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Electric power grids can operate normally only if the total electricity demand matches the total supply from all the power plants in the grid. All generators of the network have to be stabilized at the same frequency even after a perturbation. A disruption in synchronization may cause the malfunction of generators and the outages of power grids with cascading catastrophic failures of power plants, as have been observed at New York in 1965 and at the Western American network in 19961.
Power grid networks are often composed of a number of areas, which are densely connected internally and weakly interconnected with each other. This is because generators and loads are often spatially connected and the lengths of transmission lines are usually limited. The dynamical processes such as synchronization13,14,15,16 and diffusion processes17 on local subnetworks can further affect the dynamics on the entire system. For instance, phenomena of breathing synchronization where two groups synchronize at different frequencies can also emerge15.
A typical swing equation is often used to describe the dynamics in a power grid and can be taken as a second-order Kuramoto model with inertia24. The swing equation that governs the mechanical dynamics of generator i is given by
where i = 1, , n, and n is the number of machines in the network; Hi and Di are the inertia and damping coefficients of generator i, respectively. Pm,i is the mechanical power injected in i and Pe,i is the electric power output of i; θi is the rotor angle of generator i in respect to a synchronously rotating reference frame in radians. Equation (1) can be converted to a set of first-order differential equations as follows:
Maintaining the rotator frequency is a prerequisite for the stable operation of power systems. Usually, a self-feedback control of rotator is often implemented by governors9. Thus, the mechanical power input into generator i, Pm,i, for, is adjusted in order to keep the frequency close to the standard frequency. Assume that the mechanical power input at generator i in subnetwork ''a'' is controlled with the derivative of the phase frequency, that is,
We denote the equilibrium solution of equation (3) as for and, and is the state obtained by the perturbation around the equilibrium expressed as,,, (see Supplementary Information S3 for the details). By introducing vectors X1 and X2 defined as
where 0 is the zero matrix and I is the identity matrix; the matrices K and M are the self-feedback control matrix and the damping matrix (see Supplementary Information S4). The matrix C is an (na + nb) × (na + nb) Laplacian matrix representing the topology of subnetwork ''a'', subnetwork ''b'', and the network interlinks between them, which relate to the synchronized state, defined as
The matrix Cbb can be defined in a similar way as Caa. We also assume that the network is undirected, so we have Cba = (Cab)T. Since C is the Laplacian matrix, it can be further diagonalized as J = QCQ−1, where Q is composed of the eigenvectors of C, and J is the diagonal matrix of the corresponding eigenvalues, . With the transformation Z1 = Q−1X1 and Z2 = Q−1X2, equation (7) is equivalent to
Let us denote, for i = 1, , na + nb. If Δi < 0, the stability condition Λi < 0 for, is trivial, since λK < 0 is always satisfied. The maximum value of Λi, Λmax, is given by and does not change even if λK is further decreased by tuning the parameters γa,i and Ha,i for .
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