Transformers#
One of the 24 fragments of examples/pypsa.yaml: PyPSA's Transformer. It adds a term to Bus_injection, Cycle_angle_sum. It reads scenario_weight, transmission_losses 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
transformer:
description: passive branches between two buses, their flow set by impedance and tap ratio, with a phase shift fixed or optimised
cycle:
description: independent cycles of the passive network graph — the cycle basis, data prep
segment:
description: >-
the cuts a passive branch's loss curve is held above — PyPSA's tangents,
as many as its `segments` count, or its secants, as many as its tolerance
loop places; none in a lossless run
dtype: int
relations:
Transformer_bus0:
description: the bus a transformer's flow is measured at
key: transformer
values: bus
Transformer_bus1:
description: the bus at a transformer's other end
key: transformer
values: bus
parameters:
Transformer_active:
description: whether a transformer stands in a snapshot's period — PyPSA's `active`, data prep
dims: [snapshot, transformer]
dtype: bool
Transformer_capital_weight:
description: the sum of period weights a transformer stands in — PyPSA's `active * period_weighting`, summed, data prep
dims: [transformer]
Transformer_s_nom:
description: nominal apparent power
dims: [scenario, transformer]
Transformer_s_nom_extendable:
description: whether the nominal apparent power is a decision
dims: [transformer]
dtype: bool
Transformer_s_max_pu:
description: most flow either way, per unit of nominal apparent power
dims: [scenario, snapshot, transformer]
Transformer_s_nom_min:
description: least nominal apparent power an extendable transformer may be built at
dims: [scenario, transformer]
Transformer_s_nom_max:
description: most nominal apparent power an extendable transformer may be built at
dims: [scenario, transformer]
Transformer_capital_cost:
description: cost of one unit of nominal apparent power — PyPSA's `capital_cost`, periodized as an annuity in data prep
dims: [scenario, transformer]
Transformer_s_nom_set:
description: a given nominal apparent power for an extendable transformer; one without a value has no row here
dims: [scenario, transformer]
Transformer_s_set:
description: a given flow schedule; a transformer without one has no row here
dims: [scenario, snapshot, transformer]
Transformer_cycle_weight:
description: >-
the transformer's effective series reactance, `x` times its tap ratio,
signed by its orientation in the cycle — PyPSA's `x_pu_eff`, the cycle
basis, data prep; a transformer in no cycle has no row. From the first
scenario only, as a line's
dims: [transformer, cycle]
Transformer_phase_shift_weight:
description: >-
a fixed transformer's phase shift in radians at each snapshot, signed by
its orientation in the cycle — a constant added to the cycle sum, data prep; zero for a
varying transformer, whose shift is a decision instead, so the constant and
the variable term never both count a shift. A transformer with no shift or
in no cycle has no row
dims: [snapshot, transformer, cycle]
Transformer_phase_shift_varying:
description: >-
whether a transformer's phase shift is a decision — PyPSA's
`phase_shift_min < phase_shift_max`, read as a flag in data prep; false is a
fixed shift carried by `phase_shift`. The shift parameters carry no
scenario: only a cycle row reads them, and PyPSA fails on a transformer in
a cycle on a network with scenarios (`constraints.py:1654`)
dims: [transformer]
dtype: bool
Transformer_phase_shift_min:
description: >-
the least a varying transformer's phase shift may take, in degrees —
PyPSA's `phase_shift_min`; where it is below `phase_shift_max` the shift is
a decision, otherwise the transformer keeps its fixed `phase_shift`
