Establishing the Power Factor Limitations for ... - IEEE Xplore

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TMAG.2015.2443713, IEEE Transactions on Magnetics

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Establishing the Power Factor Limitations for Synchronous Reluctance Machines Yi Wang1, Dan Ionel1,2, Fellow, IEEE, David G. Dorrell3, Senior Member, IEEE, and Steven Stretz2, Member, IEEE 1

University of Wisconsin-Milwaukee, 115 East Reindl Way, Milwaukee, WI 53212-1255, USA 2 Regal Beloit Corp., 1051 Cheyenne Av., Grafton, WI 53024, USA 3 University of Technology Sydney, Broadway, Sydney 2007, Australia

This paper discusses a “Goodness” index and introduces a “Badness” factor for electric machines with special reference to synchronous reluctance motors. This factor is then used in conjunction with computationally-efficient finite element electromagnetic analysis and a differential evolution optimization algorithm in order to carry out a large-scale design study under comparable specifications with an induction motor. It is shown that for synchronous reluctance machines, careful rotor geometry optimization can improve the performance and that the inherent limitations, including those for the power factor, can be systematically established. The correlations for specific average torque, copper and core losses, and efficiency are also discussed. Index Terms—Synchronous reluctance machine, design, finite element analysis, FEA, optimization, differential evolution.

I. INTRODUCTION

N

drives with improved-design synchronous reluctance machines are now be capable of reaching high efficiency, even up to the IE4 level [1]. Such machines have been studied for many years and significant progress was made in the 1990s using axial laminations in the rotor [2]. At that time, the technology was explored and identified, in principle, as a viable alternative [3]. However, the axially-laminated rotor was very difficult to manufacture and more recent designs use radial laminations with ducts, also called flux barriers. This paper makes a further contribution to the subject by introducing, in the following section, a new global performance index. Based on this, a systematic optimization method for practical design is proposed, this employs a differential evolution algorithm. This establishes the natural limitations of the machine when using topologies that can be manufactured in high volumes and which utilize highperformance materials. EW

VARIABLE-SPEED

II. GLOBAL PERFORMANCE – POWER, GOODNESS AND BADNESS FACTORS There are several benchmarks that can be applied to assess the effectiveness of an electrical machine. Generally, such indices include energy efficiency, torque and power output density, torque ripple, and power factor. In order to establish a basis for comparison between different machines and designs, various global factors have been proposed over the years. For example, Laithwaite has defined a “Goodness” factor based on the ratio of the mechanical work done and the input power [4]. An electrical machine is subject to both electromagnetic and thermal stress; the latter being related to the power losses required by the electromechanical energy conversion and to the volume/mass and to the cooling system. Hence, an appropriate global factor of “Badness” is introduced in this study. This is defined by: Manuscript received March 20, 2015, revised 24 May, 2015. David Dorrell (e-mail: [email protected]).

B

WLoss



2 3I ph Rph

(1) T where WLoss is the total motor loss, T is the shaft torque, and Iph is the phase current. For an electric machine excited through the supply current, such as a synchronous reluctance machine, the magnetic field is non-linearly dependent on the current. Since the core losses are approximately proportional to the flux density squared. Therefore, in principle, the effect of both core and copper losses can be modeled through an equivalent phase resistance Rph. However in this optimization study such an equivalent resistance is not actually employed, since all calculations are based on estimations of actual power losses; however, it can be calculated if required. It is important to understand the conceptual equivalence since it relates badness to another more frequently used performance index (i.e., the Goodness factor). The torque is generally proportional to the current and, according to (1), the total losses are proportional to the current squared. This establishes a relationship between torque and the square root of the losses. The ratio between the rated output torque and current is typically defined as the “torque constant”. For a synchronous reluctance machine, the Badness factor (1) is similar to the inverse of the “motor constant”. This is also referred to as “motor Goodness” in design practice. This “Goodness”, which should not to be confused with Laithwaite’s original definition [4], is equal to the ratio between the torque constant and the square root of an equivalent resistance.

