# Multirate PWM balance method for the efficient field-circuit coupled simulation of power converters

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## Abstract

The field-circuit coupled simulation of switch-mode power converters with conventional time discretization is computationally expensive since very small time steps are needed to appropriately account for steep transients occurring inside the converter, not only for the degrees of freedom (DOFs) in the circuit, but also for the large number of DOFs in the field model part. An efficient simulation technique for converters with idealized switches is obtained using multirate partial differential equations, which allow for a natural separation into components of different time scales. This paper introduces a set of new PWM eigenfunctions which decouple the systems of equations and thus yield an efficient simulation of the field-circuit coupled problem. The resulting method is called the multirate PWM balance method.

## Keywords

Finite element methods Numerical analysis Partial differential equations Linear circuits DC-DC power conversion## Abbreviations

- DOFs
degrees of freedom

- PWM
pulse-width modulation

- MPDE
multirate partial differential equation

## 1 Introduction

The paper is structured as follows. Section 2 introduces the concept of MPDEs and explains the solving procedure using Galerkin approach and conventional time discretization. Subsequently Sect. 3 presents the original PWM basis functions as described in [5]. In Sect. 4 the PWM eigenfunctions are developed and their advantageous properties for the solving process are highlighted. Finally Sect. 5 summarizes numerical results and compares the three different solution approaches, i.e., conventional time discretization and the MPDE approach with PWM basis functions on the one hand and PWM eigenfunctions on the other hand. The paper is concluded by summarizing its content in Sect. 6.

## 2 Multirate formulation

*D*is the duty cycle.

**h**with \(\mathbf {h}(0)=\mathbf {x}_{0}\) is a function defining the initial values for all \(t_{2}\). The solution along the fast time scale \(t_{2}\) is periodic, i.e., \(\mathbf {\widehat {x}}(t_{1}, t_{2}+T_{\mathrm {s}}) = \mathbf {\widehat {x}}(t_{1}, t_{2})\). The multivariate right-hand side is chosen as \(\mathbf {\widehat {c}}(t_{1}, t_{2})=\mathbf {c}(t_{2})\), i.e., the pulses of the excitation occur along the fast time scale. It is possible to use MPDEs with more than two time scales. However, in the applications of this paper, it is not necessary and furthermore often not feasible since the dimension of the computation domain increases and thus also the computational effort to calculate the solution.

*j*-th solution component \({\widehat {x}}_{j}(t_{1}, t_{2})\) is approximated by expanding it into periodic basis functions \(p_{k}\) depending on the fast time scale \(t_{2}\) and coefficients \(w_{j,k}\) depending on the slow time scale \(t_{1}\)

**p**and ⊗ denotes the Kronecker product.

## 3 PWM basis functions

*D*of the excitation by construction. The higher-order basis functions \(p_{k}(\tau )\), \(2\leq k \leq N_{\mathrm {p}}\) are recursively obtained by integrating the basis functions of lower order \(p_{k-1}(\tau )\) and orthonormalizing them using the Gram–Schmidt algorithm. The generated basis functions are depicted in Fig. 2.

For the PWM basis functions, the matrices \(\boldsymbol {\mathcal {J}}\) and \(\boldsymbol {\mathcal {Q}}\) from (6) are given by \(T_{\mathrm {s}}\) multiplied by the identity matrix (due to the orthonormality of the basis functions) and a square matrix with around 25% of non-zero entries, respectively. Solving the problem requires time stepping of the entire system (5).

