Stability of sliding-mode current control for high performance induction motor position drives

L.-G.Shiau and J.-L.Lin

Abstract: A novel inner-loop sliding-mode current control scheme is proposed based on a nonlinear mathematical model of induction motor position drives. The parameters of position and speed controllers are taken into consideration in the inner-loop sliding-mode current control. After deriving the error dynamic equations, a rigorous stability verification for the overall system is provided. The proposed scheme is implemented by a PC-based controller. Agreement between analytical studies and experimental results confirms the validity of the proposed control system. The overall system exhibits robust stability and robust performance despite the presence of motor parameters and shaft moment of inertia variations as well as exogenous load disturbances.

1 Introduction

A typical servo drive requires quick response, high accu- racy and robust performance. Due to relative simple con- trol, DC drives are traditionally considered as the preferable choices. Nowadays, AC drives become popular in many applications because of the advances in power electronics and microelectronics technology. Among the various types of AC drives, permanent magnet synchro- nous motors (PMSMs) are generally used owing to their compactness and light weight. However, PMSMs are very expensive because they use costly magnetic materials. In addition, degaussing problems exist in some circumstances. For these reasons, induction motors (IMs) are alternatively considered in AC drives when weight is not a crucial factor and low cost and free maintenance are taken into account.

It is well known that field-oriented control (FOC) is an effective scheme for the variable speed control of IM drives. However, difficulties arise from the modelling uncer- tainties due to parameter variations, magnetic saturation, load disturbances and unmodelled dynamics. To ensure good dynamic performance various robust control strate- gies for IM drives have been reported in the literature. In particular, sliding-mode control (SMC) has gained wide attention owing to simple design, easy implementation, fast dynamic response, and robustness to parameters variations and load disturbances. It has been applied to the position and velocity control of IM drives [I-51.

A new SMC method was proposed [2] for IM speed control based on the model in the frame rotating synchro- nously with the stator current vector. Experimental results showed that the overall system was insensitive to motor parameter variations. However, the reference torque signal was treated as a constant in the SMC design procedure and thus the stability of the overall system cannot be guaran- teed.

0 IEE 2001 ZEE Proceedhgs online no. 20010035 DOL 10.1049/ipepa:20010035 Paper first received 2 I st February and in revised form 13th September ZOO0 The authors are with the Department of Engineering Science, National Cheng Kung University, Tainan, Taiwan 701, Republic of China

Based on the nonlinear model of an induction motor, a novel SMC scheme for the current tracking is proposed for high performance IM position servo drives. The parameters of the position and speed controllers are taken into consid- eration in the inner-loop sliding-mode current controller design procedure. The reference torque signal is recognised as an additional state variable of the overall system. After the error dynamic equations are derived, a rigorous stabil- ity verification for the overall system is provided. In addi- tion, owing to faster dynamics of the inner current loop than that of the outer position and speed loops, a reduced linear model is derived after current tracking. The behav- iour of the overall system can be thereby designed based on linear control theory. The proposed scheme is implemented in laboratory by a PC-based controller with a voltage- source PWM inverter. Experimental results show that the overall system exhibits robust stability and robust perform- ance despite the presence of motor parameters and shaft moment of inertia variations as well as load disturbances

2 Dynamic model

The dynamics of induction motors are described by a set of highly nonlinear differential equations. Under the assump- tion of identical mutual inductances, linear magnetic circuits, and neglecting the iron losses and mechanical damping effects, the state space representation of the sixth- order dynamic model [5] of an induction motor in the synchronous rotating reference frame can be described by

G, = % ( T ~ J - T ~ ) (2)

( 3 1

T, = kTiT J2Xr (4)

. UT om = - n P

with the developed electromagnetic torque

where

69 1EE Proc.-Elertr. POIVL'I. Appl.. Vol. 148, No. 1, Juniiary 2001

(5)

All parameters are denoted by Rs, R,. = stator and rotor resistances L,, L,, L,,, = stator, rotor and mutual inductances v,, is, Ar = stator voltage, current and rotor flux vectors U,, U,. = synchronous and rotor electrical angular

e,,, = rotor mechanical angular position T,, T, = electromagnetic and load torques np, J = number of pole pairs of motor and shaft

