Bus

Bus voltage in Cartesian coordinates, power balance, and shunt admittance to ground. A port to the bus reads its voltage and contributes to its balance.

Model Parameters

Symbol	Units	Data	Description	Typical Value	Note
\(V^{\min}\)	[p.u.]	Vmin	Voltage magnitude lower limit	0.95	Default 0
\(V^{\max}\)	[p.u.]	Vmax	Voltage magnitude upper limit	1.05	Default unbounded
\(G\)	[p.u.]	G	Shunt conductance to ground		Default 0
\(B\)	[p.u.]	B	Shunt susceptance to ground		Default 0
\(\mathrm{ref}\)	[-]	reference	Angle reference, with zero imaginary voltage		Default false

Parameter Validation

• \(V^{\min} \ge 0\)

• \(V^{\max} \ge V^{\min}\)

Model Derived Parameters

None.

Model Ports

None.

Model Variables

Internal Variables

Symbol	Units	Description	Note
\(V_\mathrm{r}\)	[p.u.]	Real voltage
\(V_\mathrm{i}\)	[p.u.]	Imaginary voltage

External Variables

None.

Model Equations

Objective

None.

Internal Constraints

Equality

Power balance:

\[\begin{split}\begin{aligned} 0 &= -G (V_\mathrm{r}^2 + V_\mathrm{i}^2) + \Delta P^\mathrm{bus} \\ 0 &= B (V_\mathrm{r}^2 + V_\mathrm{i}^2) + \Delta Q^\mathrm{bus} \end{aligned}\end{split}\]

At a reference bus:

\[0 = V_\mathrm{i}\]

Inequality

\[(V^{\min})^2 \le V_\mathrm{r}^2 + V_\mathrm{i}^2 \le (V^{\max})^2\]

External Constraints

None.

Initialization

\[\begin{split}\begin{aligned} V_\mathrm{r} &\leftarrow 1 \\ V_\mathrm{i} &\leftarrow 0 \end{aligned}\end{split}\]