Modeling of Deadtime Events in Power Converters with Half-Bridge Modules for a Highly Accurate Hardware-in-the-Loop Fixed Point Implementation in FPGA

Hardware-in-the-loop (HIL) simulations of power converters must achieve a truthful representation in real time with simulation steps on the order of microseconds or tens of nanoseconds. The numerical solution for the differential equations that model the state of the converter can be calculated using the fourth-order Runge–Kutta method, which is notably more accurate than Euler methods. However, when the mathematical error due to the solver is drastically reduced, other sources of error arise. In the case of converters that use deadtimes to control the switches, such as any power converter including half-bridge modules, the inductor current reaching zero during deadtimes generates a model error large enough to offset the advantages of the Runge–Kutta method. A specific model is needed for such events. In this paper, an approximation is proposed, where the time step is divided into two semi-steps. This serves to recover the accuracy of the calculations at the expense of needing a division operation. A fixed-point implementation in VHDL is proposed, reusing a block along several calculation cycles to compute the needed parameters for the Runge–Kutta method. The implementation in a low-cost field-programmable gate arrays (FPGA) (Xilinx Artix-7) achieves an integration time of 1μs. The calculation errors are six orders of magnitude smaller for both capacitor voltage and inductor current for the worst case, the one where the current reaches zero during the deadtimes in 78% of the simulated cycles. The accuracy achieved with the proposed fixed point implementation is very close to that of 64-bit floating point and can operate in real time with a resolution of 1μs. Therefore, the results show that this approach is suitable for modeling converters based on half-bridge modules by using FPGAs. This solution is intended for easy integration into any HIL system, including commercial HIL systems, showing that its application even with relatively high integration steps (1μs) surpasses the results of techniques with even faster integration steps that do not take these events into account.

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Application of the piecewise constant method in nonlinear dynamics of drill string

International Journal of Nonlinear Sciences and Numerical Simulation ◽

10.1515/ijnsns-2019-0167 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Jialin Tian ◽

Jie Wang ◽

Yi Zhou ◽

Lin Yang ◽

Changyue Fan ◽

...

Keyword(s):

Nonlinear Dynamics ◽

Dynamic Model ◽

Dynamic Characteristics ◽

Drill String ◽

Vibration Velocity ◽

Kutta Method ◽

Time Step ◽

Piecewise Constant ◽

Runge Kutta Method ◽

Runge Kutta

Abstract Aiming at the current development of drilling technology and the deepening of oil and gas exploration, we focus on better studying the nonlinear dynamic characteristics of the drill string under complex working conditions and knowing the real movement of the drill string during drilling. This paper firstly combines the actual situation of the well to establish the dynamic model of the horizontal drill string, and analyzes the dynamic characteristics, giving the expression of the force of each part of the model. Secondly, it introduces the piecewise constant method (simply known as PT method), and gives the solution equation. Then according to the basic parameters, the axial vibration displacement and vibration velocity at the test points are solved by the PT method and the Runge–Kutta method, respectively, and the phase diagram, the Poincare map, and the spectrogram are obtained. The results obtained by the two methods are compared and analyzed. Finally, the relevant experimental tests are carried out. It shows that the results of the dynamic model of the horizontal drill string are basically consistent with the results obtained by the actual test, which verifies the validity of the dynamic model and the correctness of the calculated results. When solving the drill string nonlinear dynamics, the results of the PT method is closer to the theoretical solution than that of the Runge–Kutta method with the same order and time step. And the PT method is better than the Runge–Kutta method with the same order in smoothness and continuity in solving the drill string nonlinear dynamics.

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Numerical calculation for coupling vibration system by Piecewise-Laplace method

Science Progress ◽

10.1177/0036850420938555 ◽

2020 ◽

Vol 103 (3) ◽

pp. 003685042093855

Author(s):

Pan Fang ◽

Kexin Wang ◽

Liming Dai ◽

Chixiang Zhang

Keyword(s):

Numerical Computation ◽

Laplace Transformation ◽

Original System ◽

Continuity Condition ◽

Kutta Method ◽

Laplace Method ◽

Time Step ◽

Coupling Vibration ◽

Runge Kutta Method ◽

Runge Kutta

To improve the reliability and accuracy of dynamic machine in design process, high precision and efficiency of numerical computation is essential means to identify dynamic characteristics of mechanical system. In this paper, a new computation approach is introduced to improve accuracy and efficiency of computation for coupling vibrating system. The proposed method is a combination of piecewise constant method and Laplace transformation, which is simply called as Piecewise-Laplace method. In the solving process of the proposed method, the dynamic system is first sliced by a series of continuous segments to reserve physical attribute of the original system; Laplace transformation is employed to separate coupling variables in segment system, and solutions of system in complex domain can be determined; then, considering reverse Laplace transformation and residues theorem, solution in time domain can be obtained; finally, semi-analytical solution of system is given based on continuity condition. Through comparison of numerical computation, it can be found that precision and efficiency of numerical results with the Piecewise-Laplace method is better than Runge-Kutta method within same time step. If a high-accuracy solution is required, the Piecewise-Laplace method is more suitable than Runge-Kutta method.