dims: [transformer]
Transformer_phase_shift_max:
description: >-
the most a varying transformer's phase shift may take, in degrees —
PyPSA's `phase_shift_max`; equal to `phase_shift_min` for a fixed transformer
dims: [transformer]
Transformer_phase_shift_cycle_weight:
description: >-
the cycle sign for a varying transformer's phase shift, times π/180 so a
shift in degrees enters the cycle sum in radians — data prep; zero for a
fixed transformer or one in no cycle
dims: [transformer, cycle]
Transformer_loss_max:
description: >-
the loss at a transformer's rating — PyPSA's `r_pu_eff * (s_max_pu *
s_nom_max)**2`, its `r_pu_eff` the resistance over the given `s_nom` times
the tap ratio, data prep
dims: [scenario, snapshot, transformer]
Transformer_loss_slope:
description: >-
the slope of a cut to a transformer's loss curve — a tangent's
`2 * r_pu_eff * p_k`, a secant's `r_pu_eff * (p_k + p_k+1)`, as a line's,
over the transformer's own `r_pu_eff` and rating, data prep
dims: [scenario, snapshot, transformer, segment]
Transformer_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, transformer, segment]
variables:
Transformer_s:
description: >-
`Transformer-s` — PyPSA's `p0`, the flow measured at the
`Transformer_bus0` end: a positive value withdraws there and injects at
`Transformer_bus1`, lossless
dims: [scenario, snapshot, transformer]
where: Transformer_active
Transformer_loss:
description: >-
`Transformer-loss` — what a transformer dissipates carrying its flow, as
a line does; absent, and zero in the balance, where the network is
lossless
dims: [scenario, snapshot, transformer]
where: transmission_losses AND Transformer_active
absence: zero
bounds:
lower: 0
Transformer_phase_shift:
description: >-
`Transformer-phase_shift` — a phase-shifting transformer's voltage angle
shift in degrees, chosen per snapshot to redistribute the flows around its
cycles without moving active power; absent, and zero in the cycle sum,
where the shift is fixed
dims: [scenario, snapshot, transformer]
where: Transformer_phase_shift_varying AND Transformer_active
absence: zero
bounds:
lower: Transformer_phase_shift_min
upper: Transformer_phase_shift_max
Transformer_s_nom_ext:
description: >-
`Transformer-s_nom` — nominal apparent power where it is a decision; the
parameter of the same PyPSA name carries the fixed regime
dims: [transformer]
where: Transformer_s_nom_extendable
given:
parameters:
scenario_weight: { dims: [scenario] }
transmission_losses: { dims: [], dtype: bool }
expressions:
Bus_injection: { dims: [scenario, snapshot, bus] }
Cycle_angle_sum: { dims: [scenario, snapshot, cycle] }
expressions:
Transformer_s_monitored:
description: the flow a transformer's post-contingency rows read, as a line's
dims: [scenario, snapshot, transformer]
cases:
standing: { when: Transformer_active, expression: Transformer_s }
otherwise: 0
Transformer_injection:
expression: >-
-sum(Transformer_s, by=Transformer_bus0, over=transformer, into=bus)
+ sum(Transformer_s, by=Transformer_bus1, over=transformer, into=bus)
- (0.5 * sum(Transformer_loss, by=Transformer_bus0, over=transformer, into=bus))
- (0.5 * sum(Transformer_loss, by=Transformer_bus1, over=transformer, into=bus))
adds_to: Bus_injection
Transformer_angle_sum:
expression: >-
sum(Transformer_s * Transformer_cycle_weight, over=transformer)
+ sum(Transformer_phase_shift_weight, over=transformer)
+ sum(Transformer_phase_shift * Transformer_phase_shift_cycle_weight, over=transformer)
adds_to: Cycle_angle_sum
constraints:
Transformer_fix_s_lower:
description: "`Transformer-fix-s-lower` — a fixed transformer carries at least the negative of its rating, the loss counted against it"
dims: [scenario, snapshot, transformer]