## 4 PWM eigenfunctions

*is a diagonal matrix with diagonal entries \({\lambda _{0}, \lambda _{1}, \dots , \lambda _{N_{\mathrm {p}}}}\). Thus the resulting matrices in (13) are block-diagonal and the degrees of freedom can be block-wisely decoupled. This leads to \(N_{\mathrm {p}}+1\) independent systems of equations given by*

**Λ***is complex, there is also a complex conjugate counterpart. The solutions of the decoupled system of equations corresponding to this complex eigenvalue and its conjugate complex counterpart, are, as a result, complex conjugate to each other. Therefore it is sufficient to solve one of them. This is similar to harmonic balance methods in which the harmonic basis functions are given by pairs of complex conjugates leading to similar systems of equations. In analogy to “harmonic balance method”, we call the developed method the “multirate PWM balance method”.*

**Λ**## 5 Test case and numerical results

**r**is the position vector,

*t*is the time, \(\mathbf {A}_{ \mathrm{m}}\) is the modified magnetic vector potential [4], \(\mathbf {J}_{\mathrm {s}}\) are the imposed currents, \(\mu =4 \pi \times 10^{-7} \) H/m is the magnetic permeability and

*σ*is the conductivity which is only non-zero in the ferrite core (\(\sigma _{\mathrm{fe}}\)). The problem is considered on a 2D planar domain with homogeneous Dirichlet boundary conditions.

**K**is the stiffness matrix, \(\mathbf {a}(t)\) gathers the degrees of freedom (DOFs) related to the magnetic vector potential,

**P**is the discretization of the winding function [13] and \(i_{\mathrm{L}}(t)\) is the current through the inductor. The field-circuit coupling is expressed as follows. An additional variable is introduced for the magnetic flux linkage \(\varPhi (t)=\mathbf {P}^{\top }\mathbf {a}(t)\). All equations are coupled monolithically into the index-1 differential-algebraic system of equations [1]

*j*-th element of

**h**. It only has to satisfy the condition \(\mathbf {h}(0) = \mathbf {\widehat {x}}(0,0)=\mathbf {x}_{0}\). Consequently there is a high degree of freedom in choosing the initial values \(\mathbf {w}(0)\) for the system of equations (5). However not all choices lead to an efficient simulation, i.e., low dynamics in the slow time scale. The following choice of initial values has proven advantageous. First, the steady-state solution is calculated, i.e.,

*j*. The remaining coefficients are calculated by solving the solution expansion (4) for \(w_{j,0}(0)\) and using the condition \(\mathbf {\widehat {x}}(0,0)=\mathbf {x}_{0}\). In summary the initial coefficients are given by

To calculate the reference solution with a conventional adaptive time discretization, the MATLAB solver ode15s is used. It is modified to restart the simulation at the known switching instances. Consistent initial values for the restart of the solver are calculated by using a Newton–Raphson algorithm to solve the set of algebraic equations. The required differential variables are taken from the solution at the end of the prior solution interval. After finding the new set of initial values, the initial slopes of the differential variables are calculated by solving the subsystem of ordinary differential equations for the slope.

**E**is the electric field strength,

*Ω*is the spatial computation domain, the superscript H denotes the Hermitian, i.e., the complex conjugate transposed, and \(\mathbf {e}(t)=-\frac {{\mathrm {d}}}{{\mathrm {d}}t}\mathbf {a}(t)\) is the line-integrated discrete electric field. The Joule losses are plotted as well in Fig. 5. Figure 6 depicts the solution of (13), i.e., the coefficients \(\mathbf {w}(t_{1})\), exemplary for the current through the inductor \(i_{\mathrm{L}}\). As one can see, using the initial values (25), only the coefficient \(w_{j,0}\) corresponding to the zero-th basis function varies and the others stay constant. To quantify the accuracy and efficiency of the multirate PWM balance method, it is compared to conventional time discretization and to the MPDE approach with the original PWM basis functions. Different settings are considered: To analyze the performance of the conventional time discretization, the relative and absolute tolerance setting of the solver is changed, i.e., \(\mathrm{abstol}=\mathrm{reltol} \in [10^{-6}, 10^{-1}]\); For the case of the multirate PWM balance method and the MPDE approach with the original PWM basis functions, relative and absolute tolerances are fixed at \(\mathrm{abstol}=\mathrm{reltol}=10^{-7}\) and the number of basis functions \(N_{\mathrm {p}}\in \{1,\dots,10\}\) is changed. The accuracy is measured for the voltage output of the converter, i.e., the voltage at the capacitor. The relative \({\mathrm {L}^{2}}\) error is given by

