Harmonic Linearization Method Research Articles

Stability Comparison of Grid-Connected Inverters Considering Voltage Feedforward Control in Different Domains

Under the background of high permeability, voltage feedforward control may further weaken the stability of grid-connected inverter (GCI) systems and may cause sub-synchronous oscillation in extreme cases. To solve this problem, this paper firstly considers the influence of the frequency coupling effect and voltage feedforward control, and adopts the harmonic linearization method to construct the L-type GCI sequence admittance model with PI (proportional integral) control and PR (proportional resonant) control, respectively. By comparing the sequence admittance characteristics of the GCI under two control strategies, combined with the sequence admittance model and Nyquist criterion, this paper analyzes the influence of voltage feedforward and control parameters on the stability of the GCI under two control strategies. The results show that the stability of GCI under PR control is slightly better than that under PI control. At the same time, the voltage feedforward control does reduce the stability of the GCI system under the two control strategies. Finally, the accuracy of the theoretical analysis is verified by simulation and experiment.

Improved sequential impedance modeling and stability analysis of virtual synchronous generators considering frequency coupling effects

With the increasing penetration rate of distributed power supply, the interaction between grid-connected inverters and power grid is prone to harmonic oscillation, which will seriously threaten the stable operation of power grid. At present, impedance analysis has been proved to be an effective tool to study the stability of grid-connected systems. However, few studies have considered the influence of frequency coupling effect on the output impedance characteristics of grid-connected inverters. To solve this problem, the sequence impedance model of a three-phase grid-connected inverter controlled by a virtual synchronous generator is established by harmonic linearization method based on the frequency coupling effect. In addition, an improved method of sequence impedance modeling based on grid impedance is proposed to effectively eliminate the influence of coupling terms. Finally, the correctness and practicability of the proposed simplified sequence impedance model are verified by simulation and experiment.

Harmonic Resonance Mechanisms and Influencing Factors of Distributed Energy Grid-Connected Systems

With the rapid development of global energy transformation and new power system, ensuring the stability of distributed energy grid connections is the key to maintaining the reliable operation of the whole power system. This paper constructs detailed impedance models of grid-following (GFL) and grid-forming (GFM) inverters using a harmonic linearization method and thoroughly investigates the mechanisms of resonance when inverters are connected to the grid, as well as the impact of model parameters on the stability of the grid system. This paper also briefly analyzes the scenario where distributed energy and electric vehicles are integrated into the grid simultaneously, demonstrating that grid system stability can be ensured in complex grid situations through reasonable parameter design. Lastly, the accuracy of the proposed impedance models and analysis is verified through MATLAB/Simulink simulations.

VSG Sequence Impedance Modeling and Stability Analysis of Dead-Beat Control

Virtual Synchronous Generator uses dead-beat predictive control in the voltage and current loop to improve the dynamic response speed of the system, but the presence of zero-order hold and computation time can lead to system instability. To address this issue, this paper adopts an improved dead-beat predictive control strategy for VSG. This is achieved by linear interpolation to forward predict voltage-current reference values to reduce the impact of delay, while introducing error compensation coefficients to enhance the stability of the voltage-current inner loop of VSG. The selection of error compensation coefficients is based on closed-loop impulse response functions. A harmonic linearization method is used to construct a complete sequence impedance model of VSG, and it's stability is analyzed. The results indicate that VSG's negative sequence impedance exhibits instability in the low-frequency range. To address this, a method of introducing a virtual resistor into VSG is proposed to improve stability. By constructing a sequence impedance model of VSG that includes virtual resistance, the variations in system stability under different virtual resistances and grid impedances are analyzed. The effectiveness of the proposed method is verified by building a VSG model.

