# Passivity-based design and analysis of phase-locked loops

Daniele Zonetti⋆ Oriol Gomis-Bellmunt  Eduardo Prieto-Araujo  Marc Cheah-Mañe
All authors are with the Centre d’Innovació Tecnològica en Convertidors
Estàtics i Accionaments (CITCEA-UPC), Departament d’Enginyeria Elèctrica,
Universitat Politècnica de Catalunya, 08028 Barcelona, Spain. ⋆Corresponding
author<EMAIL_ADDRESS>
###### Abstract

We consider a grid-connected voltage source converter (VSC) and address the
problem of estimating the grid angle and frequency—information that is
essential for an appropriate operation of the converter. We design phase-
locked loop (PLL) algorithms with guaranteed stability properties, under the
assumption that the grid is characterized by relatively high short-circuit-
ratio (SCR) and inertia. In particular we derive, using passivity arguments,
generalization of the conventional synchronous reference frame (SRF) and
arctangent (ATAN) PLL, which are the standard solutions in power applications.
The analysis is further extended to the case of connection of the VSC to a
low-inertia grid, ensuring robustness of the algorithms in case of frequency
variations. The results are validated on a benchmark including the grid, the
VSC and related controllers.

##  I Introduction

The ever increasing penetration of renewable energy sources (RES) in the
existing power grid is driving dramatic changes in the way power systems are
designed and operated. In this continuously evolving scenario, since the
integration of RES is enabled by voltage source converters (VSCs), many
questions and challenges arise about their operation and control [1]. An
appropriate operation of VSCs can be enforced by a variety of controllers,
typically consisting of hierarchical control loops operated at different time-
scales. The majority of these architectures rely on the definition of a
suitable rotating reference frame that is locked to the angle of the voltage
at the point of common coupling (PCC)—an information that, unfortunately, is
not available to the designer. As a result, an appropriate algorithm that
estimates the grid angle must be designed to enforce synchronization of such
controllers and then of the converter.
Phase-locked loop (PLL) algorithms are the conventional schemes employed to
achieve this objective and the analysis of their stability properties in
different, perturbed scenarios, has considerably attracted the interest of the
power systems and control community[2, 3]. To facilitate the analysis, it is
usually assumed that PLL algorithms can be operated at a time-scale much
slower than the time-scale at which the VSC controllers operate and physical
dynamics evolve. Fundamental stability properties of the conventional
synchronous reference frame (SRF)-PLL under this assumption have been already
studied in [4] and [5]. Yet, many questions arise about the ability of the
VSCs to synchronize under especially degraded conditions resulting, for
example, from the connection of the VSC to grids characterized by low short-
circuit-ratio (SCR) [6, 7], or low inertia [8, 9]. However, in all these works
only the traditional SRF-PLL mechanism is analysed and no constructive
procedure are provided for the design of these schemes.

In this paper we use passivity arguments [10] to constructively design
generalized PLL mechanisms that guarantee—under the aforementioned time-scale
separation assumption—a correct estimation of the grid angle and frequency,
paving the road to a rigourous stability analysis and potential redesign for
grid-connected VSCs in weak and low-inertia grid scenarios. Interestingly
enough, specific instances of such schemes correspond to the conventional SRF-
PLL and arctangent (ATAN)-PLL [11], which can be eventually recovered by
appropriate selection of some free parameters. Two scenarios are considered.
First, we assume that the VSC is connected to a strong grid with a relatively
high inertia. Hence, the grid angle can be represented as a balanced three-
phase sinusoidal voltage source at constant frequency. Next, we extend our
results to the case where the frequency is subject to bounded variations,
similar to [9], where an observer-based framework was employed to analyse low-
inertia scenarios. Finally, we validate our results by simulations on a
benchmark including the grid, converter and low-level controllers dynamics.

The remainder of the paper is organised as follows. Fundamental modeling
assumptions and the definition of the estimation problem in suitable
coordinates are elaborated in Section II. Section III and Section IV are then
dedicated to the passivity-based design of generalized SRF- and ATAN-
PLL—assuming that the grid is characterized by a large inertia. Robustness of
these estimation mechanisms to frequency variations is then analysed in
Section V. The results are validated in Section VI and next summarized in
Section VII, where we further provide guidelines for future investigation.

Notation. The set $\mathbb{R}_{>0}$ (respectively $\mathbb{R}_{\geq 0})$
denotes the set of real positive (respectively non-negative) real numbers. The
set $2\mathbb{Z}$ (respectively $2\mathbb{Z}+1$) denotes the subset of even
(respectively odd) integers. The set $\mathcal{S}$ denotes the unit circle in
$\mathbb{R}^{2}$. For given vectors ${x_{1}\in\mathbb{R}^{n}}$,
$x_{2}\in\mathbb{R}^{m}$, $x:=(x_{1},x_{2})\in\mathbb{R}^{n+m}$ denotes the
stacked collection of such vectors and $\|x\|$ denotes its Euclidean norm.
$\mathrm{Id}:\mathbb{R}\rightarrow\mathbb{R}$ denotes the identity function,
$\operatorname{atan2}:\mathcal{S}\rightarrow[-\pi\;\pi)$ denotes the
2-arguments arctangent function. For a given angle $\vartheta\in\mathbb{R}$,
we define the three-phase vectors:

$\overline{\sin}(\vartheta):=\begin{bmatrix}\sin(\vartheta)\\\
\sin(\vartheta-\frac{2}{3}\pi)\\\
\sin(\vartheta+\frac{2}{3}\pi)\end{bmatrix}\in\mathbb{R}^{3},\quad\overline{\cos}(\vartheta):=\begin{bmatrix}\cos(\vartheta)\\\
\cos(\vartheta-\frac{2}{3}\pi)\\\
\cos(\vartheta+\frac{2}{3}\pi)\end{bmatrix}\in\mathbb{R}^{3}.$

(I.1)

##  II Problem setup

In this section we discuss the model considered for the analysis and formulate
the problem of estimation of the grid angle and frequency. We make the
following assumptions.

