Lines#
This file states PyPSA's Line. It is one of the 25 whole specs of one topic each (fragments) that merge composes into the spec of examples/pypsa.yaml. It adds a term to each of these sums: transmission_volume_expansion, transmission_expansion_cost, tech_capacity_expansion, total_cost, Carrier_additions, Bus_injection and Cycle_angle_sum. Line_loss_tangents_forward states PyPSA's tangent and secant loss rows over one segment dimension. Line_s_monitored is the flow that the security rows contain.
given:
parameters:
scenario_weight: { dims: [scenario] }
transmission_losses: { dims: [], dtype: bool }
expressions:
transmission_volume_expansion: { dims: [global_constraint, scenario] }
transmission_expansion_cost: { dims: [global_constraint, scenario] }
tech_capacity_expansion: { dims: [global_constraint] }
total_cost: { dims: [] }
Carrier_additions: { dims: [carrier, period] }
Bus_injection: { dims: [bus, scenario, snapshot] }
Cycle_angle_sum: { dims: [cycle, scenario, snapshot] }
dimensions:
scenario:
description: the futures dispatch is chosen in, each with a weight
snapshot:
description: dispatch periods
dtype: datetime
ordered: true
bus:
description: network nodes
line:
description: passive branches, each between two buses, their flow set by impedance
cycle:
description: >-
independent cycles of the passive network graph, the cycle basis, data
prep. Each period has its own basis, of the branches that are active in it. A
label is a position in that period's basis, so one label is a different
cycle in another period
segment:
description: >-
the cuts a passive branch's loss curve is held above. They are PyPSA's
tangents, as many as its `segments` count, or its secants, as many as
its tolerance loop places. A lossless run has none
dtype: int
global_constraint:
description: PyPSA's `GlobalConstraint` rows, one label per declared limit
period:
description: investment periods. It is PyPSA's `investment_periods`
dtype: int
ordered: true
carrier:
description: energy carriers, what a growth limit is set per
maps:
snapshot:
period:
description: the investment period a snapshot falls in
refers_to: period
line:
carrier:
description: the carrier a line carries
refers_to: carrier
bus0:
description: the bus a line's flow is measured at
refers_to: bus
bus1:
description: the bus at a line's other end
refers_to: bus
parameters:
Line_active:
description: whether a line is active in a snapshot's period. It is PyPSA's `active`, data prep
dims: [snapshot, line]
dtype: bool
Line_active_asset:
description: >-
whether a line is one of PyPSA's `active_assets`, by its `active` flag
alone, without build year and lifetime. A unit that is active in no
snapshot of the horizon still counts. PyPSA refuses one that differs by
scenario (`constants.py:54`)
dims: [line]
dtype: bool
Line_capital_weight:
description: the sum of period weights a line is active in. It is PyPSA's `active * period_weighting`, summed, data prep
dims: [line]
Line_first_active:
description: >-
one in the first period a line is active in, zero elsewhere, data prep.
PyPSA takes `active & (active.cumsum() == 1)` (`global_constraints.py:278`)
dims: [period, line]
Line_s_nom:
description: nominal apparent power
dims: [scenario, line]
Line_s_nom_extendable:
description: whether the nominal apparent power is a decision
dims: [line]
dtype: bool
Line_s_nom_mod:
description: the module size a build comes in whole numbers of. No value means the build is continuous
dims: [line]
missing: absent
Line_s_max_pu:
description: most flow either way, per unit of nominal apparent power
dims: [scenario, snapshot, line]
Line_s_nom_min:
description: least nominal apparent power an extendable line may be built at
dims: [scenario, line]
Line_s_nom_max:
description: most nominal apparent power an extendable line may be built at
