Conventional Eigenvalue Analysis Research Articles

Predicting stability of synchronous nonlinear vibration in vertical rotating shaft system with journal bearing

Journal bearings (JBs) play a vital role in rotating machines across various fields. In vertical rotating shaft systems with JB, synchronous vibration experiences stages of destabilization and stabilization as rotational speed increases due to self-excited vibration. The onset speed of instability (OSI1) can be predicted via conventional eigenvalue analysis around static equilibrium positions. However, conventional analysis cannot determine the stabilizing and destabilizing speeds (OSS and OSI2) at which self-excited vibration disappears and reappears; numerical and experimental methods suffice. No other theory adequately explains these phenomena. This paper proposes an analytical method based on linearizing JB force and generalized eigenvalue analysis to efficiently predict synchronous vibration stability in vertical rotating shaft systems. The extended eigenvalue analysis directly determines synchronous orbit stability. Demonstrating accuracy and efficiency, a characteristic curve representing stability changes within an unbalance range is calculated, compared with numerical results. Our method serves as an input to rotor dynamic software, calculating rotating shaft-bearing stability thresholds.

Vector‐fitting‐based quantitative SSCI analysis for series‐compensated wind power systems

The occurrence of unstable subsynchronous control interaction (SSCI) between wind farms and series-compensated networks poses a considerable threat to the safety of power system. Different from the conventional eigenvalue analysis and Nyquist method, the black-box-model based RLC fitting technique becomes more attractive for the assessment of the SSCI risk. However, the method has merely been verified by radial networks whose SSCI mode can be well explained by an RLC circuit. A practical wind power system, however, can be meshed networks comprising multiple geographically distributed wind farms and series capacitors. Thus, it is questionable whether the application of the RLC fitting technique to such systems can still yield a sufficient accuracy. To clarify this aspect, in this work, the performance of RLC fitting in various test systems was examined. The results indicated that the aggregation technique considerably influences the accuracy of the assessment, and the RLC fitting cannot be applied when multiple series capacitors are involved in the SSCI mode. To address this problem, a more advanced frequency-domain fitting technique known as vector fitting was adopted, and its robustness and effectiveness were verified considering a modified The Electric Reliability Council of Texas (ERCOT) system.

Estimation of the resonance between forcing and weakly stable oscillations in power systems

Using a simple test example, this paper demonstrates that a weakly damped eigen-oscillation of an electric power system can lead to a loss of stability in the presence of its resonance with forcing oscillation. Such resonant interactions can risk the power system security, as confirmed by recent events in the Western American and Nordic power systems. Conventional eigenvalue analysis cannot detect the observed instability due to its non-linear nature. We show that the critical condition for the appearance of this type of instability can be the value of the perturbation energy accumulated in the system over time. To quantify the risk of stability loss, we examine the model of forcing oscillations and the resonance response of power system in a linear approximation and compare it with the simulation results for the two-area Kundur test system.

Characterisation of interaction in an offshore AC grid with large WPPs and VSC‐HVDCs

To exploit the full and flexible capability of offshore voltage-sourced converter high-voltage DC (VSC-HVDC) lines, interaction studies in the offshore AC grid are required but are not well understood or reported. This study examines the interaction dynamics of an offshore AC grid for interconnecting large wind power plants (WPPs). The conventional eigenvalue analysis method has limitations which make the interaction analysis of such systems difficult. Hence, in this study, an impedance-based analytical approach is employed to investigate the interaction phenomena. The impedance model of a VSC-HVDC converter for both direct and vector control with outer and droop controls are derived along with the impedance model of the full-converter wind generator. The interaction dynamics of the offshore grid is predicted through the well-established Nyquist criteria and is validated using time-domain simulations. The analysis shows that the system stability is decidedly influenced by the control configurations and tuning of the VSC-HVDC lines.

Structural stability of cable-stayed bridges

This paper presents an investigation of the structural stability of cable-stayed bridges based on nonlinear analysis. In general, girders and masts of cable-stayed bridges are always subjected to compressive forces due to pre-tensioned stay cables. Cable-stayed bridges exhibit various geometric nonlinearities, such as the cable-sag effect, beam-column effect of girders and masts, and large displacement effect. In this study, the characteristics of the structural stability of cable-stayed bridges are investigated. Because of various geometric nonlinearities, nonlinear finite element analysis should be performed for stability analysis instead of conventional eigenvalue analysis. Furthermore, the initial shape analysis should be performed prior to live load analysis in order to consider the dead load condition rationally. A two-step analysis method, developed based on the theory of nonlinear finite element method, is presented in this study for stability analysis under live load cases. An intensive series of parametric studies is subsequently performed using three-span cable-stayed bridges. The main buckling modes are classified depending on the location of the critical members. Also, the effects of cable arrangement, girder-mast stiffness ratio, area, and the number of cables on the structural stability are extensively investigated.

Complex harmonic modal analysis of rotor systems

Complex harmonic analysis for rotor systems has been proposed from the strict complex modal analysis based upon Floquet theory. In this process the harmonic balance method is adopted, effectively associated with conventional eigenvalue analysis. Also, the harmonic coefficients equivalent to dFRFs in harmonic mode has been derived in practice. The modes are classified from identifying the modal characteristics, and the adaptation of harmonic balance method has been proven by comparing the results of the stability analyses from Floque theory and the eigen analysis. The modal features of each critical speed are depicted in quantitatively and qualitatively by showing that the strengths of each component of the harmonic coefficients are estimated from the order of magnitude analysis according to their harmonic patterns. This effectiveness has been verified by comparing with the numerical solutions.