where: not Transformer_s_nom_extendable AND Transformer_active
expression: Transformer_s - Transformer_loss >= -Transformer_s_max_pu * Transformer_s_nom
Transformer_fix_s_upper:
description: "`Transformer-fix-s-upper` — a fixed transformer carries at most its rating, the loss included"
dims: [scenario, snapshot, transformer]
where: not Transformer_s_nom_extendable AND Transformer_active
expression: Transformer_s + Transformer_loss <= Transformer_s_max_pu * Transformer_s_nom
Transformer_ext_s_lower:
description: "`Transformer-ext-s-lower` — an extendable transformer carries at least the negative of its rating of the chosen build, the loss counted against it"
dims: [scenario, snapshot, transformer]
where: Transformer_s_nom_extendable AND Transformer_active
expression: Transformer_s - Transformer_loss >= -Transformer_s_max_pu * Transformer_s_nom_ext
Transformer_ext_s_upper:
description: "`Transformer-ext-s-upper` — an extendable transformer carries at most its rating of the chosen build, the loss included"
dims: [scenario, snapshot, transformer]
where: Transformer_s_nom_extendable AND Transformer_active
expression: Transformer_s + Transformer_loss <= Transformer_s_max_pu * Transformer_s_nom_ext
Transformer_ext_s_nom_lower:
description: "`Transformer-ext-s_nom-lower` — the chosen build is at least its floor in every scenario"
dims: [scenario, transformer]
where: Transformer_s_nom_extendable
expression: Transformer_s_nom_ext >= Transformer_s_nom_min
Transformer_ext_s_nom_upper:
description: "`Transformer-ext-s_nom-upper` — the chosen build is at most its cap in every scenario; a cap of infinity is no row"
dims: [scenario, transformer]
where: Transformer_s_nom_extendable AND Transformer_s_nom_max
expression: Transformer_s_nom_ext <= Transformer_s_nom_max
Transformer_s_nom_set:
description: "`Transformer-s_nom_set` — the chosen build pinned, wherever a value is given"
dims: [scenario, transformer]
where: Transformer_s_nom_extendable AND Transformer_s_nom_set
expression: Transformer_s_nom_ext == Transformer_s_nom_set
Transformer_s_set:
description: "`Transformer-s_set` — flow pinned to the given schedule, wherever one is given"
dims: [scenario, snapshot, transformer]
where: Transformer_s_set AND Transformer_active
expression: Transformer_s == Transformer_s_set
Transformer_loss_upper:
description: "`Transformer-loss_upper` — a transformer dissipates at most the loss at its rating"
dims: [scenario, snapshot, transformer]
where: transmission_losses AND Transformer_active
expression: Transformer_loss <= Transformer_loss_max
Transformer_loss_tangents_forward:
description: >-
`Transformer-loss_tangents-{k}-1`, `Transformer-loss_secants-pos` — the
loss sits above every cut to its curve for flow one way, as a line's
does, over the segment dimension
dims: [scenario, snapshot, transformer, segment]
where: transmission_losses AND Transformer_active
expression: Transformer_loss + Transformer_loss_slope * Transformer_s >= Transformer_loss_offset
Transformer_loss_tangents_reverse:
description: >-
`Transformer-loss_tangents-{k}--1`, `Transformer-loss_secants-neg` — the
same fan mirrored, the loss depending on the flow's magnitude
dims: [scenario, snapshot, transformer, segment]
where: transmission_losses AND Transformer_active
expression: Transformer_loss - Transformer_loss_slope * Transformer_s >= Transformer_loss_offset
objective:
sense: minimize
expression: >-
sum(((scenario_weight * Transformer_s_nom_ext) * Transformer_capital_cost) * Transformer_capital_weight)
Sets#
| Symbol | Meaning |
|---|---|
| \(\Xi\) | index \(\xi\) — scenario — the futures dispatch is chosen in, each with a weight |
| \(\mathcal{T}\) | index \(t\) — snapshot — dispatch periods |
