSA's
`bus1`, `bus2`, … columns read long, so a link of any number of output
ports is one term in the balance, data prep
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
Link_carrier:
description: the carrier a link converts from
key: link
values: carrier
Link_bus0:
description: the bus a link leaves
key: link
values: bus
Link_output_link:
description: the link an output port belongs to
key: link_output
values: link
Link_output_bus:
description: >-
the bus an output port delivers to — PyPSA's `bus1`, `bus2`, … columns.
A link of three output ports is three labels here rather than a third
relation, so the file states any number of them
key: link_output
values: bus
parameters:
Link_p_nom:
description: nominal power
dims: [scenario, link]
Link_p_nom_extendable:
description: whether the nominal power is a decision
dims: [link]
dtype: bool
Link_p_min_pu:
description: least flow, per unit of nominal power — negative for a link that carries both ways
dims: [scenario, snapshot, link]
Link_p_max_pu:
description: most flow, per unit of nominal power
dims: [scenario, snapshot, link]
Link_efficiency:
description: >-
share of the flow that arrives at an output port, PyPSA's `efficiency`,
`efficiency2`, … read long — negative where that port consumes rather
than delivers. Read at the snapshot the flow arrives, so a delayed port
delivers at its arrival snapshot's efficiency (`constraints.py:1522`)
dims: [scenario, snapshot, link_output]
Link_output_delay:
description: >-
snapshots a port's delivery lags its link's flow — PyPSA's `delay`,
`delay2`, … read long, in `snapshot_weightings.generators` units, which
the file states as whole snapshots; zero for a port that delivers at once.
Each scenario takes its own. PyPSA `1.3.0` groups the ports by delay over
all scenarios and shifts each group in every one, so a delay that differs
by scenario delivers the flow twice (`constraints.py:1269-1276`,
PyPSA/PyPSA#1941)
dims: [scenario, link_output]
dtype: int
Link_output_cyclic_delay:
description: >-
whether a delayed port's flow wraps from the end of its investment
period — PyPSA's `cyclic_delay`, `cyclic_delay2`, …; where it does not,
the flow still in transit at each period's first snapshots is lost. Each
scenario takes its own, as the delay
dims: [scenario, link_output]
dtype: bool
Link_marginal_cost:
description: cost of one unit of flow
dims: [scenario, snapshot, link]
Link_marginal_cost_quadratic:
description: cost of the square of one unit of flow
dims: [scenario, snapshot, link]
Link_p_nom_mod:
description: the module size a build comes in whole numbers of; no value means the build is continuous
dims: [link]
Link_modules_installed:
description: >-
how many whole modules a committable build has in place: `Link_p_nom
/ Link_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, link]
Link_p_min_pu_nonneg:
description: >-
true where none of the link's own minimums-per-unit is negative —
PyPSA's per-unit `(p_min_pu >= 0).all()` over every snapshot and scenario, data prep
dims: [link]
dtype: bool
Link_active:
description: whether a link stands in a snapshot's period — PyPSA's `active`, data prep
dims: [snapshot, link]
dtype: bool
Link_capital_weight:
description: the sum of period weights a link stands in — PyPSA's `active * period_weighting`, summed, data prep
dims: [link]
Link_first_active:
description: >-
one in the first period a link stands in, zero elsewhere, data prep.
