Multirate Method Research Articles

Efficient simulation of DC‐AC power converters using multirate partial differential equations

AbstractSwitch‐mode power converters are used in various applications to convert between different voltage (or current) levels. They use transistors to switch on and off the input voltage to generate a pulsed voltage whose arithmetic average is the desired output voltage of the converter. After smoothening by filters, the converter output is used to supply devices. The simulation of these switch‐mode power converters by conventional time discretization is computationally expensive since a high number of time steps is necessary to properly resolve the unknown state variables and detect switch events of the excitation. This paper proposes a multirate method based on the concept of multirate partial differential equations (MPDEs), which splits the solution into fast varying and slowly varying parts. The method is designed to work with pulse‐width modulated (PWM) excitation with a constant switching cycle and varying duty cycle. It is applicable to problems in which the circuit part generating the pulsed voltage, i.e., consisting of transistors, diodes, and voltage sources, can be approximated by an ideal PWM excitation. Switching‐event detection is no longer necessary and a much smaller number of time steps are required for a decent resolution, thus leading to a highly efficient method. The important case of varying duty cycles in the MPDE framework is addressed for the first time.

A time‐adaptive Dirichlet‐Neumann waveform relaxation method for coupled heterogeneous heat equations

AbstractWe introduce a time adaptive multirate method based on the Dirichlet‐Neumann waveform relaxation (DNWR) algorithm for the simulation of two coupled linear heat equations with strong jumps in the material coefficients across the interface. Numerical results are included to illustrate the advantages of the time adaptive approach over the multirate approach and the robustness of the multirate DNWR method with respect to its sibling, the multirate Neumann‐Neumann waveform relaxation (NNWR) method introduced in a previous work [3].

Extended multirate infinitesimal step methods: Derivation of order conditions

Multirate methods are specially designed for problems with multiple time scales. The multirate infinitesimal step method (MIS) was developed as a generalization of the so called split-explicit Runge–Kutta methods, where the integration of the fast part is conducted analytically. The MIS method was originally evolved for applications related to numerical weather prediction, i.e. the integration of the compressible Euler equation.In this work, an extension to MIS methods will be presented where an arbitrary Runge–Kutta method (RK) is applied for the integration of the fast component. Furthermore, the order convergence from the original MIS method will be reinvestigated including the derivation of conditions up to order four. Additionally will be presented how well-known methods such as recursive flux splitting multirate method, (Schlegel et al., 2012) partitioned Runge–Kutta method, (Jackiewicz and Vermiglio, 2000) and generalized additive Runge–Kutta method, (Sandu and Günther, 2015) are related to or can be cast as an extended MIS method. An exemplary MIS method of order four with five stages will show that the convergence behaviour not only depends on the applied method for the integration of the fast component. The method will further indicate that the used fast time step plays a significant role.

Multirate finite step methods

Many physical phenomena contain different scales. These phenomena can be modeled using partial differential equations (PDEs). Often, these PDEs can be split additively in a fast and a slow part. We extend the multirate infinitesimal steps methods (MIS) to multirate finite step methods (MFS). Both methods resolve the fast scale with an auxiliary differential equation with a fixed slow part. The order conditions of the MIS are derived under the assumption of an exactly resolved fast scale. In contrast, the MFS methods take numerical error of the (numerical) fast , solution into account. We introduce the MFS methods and derive their order conditions for different fast scale integrators. Finally, we give some numerical experiments and compare their stability areas.

Model predictive control for cloud-integrated networked multiagent systems under bandwidth allocation

In this paper, a model predictive control (MPC) strategy is investigated for a cloud-integrated networked multiagent system (NMAS) under bandwidth allocation. The cloud-integrated NMAS is established in cloud computing and network environments. A Stackelberg game is exploited to describe the bandwidth allocation on the cloud-integrated NMAS. With effects on the bandwidth allocation, time delays are compensated by the MPC strategy which is based on a multirate method. Equilibrium solutions and stability conditions are given for the Stackelberg game and the cloud-integrated NMAS, respectively. Effectiveness of the proposed strategy is demonstrated by simulation results.

