Parallel-in-Time Simulation of Power Converters Using Multirate PDEs
aa r X i v : . [ m a t h . NA ] J u l Parallel-in-Time Simulation of PowerConverters Using Multirate PDEs
Andreas Pels, Iryna Kulchytska-Ruchka, and Sebastian Sch¨ops
Abstract
This paper presents a numerical algorithm for the simulation of pulse-width modulated power converters via parallelization in time domain. The methodapplies the multirate partial differential equation approach on the coarse grid ofthe (two-grid) parallel-in-time algorithm Parareal. Performance of the proposed ap-proach is illustrated via its application to a DC-DC converter.
Switch-mode power converters are devices which convert electric voltages or cur-rents between different levels. For this purpose they use transistors to switch on andoff the input voltage or current to obtain the desired average voltage or current at theoutput of the converter. A technique called pulse-width modulation (PWM) is oftenutilized to control the transistors, i.e., to generate the pulsed voltage from a givencarrier and reference. An exemplary circuit of a buck converter (DC-DC converter)is depicted in Fig. 1a along with its solution in Fig. 2. It consists of fast periodicallyvarying ripples and a slowly varying envelope. The simulation of these power con-verters with conventional time stepping is computationally expensive since a highnumber of time steps is necessary to resolve the fast variations induced by the tran-sistor switching.This paper proposes the simulation of power converters using a combination oftwo methods, namely the parallel-in-time algorithm Parareal [7] and a multirateapproach based on Multirate Partial Differential Equations (MPDEs) [8]. This is ac-complished via the application of the MPDE approach on the coarse grid of Parareal.It allows the coarse propagator to obtain a more precise solution given the PWM in-
Andreas Pels, Iryna Kulchytska-Ruchka, Sebastian Sch¨opsComputational Electromagnetics Group, Technical University of Darmstadt, Schlogartenstr. 8,64289 Darmstadt, Germany, e-mail: { pels, kulchytska, schoeps } @temf.tu-darmstadt.de 1 Andreas Pels, Iryna Kulchytska-Ruchka, and Sebastian Sch¨ops RCv C L i L R L v i (a) T s △ t h D = △ t h T s time (ms) vo lt a g e v i ( V ) (b) Fig. 1: Power converter model with pulsed voltage source: (a) Circuit of a simplifiedbuck converter. Transistor switching is modeled as pulsed voltage source. (b) PWMgenerated pulsed voltage.put signal, in contrast to the standard coarse propagator when using a large time stepon the original system of equations.The paper is organized as follows: first we introduce our model problem withpulsed excitation in Section 2, then in Section 3 the Parareal method is summarized,Section 4 proposes the usage of MPDEs as coarse propagators for Parareal thatcan deal with pulsed right-hand sides and finally Section 5 discusses a numericalexample before concluding the paper.
Switch-mode power converters, which convert AC to DC, DC to AC, AC to AC, orDC to DC voltages, are frequently used devices. They use power electronic switchesto periodically switch the input voltage on and off to regulate the output voltage.For example a buck converter (DC-DC converter) transforms a given voltage to alower output voltage. It consists of a part that generates a pulsed voltage v i anda filter circuit. The latter is shown in Fig. 1a. The pulsed voltage, see Fig. 1b, isoften generated using PWM. Important quantities defining the pulsed signal are theswitching period T s and the duty cycle D which is the relation between the “on”-time and the switching period. Given a reference signal r ( t ) and a carrier signal s ( t ) the pulsed voltage is generated by v i ( t ) = V i (cid:0) sgn ( r ( t ) − s ( t )) + (cid:1) , (1)where sgn denotes the sign function and V i is the amplitude. The converter circuit ismathematically described by a system of ordinary or differential-algebraic equations(DAEs), e.g., arallel-in-Time Simulation of Power Converters Using Multirate PDEs 3 C u rr e n t i L ( A ) a ndvo lt a g e v C ( V ) i L (A) v C (V) Fig. 2: Exemplary solution of