###### Assumption II.1

The VSC is characterized by a stable current control scheme, which guarantees
instantaneous and exact tracking of the desired currents.

###### Assumption II.2

The grid dynamics is characterized by a sufficiently high short-circuit-ratio
(SCR).

Assumption II.1 is usually justified by the fast operation of the current
controller as compared to the dynamics of the estimation scheme—a property
that can be enforced by appropriate tuning of the gains. Assumption II.2 is
verified whenever the grid is characterized by a sufficiently small impedance,
so that its dynamics evolve at a time-scale much faster than the time-scale at
which the estimation scheme operates.
An immediate consequence of these assumptions is that the grid and converter
dynamics can be approximated by their steady-state model and the voltage at
the point of common coupling (PCC) can be modeled by a balanced, three-phase,
purely sinusoidal signal

$v(t):=V\;\overline{\sin}(\theta(t))\in\mathbb{R}^{3},\qquad\theta(t):=\omega
t+\phi,$ (II.1)

where $V\in\mathbb{R}$ and $\theta\in\mathbb{R}$ denote the voltage amplitude
and angle, with $\omega\in\mathbb{R}_{>0}$ and $\phi\in\mathbb{R}$ denoting
the frequency and phase shift respectively. The estimation problem consists
then in the design of a sufficiently fast mechanism able to recover the angle
and the frequency of (II.1), based solely on instantaneous measurements of
such a signal.
A simple, static approach to solve this problem consists in considering the
so-called $\alpha\beta$ transformation111Since the signal (II.1) is balanced,
the zero component can be neglected with no loss of generality. of (II.1),
given by:

$v_{\alpha\beta}=(V_{\alpha},V_{\beta})=T_{\alpha\beta}v,\quad
T_{\alpha\beta}:=\frac{1}{\gamma}\begin{bmatrix}0&-\frac{\sqrt{3}}{2}&\frac{\sqrt{3}}{2}\\\
0&-\frac{1}{2}&-\frac{1}{2}\end{bmatrix},$ (II.2)

with $\gamma:=\sqrt{3/2}$, and from which immediately follows:

$v_{\alpha\beta}=\gamma V\begin{bmatrix}\cos(\theta)\\\
\sin(\theta)\end{bmatrix}$ (II.3)

The exact value of the angle can be then recovered via:

$\theta=\operatorname{atan2}(V_{\beta},V_{\alpha}),$ (II.4)

and the frequency obtained by computing its first time-derivative. However,
this solution has two major drawbacks that stymie its practical
implementation: first, the computation of the angle might be ill-defined for
some $v_{\alpha\beta}$; second, frequency estimation relies on exact
derivation, an operation that is extremely sensitive to noise and harmonics.
For this reason, dynamic approaches are preferred and consist in the design of
a (possibly dynamic) frequency estimate $\hat{u}(t)$ for the estimator:

$\displaystyle\dot{\hat{\theta}}(t)$ $\displaystyle=\hat{u}(t),$ (II.5)

with $\hat{\theta}\in\mathbb{R}$, $\hat{\theta}_{0}\in\mathbb{R}$ denoting the
angle estimate and its initial guess respectively. The objective of such
design is to ensure that the angle estimation error
$\delta:=\hat{\theta}-\theta\in\mathbb{R}$, captured by the dynamics

$\displaystyle\dot{\delta}(t)$ $\displaystyle=-\omega+\hat{u}(t),$ (II.6)

converges to zero after reasonable transients—a fact that also implies
convergence of $\hat{u}(t)$ to $\omega$. Hence, the grid actual frequency is
ultimately provided by $\hat{u}(t)$. Based on these considerations, we can
recast the estimation problem as the following stabilization problem—also
referred as synchronization problem—where the function $\hat{u}(t)$ can be
interpreted as a control signal to be designed.

###### Problem II.3

Consider the system (II.6). Determine $\hat{u}(t)\in\mathbb{R}$ such that

$\lim_{t\to\infty}\hat{\theta}(t)=\theta(t).$ (II.7)

Unfortunately, this problem is complicated by the fact that the signal
$\delta\in\mathbb{R}$ is not directly measurable. To overcome this issue,
consider the following coordinates change:

$\displaystyle\hat{v}_{dq}=(\hat{V}_{d},\hat{V}_{q})=T_{dq}(\vartheta)v,\quad
T_{dq}(\vartheta):=\frac{1}{\gamma}\begin{bmatrix}\overline{\sin}(\vartheta)\\\
\overline{\cos}(\vartheta)\end{bmatrix},$ (II.8)

where $\vartheta:=\hat{\theta}-\frac{\pi}{2}$, which is referred as the $dq$
transformation of the signal (II.1) with transformation angle
$\vartheta\in\mathbb{R}$. We have then:

$\hat{v}_{dq}=\gamma V\begin{bmatrix}\cos(\delta)\\\
\sin(\delta)\end{bmatrix}:=\gamma V\mathsf{T}(\delta),$ (II.9)

where we have further defined the operator
$\mathsf{T}:\mathbb{R}\rightarrow\mathcal{S}$. By calculating the first time-
derivative of (II.9) and using (II.6) we obtain the following dynamical system

