- Research
- Open Access
Shape and topology optimization of a permanent-magnet machine under uncertainties
- Piotr Putek^{1, 2}Email authorView ORCID ID profile,
- Roland Pulch^{2},
- Andreas Bartel^{1},
- E Jan W ter Maten^{1},
- Michael Günther^{1} and
- Konstanty M Gawrylczyk^{3}
https://doi.org/10.1186/s13362-016-0032-6
© Putek et al. 2016
Received: 2 March 2016
Accepted: 31 October 2016
Published: 8 November 2016
Abstract
Our ultimate goal is a topology optimization for a permanent-magnet (PM) machine, while including material uncertainties. The uncertainties in the output data are, e.g., due to measurement errors in the non-/linear material laws. In the resulting stochastic forward problem, these uncertainties are stochastically modeled by random fields. The solution of the underlying PDE, which describes magnetostatics, is represented using the generalized polynomial chaos expansion. As crucial ingredient we exploit the stochastic collocation method (SCM). Eventually, this leads to a random-dependent bi-objective cost functional, which is comprised of the expectation and the variance. Subject to the optimization of the PM machine are the shapes of the rotor poles, which are described by zero-level sets. Thus, the optimization will be done by redistributing the iron and magnet material over the design domain, which allows to attain an innovative low cogging torque design of an electric machine. For this purpose, the gradient directions are evaluated by using the continuous design sensitivity analysis in conjunction with the SCM. In the end, our numerical result for the optimization of a two-dimensional model demonstrates that the proposed approach is robust and effective.
Keywords
- robust low cogging torque design
- topology and shape optimization
- random partial differential equation
- stochastic collocation method
- level set method
- continuous design sensitivity analysis
- weighted average method
- trade-off method
1 Introduction
Due to high performance, high efficiency and high power density, permanent-magnet (PM) machines are becoming more and more popular [1–3]. Consequently, these devices are currently broadly used in applications as robotics, hybrid vehicles, computer peripherals and so on, see, e.g., [4–8]. However, the PM machines suffer by construction from a considerable level of mechanical vibration and acoustic noise. More precisely, the interaction of the air-gap harmonics (stator slot driven) and the magnetomotive force harmonics (magnet driven) produces a high cogging torque (CT). On the other hand, the torque ripple is primarily provoked by the CT and higher harmonics of back-electromotive force (EMF). Also the magnetic saturation in the stator and the rotor cores [9] as well as the controller-induced parasitic torque ripples [10] might further disturb the electromagnetic torque [11].
From this perspective, especially the mitigation of the torque fluctuations is a key issue for the design of a PM machine because its result may simultaneously affect the machine performance. Especially, in the context of the deterministic/stochastic topology optimization of a PM machine, the consideration of more than one competing objective into a cost functional seems to be still a challenging problem [3, 12]. More specifically, when a multi-objective approach is involved in the designing process, a certain trade-off between conflicting criteria needs to be fulfilled. For this reason, often a Pareto front technique is accepted as an alternative. Furthermore, under assumption that the Pareto front is convex, it can be approximated by the weighted average method (WAM) or the ϵ-method [13–15]. However, in practice it is rather hard to verify this assumption, especially when nonlinear problems are considered. In such a situation, objective functions are approximated numerically, whereas a genetic or ant colony-based algorithm is recommended for the identification of the global front Pareto [3, 16]. It should be also noticed that in some cases, when a periodic functional may be applied, e.g., for a lumped model of electric machine [17] or the periodicity of objectives is a result of the geometrical structure of electric machines [12], it is also possible to prove the convexity in a rigorous mathematical way.
