Figure 1 presents the proposed FFSOGI-PLL, which adopts a fixed-frequency concept to reduce implementation complexity, enhance relative stability, and simplify the control design following the recommendations in [14]. As shown in Fig. 1, an ADSC operator is used to cancel the DC offset from the orthogonal signals, *v*_{i} is the grid voltage, *ω*_{n} is the nominal grid frequency, and \({\widehat{\omega }}_{g}\) and \(\widehat{\theta }\) are the estimated grid frequency and phase angle, respectively. *τ* is the delay length of the ADSC, and *k* is the SOGI block gain factor. As shown in Fig. 1, the estimated frequency from the SRF-PLL is fed back to the SOGI block to make it frequency adaptive.

The transfer functions of the fixed-frequency SOGI block, as shown in Fig. 1, are:

$$D\left(s\right)=\frac{{v}_{\alpha }\left(s\right)}{{v}_{i}\left(s\right)}=\frac{k{\omega }_{n}s}{{s}^{2}+k{\omega }_{n}s+{{\omega }_{n}}^{2}}$$

(1)

$$Q\left(s\right)=\frac{{v}_{\beta }\left(s\right)}{{v}_{i}\left(s\right)}=\frac{k{{\omega }_{n}}^{2}}{{s}^{2}+k{\omega }_{n}s+{{\omega }_{n}}^{2}}$$

(2)

Assuming a grid voltage of \({v}_{i}\left(t\right)=V\mathrm{sin}\left({\omega }_{g}t\right)\), where *V* is the voltage amplitude and it is assumed to be 1 pu for simplicity, the Laplace transform for the grid voltage is obtained as:

$${v}_{i}\left(s\right)=\frac{{\omega }_{g}}{{s}^{2}+{{\omega }_{g}}^{2}}$$

(3)

The output voltage in the s-domain of the SOGI, assuming fixed frequency for a given input voltage, is written as:

$${v}_{\alpha }\left(s\right)=\frac{k{\omega }_{n}s}{{s}^{2}+k{\omega }_{n}s+{{\omega }_{n}}^{2}}\left(\frac{{\omega }_{g}}{{s}^{2}+{{\omega }_{g}}^{2}}\right)$$

(4)

$${v}_{\beta }\left(s\right)=\frac{k{{\omega }_{n}}^{2}}{{s}^{2}+k{\omega }_{n}s+{{\omega }_{n}}^{2}}\left(\frac{{\omega }_{g}}{{s}^{2}+{{\omega }_{g}}^{2}}\right)$$

(5)

Simplifying the partial fraction expansion, the time domain *αβ-*signals \({v}_{\alpha }\left(t\right)\) and \({v}_{\beta }\left(t\right)\) are obtained as:

$${v}_{\alpha }\left(t\right)=\frac{k{{\omega }_{g}\omega }_{n}}{\sqrt{{({{\omega }_{n}}^{2}-{{\omega }_{g}}^{2})}^{2}+{k}^{2}{{\omega }_{n}}^{2}{{\omega }_{g}}^{2}}}\mathrm{sin}\left({\omega }_{g}t-\delta \right)+A\mathrm{sin}\left({\omega }_{d}t+{\varphi }_{1}\right){e}^{\frac{-k{\omega }_{n}}{2}t}$$

(6)

$${v}_{\beta }\left(t\right)=-\frac{k{{\omega }_{n}}^{2}}{\sqrt{{({{\omega }_{n}}^{2}-{{\omega }_{g}}^{2})}^{2}+{k}^{2}{{\omega }_{n}}^{2}{{\omega }_{g}}^{2}}}\mathrm{cos}\left({\omega }_{g}t-\delta \right)+{B \mathrm{cos}({\omega }_{d}t+{\varphi }_{2}) e}^{\frac{-k{\omega }_{n}}{2}t}$$

(7)

where \(\delta \) is the phase offset error, \(\mathrm{sin}\left(\delta \right)=\frac{{{\omega }_{g}}^{2}-{{\omega }_{n}}^{2}}{\sqrt{{({{\omega }_{n}}^{2}-{{\omega }_{g}}^{2})}^{2}+{k}^{2}{{\omega }_{n}}^{2}{{\omega }_{g}}^{2}}}\), \(A\), \(B\), \({\varphi }_{1}\), \({\varphi }_{2}\), and \({\omega }_{d}\) are functions of \({\omega }_{g}\), \({\omega }_{n}\), and \(k\).