As one can see the MPDE approach with the original PWM basis functions is considerably slower than conventional time stepping. This is due to the fact that the already large systems of equations (1) (due to field-circuit coupling) are even further increased in size through the Galerkin approach. The stagnation of the error at 10^{−6} in Fig. 7 for values larger than \(N_{\mathrm {p}}=7\) is caused by the chosen accuracy of ode15s. Furthermore one can see that when adding another basis function the error does decrease with every second basis function. This was already observed in [5, 10]. For this reason the error for the PWM eigenfunctions is only plotted for \(N_{\mathrm {p},\mathrm {pwmbal}}\in\{1,2,4,6,8,10\}\). Since the systems of equations resulting from the multirate PWM balance method are decoupled, they can be solved efficiently in parallel. For each basis function \(g_{k}\) with \(k=0,\dots ,N_{\mathrm {p}}\), a complex-valued initial value problem of the form (16) has to be solved. The size of these systems of equations is the same as that of the original system of equations (1). However, the time for solving is considerably smaller since less time steps are necessary for the same solution accuracy. Note that due to the choice of the initial values (25) most coefficients in (13) for this numerical example do not change and only those corresponding to the zero-th basis function vary. This means that only the decoupled system of equations which corresponds to the zero-th basis function takes considerable computational effort to solve. In a parallel computing environment one would choose as many processor cores as basis functions (\(N_{\mathrm {p}}+1\)). The overall runtime is then determined only by the initial value problem that takes the longest to integrate. For this numerical example it is \(k=0\). The communication overhead between processors is not taken into account since it is highly implementation and machine dependent. The slightly decreasing time to solution when \(N_{\mathrm {p},\mathrm {pwmbal}}>1\) is owed to the fact that initial values according to (25) take more a-priori information into account which leads to smaller number of time steps and faster simulation. The overall accuracy of the method is problem-specific and always depends on both the tolerance for the solver and the number of basis functions. An a-priori determination of the number of basis functions and the solver tolerance is not yet available. An a-posteriori estimator can be constructed by increasing the number of basis functions and comparing the solutions. The resulting error is also related to the time stepping error.

The MPDE approach works also for nonlinear problems. However, similarly as for the harmonic balance case, the decoupling is not straightforward anymore. Furthermore the PWM basis functions and thus also the PWM eigenfunctions might not be able to represent the solution of problems with nonlinear elements [10]. If the amplitude of the ripples is small compared to the amplitude of the envelope, the particular efficient approach described in [9] can be applied. It uses only the slowly varying envelope to evaluate the nonlinearities. Although the assembly of the field model matrices for a new envelope cannot be parallelized, the matrices in (13) can still be decoupled and calculations to obtain the following time step can be run in parallel.

## 6 Conclusion

A new efficient technique was presented for field-circuit coupled models of DC-DC power converters, in which the switches are idealized and the filtering circuit is linear. The already existing MPDE technique with PWM basis functions splits the solution into fast varying and slowly varying parts. In this paper this method has been improved by introducing a new set of PMW basis functions which decouple the systems of equations similar as in the harmonic balance method. The new method, now called multirate PWM balance method, enables a parallel solution of all PWM modes resulting in a speed-up amounting to a factor 4 for the test example.

## Notes

### Acknowledgements

This work is supported by the “Excellence Initiative” of German Federal and State Governments and the Graduate School CE at TU Darmstadt. The authors thank Johan Gyselinck for fruitful discussions. Further thanks go to Jonas Bundschuh and Erik Skär for their contribution to the first implementation of the PWM eigenfunctions.

### Availability of data and materials

Not yet available publicly. It is, however, planned to make the code, which is used to generate the results, publicly available in the near future.

### Authors’ contributions

All authors have jointly carried out the research and worked together on the manuscript. The numerical tests have been conducted by the first author. All authors read and approved the final manuscript.

### Funding

No funding to report.

### Competing interests

There are no competing interests to report.

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