*4.1. Mathematics Model*

After Clark transformation, the FFPS component in αβ-frame can be written as:

$$\begin{array}{l}v\_a(t) = V\_1^+ \cos(\theta\_1^+) \\ v\_\beta(t) = V\_1^+ \sin(\theta\_1^+) \end{array} \tag{9}$$

By using Park transformation, the FFPS component in the *dq*-frame becomes:

$$
\begin{bmatrix} v\_d(t) \\ v\_\theta(t) \end{bmatrix} = \begin{bmatrix} \cos(\hat{\theta}) & \sin(\hat{\theta}) \\ -\sin(\hat{\theta}) & \cos(\hat{\theta}) \end{bmatrix} \begin{bmatrix} v\_d(t) \\ v\_\theta(t) \end{bmatrix} = V\_1^+ \begin{bmatrix} \cos(\theta\_1^+ - \hat{\theta}) \\ \sin(\theta\_1^+ - \hat{\theta}) \end{bmatrix} \tag{10}
$$

According to Figure 5, after MDSC*<sup>m</sup>*=4,*<sup>n</sup>*=8, the FFPS component turns out to be:

$$
\begin{bmatrix} v\_{dm}(t) \\ v\_{qm}(t) \end{bmatrix} = 0.5 \begin{bmatrix} 1 & -\varepsilon^{-\frac{T}{8}s} \\ \varepsilon^{-\frac{T}{8}s} & 1 \end{bmatrix} \begin{bmatrix} v\_d(t) \\ v\_q(t) \end{bmatrix} = 0.5 \begin{bmatrix} v\_d(t) - v\_q(t - \frac{T}{8}) \\ v\_q(t) + v\_d(t - \frac{T}{8}) \end{bmatrix} \tag{11}
$$

Since two MAFs are arrange at each control path, the arc tangent function can be considered as arranged after MDSC*<sup>m</sup>*=4,*<sup>n</sup>*=8. Then, the arc tangent operation can be expressed as:

$$\arctan\left(\frac{v\_{qm}(t)}{v\_{dm}(t)}\right) = \arctan\left(\frac{\cos\left(\theta\_1^+ - \hat{\theta} - \frac{\pi}{2}\right) + \cos\left(\theta\_1^+ \left(t - \frac{T}{8}\right) - \hat{\theta}\left(t - \frac{T}{8}\right)\right)}{-\sin\left(\theta\_1^+ - \hat{\theta} - \frac{\pi}{2}\right) - \sin\left(\theta\_1^+ \left(t - \frac{T}{8}\right) - \hat{\theta}\left(t - \frac{T}{8}\right)\right)}\right) \tag{12}$$

By applying trigonometric operation, Equation (12) turns out to be:

$$\arctan\left(\frac{\upsilon\_{qm}(t)}{\upsilon\_{dm}(t)}\right) = \arctan\left(\tan\left(\frac{\Theta\_1^+ - \theta - \frac{\pi}{2} + \Theta\_1^+ \left(t - \frac{T}{8}\right) - \theta \left(t - \frac{T}{8}\right)}{2} - \frac{\pi}{2}\right)\right) \tag{13}$$

Therefore, Equation (13) can be written as:

$$\arctan\left(\frac{\upsilon\_{qm}(t)}{\upsilon\_{dm}(t)}\right) = \frac{\partial\_1^+ + \partial\_1^+\left(t - \frac{T}{8}\right) - \hat{\theta} - \hat{\theta}\left(t - \frac{T}{8}\right)}{2} + \frac{\pi}{4} \tag{14}$$

*Energies* **2019**, *12*, 4040

According to the derivation of Equations (9)–(14), the mathematics model of the proposed PLL is depicted in Figure 12. *D*'(s) is the disturbance components injected into the input voltages. *R*(*s*) is defined as follows:

$$R(s) = \frac{1}{2} + \frac{1}{2}e^{-\frac{r}{8}s} \tag{15}$$

Compared with the model of other existing PLLs, our mathematics model is not a small-signal model since the arc tangent function extracts phase information directly without any linearization procedure.

**Figure 12.** Mathematics model of the proposed PLL.

**Figure 13.** Simplified model of the proposed PLL.

### *4.2. Parameter Design Guidelines and Stability Analysis*

To transform the proposed PLL into a traditional form of a closed-loop feedback system, block diagram algebra is utilized. The block diagram in Figure 12 is transformed to a simplified schematic shown in Figure 13. Hence, the open-loop transfer function can be written as:

$$G\_{ols}(s) = \frac{\theta\_1^+(s)}{\theta\_1^+(s) - \theta\_1^+(s)} = \left(\frac{R(s)\text{MAF}(s)}{1 - R(s)\text{MAF}(s)}\right)\left(\frac{s+k}{s}\right) \tag{16}$$

Then, the transfer function of phase-error can be expressed as:

$$G\_{\mathfrak{k}}(s) = \frac{\theta\_{\mathfrak{k}}}{\theta\_{\mathfrak{1}}^{+}} = \frac{1}{1 + G\_{\text{obs}}(s)}\tag{17}$$

In this paper, a parameter is chosen by its impact on the settling time of the phase-error transfer function. With different values of *k*, the settling time of *Ge*(*s*) is examined. The phase-error transfer function under these two conditions is expressed by:

$$
\Theta\_{\varepsilon}^{\Lambda\partial}(\mathbf{s}) = \frac{\Lambda\theta}{\mathbf{s}} \mathcal{G}\_{\varepsilon}(\mathbf{s}) \tag{18}
$$

$$\Theta\_{\varepsilon}^{\Lambda\omega}(\mathbf{s}) = \frac{\Delta\omega}{s^2} \mathbf{G}\_{\mathbf{c}}(\mathbf{s}) \tag{19}$$

To calculate the settling time, inverse Laplace transformation is applied to Equations (18) and (19). Two curves of settling time as a function of *k* are depicted in Figure 14. When phase error is less than 2% of step change, transient response is considered to be over. To make a trade-off under both conditions, *k* is selected to be 148 to achieve an optimal dynamic performance for both conditions. The settling time is around one grid period.

**Figure 14.** The settling time of the proposed PLL with different values of *k* under. phase jump (solid line) and frequency step-change (dashed line) conditions.

Since the model of the proposed PLL contains a time delay unit, the system turns out to be a non-minimum phase system. To examine the stability, nyquist stabilization criterion is employed in this paper instead of using a bode diagram. The nyquist diagram of *Gols*(*s*) is depicted in Figure 15. The nyquist curve does not surround the (−1, j0) point, which means the closed-loop feedback system of *Gols*(*s*) is stable. The gain stability margin (GM) is 16.5 dB at 162 Hz. The phase stability margin (PM) is 45◦ at 56.6 Hz.

**Figure 15.** The Nyquist diagram of *Gols*(*s*).

The bode diagram of the proposed PLL and QT1-PLL is depicted in Figure 16. It can be seen that the crossover frequency of the proposed PLL is larger than that in the QT1-PLL. This yields faster transient behavior for the proposed method. It is noted that the 100 Hz component is only attenuated in the bode diagram. This does not reveal the filtering capability at 100 Hz, since the diagram is based on the model whose input is grid phase, not grid voltages. The filtering performance is already analyzed in Section 3. Experiments are also carried out to verify the filtering capability in the next section.

**Figure 16.** Bode diagram of open-loop system: Proposed PLL (solid lines), QT1-PLL (dashed lines).
