Load
This report provides a detailed description of a load model undertaken in this project consisting of resistive \(R\), inductive \(L\), and capacitive \(C\) components in series for both single-phase and three-phase systems. The objective is to model the admittance matrix (Y) for a three-phase system, which characterizes the relationship between currents and voltages in the system.
Mathematical Modeling
Single-Phase Capacitor
We will first start with the single-phase model, derive the Y parameters, and extend to three-phase models. In a single-phase series RLC circuit, the total impedance \(Z\) is given by the sum of the resistive, inductive, and capacitive impedances:
\[\begin{equation} Z = R + j\omega L - \frac{j}{\omega C} \end{equation}\]
The admittance (Y) is the inverse of the impedance:
\[\begin{equation} Z = R + j\omega L - \frac{j}{\omega C} \end{equation}\]
For a balanced three-phase system, we assume that each phase has its own resistive, inductive, and capacitive components, and these components are connected in series. For the three-phase case, it is assumed that there exists no mutual couplings between the loads of the three phases.
For phase \(a\), phase \(b\), and phase \(c\), the impedance for each phase is defined as:
\[\begin{equation} Z_a = R_a + j\omega L_a - \frac{j}{\omega C_a} \end{equation}\]
\[\begin{equation} Z_b = R_b + j\omega L_b - \frac{j}{\omega C_b} \end{equation}\]
\[\begin{equation} Z_c = R_c + j\omega L_c - \frac{j}{\omega C_c} \end{equation}\]
Y Parameters
The admittance matrix \(Y\) for a three-phase system is a \(3 \times 3\) matrix that relates the current vector \(I = [I_a, I_b, I_c]^T\) and the voltage vector \(V = [V_a, V_b, V_c]^T\) as follows:
\[\begin{equation} \begin{bmatrix} I_a \\ I_b \\ I_c \end{bmatrix} = \begin{bmatrix} Y_{aa} & Y_{ab} & Y_{ac} \\ Y_{ba} & Y_{bb} & Y_{bc} \\ Y_{ca} & Y_{cb} & Y_{cc} \end{bmatrix} \begin{bmatrix} V_a \\ V_b \\ V_c \end{bmatrix} \end{equation}\]
For an uncoupled three-phase system, the off-diagonal elements \(Y_{ab}, Y_{ac}, Y_{ba},\) etc., are zero, and the diagonal elements are the admittance of the respective phases.
Balanced Three-Phase System
For a balanced three-phase system, we assume that the impedance of each phase is the same:
\[\begin{equation} R_a = R_b = R_c = R, \quad L_a = L_b = L_c = L, \quad C_a = C_b = C_c = C \end{equation}\]
The diagonal elements of the admittance matrix are:
\[\begin{equation} Y_{aa} = Y_{bb} = Y_{cc} = \frac{1}{R + j\omega L - \frac{j}{\omega C}} \end{equation}\]
This simplifies the admittance matrix for a balanced three-phase system to:
\[\begin{equation} Y = \begin{bmatrix} Y_{aa} & 0 & 0 \\ 0 & Y_{aa} & 0 \\ 0 & 0 & Y_{aa} \end{bmatrix} = \begin{bmatrix} \frac{1}{R + j\omega L - \frac{j}{\omega C}} & 0 & 0 \\ 0 & \frac{1}{R + j\omega L - \frac{j}{\omega C}} & 0 \\ 0 & 0 & \frac{1}{R + j\omega L - \frac{j}{\omega C}} \end{bmatrix} \end{equation}\] It must be noted here that for most of the cases, this Y matrix for the three-phase load is diagonal as the mutual coupling between phases is negligible.
Code Explanation
For detailed code information, see the Harmony manual.