VoltageSource

Source: GridKit/Model/EMT/Component/VoltageSource/README.md

VoltageSource Model

VoltageSource represents a three-phase voltage source in instantaneous abc coordinates. The source waveform is configurable by phase magnitude and phase offset for each phase and is otherwise constant. Each source port is connected to the EMT bus through a phase resistance.

Model Parameters

Symbol	Units	Description	Note
\(E_a\)	[V]	Source voltage magnitude, phase a	RMS
\(E_b\)	[V]	Source voltage magnitude, phase b	RMS
\(E_c\)	[V]	Source voltage magnitude, phase c	RMS
\(\phi_a\)	[rad]	Source phase offset, phase a
\(\phi_b\)	[rad]	Source phase offset, phase b
\(\phi_c\)	[rad]	Source phase offset, phase c
\(\omega_0\)	[rad/s]	Source angular frequency
\(R_a\)	[\(\Omega\)]	Terminal resistance, phase a
\(R_b\)	[\(\Omega\)]	Terminal resistance, phase b
\(R_c\)	[\(\Omega\)]	Terminal resistance, phase c

Model Derived Parameters

None.

Model Variables

Internal Variables

Differential

None.

Algebraic

None.

External Variables

External variables enter component model equations but are owned by other components. The EMT bus at the source port owns the voltage variable and provides the equation needed to have a balanced system of equations.

Differential

Symbol	Units	Description	Note
\(\mathbf{v}\)	[V]	Port voltage vector, owned by EMT bus	\(\mathbf{v} = [v_a, v_b, v_c]^T \in \mathbb{R}^3\)

Algebraic

None.

Model Equations

Differential Equations

None.

Algebraic Equations

None.

Bus Residual Contributions

The source contributes current to the KCL residual at its port bus. The injection vector is accumulated into the owning bus residual. Given source angular frequency \(\omega_0\), the source waveform is:

\[\begin{split}\begin{aligned} e_a(t) &= \sqrt{2}\,E_a\cos(\omega_0 t + \phi_a) \\ e_b(t) &= \sqrt{2}\,E_b\cos(\omega_0 t + \phi_b) \\ e_c(t) &= \sqrt{2}\,E_c\cos(\omega_0 t + \phi_c) \end{aligned}\end{split}\]

The current contribution is positive into the bus:

\[\begin{split}\mathbf{i}^\text{inj} := \begin{bmatrix} \dfrac{e_a(t)-v_a}{R_a} \\ \dfrac{e_b(t)-v_b}{R_b} \\ \dfrac{e_c(t)-v_c}{R_c} \end{bmatrix}\end{split}\]

Initialization

No internal state is initialized. At \(t = 0\), the source waveform is:

\[\begin{split}\begin{aligned} e_a(0) &= \sqrt{2}\,E_a\cos(\phi_a) \\ e_b(0) &= \sqrt{2}\,E_b\cos(\phi_b) \\ e_c(0) &= \sqrt{2}\,E_c\cos(\phi_c) \end{aligned}\end{split}\]

Model Outputs

Candidate monitorable outputs include the source waveform components \(e_a(t)\), \(e_b(t)\), and \(e_c(t)\).

The port current injection expression is documented above as \(\mathbf{i}^\text{inj}\).