Subscripts d, q, s and r are referred to as the d-axis, q-axis, stator and rotor variables throughout this paper, respec- tively. Assume that the d-axis coincides with the rotor flu vector A,. with Aqr = 0. Let A,. = Ad,. for notational brevity, then synchronous electrical angular speed

speeds

moment of inertia

and electromagnetic torque

Te = h - ~ i , , X , (7) are obtained from eqns. 1 and 4, respectively. Combining eqns. 1-3 and eqns. 6 and 7 yields

( 8 )

(9)

1 i d s = f l + -uds

LL7

L , 1

2,s = f 2 + -'Uqs

. n,h-~. n t q S X r - J T L

J wr = -

J . WT 8, = -

nP where

Eqns. 8-12 constitute a fifth-order nonlinear model of an IM in the frame rotating synchronously with the rotor flux vector.

3 Sliding-mode current coritrol design

Induction motors are inheremly highly nonlinear. The primary difficulty in the control of IM drives arises from the multiplicative nonlinearity ol' the developed electromag- netic torque. However, if the current control problem is tackled, the speed and position regulations can be easily achieved by outer-loop controllers. To this end, the SMC scheme is first applied to the inner-loop current control.

The early work of SMC was proposed and elaborated by several researchers in the Sovilct Union. A recent survey given by Hung et al. [6] introduced the fundamental theory and practical applications. From a sliding-mode control point of view, the system trajectories must be required to approach the specified manifold from any initial state in the state plane. Then the system behaviour is goveined by the dynamics of the manifold on which the system trajecto- ries remain. Then the system trajectories will move to the origin of the state plane along Ihe given manifold.

oosition soeed - induction

Fig. 1 Overull system of induction nmor position drive

The block diagram of an IM drive system with a propor- tional position controller and a proportional-plus-integral speed controller is depicted schematically in Fig. 1. The reference speed U,* and reference torque signal T,' can be described by

w,*( t ) = K p l [ O T e , ( t ) - em@)] (15) and

Te*(t) 1 Iqpa [U," ( t ) - w r ( t ) ] +K,, [U,* ( r ) -w, (r)]dr

(16) I'

with time derivative

T,* ( t ) = I<$ (b;C - Lj,) + K 2 2 (w,* - U,)

n K - - f:< + ~ T L J + I t rp ,Kp2$lre f + K ~ ~ K ~ z S T ~ , (17)

where the superscript * denotes the desired reference signal of the corresponding variable and

- (y + K i 2 1 w, - Kp,KcLBm(18j

Recognising T,* as an additional state variable, eqns. 8-12 and 17 constitute a sixth-order state-variable dynamic equation. To proceed, define a switching function vector as

The following is to decide the desired reference signals ids* and .zqs*. In the steady stai.e, td, = &/L,,* from eqn. 10 by letting A,. = 0. If the rotor flw maintains its rated value AIrcf( for constant torque operation, the reference stator currents ids* and iqs* can be determined from eqn. 7 and gven by

IEE Pioc - E k t r Power Appl Vol 148 No I Jrrnuury 2001 70

After determining the desired reference currents id' and iys , proceed to find the control law to satisfy the sliding condition sTS < 0 [7]. From eqns. 19, S,9, 20 and 17, . *

1 Lu

S = F + HITL + H20ref + HSOTef + -U (21)

where

and

The control law for the inner-loop sliding-mode current tracking is thereby established in the following theorem. Theorem 1: Consider the sliding manifold s = [sl s2IT = 0 defined in eqn. 19 with the control law

U = -Lu(k+ H2QTef + H,B,,f + Q sgn(s) + Ks) (24)

where

41 > J f l - f l l + r l l , 4 2 > l f 4 - f 4 l + r l 2 ,

171 > 0,172 > 0, k-I > 0 , k.2 > 0 (25)

and f, and f 4 denote the estimations off, andh , respec- tively. Then the sliding manifold s = 0 is a globally attrac- tive and invariant set. Proof See Appendix (Section 9. I ) Remark I : As shown in .the proof of theorem 1 in the Appendix, it is seen that V(sj < ~ q l j s l ~ + q2(s21) - (klsL2 + k2s$). This inequality shows that V(s) is dominated by -(klsI2 + k2s22) when the system trajectories are far from the sliding manifold. Hence the larger values of kl and k2 will shorten the reaching time. On the contrary, V(s) is dominated by -(q,/.~~l + q21s21) when the system trajectories are near the sliding manifold. In this case, the smaller values of ql and q2 will attenuate the chattering. Reimrk 2 In view of the inequalities in eqn. 25, it is seen that the parameters q1 and q2 are used to deal with the var- iations of electrical parameters.

