Model Reduction Methods for Magnetic Fields Based on ... - IEEE Xplore

IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 11, NOVEMBER 2014

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Model Reduction Methods for Magnetic Fields Based on Modal Analysis Michael Kirschneck1, Daniel Rixen2 , and Henk Polinder3 1 Department

of Precision and Microsystems Engineering, Delft University of Technology, Delft 2628 CD, The Netherlands 2 Institute of Applied Mechanics, Technische Universität München, München 80333, Germany 3 Department of Electrical Power Processing Delft University of Technology, Delft 2628 CD, The Netherlands

This paper introduces model reduction techniques based on various methods known from mechanical engineering that are used in calculation for structural analysis. These techniques are applied to electrical machines reducing the number of degrees of freedom and, hence, computation time for transient analysis. It further introduces magnetic superelements for the stator and the rotor of these machines and combines them with suitable coupling methods from literature. Index Terms— Magnetic field calculation, modal analysis, model reduction, substructuring, superelements.

I. I NTRODUCTION

F

INITE element (FE) magnetic field calculations in 3-D can be computationally expensive. However, 2-D calculations are not sufficient for certain types of analysis, e.g., magneto-mechanical coupled problems or problems considering end winding effects. To accelerate transient analysis of these 3-D problems, model reduction methods can be applied. In the past, there were some efforts to reduce calculation time for magnetic field problems. However, none of them were based on modal analysis. In structural dynamics, several reduction methods have been developed [1]–[3]. The last two of these methods are based on modal analysis per subsystem using a basis transformation to the modal space reducing degrees of freedom (DoFs). These techniques can be adapted for transient magnetic field problems or magnetic field calculation. This paper will investigate to what extend the above methods are suitable for transient magnetic field FE calculations of electric machines. For simplification, a 2-D model is used. However, the methods work the same way for 3-D models. In Section II, the various model reduction methods are introduced. Afterward, the results of the calculation are presented and compared with a reference solution including all DoFs of the system. In the last two sections, the results and short comings of the methods are discussed. Further, the conclusion is drawn that the general idea of the reduction methods, i.e., using modal analysis to reduce the number of DoFs, works for magnetic fields, but that in the way presented here the reduction methods are not yet applicable to non-academic problems and therefore still need some further development. II. R EDUCTION M ETHODS Magnetic fields can be described by the partial differential equation

Manuscript received March 7, 2014; revised May 13, 2014; accepted May 21, 2014. Date of current version November 18, 2014. Corresponding author: M. Kirschneck (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMAG.2014.2328863

∂A 1 (∇ × A) = J − γ (1) μ ∂t where A denotes the magnetic vector potential, J denotes the external current densities, gamma the conductivity, and μ denotes the permeability. This equation can be discretized using the FE method yielding ∇×

Dq A + K q A = J

(2)

where D denotes the magnetic mass matrix, K denotes the magnetic stiffness matrix, J denotes the external sources, such as currents in coils, and q A denotes the DoFs of the system. (see for instance [4] for an in-depth explanation of the FE method in magnetic fields). All here introduced reduction methods split up the system in subsystems. Then, they calculate a reduction basis that has less DoFs than the original model, but which describes the system relative accurately. This reduced model is called a superelement. For an electrical machine, a natural choice for substructures are the rotor and the stator of the machine. This is also practical because for a rotating machine, these two subsystems need to move relative to each other. The subsystems need to be assembled to the total system of the machine. There are different methods to assemble them. Which method needs to be used depends on the reduction method applied to the subsystems. A nice overview over the various assembly methods and how they are linked to different reduction methods can be found in [6]. A. Guyan Method The Guyan method (also called static condensation method) [1] projects the internal DoFs on the boundary DoFs using the static response related to boundary unit inputs. Boundary DoFs are considered to be all DoFs on the interface to another subsection of the model and all DoFs where an external current is applied. Therefore, all DoFs on the interface between the rotor and the stator as well as DoFs in the coil domains of the stator model are considered as boundary DoFs. After applying the Guyan method, the model consists only of the boundary DoFs of the system.

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Fig. 1. Plots of the magnetic vector potential A. Depicted are two fixed interface modes and two free interface modes. (a) 12th fixed interface mode of the stator. (b) 13th fixed interface mode of the stator. (c) 7th fixed interface mode of the rotor. (d) 8th fixed interface mode of the rotor.

one MTA vector. In this paper, MTA is only used in combination with the Craig–Bampton method. In this combination, the MTA reduces the number of boundary DoFs significantly. By applying the MTA method, all DoFs in domains of coils become internal DoFs. Then, these DoFs can be part of the fixed interface modes calculated for the Craig–Bampton method. Fig. 2(a) and (b) shows the stator fixed interface modes for the ACB method. These modes get directly excited by the currents because the DoFs where the currents are applied are internal DoFs of the modes. Fig. 2. Plots of the magnetic vector potential A. Depicted are two fixed interface modes for the ACB method. (a) 12th fixed interface mode for the ACB method. (b) 13th fixed interface mode for the ACB method.

