Synchronisation problem of high performance reluctance motors

M.A. Badr R.M.Hamouda A.I.Alolah

Indexing terms: Three phase reluctance motor, Synchronisation, Reactance ratio, Pull-out torque, State space electrical model

Abstract: The paper reports an investigation of the synchronisation problem of three phase high performance reluctance motors. This problem stems from the fact that these motors can have extremely high reactance ratio and hence a significant pull-out torque. While this is a welcomed property from a steady state stability point of view, it can make the process of synchronisation a difficult task. To carry out this investigation a rigorous state space model in the original phase values reference frame was developed and simulated. The developed model takes into consideration the effect of torsional dynamics. This is augmented by a steady state analysis based on Park’s two-axis reference frame.

List of symbols

v, i p.u instantaneous value of voltage and cur- rent, respectively

H, HL motor and load inertia in seconds, respectively D,, H, p.u damping coefficient of mass, and its iner-

tia in seconds, respectively L, r , X p.u inductance, resistance and reactance,

respectively X,,{, X,, p.u direct and quadrature axis magnetising

reactance, respectively

T, TL p.u motor and load torque, respectively s, t slip and time in seconds, respectively P dldm t 6 load angle Y p.u flux linkage

01 cos, or

angle between the axis of phase a and d-axis supply frequency in radls and p.u rotor speed, respectively

0 IEE, 1997 IEE Proceedings online no. 19971252 Paper first received 30th September 1996 and in revised form 26th Febru- ary 1997 M.A. Badr is with the Electrical Engineering Department, College of Engineering, An-Shams University, Cairo, Egypt R.M. Hamouda and A.I. Alolah are with the Electrical Engineering Department, College of Engineering, King Saud University, P.O. Box 800, Riyadh 11421, Saudi Arabia

Subscripts d, q kd, kq

a, 6, c stator phases

0 value before modification

1 Introduction

Three phase high performance reluctance motors lave been in demand in recent years for applications requir- ing constant speed. This has been dictated by their inherent advantages of high pull-out torque, robustness and almost maintenance free. Due to the high pull-out torque of such motors, originating from their extremely high reactance ratio, they enjoy a good steady state sta- bility margin [l-91. This has nominated them for assuming the role long played by three phase synchro- nous motors. However, their main advantage of high pull-out torque can represent an obstacle during the synchronisation process. Pulling into step could be impossible for certain designs particularly when the motor starts against high inertia loads. Although many publications have dealt with the analysis of these motors in one way or another, no real effort has been spent in investigating the problem of synchronisation under loading conditions in the presence of torsional dynamics.

direct and quadrature axis, respectively direct and quadrature axis damper windings, respectively

S supply

m=l 01, Wl 0 2 , w2

motor Load

Fig. 1 System under study

IEE Proc-Electr. Power Appl., Vol. 144, No. 6, November 1997 456

The main objective of this paper is to report an investigation of the synchronisation problem of three phase high performance reluctance motors under the effect of different types of loads and with different combinations of parameters in the presence of torsional dynamics. To carry out this investigation a rigorous state space model in the original phase values reference frame is developed and simulated. This is augmented by a steady state analysis based on Park's two-axis ref- erence frame.

2 Mathematical approach

2. I System under study The system under consideration consists of a three phase high performance reluctance motor fed from a three phase balanced supply as shown in Fig. 1. The motor design data are given in the Appendix [lo].

2.2 State space model electrical system The electrical system will be described by a current state space model written in direct phase quantities. The motor is assumed to have one equivalent damper winding circuit on both the d- and q-axis of the rotor.

2.2. I Voltage equations: Considering the three phase reluctance motor system under study, the follow- ing voltage equations in the original phase values frame can be written:

v = pLi + Ri (1) where

v = [U, ' ub V c 0

i = [ i , i b ic a k d i k q l T

R = diag[r, r a r , T k d T k q ]

1 L a a L a b L a c L a k d L a k q

L a b L b b L b c L b k d L b k q

L L a c L b c L c c L c k d L c k q

Lalcd L b k d L c k d L k k d 0 1 L a k q L b k q L c k q 0 L k k q

The elements of the inductance matrix 'L' are functions of the rotor position and are accordingly time depend- ent, as given in [8].