| \(\mathcal{N}\) | index \(n\) — bus with \(\mathrm{Transformer\_bus0}: \mathcal{M} \to \mathcal{N},\ \mathrm{Transformer\_bus1}: \mathcal{M} \to \mathcal{N}\) — network nodes |
| \(\mathcal{M}\) | index \(m\) — transformer with \(\mathrm{Transformer\_bus0}: \mathcal{M} \to \mathcal{N},\ \mathrm{Transformer\_bus1}: \mathcal{M} \to \mathcal{N}\) — passive branches between two buses, their flow set by impedance and tap ratio, with a phase shift fixed or optimised |
| \(\mathcal{C}\) | index \(c\) — cycle — independent cycles of the passive network graph — the cycle basis, data prep |
| \(\mathcal{S}\) | index \(s\) — segment — the cuts a passive branch's loss curve is held above — PyPSA's tangents, as many as its segments count, or its secants, as many as its tolerance loop places; none in a lossless run |
Parameters#
| Symbol | Meaning |
|---|---|
| \(\mathrm{on}^{\sigma}\) | Transformer_active over \(\mathcal{T} \times \mathcal{M}\) — whether a transformer stands in a snapshot's period — PyPSA's active, data prep |
| \(\mathrm{W}^{\sigma}\) | Transformer_capital_weight over \(\mathcal{M}\) — the sum of period weights a transformer stands in — PyPSA's active * period_weighting, summed, data prep |
| \(\sigma^{\mathrm{nom}}\) | Transformer_s_nom over \(\Xi \times \mathcal{M}\) — nominal apparent power |
| \(\mathrm{ext}^{\sigma}\) | Transformer_s_nom_extendable over \(\mathcal{M}\) — whether the nominal apparent power is a decision |
| \(\overline{\sigma}\) | Transformer_s_max_pu over \(\Xi \times \mathcal{T} \times \mathcal{M}\) — most flow either way, per unit of nominal apparent power |
| \(\underline{\sigma}^{\mathrm{nom}}\) | Transformer_s_nom_min over \(\Xi \times \mathcal{M}\) — least nominal apparent power an extendable transformer may be built at |
| \(\overline{\sigma}^{\mathrm{nom}}\) | Transformer_s_nom_max over \(\Xi \times \mathcal{M}\) — most nominal apparent power an extendable transformer may be built at |
| \(\mathrm{c}^{\mathrm{cap},\sigma}\) | Transformer_capital_cost over \(\Xi \times \mathcal{M}\) — cost of one unit of nominal apparent power — PyPSA's capital_cost, periodized as an annuity in data prep |
| \(\sigma^{\mathrm{nom,set}}\) | Transformer_s_nom_set over \(\Xi \times \mathcal{M}\) — a given nominal apparent power for an extendable transformer; one without a value has no row here |
| \(\sigma^{\mathrm{set}}\) | Transformer_s_set over \(\Xi \times \mathcal{T} \times \mathcal{M}\) — a given flow schedule; a transformer without one has no row here |
| \(\mathrm{x}^{\sigma}\) | Transformer_cycle_weight over \(\mathcal{M} \times \mathcal{C}\) — the transformer's effective series reactance, x times its tap ratio, signed by its orientation in the cycle — PyPSA's x_pu_eff, the cycle basis, data prep; a transformer in no cycle has no row. From the first scenario only, as a line's |
| \(\vartheta\) | Transformer_phase_shift_weight over \(\mathcal{T} \times \mathcal{M} \times \mathcal{C}\) — a fixed transformer's phase shift in radians at each snapshot, signed by its orientation in the cycle — a constant added to the cycle sum, data prep; zero for a varying transformer, whose shift is a decision instead, so the constant and the variable term never both count a shift. A transformer with no shift or in no cycle has no row |
| \(\mathrm{Transformer\_phase\_shift\_varying}\) | Transformer_phase_shift_varying over \(\mathcal{M}\) — whether a transformer's phase shift is a decision — PyPSA's phase_shift_min < phase_shift_max, read as a flag in data prep; false is a fixed shift carried by phase_shift. The shift parameters carry no scenario: only a cycle row reads them, and PyPSA fails on a transformer in a cycle on a network with scenarios (constraints.py:1654) |
