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, vi is the grid voltage, ωn 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 ωN of the closed-loop control system. These have a specific desired transient response and bandwidth. Hence, based on \(\zeta \) and ωN, 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.