From (6) and (7), it can be seen that \({v}_{\alpha }\left(t\right)\) and \({v}_{\beta }\left(t\right)\) have different amplitudes if \({\omega }_{g}\ne {\omega }_{n}.\) As \(\left|{{\omega }_{n}}^{2}-{{\widehat{\omega }}_{g}}^{2}\right|\ll k{\omega }_{n}{\omega }_{g}\), (6) and (7) can be simplified as:

$${v}_{\alpha }\left(t\right)=\mathrm{sin}\left({\omega }_{g}t-\delta \right)+A\mathrm{sin}\left({\omega }_{d}t+{\varphi }_{1}\right){e}^{\frac{-k{\omega }_{n}}{2}t}$$

(8)

$${v}_{\beta }\left(t\right)=-\frac{{\omega }_{n}}{{\omega }_{g}}\mathrm{cos}\left({\omega }_{g}t-\delta \right)+{B \mathrm{cos}({\omega }_{d}t+{\varphi }_{2}) e}^{\frac{-k{\omega }_{n}}{2}t}$$

(9)

From (8) and (9), the \({v}_{\alpha }\left(t\right)\) amplitude is equal to 1, while the \({v}_{\beta }\left(t\right)\) amplitude is scaled by \({\omega }_{n}/{\omega }_{g}\). The signals after the ADSC operator are given as:

$$\Delta {v}_{\alpha }\left(t\right)={v}_{\alpha }\left(t\right)-{v}_{\alpha }\left(t-\tau \right)$$

(10)

$$\Delta {v}_{\beta }\left(t\right)={v}_{\beta }\left(t\right)-{v}_{\beta }\left(t-\tau \right)$$

(11)

Substituting (8) and (9) into (10) and (11) yields:

$$ \begin{aligned} \Delta v_{\alpha } \left( t \right) & = \sin \left( {\omega_{g} t - \delta } \right) - \sin \left( {\omega_{g} t - \delta - \omega_{g} \tau } \right) + A\left( {\sin \left( {\omega_{d} t + \varphi_{1} } \right) - \sin \left( {\omega_{d} t + \varphi_{1} - \omega_{d} \tau } \right)} \right)e^{{\frac{{ - k\omega_{n} }}{2}\left( {t - \tau } \right)}} \\ & = \sin \left( {\omega_{g} t - \delta } \right) - \sin \left( {\omega_{g} t - \delta - \omega_{g} \tau } \right) + D\left( t \right)e^{{\frac{{ - k\omega_{n} }}{2}t}} \\ \end{aligned} $$

(12)

where \(D\left(t\right)=A\left(\mathrm{sin}\left({\omega }_{d}t+{\varphi }_{1}\right)-\mathrm{sin}\left({\omega }_{d}t+{\varphi }_{1}-{\omega }_{d}\tau \right)\right){e}^{\frac{k{\omega }_{n}}{2}\tau }\).

$$ \begin{aligned} \Delta v_{\beta } \left( t \right) & = - \cos \left( {\omega_{g} t - \delta } \right) + \cos \left( {\omega_{g} t - \delta - \omega_{g} \tau } \right) + B\left( {\cos \left( {\omega_{d} t + \varphi_{1} } \right) - \cos \left( {\omega_{d} t + \varphi_{1} - \omega_{d} \tau } \right)} \right)e^{{\frac{{ - k\omega_{n} }}{2}\left( {t - \tau } \right)}} \\ & = \cos \left( {\omega_{g} t - \delta - \omega_{g} \tau } \right) - \cos \left( {\omega_{g} t - \delta } \right) + Q\left( t \right)e^{{\frac{{ - k\omega_{n} }}{2}t}} \\ \end{aligned} $$

(13)

where \(Q\left(t\right)=B\left(\mathrm{cos}\left({\omega }_{d}t+{\varphi }_{1}\right)-\mathrm{cos}\left({\omega }_{d}t+{\varphi }_{1}-{\omega }_{d}\tau \right)\right){e}^{\frac{k{\omega }_{n}}{2}\tau }\).

