IEEE Circuits and Systems Magazine - Q3 2020 - 53

Based on the second-order Kuramoto-type model,
the gradient inequality and Łojasiewicz exponent have
been employed to study the transient stability of power
networks for estimating the trapping region of the synchronous state [141]. The problem of seeking such an
attraction region is also investigated by [142] for the
frequency synchronization and the locked phase agreement in power networks, where the explicit estimation
of the synchronized trapping region is provided for the
uniform and nonuniform ratios of the damping coefficient and the inertia strength, respectively. Both efforts
of [141], [142] have tried to fill the gaps pointed by [11].
To overcome the deficiency that the conventional linearized methods can only guarantee the local stability
in or near an equilibrium state [139]-[144], the nonlinear coordinating switching control protocols are proposed to analyze the frequency stability [140]. The designed controllers compute the monitored information
from the generator-side and load-side simultaneously,
and transmit the processed signals to steer the generators and load buses for maintaining frequency stability
[140]. The global stability region is rigorously proven, in
which the sufficient conditions are derived for achieving global frequency synchronization of complex power
networks [140].
Apart from modelling the synchronous generators
as the second-order Kuramoto-type dynamics, Dörfler
and Grammatico [145] model the buses with frequencyresponsive devices (e.g., frequency-sensitive loads or inverter sources performing droop control) as the first-order Kuramoto-type dynamics and the passive buses (e.g.,
static loads or inverters performing maximum powerpoint tracking) as algebraic equation. By adopting a novel
frequency control architecture, which is based on a semidecentralized gather-and-broadcast scheme, the work of
[145] achieves the local asymptotic stability of closedloop equilibria and the economic-dispatch optimality.
Furthermore, a network of loads and DC/AC inverters equipped with power-frequency droop controllers
in islanded microgrids can also be cast as a Kuramoto
model of phase-coupled oscillators [146], [147]. Other
models of power grids related to the Kuramoto-oscillator dynamics can refer to the survey [148].
3.3. Collective Motion of Self-Propelled Vehicle
Consider a continuous-time kinematic model of identical particles (of unit mass) moving in the plane at
unit speed
	

ro k = e iik
)o
i k = u k,

k = 1, 2, f, N,

(12)

where rk = x k + iy k ! C is the position of p
- article k,
i k is the orientation of the velocity vector e ii k =
THIRD QUARTER 2020 		

Bus 8
Load C

Load D

Load E
Bus 9

Bus 7

G2

G3

Bus 2

Bus 3
Bus 6

Bus 5
Bus 4

Load A

Load B
Bus 1

G1
Figure 7. The Western System Coordinating Council (WSCC)
9-bus system with generators G1, G2, G3 and loads A-E [138].

cos i k + i sin i k, u k denotes the steering control, and
i = (i 1, i 2, f, i N ) T , u = (u 1, u 2, f, u N ) T .
The average linear momentum Ro of a group of particles satisfying (12) is the centroid of the phase particles
p i, that is
	

Ro = 1
N

N

/ ro k =

k=1

1
N

N

/ ei

ik

_ p i = p i e iW . (13)

k=1

Similar to the order parameter defined in (2), |p i| measures the synchrony of phases. When the phases balance to result in a vanishing centroid, it is minimal, i.e.,
|p i| = 0. When the phases are synchronized, it is maximal, i.e., |p i| = 1, which means all particles move in the
same direction. Control of the group linear momentum
is achieved by minimizing or maximizing the potential
U 1 (i) = N/2 |p i|2, which suggests the gradient control
u = - K gradU 1, i.e.,
	 u k = - K 2U 1 = - K G p i, ie iik H = - K
2i k
N

N

/ sin

(i j - i k) . (14)

j=1

The right-hand side of the above Eq. (14) is an all-to-all
sinusoidal coupling, which is a special case of the classical Kuramoto model (1) for ~ i = 0 and the negative
coupling form. Sepulchre et al. pioneeringly propose a
methodology to stabilize isolated relative equilibria in
an all-to-all coupled model of identical particles moving in the plane at unit speed [14]. Furthermore, for the
steering control law
	

u k = io k = ~ 0 - K
N

N

/ sin (i j - i k), ~ 0 ! R, (15)

j=1

there are four different types of collective motions associated with the particle model (12): (a) converge to a uniform linear motion with the same direction for ~ 0 = 0
and K 1 0; (b) a uniform linear motion with separate directions for ~ 0 = 0 and K 2 0; (c) converge to a
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