the buck converter depicted in Fig. 1a. Switchingfrequency f s = / T s = A dd t x ( t ) + B x ( t ) = c ( t ) , t ∈ ( t , T ] , (2)with given initial value x ( t ) = x , where x ( t ) ∈ R N s is the unknown solution vectorconsisting for example of currents and voltages, A , B ∈ R N s × N s are matrices, and c ( t ) ∈ R N s is the right-hand side containing current and voltage sources, e.g., thepulsed voltage v i ( t ) . The system may be assembled from lumped element descrip-tions based on loop or (modified) nodal analysis as described in [2]. Please note, thatwe focus on the linear case but the approach can be straight-forwardly generalized,e.g., considering B = B ( x ) .The solution of power converters, e.g., shown in Fig. 2, exhibits the multirate phe-nomenon: slow variations in the solution require large time intervals until a steadystate is reached, i.e., a large end time point T while the fast dynamics due to theswitching enforce small time steps. This is the motivation to turn to (parallel) meth-ods that can exploit this multirate behavior. In the following, we focus on the settlingprocess until the steady state is reached. If one is interested only in the latter, thenother methods may also be used, for example the application of Parareal for time-periodic problems is a natural generalization of this work, see, e.g., [5]. Parareal is an iterative algorithm which is able to accelerate the solution of (2) viaparallelization in time. The method originates from [7] and its superlinear conver-gence is proven in [3]. The two main ingredients of Parareal are the fine and thecoarse propagators. We denote by F ( t , t , x ) and G ( t , t , x ) the solutions of theinitial value problem (IVP) (2) at t ∈ ( t , T ] obtained with sequential time steppingusing fine and coarse time steps, respectively. Andreas Pels, Iryna Kulchytska-Ruchka, and Sebastian Sch¨ops
Partitioning the time interval t = T < T < · · · < T N = T we write the Pararealiteration: for k = , , . . . and n = , . . . , N solve X ( k + ) = x , (3) X ( k + ) n = F (cid:0) T n , T n − , X ( k ) n − (cid:1) + G (cid:0) T n , T n − , X ( k + ) n − (cid:1) − G (cid:0) T n , T n − , X ( k ) n − (cid:1) . (4)The solution operator F is assumed to deliver a very accurate solution (e.g., using anumerical time-integration method with small time steps δ T ) and can be executed inparallel, while G gives rough information about the solution using a cheap method(e.g., using a numerical method with large time steps ∆ T i = T i + − T i ) and has to becalculated sequentially, cf. (4).A difficulty in applying Parareal to solve problems with PWM input is that anaive implementation of a coarse propagator using a time-integrator with large timesteps will not capture the high-frequency dynamics and may also fail to propa-gate low-frequency components. A modified Parareal algorithm which still approx-imately captures the high-frequency behavior was introduced in [4]. The idea is toseparate the high-frequency (pulsed) components from the low-frequency compo-nents, i.e., A dd t x ( t ) + B x ( t ) = ¯ c ( t ) + ˜ c ( t ) | {z } = c ( t ) , (5)where ¯ c can be given as a few low-frequency sinusoids from a (fast) Fourier trans-form and ˜ c ( t ) : = c ( t ) − ¯ c ( t ) is the remainder. This allows to define a reduced coarsepropagator ¯ G fft which solves A dd t x ( t ) + B x ( t ) = ¯ c ( t ) (6)and gives rise to a modified Parareal update formula with coarse propagator ¯ G fft in(3)-(4). This modified method converges reliably but possibly with reduced order[4]. In this paper we propose an alternative method to perform time integration byusing the MPDE approach as the coarse propagator. The MPDE approach, which is used for obtaining the coarse solution in Pararealuses the MPDE concept [1]. For the given problem the solution can be convenientlydecomposed into a slowly varying envelope and fast periodically varying ripplesusing the solution expansion [8] b x j ( t , t ) . = N p ∑ k = y j , k ( t ) w k ( τ ( t )) = w ⊤ ( τ ( t )) y j ( t ) , (7) arallel-in-Time Simulation of Power Converters Using Multirate PDEs 5 where