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Delay master stability of inertial oscillator networks

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Time lags occur in a vast range of real-world dynamical systems due to finite reaction times or propagation speeds. Here we derive an analytical approach to determine the asymptotic stability of synchronous states in networks of coupled inertial oscillators with constant delay. Building on the master stability formalism, our technique provides necessary and sufficient delay master stability conditions. We apply it to two classes of potential future power grids, where processing delays in control dynamics will likely pose a challenge as renewable energies proliferate. Distinguishing between phase and frequency delay, our method offers an insight into how bifurcation points depend on the network topology of these system designs.
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Delay master stability of inertial oscillator networks
Reyk B¨orner,1, 2 Paul Schultz,1Benjamin ¨
Unzelmann,2, 1 Deli Wang,3Frank Hellmann,1, and J¨urgen Kurths1, 4
1Potsdam Institute for Climate Impact Research (PIK),
Member of the Leibniz Association, P.O. Box 60 12 03, D-14412 Potsdam, Germany
2Department of Physics, Freie Universit¨at Berlin, Arnimallee 14, 14195 Berlin, Germany
3School of Science, Xi’an University of Architecture and Technology, Xi’an 710055, China
4Department of Physics, Humboldt University of Berlin, Newtonstr. 15, 12489 Berlin, Germany
(Dated: November 25, 2019)
Time lags occur in a vast range of real-world dynamical systems due to finite reaction times or
propagation speeds. Here we derive an analytical approach to determine the asymptotic stability of
synchronous states in networks of coupled inertial oscillators with constant delay. Building on the
master stability formalism, our technique provides necessary and sufficient delay master stability
conditions. We apply it to two classes of potential future power grids, where processing delays
in control dynamics will likely pose a challenge as renewable energies proliferate. Distinguishing
between phase and frequency delay, our method offers an insight into how bifurcation points depend
on the network topology of these system designs.
Introduction. The study of nonlinear dynamics on
complex networks has received full interdisciplinary at-
tention in past years due to its potential for modeling the
complexity of real-world dynamical systems. An intrin-
sic feature of such systems is that their time evolution
generally depends on past states. Time delays, caused
by finite propagation speeds or processing times, induce
retarded reactions of variables to changes in the system.
For example, delays occur in lasers because of the finite
speed of light [1, 2]; population dynamics depend on mat-
uration and gestation times [3], and the exchange of in-
formation between neurons requires time for both signal
transmission as well as processing [4].
Mathematically, continuous delay problems are de-
scribed by delay differential equations (DDEs) [5, 6].
From their analysis it is known that delays can substan-
tially alter a system’s asymptotic behavior [7]. However,
asymptotic stability analysis of DDEs is challenging be-
cause the corresponding spectrum contains an infinite
number of complex roots. In fact, exact conditions for
stability pose an open problem in research, especially re-
garding networks. Most previous studies have been lim-
ited to numerical investigations of characteristic equa-
tions or restricted to simple network topologies, often
yielding only sufficient stability criteria.
Recent work has led to a thorough analytical under-
standing of the spectrum in the limit of large delay, with
applications in e.g. optoelectronics [8–11]. In many
cases, however, time lags may match the system’s dy-
namical timescales and may play a critical role for sta-
bility. Particularly in systems of coupled oscillators like
the paradigmatic Kuramoto model [12], delays often be-
come comparable to the oscillation period. There, the
asymptotic stability of a synchronous regime is a central
property with crucial implications for applications.
Pecora and Carroll have developed a powerful method
known as the master stability formalism to determine the
Email: hellmann@pik-potsdam.de
stability of synchronization for identical oscillators with-
out delay [13]. The main idea is to project the state vec-
tor into the eigenspace of the coupling matrix, yielding
a block diagonal form that defines the associated master
stability function (MSF). This way, dynamical parame-
ters of the system are separated from topological infor-
mation about the network.
Several studies have calculated MSFs for specific mod-
els with time-delayed couplings [14–17]. Here, for the
first time, we generalize the formalism to DDE inertial
oscillator models containing an arbitrary constant dis-
crete delay τ > 0 that may appear in the local dynamics
as well as in a diffusive coupling term. While the mas-
ter stability formalism requires complete synchronization
of oscillators, we merely assume phase synchronization
where oscillators may have constant phase differences
[18]. Our analytic approach leads to a decomposition
into second-order DDEs in terms of the eigenvalues of the
graph Laplacian matrix. Based on results from Bhatt and
Hsu [19], we derive necessary and sufficient conditions for
the asymptotic stability of synchronized inertial oscilla-
tor networks with delay. The corresponding delay master
stability function (dMSF) is given in terms of the graph
Laplacian spectrum, the delay τ, as well as dynamical
parameters of the model.