Design of High-Order Decoupled Multirate GARK Schemes

Multirate time integration methods apply different step sizes to resolve different components of a system based on local activity levels. This local selection of step sizes allows increased computational efficiency while achieving the desired solution accuracy. While the multirate idea is elegant and has been around for decades, multirate methods are not yet widely used in applications. This is due, in part, to the difficulties created by the construction of high-order multirate schemes. Seeking to overcome these challenges, this work focuses on the design of practical high-order multirate methods using the theoretical framework of multirate general-structure additive Runge--Kutta (MrGARK) methods [M. Günther and A. Sandu, Numer. Math., 133 (2016), pp. 497--524], which provides generic order conditions and linear and nonlinear stability analyses. A set of design criteria for practical multirate methods is defined herein: method coefficients should be generic in the step size ratio but should not depend strongly on this ratio, unnecessary coupling between the fast and the slow components should be avoided, and the step size controllers should adjust both the micro- and the macrosteps. Using these criteria, we develop MrGARK schemes of up to order four that are explicit-explicit (both the fast and slow component are treated explicitly), implicit-explicit (implicit in the fast component and explicit in the slow one), and explicit-implicit (explicit in the fast component and implicit in the slow one). Numerical experiments verify the accuracy of these new schemes.

A cycle-based formulation for the simulation of multi time-scale systems — Application to the modeling of the storage system of a fully electric ferry

This paper addresses the simulation of complex systems which consider phenomena with different time scales. Such problems are encountered on studies of electrical systems which try to take into account the simulation of the power converter and its control laws, over representative operating cycles of several hours. Moreover, when storage elements are integrated into a power chain, their aging may need the designer to consider larger time scales, which can then exceed a few years. It is the reason why most studies separate the time scales between the slow dynamics for the energetic, thermal and aging phenomena, and the fast ones to study the power converter and its control laws. This paper presents an original cycle-based and multi-rate method for the simulation of power systems with a wide range of time scales and with high mutual dependency between the fast and slow state variables. This method is applied to the supercapacitor energy storage system of a full-electric ferry. The proposed simulation results take into account at the same time the switching of the power converter and the aging of the supercapacitor, with a reduction of the computational effort greater than 105. In other words, while a full calculation of the problem takes 10 centuries on a personal computer, the proposed method permits to have the same result in only 15 days.

A conservative implicit multirate method for hyperbolic problems

This work focuses on the improvement of a self-adjusting multirate strategy based on an implicit time discretization for the numerical solution of hyperbolic equations that allows to employ different time steps in different areas of the spatial domain. We propose a novel mass conservative multirate approach that can be generalized to various implicit time discretization methods. Mass conservation is achieved by flux partitioning, so that mass exchanges between a cell and its neighbors are exactly balanced. A number of numerical experiments on both nonlinear scalar problems and systems of hyperbolic equations have been carried out to test the efficiency and accuracy of the proposed approach.

Systematic evaluation for multi-rate simulation of DC grids

With wide applications of power electronic devices in modern power systems, simulation using traditional electromechanical and electromagnetic tools suffers low speed and imprecision. Multi-rate methods can enhance efficiency of simulation by decreasing the scale of systems in small time-steps. However, the existing traditional methods for multi-rate simulation suffer the problems of instability and simulation errors. These have hindered the application of multi-rate simulation in power industry. Therefore theoretical evaluation on different multi-rate simulation methods is crucial to understand the feasibility and limitation of the methods, and to contribute to overcome the drawbacks of the traditional methods. In this paper, the multi-rate simulation performance based on two traditional technologies and a Modified Thevenin Interface are evaluated to provide an overall feasibility of multi-rate algorithms in the power simulation. The Modified Thevenin Interface is proposed to overcome the drawbacks in synchronization. Three theorems are proposed and proved for theoretically analyzing the stability of the simulation methods. Error analyses of the multi-rate methods are performed to identify the relationships between errors and simulation conditions. Besides, the accuracy and efficiency performance in a practical project of VSC-MTDC shows the feasibility and necessity by using multi-rate simulation. Through the theoretical analysis, the issues of stability and accuracy of multi-rate simulation for the DC grids have been better understood, based on which an improved simulation algorithm has been proposed to overcome these issues. Long-term system dynamics of large-scale systems containing DC grids and fast transients of HVDC converters can be investigated simultaneously with high speed and sufficient accuracy.

