Nyquist Stability Criterion and its Application to Power Electronics Systems

With the rapid growing of the power electronic devices in the distributed grid, the impedance stability analysis has been paid more attentions, which is the main method of analyzing the oscillation stability of the power electronic system. The harmonic source, which comes from these power electronic devices, will influence the impedance measurement when the same frequency disturbance is injected. An improved impedance stability analysis method of the DC power electronic system considering the influence of power side harmonics is proposed in the paper. This method eliminates the impact of the harmonics by considering the initial inherent background harmonics of source side. The simulation results show the accuracy of impedance stability analysis criterion can be improved with the proposed method. And the stability misjudgment caused by inaccurate impedance characteristics, impacted by the harmonics, could be avoided.

At present, power electronic converters are widely used in systems such as shipboard power systems, distributed generation systems and DC microgrids. These devices allow for direct and fast control of the system states: voltages and currents. This is a desirable characteristic, especially when considering smart grid applications. However, the rise in the number of such devices connected to the grid, adds to the inherent nonlinearity of the power system leading to phenomena such as bifurcation and chaos. Further complications arise due to the interactions between the power system and the communication system; two systems that have been historically considered as detached from one another. Due to the growing incentives for smart grid development, the interdependence of these two systems must be recognized and incorporated into existing power electronic converter system models. This work presents models for DC-DC converter systems inclusive of both power system variables and information variables. The effects of network delay on the steady-state and dynamic stability of the overall information-embedded power electronic converter system are analyzed and quantified through proposed metrics. Additionally, uncertainty is introduced into the system model in the form of perturbations in network delay, the effects of which on system stability as well as the first exit times are investigated.

Evolution of today’s predominantly alternating current (ac) transmission grid in to mixed ac transmission – high-voltage direct current (HVdc) systems is expected with the decreasing costs of and advancements in power electronics technologies. The recent technological advancements include the development of more efficient and compact voltage source converters (VSCs) that can provide fast response. Because of their controllability and fast response, HVdc systems provide the ability to enhance reliability of the ac transmission grid through frequency support, voltage control, and congestion relief. The HVdc systems can also provide other ancillary services to enhance the economics and reliability of the ac transmission grids, and to optimize their performance. The tools, models, and methods developed to evaluate their benefits will be key enablers to understanding the value proposition introduced by the fast-acting HVdc systems. The main motivation of this work was to develop models and methods to explore and quantify the technical and economic benefits of HVdc systems and fast response VSC technologies in the future grid of United States. The models and methods explore the benefits of HVdc systems through provision of controlled active/reactive power and fast response. The fast response can provide much-needed multiobjective services to the grid (e.g., frequency response, voltage control) while moving blocks of energy. The models and methods can be applied to study other power electronics technologies and scenarios such as those undertaken in ongoing studies led by the North American Electric Reliability Corporation (NERC) Inverter-based Resource Performance Task Force (IRPTF). The IRPTF studies identify the problems associated with power electronics-based technologies as they are integrated in to the bulk power system (or the ac transmission system).

After a long period of development, stability criteria are gradually improved. Among them, Nyquist-Bode stability criterion plays an important role. However, traditional Nyquist stability criterion is only applicable to minimum phase system. With the development and wide application of power electronic technology, minimum phase system is just a small portion of all the power electronic systems. To analyze the stability of non-minimum phase system, a more general Nyquist-Bode stability criterion is essentially needed.

In this study, external cavity laser diode has been modeled and non-linear equations of system are being solved by using Volterra series. In the study, linearized laser diodes rate equations are reorganized for external cavity and these equations are solved with harmonic input method using Volterra series. The transfer functions brought about for harmonics are obtained using Volterra-Weiner series approach. The stability analysis of system is compared from output to input using these transfer functions. Nyquist stability criteria have been used in the analysis.

Silva, Datta, and Bhattacharyya (in the above paper) have studied the feedback control system, provided with a proportional-integral-derivative (PID) controller and relative to a first-order plant with a time delay, and have determined the range of the PID controller parameters suitable for the stability by means of a version of the Hermite-Biehler theorem. The purpose of this note is to find the same results in a simpler way by means of the classical Nyquist's stability criterion.