PyPSA `1.3.0` takes `active.cumsum() == 1`, which also counts a link
that has retired in every later period (`global_constraints.py:276`,
PyPSA/PyPSA#1938)
dims: [period, link]
Link_p_set:
description: a given flow schedule; a link without one has no row here
dims: [scenario, snapshot, link]
Link_p_nom_min:
description: least nominal power an extendable link may be built at
dims: [scenario, link]
Link_p_nom_max:
description: most nominal power an extendable link may be built at
dims: [scenario, link]
Link_capital_cost:
description: cost of one unit of nominal power — PyPSA's `capital_cost`, periodized as an annuity in data prep
dims: [scenario, link]
Link_p_nom_set:
description: a given nominal power for an extendable link; one without a value has no row here
dims: [scenario, link]
Link_volume_weight:
description: >-
the link's length where its carrier is in the row's set, the first
scenario's length as PyPSA reads it (`global_constraints.py:835-836`) —
data prep; a
link outside it, or one that does not stand in the row's
`investment_period`, has no row
dims: [scenario, global_constraint, link]
Link_expansion_cost_weight:
description: >-
the link's capital cost where its carrier is in the row's set, times
the objective weights of the periods it stands in where the row names
no `investment_period` under `multi_investment_periods` — data prep; a
link outside the set, or one that does not stand in the row's period,
has no row
dims: [scenario, global_constraint, link]
Link_tech_capacity_weight:
description: >-
one where the link 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, link]
variables:
Link_p:
description: >-
`Link-p` — PyPSA's `p0`, the flow measured at the `Link_bus0` end: a
positive value withdraws there and injects at every bus the link's
output ports deliver to
dims: [scenario, snapshot, link]
where: Link_active
Link_n_mod:
description: "`Link-n_mod` — how many modules of an extendable modular build"
dims: [link]
where: Link_p_nom_extendable AND Link_p_nom_mod > 0
domain: integer
bounds:
lower: 0
Link_p_nom_ext:
description: >-
`Link-p_nom` — nominal power where it is a decision; the parameter of
the same PyPSA name carries the fixed regime
dims: [link]
where: Link_p_nom_extendable
given:
parameters:
snapshot_weightings_objective: { dims: [snapshot] }
Link_committable: { dims: [link], dtype: bool }
Link_maintenance_pu: { dims: [scenario, link] }
scenario_weight: { dims: [scenario] }
CVaR_omega: { dims: [] }
period_weight_objective: { dims: [period] }
variables:
Link_maintenance: { dims: [scenario, snapshot, link] }
Link_maintenance_capacity: { dims: [scenario, snapshot, link] }
expressions:
transmission_volume_expansion: { dims: [scenario, global_constraint], term: Link_transmission_volume_expansion }
transmission_expansion_cost: { dims: [scenario, global_constraint], term: Link_transmission_expansion_cost }
tech_capacity_expansion: { dims: [global_constraint], term: Link_tech_capacity_expansion }
scenario_opex: { dims: [scenario], term: Link_opex }
Carrier_additions: { dims: [period, carrier], term: Link_additions }
Bus_injection: { dims: [scenario, snapshot, bus], term: Link_injection }
expressions:
Link_p_nom_effective:
description: the build a link's limits are taken against — the chosen one where it is extendable, the given one otherwise
dims: [scenario, link]
cases:
extendable: { when: Link_p_nom_extendable, expression: Link_p_nom_ext }
otherwise: Link_p_nom
Link_p_nom_committed:
description: >-
the build a committed link's ramp rows are taken against — one module
where the build is extendable and modular, the given build otherwise
dims: [scenario, link]
cases:
modular_build: { when: Link_p_nom_extendable AND Link_p_nom_mod > 0, expression: Link_p_nom_mod }
otherwise: Link_p_nom
Link_output_arrival:
description: >-
what a link delivers to an output port at a snapshot — its flow delayed
by the port's `delay` within its investment period, times the port's
efficiency at the snapshot the flow arrives; where the port is
`cyclic_delay` the delayed flow wraps from the period's end, and where it
is not the flow still in transit at the period's first snapshots is
lost. A port that does not delay (`delay`
zero) delivers its flow unshifted, cyclic or not
dims: [scenario, snapshot, link_output]
cases:
wrapping:
when: Link_output_cyclic_delay
expression: shift(at(Link_p, by=Link_output_link, over=link, into=link_output), along=snapshot, offset=Link_output_delay, edge='wrap', by=snapshot_period, within=period) * Link_efficiency
otherwise: shift(at(Link_p, by=Link_output_link, over=link, into=link_output), along=snapshot, offset=Link_output_delay, edge=0, by=snapshot_period, within=period) * Link_efficiency
Link_transmission_volume_expansion:
expression: sum(Link_p_nom_ext * Link_volume_weight, over=link)
Link_transmission_expansion_cost:
expression: sum(Link_p_nom_ext * Link_expansion_cost_weight, over=link)
Link_tech_capacity_expansion:
expression: sum(Link_p_nom_ext * Link_tech_capacity_weight, over=link)
Link_opex:
expression: >-
sum(sum(((Link_p * Link_marginal_cost) * snapshot_weightings_objective) * at(period_weight_objective, by=snapshot_period, over=period, into=snapshot), over=link), over=snapshot)
+ sum(sum((((Link_p * Link_p) * Link_marginal_cost_quadratic) * snapshot_weightings_objective) * at(period_weight_objective, by=snapshot_period, over=period, into=snapshot), over=link), over=snapshot)
Link_additions:
expression: >-
sum(Link_p_nom_ext * Link_first_active, by=Link_carrier, over=link, into=carrier)
Link_injection:
expression: >-
-sum(Link_p, by=Link_bus0, over=link, into=bus)
+ sum(Link_output_arrival, by=Link_output_bus, over=link_output, into=bus)
constraints:
Link_fix_p_lower:
description: "`Link-fix-p-lower` — a fixed link carries at least its minimum, negative for the other way"
dims: [scenario, snapshot, link]
where: not Link_p_nom_extendable AND not Link_committable AND Link_active
expression: Link_p >= Link_p_min_pu * Link_p_nom * (1 - Link_maintenance_pu * Link_maintenance)
Link_fix_p_upper:
description: "`Link-fix-p-upper` — a fixed link carries at most its nominal power"
dims: [scenario, snapshot, link]
where: not Link_p_nom_extendable AND not Link_committable AND Link_active
expression: Link_p <= Link_p_max_pu * Link_p_nom * (1 - Link_maintenance_pu * Link_maintenance)
Link_ext_p_lower:
description: "`Link-ext-p-lower` — an extendable link carries at least its minimum of the chosen build, negative for the other way"
dims: [scenario, snapshot, link]
where: Link_p_nom_extendable AND not Link_committable AND Link_active
expression: Link_p >= Link_p_min_pu * (Link_p_nom_ext - Link_maintenance_pu * Link_maintenance_capacity)
Link_ext_p_upper:
description: "`Link-ext-p-upper` — an extendable link carries at most the chosen build"
dims: [scenario, snapshot, link]
where: Link_p_nom_extendable AND not Link_committable AND Link_active
expression: Link_p <= Link_p_max_pu * (Link_p_nom_ext - Link_maintenance_pu * Link_maintenance_capacity)
Link_ext_p_nom_lower:
description: "`Link-ext-p_nom-lower` — the chosen build is at least its floor in every scenario"
dims: [scenario, link]
where: Link_p_nom_extendable
expression: Link_p_nom_ext >= Link_p_nom_min
Link_ext_p_nom_upper:
description: "`Link-ext-p_nom-upper` — the chosen build is at most its cap in every scenario; a cap of infinity is no row"
dims: [scenario, link]