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Spectral Fixed Point Method for Nonlinear Oscillation Equation with Periodic Solution

Mathematical Problems in Engineering ◽

10.1155/2013/538716 ◽

2013 ◽

Vol 2013 ◽

pp. 1-9

Author(s):

Ding Xu ◽

Xian Wang ◽

Gongnan Xie

Keyword(s):

Periodic Solution ◽

Fixed Point ◽

Nonlinear Oscillation ◽

Fixed Point Method ◽

Point Method ◽

Kutta Method ◽

Runge Kutta Method ◽

Runge Kutta ◽

Linear Oscillation ◽

Oscillation Equation

Based on the fixed point concept in functional analysis, an improvement on the traditional spectral method is proposed for nonlinear oscillation equations with periodic solution. The key idea of this new approach (namely, the spectral fixed point method, SFPM) is to construct a contractive map to replace the nonlinear oscillation equation into a series of linear oscillation equations. Usually the series of linear oscillation equations can be solved relatively easily. Different from other existing numerical methods, such as the well-known Runge-Kutta method, SFPM can directly obtain the Fourier series solution of the nonlinear oscillation without resorting to the Fast Fourier Transform (FFT) algorithm. In the meanwhile, the steepest descent seeking algorithm is proposed in the framework of SFPM to improve the computational efficiency. Finally, some typical cases are investigated by SFPM and the comparison with the Runge-Kutta method shows that the present method is of high accuracy and efficiency.

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Trajectory errors of different numerical integration schemes diagnosed with the MPTRAC advection module driven by ECMWF operational analyses

Geoscientific Model Development ◽

10.5194/gmd-11-575-2018 ◽

2018 ◽

Vol 11 (2) ◽

pp. 575-592 ◽

Cited By ~ 6

Author(s):

Thomas Rößler ◽

Olaf Stein ◽

Yi Heng ◽

Paul Baumeister ◽

Lars Hoffmann

Keyword(s):

Integration Scheme ◽

Kutta Method ◽

Time Step ◽

Third Order ◽

Truncation Errors ◽

The Third ◽

Trajectory Calculations ◽

Integration Schemes ◽

Runge Kutta Method ◽

Runge Kutta

Abstract. The accuracy of trajectory calculations performed by Lagrangian particle dispersion models (LPDMs) depends on various factors. The optimization of numerical integration schemes used to solve the trajectory equation helps to maximize the computational efficiency of large-scale LPDM simulations. We analyzed global truncation errors of six explicit integration schemes of the Runge–Kutta family, which we implemented in the Massive-Parallel Trajectory Calculations (MPTRAC) advection module. The simulations were driven by wind fields from operational analysis and forecasts of the European Centre for Medium-Range Weather Forecasts (ECMWF) at T1279L137 spatial resolution and 3 h temporal sampling. We defined separate test cases for 15 distinct regions of the atmosphere, covering the polar regions, the midlatitudes, and the tropics in the free troposphere, in the upper troposphere and lower stratosphere (UT/LS) region, and in the middle stratosphere. In total, more than 5000 different transport simulations were performed, covering the months of January, April, July, and October for the years 2014 and 2015. We quantified the accuracy of the trajectories by calculating transport deviations with respect to reference simulations using a fourth-order Runge–Kutta integration scheme with a sufficiently fine time step. Transport deviations were assessed with respect to error limits based on turbulent diffusion. Independent of the numerical scheme, the global truncation errors vary significantly between the different regions. Horizontal transport deviations in the stratosphere are typically an order of magnitude smaller compared with the free troposphere. We found that the truncation errors of the six numerical schemes fall into three distinct groups, which mostly depend on the numerical order of the scheme. Schemes of the same order differ little in accuracy, but some methods need less computational time, which gives them an advantage in efficiency. The selection of the integration scheme and the appropriate time step should possibly take into account the typical altitude ranges as well as the total length of the simulations to achieve the most efficient simulations. However, trying to summarize, we recommend the third-order Runge–Kutta method with a time step of 170 s or the midpoint scheme with a time step of 100 s for efficient simulations of up to 10 days of simulation time for the specific ECMWF high-resolution data set considered in this study. Purely stratospheric simulations can use significantly larger time steps of 800 and 1100 s for the midpoint scheme and the third-order Runge–Kutta method, respectively.

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Application of Differential Transformation Method for Solving HIV Model with Anti-Viral Treatment

BAREKENG JURNAL ILMU MATEMATIKA DAN TERAPAN ◽

10.30598/barekengvol14iss3pp378-388 ◽

2020 ◽

Vol 14 (3) ◽

pp. 378-388

Author(s):

Esther Y. Bunga ◽

Meksianis Z. Ndii

Keyword(s):

Finite Difference ◽

Differential Equations ◽

Transformation Method ◽

Differential Transformation Method ◽

Kutta Method ◽

System Of Differential Equations ◽

Time Step ◽

Differential Transformation ◽

Runge Kutta Method ◽

Runge Kutta

Mathematical models have been widely used to understand complex phenomena. Generally, the model is in the form of system of differential equations. However, when the model becomes complex, analytical solutions are not easily found and hence a numerical approach has been used. A number of numerical schemes such as Euler, Runge-Kutta, and Finite Difference Scheme are generally used. There are also alternative numerical methods that can be used to solve system of differential equations such as the nonstandard finite difference scheme (NSFDS), the Adomian decomposition method (ADM), Variation iteration method (VIM), and the differential transformation method (DTM). In this paper, we apply the differential transformation method (DTM) to solve system of differential equations. The DTM is semi-analytical numerical technique to solve the system of differential equations and provides an iterative procedure to obtain the power series of the solution in terms of initial value parameters.. In this paper, we present a mathematical model of HIV with antiviral treatment and construct a numerical scheme based on the differential transformation method (DTM) for solving the model. The results are compared to that of Runge-Kutta method. We find a good agreement of the DTM and the Runge-Kutta method for smaller time step but it fails in the large time step.

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