Various methods for suppressing the CT have been proposed in the literature. For example in [18] the authors employed the auxiliary slots for this purpose. Some solutions apply an appropriately chosen combination of slot/pole number [19] or the optimized ratio of pole arc to pole pinch [9] in order to reduce the CT. Other efficient methods for mitigating the CT involve shaping the rotor magnets and/or stator teeth [20] including redistribution of a PM and iron material within the domain of interest using topological methods, as proposed in [6, 8, 21–24]. Moreover, the statistics-based approach such as the Taguchi method [25], or its generalization called the regression-based surface response methods, are proposed in [11, 26], especially in industrial applications to reduce the noise to signal ratio. To this last group, also techniques based on the perturbation method for calculating the first and the second derivative may be included. Based on the sensitivity information they intend to estimate deterministically the impact of the input variability on output characteristics and in consequence on the result of optimization [27]. On the one hand, this ’deterministic’ estimation of uncertainties is limits to the range of the perturbation \(|\delta| = 6-8\%\), see, e.g., [27–29]. On the other hand, the load related to the calculation of the second derivative might be really large, especially when a cost functional involves a standard deviation. Clearly, the advantage of this approach is the immediate availability of a gradient which allows to carry out the strategy, whose aim is to reduce the influence of the deterministic uncertainty onto the optimization result [28]. More recently, the efficient approach, which benefits from both the perturbation technique and the stochastic-based method has been developed [29] to estimate the statistical moments.
The topology is a major contributor to the electromagnetic torque fluctuations. Therefore in this paper, we address the topology optimization of a PM machine. Since the results of the design procedure are highly influenced by unknown material characteristics [37], these uncertainties have to be taken into account in the course of a robust optimization. Thus the soft ferromagnetic material should be modeled using uncertainty. In particular, the relative permeability/reluctivity of the magnetic material needs an accurate model in order to improve the accuracy of the magnetic flux density of permanent magnets (for certain applications [38, 39]). Therefore, in our optimization model, we take the reluctivity as uncertain.
In our case of the topology optimization, we have to trace two interfaces between different materials with some assumed variations such as air, iron and PM poles of rotor, the modified multilevel set method (MLSM) has been used [40, 41]. The level set method [42] has found a wide range of applications also in electrical engineering. For instance, it is used to address shape or topology optimization problems [6, 22].
The proposed approach is innovative, since stochastic modeling of uncertainties are combined with a topology optimization for minimal electromagnetic torque fluctuations (of the CT) and at the same time allowing for a robust optimization on the basis of a suitable optimization criterion. The current paper is an extended version of [35] with many more details and new results for an analysis of a PM electric machine in on-load state (with excitation currents involved). More precisely, we paid a lot of attention to investigate the impact of the optimization methodology on other machine parameters such as the electromagnetic torque, the torque ripple, the back electromotive force and an analysis of its frequency spectrum as the important source of noise and vibrations. However, it should be noticed that in this paper we focus mainly on a robust low cogging torque design under uncertainty which resulted in no-load steady state analysis of stochastic curl-curl equation (a density of excitation currents \(J(\mathbf {x}) = 0\)). Consequently, the machine analysis in on-load state can be considered here as a post-processing procedure, since the electromagnetic (average) torque itself was not involved in the optimization task. Moreover, we restricted ourselves to the minimization of noise and vibrations caused predominantly by the cogging torque, the torque ripple and the back EMF, however, without coupling a vibro-acoustic modeling as in, e.g., [43] with the curl-curl equation. Instead the air-gap flux density (as equivalence of the back EMF) is considered in the optimization procedure as a second objective. It yields a considerably improvement of the wave form of the back electromotive which, in turn, directly leads to the reduction of noise and vibrations. Furthermore, to deal with a stochastic multi-objective problem, the trade-off method, incorporated in the level set method (LSM) as well as the AWM, involved in a robust functional, have been applied.
The paper is organized as follows: first we describe the PM machine, which we use as test case (Section 2). Then the deterministic model is set up (Section 3). Based on that, the stochastic forward problem is formulated in Section 4. Section 5 describes the optimization problem with the needed objective functions and the constraints. Then we combine topology optimization and uncertainties (Section 6). After a short description of the simulation in Section 7, we discuss numerical results (Section 8).
2 Test case description
Main parameters of the ECPSM (an electric controlled permanent magnet excited synchronous machine) design [ 6 ]
Parameter (unit) | Symbol | Value |
---|---|---|
Pole number | 2p | 12 |
Stator outer radius (mm) | \(r_{\mathrm{ostat}}\) | 67.50 |
Stator inner radius (mm) | \(r_{\mathrm{istat}}\) | 41.25 |
One part stator axial length (mm) | \(l_{\mathrm{as}}\) | 35.0 |
Slot opening width (mm) | \(w_{\mathrm{oslot}}\) | 4.0 |
Number of slots | ns | 36 |
Number of phases | m | 3 |
Permanent magnet pole | NdFeB | 12 |
PM thickness (mm) | \(t_{\mathrm{m}}\) | 3.0 |
Remanent flux density (T) | \(B_{\mathrm{r}}\) | 1.2 |
3 Mathematical model
In the following, we set up a deterministic model for the PM machine, which is suitable for optimization. This comprises a discussion of the domain, a strong and weak formulation and the objective functions.