The terms \(D\left(t\right)\) and \(Q(t)\) decay to zero with the time constant \({\tau }_{p}=2/k{\omega }_{n}\). Therefore, (12) and (13) are simplified as:

$$\Delta {v}_{\alpha }\left(t\right)= \mathrm{sin}\left(\theta -\delta \right)-\mathrm{sin}\left(\theta -\delta -{\omega }_{g}\tau \right)$$

(14)

$$\Delta {v}_{\beta }\left(t\right)=-\mathrm{cos}\left(\theta -\delta \right)+\mathrm{cos}\left(\theta -\delta -{\omega }_{g}\tau \right)$$

(15)

where \(\theta ={\omega }_{g}t\). Using a fixed frequency in the SOGI block, \({v}_{\alpha }\left(t\right)\) is the orthogonal signal to \({v}_{\beta }\left(t\right){\widehat{\omega }}_{g}/{\omega }_{g}\) in the frequency-locked state (\({\omega }_{g}={\widehat{\omega }}_{g}\)). Hence, any variation in the grid frequency will result in a small phase difference \(\delta \) between the actual phase angle \(\theta \) and that of \({v}_{\alpha }\left(t\right)\).

If \({\theta }^{*}\) is the net phase angle difference at the grid side and \({\theta }^{*}=\theta -\delta \), the transfer function of the SOGI block can be written as:

$${G}_{SOGI}\left(s\right)=\frac{{\theta }^{*}(s)}{\theta (s)}=\frac{1}{{\tau }_{p}s+1}$$

(16)

Equations (14) and (15) can be expressed using \({\theta }^{*}\) as:

$$\Delta {v}_{\alpha }\left(t\right)= \mathrm{sin}\left({\theta }^{*}\right)-\mathrm{sin}\left({\theta }^{*}-{\omega }_{g}\tau \right)$$

(17)

$$\Delta {v}_{\beta }\left(t\right)= -\mathrm{cos}\left({\theta }^{*}\right)+\mathrm{cos}\left({\theta }^{*}-{\omega }_{g}\tau \right)$$

(18)

Using ADSC to reject the DC offset will cause a phase error, so a phase correction of \({\theta }_{0}=-\frac{{\widehat{\omega }}_{g}\tau }{2}\) is needed [9, 29]. The \({v}_{q}\left(t\right)\) signal with the phase correction can be obtained as:

$${v}_{q}\left(t\right)=-\mathrm{sin}\left(\widehat{\theta }-\frac{{\widehat{\omega }}_{g}\tau }{2}\right)\Delta {v}_{\alpha }\left(t\right)+\mathrm{cos}\left(\widehat{\theta }-\frac{{\widehat{\omega }}_{g}\tau }{2}\right) \Delta {v}_{\beta }\left(t\right)$$

(19)

which can be simplified to:

$${v}_{q}\left(t\right)=2\mathrm{sin}\left({\theta }^{*}-\widehat{\theta }+\frac{{\widehat{\omega }}_{g}\tau }{2}-\frac{{\omega }_{g}\tau }{2}\right)\mathrm{sin}(\frac{{\omega }_{g}\tau }{2})$$

(20)

### 2.1 PLL small-signal model

In this section, a small-signal model for the proposed PLL is derived. The term \(\mathrm{sin}\left(\frac{{\omega }_{g}\tau }{2}\right)\) can be written as:

$$\mathrm{sin}\left(\frac{{\omega }_{g}\tau }{2}\right)=\mathrm{sin}\left(\frac{{\omega }_{n}\tau }{2}\right)\mathrm{cos}\left(\frac{{\Delta \omega }_{g}\tau }{2}\right)+\mathrm{cos}\left(\frac{{\omega }_{n}\tau }{2}\right)\mathrm{sin}\left(\frac{{\Delta \omega }_{g}\tau }{2}\right)$$

(21)

In the small-signal analysis, \(\mathrm{cos}\left(\frac{{\Delta \omega }_{g}\tau }{2}\right)\approx 1\) and \(\mathrm{sin}\left(\frac{{\Delta \omega }_{g}\tau }{2}\right)\approx \frac{{\Delta \omega }_{g}\tau }{2}\). Hence, (21) can be simplified to:

$$\mathrm{sin}\left(\frac{{\omega }_{g}\tau }{2}\right)=\mathrm{sin}\left(\frac{{\omega }_{n}\tau }{2}\right)+\frac{{\Delta \omega }_{g}\tau }{2}\mathrm{cos}\left(\frac{{\omega }_{n}\tau }{2}\right)$$