Frequency criterion for stability «as a whole» of modulation-type pulse voltage converters according to Lyapunov

Background. The use of methods for researching and designing stable nonlinear dynamic systems by constructing the frequency characteristics of the loop gain of the negative feedback of a pulsed voltage converter through time characteristics obtained by numerical methods for solving nonlinear differential equations that describe electromagnetic processes in the converter takes a lot of time and has low accuracy in calculating optimal modes, when you need to calculate hundreds of options.
 Aim. Therefore, it seems necessary to further develop the theory of stability of modulation-type pulse voltage converters using frequency stability criteria, Lyapunov functions, amplitude and phase frequency characteristics of the converter loop gain.
 Methods. Stability in the «small» is solved using linearized models, which are constructed using the so-called first Lyapunov method - the «first approximation» method. The second method for studying the stability of nonlinear dynamic systems is the harmonic linearization method, when the nonlinear element is replaced by a harmonically linearized link with a transmission coefficient only for the first harmonic.
 Results. The Lyapunov function for a pulse voltage converter with nonlinear relay-type elements was found. Conclusion. The results obtained in the work make it possible to simplify the design of Lyapunov-stable pulse voltage converters through the use of the method of amplitude-frequency characteristics used for stationary linear systems.

Dynamics of a beam with variable cross-section protected from vibration

In this work, the issue of transverse vibrations of a hysteresistype elastic dissipative characteristic beam with a dynamic absorber is considered. In order to evaluate the effectiveness of the dynamic absorber, i.e., to select its optimal parameters, analytical expression of transfer function is determined. The problem of dynamic damping of nonlinear transverse vibrations of an elastic beam with variable cross-section is solved. In this case, the dissipative properties of the beam material and the elastic damping element of the dynamic absorber are hysteresis type and are taken into account by a nonlinear function of one value. Using the method of harmonic linearization

INFLUENCE OF EXTERNAL CUBIC VISCOSITY ON THE DYNAMICS OF A CANTILEVER ROD LOADED WITH A TRACKING FORCE

The analysis of the influence of external nonlinear cubic viscosity on the behavior of a cantilever rod loaded with a compressive tracking force is performed. The Bernoulli – Euler beam model is used for the describing of the rod bending deformations. After the transition to dimensionless variables, we obtained the form of notation in which the coefficients of the equation do not depend on the coefficient of cubic viscous friction. This indicates the independence of the critical load and system frequency characteristics from the level of nonlinear attenuation. Using the standard procedure of the Bubnov – Galerkin method, taking into account the first two forms of eigen oscillations, the problem is reduced to the analysis of a fourth order nonlinear system of ordinary differential equations with respect to generalized coordinates and generalized velocities. For the linear version of the equations with low attenuation, the values of the critical load are obtained in dependence on the ratio of the damping coefficients of the eigenforms. The behavior of the system is analyzed taking into account the considered cubic nonlinearity using the method of nonlinear normal forms. The problem is reduced to the analysis of a nonlinear dynamical system with one degree of freedom. There are constructive analytical methods of qualitative and approximate quantitative research for this class of problems. The critical value of the compressive force was determined by the analysis of the dependence of the first Lyapunov magnitude on the load parameter. It turned out to be less than the value corresponding to the system without damping (the destabilizing effect of external nonlinear friction). A bifurcation occurs in the system at the critical load value. Therefore, the zero equilibrium becomes an unstable complex focus of the first order and a limit cycle is born. The function of the self-oscillations amplitude of the load magnitude is constructed using the harmonic linearization method.

Vibroconverter mathematical model based on the levitation effect

Modern requirements for the safe operation of nuclear power plants dictate the need to introduce systems for early reactor plants state diagnosis. Vibrations have always been a threat to the safe nuclear power plants operation, therefore, solving the issues of improving the nuclear power plants vibronoise diagnostics accuracy, especially in the field of ultra-low frequencies, is an urgent task. The most promising way to solve the problem of flexible guides in electromechanical vibration transducers is the levitation effect, which makes it possible to completely eliminate mechanical contact and, accordingly, reduce the sensitivity threshold of the transducer. The article presents an analysis of existing vibration transducers and offers analogue and digital mathematical models of the magnetic levitation system, taking into account the influence of the electromagnetic force nonlinearity. The characteristics of such a nonlinear system are studied using the method of harmonic linearization, which makes it possible to obtain an equivalent linear system. At the same time, the third order terms of the electromagnetic force expansion in the Taylor series are taken into account, which made it possible to obtain a more accurate analogue the vibration transducer model. For the developed vibration transducer digital model, using the z-form method, a system function and the corresponding difference equation were obtained. Expressions and graphs of transient, pulse and frequency characteristics are derived for both analogue and digital vibration converter models, which allow us to conclude about the accuracy and adequacy of the developed digital model. Using obtained vibration transducer digital models will make it easier to model its operation and reasonably select the vibration transducer initial parameters.