For delays caused by processing times, our results offer
a complete analytic solution to the question of asymp-
totic stability. The dMSF comprises a finite number of
easily evaluated critical conditions that hold for any net-
work topology. Particularly, in an important case which
covers our central application of renewable inverter-based
power grids, the conditions further simplify to a sin-
gle stability criterion involving just the maximum graph
Laplacian eigenvalue.
Main application. We begin with an inverter-based
power grid model to exemplify how we obtain a concise
condition for a major application of oscillator networks.
Due to the energy transition, power grids currently
undergo substantial structural and dynamical changes,
threatening stable synchronization of the AC voltage fre-
arXiv:1911.09730v1 [math.DS] 21 Nov 2019
2
quency [20, 21]. Characterized by a large share of volatile
distributed generation units, e.g. solar or wind power
plants, future energy networks will require novel control
approaches like grid-forming power inverters to maintain
stability [22, 23]. As this involves measurements and pro-
cessing, delays are expected to play a critical role [24–26].
Understanding their influence on stability is thus vital to
ensure security of supply and prevent blackouts.
Specifically, we consider frequency dynamics in a
droop-controlled inverter grid [27] where the steady-state
power flow between two nodes depends on the sine of
their phase difference,
¨ϕi=˜α˙ϕi+˜
βPd
i
N
X
j=1
Kij sin ϕτ
ji .(1)
Here ϕi(t) denotes the phase angle of the i-th inverter
(oscillator) and ∆ϕτ
ji := ϕi(tτ)ϕj(tτ). ˜α > 0
and ˜
β > 0 are the inertia-specific damping and droop
constants, respectively; Pd
irepresents the desired active
power set points. Elements of the weighted adjacency
matrix (Kij ) may be interpreted as the maximally trans-
mittable power values along transmission lines in the net-
work [28] (details in SI [29]).
We find that a synchronous state of Eq. (1) is asymp-
totically stable if and only if
λN<1
˜
βry1
τ4+ ˜α2y1
τ2, y1= ˜ατ cot y1,(2)
where y1(0, π]. This exact stability condition depends
on network structure only via the largest eigenvalue λN
of the effective Laplacian matrix L(see below). Notably,
it suffices to compute precisely one unique characteristic
root y1of the linearized spectrum associated with Eq.
(1). We discuss this result further after deriving the gen-
eral approach.
Derivation. We consider a nonlinear dynamical sys-
tem of Ncoupled oscillators on a network. All oscilla-
tors (nodes) have inertia, obeying a Newtonian law of
motion. The state of the i-th oscillator at time tis given
by the phase angle ϕi(t) and angular frequency devia-
tion ωi(t)˙ϕi(t) in a reference frame co-rotating with
a coherent frequency Ω. Let the time evolution of the
global state be governed by a set of second-order DDEs
containing a discrete, constant delay τ > 0,
¨ϕi=fi(ϕi,˙ϕi) + fτ(ϕτ
i,˙ϕτ
i)
+
N
X
j=1
Aij g(∆ϕij ,∆ ˙ϕij) + gτ(∆ϕτ
ij ,∆ ˙ϕτ
ij ),(3)
for i∈ {1, . . . , N }. Here time arguments are abbrevi-
ated as ϕiϕi(t) and ϕτ
iϕi(tτ); furthermore
ϕij ϕjϕiand ∆ϕτ
ij ϕτ
jϕτ
i. The real scalar
functions fiand fτrepresent undelayed and delayed iso-
lated dynamics, respectively. Unlike identical oscillators,
fimay differ from node to node by an additional con-
stant ciR, which accounts for heterogeneous driving
forces. In the interaction term, gdenotes undelayed cou-
pling dynamics, whereas gτdescribes interactions with
a coupling processing delay. The strength of the cou-
pling as well as the network topology are stored in the
weighted adjacency matrix ARN×N, with Aij >0 if
nodes iand jare connected and 0 otherwise. Here, we
consider undirected graphs without self-loops.
The delay considered in Eq. (3) is a processing delay
which arises, for example, in engineered systems with
feedback control due to measurement and processing
times. Contrarily, transmission or communication de-
lays in diffusive coupling of the form gτ(xτ
jxi) require
separate treatment (see SI).
To assess asymptotic stability, we linearize our DDE
model near the phase synchronization manifold Z, de-
fined by Z:= {(ϕi, ωi)R2:ωi= ˙ωi= 0 i}. Phys-
ically, this means that all oscillators are entrained to a
coherent frequency Ω but possibly with fixed phase dif-
ferences between them. A synchronous solution ϕ=
(ϕ
1, . . . , ϕ
N) with ω= (0,...,0) lies on Zand corre-
sponds to a fixed point of Eq. (3).