(22)

Equation (20) can then be rewritten using (22), as:

$${v}_{q}\left(t\right)=2\mathrm{sin}\left({\theta }^{*}-\widehat{\theta }+\frac{{\widehat{\omega }}_{g}\tau }{2}-\frac{{\omega }_{g}\tau }{2}\right)\left(\mathrm{sin}\left(\frac{{\omega }_{n}\tau }{2}\right)+\frac{{\Delta \omega }_{g}\tau }{2}\mathrm{cos}\left(\frac{{\omega }_{n}\tau }{2}\right)\right)$$

(23)

The term \({\Delta \omega }_{g}\tau \left(\Delta {\theta }^{*}-\Delta \widehat{\theta }+\frac{\Delta {\widehat{\omega }}_{g}\tau }{2}-\frac{{\Delta \omega }_{g}\tau }{2}\right)\) in (23) equals zero in the small-signal analysis, and \({v}_{q}\left(t\right)\) can be simplified as:

$${v}_{q}\left(t\right)=2\mathrm{sin}(\frac{{\omega }_{n}\tau }{2})\left(\Delta {\theta }^{*}-\Delta \widehat{\theta }+\frac{\Delta {\widehat{\omega }}_{g}\tau }{2}-\frac{{\Delta \omega }_{g}\tau }{2}\right)$$

(24)

Rearranging (24) yields:

$${v}_{q}\left(t\right)=2\mathrm{sin}\left(\frac{{\omega }_{n}\tau }{2}\right)\left(\frac{\Delta {\theta }^{*}+\Delta {\theta }^{*}-{\Delta \omega }_{g}\tau }{2}-\Delta \widehat{\theta }+\frac{\Delta {\widehat{\omega }}_{g}\tau }{2}\right)$$

(25)

Applying the Laplace transform to (25) yields:

$${v}_{q}\left(s\right)=2\mathrm{sin}\left(\frac{{\omega }_{n}\tau }{2}\right)\left(\frac{1+{e}^{-s\tau }}{2}\Delta {\theta }^{*}\left(s\right)-\Delta \widehat{\theta }\left(s\right)+\frac{\Delta {\widehat{\omega }}_{g}\left(s\right)\tau }{2}\right)$$

(26)

Substituting the value of \(\Delta {\theta }^{*}(s)\) from (16) into (26) yields:

$${v}_{q}\left(s\right)={k}_{v}\left(\frac{1+{e}^{-s\tau }}{2}\frac{1}{{\tau }_{p}s+1}\Delta \theta (s)-\Delta \widehat{\theta }\left(s\right)+\frac{\Delta {\widehat{\omega }}_{g}\left(s\right)\tau }{2}\right)$$

(27)

where \({k}_{v}=2\mathrm{sin}\left(\frac{{\omega }_{n}\tau }{2}\right)\) is the amplitude scaling factor.

The derived small-signal model does not consider the dynamic of the phase offset error, so to enhance its accuracy, compensation for the phase offset error dynamic is calculated following the guidelines in [1], where \(\delta \approx \mathrm{sin}\left(\delta \right)\approx \frac{{{\widehat{\omega }}_{g}}^{2}-{{\omega }_{n}}^{2}}{k{\omega }_{n}{\widehat{\omega }}_{g}}\). Substituting the values of \({\omega }_{g}={\omega }_{n}+\Delta {\omega }_{g}\) and \({\widehat{\omega }}_{g}= {\omega }_{n}+ \Delta {\widehat{\omega }}_{g}\), \(\delta \) can be simplified to:

$$\delta \approx \frac{{{{\omega }_{n}}^{2}+\Delta {\widehat{\omega }}_{g}}^{2}+2\Delta {\widehat{\omega }}_{g}{\omega }_{n}-{{\omega }_{n}}^{2}}{k{\omega }_{n}({\omega }_{n}+ \Delta {\widehat{\omega }}_{g})}\approx \frac{\Delta {\widehat{\omega }}_{g}(\Delta {\widehat{\omega }}_{g}+2{\omega }_{n})}{k{\omega }_{n}({\omega }_{n}+ \Delta {\widehat{\omega }}_{g})}\approx \frac{2\Delta {\widehat{\omega }}_{g}}{k{\omega }_{n}}\approx {\tau }_{p}\Delta {\widehat{\omega }}_{g}$$