T

III. LARGE SCALE AUTOMATIC OPTIMIZATION STUDY A parametric model for a ducted-rotor geometry was employed in this study (Fig. 1) and finite element analysis was used to simulate the machine (Fig. 2). More details of this study are available in [5]. This contribution focuses on how the operation of these machines can be improved by careful design of the rotor ducting. Only the rotor geometry is modified and analysis is conducted using a fast 2D finite element technique [6]. This allows the variation of the current and of the current angle

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with respect to the rotor q axis to be studied.

Multi-objective Optimization with Differential Evolution (CMODE) [7] - was utilized to solve constrained optimization problems. In the CMODE method, the mutation and crossover operations of the DE are applied in order to generate the offspring – the next population/generation. In CMODE the selection operation is not utilized. The process is as described by the following pseudo-code: While ( N < Gmax) Set Q: λ individuals randomly selected from 𝑃(𝑁) For (each individual in Q set) 

 

Set C: Cross over and mutation uj,i,g =vj,i,g , if randj (0,1)≤Cr or j=jrand { xj,i,g , otherwise Set R: non-dominated individuals in C 𝑃(𝑁) = 𝑃(𝑁) ∪ 𝑄

End For (each individual in Q set) Infeasible solutions archiving and replacement  Set A: infeasible individuals in R  Replace 𝑃(𝑁) with individuals in A

Fig. 1. Variables for the generic and parametric synchronous reluctance machine geometry considered in the large scale optimization study.

Essentially, the d axis current magnetizes the machine and the q axis current is the electric loading which produces torque. The key to optimizing the performance is to use a small airgap and carefully chosen duct widths. If the ducts are too narrow the d axis inductance will be high but so will be the q axis inductance. If the ducts are too wide then the ratio will be high but the d axis inductance will be significantly reduced; saturation will onset earlier. There is obviously an optimum design for a range of duct sizes which also accounts for strong non-linearity due to the laminated steel characteristics.

End End Computationally-efficient finite element analysis (CEFEA) was employed for calculating the motor performance. This includes the torque profile, the terminal emf, the induced voltage, and losses (stator iron and copper), etc. [8]-[10]. In order to speed up the optimization process, the distributed solve option (DSO) of ANSYS Maxwell software package was employed. The rather low power factor is widely recognized as an inherent and significant limitation for synchronous reluctance machines [11]. In order to make sure that its potential is fully exploited, an optimization process was performed with an emphasis on the power factor. In the case of single-objective problems, the DE algorithm aims to find a global maximum power factor through simple comparison to other designs until the stopping criterion is satisfied. TABLE I. GEOMETRY RATIOS USED IN THE OPTIMIZATION Wt  Wt1  Wt 2  Wt 3

Fig. 2. Flux plot for a synchronous reluctance machine (286 frame 36-slot 4pole) with 4 rotor layers per pole analyzed using ANSYS/Maxwell.

Optimization has been recently used for a variety of machine design studies [13]-[15]. Differential evolution (DE), which includes mutation, crossover, and possibly selection, is a highly recommended method for multi-objective design. In this paper, an algorithm proposed by Wang and Cai - Combing

hct  hc1 2  hc 2  hc3  hc4 Lbt  Wt  hct

K_b 

hct Lbt

K_b2 

hc2 hct  hc1 2

K_t1 

Wt1 Wt

K_bt 

Lbt Rc sin  p 2 

K_b3 

hc3 hct  hc1 2  hc2

K _t 2 

Wt 2 Wt  Wt1

K_b1 

hc1 2hct

K_b4 

hc 4  Lbt  hc1 2  hc 2      hc3  Wt1  Wt 2  Wt 3 

K_t 3 

Wt 2 Wt  Wt1

According to Cupertino and Pellegrino [12], in order to reduce the computational time per candidate machine design, the current angle, _0, can be considered as an independent variable in this optimization problem. Eight rotor geometrical parameters (K_b, K_bt, K_b4, K_b3, K_b2, K_b1, K_t3, K_t2, and K_t1) were also considered as independent variables (Fig. 1 and Table I). These ratios were used to obtain a robust