Traditional MSFs require complete synchronization,
i.e. all oscillators move in phase with frequency Ω.
Then, Jacobians evaluated on the synchronization man-
ifold are identical for all nodes [13]. To achieve a block
decomposition similar to MSF for phase synchroniza-
tion, we assume: 1) The Jacobian matrices of all local
functions fiand fτ, evaluated on Z, are identical. 2)
The Jacobians of all coupling functions gand gτ, evalu-
ated on Z, are edge-independent except for a pre-factor
wij (∆ϕ
ij )=wji (∆ϕ
ji )Rwhich may depend on the
fixed point ϕ. We note that the following procedure also
holds for more general coupling functions g(ϕi,˙ϕi, ϕj,˙ϕj)
and gτ(ϕτ
i,˙ϕτ
i, ϕτ
j,˙ϕτ
j) if their first partial derivatives are
antisymmetric with respect to the exchange of iand j.
Nonetheless, we present the widely applied diffusive form
here and refer to the SI for more information.
We define the effective Laplacian matrix Lof the
linearized network model such that Lij := wijAij +
δij Pjwij Aij . This matrix is symmetric, positive-
semidefinite, and consequently diagonalizable. In the
spirit of MSF, we now transform coordinates into the
space spanned by the eigenvectors of L, with correspond-
ing eigenvalues λk,k∈ {1, . . . , N }. Diagonalization does
not affect the Jacobians (which are node-independent by
assumption after absorbing wij in the adjacency matrix),
such that the system of DDEs decomposes into Nblocks
given in terms of λk,
¨
θk=ak(λk)˙
θkbk(λk)θkaτ
k(λk)˙
θτ
kbτ
k(λk)θτ
k.
(4)
Here the set of θkdenotes (small) phase angle deviations
from ϕexpressed in the transformed coordinates. The
coefficients are given by elements of the Jacobian matri-
ces (see SI); in the following we suppress their dependence
on λk.
The stability of a synchronous state ϕdepends on
the real parts of the roots of the characteristic equation
3
associated with Eq. (4),
H(z) := (z2+akτz +bkτ2)ez+aτ
kτz +bτ
kτ2= 0 .(5)
The exponential polynomial Hfeatures an infinite num-
ber of complex roots; all must have negative real parts
for asymptotic stability. We now assume that the delay
τappears either in the time argument of the phases ϕi
or of the frequencies ωibut not in both. For these cases,
Bhatt and Hsu [19] derive necessary and sufficient stabil-
ity conditions for scalar second-order DDEs, determined
by a finite number of decisive roots within the infinite
spectrum of H(see also [30, 31]). After the block decom-
position outlined above, we may transfer these conditions
to inertial oscillator networks to obtain delay master sta-
bility conditions.
If only the coupling is delayed (fτ= 0), the longitu-
dinal eigenvalue λ1= 0 describes dynamics within the
synchronization manifold and asymptotic stability is de-
termined by the N1 transversal directions [13]. Con-
trarily, if fτ6= 0, all kmust be considered for stabil-
ity analysis. We define the transversal set N, which is
{2, . . . , N }for fτ= 0 and {1, . . . , N }otherwise.
Substituting z=iy in Eq. (5), Hseparates into a real
and an imaginary part. First, we consider the case of
phase delay (aτ
k= 0). Let ak>0 and ak< bkτ0.
Then, for each k, there exists one decisive root y1,k
(0, π] which solves the imaginary part of Eq. (5) [19]. A
synchronous fixed point of Eq. (3) with phase delay is
asymptotically stable if and only if, for all k∈ N,
bk< bτ
k<Rk(y1,k)
τ2,(6)
with Rk(y) := p(y2bkτ2)2+ (akτy)2.
In the frequency delay case (bτ
k= 0), we assume ak>0
and bk>0. Here we examine positive solutions ykof the
real part of Eq. (5). The first positive root yk,0lies in the
interval (0, π/2), and one root yk,m is situated in each π-
interval (π/2, mπ +π/2) for m= 1,2, . . . . Of these
roots, the decisive roots y
kand y∗∗
kare found according to
y
k= min
moddyk,m τbkand min
mevenyk,m τbk. Then,
it is necessary and sufficient for asymptotic stability of a
synchronous state that, for all k∈ N,
Rk(y∗∗
k)
y∗∗
k
< aτ
kτ < Rk(y
k)
y
k
,(7)
where Rk(y) is defined beneath Eq. (6). Though finding
decisive roots appears more complicated than in the pre-
vious case, it turns out that we must find at maximum
2Nroots in total, within known intervals. Details are
provided in the SI.
