The synchronization of power generators is an important condition for the proper functioning of a power system, in which the fluctuations in frequency and the phase angle differences between the generators are sufficiently small when subjected to stochastic disturbances. Serious fluctuations can prompt desynchronization, which may lead to widespread power outages. Here, we model the stochastic disturbance by a Brownian motion process in the linearized system of the non-linear power systems and characterize the fluctuations by the variances of the frequency and the phase angle differences in the invariant probability distribution. We propose a method to calculate the variances of the frequency and the phase angle differences. For the system with uniform disturbance-damping ratio, we derive explicit formulas for the variance matrices of the frequency and the phase angle differences. It is shown that the fluctuation of the frequency at a node depends on the disturbance-damping ratio and the inertia at this node only, and the fluctuations of the phase angle differences in the lines are independent of the inertia. In particular, the synchronization stability is related to the cycle space of the network. We reveal the influences of constructing new lines and increasing capacities of lines on the fluctuations in the phase angle differences in the existing lines. The results are illustrated for the transmission system of Shandong Province of China. For the system with non-uniform disturbance-damping ratio, we further obtain bounds of the variance matrices.@en

The synchronization stability of a complex network system of coupled phase oscillators is discussed. In case the network is affected by disturbances, a stochastic linearized system of the coupled phase oscillators may be used to determine the fluctuations of phase differences in the lines between the nodes and to identify the vulnerable lines that may lead to desynchronization. The main result is the derivation of the asymptotic variance matrices of the phase differences which characterizes the severity of the fluctuations. It is found that the cycle space of the graph of the system plays a role in this characterization. With theory of the cycle space, the effect of forming small cycles on the fluctuations is evaluated. It is proven that adding a new line or increasing the coupling strength of a line affects the fluctuations in the lines in any cycle including this line, while it does not affect the fluctuations in the other lines. In particular, if the phase differences at the synchronous state are not changed by these actions, then the affected fluctuations reduce.@en

The power system is one of the most complicated man-made non-linear systems which plays an important role for human being since it was first made in the 19th century. In the past decade, the integration of renewable power sources such as wind energy and solar energy has increased rapidly due to their sustainability. However, these energy sources are weather dependent which cannot be controlled or even predicted precisely. A challenge brought by this transition to renewable power generation is the uncertain fluctuations that negatively affects the stability of the power system, which leads to the important problem: how to improve by control the stability of the system such that it remains stable when subjected to considerable fluctuations in the energy supply? Hence, research is needed into the stability metrics of the non-linear power system and control strategies for the stability improvement. In this chapter, we describe the linear and non-linear stability analysis of power systems and summarize the corresponding control strategies for stability improvement.@en

We investigate the performance and robustness to noise of a centralized control called Power-Imbalance Allocation Control (PIAC) for secondary frequency control of a power system. The noise affects the frequency measurements and the communications. The impact of the noise on the synchronous state of the system is investigated. Analysis shows that the synchronized frequency deviation is proportional to the bias of the noise in the frequency measurements and gathering procedure of these measurements. With the noise considered as an input, the Input-to-State Stability (ISS) is proven for the closed-loop system.@en

We investigate the performance and robustness to noise of a centralized control called Power-Imbalance Allocation Control (PIAC) for secondary frequency control of a power system. The noise affects the frequency measurements and the communications. The impact of the noise on the synchronous state of the system is investigated. Analysis shows that the synchronized frequency deviation is proportional to the bias of the noise in the frequency measurements and gathering procedure of these measurements. With the noise considered as an input, the Input-to-State Stability (ISS) is proven for the closed-loop system.@en