## 2 Basic Concepts of DQ0-Based Models
### Basic Definitions
#### 2.1.1 Modeling Resistors, Inductors, and Capacitors  
We will now develop the basic \(dq0\) models of linear passive components. These will help us understand the models of more complex units [30; 31; 32]. To begin, we need the following definitions:  
* Recall that _balanced_ three-phase signals are sinusoidal signals with equal magnitudes, phase shifts of \(\pm 120^{\circ}\), and a sum of zero. * We say that a power network is balanced or symmetrically configured if balanced three-phase voltages at its ports result in balanced three-phase currents, and vice versa. Two examples are shown in Figure 2.  
In this paper we use the term _balanced_ when referring to signals, and _symmetrically configured_ when referring to networks and systems. The following discussion focuses on networks that are symmetrically configured. Networks which are not symmetrically configured will be discussed next. Figure 1: Mapping of sinusoidal \(abc\) signals to constant \(dq0\) signals. Assume a symmetrically configured three-phase resistor \(R\), which is modeled as  
\[ \begin{bmatrix}v_{a}\\ v_{b}\\ v_{c}\end{bmatrix} =R \begin{bmatrix}i_{a}\\ i_{b}\\ i_{c}\end{bmatrix} . \tag{5}\]  
Multiply both sides of the equation by the \(dq0\) transformation \(T_{\theta}\) (from the left) to obtain  
\[ \begin{bmatrix}v_{d}\\ v_{q}\\ v_{0}\end{bmatrix} =R \begin{bmatrix}i_{d}\\ i_{q}\\ i_{0}\end{bmatrix} . \tag{6}\]  
This is the \(dq0\) model of a symmetrically configured three-phase resistor. Now assume a symmetrically configured three-phase inductor \(L\), which is modeled as  
\[ \begin{bmatrix}v_{a}\\ v_{b}\\ v_{c}\end{bmatrix} =L\frac{\mathrm{d}}{\mathrm{d}t} \begin{bmatrix}i_{a}\\ i_{b}\\ i_{c}\end{bmatrix} . \tag{7}\]  
The relation \([i_{a},i_{b},i_{c}]^{\mathsf{T}}=T_{\theta}^{-1}[i_{d},i_{q},i_{0}]^{\mathsf{T}}\) provides  
\[ \begin{bmatrix}v_{a}\\ v_{b}\\ v_{c}\end{bmatrix} =L\frac{\mathrm{d}}{\mathrm{d}t} \begin{pmatrix}T_{\theta}^{-1 }\begin{bmatrix}i_{d}\\ i_{q}\\ i_{0}\end{bmatrix}\end{pmatrix} . \tag{8}\]  
Using the derivative product rule yields  
\[ \begin{bmatrix}v_{a}\\ v_{b}\\ v_{c}\end{bmatrix} =L\left(\frac{\mathrm{d}}{\mathrm{d}t}T_{\theta}^{-1}\right) \begin{bmatrix}i_{d}\\ i_{q}\\ i_{0}\end{bmatrix} +LT_{\theta}^{-1}\frac{\mathrm{d}}{\mathrm{d}t} \begin{bmatrix} i_{d}\\ i_{q}\\ i_{0}\end{bmatrix} . \tag{9}\]  
By direct computation, it can be verified that the derivative of the inverse \(dq0\) transformation is  
\[\frac{\mathrm{d}}{\mathrm{d}t}T_{\theta}^{-1}=-T_{\theta}^{-1}\mathcal{W} \tag{10}\]  
with  
\[\mathcal{W}= \begin{bmatrix}0&\frac{\mathrm{d}}{\mathrm{d}t}\theta&0\\ -\frac{\mathrm{d}}{\mathrm{d}t}\theta&0&0\\ 0&0&0\end{bmatrix} . \tag{11}\]  
Figure 2: Example of symmetric and non-symmetric configurations. (**a**) Symmetrically configured network; (**b**) Network with a non-symmetric configuration. Substitute these expressions in (9) to obtain  
\[ \begin{bmatrix}v_{a}\\ v_{b}\\ v_{c}\end{bmatrix} =-LT_{\theta}^{-1}\mathcal{W} \begin{bmatrix}i_{d}\\ i_{q}\\ i_{0}\end{bmatrix} +LT_{\theta}^{-1}\frac{\mathrm{d}}{\mathrm{d}t} \begin{bmatrix} i_{d}\\ i_{q}\\ i_{0}\end{bmatrix} . \tag{12}\]  
Now multiply from the left by \(T_{\theta}\) and rearrange terms to get  
\[\frac{\mathrm{d}}{\mathrm{d}t} \begin{bmatrix}i_{d}\\ i_{q}\\ i_{0}\end{bmatrix} =\mathcal{W} \begin{bmatrix}i_{d}\\ i_{q}\\ i_{0}\end{bmatrix} +\frac{1}{L} \begin{bmatrix}v_{d}\\ v_{q}\\ v_{0}\end{bmatrix} . \tag{13}\]  
This is the \(dq0\) model of a symmetrically configured inductor. The explicit expressions are  
\[ \begin{split}\frac{\mathrm{d}}{\mathrm{d}t}i_{d}& =\frac{\mathrm{d}\theta}{\mathrm{d}t}i_{q}+\frac{1}{L}v_{d},\\ \frac{\mathrm{d}}{\mathrm{d}t}i_{q}&=-\frac{\mathrm{d}\theta}{ \mathrm{d}t}i_{d}+\frac{1}{L}v_{q},\\ \frac{\mathrm{d}}{\mathrm{d}t}i_{0}&=\frac{1}{L}v_{0}.\end{split}  \tag{14}\]  
Note that the quadrature component affects the direct component, and vice versa. Similarly, the dynamic model of a symmetrically configured capacitor \(C\) is  
\[\frac{\mathrm{d}}{\mathrm{d}t}