2.2.2 Current state space model: The current state space model can be obtained by manipulating eqn. 1 as follows:

v = Lpi+ Gi (2) where

In state space form, eqn. 2 can be rewritten as

pi = A i + B v ( 3 ) where

2.3 Electromagnetic torque equation The developed electromagnetic torque can be obtained as the partial derivative of the energy stored in the mutually coupled inductive circuits with respect to the

IEE Proc.-Electr. Power Appl., Vol. 144, No. 6, November 1997

angle 0. Accordingly, the electromagnetic torque can be expressed as

a T, = i''-(Li)/3.0 801 (4)

2.4 Mechanical system The mechanical system consists of two masses as shown in Fig. 1. This system is represented as two iner- tias interconnected by a torsional spring (the connect- ing shaft), and can be described by a set of second- order differential equation of the form [l I]:

where T = { H*p2 + Dp + K}B ( 5 )

H,* is the normalised inertia of mass i and is given by: H," = 2w,H,

The transmitted torque Tv is related to the difference in crank angles by the following equation:

T%j = K,, (0% - 0,) (6) The equations describing the unforced, undamped mechanical system may be rearranged into the follow- ing form:

( 7 ) -1

p'6J = -lH* KO = A,Q The eigenvalues and eigenvectors of A, yield the mechanical system modal (natural) frequencies.

The overall state space model representing the tran- sients of the system under study can be obtained by augmenting eqns. 3, 4 and 5 .

2.5 Steady state analysis The main intention of this Section is to develop an expression for the avera,pe electromagnetic torque as a function of slip.

2.5.1 General voltage equations: The voltage equations of a reluctance motor in Park's two-axis ref- erence frame can be arranged as follows [12]:

v d = p91-1 - p B q q + r a i d ( 8 )

vq p*q + p09d + T a i q (9)

T = iq'€'d - id'$'q (10)

q d = X d ( P ) i d (11)

Q q = Xq(P)iq (12) The convention adopted here is the motor convention. Positive stator current stands for generation. The q-axis is assumed leading the 11-axis in the direction of rota- tion.

2.5.2 Steady state ac representation: If the rotor rotates with a p.u speed of p B = (1 ~ s), while a sinusoidal voltage is applied to the three phase arma- ture, the axes components will alternate with slip fre- quency. The rotor angle 6 decreases with time. The following relations can then be written:

-Vsin(st) = - p X d ( ; ~ ) i d + (1 - s ) X q ( p ) i q - r a i d

(13)

451

v cos(st) IZ - (1-s)Xd(p)id-PXq(p)i , -r , i , (14) For steady state ac operation the operator p can be replaced by j s and the classical phasor notation can be used as shown below:

_ _ _ _ jV [-T, - J s X ~ I I ~ + (I - s)X, I, (15)

_ _ _ _ v - ( s - 1)xd Id + [-T, - j sXq]Iq (16) where X, and x4 are phasors representing the opera- tional impedances as functions of slip. Expressions for

can be derived from the equivalent circuit of Fig. 2.

and

Fig.2 Equivalent circuit of the motor

Solving eqns. 15 and 16 yields the following expres- sions:

2.5.3 Electromagnetic torque components: The average value of the asynchronous torque compo- nent can be expressed by [ 121

-- _- Teasyn = 0.5Real[Iq Q i - I d Q;] (20)

The expression for the synchronous torque component of a reluctance motor can be given by:

where gl = 0.5(Xd - Xq) sin 26 + Y,

g2 = (r,)[r,2+ Y,(X~ - X,) sin 26]/(r,2+ XdXq) g3 = (ra)[X,2 sin2 6 +x; cos2 S]/(Y; + x,X,> 6 = 6, f st The average value of this component is zero as long as the motor is not synchronised.

458

3 Results and discussion

3. I Motor with the original design data The starting transients of the motor under study have been obtained by solving numerically the state space model given by eqns. 3 and 4 using the standard Runge-Kutta-Merson integration routine.

Fig. 3 shows clearly that this motor cannot reach its synchronous speed even when it starts against no load. The situation is of course worse when the motor starts

3

?

I-&

a

If

-U ' U a

1 ?

U* l a ,

a a, a torque ffl

-3 0 b

1 31 speed r - ? d

U-

I-"

Q

a, a, Q ffl

-3 0 0 2.5 5.0-

t,s C

Fig.3 a No load; b TL = 0.4~0, p.u; c T, = 0.4p.u.

Starting transients of the motor with its original design data

3 ? Q

I-"

-3 a

3 1 ?

U. Q ?

Q

torque

0 2.5 t,s

5.0

b Fig.4 transients

a No load; b TL = 0.4~0 I

Effect of reducing the d-axis rotor resistance on the starting

( I k d = 'kdo12)

3 1 ?