3.1 Field quantity and simulation domain
3.2 Strong formulation
This quasi-linear elliptic problem (1) is equipped with periodic boundary conditions on the radii of D and it is also equipped with a homogeneous Dirichlet condition on the outer arc: \(A(\mathbf {x})=0\).
3.3 Weak form
4 Stochastic forward problem
The characteristics of ferromagnetic materials are usually deduced from measurements. In our case, this applies to the reluctivity υ, which suffers from measurement uncertainties. Now, the result for the electrical machine design is strongly affected by the unknown material characteristics and geometric uncertainties, for example, the air-gap thickness, see in [28, 37]. The latter effect can be also simulated by changing the material parameter using the level set methodology (see below). This leaves us to include these material uncertainties into the mathematical model and thus into the optimization procedure in order to achieve a robust design. Therefore, in our work, the reluctivity becomes a random field, which allows us to quantify these uncertainties. The respective modeling is our next subject.
4.1 Stochastic modeling of uncertain reluctivities
4.2 Polynomial chaos expansion
Our uncertainty quantification is based on the concept of the polynomial chaos expansion (PCE). The homogeneous polynomial chaos was introduced in [46] for Gaussian probability distributions. Later this concept was extended to other probability distributions, which resulted in the generalized polynomial chaos, see [33, 47].
To enable a numerical realization, the series (9) has to be truncated at some integer \(N_{\max}\). Often all multivariate polynomials up to some total degree are included in the truncated expansion. This truncation causes that the associated variance is just an approximation of the exact variance in (11). Furthermore, the coefficients (10) are typically not given exactly, since they are influenced by numerical errors. If a function f is also space-dependent, then the PCE (9) is considered pointwise for each \({\mathbf{x}} \in{\mathrm{D}}\).
4.3 Stochastic PDE model and PCE approximation
4.4 Stochastic collocation
There are mainly two classes of numerical methods for the approximative computation of the coefficient functions \(a_{i}\): stochastic collocation methods (SCM) and stochastic Galerkin techniques, see, e.g., [33, 49]. We apply the SCM for our problem, since this strategy represents a non-intrusive approach, where the codes for the simulation of the deterministic case can be reused in the stochastic case.
As multi-dimensional quadrature on \([-1,1]^{Q}\), we apply the Stroud formulas with constant weight function, see [50, 51]. This type of quadrature methods exhibits an optimality property with respect to the number of required nodes to calculate the integral exactly for all multivariate polynomials up to a total degree R. For example, it holds that \(K=2Q\) for \(R=3\) and \(K=2Q^{2}+1\) for \(R=5\).
5 Modeling of the optimization problem
We need to define how the quality of the machine design is measured and the optimization variables as well as their variations.
5.1 Objective functions
For the optimization, one has to assess the quality of the design of a PM motor. Here we will have two ingredients, i.e., we set up a bi-objective optimization problem.
From engineering viewpoint, both objectives, which compete each other, are very important. The first of them as a main component of the torque ripple is responsible for minimizing the noise and vibrations, which are crucial for the low-speed application. While the second function allows for ensuring possibly the bigger value of the flux density calculated in air-gap or, equivalently, the highest value of \(U_{\mathrm{back}}\). The spectrum of the latter has also impact on vibrations. As a result, it influences the electromagnetic torque as well. For a solution of the multi-objective problem the ϵ-method [13], incorporated in the LSM scheme has been applied. It means that a second criterion serves as a constraint bounded by some allowable range of parameters ϵ. On the one hand, this method requires some technical information about objectives preferences as well as the convexity of a Pareto front, what for the periodic functions is often fulfilled. On the other hand, the obtained solution might not necessary be globally non-dominated [53] due to the treatment of the \(B_{\mathrm{r}}\) objective as the fraction of areas. Therefore, in order to find a non-dominated globally solution a Pareto front technique need to be applied for a robust optimization.