where: Link_p_nom_extendable AND Link_p_nom_max
expression: Link_p_nom_ext <= Link_p_nom_max
Link_p_nom_set:
description: "`Link-p_nom_set` — the chosen build pinned, wherever a value is given"
dims: [scenario, link]
where: Link_p_nom_extendable AND Link_p_nom_set
expression: Link_p_nom_ext == Link_p_nom_set
Link_p_nom_modularity:
description: "`Link-p_nom_modularity` — the chosen build is a whole number of modules"
dims: [link]
where: Link_p_nom_extendable AND Link_p_nom_mod > 0
expression: Link_p_nom_ext == Link_p_nom_mod * Link_n_mod
Link_p_set:
description: "`Link-p_set` — flow pinned to the given schedule, wherever one is given"
dims: [scenario, snapshot, link]
where: Link_p_set AND Link_active
expression: Link_p == Link_p_set
assumptions:
Link_marginal_cost_quadratic_without_risk_preference:
holds: "Link_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 * Link_p_nom_ext) * Link_capital_cost) * Link_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 \(\mathrm{Link\_bus0}: \mathcal{L} \to \mathcal{N},\ \mathrm{Link\_output\_bus}: \mathcal{O} \to \mathcal{N}\) — network nodes |
| \(\mathcal{L}\) | index \(l\) — link with \(\mathrm{Link\_carrier}: \mathcal{L} \to \mathcal{I},\ \mathrm{Link\_bus0}: \mathcal{L} \to \mathcal{N},\ \mathrm{Link\_output\_link}: \mathcal{O} \to \mathcal{L}\) — controllable connections, each from one bus to the buses it delivers to |
| \(\mathcal{O}\) | index \(o\) — link_output with \(\mathrm{Link\_output\_link}: \mathcal{O} \to \mathcal{L},\ \mathrm{Link\_output\_bus}: \mathcal{O} \to \mathcal{N}\) — a link's output ports, one label per port a link declares — PyPSA's bus1, bus2, … columns read long, so a link of any number of output ports is one term in the balance, data prep |
| \(\mathcal{G}\) | index \(g\) — 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 \(\mathrm{Link\_carrier}: \mathcal{L} \to \mathcal{I}\) — energy carriers, what a growth limit is set per |
Parameters#
| Symbol | Meaning |
|---|---|
| \(\mathrm{f}^{\mathrm{nom}}\) | Link_p_nom over \(\Xi \times \mathcal{L}\) — nominal power |
| \(\mathrm{ext}^{f}\) | Link_p_nom_extendable over \(\mathcal{L}\) — whether the nominal power is a decision |
| \(\underline{\mathrm{f}}\) | Link_p_min_pu over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — least flow, per unit of nominal power — negative for a link that carries both ways |
| \(\overline{\mathrm{f}}\) | Link_p_max_pu over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — most flow, per unit of nominal power |
| \(\eta\) | Link_efficiency over \(\Xi \times \mathcal{T} \times \mathcal{O}\) — share of the flow that arrives at an output port, PyPSA's efficiency, efficiency2, … read long — negative where that port consumes rather than delivers. Read at the snapshot the flow arrives, so a delayed port delivers at its arrival snapshot's efficiency (constraints.py:1522) |
| \(\mathrm{d}^{f}\) | Link_output_delay over \(\Xi \times \mathcal{O}\) — snapshots a port's delivery lags its link's flow — PyPSA's delay, delay2, … read long, in snapshot_weightings.generators units, which the file states as whole snapshots; zero for a port that delivers at once. Each scenario takes its own. PyPSA 1.3.0 groups the ports by delay over all scenarios and shifts each group in every one, so a delay that differs by scenario delivers the flow twice (constraints.py:1269-1276, PyPSA/PyPSA#1941) |
| \(\mathrm{cyc}^{f}\) | Link_output_cyclic_delay over \(\Xi \times \mathcal{O}\) — whether a delayed port's flow wraps from the end of its investment period — PyPSA's cyclic_delay, cyclic_delay2, …; where it does not, the flow still in transit at each period's first snapshots is lost. Each scenario takes its own, as the delay |