dims: [scenario, line]
Line_capital_cost:
description: >-
cost of one unit of nominal apparent power for the modelled horizon
(PyPSA's `periodized_cost`), data prep. It is `overnight_cost` as an
annuity over `lifetime` at `discount_rate`, times `nyears`, where
`overnight_cost` is given, and `capital_cost` where it is not. Both add
`fom_cost` (`components.py:1157-1178`, `costs.py:102-203`)
dims: [scenario, line]
Line_s_nom_set:
description: a given nominal apparent power for an extendable line. One without a value has no row here
dims: [line]
missing: absent
Line_s_set:
description: a given flow schedule. A line without one has no row here
dims: [scenario, snapshot, line]
missing: absent
Line_v_ang_max:
description: >-
the most the voltage angle difference across a line may be either way,
in degrees. It is PyPSA's `v_ang_max`. Infinite, and so no row, by
default. A line whose carrier is not AC has no row either. The
deprecated `v_ang_min` is ignored, as PyPSA ignores it with a
`DeprecationWarning` (`constraints.py:1797-1804`)
dims: [scenario, line]
missing: absent
Line_x_pu_eff:
description: >-
the line's effective series reactance. It is PyPSA's `x_pu_eff`, `x`
over the square of its bus's nominal voltage, data prep
dims: [scenario, line]
Line_cycle_weight:
description: >-
the line's series impedance, signed by its orientation in the cycle: the
cycle basis, data prep. A line in no cycle has no row. PyPSA builds the
cycle basis from the first scenario (`networks.py:1430-1437`), and
refuses a `v_nom`, `x_pu_eff` or `r_pu_eff` that differs by scenario
(`constants.py:33-35`). So the weight is the same in every scenario
dims: [period, line, cycle]
missing: absent
Line_loss_max:
description: the loss at a line's rating. It is PyPSA's `r_pu_eff * (s_max_pu * s_nom_max)**2`, data prep
dims: [scenario, snapshot, line]
Line_loss_slope:
description: >-
the slope of a cut to the loss curve, data prep. It is a tangent's `2 *
r_pu_eff * p_k` at its segment's flow, or a secant's `r_pu_eff * (p_k +
p_k+1)` between consecutive breakpoints
dims: [scenario, snapshot, line, segment]
Line_loss_offset:
description: >-
where that cut meets the loss axis: a tangent's `loss_k - slope_k *
p_k`, a secant's `-r_pu_eff * p_k * p_k+1`, negative, data prep
dims: [scenario, snapshot, line, segment]
Line_volume_weight:
description: >-
the line's length in each scenario where its carrier is in the row's set
(`global_constraints.py:882-883`), data prep. A line outside it, or one
that is not active in the row's `investment_period`, has no row
dims: [scenario, global_constraint, line]
missing: absent
Line_expansion_cost_weight:
description: >-
the line's capital cost where its carrier is in the row's set, data
prep. Where the row states no `investment_period` under
`multi_investment_periods`, it is times the objective weights of the
periods the line is active in. A line outside the set, or one that is not
active in the row's period, has no row. The capital cost is PyPSA's
`capital_cost` property, which is `Line_capital_cost` without `fom_cost`
(`components.py:1182-1200`, `global_constraints.py:905-914`)
dims: [scenario, global_constraint, line]
missing: absent
Line_tech_capacity_weight:
description: >-
one where the line is in the row's carrier-and-bus set, data prep. One
outside it, or one that is not active in the row's `investment_period`,
has no row
dims: [global_constraint, line]
missing: absent
variables:
Line_s:
description: >-
`Line-s` — PyPSA's `p0`, the flow measured at the `line[bus0]` end: a
positive value withdraws there and injects at `line[bus1]`, lossless
dims: [scenario, snapshot, line]
where: Line_active
Line_loss:
description: >-
`Line-loss` — what a line dissipates carrying its flow, pushed down by
the cost and held up by the cuts. Absent, and zero in the balance, where