| \(\mathrm{Transformer\_phase\_shift\_min}\) | Transformer_phase_shift_min over \(\mathcal{M}\) — the least a varying transformer's phase shift may take, in degrees — PyPSA's phase_shift_min; where it is below phase_shift_max the shift is a decision, otherwise the transformer keeps its fixed phase_shift |
| \(\mathrm{Transformer\_phase\_shift\_max}\) | Transformer_phase_shift_max over \(\mathcal{M}\) — the most a varying transformer's phase shift may take, in degrees — PyPSA's phase_shift_max; equal to phase_shift_min for a fixed transformer |
| \(\mathrm{Transformer\_phase\_shift\_cycle\_weight}\) | Transformer_phase_shift_cycle_weight over \(\mathcal{M} \times \mathcal{C}\) — the cycle sign for a varying transformer's phase shift, times π/180 so a shift in degrees enters the cycle sum in radians — data prep; zero for a fixed transformer or one in no cycle |
| \(\overline{\ell}^{\sigma}\) | Transformer_loss_max over \(\Xi \times \mathcal{T} \times \mathcal{M}\) — the loss at a transformer's rating — PyPSA's r_pu_eff * (s_max_pu * s_nom_max)**2, its r_pu_eff the resistance over the given s_nom times the tap ratio, data prep |
| \(\mathrm{a}^{\sigma}\) | Transformer_loss_slope over \(\Xi \times \mathcal{T} \times \mathcal{M} \times \mathcal{S}\) — the slope of a cut to a transformer's loss curve — a tangent's 2 * r_pu_eff * p_k, a secant's r_pu_eff * (p_k + p_k+1), as a line's, over the transformer's own r_pu_eff and rating, data prep |
| \(\mathrm{b}^{\sigma}\) | Transformer_loss_offset over \(\Xi \times \mathcal{T} \times \mathcal{M} \times \mathcal{S}\) — 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 |
Variables#
| Symbol | Meaning |
|---|---|
| \(\sigma\) | Transformer_s over \(\Xi \times \mathcal{T} \times \mathcal{M}\) — Transformer-s — PyPSA's p0, the flow measured at the Transformer_bus0 end: a positive value withdraws there and injects at Transformer_bus1, lossless |
| \(\ell^{\sigma}\) | Transformer_loss over \(\Xi \times \mathcal{T} \times \mathcal{M}\) — Transformer-loss — what a transformer dissipates carrying its flow, as a line does; absent, and zero in the balance, where the network is lossless |
| \(\mathit{Transformer\_phase\_shift}\) | Transformer_phase_shift over \(\Xi \times \mathcal{T} \times \mathcal{M}\) — Transformer-phase_shift — a phase-shifting transformer's voltage angle shift in degrees, chosen per snapshot to redistribute the flows around its cycles without moving active power; absent, and zero in the cycle sum, where the shift is fixed |
| \(\Sigma\) | Transformer_s_nom_ext over \(\mathcal{M}\) — Transformer-s_nom — nominal apparent power where it is a decision; the parameter of the same PyPSA name carries the fixed regime |
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 \(\Xi \times \mathcal{T} \times \mathcal{N}\), an expression this file adds Transformer_injection to |
| \(\mathit{Cycle\_angle\_sum}\) | Cycle_angle_sum over \(\Xi \times \mathcal{T} \times \mathcal{C}\), an expression this file adds Transformer_angle_sum to |
Definitions#
| Symbol | Meaning |
|---|---|
| \(\check{\sigma}\) | Transformer_s_monitored over \(\Xi \times \mathcal{T} \times \mathcal{M}\) — the flow a transformer's post-contingency rows read, as a line's |
| \(\mathit{Transformer\_injection}\) | Transformer_injection over \(\Xi \times \mathcal{T} \times \mathcal{N}\) |
| \(\mathit{Transformer\_angle\_sum}\) | Transformer_angle_sum over \(\Xi \times \mathcal{T} \times \mathcal{C}\) |
Upright is what the data supplies — a parameter such as \(\mathrm{Transformer\_phase\_shift\_varying}\), a coordinate map, a label — and italic is what the solver chooses, such as \(\mathit{Transformer\_phase\_shift}\). An index is italic too, being what a quantifier chooses, and a set is script.