(28)

According to (27) and (28) and based on Fig. 1, the small-signal model of the proposed FFSOGI-PLL is shown in Fig. 2, and the closed-loop transfer function is obtained as:

$${G}_{cl}\left(s\right)=\frac{\Delta {\widehat{\theta }}_{c}}{\Delta \theta }=\frac{1}{{\tau }_{p}s+1}\frac{1+{e}^{-\tau s}}{2}\frac{{k}_{v}({k}_{p}s+{k}_{i})}{{s}^{2}+{k}_{v}\left({k}_{p}-\frac{\tau }{2}{k}_{i}\right)s+{k}_{v}{k}_{i}}$$

(29)

The transfer function in (29) contains a dominant second-order system and a nondominant first-order system. The dominant roots capture the dynamic performance of the system, so the small-signal model can be reduced to a second-order system, as:

$${G}_{cl}\left(s\right)=\frac{\Delta {\widehat{\theta }}_{c}}{\Delta \theta }\approx \frac{1+{e}^{-\tau s}}{2}\frac{{k}_{v}({k}_{p}s+{k}_{i})}{{s}^{2}+{k}_{v}\left({k}_{p}-\frac{\tau }{2}{k}_{i}\right)s+{k}_{v}{k}_{i}}$$

(30)

### 2.2 PI gains design

From the small-signal model represented by the dominant second-order system as in (30), the following characteristic equation (CE) is obtained:

\({s}^{2}+{k}_{v}\left({k}_{p}-\frac{\tau }{2}{k}_{i}\right)s+{k}_{v}{k}_{i}=0\). The second-order system can be designed using linear control theory. The most straightforward method to design the PI-controller gains is to specify the desired damping ratio \(\zeta \) and the natural damping ω_{N} of the closed-loop control system. These have a specific desired transient response and bandwidth. Hence, based on \(\zeta \) and ω_{N}, the closed-loop CE is obtained as \({s}^{2}+2{\omega }_{N}\zeta s+{{\omega }_{N}}^{2}=0\), and the PI-controller gains are designed by comparing the actual CE with the desired CE. This yields \({k}_{v}{k}_{i}={{\omega }_{N}}^{2}\) and \({k}_{v}\left({k}_{p}-\frac{\tau }{2}{k}_{i}\right)=2{\omega }_{N}\zeta\), from which

$${k}_{i}=\frac{{{\omega }_{N}}^{2} }{{k}_{v}}$$

(31)

$${k}_{p}=\frac{2{\omega }_{N}\zeta }{{k}_{v}}+\frac{\tau {k}_{i}}{2}$$

(32)

If \(\tau =0.002\) s, \(\zeta =0.707\), and \({\omega }_{N}=41\pi \) rad/s, the PI gains are calculated using (31) and (32) as \({k}_{p}=\) 325.1547 and \({k}_{i}=\) 27,397. The SOGI gain factor \(k\) should be as large as possible. However, the related PLL small-signal model reveals that a lower value for \(k\) leads to better filtering capability but at the cost of a slower dynamic response. Therefore, \(k\) should be selected to achieve an acceptable trade-off between the disturbance rejection and response speed. To make a fair comparison with other PLLs, \(k=2\) is selected.

Figure 3 shows the actual and small-signal model responses of the proposed PLL under a phase jump of 20° at 0.02 s, while the actual and estimated voltages are shown in Fig. 4. The results in Figs. 3 and 4 validate the accuracy of the derived small-signal model in predicting the dynamic behavior of the proposed PLL.

The performance of the proposed FFSOGI-PLL is also tested under the following case studies:

**Case Study 1:** A 20% voltage sag is applied at 0.1 s and recovered to 1 pu at 0.2 s. At 0.3 s, a 20° phase jump occurs, while grid DC voltage offset is imposed at 0.4 s. The grid frequency is fixed at 50 Hz. The results are shown in Fig. 5.

**Case Study 2:** At 0.1 s, a 3 Hz frequency variation occurs in the grid, while a grid DC voltage offset is added at 0.3 s. The results are shown in Fig. 6.