B. Craig–Bampton Method In addition to the boundary DoFs, the Craig–Bampton method introduces DoFs describing the participation of certain fixed-interface modes [2]. The modes consist of the eigenvectors and eigenvalues of the dynamic system describing the magnetic field. Fixed interface modes are characterized by the fact that all boundary DoFs are set to zero. The dynamical behavior of the system is represented by the fixed-interface modes and is not neglected compared with the Guyan method. Fig. 1(a) and (b) shows two fixed interface modes of the stator and Fig. 1(c) and (d) shows two fixed interface modes of the rotor. C. Augmented–Craig–Bampton Method The augmented Craig–Bampton (ACB) method [5] uses modal truncation augmentation (MTA) vectors to extend the reduction basis of reduction methods based on modal analysis. It further reduces the number of DoFs of the reduced model. The MTA method computes load-depended vectors that capture the response of the system to the load not captured in the retained modes of the reduction basis. This method is especially suitable for electric machines because the load is represented by the current density in the coil domain and has thus a well-defined spatial distribution. It is therefore possible to capture the whole load generated by the coils by three load vectors. Each phase is represented by

D. Mode Selection Based on Coil Participation Factors The Craig–Bampton method and the ACB method yield a reduction of DoFs by including only a reduced amount of fixed-interface modes. The other modes are neglected. To yield the best accuracy, the modes with the largest impact to the dynamics should be chosen. Which modes have the largest impact on the dynamics is determined by calculating to what extend the modes are excited by the coils in the machine, i.e., the coil participation factor, for each identified mode (each solution of the solved eigenvalue problem). This factor indicates how important a mode is for the dynamics of the system. Assuming a harmonic excitation, it can be shown that the participation factor ψr can be calculated by 1 r · q j (3) m r i ω + kr where m r denotes the modal projection of the magnetic mass matrix D on the r th mode, kr denotes the modal stiffness of the r th mode, and r denotes the eigenvector of the r th mode. ω denotes the excitation frequency, i denotes the imaginary number, and q j denotes the excitation created by the currents in the coils. The higher ψr the larger the excitation of that mode by the coils. ψr =

III. R ESULTS As an example, the step response and the rotation of a 2-D model of an electric machine were calculated. The current density in the coils was used, as shown in Fig. 3. This corresponds to the current density distribution of a three phase current at the moment when phase a is at its peak current value

KIRSCHNECK et al.: MODEL REDUCTION METHODS FOR MAGNETIC FIELDS

TABLE I C OIL PARTICIPATION FACTORS OF THE VARIOUS

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TABLE II R ESULTS OF THE R EDUCED M ODELS FOR THE S TEP R ESPONSE

S TATOR M ODES U SED IN THE R EDUCTION M ETHODS

and thus phase b and phase c at minus half their peak current values. Table I shows the current participation factors of the first 14 modes for the various reduction methods. The columns for the participation factors of the Craig–Bampton method are all zero. The reason is that the fixed-interface modes used for the Craig–Bampton method are calculated with the constraint that the DoFs in the coil domains are zero. Therefore, the inner product of the load vector J and the mode shapes is also zero for all modes. Although, these modes are not directly excited by the current in the coils, they still participate in the description of the dynamic response of the systems. However, their contribution is very small. It can be concluded that the criteria (3) to determine important modes is not suitable for the Craig–Bampton method. For the reduction basis, the 14 modes with the lowest associated eigenvalue were chosen for the reduction basis. For the ACB method, where the DoFs of the coil domains are not set to zero during the mode calculation, this is not the case and the modes were chosen according to (3). For the rotor where no coils are present a coil participation factor cannot be calculated. Here, the first, seventh, eighth, 10th, 11th, and 13th eigenmodes are used because these are cyclic symmetric. Because the machine is a 12-pole machine, these modes are most likely to be excited in this configuration. For the stator according to Table I, the modes 10–13 were used. Like the 12th and 13th modes shown in Fig. 2, the 10th and 11th are the same mode rotated, respectively, to each other. Such a mode couple can reproduce that same mode for any rotation. To compare the time integration results of the reduced and the reference solutions, the proper orthogonal decomposition (POD) method was used [7]. This method calculates the proper orthogonal modes (POMs) and the proper orthogonal values (POVs). The first POM represents the mode that captures the most energy of the time series it is calculated from.