Objective#
\[
\min \sum_{\xi \in \Xi,\ m \in \mathcal{M}} \pi_{\xi} \cdot \Sigma_{m} \cdot \mathrm{c}^{\mathrm{cap},\sigma}_{\xi,m} \cdot \mathrm{W}^{\sigma}_{m}
\]
Subject to#
Transformer_fix_s_lower
\[
\sigma_{\xi,t,m} - \ell^{\sigma}_{\xi,t,m} \ge -\overline{\sigma}_{\xi,t,m} \cdot \sigma^{\mathrm{nom}}_{\xi,m} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ m \in \mathcal{M} \,:\, \neg \mathrm{ext}^{\sigma}_{m} \wedge \mathrm{on}^{\sigma}_{t,m}
\]
Transformer_fix_s_upper
\[
\sigma_{\xi,t,m} + \ell^{\sigma}_{\xi,t,m} \le \overline{\sigma}_{\xi,t,m} \cdot \sigma^{\mathrm{nom}}_{\xi,m} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ m \in \mathcal{M} \,:\, \neg \mathrm{ext}^{\sigma}_{m} \wedge \mathrm{on}^{\sigma}_{t,m}
\]
Transformer_ext_s_lower
\[
\sigma_{\xi,t,m} - \ell^{\sigma}_{\xi,t,m} \ge -\overline{\sigma}_{\xi,t,m} \cdot \Sigma_{m} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ m \in \mathcal{M} \,:\, \mathrm{ext}^{\sigma}_{m} \wedge \mathrm{on}^{\sigma}_{t,m}
\]
Transformer_ext_s_upper
\[
\sigma_{\xi,t,m} + \ell^{\sigma}_{\xi,t,m} \le \overline{\sigma}_{\xi,t,m} \cdot \Sigma_{m} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ m \in \mathcal{M} \,:\, \mathrm{ext}^{\sigma}_{m} \wedge \mathrm{on}^{\sigma}_{t,m}
\]
Transformer_ext_s_nom_lower
\[
\Sigma_{m} \ge \underline{\sigma}^{\mathrm{nom}}_{\xi,m} \qquad \forall\, \xi \in \Xi,\ m \in \mathcal{M} \,:\, \mathrm{ext}^{\sigma}_{m}
\]
Transformer_ext_s_nom_upper
\[
\Sigma_{m} \le \overline{\sigma}^{\mathrm{nom}}_{\xi,m} \qquad \forall\, \xi \in \Xi,\ m \in \mathcal{M} \,:\, \mathrm{ext}^{\sigma}_{m} \wedge \overline{\sigma}^{\mathrm{nom}}_{\xi,m} \text{ is defined}
\]
Transformer_s_nom_set
\[
\Sigma_{m} = \sigma^{\mathrm{nom,set}}_{\xi,m} \qquad \forall\, \xi \in \Xi,\ m \in \mathcal{M} \,:\, \mathrm{ext}^{\sigma}_{m} \wedge \sigma^{\mathrm{nom,set}}_{\xi,m} \text{ is defined}
\]
Transformer_s_set
\[
\sigma_{\xi,t,m} = \sigma^{\mathrm{set}}_{\xi,t,m} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ m \in \mathcal{M} \,:\, \sigma^{\mathrm{set}}_{\xi,t,m} \text{ is defined} \wedge \mathrm{on}^{\sigma}_{t,m}
\]
Transformer_loss_upper
\[
\ell^{\sigma}_{\xi,t,m} \le \overline{\ell}^{\sigma}_{\xi,t,m} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ m \in \mathcal{M} \,:\, \mathrm{lossy} \wedge \mathrm{on}^{\sigma}_{t,m}
\]
Transformer_loss_tangents_forward
\[