Investigation of the Feasibility of the Dynamic Equivalent Model of Large Photovoltaic Power Plants in a Harmonic Resonance Study

In recent years, there have been several harmonic resonance accidents around the world that involve renewable energy power plants. The frequency scanning method is the most widely used technique in engineering practice to evaluate the severity of resonance due to its simple operation and clear physical meaning. However, when establishing electromagnetic transient simulation models and conducting frequency scans, a single generation unit or a few renewable generation units are usually used to represent the original power plant for the purpose of reducing model complexity and improving the simulation efficiency. Such a practice has been found to be effective in dynamic studies around the fundamental frequency. However, its feasibility in harmonic resonance studies has not yet been fully investigated. Because of this research gap, a feasibility study is conducted in this paper by using a real-life photovoltaic power plant. The detailed harmonic model of the plant is first established using the harmonic linearization method, and the equivalent harmonic model is then developed using the power loss conservation method. The feasibility of the equivalent model was investigated in detail, and the impact of the different impedance models on the resonance analysis was analyzed. The results indicate that the conventional dynamic equivalent model can effectively reflect the harmonic resonance characteristics of photovoltaic power plants. Furthermore, a more simplified model that ignores the inductance of the collector line is recommended in this paper to further reduce the modeling complexity.

ОБГРУНТУВАННЯ ВИБОРУ РЕГУЛЯТОРА ШВИДКОСТІ ЕЛЕКТРОПРИВОДА ОДНОЦИЛІНДРОВОГО ПОРШНЕВОГО КОМПРЕСОРА

The peculiarities of the operation of closed systems of speed regulation of electric drives of single-cylinder piston com-pressors due to the presence of a non-linear dependence between the load torque and the angular movement of its rotor are considered. It is shown that the presence of an internal nonlinear connection, which forms a load torque, in a closed control systems can lead to subharmonic mechanical self-oscillations, which worsen the power and performance indicators of the electric drive. The use of static controllers is proposed to eliminate subharmonic self-oscillations and increase the efficiency of the electric drive. Ref. 10, fig. 6. Keywords: piston compressor, self-oscillation, method of harmonic linearization, static controller.

Investigation of vibration natural characteristics and response for rotating beam with tenon jointed structure under thermal environment

The dynamic model of a rotating beam with a tenon structure under the thermal environment was established based on the Assumed Mode Method. The Harmonic Linearization Method was used to linearize the frictional force. The function of friction coefficient and temperature was fitted based on the experimental data in the literature. The accuracy of the proposed model was verified by comparing the dynamic properties with the finite element model and the models in the literature. Natural Frequencies and vibration responses under different conditions were solved to investigate the effects of rotational speed and temperature on the dynamic characteristics of the rotating blade. The vibration characteristics of a rotating beam by considering the combined effects of temperature, rotational speed, and friction were compared with those of a single factor. The results demonstrate that the temperature variation will affect the tenon structure's friction characteristics and the rotating beam's dynamic behavior. The comprehensive influence of temperature, friction, and rotational speed is distinct from that of a single factor.

The behavior of hidden bifurcation in 2D scroll via saturated function series controlled by a coefficient harmonic linearization method

Abstract In this article, the behavior of hidden bifurcation in a two-dimensional (2D) scroll via saturated function series controlled by the coefficient harmonic linearization method is presented. A saturated function series approach for chaos generation. The systematic saturated function series methodicalness improved here can make multi-scroll and grid scroll chaotic attractors from a 3D linear autonomous system with a plain saturated function series supervisor. We have used a hidden bifurcation method in grid scroll., where the method of hidden bifurcation presented by Menacer, et al. in (2016) for Chua multi-scroll attractors. This additional parameter, which is absent from the initial problem, is perfectly adapted to unfold the structure of the multispiral chaotic attractor. The novelty of this article is twofold: first, the saturated function series model for hidden bifurcation in a 2 – D scroll; and second, the control of hidden bifurcation behavior by the value of the harmonic coefficient k 3 {k}_{3} .