Phase delay. Our motivating example introduced in
Eq. (1) illustrates the case of phase delay. Here the
root y1,k is k-independent and we must only consider the
largest eigenvalue λNof L. We may write the resulting
stability condition (Eq. (2)) as a dMSF σ,
σ(λN, τ ) = λN1
˜
βry1
τ4+ ˜α2y1
τ2.(8)
A combination of λNand τis stable if and only if σ < 0.
Monotonicity arguments for y1(τ) prove the existence of
precisely one critical delay τc=τcα, ˜
β, λN). A syn-
chronous state that is stable without delay will remain
asymptotically stable for all τ < τcand is unstable for
all ττc.
To visualize σ, we first choose a star topology as in
Ref. [32]. A producer in the center with steady-state
power production P+= 3P0is connected to three con-
sumers with P=P0via transmission lines of equal
capacity K0(Fig. 1a). Due to the symmetry of the
configuration, we obtain three distinct Laplacian eigen-
values. The phase delay case depends on λNonly; thus
we get a single curve with σ= 0 at τc(Fig. 1b). For
typical parameter values, τc40 ms is about twice the
50 Hz oscillation period.
Frequency delay. Stability in the presence of a fre-
quency delay is qualitatively different. We show this
by applying our approach to the decentral smart grid
control (DSGC) scheme [32, 33]. The model incorpo-
rates electricity price dynamics by locally relating the
price to the current grid frequency, motivating produc-
ers/consumers to adapt their feed-in/consumption to the
currently available power supply. The continuous mea-
surements required for this smart grid regulation induce
a local processing delay at each node. The model reads
K0
3
-1 -1
-1
FIG. 1. (Color online). Delay master stability functions for a
four-node star topology (a) with three consumers connected
to one producer. (b) dMSF as a function of delay τfor the
inverter model with phase delay. The system is asymptoti-
cally stable for all τ < τc45 ms (grey area). (c) dMSF
for the DSGC model with frequency delay. Each Laplacian
eigenvalue λkcontributes a curve σ(λk, τ ); the system is sta-
ble in regimes where σmax <0 (grey areas). Parameter values
are P0= 1 s2,K0= 8 s2, ˜α=α= 0.1 s1,˜
β= 0.07, and
γ= 0.25 s1.
4
FIG. 2. (Color online). Critical delay τcon Watts-Strogatz
networks. (a) shows a network example for N= 100 nodes,
degree k= 4, and rewiring proabability p= 0.5. Con-
sumers/producers with equal power input/output, connected
via lines of equal admittance, are placed alternately on the
original ring. Plots (b-d) illustrate how τcdepends on p, k, N
for the inverter model (phase delay, green circles) and for the
DSGC model (blue triangles). Parameter values not explic-
itly given are N= 100, k= 4, p= 0.5; model parameters are
as in Fig. 1. Vertical axes are normalized for each series; data
points are averaged over 10 realizations.
[32]
¨ϕi=Piα˙ϕiγ˙ϕτ
i+X
j
Kij sin(ϕjϕi),(9)
where Piis the produced/consumed power at node iand
K= (Kij ) denotes the weighted adjacency matrix as
before. Next to the damping α > 0, the price elasticity
γ > 0 acts as a second, delayed damping term.
For the DSGC model we obtain Ndelay master sta-
bility conditions
0< γτ < 1
y
kq(y
k
2λkτ2)2+ (ατy
k)2.(10)
Here it is not a priori identifiable which root y
k=
y
k(λk, τ ) determines stability for a given τ; we must
regard all Laplacian eigenvalues λk. The root y∗∗
kis
not relevant because γτ > 0. Analogous to the pre-
vious case, we may formulate Eq. (10) as a dMSF
σ(λk, τ ). Then, the system is stable in all regions where
σmax(τ) := max
λk
σ(λk, τ )<0.
Calculating σmax for the star topology (Fig. 1a), we
now have contributions from all three distinct eigenval-
ues λkas depicted in Fig. 1c. In addition to a stable
regime beginning at τ= 0, there exist further windows
of stability for larger delays, corroborating prior results
based on numerical analysis [32].
Networks. For frequency delays, we conjecture that
the length of the first stability window, extending from
τ= 0 to a critical delay τc, is determined by λN. In
the limit of an infinite, heterogenous graph, its Laplacian
spectrum may be expected to become quasi-continuous,
such that further stability windows vanish. Thus, in both
frequency and phase delay, the maximum eigenvalue of
the effective graph Laplacian plays a crucial role. Several
bounds and estimates in terms of network characteristics
have been published for the largest Laplacian eigenvalue
of a weighted graph, e.g. [34]. Therefore, our method
may provide insight even without explicitly diagonalizing
L, which could be practical particularly for large systems.
