Synchronization of cyclic power grids: Equilibria and stability of the synchronous state

Abstract

Synchronization is essential for the proper functioning of power grids; we investigate the synchronous states and their stability for cyclic power grids. We calculate the number of stable equilibria and investigate both the linear and nonlinear stabilities of the synchronous state. The linear stability analysis shows that the stability of the state, determined by the smallest nonzero eigenvalue, is inversely proportional to the size of the network. We use the energy barrier to measure the nonlinear stability and calculate it by comparing the potential energy of the type-1 saddles with that of the stable synchronous state. We find that the energy barrier depends on the network size (N) in a more complicated fashion compared to the linear stability. In particular, when the generators and consumers are evenly distributed in an alternating way, the energy barrier decreases to a constant when N approaches infinity. For a heterogeneous distribution of generators and consumers, the energy barrier decreases with N. The more heterogeneous the distribution is, the stronger the energy barrier depends on N. Finally, we find that by comparing situations with equal line loads in cyclic and tree networks, tree networks exhibit reduced stability. This difference disappears in the limit of N→∞. This finding corroborates previous results reported in the literature and suggests that cyclic (sub)networks may be applied to enhance power transfer while maintaining stable synchronous operation.

Synchronization of Power Systems under Stochastic Disturbances

“…Similarly, for networks with a non-uniform disturbance-damping ratio, the newly connected nodes with disturbance-damping ratios in the set [𝜂, 𝜂] will not change the bounds of the variance, as follows from the formulas ( 6) and ( 7). This conclusion is different from that obtained by a study of linear stability 27 , where the linear stability decreased if the scale of the network increased.…”

“…The benefit of forming small cycles is that the fluctuations in the phase angle differences decrease by 𝑂 (1/𝑁), where 𝑁 denotes the length of the cycle. This is consistent with the findings obtained by studying the energy barrier of a nonlinear system with a cyclic network in Xi et al 27 . Future power grids will comprise many small distributed generation sites, such as rooftop solar panels and windmills at farms.…”

Multilevel Power-Imbalance Allocation Control for Secondary Frequency Control of Power Systems

“…where θ i j = θ i − θ j for (i, j) ∈ E, i and j are the node indices, and r, q are the area indices. As in the Kuramoto-model [6], the closed-loop system (20) may not have a synchronous state if the power injections {P i , ∈ V} are much larger that the line capacity {B i j , (i, j) ∈ E} [8] and [38]. We assume there exists a synchronous state for the power system, which can be satisfied by reserving some margin in the line capacity by tertiary control.…”

On distributed coordination in networks of cyber-physical systems

“…Synchronization 7 and consensus 8 are two remarkable instances of network coordination relevant to a number of applications as the study of opinion dynamics, 9 of bacterial quorum-sensing, 10 and of distributed generation in power grids. 11,12 The emergence of this coordinated behavior in a network can be studied in terms of convergence of the agents' states toward some synchronization manifold in state space. 13 Analogously, phenomena such as the formation of synchronized clusters of agents 14 in a network (or clustering) can be linked to agents' dynamics away from such a manifold.…”