Multirate method for co-simulation of electrical-chemical systems in multiscale modeling

Multiscale modeling by means of co-simulation is a powerful tool to address many vital questions in neuroscience. It can for example be applied in the study of the process of learning and memory formation in the brain. At the same time the co-simulation technique makes it possible to take advantage of interoperability between existing tools and multi-physics models as well as distributed computing. However, the theoretical basis for multiscale modeling is not sufficiently understood. There is, for example, a need of efficient and accurate numerical methods for time integration. When time constants of model components are different by several orders of magnitude, individual dynamics and mathematical definitions of each component all together impose stability, accuracy and efficiency challenges for the time integrator. Following our numerical investigations in Brocke et al. (Frontiers in Computational Neuroscience, 10, 97, 2016), we present a new multirate algorithm that allows us to handle each component of a large system with a step size appropriate to its time scale. We take care of error estimates in a recursive manner allowing individual components to follow their discretization time course while keeping numerical error within acceptable bounds. The method is developed with an ultimate goal of minimizing the communication between the components. Thus it is especially suitable for co-simulations. Our preliminary results support our confidence that the multirate approach can be used in the class of problems we are interested in. We show that the dynamics ofa communication signal as well as an appropriate choice of the discretization order between system components may have a significant impact on the accuracy of the coupled simulation. Although, the ideas presented in the paper have only been tested on a single model, it is likely that they can be applied to other problems without loss of generality. We believe that this work may significantly contribute to the establishment of a firm theoretical basis and to the development of an efficient computational framework for multiscale modeling and simulations.

Implementation of a Multirate Resampler for Multi-carrier Systems on GPUs

Efficient sample rate conversion is of widespread importance in modern communication and signal processing systems. Although many efficient kinds of polyphase filterbank structures exist for this purpose, they are mainly geared toward serial, custom, dedicated hardware implementation for a single task. There is, therefore, a need for more flexible sample rate conversion systems that are resource-efficient, and provide high performance. To address these challenges, we present in this paper an all-software-based, fully parallel, multirate resampling method based on graphics processing units (GPUs). The proposed approach is well-suited for wireless communication systems that have simultaneous requirements on high throughput and low latency. Utilizing the multidimensional architecture of GPUs, our design allows efficient parallel processing across multiple channels and frequency bands at baseband. The resulting architecture provides flexible sample rate conversion that is designed to address modern communication requirements, including real-time processing of multiple carriers simultaneously.

Simulation model of a gear synchronisation unit for application in a real-time HiL environment

ABSTRACTGear shifting simulations using the multibody system approach and the finite-element method are standard in the development of transmissions. However, the corresponding models are typically large due to the complex geometries and numerous contacts, which causes long calculation times. The present work sets itself apart from these detailed shifting simulations by proposing a much simpler but powerful synchronisation model which can be computed in real-time while it is still more realistic than a pure rigid multibody model. Therefore, the model is even used as part of a Hardware-in-the-Loop (HiL) test rig. The proposed real-time capable synchronization model combines the rigid multibody system approach with a multiscale simulation approach. The multibody system approach is suitable for the description of the large motions. The multiscale simulation approach is using also the finite-element method suitable for the analysis of the contact processes. An efficient contact search for the claws of a car transmission synchronisation unit is described in detail which shortens the required calculation time of the model considerably. To further shorten the calculation time, the use of a complex pre-synchronisation model with a nonlinear contour is presented. The model has to provide realistic results with the time-step size of the HiL test rig. To reach this specification, a particularly adapted multirate method for the synchronisation model is shown. Measured results of test rigs of the real-time capable synchronisation model are verified on plausibility. The simulation model is then also used in the HiL test rig for a transmission control unit.