Finally, we explore how the critical delay τc(end of first
stability window) depends on the network structure in
both models. As a versatile example, we generate Watts-
Strogatz networks [35] with different rewiring probabili-
ties p, mean degrees k, and number of nodes N(e.g. Fig.
2a). Consumers and producers are placed alternately on
the original ring graph. The results for varying p, k, and
N(Figs. 2b-d) show that the critical delay decreases
with an increasing number of nodes and edges as well as
randomness of the system, which emphasizes the signif-
icance of delays for the design and control of real-world
dynamical systems. However, the decline is smaller for
the DSGC model compared to the droop-controlled in-
verter model.
Conclusion. In this Letter, we present an analytical
approach to assess the asymptotic stability of synchro-
nized inertial oscillator networks with delayed dynamics.
Specifically, we consider processing delays either in the
phase or in the frequency. We show how to extend the
master stability formalism to an arbitrary lag time, ob-
taining dMSFs in terms of the delay τand eigenvalues of
the effective graph Laplacian L. Unlike MSF, we more
generally consider phase synchronization and allow for
constant inhomogeneities in the local dynamics of other-
wise identical oscillators. A block decomposition of the
linearized model yields necessary and sufficient stability
conditions which deliver an analytic expression for the
dependence on network structure. These criteria involve
maximally 2Ndecisive roots. In contrast, previous nu-
merical asymptotic stability analyses rely on randomly
computing a significantly larger number of characteristic
roots without being certain that all decisive roots have
been found. Illustrating our approach, we consider two
concrete models for renewable power grids as our main
applications. Notably, we are able to boil down the prob-
lem of stability to a single condition in the case of the
droop-controlled inverter model. We therefore believe
that our method could contribute to the development of
design criteria for future energy systems. Generally, our
results advance the stability analysis of dynamical sys-
tems on complex oscillator networks complying with Eq.
(3).
5
Due to its increasing importance in real-world appli-
cations, delay stability in complex systems remains an
important topic with many open challenges. Our work
opens a new analytic approach on this subject. From
the perspective of power grids, it is crucial to also tackle
non-identical oscillators, to consider multiple delays, and
combined phase and frequency delays. In the wider con-
text of physical systems, it is also highly interesting to
study more general stability criteria for diffusive coupling
with transmission delays.
The authors acknowledge the support of BMBF, Con-
dynet2 FK. 03EK3055A. This work was funded by the
Deutsche Forschungsgemeinschaft (DFG, German Re-
search Foundation) – KU 837/39-1 / RA 516/13-1. All
authors gratefully acknowledge the European Regional
Development Fund (ERDF), the German Federal Min-
istry of Education and Research and the Land Branden-
burg for supporting this project by providing resources
on the high performance computer system at the Pots-
dam Institute for Climate Impact Research.
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6
SUPPLEMENTAL INFORMATION (SI)
I. COEFFICIENTS OF SECOND-ORDER DDE BLOCKS
The coefficients in Eq. (4) of the main text are given by
ak=FωλkGωaτ
k=Fτ
ωλkGτ
ω
bk=FϕλkGϕbτ
k=Fτ
ϕλkGτ
ϕ,(S.1)
where
Fϕ:= ∂fi(ϕi,˙ϕi)
∂ϕiϕ
Gϕ:= 1
wij (ϕ)
∂g(∆ϕij ,∆ ˙ϕij )
(∆ϕij )ϕ
Fω:= ∂fi(ϕi,˙ϕi)
˙ϕiϕ
Gω:= 1
wij (ϕ)
∂g(∆ϕij ,∆ ˙ϕij )
(∆ ˙ϕij )ϕ
.(S.2)
Elements of the delayed Jacobians are written analogously as Fτ
ϕetc.
II. GENERAL DERIVATION
In the Letter, we outline the derivation of our approach based on the inertial oscillator model described by Eq.
(3) of the main text. Here we present a more elaborate variant (discussed in detail in Ref. [36]). This allows us
to illuminate the underlying assumptions of our method and discuss why including communication delays involves a
strong restriction.
Recall that we formulate the oscillators’ dynamics in a reference frame co-rotating with the frequency Ω. A
synchronous state where all oscillators are entrained to frequency Ω corresponds to a fixed point characterized by
ωi= ˙ωi= 0 i.