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parametric model and avoid the geometric overlapping in the automated design optimization procedure. For instance, the width of the first and second flux tubes, Wt1 and Wt2, were defined using the ratio expression of K_t1 and K_t2 as defined in Table I. The current density through the stator winding and the core length were considered constant. In this study, a single objective of maximum power factor and a constraint of maximum Badness of 0.65 [(W)/Nm], which is a value typical for an induction machine of comparable 10 HP rating, were employed. IV. RESULTS AND DISCUSSION Optimization was performed with a DE algorithm with 51 generations, each comprising 100 candidate designs. This study involved 5100 motor configurations, and was possible through an ultra-fast computationally-effective 2D electromagnetic FE technique [6]. On a state of the art PC workstation, the optimization procedure can be run in less than two days. A further significant reduction in the computational time can be achieved using parallel processing on multi-core HPC systems, so that the optimization can be completed in just a few hours. Scatter plot clustering and Pareto fronts were used to identify best candidate designs as shown in Figs. 3 and 6. A maximum power factor of 0.780 is exhibited by design 2103, which also has an undesirable high Badness. Furthermore, as shown in Fig. 4, the manufacturing of the rotor can be a challenge in this case due to the relatively thin flux guides, one of them being smaller than 3 mm. Designs with comparable power factor and lower Badness include 441, 1621, and 4313. Finally, design 3092, which achieves a satisfactory compromise between power factor and Badness is identified. It should be noted that this 3092 design is also satisfactory in terms of output power, torque ripple and efficiency. The cross-sections of motor designs 2103 and 3092 are shown in Fig. 4. It can be seen that the changes in rotor geometry are very subtle, but these can make a substantial difference in performance. This shows the need for careful design in synchronous reluctance machines, and highlights the role that this optimization method plays in realizing an improved design. Motor designs were prototyped and tested on an active dynamometer set-up equipped with a computer data acquisition system. Example rated load and rated speed experimental results for a reference design with the crosssection shown in Figs. 2 and 5 are provided in Fig. 6. Satisfactory model validation was also noted for the torque and torque ripple. For high values of the torque angle, the saturation of the flux channels is significant. This is an effect that may explain the differences between calculations and measurements under those conditions. This illustrates the need for improved models for the laminated steel characteristics taking into account any possible variations due to the manufacturing process. Nevertheless, from a practical engineering point of view, the torque angle for high performance designs does not typically exceed 150 deg.

Fig. 3. Scatter plot with the 5,100 candidate designs considered in the differential evolution optimization and studied with electromagnetic FEA.

Fig. 4. Cross section of motor designs with relatively high power factor and low badness (see also Fig. 3).

Fig.5. Example prototype lamination (left) and a cut-way through a NEMA frame motor.

Fig. 6. Example validation of FEA versus experimental power factor (PF) and efficiency at rated load and rated speed operation.



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4 illustrate the superiority of design 3092 over 2103 in terms of higher output power, torque, and efficiency. V. CONCLUSIONS This paper documents through a large scale automatic optimization study the relatively low values of the power factor as a natural limitation of the synchronous reluctance machine. The results also point out the conflicting nature of high power factor and low “Badness” requirements, as well as the fact that relatively small modifications in the dimensions of the ducts may result in relatively large changes in output performance. The proposed optimization procedure represents a useful tool for design engineers as it enables the systematic exploration of the design space and the identification of “best compromise” designs that perform well in terms of multiple indices, such as specific torque, torque ripple, and efficiency. REFERENCES [1]

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[13] Fig. 7. Scatter plots of performance for 5,100 candidate designs.

Scatter plots for the 5100 designs considered in the optimization study are shown in Fig. 6. These are for key performance parameters that are used to benchmark the operation of a machine and include the torque ripple, output power, core losses, torque, and efficiency. The graphs

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