Disturbance-Observer-Based PLL#
A phase-locked loop (PLL) is commonly used in grid-following converters to synchronize the converter output with the grid [1]. Here, we represent the PLL using the disturbance observer structure [2], 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 [3].
Disturbance Model#
Consider the positive-sequence voltage at the point of common coupling (PCC) in general coordinates, rotating at the angular speed \(\omega_\mathrm{c}\). The dynamics of the PCC voltage \(\boldsymbol{u}_\mathrm{g}\) can be modeled using a disturbance model as
where \(\omega_\mathrm{g}\) 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.
PLL in General Coordinates#
Based on (1), the disturbance observer containing the regular PLL functionality can be formulated as
where \(\hat{\boldsymbol{u}}_\mathrm{g}\) is the estimated PCC voltage, \(\hat{\omega}_\mathrm{g}\) is the grid angular frequency estimate (either constant corresponding to the nominal value or dynamic from grid-frequency tracking), and \(\alpha_\mathrm{g}\) is the bandwidth. If needed, the disturbance observer can be extended with the frequency tracking as
where \(k_\omega\) is the frequency-tracking gain. Notice that (2) and (3) are both driven by the estimation error \(\boldsymbol{u}_\mathrm{g} - \hat{\boldsymbol{u}}_\mathrm{g}\) of the PCC voltage.
PLL in Estimated PCC Voltage Coordinates#
The disturbance observer (2) can be equivalently expressed in estimated PCC voltage coordinates, where \(\hat{\boldsymbol{u}}_\mathrm{g} = \hat{u}_\mathrm{g}\) and \(\omega_\mathrm{c} = \mathrm{d} \hat{\vartheta}_\mathrm{g}/ \mathrm{d} t\), resulting in
It can be seen that the first equation in (4) 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 (3) reduces to
It can be noticed that the disturbance observer with the frequency tracking equals the conventional frequency-adaptive PLL [1], 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 [4].
Linearized Closed-Loop System#
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 \(\Delta \boldsymbol{u}_\mathrm{g} = \boldsymbol{u}_\mathrm{g} - \boldsymbol{u}_\mathrm{g0}\), where \(\boldsymbol{u}_\mathrm{g0}\) is the operating-point quantity. From (1)–(3), the linearized model for the estimation-error dynamics is obtained as
where \(\Delta \tilde{\boldsymbol{u}}_\mathrm{g} = \Delta\boldsymbol{u}_\mathrm{g} - \Delta \hat{\boldsymbol{u}}_\mathrm{g}\) is the estimation error.
First, assume that the grid frequency \(\omega_\mathrm{g}\) is constant and the frequency tracking is disabled. From (6), the closed-loop transfer function from the PCC voltage to its estimate becomes
It can be realized that both the angle and magnitude of the PCC voltage estimate converge with the bandwidth \(\alpha_\mathrm{g}\).
Next, the frequency-tracking dynamics are also considered. From (6), the closed-loop transfer function from the grid angular frequency to its estimate becomes
Choosing \(k_\omega = \alpha_\mathrm{pll}^2\) and \(\alpha_\mathrm{g} = 2\alpha_\mathrm{pll}\) yields the double pole at \(s = -\alpha_\mathrm{pll}\), where \(\alpha_\mathrm{pll}\) 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.
References