5.2 Optimization variables and multi-level representation
5.3 Optimization and uncertainty
- (a)
First, after a model initialization, using, e.g., a gradient topological method, the signed distance functions \(\phi_{i}\), \(i=1,2\) need to be calculated. In particular, it means that the shapes under colorred the consideration are described by the zero-level sets, shown on Figure 4 (without utilizing any additional parametrization function, besides a model discretization).
- (b)
Based on the knowledge of the zero-level set velocity \(V_{n,i}\), modified by area constraints, for which the adjoint variable method or the continuum design sensitivity analysis might be applied, the corrections of the distribution of signed distance functions are calculated and then introduced into the model in every iteration. Here also the distribution of level sets can be modified based on the topological information. Additionally, the Tikhonov regularization or the total validation technique can be used in order to control the complexity/smoothness of the optimized shapes [6, 23].
- (c)
Finally, stops criteria are checked and the optimization process is continued until they will be fulfilled.
6 Topology optimization under uncertainties
We have setup a shape optimization problem constrained by the elliptic PDEs (12) with random material variations. Now we need efficient and robust computation strategies.
6.1 Dual problem
In the steady-state analysis and using a Newton-Raphson algorithm, the adjoint variable ζ can be computed directly. This is due to the fact that the converged system of the direct problem (4) and the adjoint problem (23) are the same [56]. This technique, the so-called frozen method was successfully applied for calculating the electromagnetic force in the nonlinear magnetostatic system [55] and for providing the on-load CT [57].
6.2 Robust topology optimization problem
The minimization of the cogging torque in our 2D magnetostatic case can be equivalently represented as the minimization of the magnetic energy \(W_{\mathrm{r}}\) variation [21, 41].
Here, one can compute the total derivative of the magnetic energy (24) on the basis of the forward analysis, only. That is, the forward model is calculated in the collocation points, while using the models for υ (21), \(b_{r}\) (22) and the sensitivity (25) as well as the coefficients of the PCE (14) and the moments (15). We remark that a similar approach was used in [60, 61] for the solution of stochastic identification/control problems for constrained PDEs with random input data. However, their type of cost functional was different. It should be also emphasized that in contrary to the work by [22] about the deterministic low ripple torque design, in our paper we deal with the low cogging torque design of the ECPSM machine under uncertainties.
7 Simulation procedure
Furthermore, the algorithm for the robust topology optimization has been implemented using the Comsol (COMSOL 3.5a, The COMSOL Inc., Burlington, MA, 2008) and Matlab (MATLAB 7.10, The MathWorks Inc., Natick, MA, 2010) scripts. Thus the Finite Element method is used to solve both the weak formulations of primary and dual system, defined by (4) and (23), respectively. The respective triangular mesh consists of 81,016 elements with each second order Lagrange polynomials for the A-formulation. As a reluctivity model for iron parts, the soft iron material without losses has been applied and a standard spline interpolation of measured data are used for the nonlinear dependence on \(|\nabla A|^{2}\). The computation time (wall clock time) for a fixed position of rotor and stator in our configuration was about 30.31 s. This did involve \(163\text{,}283\) degrees of freedom. Additionally, every rotor pole has been divided into 468 voxels. This applies to PM and iron pole, separately. The UQ analysis has been performed using the software implemented by [62]. The Stroud-5 formula has been used for this purpose. The optimized shapes of rotor poles have been found in the 10th iteration of the optimization process. We have applied the Stroud-3 formula in order to obtain the final results for the UQ analysis of the CT, the back EMF, the electromagnetic torque and the magnetic flux density in the air-gap, respectively.
