**3. DPC and PVMT-Based PV-VSI Modelling**

In this section, the mathematical modelling of PV-VSI based on DPC and PVMT is presented. L-filters were used at the output of PV-VSI to reduce the harmonics in current and voltage. In Figure 2a,b, the schematics of the dq CCS-based control method with PLL and the proposed PVMT-based FLDPC method without PLL are presented respectively.

**Figure 2.** Power controllers for grid-tied AC-MG's PV-VSI, based on (**a**) PLL-PI-integrated dq CCS and (**b**) the proposed PVMT-based FLDPC.

The dynamic realtionships between VSI's output voltages, currents and PCC voltages can be represented using (10):

$$\begin{aligned} L\frac{di\_{pa}}{dt} &= -\upsilon\_{pa} + \iota\_{a} - \mathrm{Ri}\_{pa} \\ L\frac{di\_{pb}}{dt} &= -\upsilon\_{b\mathcal{g}} + \iota\_{b} - \mathrm{Ri}\_{pb} \\ L\frac{di\_{pc}}{dt} &= -\upsilon\_{\mathcal{E}\mathcal{g}} + \iota\_{c} - \mathrm{Ri}\_{pc} \end{aligned} \tag{10}$$

where, *vpabc, ipabc,* and *uabc* are PCC voltages, VSI output currents and voltages, respectively. *R* and *L* are the resistance and inductance of filter, respectively.

The stationary reference frame of the equations presented in (10) can be transformed to (11) using Clarke's transformation:

$$\begin{aligned} \mathcal{L}\frac{di\_{\text{pa}}}{dt} &= \boldsymbol{\mu}\_{\text{a}} - \boldsymbol{\upsilon}\_{\text{pa}} - \mathcal{R}\boldsymbol{i}\_{\text{pa}} \\ \mathcal{L}\frac{di\_{\text{p\S}}}{dt} &= \boldsymbol{\mu}\_{\text{\S}} - \boldsymbol{\upsilon}\_{\text{p\S}} - \mathcal{R}\boldsymbol{i}\_{\text{p\S}} \end{aligned} \tag{11}$$

where PCC voltages are *uαβ*, and VSI currents and voltages are *ipαβ* and *vpαβ*, respectively, in *α*–*β* frame.

The stationary reference frame representation of instant reactive and real power flow between the utility grid and VSI can be presented as (12):

$$P = \frac{3}{2} \left( i\_{pa} \upsilon\_{pa} + i\_{p\beta} \upsilon\_{p\beta} \right)$$

$$Q = \frac{3}{2} \left( -i\_{p\beta} \upsilon\_{pa} + i\_{p\alpha} \upsilon\_{p\beta} \right) \tag{12}$$

where instant real and reactive powers supplied/injected by the grid are *P* and *Q*, respectively. By differentiating (12), *P* and *Q* dynamic equations can be obtained as follows:

$$\frac{dP}{dt} = \frac{3}{2} \left( \upsilon\_{pa} \frac{di\_a}{dt} + i\_{pa} \frac{dv\_{pa}}{dt} + \upsilon\_{p\beta} \frac{di\_{p\beta}}{dt} + i\_{p\beta} \frac{dv\_{p\beta}}{dt} \right)$$

$$\frac{dQ}{dt} = \frac{3}{2} \left( -\upsilon\_{pa} \frac{di\_{p\beta}}{dt} - i\_{p\beta} \frac{dv\_{pa}}{dt} + \upsilon\_{p\beta} \frac{di\_{pa}}{dt} + i\_{a} \frac{dv\_{p\beta}}{dt} \right) \tag{13}$$

For simplifying the dynamics of *P* and *Q* in the balanced grid condition, the relationship of the PCC *α*–*β* voltage can be obtained as given in (14):

$$v\_{p\mathfrak{K}} = V\_{pcc} \cos(\omega t)$$

$$v\_{p\mathfrak{F}} = V\_{pcc} \sin(\omega t) \tag{14}$$

where:

$$\begin{aligned} V\_{\text{pcc}} &= \sqrt{v\_{p\alpha}^2 + v\_{p\beta}^2} \\ \omega &= 2 \prod f \end{aligned} \tag{15}$$

where *PCC* voltage amplitude is *Vpcc*, angular frequency is *ω* and grid voltage frequency is *f*. The dynamic equations of *PCC* voltages are obtained as (16) by differentiating (14).

$$\begin{split} \frac{dv\_{p\text{a}}}{dt} &= -\omega V\_{p\text{cc}} \sin(\omega t) = -v\_{p\beta}\omega\\ \frac{dv\_{p\beta}}{dt} &= \omega V\_{p\text{cc}} \cos(\omega t) = v\_{p\text{a}}\omega \end{split} \tag{16}$$

By substituting (10) and (16) in (13), the dynamic expression of real and reactive powers can be obtained as (17):

$$\begin{split} \frac{dp}{dt} &= \frac{3}{2L} \left( -V\_{pcc}{}^2 + \mu\_a v\_{p\alpha} + \mu\_\beta v\_{p\beta} \right) - \omega q - p\frac{R}{L} \\ \frac{dq}{dt} &= \frac{3}{2L} \left( -\mu\_\beta v\_{p\alpha} + \mu\_a v\_{p\beta} \right) - \omega q - q\frac{R}{L} \end{split} \tag{17}$$

where, dynamic real and reactive power control inputs and outputs are (*p* and *q*) and (*u<sup>α</sup>* and *uβ*), respectively.

Since both the control inputs in (17) are coupled in *P* and *Q* states, by using voltage modulation theory [34], the dynamics of (17) can be simplified as (18) to define new voltage modulated control inputs:

$$u\_P := u\_\alpha v\_{p\alpha} + u\_\beta v\_{p\beta}$$

$$u\_Q := u\_\beta v\_{p\alpha} - u\_\alpha v\_{p\beta} \tag{18}$$

where the new control inputs are *u<sup>P</sup>* and *uQ*, and they are transformed into DC components as they satisfy (19):

$$
\begin{bmatrix} u\_P \\ u\_Q \end{bmatrix} = V\_{pcc} \begin{bmatrix} \cos(\omega t) & \sin(\omega t) \\ -\sin(\omega t) & \cos(\omega t) \end{bmatrix} \begin{bmatrix} u\_d \\ u\_\beta \end{bmatrix} = V\_{pcc} \begin{bmatrix} u\_d \\ u\_q \end{bmatrix} \tag{19}
$$

where *u<sup>d</sup>* and *u<sup>q</sup>* are the *d*-*q* frame VSI voltages. Though the proposed method has no PLL system, the system is still presented in dq axis frame.

The dynamic expression of real and reactive powers presented in (17) can be expressed as (20), by substituting the control inputs of (17) with the new control inputs (*u<sup>P</sup>* and *uQ*).

$$\begin{split} \frac{dP}{dt} &= \frac{3}{2L} \left( -V\_{pcc}{}^2 + \mu\_P \right) - \omega Q - P \frac{R}{L} \\ \frac{dQ}{dt} &= \frac{3}{2L} \mu\_Q - \omega Q - P \frac{R}{L} \end{split} \tag{20}$$