the network is lossless
dims: [scenario, snapshot, line]
where: Line_lossy
missing: 0
bounds:
lower: 0
Line_s_nom_ext:
description: >-
`Line-s_nom` — nominal apparent power where it is a decision, for a line
that is active. A fixed build is the parameter of the same PyPSA name
dims: [line]
where: Line_s_nom_extendable AND Line_active_asset
Line_n_mod:
description: "`Line-n_mod` — how many modules of an extendable modular build. A unit that is not active has none"
dims: [line]
where: Line_s_nom_extendable AND Line_s_nom_mod > 0 AND Line_active_asset
domain: integer
bounds:
lower: 0
expressions:
Line_transmission_volume_expansion:
expression: sum(Line_s_nom_ext * Line_volume_weight, over=line)
adds_to: transmission_volume_expansion
Line_transmission_expansion_cost:
expression: sum(Line_s_nom_ext * Line_expansion_cost_weight, over=line)
adds_to: transmission_expansion_cost
Line_tech_capacity_expansion:
expression: sum(Line_s_nom_ext * Line_tech_capacity_weight, over=line)
adds_to: tech_capacity_expansion
Line_capex:
expression: sum(scenario_weight * Line_s_nom_ext * Line_capital_cost * Line_capital_weight)
adds_to: total_cost
Line_additions:
expression: >-
sum(Line_s_nom_ext * Line_first_active, over=line, by=line[carrier])
adds_to: Carrier_additions
Line_s_monitored:
description: >-
the flow in a line's post-contingency rows. It is its flow where it
is active, and nothing where it does not. PyPSA builds those rows for every
branch of the sub-network in every snapshot
dims: [scenario, snapshot, line]
cases:
standing: { when: Line_active, expression: Line_s }
otherwise: 0
Line_injection:
expression: >-
-sum(Line_s, over=line, by=line[bus0])
+ sum(Line_s, over=line, by=line[bus1])
- (0.5 * sum(Line_loss, over=line, by=line[bus0]))
- (0.5 * sum(Line_loss, over=line, by=line[bus1]))
adds_to: Bus_injection
Line_angle_sum:
expression: sum(Line_s * at(Line_cycle_weight, by=snapshot[period]), over=line)
adds_to: Cycle_angle_sum
conditions:
Line_lossy:
description: >-
a line that is active in the snapshot's period, where the run models
transmission losses
condition: transmission_losses AND Line_active
constraints:
Line_fix_s_lower:
description: "`Line-fix-s-lower` — a fixed line carries at least the negative of its rating, the loss counted against it"
dims: [scenario, snapshot, line]
where: not Line_s_nom_extendable AND Line_active
expression: Line_s - Line_loss >= -Line_s_max_pu * Line_s_nom
Line_fix_s_upper:
description: "`Line-fix-s-upper` — a fixed line carries at most its rating, the loss included"
dims: [scenario, snapshot, line]
where: not Line_s_nom_extendable AND Line_active
expression: Line_s + Line_loss <= Line_s_max_pu * Line_s_nom
Line_ext_s_lower:
description: "`Line-ext-s-lower` — an extendable line carries at least the negative of its rating of the chosen build, the loss counted against it"
dims: [scenario, snapshot, line]
where: Line_s_nom_extendable AND Line_active
expression: Line_s - Line_loss >= -Line_s_max_pu * Line_s_nom_ext
Line_ext_s_upper:
description: "`Line-ext-s-upper` — an extendable line carries at most its rating of the chosen build, the loss included"
dims: [scenario, snapshot, line]
where: Line_s_nom_extendable AND Line_active
expression: Line_s + Line_loss <= Line_s_max_pu * Line_s_nom_ext
Line_ext_s_nom_lower:
description: "`Line-ext-s_nom-lower` — the chosen build is at least its floor in every scenario"
dims: [scenario, line]
where: Line_s_nom_extendable AND Line_active_asset
expression: Line_s_nom_ext >= Line_s_nom_min
Line_ext_s_nom_upper:
description: "`Line-ext-s_nom-upper` — the chosen build is at most its cap in every scenario. A cap of infinity is no row"
dims: [scenario, line]
where: Line_s_nom_extendable AND Line_active_asset AND Line_s_nom_max
expression: Line_s_nom_ext <= Line_s_nom_max
Line_s_nom_set:
description: "`Line-s_nom_set` — the chosen build pinned, wherever a value is given"
dims: [line]
where: Line_s_nom_extendable AND Line_active_asset AND Line_s_nom_set
expression: Line_s_nom_ext == Line_s_nom_set
Line_s_nom_modularity:
description: "`Line-s_nom_modularity` — the chosen build is a whole number of modules"
dims: [line]
where: Line_s_nom_extendable AND Line_s_nom_mod > 0 AND Line_active_asset
expression: Line_s_nom_ext == Line_s_nom_mod * Line_n_mod
Line_s_set:
description: "`Line-s_set` — flow pinned to the given schedule, wherever one is given"
dims: [scenario, snapshot, line]
where: Line_s_set AND Line_active
expression: Line_s == Line_s_set
Line_v_ang_lower:
description: >-
`Line-v_ang-lower` — an AC line carries at least the flow at which the
voltage angle difference across it is the negative of its limit. That
difference is `x_pu_eff` times the flow in radians
dims: [scenario, snapshot, line]
where: Line_v_ang_max AND line[carrier] == 'AC' AND Line_active
expression: Line_s >= -Line_v_ang_max * (3.141592653589793 / 180) / Line_x_pu_eff
Line_v_ang_upper:
description: >-
`Line-v_ang-upper` — an AC line carries at most the flow at which the
voltage angle difference across it reaches its limit
dims: [scenario, snapshot, line]
where: Line_v_ang_max AND line[carrier] == 'AC' AND Line_active
expression: Line_s <= Line_v_ang_max * (3.141592653589793 / 180) / Line_x_pu_eff
Line_loss_upper:
description: "`Line-loss_upper` — a line dissipates at most the loss at its rating"
dims: [scenario, snapshot, line]
where: Line_lossy
expression: Line_loss <= Line_loss_max
Line_loss_tangents_forward:
description: >-
`Line-loss_tangents-{k}-1`, `Line-loss_secants-pos` — the loss sits
above every cut to its curve for flow one way. PyPSA names one row per
tangent `k`, or one row stacked over its `secant` axis, and this entry
states them all over the segment dimension
dims: [scenario, snapshot, line, segment]
where: Line_lossy
expression: Line_loss + Line_loss_slope * Line_s >= Line_loss_offset
Line_loss_tangents_reverse:
description: >-
`Line-loss_tangents-{k}--1`, `Line-loss_secants-neg` — the same fan
mirrored, the loss depending on the flow's magnitude
dims: [scenario, snapshot, line, segment]
where: Line_lossy
expression: Line_loss - Line_loss_slope * Line_s >= Line_loss_offset
Given#
| Symbol | Meaning |
|---|---|
| \(\pi\) | scenario_weight over \(\Xi\), data another file declares |
| \(\mathrm{lossy}\) | transmission_losses (scalar), data another file declares |
| \(\mathit{Bus\_injection}\) | Bus_injection over \(\mathcal{N} \times \Xi \times \mathcal{T}\), an expression this file adds Line_injection to |
| \(\mathit{Carrier\_additions}\) | Carrier_additions over \(\mathcal{I} \times \mathcal{Y}\), an expression this file adds Line_additions to |
| \(\mathit{Cycle\_angle\_sum}\) | Cycle_angle_sum over \(\mathcal{C} \times \Xi \times \mathcal{T}\), an expression this file adds Line_angle_sum to |
| \(\mathit{tech\_capacity\_expansion}\) | tech_capacity_expansion over \(\mathcal{G}\), an expression this file adds Line_tech_capacity_expansion to |
| \(\mathit{total\_cost}\) | total_cost (scalar), an expression this file adds Line_capex to |
| \(\mathit{transmission\_expansion\_cost}\) | transmission_expansion_cost over \(\mathcal{G} \times \Xi\), an expression this file adds Line_transmission_expansion_cost to |
| \(\mathit{transmission\_volume\_expansion}\) | transmission_volume_expansion over \(\mathcal{G} \times \Xi\), an expression this file adds Line_transmission_volume_expansion to |
Sets#
| Symbol | Meaning |
|---|---|
| \(\mathcal{N}\) | index \(n\) — bus with \(\mathrm{bus0}: \mathcal{K} \to \mathcal{N},\ \mathrm{bus1}: \mathcal{K} \to \mathcal{N}\) — network nodes |
| \(\mathcal{I}\) | index \(i\) — carrier with \(\mathrm{carrier}: \mathcal{K} \to \mathcal{I}\) — energy carriers, what a growth limit is set per |