| \(\mathrm{c}^{f}\) | Link_marginal_cost over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — cost of one unit of flow |
| \(\mathrm{c}^{f,(2)}\) | Link_marginal_cost_quadratic over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — cost of the square of one unit of flow |
| \(\mathrm{f}^{\mathrm{mod}}\) | Link_p_nom_mod over \(\mathcal{L}\) — the module size a build comes in whole numbers of; no value means the build is continuous |
| \(\mathrm{N}^{f,\mathrm{fix}}\) | Link_modules_installed over \(\Xi \times \mathcal{L}\) — how many whole modules a committable build has in place: Link_p_nom / Link_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 |
| \(\mathrm{nonneg}^{f}\) | Link_p_min_pu_nonneg over \(\mathcal{L}\) — true where none of the link's own minimums-per-unit is negative — PyPSA's per-unit (p_min_pu >= 0).all() over every snapshot and scenario, data prep |
| \(\mathrm{on}^{f}\) | Link_active over \(\mathcal{T} \times \mathcal{L}\) — whether a link stands in a snapshot's period — PyPSA's active, data prep |
| \(\mathrm{W}^{f}\) | Link_capital_weight over \(\mathcal{L}\) — the sum of period weights a link stands in — PyPSA's active * period_weighting, summed, data prep |
| \(\mathrm{new}^{f}\) | Link_first_active over \(\mathcal{Y} \times \mathcal{L}\) — one in the first period a link stands in, zero elsewhere, data prep. PyPSA 1.3.0 takes active.cumsum() == 1, which also counts a link that has retired in every later period (global_constraints.py:276, PyPSA/PyPSA#1938) |
| \(\mathrm{f}^{\mathrm{set}}\) | Link_p_set over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — a given flow schedule; a link without one has no row here |
| \(\underline{\mathrm{f}}^{\mathrm{nom}}\) | Link_p_nom_min over \(\Xi \times \mathcal{L}\) — least nominal power an extendable link may be built at |
| \(\overline{\mathrm{f}}^{\mathrm{nom}}\) | Link_p_nom_max over \(\Xi \times \mathcal{L}\) — most nominal power an extendable link may be built at |
| \(\mathrm{c}^{\mathrm{cap},f}\) | Link_capital_cost over \(\Xi \times \mathcal{L}\) — cost of one unit of nominal power — PyPSA's capital_cost, periodized as an annuity in data prep |
| \(\mathrm{f}^{\mathrm{nom,set}}\) | Link_p_nom_set over \(\Xi \times \mathcal{L}\) — a given nominal power for an extendable link; one without a value has no row here |
| \(\mathrm{len}^{f}\) | Link_volume_weight over \(\Xi \times \mathcal{G} \times \mathcal{L}\) — the link's length where its carrier is in the row's set, the first scenario's length as PyPSA reads it (global_constraints.py:835-836) — data prep; a link outside it, or one that does not stand in the row's investment_period, has no row |
| \(\mathrm{cc}^{f}\) | Link_expansion_cost_weight over \(\Xi \times \mathcal{G} \times \mathcal{L}\) — the link's capital cost where its carrier is in the row's set, times the objective weights of the periods it stands in where the row names no investment_period under multi_investment_periods — data prep; a link outside the set, or one that does not stand in the row's period, has no row |
| \(\mathrm{m}^{f}\) | Link_tech_capacity_weight over \(\mathcal{G} \times \mathcal{L}\) — one where the link 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 |
|---|---|
| \(f\) | Link_p over \(\Xi \times \mathcal{T} \times \mathcal{L}\) — Link-p — PyPSA's p0, the flow measured at the Link_bus0 end: a positive value withdraws there and injects at every bus the link's output ports deliver to |
| \(N^{f}\) | Link_n_mod over \(\mathcal{L}\) — Link-n_mod — how many modules of an extendable modular build |
| \(F\) | Link_p_nom_ext over \(\mathcal{L}\) — Link-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 |