It is known that self-excited vibration in cylindrical grinding occurs if the work speed is high. In this paper, the effect of the work speed on the occurrence of self-excited vibration is investigated analytically and numerically. In analytical study, Nyquist's stability criterion is used. The results are as follows: If the work speed is low, self-excited vibration does not occur. On the other hand, if the work speed is high, self-excited vibration always occurs. It is also shown numerically that the phase of the work displacement is delayed in phase by ;π/2 to that of previous grinding. The theoretical result about the critical speed of the work is in good agreement with the experimental one.

The tracking/rejection of periodic signals constitutes a wide field of research in the control theory and applications area and Repetitive Control has proven to be an efficient way to face this topic; however, in some applications the period of the signal to be tracked/rejected changes in time or is uncertain, which causes and important performance degradation in the standard repetitive controller. This thesis presents some contributions to the open topic of repetitive control working under varying frequency conditions. These contributions can be organized as follows: One approach that overcomes the problem of working under time varying frequency conditions is the adaptation of the controller sampling period, nevertheless, the system framework changes from Linear Time Invariant to Linear Time-Varying and the closed-loop stability can be compromised. This work presents two different methodologies aimed at analysing the system stability under these conditions. The first one uses a Linear Matrix Inequality (LMI) gridding approach which provides necessary conditions to accomplish a sufficient condition for the closed-loop Bounded Input Bounded Output stability of the system. The second one applies robust control techniques in order to analyse the stability and yields sufficient stability conditions. Both methodologies yield a frequency variation interval for which the system stability can be assured. Although several approaches exist for the stability analysis of general time-varying sampling period controllers few of them allow an integrated controller design which assures closed-loop stability under such conditions. In this thesis two design methodologies are presented, which assure stability of the repetitive control system working under varying sampling period for a given frequency variation interval: a mu-synthesis technique and a pre-compensation strategy. On a second branch, High Order Repetitive Control (HORC) is mainly used to improve the repetitive control performance robustness under disturbance/reference signals with varying or uncertain frequency. Unlike standard repetitive control, the HORC involves a weighted sum of several signal periods. With a proper selection of the associated weights, this high order function offers a characteristic frequency response in which the high gain peaks located at harmonic frequencies are extended to a wider region around the harmonics. Furthermore, the use of an odd-harmonic internal model will make the system more appropriate for applications where signals have only odd-harmonic components, as in power electronics systems. Thus an Odd-harmonic High Order Repetitive Controller suitable for applications involving odd-harmonic type signals with varying/uncertain frequency is presented. The open loop stability of internal models used in HORC and the one presented here is analysed. Additionally, as a consequence of this analysis, an Anti-Windup (AW) scheme for repetitive control is proposed. This AW proposal is based on the idea of having a small steady state tracking error and fast recovery once the system goes out of saturation. The experimental validation of these proposals has been performed in two different applications: the Roto-magnet plant and the active power filter application. The Roto-magnet plant is an experimental didactic plant used as a tool for analysing and understanding the nature of the periodic disturbances, as well as to study the different control techniques used to tackle this problem. This plant has been adopted as experimental test bench for rotational machines. On the other hand, shunt active power filters have been widely used as a way to overcome power quality problems caused by nonlinear and reactive loads. These power electronics devices are designed with the goal of obtaining a power factor close to 1 and achieving current harmonics and reactive power compensation.

Accurate modeling of magnetic components is critical for predicting the high-frequency behavior of power electronic converters. Parasitic elements such as leakage inductance, winding capacitance, and magnetizing inductance significantly affect converter performance, especially in resonant and high-speed switching applications. This paper presents a Genetic Algorithm (GA)-based approach for estimating the parasitic parameters of a transformer by fitting an analytical impedance model to frequency response data obtained via experimental Frequency Response Analysis (FRA). The proposed method eliminates the need for complex geometric modeling or direct measurement techniques, offering a data-driven and optimization-based solution. The equivalent circuit consists of six parameters: <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$L_{1}, C_{1},\ L_{2}, C_{2}, L_{m}$</tex>, and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$C_{3}$</tex>, and their values are optimized to minimize the magnitude error between measured and modeled responses. The methodology is implemented in MATLAB, and the results demonstrate high accuracy, with the optimized model achieving an error below 0.01 dB. The approach is robust, scalable, and adaptable for all type of magnetic structures across power electronic systems even for the integrated inductor-transformer systems.

