LoadZ

LoadZ represents an \(N\)-phase impedance load. Current \(\mathbf{i}\) is injected from the load into the EMT bus.

Block Diagram

Figure 1: LoadZ model

Model Parameters

Symbol	Units	JSON	Description	Note
\(N\)	[-]	N	Number of phases	Required, positive integer

Parameter Validation

\[N \in \mathbb{Z}_{>0}\]

Model Derived Parameters

None.

Model Ports

Symbol	Port	Type	Units	Description	Note
\(\mathbf{v}\)	v	Input	[V]	Bus voltage at load port	\(\mathbf{v} \in \mathbb{R}^N\)
\(\mathbf{i}\)	i	Output	[A]	Current injection at load port	\(\mathbf{i} \in \mathbb{R}^N\)

Submodels

Symbol	Description	Type	Order	JSON	Inputs	Outputs
\(\mathbf{z}\)	Impedance	VectorFit	\(NQ_{\mathbf{z}}\)	Z	\(\mathbb{R}^N\)	\(\mathbb{R}^N\)

Submodel Validation

The current is differential when \(\mathbf{E}^{\mathbf{z}}\) is nonsingular and algebraic when \(\mathbf{E}^{\mathbf{z}}=\mathbf{0}\).

\[\mathbf{E}^{\mathbf{z}}=\mathbf{0} \qquad \text{or} \qquad \mathrm{rank}(\mathbf{E}^{\mathbf{z}})=N\]

Submodel Wiring

None.

Model Variables

Internal Variables

Differential

Symbol	Units	Description	Note
\(\mathbf{i}\)	[A]	Current injection from load into EMT bus	\(\mathbf{i} \in \mathbb{R}^N\), \(\mathrm{rank}(\mathbf{E}^{\mathbf{z}})=N\)

Algebraic

Symbol	Units	Description	Note
\(\mathbf{i}\)	[A]	Current injection from load into EMT bus	\(\mathbf{i} \in \mathbb{R}^N\), \(\mathbf{E}^{\mathbf{z}}=\mathbf{0}\)

External Variables

Differential

Symbol	Units	Description	Note
\(\mathbf{v}\)	[V]	Bus voltage vector owned by EMT bus	\(\mathbf{v} \in \mathbb{R}^N\)

Algebraic

None.

Model Equations

Internal Equations

Differential

For \(\mathrm{rank}(\mathbf{E}^{\mathbf{z}})=N\),

\[0 = \mathbf{z}[\mathbf{i}] + \mathbf{v}\]

Algebraic

For \(\mathbf{E}^{\mathbf{z}}=\mathbf{0}\), the same residual is algebraic.

External Equations

The load supplies input \(d\) of the bus current balance:

\[\mathbf{i}_d \leftarrow \mathbf{i}\]

Initialization

None.

Monitors

Monitor	Units	Description	Note
i	[A]	Load current injection	\(\mathbf{i} \in \mathbb{R}^N\)