| \(\mathcal{C}\) | index \(c\) — cycle — independent cycles of the passive network graph, the cycle basis, data prep. Each period has its own basis, of the branches that are active in it. A label is a position in that period's basis, so one label is a different cycle in another period |
| \(\mathcal{G}\) | index \(g\) — global_constraint — PyPSA's GlobalConstraint rows, one label per declared limit |
| \(\mathcal{K}\) | index \(k\) — line with \(\mathrm{bus0}: \mathcal{K} \to \mathcal{N},\ \mathrm{bus1}: \mathcal{K} \to \mathcal{N},\ \mathrm{carrier}: \mathcal{K} \to \mathcal{I}\) — passive branches, each between two buses, their flow set by impedance |
| \(\mathcal{Y}\) | index \(y\) — period with \(\mathrm{period}: \mathcal{T} \to \mathcal{Y}\) — investment periods. It is PyPSA's investment_periods |
| \(\Xi\) | index \(\xi\) — scenario — the futures dispatch is chosen in, each with a weight |
| \(\mathcal{E}\) | index \(e\) — segment — the cuts a passive branch's loss curve is held above. They are PyPSA's tangents, as many as its segments count, or its secants, as many as its tolerance loop places. A lossless run has none |
| \(\mathcal{T}\) | index \(t\) — snapshot with \(\mathrm{period}: \mathcal{T} \to \mathcal{Y}\) — dispatch periods |
Parameters#
| Symbol | Meaning |
|---|---|
| \(\mathrm{on}^{s}\) | Line_active over \(\mathcal{K} \times \mathcal{T}\) — whether a line is active in a snapshot's period. It is PyPSA's active, data prep |
| \(\mathrm{act}^{s}\) | Line_active_asset over \(\mathcal{K}\) — whether a line is one of PyPSA's active_assets, by its active flag alone, without build year and lifetime. A unit that is active in no snapshot of the horizon still counts. PyPSA refuses one that differs by scenario (constants.py:54) |
| \(\mathrm{c}^{\mathrm{cap},s}\) | Line_capital_cost over \(\mathcal{K} \times \Xi\) — cost of one unit of nominal apparent power for the modelled horizon (PyPSA's periodized_cost), data prep. It is overnight_cost as an annuity over lifetime at discount_rate, times nyears, where overnight_cost is given, and capital_cost where it is not. Both add fom_cost (components.py:1157-1178, costs.py:102-203) |
| \(\mathrm{W}^{s}\) | Line_capital_weight over \(\mathcal{K}\) — the sum of period weights a line is active in. It is PyPSA's active * period_weighting, summed, data prep |
| \(\mathrm{x}\) | Line_cycle_weight over \(\mathcal{C} \times \mathcal{K} \times \mathcal{Y}\), absent where the data has no row — the line's series impedance, signed by its orientation in the cycle: the cycle basis, data prep. A line in no cycle has no row. PyPSA builds the cycle basis from the first scenario (networks.py:1430-1437), and refuses a v_nom, x_pu_eff or r_pu_eff that differs by scenario (constants.py:33-35). So the weight is the same in every scenario |
| \(\mathrm{cc}\) | Line_expansion_cost_weight over \(\mathcal{G} \times \mathcal{K} \times \Xi\), absent where the data has no row — the line's capital cost where its carrier is in the row's set, data prep. Where the row states no investment_period under multi_investment_periods, it is times the objective weights of the periods the line is active in. A line outside the set, or one that is not active in the row's period, has no row. The capital cost is PyPSA's capital_cost property, which is Line_capital_cost without fom_cost (components.py:1182-1200, global_constraints.py:905-914) |
| \(\mathrm{new}^{s}\) | Line_first_active over \(\mathcal{K} \times \mathcal{Y}\) — one in the first period a line is active in, zero elsewhere, data prep. PyPSA takes active & (active.cumsum() == 1) (global_constraints.py:278) |
| \(\overline{\ell}\) | Line_loss_max over \(\mathcal{K} \times \Xi \times \mathcal{T}\) — the loss at a line's rating. It is PyPSA's r_pu_eff * (s_max_pu * s_nom_max)**2, data prep |