$\displaystyle\dot{\hat{V}}_{d}$ $\displaystyle=-\dot{\delta}\gamma
V\sin(\delta)=-(\hat{u}-\omega)\hat{V}_{q}$ (II.10)
$\displaystyle\dot{\hat{V}}_{q}$ $\displaystyle=\dot{\delta}\gamma
V\cos(\delta)=(\hat{u}-\omega)\hat{V}_{d},$

that in compact form reads:

$\displaystyle\dot{\hat{v}}_{dq}$
$\displaystyle=J_{2}(\hat{u}-\omega)\hat{v}_{dq},\quad
J_{2}:=\begin{bmatrix}0&-1\\\ 1&0\end{bmatrix}\in\mathbb{R}^{2\times 2}.$
(II.11)

Note that in contrast with the scalar system (II.6), the state of (II.11) is
measurable and defined on the set

$\mathcal{V}:=\gamma V\mathcal{S}\subset\mathbb{R}^{2}.$

We make now the following observation. The set of assignable equilibria for
this system, i.e. the set of equilibria that can be assigned via a constant
(equilibrium) control, is given by the whole state space $\mathcal{V}$.
Moreover, for a given $v_{dq}\in\mathcal{V}$, the following relation holds:

$v_{dq}=\gamma V\mathsf{T}(\delta^{h}),$ (II.12)

where

$\delta^{h}:=\delta_{\mathrm{ref}}+h\pi,\qquad h\in 2\mathbb{Z},$

for some $\delta_{\mathrm{ref}}\in[-\pi\;\pi)$. As a result, stabilization of
$\hat{v}_{dq}$ to $v_{dq}$ further implies convergence of the angle estimation
error $\delta$ to $\delta_{\mathrm{ref}}$ (modulo $2\pi$). Based on these
considerations, Problem II.3 can be then reformulated as the following
stabilization problem in $dq$ coordinates, where correct estimation of the
grid angle is imposed by picking a $\delta_{\mathrm{ref}}=0$.

###### Problem II.4

Consider the system (II.11) and let

$v_{dq\star}:=\gamma V\mathsf{T}(h\pi)=(\gamma V,0).$ (II.13)

Determine $\hat{u}(t)\in\mathbb{R}$ such that

$\lim_{t\to\infty}\hat{v}_{dq}(t)=v_{dq\star}.$ (II.14)

##  III SRF-PLL

We now derive a synchronization mechanism with guaranteed stability
properties, referred in the sequel as generalized synchronous reference frame
(gSRF)-PLL. We have the following preliminary proposition.

###### Proposition III.1

Consider the system (II.11), let $v_{dq}\in\mathcal{V}$ and define

$\tilde{u}:=\hat{u}-\omega,\quad\tilde{y}_{1}:=-v_{dq}^{\top}J_{2}\hat{v}_{dq}.$
(III.1)

Then the map $\tilde{u}\rightarrow\tilde{y}_{1}$ is passive with storage
function

$\mathcal{H}_{1}(\hat{v}_{dq}):=\frac{1}{2}\|\hat{v}_{dq}-v_{dq}\|^{2}.$
(III.2)

###### Proof:

Consider the energy function (III.2). By calculating its derivative along the
system’s trajectories we get:

$\displaystyle\dot{\mathcal{H}}_{1}$
$\displaystyle=(\hat{v}_{dq}-v_{dq})^{\top}\dot{\hat{v}}_{dq}=(\hat{v}_{dq}-v_{dq})^{\top}J_{2}\tilde{u}\hat{v}_{dq}=\tilde{y}_{1}\tilde{u}$
(III.3)

where in the second equivalence we simply replaced (II.11), while the last one
follows from skew-symmetry of $J_{2}$ the definition of $\tilde{y}_{1}$, thus
completing the proof. ∎

The passive property of the system can be then employed to constructively
design a class of estimation algorithms in the form of PI controllers driven
by the ouput $\tilde{y}_{1}$, as explained in the following proposition.

###### Proposition III.2

Let $v_{dq}\in\mathcal{V}$. Consider the system (II.11) in closed-loop with
the PLL:

$\displaystyle\hat{u}$ $\displaystyle=-K_{P}\Phi(\tilde{y}_{1})+\hat{\omega}$
(III.4) $\displaystyle\dot{\hat{\omega}}$ $\displaystyle=-K_{I}\tilde{y}_{1},$

with state $\hat{\omega}\in\mathbb{R}$, $\tilde{y}_{1}$ given by (III.1),
$K_{I}$, $K_{P}$ positive scalars and $\Phi$ is a scalar function verifying
$\Phi(0)=0$, $\Phi(s)s>0$ for any $s\neq 0$. Then for any initial condition
$(v_{dq0},\omega_{0})\in\mathcal{V}\times\mathbb{R}$ the system’s trajectories
converge to the set

$\mathcal{E}_{dq}:=(v_{dq},\omega)\cup(-v_{dq},\omega).$

###### Proof:

First, note that at the equilibrium $\hat{\omega}=\omega.$ Then, consider the
augmented energy function

