Study Of Time-varying Nonlinear Dynamic Modes Along Disturbed Trajectories Based On EEAC Theory

When the generators are synchronized to the system through the transmission lines, their rotors will swing against each other after disturbances, and even oscillate sustainedly due to inadequate damping. Furthermore, the power on the transmission lines will hunt sustainedly. Because the oscillation frequency (normally between 0.1～2.5Hz) is much low, it is called low frequency oscillation. Power system low frequency oscillations are categorized of local modes and inter-area modes. The local mode has a frequency of 0.7～2.5Hz, and the inter-area mode has a frequency of 0.1～0.7Hz. The local mode represents the oscillation of a generator against the rest of the system, and the inter-area mode is concerned with the oscillations of a group of generators in one area against the rest of the system. In recent years, low frequency oscillations occurred frequently, and they would endanger the power system security and stability, even lead to large-scale blackouts.The present methods of low frequency oscillations are reviewed and it is pointed out that the classical eigenvalue method can only reflect the dynamic performances near the equilibrium point. The normal forms of vector field technique and modal series analysis expand the Taylor series at the equilibrium point and use the higher terms to show the interactions of fundamental oscillation modes. The equilibrium point eigenvalues are independent of the disturbance location and disturbance degree, so they can not reflect the oscillation modes which have been trigged during the disturbed process. If the post-fault SEP (Stable Equilibrium Points) is different from the pre-fault one, the power system dynamic characteristics depend not only on the eigenvalues of the two SEPs, but also on the dynamic evolution.The disturbed trajectory is the direct expression of system dynamic performances and it contains time response informations of the specified disturbance. The oscillation frequency and damping series, which have been distilled from sliding time windows of corresponding disturbed trajectories, are called trajectory eigenvalues. This type of eigenvalues is capable of assessing the effects of nonlinearity and non-autonomy, and can reflect the oscillation cluster and the evolution of dominate interface.There are two steps to calculate trajectory eigenvalues, (1) get the time response curves from digital simulation of actual power system models or data acquisition of physical systems; (2) extracts eigenvalue series from sliding time windows or from successive time-profile.There are two ways to calculate trajectory eigenvalues along the actual/simulation trajectories, one way is to extracting oscillation mode information from a series of sliding windows using signal processing techniques, these eigenvalue series are named as "trajectory window eigenvalues". It is applicable for the actual measurement curves, but not suitable for either fast time-varying systems or strong nonlinear systems. The other one is called "trajectory profile eigenvalues", it is based on the piecewise-linearized system models along the disturbed trajectories and the eigenvalue calculation is performed at the beginning of every time step for numerical integration. This type of eigenvalues can cope with the effects of fast time-variance or strong nonlinearity, but it needs to know the system models and parameters beforehand.In this paper, the wavelet ridge algorithm is used to extracting oscillation modes information along the disturbed trajectories, and the trajectory eigenvalues are used to analyze the effects of nonlinearity and non-autonomy on oscillation performances of an OMIB system and a 3-machine 9-node system. Furthermore, simulations on the 3-machine 9-node system show that its dynamic behaviors may be fully different from eigenvalue results at the equilibrium point, and the latter may even miss the most critical nonlinear modes.The nonlinearity and time- variance factors can influence the eigenvalue results at the equilibrium point. Because the classical eigenvalue method can not evaluate their influence degree quantitatively, it is difficult to study the mechanism of actual engineering phenomena. The paper proposes an index to evaluate the influence on oscillation behavior of both nonlinearity and time-variance. This method substitutes the full nonlinear equations with successive time-profile data of linear system disturbed trajectory, taking the mechanical power or the damping coefficient as the sole parameter to be identified, then the error of approximate models can be evaluated by the differences between the identified value and the actual one in full nonlinear models. The index increases with the errors of the approximate models. Therefore, when the evaluated models are equal to the full nonlinearity models, the index is zero. Based on the index, the shortcomings of the conventional eigenvalue method can be studied.Based on the piecewise-linearized system models along the disturbed trajectories, the trajectory profile eigenvalue calculation is performed at the beginning of every time step for numerical integration. The method is applied to an OMIB and a 3-machine 9-node test system and the effects of nonlinear factors on the eigenvalues at the equilibrium point are investigated.Based on the formula of the analytical calculation eigenvalues, the influences of the parameters on the real part and the image part of eigenvalues are analyzed. Simulations on the national power grid show the critical oscillation interface and the reason of occurrence of ultra-low frequency oscillation in the large scale power systems. It is also pointed out that this type of analytical eigenvalue method, which is the same as the eigenvalue technique at the equilibrium point, can not applicable for strong nonlinear and time-varying systems.To overcome the shortcoming, the method of conventional analytical calculation eigenvalues is expanded to the time-varying eigenvalues series along the disturbed trajectories and it makes the estimation of trajectory eigenvalues quickly and suitable for all kinds of disturbances. The piecewise-linearized dynamic equations are mapped into a series of time-varying OMIB systems and the oscillation mode within each integration-step can be identified with analytical formulas of EEAC. The method is applied to a 3-machine 9-node test system and the accuracy of the analytical estimation results is investigated.EEAC is the quantitative stability theory and algorithm along disturbed trajectories without any approximation. Based on EEAC theory, simulations on the national interconnected power grid show that remote units can also become the critical ones in some cases. Furthermore, it is also found that variation of the loads near the disturbance can not only change the critical cluster or the instability swing, but also can change both aspects simultaneously.In large scale power grids, the complexity of the instability modes often lead to the negative effect of emergency controls, it is important to understand its internal mechanism. Based on theoretical analysis and simulation results, it has been verified that control effects are opposite not only between the critical cluster and the remaining cluster, but also between two neighboring swings if the same kind of control actions, such as generator tripping, are adopted. Further studies proved that the bifurcation on unstable modes may take place when the value of equivalent mechanical power is close to that of the aperiodic component of the post-fault electrical power. So the control strategies of transient stability should take into account all kinds of instability modes, comprehensively coordinate time-space effects based on the optimization of time and space separately.This work is jointly supported by National Natural Science Foundation of China (No. 50595413) and State Grid Corporation of China (SGKJ[2007]98&187). The research work is of significance both on theory and in practice. Based on the time series information of trajectory eigenvalues, it can be used to analyze low frequency oscillation problems of the weak damping or negative damping, improve the PSS which has been designed using conventional eigenvalue theory and damp time-varying oscillations effectively.