U- Q

a,

Q ffl

? Q

+a a,

-3 0 a 3 1

? ? Q

U- +@ Q

a, a, Q ffl

- 3 - f I.0 0 2.5 5.0

t . 5 b

Fig.5 transients

U No load; b T, = 0.4~0,

EfSect of reducing the q-axis rotor resistance on the starting

(rky = rkqo12)

IEE Proc.-Elect?. Power Appl., Vol. 144, No. 6, November 1997

against either a constant torque load or a load torque proportional to the motor speed. It should be noted that the pull-out torque of this motor at steady state is 0.506p.u.

3.2 Motor with modified parameters Figs. 4-6 display the starting transient response curves for the motor under study when its parameters are modified one at a time. From these Figures it can be seen that reduction of either the stator or the d-axis rotor resistances improves the synchronisation capabil- ity of this motor at no load. The motor can now pull into step if unloaded. It also allows the motor to syn- chronise while starting against light loads. However, the maximum loads that can be synchronised are still far below the pull-out torque of this motor. In this case, it is recommended that the motor starts against no load and is loaded after it reaches synchronous speed. A typical illustration of this case is given by Fig. 7. In this case the motor can carry a load torque close to its pull-out torque without losing synchronism. Reduction of the q-axis rotor resistance has no signifi- cant effect in this respect.

r 1 speed 3

.. U- ? Q

torque Q +a

Q aJ aJ m

* O a 31 speed r l

? ? Q

U- cu If: Q

torque I $ -3 4 LO

0 2.5 5.0 t , s b

Fig.6

U No load; b T, = 0.4w,

Ef jc t of reducing the stator resistance on the starting transients = r O 0 4

-3 P 0 2.5 5.0

t , s Transients of the motor under study with reduced stator resist- Fig. 7

ance (ru = r /2). The motor starts against no load and then a 0.4 p.u load torque is applyld at f = 3.0s

3

? Q

IS -3 Y

0 2.5 5.0 t,s

Ejject of inertia constant on the synchronisation pvoces.~ Fig. 8 ( r , = rc,(l12); TI- = 0.0; H = 2.4Ho

It has been found that variation of the rotor leakage reactances has no significant effect on the synchronisa- tion capability. The inertia constants of the motor and its load are important parameters that decide the suc- cess or failure of the synchronisation process. As can

IEE Proc-Electr Power Appl , Vol. 144, No. 6, November. 1997

be seen from Fig. 8, the motor with modified parame- ters failed to pull into ;step even at no load when the motor inertia was increased by almost 140%.

3.3 Physical understanding For easy synchronisation, the asynchronous torque-slip curve near synchronous speed should be steep. This calls for the production of a high maximum torque at low slip. The d-axis rotor resistance plays an important role in achieving this requirement. Lower values for this resistance increase the steepness of the torque-slip curve near synchronous speed but do not affect the magnitude of the maxirrtum torque, Fig. 9a and b. The q-axis rotor resistance has no significant effect in this respect. Stator resistance also plays an important role in securing the synchronisation process. As can be seen from Fig. 9c, higher stator resistance reduces the asyn- chronous torque near synchronous speed. It may even result in a negative asynchronous torque in this region. This adversely affects the process of synchronisation. The reduction of the leatkage reactance in either the d- or q-axis increases the rnaximum asynchronous torque but has no effect on the steepness of the torque-slip curve near synchronous speed, as shown in Fig. 9d and e. The effect of varying the d-axis rotor parameters is more pronounced than i.hat of varying the q-axis rotor parameters. This could be attributed to the shunting effect of XUq which is much lower than XUd.

-0.4 1 a 0.81

...... e====-

h -0.4

- - - - - - . . . . .

0

-0.4 %E5, d ~ , I

-0.4 k7. 0 sllp 0.5 1 .o 0

e Effect of motor parameters on the asynchronous torque-slip Fig.9

curve a Effect of rk& b Effect of r,<<,; c Effect of ra; d Effect of Xkd; r Effect of Xkq ~ design value

0 . 5 ~ design value - 2 . 0 ~ design value -~~

It should be pointed out that improving the synchro- nisation capability through increasing the steepness of the asynchronous torque slip curve may be coupled with deterioration of the speed stability of the motor U].

