Power flow#

One of the 24 fragments of examples/pypsa.yaml: Kirchhoff's voltage law around each cycle. It declares Cycle_angle_sum as an empty sum, empty: true, which the lines and transformers add to.

dimensions:
  scenario:
    description: the futures dispatch is chosen in, each with a weight
  snapshot:
    description: dispatch periods
    dtype: datetime
  cycle:
    description: independent cycles of the passive network graph — the cycle basis, data prep

expressions:
  Cycle_angle_sum:
    dims: [scenario, snapshot, cycle]
    empty: true
    description: >-
      the voltage angle differences around a cycle: every branch flow times
      its cycle weight, and every transformer phase shift

constraints:
  Kirchhoff_Voltage_Law:
    description: >-
      `Kirchhoff-Voltage-Law` — around every independent cycle the
      impedance-weighted flows sum to nothing, which is what makes the linear
      power flow physical rather than transport. A transformer's flow weighs its
      effective reactance, and its phase shift enters the cycle sum too: a
      constant where the shift is fixed, or the shift decision times its cycle
      weight where the shift is a phase-shifting transformer's to choose
    dims: [scenario, snapshot, cycle]
    expression: Cycle_angle_sum == 0

Sets#

Symbol	Meaning
\(\Xi\)	index \(\xi\) — `scenario` — the futures dispatch is chosen in, each with a weight
\(\mathcal{T}\)	index \(t\) — `snapshot` — dispatch periods
\(\mathcal{C}\)	index \(c\) — `cycle` — independent cycles of the passive network graph — the cycle basis, data prep

Definitions#

Symbol	Meaning
\(\mathit{Cycle\_angle\_sum}\)	`Cycle_angle_sum` over \(\Xi \times \mathcal{T} \times \mathcal{C}\) — the voltage angle differences around a cycle: every branch flow times its cycle weight, and every transformer phase shift

Subject to#

Kirchhoff_Voltage_Law

\[ \mathit{Cycle\_angle\_sum}_{\xi,t,c} = 0 \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ c \in \mathcal{C} \]

Definitions#

Cycle_angle_sum

\[ \mathit{Cycle\_angle\_sum}_{\xi,t,c} = \cdots \qquad \forall\, \xi \in \Xi,\ t \in \mathcal{T},\ c \in \mathcal{C} \]