$\mathcal{W}_{1}(\hat{v}_{dq},\hat{\omega}):=\mathcal{H}_{1}(\hat{v}_{dq})+\frac{1}{2K_{I}}(\hat{\omega}-\omega)^{2},$
(III.5)

which is positive definite and radially unbounded, being $K_{I}$ positive.
Then

$\displaystyle\dot{\mathcal{W}}_{1}$
$\displaystyle=\dot{\mathcal{H}}_{1}+\frac{1}{K_{I}}(\hat{\omega}-\omega)\dot{\hat{\omega}}$
(III.6)
$\displaystyle=\tilde{y}_{1}(-K_{P}\Phi(\tilde{y}_{1})+\hat{\omega}-\omega)-(\hat{\omega}-\omega)\tilde{y}_{1}$
$\displaystyle=-K_{P}\Phi(\tilde{y}_{1})\tilde{y}_{1}$ $\displaystyle\leq 0,$

where in the second equivalence we replaced the dynamics of (III.4) and used
Proposition III.1, while the last inequality results from positivity of
$K_{P}$ and the definition of $\Phi$. The proof is completed using LaSalle
invariance principle and noting that $\dot{\mathcal{W}}_{1}=0$ implies
$(\hat{v}_{dq},\hat{\omega})=(\pm v_{dq},\omega)$. ∎

Proposition III.2 merely states that for any initial guess of the gSRF-PLL,
the trajectories converge to the set $\mathcal{E}_{dq}$, which consists of two
isolated equilibria $(\pm v_{dq},\omega)\in\mathcal{V}\times\mathbb{R}$. We
deduce then that the frequency $\omega$ is correctly estimated via (III.4). It
is also immediate to see that a specific implementation of the controller
(III.4) for $\Phi:=\frac{1}{\gamma V}\mathrm{Id}$ and $v_{dq}=v_{dq\star}$ is
given by the conventional SRF-PLL, as stated in the following corollary.

###### Corollary III.3

Consider the system (II.11) in closed-loop with the SRF-PLL:

$\displaystyle\hat{u}$ $\displaystyle=-K_{P}\hat{V}_{q}+\hat{\omega}$ (III.7)
$\displaystyle\dot{\hat{\omega}}$ $\displaystyle=-K_{I}\hat{V}_{q},$

with $K_{I}$, $K_{P}$ positive scalars. Then for any initial condition
$(v_{dq0},\omega_{0})\in\mathcal{V}\times\mathbb{R}$ the system’s trajectories
converge to the set

$\mathcal{E}_{dq\star}:=(v_{dq\star},\omega)\cup(-v_{dq\star},\omega).$

It must be noted that Corollary III.3 is inconclusive about the ability of the
SRF-PLL to correctly estimate the angle. Indeed, while such objective is
achieved if trajectories converge to $(v_{dq\star},\omega)$, convergence to
$(-v_{dq\star},\omega)$ would result into an error of $\pi$ radians. To show
that the latter is an exceptional case, let us rewrite the SRF-PLL (III.7) in
polar coordinates using (II.9), leading to the following nonlinear system:

$\displaystyle\dot{\delta}$ $\displaystyle=-\omega-K_{P}\gamma
V\sin(\delta)+\hat{\omega}$ (III.8) $\displaystyle\dot{\hat{\omega}}$
$\displaystyle=-K_{I}\gamma V\sin(\delta).$

By setting the left-hand side to zero, it is immediate to see that in such
coordinates the system has infinitely many equilibria
$\mathcal{E}_{\star}:=\cup_{n}(n\pi,\omega)$, with $n\in\mathbb{Z}$, and that
these are mapped to the elements of $\mathcal{E}_{dq\star}$. More precisely,
let us define

$\displaystyle\mathcal{E}_{\star}^{+}$
$\displaystyle:=\cup_{h}(h\pi,\omega),\quad h\in 2\mathbb{Z}$
$\displaystyle\mathcal{E}_{\star}^{-}$
$\displaystyle:=\cup_{k}(k\pi,\omega),\quad k\in 2\mathbb{Z}+1.$

Therefore, we have:

$\displaystyle\mathcal{E}_{dq\star}^{+}:$
$\displaystyle=(+v_{dq\star},\omega)=(\gamma
V\mathsf{T}(\delta^{h}),\omega),\quad\forall\delta^{h}\in\mathcal{E}_{\star}^{+}$
(III.9) $\displaystyle\mathcal{E}_{dq\star}^{-}:$
$\displaystyle=(-v_{dq\star},\omega)=(\gamma
V\mathsf{T}(\delta^{k}),\omega),\quad\forall\delta^{k}\in\mathcal{E}_{\star}^{-}.$

We are now ready for the following propositions, similar to [12].222The same
result apply, with minor differences, to the gSRF-PLL (III.4).

###### Proposition III.4

Any element of $\mathcal{E}_{\star}^{-}$ is a saddle equilibrium point for the
system (III.8).

###### Proof:

For the proof, it suffices to consider the linearization of the system (III.8)
at equilibria $\delta^{k}\in\mathcal{E}_{\star}^{-}$ and evaluate the sign of
the resulting eigenvalues. ∎

###### Proposition III.5

Any element of $\mathcal{E}_{\star}^{+}$ is a locally asymptotically stable
equilibrium of the system (III.8) and an inner estimate of the corresponding
region of attraction (RoA) $\mathcal{R}_{1}^{h}\subset\mathbb{R}^{2}$, $h\in
2\mathbb{Z}$, is given by

$\hat{\mathcal{R}}_{1}^{h}:=\left\\{(\delta,\hat{\omega})\in\mathbb{R}^{2}:\;\left(1-\cos(\delta-h\pi)\right)+\frac{(\hat{\omega}-\omega)^{2}}{2\gamma
V}<\frac{K_{I}}{\gamma V}\right\\}.$

(III.10)

###### Proof:

Consider the Lyapunov functions

$\mathcal{W}_{1}^{h}(\delta,\hat{\omega}):=\gamma
V(1-\cos(\delta-h\pi))+\frac{1}{2K_{I}}(\hat{\omega}-\omega)^{2},$ (III.11)

which are all positive definite in $[-\pi\;\pi]\times\mathbb{R}$, modulo
$2\pi$, with respect to the corresponding equilibrium point $(h\pi,\omega)$.
By computing its derivative along the system’s trajectories, we get:

$\displaystyle\dot{\mathcal{W}}_{1}^{h}=$ $\displaystyle-
K_{P}\gamma^{2}V^{2}\sin^{2}(\delta-h\pi)+\gamma
V\sin(\delta-h\pi)(\hat{\omega}-\omega)$ (III.12)
$\displaystyle+(\hat{\omega}-\omega)(-\gamma V\sin(\delta-h\pi))$
$\displaystyle=$ $\displaystyle-K_{P}\gamma^{2}V^{2}\sin^{2}(\delta-h\pi)$
$\displaystyle\leq$ $\displaystyle 0,$

which suffices to prove boundedness of the trajectories. Local asymptotic
stability can be then inferred using LaSalle invariance’s principle since
$\sin^{2}(\delta)=0$ implies $(\delta,\hat{\omega})=(h\pi,\omega)$. An
estimation of the RoA can be then computed taking the largest, compact
sublevel set of $\mathcal{W}_{1}^{h}$. ∎

Recalling (III.9), from Corollary III.2 and Proposition III.4 we can then
conclude that for any initial condition belonging to the set
$\\{\mathcal{V}\times\mathbb{R}\\}/\mathcal{E}_{dq\star}^{-}$ the solutions of
(II.11)-(III.7) converge to $\mathcal{E}_{dq\star}^{+}$. Or, equivalently,
that for any initial condition belonging to the set
$\mathbb{R}^{2}/\mathcal{E}_{\star}^{-}$, the solutions of (III.8) converge to
a point of $\mathcal{E}_{\star}^{+}$. Hence, we say that for almost any
initial conditions a correct estimation of the actual value of the angle and
of the frequency is guaranteed. Equilibria and convergence properties of the
incremental, scaled version of the SRF-PLL are illustrated in Fig. 1 and Fig.
2, for different initial guesses and both in polar and $dq$ coordinates.

Figure 1: Solutions of the SRF-PLL dynamics in polar coordinates (III.8), for
different initial guesses (square points) of the estimator. White and red
circles denote respectively equilibria of $\mathcal{E}_{\star}^{+}$ and
$\mathcal{E}_{\star}^{-}$, with blue lines denoting lower estimations of the
RoAs of $\mathcal{E}_{\star}^{+}$. Figure 2: Solutions of the SRF-PLL dynamics
in $dq$ coordinates (II.11)-(III.7), for different initial guesses (square
points) of the estimator. White and red circles denote respectively the
equilibria $(-v_{dq\star},\omega)$ and $(v_{dq\star},\omega)$.

An important observation is that the less accurate are the initial guesses,
the poorest are the performances of the estimator. In polar coordinates, this
is clearly captured by the fact that for some initial conditions the system’s
trajectories do not converge to the closest equilibrium, but rather exhibit a
transients oscillating behavior before converging to farther equilibria. This
suggests that information about the RoA of the equilibria of
$\mathcal{E}_{\star}^{+}$, given by Proposition III.5, might be helpful to
assess performances of the SRF-PLL, see also Remark IV.7.

##  IV ATAN-PLL

An alternative, widely diffused synchronization scheme for the estimation of
the angle and of the frequency of (II.1) relies on the use of a suitably
defined inverse tangent function. We have first the following preliminary
proposition.

###### Proposition IV.1

Consider the system (II.11), let $v_{dq}\in\mathcal{V}$ and define

$\tilde{u}:=\hat{u}-\omega,\quad\tilde{y}_{2}:=\operatorname{atan2}(\hat{V}_{q},\hat{V}_{d})-\operatorname{atan2}(V_{q},V_{d}).$
(IV.1)

Then, the map $\tilde{u}\rightarrow\tilde{y}_{2}$ is passive with storage
function

$\mathcal{H}_{2}(\hat{v}_{dq})=\frac{1}{2}\left[\operatorname{atan2}(\hat{V}_{q},\hat{V}_{d})-\operatorname{atan2}(V_{q},V_{d})\right]^{2}$
(IV.2)

###### Proof:

The proof is similar to the proof of Proposition III.1. Note that
$\mathcal{H}_{2}=\frac{1}{2}\tilde{y}_{2}^{2}$. By calculating its derivative
along the system’s trajectories we get:

$\dot{\mathcal{H}}_{2}=\left(-\tilde{y}_{2}\frac{\hat{V}_{q}}{\gamma^{2}V^{2}}\right)\left(-\hat{V}_{q}\tilde{u}\right)+\left(\tilde{y}_{2}\frac{\hat{V}_{d}}{\gamma^{2}V^{2}}\right)\left(\hat{V}_{d}\tilde{u}\right)=\tilde{y}_{2}\tilde{u},$

(IV.3)

where in the last equivalence we used the fact that
$\hat{v}_{dq}\in\mathcal{V}$, thus completing the proof. ∎

We can now determine a class of estimators, referred as generalized arctangent
(gATAN)-PLL, that guarantee a solution to the synchronization problem,
similarly to Proposition III.2.

###### Proposition IV.2

Let $v_{dq}\in\mathcal{V}$. Consider the system (II.11) in closed-loop with
the PLL:

$\displaystyle\hat{u}$ $\displaystyle=-K_{P}\Phi(\tilde{y}_{2})+\hat{\omega}$
(IV.4) $\displaystyle\dot{\hat{\omega}}$ $\displaystyle=-K_{I}\tilde{y}_{2},$

with state $\hat{\omega}\in\mathbb{R}$, $\tilde{y}_{2}$ given by (IV.1),
$K_{I}$, $K_{P}$ positive scalars and $\Phi$ is a scalar function verifying
$\Phi(0)=0$, $\Phi(s)s>0$ for any $s\neq 0$. Then, the equilibrium point
$(v_{dq},\omega)\in\mathcal{V}\times\mathbb{R}$ is globally asymptotically
stable.

