Large reluctance motor design—nonlinear programming approach

Conput.& Elect. EngngVol.8, No.4, pp. 283-288,1981

0045-7906/81/040283-06502.00/0 © 1981PergamonPress Ltd.

Printed in GreatBritain,

LARGE RELUCTANCE MOTOR DESIGN--NONLINEAR PROGRAMMING APPROACH M. H. NAGRIAL Faculty of Engineering, P.O. Box 9476, Garyounis University, Benghazi, Libya (Received 20 March 1980)

Abstract--The non linear programming approach is used to predict an optimum design for a large reluctance motor of the flux guided type. It is shown that for large reluctance motors, it is necessary to use more than 1 FG[pole. A rotor design with equal permeance factors (I4'7T) for each flux guide will always be required. The Powell method along with the SUMT technique is employed to optimize the design whose predicted performance is more than that achieved by an equivalent induction motor. INTRODUCTION

Reluctance motors are synchronous motors with no field windings or excitations on the rotor. They are of salient pole types or the rotors are so shaped that the reluctance is less on 1 axis (direct axis) and large on the other axis (quadrature axis). They have squirrel-cage rotor windings and start as an induction motor. Near synchronous speed the motor is pulled into synchronism with the rotating field. They have very poor performance (low output power, low efficiency and low p.f.) as compared to ordinary squirrel-cage induction motors. There have been many attempts to improve their performance by using different rotor magnetic structure configurations such as segments[l, 2], axially laminated anisotropic material[3], flux barriers, etc. [4, 5]. All these modifications have resulted in much improved performance compared to the conventional design (salient poles) but still in some respects inferior to the ordinary induction motor. It has been shown quite recently that the performance of reluctance motors can be improved using nonlinear optimization techniques [6, 7]. These techniques are iterative in nature and their application has been due to the availability of large and high speed digital computers. All the efforts, till now, have been concentrated on designs of small reluctance motors (~< 4 h.p.). There are no rotor copper losses in reluctance motors due to the synchronous nature of their operation. Hence, it is thought that reluctance motors could have more output as compared to induction motors. There has been no attempt to design large reluctance motors. The purpose of this paper is to use nonlinear optimization techniques for the design optimization of large reluctance motors. The design optimization is based on the flux-guided principle and a design with 2 flux guides per pole is attempted. A typical induction motor frame size is taken (see details in Appendix I) and the reluctance motor rotor is designed using nonlinear optimization techniques. The optimum performance obtained is compared with that of an induction motor in the same frame size. NONLINEAR PROGRAMMING APPROACH

There are two main types of nonlinear programming techniques: (i) Gradient type. (ii) Direct search type. The latter does not require the first and second derivatives of the objective function for finding the minimum or maximum and hence is more suitable for design optimization of electrical machines. It has been shown recently[8] that the Powell method[9] is more efficient than other available direct search methods. Hence the Powell method is used in the present investigation and the penalty function approach[10] is employed for handling constraints. The basic algorithms for these methods are well established, and need not be discussed further. The optimization can be attempted by defining an objective function, F(X), which is a function of the design variables and then finding the minimum or maximum of this function. Any of the following criteria can be taken as the objective function, e.g. output power, total loss, weight or cost of materials or a combination of these. The design is also constrained in some sense so that a practically feasible design is obtained. These constraints could be independent variables, usually design dimensions or implicit functions which are dependent on design variables (e.g.p.f., efficiency, loss, etc.). In the present investigation, the synchronous output is taken as the objective function and the constraints and design variables are discussed in the next section. CAEE Vol. 8, No. 4--D

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M.H. NAOglAL DESIGN O P T I M I Z A T I O N

The flux guided reluctance motor is not only attractive from a manufacturing point of view but can have better performance characteristics. It has been a usual practice to use 1 flux guide (or barrier) per pole in small reluctance motors due to manufacturing difficulties but such difficulty is not encountered in la'ige reluctance motors due to large rotor diameters and other dimensions. As shown later, the performance also improves with increasing number of flux guides per pole. The aim of the present investigation is to explore the possibility of using more than 1 flux guide per pole and then optimizing the design. Figure 1 shows a typical flux guided reluctance motor rotor with 2 flux guides per pole. Table 1 shows the optimum performance obtained from a typical design for 1, 2 and 3 flux guides, respectively. The output increases as the number of flux guides are increased but the rate of improvement decreases and is not justified due to increases in manufacturing cost. A large number of flux guides also increases the number of variables for optimization and in actual practice, the optimum position of flux guides (a.s) cannot be precisely located during manufacture due to the presence of rotor slots. Therefore, it is quite reasonable to assume that a large reluctance motor would have 2 or at the most 3 flux guides per pole. Therefore, in the present case, 2 FG/pole design is assumed and design optimization is attempted using a standard 40 kW, squirrel cage induction motor frame. Direct Axis

i

iron 8r~dg~

Fig. 1. Reluctance Motor rotor with 2 flux-guides[pole.