| \(\mathrm{com}^{f}\) | Link_committable over \(\mathcal{L}\), data another file declares |
| \(\gamma^{f}\) | Link_maintenance_pu over \(\Xi \times \mathcal{L}\), 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 |
| \(\mu^{f}\) | Link_maintenance over \(\Xi \times \mathcal{T} \times \mathcal{L}\) |
| \(\mu^{f,\mathrm{nom}}\) | Link_maintenance_capacity over \(\Xi \times \mathcal{T} \times \mathcal{L}\) |
| \(\mathit{transmission\_volume\_expansion}\) | transmission_volume_expansion over \(\Xi \times \mathcal{G}\), an expression this file adds Link_transmission_volume_expansion to |
| \(\mathit{transmission\_expansion\_cost}\) | transmission_expansion_cost over \(\Xi \times \mathcal{G}\), an expression this file adds Link_transmission_expansion_cost to |
| \(\mathit{tech\_capacity\_expansion}\) | tech_capacity_expansion over \(\mathcal{G}\), an expression this file adds Link_tech_capacity_expansion to |
| \(\mathit{scenario\_opex}\) | scenario_opex over \(\Xi\), an expression this file adds Link_opex to |
| \(\mathit{Carrier\_additions}\) | Carrier_additions over \(\mathcal{Y} \times \mathcal{I}\), an expression this file adds Link_additions to |
| \(\mathit{Bus\_injection}\) | Bus_injection over \(\Xi \times \mathcal{T} \times \mathcal{N}\), an expression this file adds Link_injection to |
Definitions#
| Symbol | Meaning |
|---|---|
| \(\widetilde{\mathrm{f}}^{\mathrm{nom}}\) | Link_p_nom_effective over \(\Xi \times \mathcal{L}\) — the build a link's limits are taken against — the chosen one where it is extendable, the given one otherwise |
| \(\widehat{\mathrm{f}}^{\mathrm{nom}}\) | Link_p_nom_committed over \(\Xi \times \mathcal{L}\) — the build a committed link's ramp rows are taken against — one module where the build is extendable and modular, the given build otherwise |
| \(\overrightarrow{f}\) | Link_output_arrival over \(\Xi \times \mathcal{T} \times \mathcal{O}\) — what a link delivers to an output port at a snapshot — its flow delayed by the port's delay within its investment period, times the port's efficiency at the snapshot the flow arrives; where the port is cyclic_delay the delayed flow wraps from the period's end, and where it is not the flow still in transit at the period's first snapshots is lost. A port that does not delay (delay zero) delivers its flow unshifted, cyclic or not |
| \(\mathit{Link\_transmission\_volume\_expansion}\) | Link_transmission_volume_expansion over \(\Xi \times \mathcal{G}\) |
| \(\mathit{Link\_transmission\_expansion\_cost}\) | Link_transmission_expansion_cost over \(\Xi \times \mathcal{G}\) |
| \(\mathit{Link\_tech\_capacity\_expansion}\) | Link_tech_capacity_expansion over \(\mathcal{G}\) |
| \(\mathit{Link\_opex}\) | Link_opex over \(\Xi\) |
| \(\mathit{Link\_additions}\) | Link_additions over \(\mathcal{Y} \times \mathcal{I}\) |
| \(\mathit{Link\_injection}\) | Link_injection over \(\Xi \times \mathcal{T} \times \mathcal{N}\) |
\(t \ominus k\) denotes cyclic translation: index \(t-k\) taken modulo the size of the dimension (roll). Plain \(t-k\) (shift) has no wraparound — terms translated past the edge are simply absent.
\(t \boxminus_{v} k\) denotes translation with \(v\) standing where index \(t-k\) leaves the dimension (shift(edge=v)), so the row at that boundary is built and carries \(v\) rather than being dropped.
\(t \ominus^{\mathrm{relation}(t)} k\) denotes a translation counted inside the group a relation puts \(t\) in (shift(by=relation)), so a term never crosses out of its own group. The two modifiers take different slots — the group above, the fill below — so \(t \boxminus_{v}^{\mathrm{relation}(t)} k\) is both at once.