Compensation Methods for Regulating Parameters of Switch Voltage Converters

Introduction. Approaches to the introduction of negative feedback in pulse control systems regulating the parameters of electricity flow with pulse-width modulation (PWM) continue to attract research attention in the field of switch power amplifiers (SPA) and switch voltage converters (SVC). Evaluation of the potential of compensation methods is of importance when selecting power electronics devices for various purposes.Aim. To review compensation methods for parameter regulation and to investigate conditions for the operational stability of SVC with feedback on output voltage and current.Materials and methods. The research methodology included the theory of pulsed automatic control systems, the method of harmonic linearization with stability assessment according to Nyquist criteria, and the generating filter method based on multiple frequencies.Results. An analysis of operational stability conditions was carried out. Analytical dependences and graphical representation of the limit feedback values from the modulation parameters and the switch transformation scheme were proposed.Conclusion. The conducted study of the feedback depth in SVC, limited by the penetration of high-frequency components into the path of a width-modulated signal and its delay, due to the peculiarities of the terminal stage of the SPA, demonstrates that the limit value of the voltage feedback depth at typical circuit parameters does not exceed 12 dB, while the depth of the current feedback of the filter choke of the low frequencies can be fundamentally (more than 20 dB) higher. At the same time, the implementation of voltage stabilization and output current limitation modes for maintaining reliable operation in start-up and overload modes is possible only with the use of combined feedback.

Адаптация стратегического планирования на промышленных предприятиях к условиям трансформационных шоков

The purpose of the article is to define the concept of systemic transformation and adaptation of the process of strategic planning of agricultural engineering enterprises to transformational shocks under external constraints. To achieve this goal, the key tasks of strategic planning of agricultural enterprises under the conditions of external constraints and transformational shocks are identified. The criteria for choosing an object of study are determined. The principal arguments for the need to adapt industrial enterprises to the conditions of transformational shocks are shown.The enterprises of agricultural engineering of the South of Russia, functioning as a cluster, were chosen as the object of the study. The internal and external environment of agricultural engineering enterprises in the South of Russia is revealed on the example of a cluster in Krasnodar Krai. The advantages of the development of agricultural engineering industry in the specified region are shown; the factors of negative shock influence and systemic effects on the development of a cluster of agricultural engineering enterprises in Krasnodar Krai are identified. With the help of the methodology for express assessment of the economic sustainability of industrial enterprises in the agricultural sector, as well as by illustrating the modernized production function, the cluster of agricultural engineering enterprises in Krasnodar Krai is defined as a protocluster. Methods and methodology. Within the framework of this study, the methods of systemic and comparative analysis, integrative planning, the author’s method of construction of a modernized production function, method of harmonic linearization, dialectical method, method of empirical observations, and survey method were used. The research methodology is presented by cognitive opportunities of the theory of system economics, theory of transformation, theory of strategic planning, and theory of creative economy. The heuristic potential of interdisciplinary research theory is used. Suggestions are made to ensure the conditions for the adaptation of industrial enterprises to changing political and economic conditions and competitive development of the cluster of agricultural engineering enterprises in Krasnodar Krai.

A Hierarchical Small-Signal Controller Stability Analysis Method for the MMCs

Differing from a two-level voltage source converter (TL-VSC), the internal dynamics of a modular multilevel converter (MMC) are much more complicated. Since the fluctuation of submodule capacitor voltage is unavoidable when transmitting power, the MMCs output voltages cannot track the references perfectly. The interaction between the time-varying capacitor dynamics and controllers may lead to self-instability of the MMCs. In order to address this self-instability issue, this paper first establishes the small-signal closed loop transfer function matrices (TFMs) of the external and internal control loop by the harmonic linearization method. Further, a novel design principle of MMC controllers is proposed. Then, the critical impact factors on the stability of circulating current controller and AC voltage controller are investigated based on the generalized Nyquist criterion(GNC). The proposed models and analysis methods are validated by time-domain simulations in PSCAD/EMTDC.

Efficiency of the use of feedback in the stabilization of motion of the rotary hydraulic drive

This article reflects a study of the operation of a hydraulic drive with nonlinear characteristics of an axial piston hydraulic motor, which operates at the lower limit of the speeds of the executive bodies. A nonlinear frictional moment arises in the end valve of the hydraulic motor connected to the actuator and leads to movement with stops. The mathematical models of the dynamics of the hydraulic drive, proposed in this study, are linearized by the method of harmonic linearization and allow engineering calculations of the parameters of the modes of non-stop motion. Evaluation of the dynamic properties of the hydraulic drive, corresponding to the onset of an unstable mode, makes it possible to determine the minimum value of the constant component of the speed of the executive body.