Fig. 3. Model of the 12 pole machine used as demonstrator. (a) Red line: boundary where the boundary condition A = 0 is applied. Blue line: interface between the two sub parts of the electric machine. The coil domains are green. (b) Source vector used for the step response.

The corresponding POV represents the energy of that POM. Comparing the first three POM/POV couples of the reduced and reference solutions gives an impression how accurately the reduced model reproduces the reference solution. The POMs are compared by a modal assurance criterion, whereas the POVs are compared by calculating the relative error between them. Table II and Fig. 4 show the reduction in time for the Craig–Bampton Method and the ACB method. The reduction in time is significant for both test cases. Table II also shows the values for the first three modes of the POD. IV. D ISCUSSION The previous section clearly shows that using the four introduced methods for rotating electric machines yield a

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The methods introduced here consider a whole machine. However, for time-dependent analysis of magnetic fields in electric machines usually only a section of the machine is computed when the machine is cyclic symmetric. This can also be done for modal analysis of cyclic structures. In the past methods based on Floquet theory have been developed to calculate modes for a cyclic structures modeling only a section of the cyclic symmetric structure [9]. This method could be adapted for the introduced methods for magnetic fields in electric machines. V. C ONCLUSION

Fig. 4. Reduction of calculation time and DoFs for the reduction methods introduced.

significant reduction of DoFs and thus calculation time for magnetic fields of electric machines. However, there are some issues associated with the methods proposed here that still need some further research. The results are less accurate for the step response than for the rotational magnetic field. The step response excites many modes, whereas the rotating field excites mainly two modes. Neglecting a considerable amount of modes, the reduction methods cannot capture the entire dynamics caused by a current step. The reference response on the other side is able to do so, because no modes are omitted. For the rotating field, this is not the case because here also in the reference solution mainly two modes are excited. Electric machines are designed to operate in saturation. The linear FE models introduced in this paper cannot consider saturation of the iron. Saturation will lead to a B–H curve that is non-linear and therefore will lead to a non-linear FE formulation resulting in a stiffness matrix that depends on the magnetic flux density. Normally that problem is overcome by reassembling the stiffness matrix for every time step. However, this method is not applicable here as the assembling of the reduction matrices is computationally very expensive. A method needs to be used that can adapt the stiffness matrix in a computational cheap manner. In mechanical engineering, there are some reduction methods that are based on modal analysis that can deal with this problem [8]. These reduction methods have to be adapted for magnetic fields.

This paper has shown that the ideas used for model reduction methods based on modal analysis can also be used for the model reduction of FE models of magnetic fields. All three investigated methods work for magnetic fields reducing the computational cost of transient analysis for magnetic fields. However, it was discussed that there are some issues before these methods can be applied in practical examples as nonlinear material behavior cannot be captured in the current form of algorithms. R EFERENCES [1] R. J. Guyan, “Reduction of stiffness and mass matrices,” AIAA J., vol. 3, no. 2, p. 380, Feb. 1965. [2] M. C. C. Bampton and R. R. Craig, Jr., “Coupling of substructures for dynamic analyses,” AIAA J., vol. 6, no. 7, pp. 1313–1319, Jul. 1968. [3] D. J. Rixen, “A dual Craig–Bampton method for dynamic substructuring,” J. Comput. Appl. Math., vol. 168, nos. 1–2, pp. 383–391, 2004. [4] K. J. Binns, P. J. Lawrenson, and C. Trowbridge, The Analytical and Numerical Solution of Electric and Magnetic Fields. New York, NY, USA: Wiley, 1993. [5] D. J. Rixen, “Generalized mode acceleration methods and modal truncation augmentation,” in Proc. Struct., Struct. Dyn. Mater. Conf. Exhibit, 2001. [6] S. N. Voormeeren, P. L. C. Van Der Valk, and D. J. Rixen, “Generalized methodology for assembly and reduction of component models for dynamic substructuring,” AIAA J., vol. 49, no. 5, pp. 1010–1020, May 2011. [7] G. Kerschen, J.-C. Golinval, A. F. Vakakis, and L. A. Bergman, “The method of proper orthogonal decomposition for dynamical characterization and order reduction of mechanical systems: An overview,” Nonlinear Dyn., vol. 41, nos. 1–3, pp. 147–169, Aug. 2005. [8] P. Tiso, R. Dedden, and D. Rixen, “A modified discrete empirical interpolation method for reducing non-linear structural finite element models,” in Proc. Volume 7B, 9th Int. Conf. Multibody Syst., Nonlinear Dyn., Control, Aug. 2013. [9] B. Lalanne and M. Touratierb, “Aeroelastic vibrations and stability in cyclic symmetric domains,” Int. J. Rotating Mach., vol. 6, no. 6, pp. 445–452, 2000.