| \(\mathrm{b}\) | Line_loss_offset over \(\mathcal{K} \times \Xi \times \mathcal{E} \times \mathcal{T}\) — where that cut meets the loss axis: a tangent's loss_k - slope_k * p_k, a secant's -r_pu_eff * p_k * p_k+1, negative, data prep |
| \(\mathrm{a}\) | Line_loss_slope over \(\mathcal{K} \times \Xi \times \mathcal{E} \times \mathcal{T}\) — the slope of a cut to the loss curve, data prep. It is a tangent's 2 * r_pu_eff * p_k at its segment's flow, or a secant's r_pu_eff * (p_k + p_k+1) between consecutive breakpoints |
| \(\overline{\mathrm{s}}\) | Line_s_max_pu over \(\mathcal{K} \times \Xi \times \mathcal{T}\) — most flow either way, per unit of nominal apparent power |
| \(\mathrm{s}^{\mathrm{nom}}\) | Line_s_nom over \(\mathcal{K} \times \Xi\) — nominal apparent power |
| \(\mathrm{ext}^{s}\) | Line_s_nom_extendable over \(\mathcal{K}\) — whether the nominal apparent power is a decision |
| \(\overline{\mathrm{s}}^{\mathrm{nom}}\) | Line_s_nom_max over \(\mathcal{K} \times \Xi\) — most nominal apparent power an extendable line may be built at |
| \(\underline{\mathrm{s}}^{\mathrm{nom}}\) | Line_s_nom_min over \(\mathcal{K} \times \Xi\) — least nominal apparent power an extendable line may be built at |
| \(\mathrm{s}^{\mathrm{mod}}\) | Line_s_nom_mod over \(\mathcal{K}\), absent where the data has no row — the module size a build comes in whole numbers of. No value means the build is continuous |
| \(\mathrm{s}^{\mathrm{nom,set}}\) | Line_s_nom_set over \(\mathcal{K}\), absent where the data has no row — a given nominal apparent power for an extendable line. One without a value has no row here |
| \(\mathrm{s}^{\mathrm{set}}\) | Line_s_set over \(\mathcal{K} \times \Xi \times \mathcal{T}\), absent where the data has no row — a given flow schedule. A line without one has no row here |
| \(\mathrm{m}^{l}\) | Line_tech_capacity_weight over \(\mathcal{G} \times \mathcal{K}\), absent where the data has no row — one where the line is in the row's carrier-and-bus set, data prep. One outside it, or one that is not active in the row's investment_period, has no row |
| \(\overline{\delta}\) | Line_v_ang_max over \(\mathcal{K} \times \Xi\), absent where the data has no row — the most the voltage angle difference across a line may be either way, in degrees. It is PyPSA's v_ang_max. Infinite, and so no row, by default. A line whose carrier is not AC has no row either. The deprecated v_ang_min is ignored, as PyPSA ignores it with a DeprecationWarning (constraints.py:1797-1804) |
| \(\mathrm{len}\) | Line_volume_weight over \(\mathcal{G} \times \mathcal{K} \times \Xi\), absent where the data has no row — the line's length in each scenario where its carrier is in the row's set (global_constraints.py:882-883), data prep. A line outside it, or one that is not active in the row's investment_period, has no row |
| \(\mathrm{x}^{\mathrm{eff}}\) | Line_x_pu_eff over \(\mathcal{K} \times \Xi\) — the line's effective series reactance. It is PyPSA's x_pu_eff, x over the square of its bus's nominal voltage, data prep |
Conditions#
| Symbol | Meaning |
|---|---|
| \(\mathrm{on}^{s,\mathrm{lossy}}\) | Line_lossy over \(\mathcal{K} \times \mathcal{T}\) — a line that is active in the snapshot's period, where the run models transmission losses |
Variables#
| Symbol | Meaning |
|---|---|
| \(\ell\) | Line_loss over \(\mathcal{K} \times \Xi \times \mathcal{T}\), \(0\) where the mask leaves it out — Line-loss — what a line dissipates carrying its flow, pushed down by the cost and held up by the cuts. Absent, and zero in the balance, where the network is lossless |
| \(N^{s}\) | Line_n_mod over \(\mathcal{K}\) — Line-n_mod — how many modules of an extendable modular build. A unit that is not active has none |