A New Method To Detect Partial Blockage in Gas Pipelines

Summary Because of its efficiency, cleanliness, and reliability, natural gas is an important sector in global energy consumption. It supplies nearly one-fourth of all energy used in the United States and is expected to increase 50% within the next 20 years. More gas-delivery infrastructure is being constructed to meet the transportation requirement of the ever-increasing demand for natural gas, while at the same time, the existing gas infrastructure is aging. Ensuring natural-gas-infrastructure reliability is one of the critical needs for the energy sector. Operators prefer to capitalize on the transportation capacity of these old pipeline systems to reduce the cost for building new pipelines, but they run a high risk of encountering partial blockage in the pipeline, which can cause operating pressure to exceed the safety specification. Therefore, the reliable and timely detection of a partial blockage in a gas pipeline is critical to ensuring the reliability of the natural-gas infrastructure. To design proper pigging tools, it is important to detect the location and size of partial blockages. Physical inspection and mathematical-model simulation are used to identify partial blockage in gas pipelines. Generally, the physical method can result in an accurate detection of the location and size of the partial blockage, but at the expense of production shutdown and high cost/long time to run the physical detection, which is a very expensive measure in a long-distance gas pipeline. The mathematical simulation detects partial blockage through numerical modeling, which could provide a quick evaluation at a much lower cost, but with higher uncertainties. Our literature review indicates that a simple, practical, and reliable method to detect partial blockage without a recorded inlet or outlet pressure is in great demand. In this study, we develop a multirate test method to detect partial blockage in a gas pipeline. By conducting multirate tests, the location and size of the partial blockage can be evaluated. The new method can be applied under the conditions of no measured inlet or outlet pressure, which have not been investigated before. It is worth locating a partial blockage under these conditions because as oil and gas exploration and production move to harsh environments, no pressure gauge being installed at the inlet or outlet of the pipeline can be a common circumstance in the fields. Even for onshore fields or fields with easy access, pressure is not transferred to the central office in real time. In addition, the metering equipment and pressure gauges installed in the pipeline may not be working. Therefore, our method provides a practical, quick, and low-computational-cost approach to estimate partial blockages corresponding to these conditions. The partial blockages in a single pipeline and in parallel/looped pipelines were evaluated in this project by use of the proposed method. Considering that most of the complicated pipeline systems under operation can be decomposed into basic units, such as single pipeline and parallel/looped pipelines, the proposed model can realistically and feasibly identify partial blockage in a complex pipeline network. Furthermore, existing studies assume only single partial blockage in the pipeline, which limits the application of available models because the detection will be misleading if there is more than one partial blockage in the pipeline. To fill this gap, we developed a model to differentiate the single-partial-blockage scenario from the multiple-partial-blockage scenario on the basis of multirate tests. The identification is critical because it guides partial-blockage detection in the right direction.

A cycle-based and multirate approach for power system simulation application to the ageing estimation of a supercapacitor-based ferry

This paper presents a general multi-rate method for the simulation of power systems with a wide range of time scales and with high mutual dependency between the fast and slow state variables. The original concept developed in this work relies on a cycle-based problem formulation, which consists in finding the repeating operation sequences of the system, at the different time scales. Next, a discrete form version of a conventional second order variable step solver is proposed and allows the extrapolation of the results beyond a large number of cycles, while maintaining and controlling a predetermined error tolerance on the state variables. This cycle-based formulation is applied to the ageing behaviour simulation of the energy storage unit of an all-electric ferry which only uses supercapacitors as energy storage. This problem needs the simulation of the electrical power operation of the boat, throughout its lifetime of 20 years. With this original cycle-based formulation, the computation effort can be reduced by several hundreds while maintaining low error tolerances.

Efficient simulation of a slow-fast dynamical system using multirate finite difference schemes

We consider a system of ordinary differential equations describing a slow-fast dynamical system, in particular, a predator-prey system that is highly susceptible to local time variations. This model exhibits coexistence of predatorprey dynamics in the case when the prey population grows much faster than that of the predators with a quite diversified time response. For particular parametric values their interactions show a stable relaxation oscillation in the positive octant. Such characteristics are difficult to mimic using conventional time integrators that are used to solve systems of ordinary differential equations. To resolve this, we design and analyze multirate time integration methods to solve a mathematical model for a slow-fast dynamical system. Proposed methods are based on using extrapolation multirate discretisation algorithms. Through these methods, we reduce the integration time by integrating the slow sub-system with a larger step length than the fast sub-system. This allows us to efficiently solve multiscale ordinary differential equations. Besides theoretical results, we provide thorough numerical experiments which confirm that these multirate schemes outperform corresponding single-rate schemes substantially both in terms of computational work and CPU times.