Table 1. v = 0.25, WIT = 10 Machine type (FG/pole) 1 2 3

Output (kw)

p.f.

14.74 0.818 18.07 0.84 19.02 0.85

Efficiency

Increase in output

0.87 0.88 0.88

18.48% 23.64%

Large reluctance motor design--nonlinear programmingapproach

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The synchronous performance of reluctance motors is dominatly influenced by the direct and quadrature axis reactances Xd and Xq, respectively, and all the analytical approaches are to find Xd and Xq for a given configuration, so that Xmd = KdXc

(1)

Xmq = gqxc

(2)

Xa = Xm,~+ X~

(3)

Xq = Xmq + XI

(4)

where Xc = cylindrical rotor magnetizing reactance; X~ = stator leakage reactance; Xmd & X,,,q = magnetizing reactance (direct & quadrature axis). The dimensionless parameters Kd and K~ are calculated using the generalized analysis/6] and then using the 2-axis theory, the synchronous performance is calculated [11]. The design optimization can be attempted by following either of the following approaches: (i) Stator and rotor design dimensions are taken as variables. (ii) Rotor dimensions are taken as variables. In most cases, the reluctance motor is manufactured using the same stators as for induction motor. It is a usual practice to have a common stator and interchangeable rotors. Moreover, the reluctance motor designs are thermally limited and the maximum output is limited due to temperature rise considerations. The most important constraint is total loss. It has been found that taking stator and rotor dimensions as variables will give the same output as given by taking rotor dimensions as variables. Therefore, the second approach, i.e. only optimizing rotor dimensions is considered here. The stator used is that of a standard 40 kW, induction motor whose details are given in Appendix 1. From Fig. 1, the following dimensions can be taken as variables X1 = channel width/pole pitch (v) X2 = flux guide span/pole pitch (FG~) (al) X3 = flux guide span/pole pitch (FG2), (as) X4 = channel depth (G) Xs, X6 = length of flux guides (W~, W2) XT, X8 = thickness of flux guides (7'1, T~) X9 = air gap length (g). The number of variables come out to be 9 and hence the optimization process will become more complex. It has been found that a small air gap length g is always desirable and hence the air gap length is taken as low as mechanically possible and, in the present case, is equal to induction motor air gap length. A deeper channel is usually beneficial but is normally selected so as to have sufficient rotor core iron. In the present case, it is taken as 30 times the air gap length. The flux guide permeance factor W~T (ratio of W and T) influences Xmd and Xmq and (WI/T1) and (W2/T2) can be taken as variables, instead of Wl, W2, TI and T2. As shown later, the saturation limits the flux guide thickness. It has been found that (WI/TO = (W2/Tz) result in improved performance as compared to (WI/TO ~ (WE/T2) (Fig 2). The flux guide thickness is selected so that it is not more than the width of 2 rotor slots. (WJTI)= (W2/T:)= 27.6 is the right proportion not to saturate the core. The effective permeance factor (W[ T) is taken as 1.4 times the above value due to presence of iron bridges. Hence, we are effectively left with the following three variables v, a~ and a2. The following constraints are applied: (i) The design variables should be positive. (ii) p.f. I> 0.75. (iii) Total loss ~<3900 W. The iron loss (in case of reluctance motors) is usually higher and here is taken equal to 11 times the induction motor iron loss. Due to large iron loss assumptions, the optimum output is rather pessimistic and if the iron loss could be reduced, then the output power would be higher than predicted. Based on above conclusions, the optimum design for a medium h.p. reluctance motor is

286

M.H. NAGRIAL

7O

A

8

60 50

n~ UJ

o

40 30 20 10 i

10

20

30

40

50

60

70

80

90

degrees

ANGLE Fig. 2. Variation of output power with equal and unequal flux-guide permeance factors.

100 CURRENT PF,EFF 90

EFF

°

0.8-

~CURREN T

0.7-

PF: 0.E,0.50.40.3.

9.20.1-

"It' = 0.25 (~d = 38.6 T

10

KW 10

20

30 OUT

40 PUT

50

60

POWER

Fig. 3. Performance characteristics of optimally designed reluctance motor.