###### Proof:

Consider the energy function

$\mathcal{W}_{2}(\hat{v}_{dq},\hat{\omega})=\mathcal{H}_{2}(\hat{v}_{dq})+\frac{1}{2K_{I}}(\hat{\omega}-\omega)^{2},$
(IV.5)

which is positive definite and radially unbounded. We have then that

$\displaystyle\dot{\mathcal{W}}_{2}=$
$\displaystyle\dot{\mathcal{H}}_{2}+\frac{\hat{\omega}-\omega}{K_{I}}\dot{\hat{\omega}}$
(IV.6) $\displaystyle=$
$\displaystyle\tilde{y}_{2}\left(-K_{P}\Phi(\tilde{y}_{2})+\hat{\omega}-\omega\right)+\frac{\hat{\omega}-\omega}{K_{I}}\left(-K_{I}\tilde{y}_{2}\right)$
$\displaystyle=$ $\displaystyle-K_{P}\Phi(\tilde{y}_{2})\tilde{y}_{2}$
$\displaystyle\leq 0.$

Global asymptotic stability follows using LaSalle invariance principle, noting
that $\tilde{y}_{2}=0$ implies $\hat{v}_{dq}=v_{dq}$. ∎

A specific implementation of the controller (IV.7) for $\Phi=\mathrm{Id}$ and
$v_{dq}=v_{dq\star}\in\mathcal{V}$ is given by the conventional ATAN-PLL [11],
as stated in the following corollary.

###### Corollary IV.3

Consider the system (II.11) in closed-loop with the ATAN-PLL:

$\displaystyle\hat{u}$
$\displaystyle=-K_{P}\operatorname{atan2}(\hat{V}_{q},\hat{V}_{d})+\hat{\omega}$
(IV.7) $\displaystyle\dot{\hat{\omega}}$
$\displaystyle=-K_{I}\operatorname{atan2}(\hat{V}_{q},\hat{V}_{d}),$

with $K_{I}$, $K_{P}$ positive scalars. Then, the equilibrium point
$(v_{dq\star},\omega)\in\mathcal{V}\times\mathbb{R}$ is globally
asymptotically stable.

An immediate consequence of Corollary IV.3 is that for any initial guess of
the estimator both the frequency and angle estimation errors converges to
zero. Similar to the case of the SRF-PLL an accurate estimation of the RoA of
the equilibria can be used to assess performances of the estimator. For this
purpose, it is convenient to transform the system in polar coordinates.
Recalling (II.9), (II.6)-(IV.7) can be rewritten as

$\displaystyle\dot{\delta}$ $\displaystyle=-K_{P}\operatorname{atan2}(\gamma
V\sin(\delta),\gamma V\cos(\delta))+\hat{\omega}-\omega$ (IV.8)
$\displaystyle\dot{\hat{\omega}}$
$\displaystyle=-K_{I}\operatorname{atan2}(\gamma V\sin(\delta),\gamma
V\cos(\delta)).$

We are now ready for the following proposition.

###### Proposition IV.4

The system (IV.8) admits infinitely many locally asymptotically stable
equilibria $(h\pi,\omega)$, with $h\in 2\mathbb{Z}$, and an inner estimate of
the corresponding RoA $\mathcal{R}_{2}^{h}\subset\mathbb{R}^{2}$ is given by

$\hat{\mathcal{R}}_{2}^{h}:=\left\\{(\delta,\hat{\omega})\in\mathbb{R}^{2}:\;4(\delta-h\pi)^{2}+(\hat{\omega}-\omega)^{2}<4\pi^{2}K_{I}\right\\}.$
(IV.9)

###### Proof:

The equilibria of (IV.8) are the solutions of

$0=\hat{\omega}-\omega,\quad 0=\operatorname{atan2}(\gamma
V\sin(\delta),\gamma V\cos(\delta)),$ (IV.10)

from which follows that there exist infinitely many equilibria
$(h\pi,\omega)$, with $h\in 2\mathbb{Z}$. Note now that for any interval
$[-\pi\;\pi)$ modulo $2\pi$ we further have

$\tilde{y}_{2}=\operatorname{atan2}(\gamma V\sin(\delta),\gamma
V\cos(\delta))=\delta-h\pi.$

Define the energy functions

$\mathcal{W}_{2}^{h}(\delta,\hat{\omega})=\frac{1}{2}(\delta-h\pi)^{2}+\frac{1}{2K_{I}}(\hat{\omega}-\omega)^{2},$
(IV.11)

which are all locally positive definite with respect to the corresponding
equilibrium point $(h\pi,\omega)$. We have then that

$\displaystyle\dot{\mathcal{W}}_{2}^{h}$
$\displaystyle=(\delta-h\pi)\left(-K_{P}\phi(\tilde{y}_{2})+\hat{\omega}-\omega\right)-(\hat{\omega}-\omega)(\delta-h\pi)$
(IV.12) $\displaystyle=-K_{P}\Phi(\tilde{y}_{2})\tilde{y}_{2}$
$\displaystyle\leq 0.$

Local asymptotic stability follows using LaSalle invariance principle, noting
that $\tilde{y}_{2}=0$ implies $\delta=h\pi$, $\hat{\omega}=\omega.$ An
estimation of the RoA can be then computed taking the largest, compact
sublevel set of $\mathcal{W}_{2}^{h}$. ∎

Figure 3: Solutions of the ATAN-PLL dynamics in polar coordinates (IV.8), for
different initial guesses (square points) of the estimator. White circles
denote the system equilibria, while blue lines denote lower estimations of the
corresponding RoAs.