y j , k ( t ) are slowly varying coefficients and w k ( τ ( t )) are a finite set of basisfunctions ( k = , . . . , N p ) whose periodicity is accounted for by the relative time τ ( t ) = t T s mod 1. Its application to (2) yields A (cid:18) ∂ b x ( t , t ) ∂ t + ∂ b x ( t , t ) ∂ t (cid:19) + B b x ( t , t ) = b c ( t , t ) , (8)where the relation between the original (2) and the MPDE (8) solution and right-hand side are given by b x ( t , t ) = x ( t ) , b c ( t , t ) = c ( t ) . (9)This implies that if a solution to (8) is found, the solution of (2) can be extractedfrom it. Applying a Galerkin approach along the fast time scale t leads to the en-larged equation system A ( t ) d y d t + B ( t ) y ( t ) = C ( t ) , (10)where the matrices are given by [8] A = A ⊗ J , with J = T s 1 Z w ( τ ) w ⊤ ( τ ) d τ , B = B ⊗ J + A ⊗ Q , with Q = − Z ∂ w ( τ ) ∂τ w ⊤ ( τ ) d τ , C = Z T s b c ( t , t ) ⊗ w ( τ ( t )) d t . Suitable basis functions, which can well represent the ripples in the power convertersolution, are, e.g., B-Splines with suitable continuity or the PWM basis functions[6]. The latter are global polynomial ansatz functions with w ( τ , D ) = w ( τ , D ) piecewise linear and w k ( τ , D ) is obtained recursively by integrating w k − ( τ ) andorthonormalizing for 3 ≤ k ≤ N p , see Fig. 3. It has been shown in [8] that they arecapable of very effectively representing the ripples in linear problems.Finally, equation (10) can be time-stepped along t by using much larger timesteps than are needed to solve (2) since the fast variations are taken into account bythe basis functions. The accuracy of the solution (reconstructed using (7)) increaseswith N p . However increasing N p also makes each time step of an implicit methodmore costly since an enlarged linear equation system has to be inverted. Neverthe-less, even with very few basis functions the reconstructed solution can be expectedto capture the main features of the exact solution. This motivates the introductionof another coarse propagator ¯ G mpde in Parareal which solves (10) and extracts after-wards the single-time solution according to (7). Andreas Pels, Iryna Kulchytska-Ruchka, and Sebastian Sch¨ops D − − τ w w w w (a) D V relative time τ (b) Fig. 3: Construction of basis functions with cusp at relative switching time D : (a)PWM basis functions on relative time interval and (b) right-hand side. The proposed approach is applied to the example of the buck converter (see Fig. 1a).Its circuit is described by the IVP (2) given by A = (cid:20) L C (cid:21) ; B = (cid:20) R L − / R (cid:21) and c ( t ) = (cid:20) v i ( t ) (cid:21) , with inductance L = − H , capacitance C = − F , resistances R L = − Ω and R = . Ω . The PWM input v i ( t ) has the amplitude of V i = s ( t ) = t f s mod 1 with switching frequency of f s = r ( t ) = . [ , ] ms is partitioned into N =
40 windows for all Parareal variants. The coarsetime step size is ∆ T = T / N = × − s and the fine propagator uses the time step δ T = − s. All solutions are obtained with the implicit Euler method.First, the classical Parareal method (3)-(4) is applied. It solves the original sys-tem (2) with the PWM input in both propagators, i.e., G and F . This method iscompared to two variants where G is changed to: 1.) ¯ G fft which solves system (6)containing only the DC component instead of the PWM signal on the right-handside (modified Parareal [4]); 2.) ¯ G mpde which solves (2) using the MPDE approachwith N p = N p = b c ( t , t ) = c ( t ) .The maximal relative mismatch of the solution (‘jump’) at the synchronizationpoints T n for n = , . . . , N − N p = − , where we need 4 vs. 2iterations. The classical Parareal converges up to the relative tolerance of 10 − in 9iterations. This corresponds to 2 700 and 360 sequential solutions of linear algebraicsystems of size N s = arallel-in-Time Simulation of Power Converters Using Multirate PDEs 7 − − − − number of iterations m a x i m a l r e l a ti v e m i s m a t c h PWM inputDC componentMPDE 1MPDE 3 (a) S e qu e n ti a l P W M D C M P D E M P D E . · U n it s o f ti m e fine solver ( δ T )coarse solver ( ∆ T ) (b) Fig. 4: Convergence of Parareal using different coarse propagators for the buck con-verter model (a) and units of time for solving the effective linear systems (b).systems in total. By the number of sequential solves we mean the number of solvercalls which cannot be carried out in parallel (communication costs are neglected).The approaches using the DC component and the MPDE approach with N p = N p = N s = N s × N p = N s requires one unit of time. Then, the classical Parareal takes 3060 unitsof time. Using MPDE 1 (i.e., N p =