| \(s\) | Line_s over \(\mathcal{K} \times \Xi \times \mathcal{T}\) — Line-s — PyPSA's p0, the flow measured at the line[bus0] end: a positive value withdraws there and injects at line[bus1], lossless |
| \(S\) | Line_s_nom_ext over \(\mathcal{K}\) — Line-s_nom — nominal apparent power where it is a decision, for a line that is active. A fixed build is the parameter of the same PyPSA name |
Definitions#
| Symbol | Meaning |
|---|---|
| \(\mathit{Line\_additions}\) | Line_additions over \(\mathcal{I} \times \mathcal{Y}\) |
| \(\mathit{Line\_angle\_sum}\) | Line_angle_sum over \(\mathcal{C} \times \Xi \times \mathcal{T}\) |
| \(\mathit{Line\_capex}\) | Line_capex (scalar) |
| \(\mathit{Line\_injection}\) | Line_injection over \(\mathcal{N} \times \Xi \times \mathcal{T}\) |
| \(\check{s}\) | Line_s_monitored over \(\mathcal{K} \times \Xi \times \mathcal{T}\) — the flow in a line's post-contingency rows. It is its flow where it is active, and nothing where it does not. PyPSA builds those rows for every branch of the sub-network in every snapshot |
| \(\mathit{Line\_tech\_capacity\_expansion}\) | Line_tech_capacity_expansion over \(\mathcal{G}\) |
| \(\mathit{Line\_transmission\_expansion\_cost}\) | Line_transmission_expansion_cost over \(\mathcal{G} \times \Xi\) |
| \(\mathit{Line\_transmission\_volume\_expansion}\) | Line_transmission_volume_expansion over \(\mathcal{G} \times \Xi\) |
Conditions#
Line_lossy
\[
\mathrm{on}^{s,\mathrm{lossy}}_{k,t} \iff \mathrm{lossy} \wedge \mathrm{on}^{s}_{k,t} \qquad \forall\, k \in \mathcal{K},\ t \in \mathcal{T}
\]
Variable domains#
Line_loss
\[
\ell_{k,\xi,t} \ge 0 \qquad \forall\, k \in \mathcal{K},\ \xi \in \Xi,\ t \in \mathcal{T} \,:\, \mathrm{on}^{s,\mathrm{lossy}}_{k,t}
\]
Line_n_mod
\[
N^{s}_{k} \ge 0, N^{s}_{k} \in \mathbb{Z} \qquad \forall\, k \in \mathcal{K} \,:\, \mathrm{ext}^{s}_{k} \wedge \mathrm{s}^{\mathrm{mod}}_{k} > 0 \wedge \mathrm{act}^{s}_{k}
\]
Line_s
\[
s_{k,\xi,t} \in \mathbb{R} \qquad \forall\, k \in \mathcal{K},\ \xi \in \Xi,\ t \in \mathcal{T} \,:\, \mathrm{on}^{s}_{k,t}
\]
Line_s_nom_ext
\[
S_{k} \in \mathbb{R} \qquad \forall\, k \in \mathcal{K} \,:\, \mathrm{ext}^{s}_{k} \wedge \mathrm{act}^{s}_{k}
\]
Definitions#
Line_additions
\[
\mathit{Line\_additions}_{i,y} = \sum_{k \in \mathcal{K} \,:\, \mathrm{carrier}(k) = i} S_{k} \cdot \mathrm{new}^{s}_{k,y} \qquad \forall\, i \in \mathcal{I},\ y \in \mathcal{Y}
\]
Line_angle_sum
\[
\mathit{Line\_angle\_sum}_{c,\xi,t} = \sum_{k \in \mathcal{K}} s_{k,\xi,t} \cdot \mathrm{x}_{c,k,\mathrm{period}(t)} \qquad \forall\, c \in \mathcal{C},\ \xi \in \Xi,\ t \in \mathcal{T}
\]
Line_capex
\[
\mathit{Line\_capex} = \sum_{k \in \mathcal{K},\ \xi \in \Xi} \pi_{\xi} \cdot S_{k} \cdot \mathrm{c}^{\mathrm{cap},s}_{k,\xi} \cdot \mathrm{W}^{s}_{k}
\]
Line_injection
\[
\mathit{Line\_injection}_{n,\xi,t} = -\left( \sum_{k \in \mathcal{K} \,:\, \mathrm{bus0}(k) = n} s_{k,\xi,t} \right) + \sum_{k \in \mathcal{K} \,:\, \mathrm{bus1}(k) = n} s_{k,\xi,t} - 0.5 \cdot \left( \sum_{k \in \mathcal{K} \,:\, \mathrm{bus0}(k) = n} \ell_{k,\xi,t} \right) - 0.5 \cdot \left( \sum_{k \in \mathcal{K} \,:\, \mathrm{bus1}(k) = n} \ell_{k,\xi,t} \right) \qquad \forall\, n \in \mathcal{N},\ \xi \in \Xi,\ t \in \mathcal{T}
\]
Line_s_monitored
\[
\check{s}_{k,\xi,t} = \begin{cases} s_{k,\xi,t} & \text{if } \mathrm{on}^{s}_{k,t} \\ 0 & \text{otherwise} \end{cases} \qquad \forall\, k \in \mathcal{K},\ \xi \in \Xi,\ t \in \mathcal{T}
\]
Line_tech_capacity_expansion
\[
\mathit{Line\_tech\_capacity\_expansion}_{g} = \sum_{k \in \mathcal{K}} S_{k} \cdot \mathrm{m}^{l}_{g,k} \qquad \forall\, g \in \mathcal{G}
\]
Line_transmission_expansion_cost
\[
\mathit{Line\_transmission\_expansion\_cost}_{g,\xi} = \sum_{k \in \mathcal{K}} S_{k} \cdot \mathrm{cc}_{g,k,\xi} \qquad \forall\, g \in \mathcal{G},\ \xi \in \Xi