Determining the propagation path of a disturbance in multi-rate process and electromechanical systems

This paper proposes a multi-rate method to identify the propagation path of a persistent disturbance in an enlarged system envelope which includes the process plant and its electromechanical equipment. The need to integrate process and equipment diagnosis has been highlighted by industrial commentators. However, process and electromechanical measurements often have different sampling rates. The multi-rate method proposed extends a state-of-the-art propagation path method so that it combines fast-sampled electromechanical measurements and slow-sampled process measurements. The method is based on non-linear mutual prediction, which yields the directionality in the relationship between two time series. The method was demonstrated and validated, giving the expected outcome in an experimental case study, in which the root cause and propagation path of the disturbance were known.

Locomotive wheel slip detection based on multi-rate state identification of motor load torque

This paper presents a locomotive slip phenomenon detection method based on the Multi-rate Extended Kalman Filter (MREKF) state identification method. The proposed method combines the multi-rate method and the EKF method to identify traction motor load torque in order to detect locomotive slip phenomenon. Unlike traditional methods, the proposed detection only based on electrical quantities, achieves a shorter detection time due to the much smaller time constants in the electrical systems. The adhesion availability, which is an important performance index of a locomotive, is improved by rapid detection method. An extended model which contains vector-controlled induction motor model, wheel-set multiple axle model and locomotive motion model is proposed. The locomotive slip phenomenon corresponding to the actual slipping is realized in the simulation. The simulation result matches with the actual data obtained from HXD2 locomotive. Moreover, the proposed locomotive slip phenomenon detection method has been applied to the extended model, and experimental results verified the accuracy and effectiveness of this model. A comprehensive analysis is proposed by compare multi-rate and single-rate EKF method. Experimental results depict that the detection time of slip phenomenon has been greatly reduced.

Multivariate Detection of Transient Disturbances for Uni- and Multirate Systems

This paper presents a method to detect transient disturbances in a multivariate context, and an extension of that method to handle multirate systems. Both methods are based on a time series analysis technique known as nearest neighbors, and on multivariate statistics implemented as a singular value decomposition. The motivation for these developments is that there is an increasing industrial requirement for the analysis of data sets comprising measurements from industrial processes together with their associated electrical and mechanical equipment. These systems are increasingly affected by transient disturbances, and their measurements are commonly sampled at different rates. This paper demonstrates superior results with the multivariate method in comparison with the univariate approach, and with the multirate method in comparison to a unirate method, for which the fast-sampled measurements had to be downsampled. The method is demonstrated on experimental and industrial case studies.

A Linearly Fourth Order Multirate Runge–Kutta Method with Error Control

To integrate large systems of locally coupled ordinary differential equations with disparate timescales, we present a multirate method with error control that is based on the Cash---Karp Runge---Kutta formula. The order of multirate methods often depends on interpolating certain solution components with a polynomial of sufficiently high degree. By using cubic interpolants and analyzing the method applied to a simple test equation, we show that our method is fourth order linearly accurate overall. Furthermore, the size of the region of absolute stability is increased when taking many "micro-steps" within a "macro-step." Finally, we demonstrate our method on three simple test problems to confirm fourth order convergence.

Signal Reconstruction with Adaptive Multi-Rate Signal Processing Algorithms

Multi-rate digital signal processing techniques have been developed in recent years for a wide range of applications, such as speech and image compression, statistical and adaptive signal processing and digital audio. Multi-rate statistical and adaptive signal processing methods provide solution to original signal reconstruction, using observation signals sampled at different rates. In this study, a signal reconstruction process by using the observation signals which are sampled at different sampling rates is presented. The results are compared with the least mean squares (LMS) and the normalized least mean squares (NLMS) methods. As the results indicate, the signal estimation obtained is much more efficient than the previous studies. Designed multi-rate scheme provides significant advantages in terms of error and estimation accuracy.