Equilibria and convergence properties of the incremental, scaled version of
the ATAN-PLL are illustrated in Fig. 3.

###### Remark IV.5

When restricted to the interval $[-\pi\;\pi)$, the ATAN-PLL estimation error
dynamics (IV.8) reduce to a linear system. This should be contrasted with the
estimation error dynamics (III.8), which are nonlinear on the same interval.

###### Remark IV.6

Note that convergence of (III.4) or (IV.4) implies that both the signal
$\hat{u}(t)$ and $\hat{\omega}(t)$ converge to the actual frequency $\omega$
and thus can be both interpreted as an estimate of such a signal.

###### Remark IV.7

By inspection of $\hat{\mathcal{R}}_{1}^{h}$ and $\hat{\mathcal{R}}_{2}^{h}$,
it is immediate to see that larger values of the integral gain $K_{I}$
determine a larger RoA (or, at least, of its estimate) for both
synchronization schemes. One should expect this to have also a relevant impact
on the performances of the estimator, whose time-constant is defined indeed by
$\tau:=1/K_{I}.$

##  V Robustness to frequency variations

In the previous section we developed a stability analysis of the SRF- and
ATAN-PLL schemes under the assumption that the voltage at the PCC can be
represented by a three-phase balanced, purely sinusoidal voltage source with a
constant frequency. In this section we lift this last assumption, by extending
such results to the case where the frequency is described via the following
dynamics

$\displaystyle\dot{\omega}(t)=\eta(t),$ (V.1)

where $\eta:\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}$ is a function
representing the rate of change of frequency (RoCoF). We now borrow a standard
theorem from [13] to prove that, under the assumption that $\eta(t)$ is
bounded, i.e. $|\eta(t)|<\eta_{\mathrm{max}}$, the trajectories of the ATAN-
PLL ultimately converge to a set that is shaped by the PLL gains (similar
conclusions can be drawn for the SRF-PLL, a result that was obtained also in
[9] using different arguments). To ease notation, let us define
$\tilde{\omega}:=\hat{\omega}-\omega$. From (IV.8) and (V.1) we obtain:

$\displaystyle\dot{\delta}$ $\displaystyle=-K_{P}\operatorname{atan2}(\gamma
V\sin(\delta),\gamma V\cos(\delta))+\tilde{\omega}$ (V.2)
$\displaystyle\dot{\tilde{\omega}}$
$\displaystyle=-K_{I}\operatorname{atan2}(\gamma V\sin(\delta),\gamma
V\cos(\delta))-\eta.$

We have then the following proposition.

###### Proposition V.1

For any initial condition
$(\delta_{0},\tilde{\omega}_{0})\in\mathcal{R}_{2}^{0}$ the trajectories of
the system (V.2) are ultimately bounded, with ultimate bound given by

$\displaystyle\min_{\begin{subarray}{c}\epsilon>0\end{subarray}}\frac{\sqrt{(\epsilon^{2}+K_{I})\lambda_{\mathrm{max}}(P_{\epsilon})}}{K_{I}\lambda_{\mathrm{min}}(P_{\epsilon})\lambda_{\mathrm{min}}(Q_{\epsilon})}\eta_{\mathrm{max}}\quad\mathrm{s.t.}\;P_{\epsilon}>0,Q_{\epsilon}>0,$
(V.3)

with

$\displaystyle P_{\epsilon}:=\begin{bmatrix}1&-\frac{\epsilon}{2}\\\
-\frac{\epsilon}{2}&K_{I}\end{bmatrix},\;Q_{\epsilon}:=\begin{bmatrix}K_{P}-\epsilon
K_{I}&-\frac{\epsilon}{2}K_{P}\\\
-\frac{\epsilon}{2}K_{P}&\epsilon\end{bmatrix}.$ (V.4)

###### Proof:

Consider the energy function

$\mathcal{W}_{2\epsilon}(\delta,\tilde{\omega}):=\frac{1}{2}\delta^{2}+\frac{1}{2K_{I}}\tilde{\omega}^{2}-\epsilon\delta\frac{\tilde{\omega}}{K_{I}},$
(V.5)

defined in $(-\pi\;\pi)\times\mathbb{R}$, which is positive being
$P_{\epsilon}>0$. By calculating the derivative of $\mathcal{W}_{2\epsilon}$
along the system’s trajectories we get:

$\displaystyle\dot{\mathcal{W}}_{2\epsilon}=$
$\displaystyle\delta\left(-K_{P}\delta+\hat{\omega}\right)+\frac{\hat{\omega}}{K_{I}}\left(-K_{I}\delta-\eta\right)+$
(V.6)
$\displaystyle-\epsilon\frac{\tilde{\omega}}{K_{I}}\left(-K_{P}\delta+\hat{\omega}\right)-\frac{\epsilon}{K_{I}}\delta\left(-K_{I}\delta-\eta\right)$
$\displaystyle=$ $\displaystyle-\begin{bmatrix}\delta\\\
\frac{\tilde{\omega}}{K_{I}}\end{bmatrix}^{\top}\begin{bmatrix}K_{P}-\epsilon
K_{I}&-\frac{\epsilon}{2}K_{P}\\\
-\frac{\epsilon}{2}K_{P}&\epsilon\end{bmatrix}\begin{bmatrix}\delta\\\
\frac{\tilde{\omega}}{K_{I}}\end{bmatrix}+$
$\displaystyle+\begin{bmatrix}\delta\\\
\frac{\tilde{\omega}}{K_{I}}\end{bmatrix}^{\top}\begin{bmatrix}\frac{\epsilon}{K_{I}}\\\
-1\end{bmatrix}\eta(t)$