1) or the DC-component as coarse propagatorrequires only 2720 units of time. Finally, MPDE 3 (i.e., N p =
3) takes 2940 unitsof time. We see that, even in this theoretical setting with an optimal solver, the in-creased accuracy of the coarse propagator, i.e, application of MPDE 3 with N p = G mpde us-ing a constant basis function, i.e., N p = G fft usingonly the DC excitation perform very similarly (if not identically). This resemblanceis not surprising since the MPDE 1 approach with N p = N p >
3) does not improve the convergence of Parareal, theyare similar to the case N p = Andreas Pels, Iryna Kulchytska-Ruchka, and Sebastian Sch¨ops
In this paper we introduced a novel parallel-in-time algorithm, able to treat systemsexcited by pulse-width modulated signals. The method extends the two-grid Pararealalgorithm by exploiting the MPDE solution approach on the coarse grid. It was ap-plied to the time-domain simulation of a buck converter supplied by a PWM voltagesource. Comparison of the proposed algorithm to the standard Parareal method andto the Parareal with reduced coarse dynamics illustrated its faster convergence. Fu-ture research will further investigate the similarity of Parareal with the MPDE coarsepropagator and the modified Parareal as well as higher order MPDE approaches ascoarse propagators.
Acknowledgements
The authors thank Ruth Vazquez Sabariego from KU Leuven for many fruit-ful discussions on the MPDE approach. This research was supported by the Excellence Initiativeof the German Federal and State Governments and the Graduate School of Computational Engi-neering at Technische Universit¨at Darmstadt, as well as by DFG grant SCHO1562/1-2 and BMBFgrant 05M2018RDA (PASIROM).
References
1. Hans Georg Brachtendorf, G¨unther Welsch, Rainer Laur, and Angelika Bunse-Gerstner. Nu-merical steady state analysis of electronic circuits driven by multi-tone signals.
Electrical En-gineering (Archiv f¨ur Elektrotechnik) , 79(2):103–112, 1996.2. Diana Est´evez Schwarz and Caren Tischendorf. Structural analysis of electric circuits andconsequences for MNA.
International Journal of Circuit Theory and Applications , 28(2):131–162, 2000.3. Martin J. Gander and Ernst Hairer. Nonlinear convergence analysis for the parareal algo-rithm. In Ulrich Langer, Marco Discacciati, David E. Keyes, Olof B. Widlund, and WalterZulehner, editors,
Domain Decomposition Methods in Science and Engineering XVII , pages45–56. Springer, 2008.4. Martin J. Gander, Iryna Kulchytska-Ruchka, Innocent Niyonzima, and Sebastian Sch¨ops. Anew parareal algorithm for problems with discontinuous sources.
SIAM Journal on ScientificComputing , 41(2):B375–B395, 2019.5. Martin J. Gander, Iryna Kulchytska-Ruchka, and Sebastian Sch¨ops. A new parareal algorithmfor time-periodic problems with discontinuous inputs. In
Domain Decomposition Methodsin Science and Engineering XXV , Lecture Notes in Computational Science and Engineering.Springer, 2019.6. Johan Gyselinck, Claudia Martis, and Ruth V. Sabariego. Using dedicated time-domain basisfunctions for the simulation of pulse-width-modulation controlled devices – application to thesteady-state regime of a buck converter. In
Electromotion 2013 , 2013.7. Jacques-Louis Lions, Yvon Maday, and Gabriel Turinici. A parareal in time discretization ofPDEs.
Comptes Rendus de l’Acad´emie des Sciences – Series I – Mathematics , 332(7):661–668,2001.8. Andreas Pels, Johan Gyselinck, Ruth V. Sabariego, and Sebastian Sch¨ops. Efficient simulationof DC-DC switch-mode power converters by multirate partial differential equations.