Impedance Modeling and Stability Analysis of All-DC Delivered Offshore Wind Farm

The all-DC delivered offshore wind farm has great potential in power engineering application due to its advantages in large capacity and long distance delivery. Considering the cost and reliability, the all-DC wind farm can be connected to the onshore grid via the Line-Commuted Converter High Voltage Direct Current (LCC-HVDC) but more interaction may also occur in the form of sub-/super-synchronous oscillations. In order to analyze the stability issue of such offshore wind farm system, the DC-port impedance are firstly established for the aggregated wind farm based on permanent magnet synchronous generator (PMSG) and LCC-inverter by using the harmonic linearization method, where the power outer loop cascaded by the current inner loop of PMSG wind farm and the phase-lock loop (PLL) of the LCC inverter are taken into consideration. Furthermore, the interaction between these two subsystems is analyzed based on the stability criteria through their impedance characteristics. Finally, simulation results are conducted by reproducing the time-domain oscillation in order to evaluate the effectiveness of impedance model for the all-DC delivered offshore wind farm system.

Impedance Modelling Mechanisms and Stability Issues of Single Phase Inverter With SISO Structure and Frequency Coupling Effect

Multi-Input and Multi-Output (MIMO) impedance model considering the Mirror Frequency Effect (MFE) has been studied for single phase systems in the past five years. However, the resulting impedance matrix is mathematically intractable without practically physical meaning. Besides, another unique High Frequency Effect (HFE) existed only in single phase systems has received less attention. To tackle these problems, this paper presents a Single-Input and Single-Output (SISO) modeling mechanism for single phase inverter considering both MFE and HFE. Firstly, the accurate response of T/4 delay PLL is derived in this paper. Then, based on the harmonic linearization method, the inverter impedance model is successfully realized in the SISO structure, where MFE and HFE are represented by two additional impedance. And consequently, the influence of HFE on inverter impedance is analysed with varying PLL bandwidth. Applying the Nyquist criteria, the proposed overall impedance model can precisely predict the sub/super-synchronous oscillation that occurs in the weak grid. Finally, all the theoretical analysis results are validated by simulation cases. The proposed work could significantly simplify the traditional MIMO modeling procedure with a more straightforward impedance structure, which might be helpful for solving potential oscillation issues of renewable energy integration.

Internal friction in the Earth' crust and transverse seismic waves

<abstract> <p>A transverse harmonic seismic wave, which propagates through the upper fractured layer of the Earth towards the Earth's surface, is considered. It is shown that the attenuation of a seismic wave in the Earth's upper crust is due to internal friction, i.e., friction between the sides of micro-fractures. Such a damping mechanism does not work in the deeper layers of the Earth where high lithostatic pressure prevents the movement along fractures. The governing equation for a brittle fractured medium is nonlinear. In the study of wave propagation, an approximate method of harmonic linearization is used. It is shown that internal friction in the upper crust leads to the transformation of a transverse harmonic wave into a shock wave.</p> </abstract>

Impedance modelling and grid integration stability evaluation of three‐phase virtual synchronous generator

Virtual synchronous generator (VSG), which can behave with analogous characteristics as the conventional synchronous generator (SG) and provide voltage and inertia/damping support, is universally regarded as a grid-friendly interface. Here, the comprehensive sequence-impedance model of the VSG is developed by using the harmonic linearization method. To improve the accuracy of impedance model, the influences of the reactive power loop (RPL) and mirror frequency effect (MFE) are discussed and evaluated for the first time. Three different impedance models of the VSG (conventional, considering RPL, and considering both MFE and RPL) are derived and compared in detail. Besides, the synchronous frequency resonance (SFR) issue and grid-interactive oscillation risk of the VSG are also investigated. The results indicate that the VSG may suffer from power oscillation when the integral coefficient of RPL is too large, but has excellent robustness and stability when connected to weak grid. This feature makes it more appealing in weak grid integration. Finally, the validity of the theoretical analysis is verified by simulations and experiments.