8 Numerical results
Values of some physical parameters of the ECPSM model before and after optimization
Quantity (unit) | Before optimization | After optimization | Decrease/increase |
---|---|---|---|
Expectation of the cogging torque (Nm) | |||
Rectified mean value | 0.072 | 0.012 | 83.70%↓ |
RMS value | 0.085 | 0.015 | 82.19%↓ |
Minimal value | −0.139 | −0.027 | 80.21%↓ |
Maximal value | 0.138 | 0.026 | 80.51%↓ |
Mean value of standard deviation | 0.004 | 0.002 | 43.19%↓ |
Expectation of the back EMF (V) | |||
Rectified mean value | 257.1 | 225.9 | 12.14%↓ |
RMS value | 268.3 | 240.6 | 10.33%↓ |
Minimal value | −330.0 | −299.8 | 9.14%↓ |
Maximal value | 330.0 | 299.8 | 9.14%↓ |
Mean value of standard deviation | 6.34 | 6.23 | 1.85%↓ |
Expectation of the air-gap magnetic flux density (T) | |||
Rectified mean value | 0.575 | 0.471 | 18.08%↓ |
RMS value | 0.592 | 0.502 | 15.11%↓ |
Minimal value | −0.647 | −0.539 | 16.73%↓ |
Maximal value | 0.733 | 0.749 | 2.29%↑ |
Mean value of standard deviation | 0.019 | 0.017 | 12.29%↓ |
Expectation of the electromagnetic torque (Nm) | |||
Rectified mean value | 2.407 | 2.014 | 16.31%↓ |
RMS value | 2.433 | 2.019 | 17.02%↓ |
Minimal value | 1.738 | 1.779 | 2.31%↑ |
Maximal value | 2.788 | 2.200 | 21.11%↓ |
Mean value of standard deviation | 0.09 | 0.063 | 20.33%↓ |
Expectation of others quantities | |||
Ripple torque (%) | 43.62 | 20.90 | 52.10%↓ |
THD of the back EMF (V/V) | 0.732 | 0.498 | 31.93%↓ |
Mass of iron pole (g) | 15.95 | 14.94 | 6.32%↓ |
Mass of PM pole (g) | 15.95 | 12.19 | 23.56%↓ |
9 Conclusion
In this paper we demonstrated how to combine the stochastic collocation method (SCM) with the multi-level set method (MLSM) and how to apply this technique efficiently for the robust topology optimization of a PM synchronous machine. In the end, the shape of rotor poles were optimized also with respect to the level of noise and vibrations. This did result in significant reductions of both the rms of the CT (82%) and the mean value of the standard deviation (43%). Thereby we were able to take variations with respect to manufacturing tolerances/imperfections into account by assuming a random field for the reluctivities. As a drawback, we reported a small decrease in the root mean square values of the electromagnetic torque and back EMF. However, it should be noticed that aims of the low cogging torque robust topology optimization (the electric machine in the no-load mode) have been completely fulfilled. Additionally, the waveform has been considerably improved about 32%, while the torque ripple has been reduced around 52%. The detailed analysis presented in Table 2 indicates that further research should be focused on the low ripple torque robust design in the on-load state (with excitation currents included). Then, the robust optimization of the electric machine could be performed when taking both the ripple torque and the average electromagnetic torque into account. This is considered as a further direction of our investigation. This work also highlights the effectiveness of the proposed methodology.
For an orthogonal system of basis polynomials a normalization can be done straightforward, e.g., [33].
Due to the used Stroud quadrature formulas [51], the same distribution had to be assumed with a relatively high variance based on [38] for the reluctivity of a PM.
A similar PM machine was also the topic of the scientific project ‘The Electrically Controlled Permanent Magnet Excited Synchronous Machine (ECPSM) with application to electro-mobiles’ under the Grant No. N510 508040, founded by Polish Government. There the topology was deterministically optimized.
The THD is defined as: \(\mathrm{THD} = \sqrt{\frac{V_{2}^{2} + V_{3}^{2} + V_{4}^{2} +\cdots+ V_{n}^{2} }{V_{1}^{2}}}\), where \(V_{k}\) is the root mean square voltage of the kth harmonic and \(k = 1\) denotes the fundamental frequency.
Declarations
Acknowledgements
The project nanoCOPS (Nanoelectronic COupled Problems Solutions) is supported by the European Union in the FP7-ICT-2013-11 Program under the grant agreement number 619166 and the SIMUROM project is supported by the German Federal Ministry of Education and Research (05M13PXB). The authors would also like to thank Dr. Piotr Paplicki, West Pomeranian University of Technology in Szczecin, for his help in the final simulations and Professor Ryszard Pałka, West Pomeranian University of Technology in Szczecin, for the opportunity to use the ECPSM model in our research. We are also thankful to Kai Gausling, MSc., University of Wuppertal, for using his software for the uncertainty quantification.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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