\ell^{\sigma}_{\xi,t,m} + \mathrm{a}^{\sigma}_{\xi,t,m,s} \cdot \sigma_{\xi,t,m} \ge \mathrm{b}^{\sigma}_{\xi,t,m,s} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ m \in \mathcal{M},\ s \in \mathcal{S} \,:\, \mathrm{lossy} \wedge \mathrm{on}^{\sigma}_{t,m}
\]
Transformer_loss_tangents_reverse
\[
\ell^{\sigma}_{\xi,t,m} - \mathrm{a}^{\sigma}_{\xi,t,m,s} \cdot \sigma_{\xi,t,m} \ge \mathrm{b}^{\sigma}_{\xi,t,m,s} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ m \in \mathcal{M},\ s \in \mathcal{S} \,:\, \mathrm{lossy} \wedge \mathrm{on}^{\sigma}_{t,m}
\]
Definitions#
Transformer_s_monitored
\[
\check{\sigma}_{\xi,t,m} = \begin{cases} \sigma_{\xi,t,m} & \text{if } \mathrm{on}^{\sigma}_{t,m} \\ 0 & \text{otherwise} \end{cases} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ m \in \mathcal{M}
\]
Transformer_injection
\[
\mathit{Transformer\_injection}_{\xi,t,n} = -\left( \sum_{m \in \mathcal{M} \,:\, \mathrm{Transformer\_bus0}(m) = n} \sigma_{\xi,t,m} \right) + \sum_{m \in \mathcal{M} \,:\, \mathrm{Transformer\_bus1}(m) = n} \sigma_{\xi,t,m} - 0.5 \cdot \left( \sum_{m \in \mathcal{M} \,:\, \mathrm{Transformer\_bus0}(m) = n} \ell^{\sigma}_{\xi,t,m} \right) - 0.5 \cdot \left( \sum_{m \in \mathcal{M} \,:\, \mathrm{Transformer\_bus1}(m) = n} \ell^{\sigma}_{\xi,t,m} \right) \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ n \in \mathcal{N}
\]
Transformer_angle_sum
\[
\mathit{Transformer\_angle\_sum}_{\xi,t,c} = \sum_{m \in \mathcal{M}} \sigma_{\xi,t,m} \cdot \mathrm{x}^{\sigma}_{m,c} + \sum_{m \in \mathcal{M}} \vartheta_{t,m,c} + \sum_{m \in \mathcal{M}} \mathit{Transformer\_phase\_shift}_{\xi,t,m} \cdot \mathrm{Transformer\_phase\_shift\_cycle\_weight}_{m,c} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ c \in \mathcal{C}
\]
Variable domains#
Transformer_s
\[
\sigma_{\xi,t,m} \in \mathbb{R} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ m \in \mathcal{M} \,:\, \mathrm{on}^{\sigma}_{t,m}
\]
Transformer_loss
\[
\ell^{\sigma}_{\xi,t,m} \ge 0 \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ m \in \mathcal{M} \,:\, \mathrm{lossy} \wedge \mathrm{on}^{\sigma}_{t,m}
\]
Transformer_phase_shift
\[
\mathrm{Transformer\_phase\_shift\_min}_{m} \le \mathit{Transformer\_phase\_shift}_{\xi,t,m} \le \mathrm{Transformer\_phase\_shift\_max}_{m} \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ m \in \mathcal{M} \,:\, \mathrm{Transformer\_phase\_shift\_varying}_{m} \wedge \mathrm{on}^{\sigma}_{t,m}
\]
Transformer_s_nom_ext
\[
\Sigma_{m} \in \mathbb{R} \qquad \forall\, m \in \mathcal{M} \,:\, \mathrm{ext}^{\sigma}_{m}
\]