Phase-Locked Loop

A phase-locked loop (PLL) is commonly used in grid-following converters to synchronize the converter output with the grid [Kaura and Blasko, 1997]. Here, we represent the PLL using the disturbance observer structure [Franklin et al., 1997], which may be simpler to extend than the classical PLL. Furthermore, this structure allows to see links to synchronization methods used in grid-forming converters, see [Nurminen et al., 2024].

Disturbance Model

Consider the positive-sequence voltage at the point of common coupling (PCC) in general coordinates, rotating at the angular speed \(\omegac\). The dynamics of the PCC voltage \(\ug\) can be modeled using a disturbance model as

(1)\[\frac{\D\ug}{\D t} = \jj(\omegag - \omegac)\ug\]

where \(\omegag\) is the grid angular frequency. If needed, this disturbance model could be extended, e.g., with a negative-sequence component, allowing to design the PLL for unbalanced grids.

Note

The disturbance model in (1) can be equivalently expressed in polar form as

(2)\[\begin{align} \frac{\D \absug}{\D t} &= 0 \\ \frac{\D \delta_\mathrm{g}}{\D t} &= \omegag - \omegac \\ \ug &= \absug \e^{\jj\delta_\mathrm{g}} \end{align}\]

where \(\delta_\mathrm{g} = \thetag - \vartheta_\mathrm{c}\) is the angle of the grid voltage in general coordinates and the last equation is the output equation.

PLL in General Coordinates

Based on (1), the disturbance observer containing the regular PLL functionality can be formulated as

(3)\[\frac{\D\hatug}{\D t} = \jj(\hatomegag - \omegac)\hatug + \alphao (\ug - \hatug )\]

where \(\hatug\) is the estimated PCC voltage, \(\hatomegag\) is the grid angular frequency estimate (either constant corresponding to the nominal value or dynamic from grid-frequency tracking), and \(\alphao\) is the bandwidth. If needed, the disturbance observer can be extended with the frequency tracking as

(4)\[\frac{\D\hatomegag}{\D t} = k_\upomega\IM\left\{ \frac{\ug - \hatug}{\hatug} \right\}\]

where \(k_\upomega\) is the frequency-tracking gain. Notice that (3) and (4) are both driven by the estimation error \(\ug - \hatug\) of the PCC voltage.

PLL in Estimated PCC Voltage Coordinates

The disturbance observer (3) can be equivalently expressed in estimated PCC voltage coordinates, where \(\hatug = \hatabsug\) and \(\omegac = \D \hatthetag/ \D t\), resulting in

(5)\[\begin{split}\frac{\D\hatabsug}{\D t} &= \alphao \left(\mathrm{Re}\{\ug\} - \hatabsug \right) \\ \frac{\D \hatthetag}{\D t} &= \hatomegag + \frac{\alphao}{\hatabsug}\IM\{ \ug \} = \omegac\end{split}\]

It can be seen that the first equation in (5) is low-pass filtering of the measured PCC voltage magnitude and the second equation is the conventional angle-tracking PLL. In these coordinates, the frequency tracking (4) reduces to

(6)\[\frac{\D \hatomegag}{\D t} = \frac{k_\upomega}{\hatabsug } \IM\{ \ug \}\]

It can be noticed that the disturbance observer with the frequency tracking equals the conventional frequency-adaptive PLL [Kaura and Blasko, 1997], with the additional feature of low-pass filtering the PCC voltage magnitude. The low-pass filtered PCC voltage can be used as a feedforward term in current control [Harnefors et al., 2021].

Closed-Loop System Analysis

The estimation-error dynamics are analyzed by means of linearization. Using the PCC voltage as an example, the small-signal deviation about the operating point is denoted by \(\Delta \ug = \ug - \ugo\), where \(\ugo\) is the operating-point quantity. From (1)–(4), the linearized model for the estimation-error dynamics is obtained as

(7)\[\begin{split}\frac{\D\Delta \tildeug}{\D t} &= -\alphao\Delta \tildeug + \jj\ugo (\Delta \omegag - \Delta \hatomegag) \\ \frac{\D\Delta \hatomegag}{\D t} &= k_\omega\IM\left\{ \frac{\Delta \tildeug}{\ugo} \right\}\end{split}\]

where \(\Delta \tildeug = \Delta\ug - \Delta \hatug\) is the estimation error.

First, assume that the grid frequency \(\omegag\) is constant and the frequency tracking is disabled. From (7), the closed-loop transfer function from the PCC voltage to its estimate becomes

(8)\[\frac{\Delta\hatug(s)}{\Delta\ug(s)} = \frac{\alphao}{s + \alphao}\]

It can be realized that both the angle and magnitude of the PCC voltage estimate converge with the bandwidth \(\alphao\).

Next, the frequency-tracking dynamics are also considered. From (7), the closed-loop transfer function from the grid angular frequency to its estimate becomes

(9)\[\frac{\Delta\hatomegag(s)}{\Delta\omegag(s)} = \frac{k_\upomega}{s^2 + \alphao s + k_\omega}\]

Choosing \(k_\upomega = \alphapll^2\) and \(\alphao = 2\alphapll\) yields the double pole at \(s = -\alphapll\), where \(\alphapll\) is the frequency-tracking bandwidth.

This PLL in estimated PCC coordinates is implemented in the class motulator.grid.control.PLL. The Grid-Following Control examples use the PLL to synchronize with the grid.