LoadZIP

Source: GridKit/Model/PhasorDynamics/LoadZIP/README.md

LoadZIP Model

This represents a static load model with impedance (Z), current (I), and power (P) components.

Model Parameters

Symbol	Units	Description
\(P_0\)	[p.u.]	Load nominal real power
\(Q_0\)	[p.u.]	Load nominal reactive power
\(V_0\)	[p.u.]	Load nominal voltage
\(\alpha_I\)	[unitless]	Fraction of load to be represented as constant current
\(\alpha_P\)	[unitless]	Fraction of load to be represented as constant power

Model Variables

Internal Variables

Differential

None.

Algebraic

Symbol	Units	Description	Note
\(I_r\)	[p.u.]	Terminal current, real component	Read by bus
\(I_i\)	[p.u.]	Terminal current, imaginary component	Read by bus

External Variables

Differential

None.

Algebraic

Symbol	Units	Description	Note
\(V_r\)	[p.u.]	Terminal voltage, real component	owned by bus object
\(V_i\)	[p.u.]	Terminal voltage, imaginary component	owned by bus object

Model Equations

Differential Equations

None.

Algebraic Equations

\[\begin{split}\begin{aligned} 0 &= I_{r} + (P_{0} V_{r} + Q_{0} V_{i}) \left(\frac{1}{V_0^2} (1 - \alpha_I - \alpha_P) + \frac{1}{V_0 \sqrt{V_r^2+V_i^2}} \alpha_I + \frac{1}{V_r^2+V_I^2} \alpha_P\right) \\ 0 &= I_{i} + (P_{0} V_{i} - Q_{0} V_{r}) \left(\frac{1}{V_0^2} (1 - \alpha_I - \alpha_P) + \frac{1}{V_0 \sqrt{V_r^2+V_i^2}} \alpha_I + \frac{1}{V_r^2+V_I^2} \alpha_P\right) \end{aligned}\end{split}\]

Initialization Procedure

Use the algebraic equations to solve for \(I_{r}\) and \(I_{i}\).

Model Outputs

Real and imaginary values of the load current are the variables \(I_{r}\) and \(I_{i}\).

Current is oriented leaving the load (i.e. entering the bus).

Current magnitude \(I_{m}\) is the phasor magnitude of the current.

\[\begin{aligned} I_{m} &= \sqrt{(I_{r})^2 + (I_{i})^2} \end{aligned}\]

Active and reactive power (\(P\) and \(Q\)) are the real and imaginary parts of the complex power, where the complex power is defined as \(S=VI^{\ast}=(V_r + j V_i)(I_r - jI_i)\)

\[\begin{split}\begin{aligned} P &= V_{r} I_{r} + V_{i} I_{i}\\ Q &= V_{i} I_{r} - V_{r} I_{i} \end{aligned}\end{split}\]

Note on Derivation

The origin of the algebraic equations is easier to understand in complex form:

\[\begin{split}\begin{aligned} S &= S_z \left(\frac{|V|}{V_0}\right)^2 + S_I \left(\frac{|V|}{V_0}\right) + S_P \\ S &= V I^* \end{aligned}\end{split}\]