The resulting block diagram of the overall system with the inner-loop sliding-mode current control is shown in Fig. 2.

4 System stability

The sliding-mode control law in eqn. 24 ensures that the system trajectories will approach the sliding manifold s = [s, sdT = 0 from any initial state in the state plane. In this Section we study the stability of the overall system shown in Fig. 2 with a proportional position controller and a proportional-plus-integral speed controller. Substituting the control law in eqn. 24 into eqn. 21, one can obtain the dynamics

(as) for the current traclung error s = [s1 sdT = [id, - id5* i,, -

Lemma I : The dynamics of the current tracking error s = [sl ,sJT, shown in eqn. 26 is bounded with sXt) -+ 0 in a finite time smaller than tl given by

S + K s = ( F - k') - Q sgn(s) + HITI,

iJT.

Pvoof.'. See Appendix (Section 9.2) Define the rotor flux tracking error as en(t) := ?+(t) -

It follows from eqn. 10 that the dynamics of the rotor flux error can be rewritten as

rTex + ex = Lmsl which leads to the following lemma. Lemma 2: The dynamics of the rotor flux error ea(tj shown in eqn. 27 are bounded with ea(t) + 0 as t + W.

Prooj See Appendix (Section 9.3). From eqns. 11, 12 and 15, the dynamic equations of the

speed tracking error edt ) := o,.*(t) - u,(t) and the position tracking error edt) := Orc,t) - O,,(t) are given by

(27)

and

after some manipulation. It is interesting to note that eqns. 2&29 describe the error dynamics, shown in Fig. 3, of the overall system of Fig. 2. After deriving the error dynamics the stability of the overall system can be guaran- teed by the following theorem.

' - T - T I

Fig. 2 Position cuntrd systein with inner-hp sliding-Fnodr rurreiit control

IEE Proc.-Elects. P o w r Appl.. t701. 148, No. 1, Junuuiy 2001 71

eqn. 27

eqn. 26 eqns. 28 29 - e,$) Fig. 3

Theorem 2: Consider the error dynamic equations

Error d y m i c s of overall system

i 1 + hs1 = (fl - f l ) - 41 sgn(s1)

= Kp,B,,f + %[TL - k?.(ex + X;at)s2] J 1

n P nP e o + %eo - -e, = e,,,

of the overall system of Fig. 2 with constant load torque TL and K > 0, Kp2 > 0, K12 0, k , > 0, k2 > 0, q1 > 0 and q2 > 0. Assume that the reference position input e r e , t ) is dif- ferentiable and 1imf+- Oreht) = 0. Then

lim eg(t) = 0 t-+m

if n,Kp~KE2 + niKp2K22 - JKP1Kz2 > 0 (30)

Prooj See Appendix (Section 9.4). A necessary condition for the stability of the overall sys-

tem has been derived in theorem 2. In view of eqn. 30 a wide range of parameters can be adjusted by outer-loop controllers.

T.

1 U

Fig. 4 Block diagram of system after current tracking

From a control point of view, the dynamics of the inner current loop are much faster than that of the outer position and speed loops. The overall system shown in Fig. 2 can be thereby reduced to Fig. 4 after current tracking is achieved by the SMC. The closed-loop transfer function from the reference position input to the shaft position e,,? in Fig. 4 is given by

em (4

- KPiKP2s + KPiKt2 e,,,o l T L = o -

Js3 + nPhTp2s2 + (Kp1Kp2 + nPK22)s + Kp1K22

To verify the results derived, the position responses with Kpl = 30, Kp2 = 0.45 and K12 = 11.25 to a step position command er, = 2n rad for the system in Fig. 2 and the reduced system in Fig. 4 are shown in Fig. 5. Agreement between two responses reveals that the overall system in Fig. 2 can be reduced to the one shown in Fig. 4. The outer-loop behaviour can thereby be designed by the linear

(31)

12

control theory. For instance, the closed-loop poles related to the integral gain K, can be determined by the root-locus technique. Fig. 6 depicts the root-locus plot for 0 5 ICl2 2 m

with proportional gains K,, = 30 and Kp2 = 0.45. It is seen that the larger values of Ki2 will lead to the smaller damp- ing ratio.