Instead of the second-order form stated in Eq. (3) of the main text, we may equivalently express our inertial
oscillator network model as a set of first-order DDEs by treating the phase angles ϕi(t) and angular frequency
deviations ωi(t) = ˙ϕi(t) as two independent variables for each node. We thus define the vector xi(ϕi, ωi)>and
write
˙xi=fi(xi) + fτ(xτ
i) +
N
X
j=1
Aij g00(xi, xj) + gτ τ (xτ
i, xτ
j) + g0τ(xi, xτ
j),(S.3)
where we have abbreviated xixi(t) and xτ
ixi(tτ). Here we denote vector-valued functions (R2R2)
by bold letters, while fi, f τ, g, and gτwill remain the scalar functions introduced in the main text; e.g. fi(xi) =
(ωi, fi(ϕi, ωi))>. In contrast to Eq. (3), this form now includes a communication delay via the function g0τ(xi(t), xj(t
τ)). Furthermore, note that the coupling functions g00,gτ τ , and g0τmay depend on xiand xjin an arbitrary fashion.
A communication delay of the type above may describe transmission or propagation lags between nodes. The
intuition is that a change of node iat time tdepends on the history of connected nodes because it takes the time τ
until a signal from a node jreaches node i.
We now linearize Eq. (S.3) around a fixed point x= (x
1, . . . , x
N)>defined by the conditions ˙x
i= (0,0) for all i.
The set of fixed points constitutes the phase synchronization manifold Z,
Z:= {xiZR2:i= 1, . . . , N and ˙xi= (0,0) i}.
Substituting xi=x
i+ηi, with ||ηi|| small for all i, this yields
˙ηiDifiηi+ Difτητ
i(S.4)
+
N
X
j=1
Aij D1
ij g00ηi+ D2
ij g00ηj+D1
ij gτ τ ητ
i+ D2
ij gτ τ ητ
j+D1
ij g0τηi+ D2
ij g0τητ
j ,
where Jacobian matrices, all evaluated at the fixed point, are written in short notation,
Dif:= f(xi)
∂xixi=x
i
,D1/2
ij g:= g(xi, xj)
∂xi/j xi=x
i, xj=x
j
7
and ητ
iηi(tτ). Due to the relation ωi= ˙ϕibetween coordinates of the vector xi, some elements of the Jacobians
are immediately zero or one. Particularly,
Difi=0 1
ϕifiωifi,Difτ=0 0
ϕifτωifτ,(S.5)
where mfdenotes the partial derivative of the function fby the argument m, evaluated at the fixed point. In the
same manner,
D1
ij g00 =0 0
ϕig ∂ωig,D2
ij g00 =0 0
ϕjg ∂ωjg,etc. (S.6)
We emphasize that these Jacobians depend on the fixed point. For the phase synchronization manifold, this implies
that the Jacobians may differ for different iand j.
A. Antisymmetric coupling
Assume now that
1. there is no communication delay (i.e. g0τ= 0) and
2. the linearized coupling between two nodes iand jis antisymmetric, that is, D1
ij g00(x
i, x
j) = D2
ij g00(x
i, x
j)
and D1
ij gτ τ (x
i, x
j) = D2
ij gτ τ (x
i, x
j).
This is fulfilled by the model discussed in the main text (Eq. (3)), where we have diffusive coupling. The antisymmetry
requirement will allow us to write the problem in terms of the effective graph Laplacian matrix L.
Our goal is to decouple local information about the dynamics of single nodes from global terms characterizing the
network as a whole. Mathematically, this is achieved when local 2 ×2 matrices and global N×Nmatrices factorize
into a Kronecker product (symbolized by ).
If we have complete synchronization (i.e. all nodes oscillate with identical frequency and phase angle), the Jacobians
are homogeneous for all i, j, resulting in immediate Kronecker factorization. This is not true for the more general case
of phase synchronization. To achieve a decomposition nonetheless, we impose the following restrictions in analogy to
the main text:
3. The Jacobians of the local functions fand fτ, respectively, evaluated on the phase synchronization manifold,
are identical for all nodes:
F:= D1f= D2f=··· = DNf
Fτ:= D1fτ= D2fτ=··· = DNfτ.
4. The Jacobians of the coupling functions g00 and gττ , respectively, evaluated on the phase synchronization
manifold, are identical for all i, j up to a prefactor wij(x
i, x
j)Rwhich contains all dependencies on the fixed
point. It has the property wij = wji >0;
wij (x
i, x
j)G00 := D2
ij g00(x
i, x
j)
wij (x
i, x
j)Gττ := D2
ij gτ τ (x
i, x
j).