where in the first equivalence we substituted the estimation error dynamics,
while the last one follows from (V.4). Using Schur’s complement it is then
possible to show that there exists always an $\epsilon>0$ such that
$P_{\epsilon}$ and $Q_{\epsilon}$ are positive definite. Hence $Q_{\epsilon}$
can be made positive so that the quadratic term eventually dominates the
linear one, while preserving positive definiteness of
$\mathcal{W}_{2\epsilon}$. More precisely, using standard norm inequalities
and recalling that $|\eta(t)|<\eta_{\mathrm{max}}$ we get:

$\dot{\mathcal{W}}_{\epsilon}<-\lambda_{\mathrm{min}}(Q_{\epsilon})\|x\|^{2}+\frac{\sqrt{\epsilon^{2}+K_{I}^{2}}}{K_{I}}\eta_{\mathrm{max}}\|x\|,$
(V.7)

where $x:=(\delta,\tilde{\omega}/K_{I})\in\mathbb{R}^{2}$, from which follows

$\dot{\mathcal{W}}_{\epsilon}<0\quad\forall\|x\|>\frac{\sqrt{\epsilon^{2}+K_{I}^{2}}}{\lambda_{\mathrm{min}}(Q)K_{I}}\eta_{\mathrm{max}}.$
(V.8)

The ultimate bound can be then obtained using Theorem 4.18 in [13]. ∎

##  VI Simulations

To validate our result via simulations we consider a controlled VSC interfaced
to a sufficiently strong, high-voltage AC grid, in accordance with Assumption
II.2. A conventional VSC current control scheme that guarantees fast
regulation of active and reactive power under limited voltage variations is
further considered, to comply with Assumption II.1. Physical parameters and
values of the reference signals are given in Table I.

TABLE I: Physical parameters and nominal operating points. $L$ | $65.20\;\textrm{mH}$ | $L_{g}$ | $32.60\;\textrm{mH}$ | $P_{\mathrm{nom}}$ | $\phantom{00}1\;\textrm{GW}$
---|---|---|---|---|---
$R$ | $\phantom{0}1.02\;\Omega$ | $R_{g}$ | $\phantom{0}0.20\;\Omega$ | $V_{\mathrm{nom}}$ | $320\;\textrm{kV}$

Based on this setup, we first illustrate performances and stability properties
of both traditional and generalized ATAN-PLL under the assumption that the
grid has relatively high inertia.333Similar results, not reported here for
sake of brevity, apply to the (g)SRF-PLL. More precisely, we consider step
variations in the VSC power references over a time span of $0.5\;s$, inducing
any $T=0.1\;s$ almost instantaneous variation of the phase, with the frequency
remaining constant. In Fig. 4 we show active power responses resulting from
synchronization of the VSC via two differently tuned ATAN-PLLs (IV.7)—referred
as ATAN-PLL1 and ATAN-PLL10—and a gATAN-PLL (IV.4). For all schemes we set the
integral gain to $K_{I}=10^{3}$, which guarantees that the estimated RoA given
by (IV.9) is sufficiently large for any of the operating points of interest.
For the ATAN-PLLs we select the proportional gains $K_{P1}=2\cdot 10^{2}$ and
$K_{P10}=10K_{P1}$. For the gATAN-PLL, we select $\Phi$ as an appropriate
piecewise proportional function, allowing to adapt the proportional gain as a
function of the output $\tilde{y}_{2}$. It is shown that all schemes guarantee
a stable behavior and are able to ensure both the required power flow and a
correct estimation of the frequency. It can be further seen that a suitable
design of adaptive gains via the gATAN-PLL can improve the PLL performances.

Figure 4: Active power and frequency responses under ATAN-PLL1, ATAN-PLL10 and
gATAN-PLL, for a grid with relatively high-inertia.

As a second scenario, we assume that the grid is characterized by a low
inertia and that an inadvertent change in the power demand generates a
deviation of the frequency from its nominal value. More precisely, similar to
[9], we suppose that the frequency is constant and equal to $f:=50\;Hz$ during
the time interval from $0$ to $1\;s$ and immediately afterwards the following
additive disturbance is considered:

$\Delta\omega(t):=-8\pi e^{-0.1(t-1)}\sin(0.2(t-1)).$ (VI.1)

Frequency estimates under SRF- and ATAN-PLL are illustrated in Fig. 5, where
it is shown that they have an almost identical response, which converges to
the actual value of the frequency after approximately $7\;s$, in accordance
with the result of Proposition V.1.

Figure 5: Frequency responses under SRF-PLL and ATAN-PLL under bounded
variation of the frequency.

##  VII Conclusions

In this paper we constructively derived, using passivity arguments, two
classes of PLL synchronization mechanisms for grid-connected VSC that ensure a
correct estimation of the angle and of the frequency at the PCC. It is shown
that (almost) global synchronization can be obtained for both classes of PLLs
using representations of the system in polar and $dq$ coordinates, and that
the conventional SRF- and ATAN-PLL correspond to particular instances of such
generalized synchronization schemes. Finally, we have shown that these schemes
are robust to variations of the frequency—a scenario particularly relevant for
VSCs connected to grids with low inertia. Future investigation will be focused
on the extension of such result to the case of weak grids, i.e. grid
characterized by a low SCR, by lifting Assumption II.1 and Assumption II.2.

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