Objective#
\[
\min \sum_{\xi \in \Xi,\ l \in \mathcal{L}} \pi_{\xi} \cdot F_{l} \cdot \mathrm{c}^{\mathrm{cap},f}_{\xi,l} \cdot \mathrm{W}^{f}_{l}
\]
Subject to#
Link_fix_p_lower
\[
f_{\xi,t,l} \ge \underline{\mathrm{f}}_{\xi,t,l} \cdot \mathrm{f}^{\mathrm{nom}}_{\xi,l} \cdot \left( 1 - \gamma^{f}_{\xi,l} \cdot \mu^{f}_{\xi,t,l} \right) \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \neg \mathrm{ext}^{f}_{l} \wedge \neg \mathrm{com}^{f}_{l} \wedge \mathrm{on}^{f}_{t,l}
\]
Link_fix_p_upper
\[
f_{\xi,t,l} \le \overline{\mathrm{f}}_{\xi,t,l} \cdot \mathrm{f}^{\mathrm{nom}}_{\xi,l} \cdot \left( 1 - \gamma^{f}_{\xi,l} \cdot \mu^{f}_{\xi,t,l} \right) \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \neg \mathrm{ext}^{f}_{l} \wedge \neg \mathrm{com}^{f}_{l} \wedge \mathrm{on}^{f}_{t,l}
\]
Link_ext_p_lower
\[
f_{\xi,t,l} \ge \underline{\mathrm{f}}_{\xi,t,l} \cdot \left( F_{l} - \gamma^{f}_{\xi,l} \cdot \mu^{f,\mathrm{nom}}_{\xi,t,l} \right) \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{ext}^{f}_{l} \wedge \neg \mathrm{com}^{f}_{l} \wedge \mathrm{on}^{f}_{t,l}
\]
Link_ext_p_upper
\[
f_{\xi,t,l} \le \overline{\mathrm{f}}_{\xi,t,l} \cdot \left( F_{l} - \gamma^{f}_{\xi,l} \cdot \mu^{f,\mathrm{nom}}_{\xi,t,l} \right) \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{ext}^{f}_{l} \wedge \neg \mathrm{com}^{f}_{l} \wedge \mathrm{on}^{f}_{t,l}
\]
Link_ext_p_nom_lower
\[
F_{l} \ge \underline{\mathrm{f}}^{\mathrm{nom}}_{\xi,l} \qquad \forall\, \xi \in \Xi,\ l \in \mathcal{L} \,:\, \mathrm{ext}^{f}_{l}
\]
Link_ext_p_nom_upper
\[
F_{l} \le \overline{\mathrm{f}}^{\mathrm{nom}}_{\xi,l} \qquad \forall\, \xi \in \Xi,\ l \in \mathcal{L} \,:\, \mathrm{ext}^{f}_{l} \wedge \overline{\mathrm{f}}^{\mathrm{nom}}_{\xi,l} \text{ is defined}
\]
Link_p_nom_set
\[
F_{l} = \mathrm{f}^{\mathrm{nom,set}}_{\xi,l} \qquad \forall\, \xi \in \Xi,\ l \in \mathcal{L} \,:\, \mathrm{ext}^{f}_{l} \wedge \mathrm{f}^{\mathrm{nom,set}}_{\xi,l} \text{ is defined}
\]
Link_p_nom_modularity
\[
F_{l} = \mathrm{f}^{\mathrm{mod}}_{l} \cdot N^{f}_{l} \qquad \forall\, l \in \mathcal{L} \,:\, \mathrm{ext}^{f}_{l} \wedge \mathrm{f}^{\mathrm{mod}}_{l} > 0
\]
Link_p_set
\[
f_{\xi,t,l} = \mathrm{f}^{\mathrm{set}}_{\xi,t,l} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{f}^{\mathrm{set}}_{\xi,t,l} \text{ is defined} \wedge \mathrm{on}^{f}_{t,l}
\]
Definitions#
Link_p_nom_effective
\[
\widetilde{\mathrm{f}}^{\mathrm{nom}}_{\xi,l} = \begin{cases} F_{l} & \text{if } \mathrm{ext}^{f}_{l} \\ \mathrm{f}^{\mathrm{nom}}_{\xi,l} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ l \in \mathcal{L}
\]
Link_p_nom_committed
\[
