All loads will be described in terms of ZIP fractions and represented by voltage and current injections into the powerflow solution. The loads will be connected in a nodal manner so that any combination of wye and delta connected loads can be attached to a single bus.

## Contents

# Parameters

The parameters required to determine the current injections due to a load connected to nodes [math]n[/math] and [math]m[/math] of a bus are listed below.

- [math]|S_{b,nm}|[/math] The magnitude of the base absolute power of the load connected to nodes [math]n[/math] and [math]m[/math] of a particular bus (VA) at [math]V_{b,nm}[/math].
- [math]V_{nm}[/math] The variable voltage difference between the node to ground voltages of nodes [math]n[/math] and [math]m[/math] of a particular bus (V).
- [math]|V_{b,nm}|[/math] The magnitude of the base voltage difference between the node to ground voltages of nodes [math]n[/math] and [math]m[/math] of a particular bus (V).
- [math]pfr_{p,nm}[/math] The constant power fraction of the load connected to nodes [math]n[/math] and [math]m[/math] of a particular bus (unitless).
- [math]pf_{p,nm}[/math] The power factor of the constant power fraction of the load connected to nodes [math]n[/math] and [math]m[/math] of a particular bus (-1.0 - 1.0).
- [math]pfr_{i,nm}[/math] The constant current fraction of the load connected to nodes [math]n[/math] and [math]m[/math] of a particular bus (unitless) at [math]V_{b,nm}[/math].
- [math]pf_{i,nm}[/math] The power factor of the constant current fraction of the load connected to nodes [math]n[/math] and [math]m[/math] of a particular bus (-1.0 - 1.0) at [math]V_{b,nm}[/math].
- [math]pfr_{z,nm}[/math] The constant impedance fraction of the load connected to nodes [math]n[/math] and [math]m[/math] of a particular bus (unitless) at [math]V_{b,nm}[/math].
- [math]pf_{z,nm}[/math] The power factor of the constant impedance fraction of the load connected to nodes [math]n[/math] and [math]m[/math] of a particular bus (-1.0 - 1.0) at [math]V_{b,nm}[/math].

# Equations

For the parameters described above the current injections due to each type of load fraction can be found below.

## Constant Impedance Loads

The constant impedance of the load can be found by the following equations

- [math]\displaystyle{}S_{z,nm}=S_{b,nm}*pfr_{z,nm}*[pf_{z,nm}+j*sign(pf_{z,nm})*(1-pf_{z,nm}^{2})^{\frac{1}{2}}][/math]
- [math]\displaystyle{}Z_{nm}=\frac{|V_{b,nm}|^{2}}{S_{z,nm}^{*}}[/math]

The current injection due to the constant impedance can then be found from the equation below.

- [math]\displaystyle{}I_{z,nm}=\frac{V_{nm}}{Z_{nm}}[/math]

## Constant Current Loads

The constant current of the load can be found by the following equations

- [math]\displaystyle{}S_{i,nm}=S_{b,nm}*pfr_{i,nm}*[pf_{i,nm}+j*sign(pf_{i,nm})*(1-pf_{i,nm}^{2})^{\frac{1}{2}}][/math]
- [math]\displaystyle{}I_{nm}=(\frac{S_{i,nm}}{|V_{b,nm}|})^{*}[/math]

The current injection due to the constant impedance can then be found from the equation below.

- [math]\displaystyle{}I_{i,nm}=I_{nm}[/math]

## Constant Power Loads

The constant Power of the load can be found by the following equation

- [math]\displaystyle{}S_{p,nm}=S_{b,nm}*pfr_{p,nm}*[pf_{p,nm}+j*sign(pf_{p,nm})*(1-pf_{p,nm}^{2})^{\frac{1}{2}}][/math]

The current injection due to the constant impedance can then be found from the equation below.

- [math]\displaystyle{}I_{p,nm}=(\frac{S_{p,nm}}{V_{nm}})^{*}[/math]

## Total Current Injection

The total current injection into a node [math]n[/math] due to loads connecting node [math]n[/math] to any number of other nodes on the same bus can be found by the following equation

- [math]\displaystyle{}I_{n}=\Sigma (I_{z,nj}+I_{i,nj}+I_{p,nj})[/math]

where [math]j[/math] are the nodes the loads connect node [math]n[/math] to on a bus.