7~

Large reluctance motor design--nonlinear programming approach

287

predicted, using the optimization routine mentioned in Section 2, whose design details and predicted p e r f o r m a n c e are summarized below. It can be seen that full load output for the same total loss is 1.9 k W more than achieved b y the induction motor. v = 0.25,

al = 0.69,

c~2 = 0.90

W1 = I4"2 = 38.64 (effective). T~ 7'2 Full load o u t p u t = 4 1 . 9 k W , full load current = 4 7 . 6 8 A , full load p.f. -- 0.7866, total loss = 3887 W, iron loss --- 2346 W. The predicted p e r f o r m a n c e characteristics for optimum design are shown in Fig. 3. CONCLUSIONS The possibility of using a nonlinear programming a p p r o a c h to the design optimization of large reluctance motors is investigated. It has been shown that a large reluctance motor would require more than 1 flux guide per pole. It has been shown that the number of variables could be reduced to an acceptable limit where the complexity, storage and time required is substantially reduced. The Powell method alongwith the S U M T technique is e m p l o y e d to predict optimum rotor design for a medium h.p. reluctance motor whose predicted output is more than an induction motor in the same frame size and similar stators. Acknowledgements--The author is grateful to Prof. P. J. Lawrenson, University of Leeds, England for guiding some of the

earlier works on reluctance motors, whose extension is the present work. The author is also indebted to Dr. D. M. Lakhder, Dean, Faculty of Engineering for providing many facilities, and Dr. M. A. Hassan for computer facilities and Mr. M. A. Hamied in assisting to run computer programs. REFERENCES 1. P. J. Lawrenson and L. A. Agu, Theory and performance of polyphase reluctance machines. Proc. lEE 111, 1435-1445 (1964). 2. P. J. Lawrenson and S. K. Gupta, Developments in the performance and theory of segmental-rotor reluctance motors. Proc. lEE 114, 645-653 (1%7). 3. A. J. O. Cruickshank, R. W. Menzies and A. F. Anderson, Theory and performance of reluctance motors with axially laminated anisotropic rotors. Proc. IEE 118, 887--894 (1971). 4. V. B. Honsinger, The inductances Ld and Lq of reluctance machines. IEEE Trans. Power App. Systems PAS-90, 298-304 (1971). 5. W. Fong and J. S. C. Htsui, New type of reluctance motor. Proc. lEE 11.7,545-551 (1970). 6. M. H. Nagrial Synthesis o/Reluctance Motors [or Optimum Transient Performance. Ph.D. Thesis, University of Leeds (1974). 7. M. Ramamoorthy and P. J. Rap, Optimization of polyphase segmental-rotor reluctance motor design: A nonlinear programming approach. IEEE Trans. Power App. Systems PAS-9$, 527-535 (1979). 8. M ~I. Nagrial and P. J. Lawrenson, Comparative performance of direct search methods of minimization for designs of electrical machines. Electric Machines & Electromechanics (U.S.A.) 3, 315-324 (1979). 9. M. J. D. Powell, An efficient method for finding the minimum of a function of several variables without calculating derivatives. Computer J. 7, 155-164 (1%4). 10. A. V. Fiacco and G. P. McCormick, Computational algorithm for the sequentially unconstrained minimization technique for non-linear programming. Management Science 10, 610-617 (1%4). 11. V. B. Honsinger, Steady state performance of reluctance machines. IEEE Trans. Power App. Systems PAS-90,305-311 (1971). APPENDIX 1 DESIGN PARAMETERS (INDUCTION MOTOR) STATOR Winding Winding factor Conductors per phase Resistance/Ph Cold/Hot Air gap length Radius at back of slot External radius Slot width Tooth width Connection

ROTOR 2 Layer Cone. 0.956 224 0.186/0.226 ft 0.000761 m 0.165 m

0.2065 m 1.15cm 1.016 cm Delta.

Diameter of rotor Length of machine Stacking factor No. of rotor Slots Shaft radius

0.252 m 0.14m 0.95 54 0.0428 m

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M. H. NAGRIAL INDUCTION MOTOR RATINGS 3 phase, 415 V, 4 pole, 50Hz, 40kW Iron loss (N.L)= 1564W Total loss (F.L) = 3900 W Carter's factor = 1.23 Leakage factor = 0.035 Saturation factor = 1.6

RELUCTANCE MOTOR ROTOR Bridge correction factor = 1.4 Number of FG/pole = 2 Saturation factor = 1.7