\]
Line_transmission_volume_expansion
\[
\mathit{Line\_transmission\_volume\_expansion}_{g,\xi} = \sum_{k \in \mathcal{K}} S_{k} \cdot \mathrm{len}_{g,k,\xi} \qquad \forall\, g \in \mathcal{G},\ \xi \in \Xi
\]
Subject to#
Line_ext_s_lower
\[
s_{k,\xi,t} - \ell_{k,\xi,t} \ge -\overline{\mathrm{s}}_{k,\xi,t} \cdot S_{k} \qquad \forall\, k \in \mathcal{K},\ \xi \in \Xi,\ t \in \mathcal{T} \,:\, \mathrm{ext}^{s}_{k} \wedge \mathrm{on}^{s}_{k,t}
\]
Line_ext_s_nom_lower
\[
S_{k} \ge \underline{\mathrm{s}}^{\mathrm{nom}}_{k,\xi} \qquad \forall\, k \in \mathcal{K},\ \xi \in \Xi \,:\, \mathrm{ext}^{s}_{k} \wedge \mathrm{act}^{s}_{k}
\]
Line_ext_s_nom_upper
\[
S_{k} \le \overline{\mathrm{s}}^{\mathrm{nom}}_{k,\xi} \qquad \forall\, k \in \mathcal{K},\ \xi \in \Xi \,:\, \mathrm{ext}^{s}_{k} \wedge \mathrm{act}^{s}_{k} \wedge \overline{\mathrm{s}}^{\mathrm{nom}}_{k,\xi} \text{ is defined}
\]
Line_ext_s_upper
\[
s_{k,\xi,t} + \ell_{k,\xi,t} \le \overline{\mathrm{s}}_{k,\xi,t} \cdot S_{k} \qquad \forall\, k \in \mathcal{K},\ \xi \in \Xi,\ t \in \mathcal{T} \,:\, \mathrm{ext}^{s}_{k} \wedge \mathrm{on}^{s}_{k,t}
\]
Line_fix_s_lower
\[
s_{k,\xi,t} - \ell_{k,\xi,t} \ge -\overline{\mathrm{s}}_{k,\xi,t} \cdot \mathrm{s}^{\mathrm{nom}}_{k,\xi} \qquad \forall\, k \in \mathcal{K},\ \xi \in \Xi,\ t \in \mathcal{T} \,:\, \neg \mathrm{ext}^{s}_{k} \wedge \mathrm{on}^{s}_{k,t}
\]
Line_fix_s_upper
\[
s_{k,\xi,t} + \ell_{k,\xi,t} \le \overline{\mathrm{s}}_{k,\xi,t} \cdot \mathrm{s}^{\mathrm{nom}}_{k,\xi} \qquad \forall\, k \in \mathcal{K},\ \xi \in \Xi,\ t \in \mathcal{T} \,:\, \neg \mathrm{ext}^{s}_{k} \wedge \mathrm{on}^{s}_{k,t}
\]
Line_loss_tangents_forward
\[
\ell_{k,\xi,t} + \mathrm{a}_{k,\xi,e,t} \cdot s_{k,\xi,t} \ge \mathrm{b}_{k,\xi,e,t} \qquad \forall\, k \in \mathcal{K},\ \xi \in \Xi,\ e \in \mathcal{E},\ t \in \mathcal{T} \,:\, \mathrm{on}^{s,\mathrm{lossy}}_{k,t}
\]
Line_loss_tangents_reverse
\[
\ell_{k,\xi,t} - \mathrm{a}_{k,\xi,e,t} \cdot s_{k,\xi,t} \ge \mathrm{b}_{k,\xi,e,t} \qquad \forall\, k \in \mathcal{K},\ \xi \in \Xi,\ e \in \mathcal{E},\ t \in \mathcal{T} \,:\, \mathrm{on}^{s,\mathrm{lossy}}_{k,t}
\]
Line_loss_upper
\[
\ell_{k,\xi,t} \le \overline{\ell}_{k,\xi,t} \qquad \forall\, k \in \mathcal{K},\ \xi \in \Xi,\ t \in \mathcal{T} \,:\, \mathrm{on}^{s,\mathrm{lossy}}_{k,t}
\]
Line_s_nom_modularity
\[
S_{k} = \mathrm{s}^{\mathrm{mod}}_{k} \cdot N^{s}_{k} \qquad \forall\, k \in \mathcal{K} \,:\, \mathrm{ext}^{s}_{k} \wedge \mathrm{s}^{\mathrm{mod}}_{k} > 0 \wedge \mathrm{act}^{s}_{k}
\]
Line_s_nom_set
\[
S_{k} = \mathrm{s}^{\mathrm{nom,set}}_{k} \qquad \forall\, k \in \mathcal{K} \,:\, \mathrm{ext}^{s}_{k} \wedge \mathrm{act}^{s}_{k} \wedge \mathrm{s}^{\mathrm{nom,set}}_{k} \text{ is defined}
\]
Line_s_set
\[
s_{k,\xi,t} = \mathrm{s}^{\mathrm{set}}_{k,\xi,t} \qquad \forall\, k \in \mathcal{K},\ \xi \in \Xi,\ t \in \mathcal{T} \,:\, \mathrm{s}^{\mathrm{set}}_{k,\xi,t} \text{ is defined} \wedge \mathrm{on}^{s}_{k,t}
\]
Line_v_ang_lower
\[
s_{k,\xi,t} \ge \frac{-\overline{\delta}_{k,\xi} \cdot \frac{3.141592653589793}{180}}{\mathrm{x}^{\mathrm{eff}}_{k,\xi}} \qquad \forall\, k \in \mathcal{K},\ \xi \in \Xi,\ t \in \mathcal{T} \,:\, \overline{\delta}_{k,\xi} \text{ is defined} \wedge \mathrm{carrier}(k) = \text{'}\mathrm{AC}\text{'} \wedge \mathrm{on}^{s}_{k,t}
\]
Line_v_ang_upper
\[
s_{k,\xi,t} \le \frac{\overline{\delta}_{k,\xi} \cdot \frac{3.141592653589793}{180}}{\mathrm{x}^{\mathrm{eff}}_{k,\xi}} \qquad \forall\, k \in \mathcal{K},\ \xi \in \Xi,\ t \in \mathcal{T} \,:\, \overline{\delta}_{k,\xi} \text{ is defined} \wedge \mathrm{carrier}(k) = \text{'}\mathrm{AC}\text{'} \wedge \mathrm{on}^{s}_{k,t}
\]
See also
- PyPSA in 25 files, for how the fragments compose
- PyPSA in one file, for the same spec as one file, rung by rung