IEEE Circuits and Systems Magazine - Q3 2020 - 52

For the case that the injection current I inj is not weak
anymore, Mirzaei et al. have developed a generalized
Adler's equation [121]:
	

di - ~ + ~ 0
0
dt
2Q

4
r

I inj sin (i inj - i)
, (8)
I + I inj cos (i inj - i)

which is a compact differential equation of phases that
can apply to any strength of injection, only assuming
that the transconductors are hard-limited. This equation
reveals all modes of the oscillators, no matter whether
a mode is stable or not. It is solved exactly to understand how the component mismatch affects quadrature
phase accuracy, and proves to be a remarkably simple
way to analyze oscillator phase noise [121]. The generalized equation (8) looks less like the classic Kuramoto
model (1) due to the term of cosine coupling contained
in denominator, which hinders their inherent connection certainly. In fact, this type of generalized models
has been widely adopted for investigating the injection
locking and pulling in electrical oscillators [122]-[130].
3.2. Frequency Synchronization in Power Grids
Back in 1996, when Hisa-Aki Tanaka et al. [131] studied
the self-organizing synchronization of the second-order
Kuramoto model with hysteresis effect, they boldly
speculated on its possible application in interconnected power grids. Subbarao et al. [132] later introduced
the order parameter of Kuramoto model into the swing
equations of power system for self-organizing synchronization. Filatrella, Nielsen and Pedersen [133] construct the dynamic model of power grids in accordance
with the energy conservation, where the generators and
users are modeled as a class of the second-order Kuramoto model. Dörfler and Bullo [134] have investigated
the synchronization problem for the network-reduced
model of a power system with nontrivial transfer conductances. By assuming over-damped generators, the
singular perturbation analysis shows the equivalence
between the classic swing equations and a nonuniform
Kuramoto model [134]. Meanwhile, the sufficient conditions for synchronization of nonuniform Kuramoto oscillators have been established, which reduces to necessary and sufficient tests for the standard Kuramoto
model [134]. Furthermore, concise and purely algebraic
conditions are derived, which relate synchronization of
a power network to the underlying network parameters
[134]. Analytical results on the role of coupling strength
in the onset of complete frequency locking are derived
by modeling the power-grids as a network of the second-order Kuramoto oscillators [135]. Recent effort of
[136] has developed the frequency boundedness, which
-effectively relaxed the exact frequency synchronization
and the assumption of overdamped generators in [134].
52 	

In power grids, Dörfler et al. [11] have modeled the
synchronous generator as the second-order generalized
Kuramoto-oscillator dynamics with inertia coefficients
Mi and viscous damping Di, which is governed by
	

M i ip i + D i io i = ~ i -

N

/ a ij

sin (i i - i j), i ! V1, (9)

j=1

and the load is modeled as the first-order generalized
Kuramoto-oscillator dynamics with time constants Di,
which comes as
	

D i io i = ~ i -

N

/ a ij

sin (i i - i j), i ! V2, (10)

j=1

where a ij = Vi · V j · 1 (Yij) , admittance matrix Y = Y T
! C N # N , V1 is the set of load buses, V2 is the set of
synchronous generators buses, and the voltage phasor
Vi = Vi e iii corresponds to the phase i i ! T and magnitude Vi $ 0 of the sinusoidal solution to the circuit
equations. A concise and sharp synchronization condition has been presented [11], namely, the coupled oscillator model (9)-(10) has a unique and stable solution
i * with synchronized frequencies and cohesive phases
i *i - i *j # c 1 r/2 for every connected pair of connected oscillators {i, j} ! E , if
	

L@ ~

E, 3

# sin (c), (11)

where L@ is the pseudoinverse of the network Laplacian
matrix L, and x E, 3 = max {i, j} ! E x i - x j is the worstcase dissimilarity for x = (x 1, f, x n) over the edges
E . For condition (11), some equivalent formulations
have been provided from the perspectives of complex
networks, Kuramoto oscillator, power network, auxiliary linear, and energy landscape, respectively, which
help to develop deeper intuition and reach insightful
conclusions [11]. By analytical and statistical methods, they have established the broad applicability of
condition (11) for various classes of networks, which
features elegant graph theoretical and physical interpretations, and improves upon the existing tests in the
synchronization literature.
Note that the above dynamics (9)-(10) constitute the
celebrated structure-preserving network model of power
systems [137], which has been analyzed in Western System Coordinating Council (WSCC) 9-bus system (see
Fig. 7) and other larger scale power grids [139], [140]. In
[29], the natural frequency wi contained in (9) is replaced
by the mechanical power input Pm, i 2 0, and wi contained
in (10) is replaced by the active power drawn consisting
of a constant term P l, i 1 0. The combination of dynamics
(9)-(10) can also be called a multirate Kuramoto-oscillator network, which is to address in Section 5.2.

IEEE CIRCUITS AND SYSTEMS MAGAZINE 		

THIRD QUARTER 2020
IEEE Circuits and Systems Magazine - Q3 2020

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