With these assumptions, all dependencies on the fixed point may be absorbed in the effective adjacency matrix A
with entries Aij := wijAij . Since we assume antisymmetric coupling (assumption 2), we may furthermore replace A
by the the effective graph Laplacian matrix Lgiven by Lij := δij PjAij −Aij . Now, the set of linearized DDEs reads
˙ηi=F ηi+Fτητ
i
N
X
j=1 Lij G00ηj+Gτ τ ητ
j,(S.7)
8
or, in vector notation for the entire system, η= (η1, . . . , ηN)>,
˙η= [INF−L⊗G00 ]η+ [INFτ−L⊗Gτ τ ]ητ.(S.8)
Here INis the N-dimensional unit matrix. According to assumption 4, Lis symmetric and therefore diagonalizable.
Switching to a basis Bof eigenvectors via the coordinate transform ξ= [TBI2]η, we diagonalize L=T1
BΛTB
to obtain the diagonal matrix of its eigenvalues, Λ = diag(λ1, . . . , λN). This leads to a block-diagonal form; each
two-dimensional block is given by the equation
˙
ξk=FλkG00ξk+FτλkGτ τ ξτ
k,(S.9)
where k1, . . . , N . (Note that we have switched index from ito kto emphasize that ξrepresents the state vector in
the eigenbasis B.)
Similar to the master stability formalism, we have thus decomposed the problem into blocks which vary only in the
effective Laplacian eigenvalue λk. All eigenvalues are non-negative and λ1= 0 because, according to the properties
of an undirected graph’s Laplacian matrix, Lis positive-semidefinite.
Recalling that the two components of the vector xiare related via ωi= ˙ϕi, we may write Eq. (S.9) in second-order
form. The vector φof all phase angles is given by the linear combination φ=Pkvkθk, where vkis an eigenvector of
L. In terms of the phase angles in the transformed coordinates, θk, we obtain a second-order DDE,
¨
θk=ak(λk)˙
θkbk(λk)θkaτ
k(λk)˙
θτ
kbτ
k(λk)θτ
k.(S.10)
The coefficients are given in Eq. (S.1) as functions of λkand the Jacobians F,Fτ,G, and Gτ τ . Following Bhatt and
Hsu [19], we state stability criteria in terms of these coefficients, as discussed in the main text.
B. Communication delays
The derivation above relies on the assumption of antisymmetric coupling between connected nodes iand j(as-
sumptions 1 and 2). This permits rewriting the problem in terms of the effective Laplacian matrix L, which leads to
a decomposition in its eigenvalues.
Let us briefly consider symmetric or asymmetric coupling which does not satisfy assumptions 1 and 2. Notably, a
communication delay immediately destroys the antisymmetry because the coupling function g0τ(xi, xτ
j) evaluates its
arguments at different times. In this case, the problem cannot be expressed in graph Laplacian form. Instead, we
consider an effective adjacency matrix Aand an effective degree matrix D. These matrices must then commute to
allow a block decomposition.
In the following, let grepresent any of the functions g00,gττ , or g0τ. When linearizing the inertial oscillator model,
we obtain one Jacobian D2
ij g(x
i, x
j) containing derivatives with respect to the j-th node, and a second Jacobian
D1
ij g(x
i, x
j) with derivatives by xi. Replacing assumption 4, we assume:
5. The adjacency matrix Aand the Jacobian D2
ij g(x
i, x
j), evaluated on the phase synchronization manifold Z,
factorize into the direct product A ⊗ G(2) of an effective N×Nadjacency matrix Aand a universal 2 ×2
Jacobian G(2), such that the local matrix G(2) is the same for all nodes and only Adepends on the indices i, j
of the network.
Similarly, Aand D1
ij g(x
i, x
j) factorize into the direct product A0G(1).
We require that Aand A0are symmetric matrices.
Since Aand A0are symmetric by assumption, they are diagonalizable. We define the effective degree matrix D,
Dij := (˜
di:= PlA0
il if j=i
0 otherwise i, j, l 1, . . . , N .
Now we assume:
6. The matrices Aand Dcommute, i.e.
[A,D]=0.
This implies that they are simultaneously diagonalizable. In that case, the DDE (Eq. (S.4)) decomposes into blocks in
terms of the eigenvalues of Aand D. Then, the coefficients in Eq. (S.10) are functions of these eigenvalues instead of
the eigenvalues λkof the graph Laplacian matrix L, and our method can be applied in analogy to the antisymmetric
case.
Regular graphs with homogeneous weights present a special case where [A,D] = 0. For more complex network
topologies, however, assumption 6 is generally not satisfied.