The failure of the high performance motor to syn- chronise when carrying a load could be attributed to the presence of considerable saddles in its torque-speed curve near synchronous speed. These saddles result

459

from the oscillatory reluctance torque component. Although such saddles exist all over the starting period, their effect in the vicinity of the instant of starting is undermined by the high frequency of the oscillatory reluctance synchronous torque component.

3.4 Asynchronous mode of operation A reluctance motor can operate asynchronously when its load torque exceeds the pull-out torque. In this case the motor experiences oscillating reluctance torque, Fig. 10. Under such a condition, the motor may be subjected to severe torsional stresses if the frequency of the reluctance torque pulsations coincides with the tor- sional natural frequency of the motor shaft assembly.

-3 Ji a

3 ?

I--

Q

c11

-3 -! 0 1 2 3 4 5

t , s b

Fig. 10 Transients of the motor under study with reduced stator resist- ance (r , = rJ2) . The motor starts against no load and then a load torque i s applied at t = 3.0s, T, = 0.54p.u.

4 Conclusions

This paper presents the results of an investigation of the problem of synchronising high performance reluc- tance motors under the effect of torsional dynamics. It has been found that improper combination of motor parameters could lead to failure of the synchronisation process, particularly when the motor drives high inertia loads. In general, these motors may not be able to pull into step while starting against a load torque. Under such a condition, the motor should be allowed to start against no load and be loaded after it reaches synchro- nous speed. Failure of synchronisation could lead to

serious problems in the presence of torsional effects. The motor may experience, in this case, a prolonged period of below synchronous speed operation associ- ated with torque and speed oscillations. If the fre- quency of such oscillations matches one of the natural frequencies of the motor-load-shaft assembly, reso- nance may occur with its damaging consequences. For this reason, the asynchronous mode of operation should be avoided.

5

1

2

3

4

5

6

7

8

9

References

LAWRENSON, P., and AGU, L.: ‘Low inertia reluctance machines’, ZEE Proc., 1964, 111, (12), pp. 2017-2025 STEPHENSON, J., and LAWRENSON, P.: ‘Average asynchro- nous torque of synchronous machines with particular reference to reluctance machines’, ZEE Proc., 1969, 116, (6), pp. 1049-1051 LAWRENSON, P., and BOWES, S.: ‘Stability of reluctance machines’, ZEE Proc., 1971, 118, (2), pp. 356-369 LAWRENSON, P., and MATHUR, R.: ‘Asynchronous perform- ance of reluctance machines allowing for irregular distribution of rotor conductors’, ZEE Proc., 1972, 119, (3), pp. 318-324 HONSINGER, V.: ‘Steady state performance of reluctance machines’, ZEEE Trans., 1971, PAS-90, (l), pp. 305-317 ALOLAH, A.I.: ‘Steady state operation of three phase reluctance motor from single phase supply’, J. ZEE (Zndia), 1989, 70, Pt.EL, (5). vv. 157-161

~ I,

ALOLAH, A.I., and BADR, M.A.: ‘Starting of three-phase reluctance motors connected to a single phase supply’, ZEEE Trans., 1992, EC-7, (2), pp. 295-301 BADR. M.A.. and ALOLAH. A.I.: ‘Transient analvsis of three phase reluctance motors fed from a single phase ;upply’, ZEE Proc. B, 1995, 142, (2), pp. 104-112 FAROUK, A., ABO-SHADY, S., and FARAHAT, K.: ‘Syn- chronisation of reluctance motors’, IEEE Trans., 1981, PAS-100. (41. uv. 1885-1892

I, A I

10 OSHEBA, S., and ABDEL-KADER, F.: ‘A comparison between conventional and segmental reluctance motors’, Electr. Mach. Power Syst., 1989, 16, pp. 249-264

11 HAMOUDA, R.M., BADR, M.A., and ALOLAH, A.I.: ‘Effect of torsional dynamics on salient pole synchronous motor-driven compressors’, ZEEE Trans., 1996, EC-11, (3), pp. 531-538

12 SARMA, M.: ‘Synchronous machines: Their theory, stability and excitation systems, 1st ed.’ (Gordon and Breach, New York, 19791

6 Appendix

The p.u parameters of the motor are: rao = 0.205, X , , = 1.296, Xqo = 0.409 X,do 1.210, Xaqo = 0.323, x k d o = 0.341 Xkqo = 0.427, R k d o = 0.177, Rkqo = 0.453 K,, = 5.0, HL* = 28.65p.u, x* = 50.52p.u v, = 220v

460 IEE Proc -Electr Power Appl, Vol. 144, No 6, November 1997