A novel stabilization approach for small signal disturbance of power system with time-varying delay

Small signal instability may cause severe accidents for power system if it can not be dealt correctly and timely. How to maintain power system stable under small signal disturbance is a big challenge for power system operators and dispatchers. Time delay existing in signal transmission process makes the problem more complex. Conventional eigenvalue analysis method neglects time delay influence and can not precisely describe power system dynamic behaviors. In this work, a modified small signal stability model considering time varying delay influence was constructed and a new time delay controller was proposed to stabilize power system under disturbance. By Lyapunov-Krasovskii function, the control law in the form of nonlinear matrix inequality (NLMI) was derived. Considering synthesis method limitation for time delay controller at present, both parameter adjustment method by using linear matrix inequality (LMI) solver and iteration searching method by solving nonlinear minimization problem were suggested to design the controller. Simulation tests were carried out on synchronous-machine infinite-bus power system. Satisfactory test results verify the correctness of the proposed model and the feasibility of the stabilization approach.

Coordinated design of probabilistic PSS and SVC damping controllers

This paper presents an application of probabilistic theory to the coordinated design of power system stabilizers (PSSs) and FACTS controllers, taking static VAr system (SVC) as an example. The aim is to enhance the damping of multi electro-mechanical modes in a multimachine system over a large and pre-specified set of operating conditions. In this work, conventional eigenvalue analysis is extended to the probabilistic environment in which the statistical nature of eigenvalues corresponding to different operating conditions is described by their expectations and variances. Probabilistic sensitivity indices (PSIs) are used for robust damping controller site selection and for optimization objective functions. A probabilistic eigenvalue-based objective function is employed for coordinated design of PSS and SVC controller parameters. The effectiveness of the proposed controllers is demonstrated on an 8-machine system.

Choice of SVC location/signal and its controller design by probabilistic method

This paper presents a probabilistic algorithm for supplementary damping control design for the static var compensator (SVC). Conventional eigenvalue analysis is first extended to the probabilistic environment in which the statistical nature of eigenvalues, corresponding to the load duration curves, is described by their expectations and variances. A probabilistic sensitivity index (PSI) is then derived and successfully applied to the location, feedback signal and parameter selection of the damping controller of the SVC. System robustness can be guaranteed because the load variation has been taken into account in probabilistic representation. The effectiveness of the proposed approach is demonstrated on a two-area four-machine system.

Probabilistic eigenvalue sensitivity analysis and pss design in multimachine systems

This paper presents an application of probabilistic theory to the selection of locations and parameters. The aim is to enhance the damping of multiple electromechanical modes in a multimachine system over a large and prespecified set of operating conditions. Conventional eigenvalue analysis is extended to the probabilistic environment in which the statistical nature of eigenvalues corresponding to different operating conditions is described by their expectations and variances. Probabilistic sensitivity indices to facilitate robust PSS site selection and a probabilistic eigenvalue-based objective function for coordinated synthesis of parameters are then proposed. The quasi-Newton technique of nonlinear programming is used to solve the objective function and its convergence properties are discussed and compared with the conventional steepest descent approach. The effectiveness of the proposed stabilizers, with a classical lead/lag structure, is demonstrated on an eight-machine system.

Dynamic stability assessment of wind turbine generators using the Lyapunov function approach

This paper presents the application of the Lyapunov direct method (LDM) for dynamic stability assessment of a single wind turbine generator (WTG) system connected to an infinite bus under widely varying loading conditions. The proposed method is used to assess the dynamic stability for a given operating condition of P and Q very easily without calculating the overall system eigenvalues. In fact, the use of dominant complex eigenvalue loci plotted in the s-plane as constant P and Q contours to obtain the dynamic stability information for a given operating condition of the WTG system may not give the true picture because of complex surface shapes. The proposed method can be used to obtain the optimum reactive power loading for a given real power loading condition, corresponding to which the transient response of the system would be the best, by calculating a parameter defined as the dynamic stability measure. The system performance curves presented ( V- Q curves for constant P) enable not only the optimum reactive power loading for a particular real power loading to be identified but also give some indication of the total optimal range in which it can be chosen from the point of view of fast transient response. Several cases are examined and the effects of the system and control parameters on the dynamic stability of the wind turbine generator are discussed. It is observed that the proposed LDM is computationally more efficient than the conventional eigenvalue analysis for dynamic stability assessment of the WTG system.

Eigenproperties of Nonclassically Damped Primary Structure and Oscillator Systems

The calculation of the combined eigenproperties of a nonclassically damped structure and a supported equipment is of practical interest. Herein an approach is developed whereby these properties can be obtained, in terms of the eigenproperties of the structure and equipment, without a conventional eigenvalue analysis of the combined system. The eigenvalues are obtained as the solutions of a nonlinear characteristic equation, easily solvable by a simple Newton-Raphson scheme. Once the eigenvalues are known, the corresponding eigenvectors can be obtained from closed-form expressions. The approach can also be used effectively to obtain exact eigenproperties for very light as well as very heavy equipment supported on structures.

Induction Machine Instability Predictons - Based on Equivalent Circuits

Induction machine instability is predicted on the basis of equivalent circuits. The circuits are derived assuming sinusoidal perturbation of the rotor speed. Classic rotating field concept is used in the analysis. It is shown that equivalent estimates of the critical supply frequencies, at which instability occurs is obtained by the method shown in the paper and conventional eigenvalue analysis. Approximate, conservative analytical expressions are also obtained by using certain simplifying assumptions. It is possible to obtain the sensitivity of the instability region to different machine parameters. This indicates the parameter which has the greatest influence on stabilizing the machine. The approach developed may be extended to study the effect of time harmonics on the stability of induction machines.