\widehat{\mathrm{f}}^{\mathrm{nom}}_{\xi,l} = \begin{cases} \mathrm{f}^{\mathrm{mod}}_{l} & \text{if } \mathrm{ext}^{f}_{l} \wedge \mathrm{f}^{\mathrm{mod}}_{l} > 0 \\ \mathrm{f}^{\mathrm{nom}}_{\xi,l} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ l \in \mathcal{L}
\]
Link_output_arrival
\[
\overrightarrow{f}_{\xi,t,o} = \begin{cases} f_{\xi,t \ominus^{\mathrm{snapshot\_period}(t)} \mathrm{d}^{f},\mathrm{Link\_output\_link}(o)} \cdot \eta_{\xi,t,o} & \text{if } \mathrm{cyc}^{f}_{\xi,o} \\ f_{\xi,t \boxminus_{0}^{\mathrm{snapshot\_period}(t)} \mathrm{d}^{f},\mathrm{Link\_output\_link}(o)} \cdot \eta_{\xi,t,o} & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ o \in \mathcal{O}
\]
Link_transmission_volume_expansion
\[
\mathit{Link\_transmission\_volume\_expansion}_{\xi,g} = \sum_{l \in \mathcal{L}} F_{l} \cdot \mathrm{len}^{f}_{\xi,g,l} \qquad \forall\, \xi \in \Xi,\ g \in \mathcal{G}
\]
Link_transmission_expansion_cost
\[
\mathit{Link\_transmission\_expansion\_cost}_{\xi,g} = \sum_{l \in \mathcal{L}} F_{l} \cdot \mathrm{cc}^{f}_{\xi,g,l} \qquad \forall\, \xi \in \Xi,\ g \in \mathcal{G}
\]
Link_tech_capacity_expansion
\[
\mathit{Link\_tech\_capacity\_expansion}_{g} = \sum_{l \in \mathcal{L}} F_{l} \cdot \mathrm{m}^{f}_{g,l} \qquad \forall\, g \in \mathcal{G}
\]
Link_opex
\[
\mathit{Link\_opex}_{\xi} = \sum_{t \in \mathcal{T}} \sum_{l \in \mathcal{L}} f_{\xi,t,l} \cdot \mathrm{c}^{f}_{\xi,t,l} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{snapshot\_period}(t)} + \sum_{t \in \mathcal{T}} \sum_{l \in \mathcal{L}} f_{\xi,t,l} \cdot f_{\xi,t,l} \cdot \mathrm{c}^{f,(2)}_{\xi,t,l} \cdot \mathrm{w}_{t} \cdot \mathrm{w}^{y}_{\mathrm{snapshot\_period}(t)} \qquad \forall\, \xi \in \Xi
\]
Link_additions
\[
\mathit{Link\_additions}_{y,i} = \sum_{l \in \mathcal{L} \,:\, \mathrm{Link\_carrier}(l) = i} F_{l} \cdot \mathrm{new}^{f}_{y,l} \qquad \forall\, y \in \mathcal{Y},\ i \in \mathcal{I}
\]
Link_injection
\[
\mathit{Link\_injection}_{\xi,t,n} = -\left( \sum_{l \in \mathcal{L} \,:\, \mathrm{Link\_bus0}(l) = n} f_{\xi,t,l} \right) + \sum_{o \in \mathcal{O} \,:\, \mathrm{Link\_output\_bus}(o) = n} \overrightarrow{f}_{\xi,t,o} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ n \in \mathcal{N}
\]
Variable domains#
Link_p
\[
f_{\xi,t,l} \in \mathbb{R} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \mathrm{on}^{f}_{t,l}
\]
Link_n_mod
\[
N^{f}_{l} \ge 0, N^{f}_{l} \in \mathbb{Z} \qquad \forall\, l \in \mathcal{L} \,:\, \mathrm{ext}^{f}_{l} \wedge \mathrm{f}^{\mathrm{mod}}_{l} > 0
\]
Link_p_nom_ext
\[
F_{l} \in \mathbb{R} \qquad \forall\, l \in \mathcal{L} \,:\, \mathrm{ext}^{f}_{l}
\]
Assumptions#
Link_marginal_cost_quadratic_without_risk_preference
\[
\mathrm{c}^{f,(2)}_{\xi,t,l} = 0 \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ l \in \mathcal{L} \,:\, \omega > 0
\]