9
III. DECISIVE ROOTS
In the case of phase delay, there is in principle one uniquely defined decisive root for each k. Thus, the stability
analysis is generally based on calculating maximally Ndecisive roots for a system of size N. In contrast, the frequency
delay case generally requires identification of two decisive roots y
k,y∗∗
kfor each k. The value of these roots depends
on the delay and on the coefficient bk.
Assume the value bkτ2is located in the interval (π/2, jπ +π/2), where jis a positive integer. If jis an odd
number, the closest decisive root y
kis in the same interval. We then find the root y∗∗
keither in the π-interval to the
right (j+ 1) or to the left (j1). To find all decisive roots for a given k, we must therefore calculate three roots and,
among them, compare the two possible candidates for y∗∗
k.
If jis an even number, then y∗∗
kis located in the same interval and y
kis one of the two roots found in each of the
adjacent intervals. In any case, all decisive roots must lie within a distance of 3π/2 from the value bkτ2(see figure
S.1).
In conclusion, for frequency delays, we must calculate at maximum 3Ncharacteristic roots to find in total 2N
decisive roots that the stability conditions are based on. In the DSGC model presented in the main text, the root y∗∗
k
turns out to be irrelevant; thus it suffices to calculate at maximum 2Ncharacteristic roots.
IV. DROOP-CONTROLLED INVERTER MODEL
In this section, we provide further detail on the renewable inverter-based power grid model with processing delay,
considered in the main text as a central application of the phase delay case. For additional information on the
theoretical study of power systems we refer to [28, 37–39].
To model a renewable power system with phase delay, we describe the dynamics of grid-forming inverters (repre-
sented by nodes of the network) using the swing equation [28, 37],
mi¨ϕi+αi˙ϕi=Pd
iPel
i.(S.11)
Here ϕ(t) denotes the phase angle, mithe inertia, aithe damping constant, and Pd
ithe desired power set point of the
i-th node. The set point is positive for production and negative for consumption. Furthermore, we write the electrical
power at node ias
Pel
i=X
j
U2
0|Bij |sin(ϕiϕj),(S.12)
where U0denotes the AC voltage amplitude which is assumed constant throughout the system, i.e. U0=Uii[39].
Bij represents the susceptance of the transmission line between nodes iand j(we may choose its value to be zero if
iand jare not directly connected). Eq. (S.12) expresses a common choice in the literature to model steady-state
power flow [28]. It follows when neglecting losses (purely inductive power lines) and assuming that all phase differences
|ϕjϕi|< π/2.
According to Eq. (S.12), the power flow along a transmission line between two nodes depends on the phase angle
difference between them. Let us suppose that the state of this power line (an edge in the network) enters the frequency
control, yet with a processing delay. One way to model this would be delayed coupling, where the phase difference is
evaluated at time (tτ).
Furthermore, we introduce the weighted adjacency matrix Kij := U2
0|Bij |and define ˜α=αi/mi,˜
β= 1/mii,
assuming homogeneous inertia-specific damping and droop constants. This leads to
¨ϕi= ˜α˙ϕi+˜
βPd
i
N
X
j=1
Kij sin ϕτ
ji ,(S.13)
which corresponds to the droop-controlled inverter model presented in Eq. (1) of the main text.
10
FIG. S.1. Scatter plot of decisive roots y
2(blue dots) and y
N(orange dots) as a function of delay τfor the DSGC model
(frequency delay) on the four-node star network. The roots only lie in odd intervals (π/2, jπ +π/2), jodd. The solid
lines show the value λkτ2for k= 2 (blue) and k=N(red). The dotted lines have a vertical distance of 3π/2 from the
corresponding solid line, thus bordering the interval where the root, for given τ, could be found. Parameter values for the
calculation are P0= 1 s2, K0= 8 s2, α = 0.1 s1, and γ= 0.25 s1.
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Preface 1. Introduction Part I. Synchronization Without Formulae: 2. Basic notions: the self-sustained oscillator and its phase 3. Synchronization of a periodic oscillator by external force 4. Synchronization of two and many oscillators 5. Synchronization of chaotic systems 6. Detecting synchronization in experiments Part II. Phase Locking and Frequency Entrainment: 7. Synchronization of periodic oscillators by periodic external action 8. Mutual synchronization of two interacting periodic oscillators 9. Synchronization in the presence of noise 10. Phase synchronization of chaotic systems 11. Synchronization in oscillatory media 12. Populations of globally coupled oscillators Part III. Synchronization of Chaotic Systems: 13. Complete synchronization I: basic concepts 14. Complete synchronization II: generalizations and complex systems 15. Synchronization of complex dynamics by external forces Appendix 1. Discovery of synchronization by Christiaan Huygens Appendix 2. Instantaneous phase and frequency of a signal References Index.