**3. The Proposed PLL Structure**

In this section, the voltage sequence component of non-ideal grid voltages is analyzed at first to provide the basis for the design procedure of the proposed PLL. To achieve our objective, a hybrid filtering stage based on MAFs and MDSCs is suggested and analyzed in this section. Then, it is incorporated into a QT1-PLL structure.

### *3.1. The Component Analysis of Distorted Grid Voltages*

Under distorted grid voltage conditions, three-phase grid voltages contain fundamental frequency positive sequence (FFPS), fundamental frequency negative sequences (FFNS) and other harmonic sequence components. FFPS components can be written as follows:

$$\begin{array}{l} v\_{a,1}(t) = V\_1^+ \sin(\omega t) \\ v\_{b,1}(t) = V\_1^+ \sin(\omega t - \frac{2\pi}{3}) \\ v\_{c,1}(t) = V\_1^+ \sin(\omega t + \frac{2\pi}{3}) \end{array} \tag{3}$$

where *V*+1 represents the amplitude of FFPS and ω is the grid frequency. Then, *n* order harmonic sequence components can be written as:

$$\begin{array}{l}v\_{\mathfrak{u},\mathfrak{n}}(t) = V\_{\mathfrak{n}}\sin(n\omega t + q\_{\mathfrak{n}})\\v\_{\mathfrak{d},\mathfrak{n}}(t) = V\_{\mathfrak{n}}\sin(n\omega t + q\_{\mathfrak{n}} - \frac{2\pi}{3})\\v\_{\mathfrak{c},\mathfrak{n}}(t) = V\_{\mathfrak{n}}\sin(n\omega t + q\_{\mathfrak{n}} + \frac{2\pi}{3})\end{array} \tag{4}$$

By using the symmetrical component method, all voltage components can be considered as the sum of positive sequences, negative sequences and zero sequences. Applying Clark transformation to three phase voltages yields:

$$\begin{bmatrix} v\_{\boldsymbol{\alpha}}(t) \\ v\_{\boldsymbol{\beta}}(t) \\ v\_{\boldsymbol{0}}(t) \end{bmatrix} = \begin{bmatrix} T\_{\boldsymbol{\alpha}\boldsymbol{\beta}} \begin{bmatrix} V\_{1,4,7,\ldots}^{+} \sin(n\omega t + \phi\_{\boldsymbol{n}}) & +V\_{2,5,8,\ldots}^{-} \sin(n\omega t + \phi\_{\boldsymbol{n}}) & +V\_{3,6,9,\ldots}^{0} \sin(n\omega t + \phi\_{\boldsymbol{n}}) \\ V\_{1,4,7,\ldots}^{+} \sin(n\omega t + \phi\_{\boldsymbol{n}} - \frac{2\pi}{3}) & +V\_{2,5,8,\ldots}^{-} \sin(n\omega t + \phi\_{\boldsymbol{n}} + \frac{2\pi}{3}) & +V\_{3,6,9,\ldots}^{0} \sin(n\omega t + \phi\_{\boldsymbol{n}}) \\ V\_{1,4,7,\ldots}^{+} \sin(n\omega t + \phi\_{\boldsymbol{n}} + \frac{2\pi}{3}) & +V\_{2,5,8,\ldots}^{-} \sin(n\omega t + \phi\_{\boldsymbol{n}} - \frac{2\pi}{3}) & +V\_{3,6,9,\ldots}^{0} \sin(n\omega t + \phi\_{\boldsymbol{n}}) \end{bmatrix} \tag{5}$$

In Equation (5), *<sup>T</sup>*αβ is the transfer matrix of Clark transformation. After applying Clark transformation, Equation (5) is as follows:

$$
\begin{bmatrix} v\_a(t) \\ v\_{\beta}(t) \\ v\_0(t) \end{bmatrix} = \begin{bmatrix} v\_{a,1,7,13,\dots}^+(t) \\ v\_{\beta,1,7,13,\dots}^+(t) \\ 0 \end{bmatrix} + \begin{bmatrix} v\_{a,5,11,\dots}^-(t) \\ v\_{\beta,5,11,\dots}^-(t) \\ 0 \end{bmatrix} + \begin{bmatrix} 0 \\ 0 \\ v\_{3,6,9,\dots}^0(t) \end{bmatrix} \tag{6}
$$

Observing Equation (6), it can be found that there is no triple odd harmonic in *v*α and *<sup>v</sup>*β. In the αβ-frame, only *n*= +1, −5, +7, −11, +13, ... order sequence components exist. By using Park transformation, the components in the αβ-frame turn out to be *n* = −2, ±6, ±12, ... order and DC components. The voltage sequence components can be summarized in Table 1. It should be noticed that the sign of frequency represents the rotating direction of the voltage sequence vector. A negative frequency means the voltage vector rotates in a counterclockwise direction.

**Table 1.** Voltage sequence components in grid voltages.