HVDC is in many respects the ultimate power electronics based FACTS device which can act within an existing AC network or even replace parts of it. In evaluating the usefulness of HVDC it is important to remember the controllability which it adds to the power system in addition to its bulk power transfer capabilities. HVDC often provides the best of both worlds support from a neighbouring network without the problems of synchronous interconnection. Power electronic based devices enable better utilisation to be achieved of all the principal attributes of the electricity network generation, transmission and distribution. Where slow acting devices such as tap changers and circuit-breakers can be used then these are usually most economic. However, if fast or controlled response is required, for example to prevent system collapse, then FACTS power electronic facilities provide a cost effective solution. In many cases power electronics is the only way to achieve some functions, particularly with HVDC. Otherwise, they can be justified where the application requires one or more of the following attributes: rapid control action; frequent variation of output; smoothly adjustable output; and precise output.

Stability analysis of discrete-time systems with arbitrary delay kernels based on kernel-related summation inequality and model transformation

With the partially power-decoupled system architecture, a doubly fed induction motor-based shipboard propulsion system (DFIM-SPS) has the potential of increasing the system safety level by decreasing the penetration of power electronics. In this article, a phase-locked-loop-free (PLL-free) direct power control (DPC) strategy is proposed for a DFIM-SPS to increase the system dynamic performance. The system small-signal stability analysis is carried out based on the source-load structure, where a salient-pole synchronous generator supplies power to the DFIM electric drivetrain. A detailed dq impedance model of DFIM-SPS is established with the power controller gains of the back-to-back power converter (BTBPC) considered. The proposed source-load dq impedance model of PLL-free DPC-based DFIM-SPS can easily reflect the equivalent impedance produced in the stator and rotor circuits. The effects on system stability caused by the equivalent impedance variations due to power controller gain changes are evaluated in detail. The frequency response of the elements in the DFIM impedance matrix is investigated first, and then, the minor loop gain matrix is obtained. The system input-output small-signal stability is analyzed based on the Nyquist stability criterion. Time-domain simulations are carried out in MATLAB/Simulink to validate the effectiveness of the proposed PLL-free DPC strategy for DFIM-SPS.

This paper presents an analysis of multi-infeed HVDC systems voltage stability using small-signal analysis techniques (eigen-structure analysis). A correlation ratio relating to voltage stability is put forward. A small test system with two HVDC infeeds is built to study the voltage stability of the multi-infeed HVDC system. Then the detailed dynamic models of the HVDC system including DC system control model, the generator, the induction motor load and the AC network are investigated. The case studies presented in this paper indicate that the voltage stability correlation ratio (VSCR) can effectively identify the critical elements which strongly correlate with voltage stability in multi-infeed HVDC systems.

This paper discusses a method for automated steady state analysis of power electronics systems using the formalism of Component Connection Models (CCMs). The method offers several advantages: power electronic system's equation can be derived automatically for any power processor configuration, different ideal semiconductor switching device models can be formally incorporated and the model is suitable for Periodic Steady State (PSS) simulations. Several application examples are included.

Robust stability analysis of polytopic linear time-invariant (LTI) systems is a basic problem in engineering. This paper analyzes the complexity of three fundamental methods used to address this problem, all of them providing a sufficient and necessary condition (with finite and known sizes) for robust stability. The first method is based on the use of a polynomially parameter-dependent Lyapunov function. The second method is a simplified version of the Routh-Hurwitz stability criterion. Lastly, the third method is based on eigenvalue combinations. It is explained that the robust stability conditions provided by these three methods require to establish positive definiteness of symmetric matrix forms (SMFs) over the simplex. Also, it is explained that a sufficient condition for the latter problem can be given in terms of a linear matrix inequality (LMI) feasibility test. Hence, the complexity of the three methods is analyzed and compared by deriving the number of scalar variables in the LMI feasibility tests. A numerical example is also presented to investigate the computational time of these tests.