7 , -

- 0.2 0.4 0.6 0.8

time. s

0 1

Fig. 5 ~ Fig. 2 ?‘6 Fig. 4

position response to step position c o m d e,.d = 2n rod

60 . ................................ ................. ............,.....

............

............ ...............................

-80 :.....

-1 00 -100 -80 -60 -40 -20 0

real axis Fig. 6 Root loa for 0 5 Ki2 wrth K,,, = 30 Luld Kp2 = 0 45

.- o o

-50

e ._

z” g 100

2 - 0 5om7q ..................... $ 0 a,

i -501 ‘2 time, s time, 6

Fig.7 Responses of overall syrtm with K,,, = 100, Kp2 = 0.1 a d KI2 = 10

Now choose Kpl = 100, Kl,2 = 0.1 and K12 = 10 such that the stability inequality in eqri. 30 is not satisfied. It is inter- esting to see that the responses in Fig. 7 show the instability of the overall system with q, = 300, q2 = 150, k, = 4500 and k2 = 1500 in the proposed sliding-mode current control.

IEE Proc -E/ecfr Piiwer Appl , Vol. 148, N o I , January 2001

5 Experimental results

In the laboratory an IM position drive system is assembled to verify the effectiveness of the proposed control scheme. The block diagram of the overall control system is depicted schematjcally in Fig. 8. The rotor flux Ar and the load torque T , used in SMC are obtained from a current-model rotor flux observer in a rotor reference frame [8] and a torque estimator [9], respectively. The overall system is composed of an IBM PC-586 personal computer with three 12-bit D/A and A D converters, a sinusoidal PWM voltage source inverter, a torque transducer and a separately excited DC machne load, a 2000-pulse optical encoder with a four times frequency multiplier and an 8OOW squir- rel cage induction motor with specifications and parameters listed in Tables 1 and 2, respectively. The three phase currents are measured by Hall-effect current sensors and converted to digital signals by A D converters. The config- uration of the experimental system is shown in Fig. 9. The

Table 1: Specifications of IM

rated power 800 W rated current 5.6A

rated speed 2000 revlmin

rated voltage 130V

rated torque 3.82N.m

no. of pole pairs 2

Table 2: Parameters of IM

RS 0.7231 ohm

R, 0.8415ohm

J 0.008233N.m.$/rad

B 0.000534N.m.s/rad

Ls 57.31 mH

Lr 57.31 mH

L, 52.43mH

sampling rate and reference rotor flux A,.rur are set to be 5kHz and 0.1928Wb, respectively. The parameters of the sliding-mode controller are chosen as q1 = 300, q2 = 150, k, = 4500 and k2 = 3000. To guarantee the stability of the overall system, the parameters of the position and speed controller are selected as Kpl = 30, Kp2 = 0.45 and Ki2 = 11.25 such that the necessary condition in eqn. 30 is satis- fied.

tal oder

Fig. 9 Conjgurulwn of experwnmtaf system

To show the robustness of the overall system with no load, the values of the motor parameters used in the proposed control scheme are purposely deviated from the nominal values as shown in cases A to C in Table 3. Case A shows the nominal values of the parameters. The varia- tions of rotor time constant 7,. exert the primary influence on the performance of IM drives. Based on this observa- tion, case B corresponds to the situation where the motor has been running for a long time with a heavy load. Hence the resulting rotor time constant 7, used in SMC is increased by 150% over the nominal value owing to magnetic saturation and temperature variation. Moreover, because the changes of rotor resistance R, are commonly considered as the main source of parameter variations, the case C is used to concern with the variations of rotor resist-

so f l w a r e .................

... ................................. ................................ 1.8 BIock &grm of expwinentul system

IEE Proc.-EIectr Power Appl. , Vol. 148, No. I , January 2001 13

ance. The responses to a step position command 8,q 27c rad for the three cases are illustrated in Fig. 10. It IS seen from Figs. loa-d that the detuning has the largest impact on the responses of the electromagnetic torque and the phase current. It has little effect on the position response. Case B is the most remarkable among three cases.

Table 3: Moter parameters used in sliding-mode controller

Rs(C2) R,(Q) L, (mH) L, (mH) L, (mH) T, (ms)

caseA 0.7231 0.8415 57.31 57.31 52.43 68.10

case B 0.4339 0.5049 85.97 85.97 81.09 170.27

case C 0.7231 0.4208 57.31 57.31 52.43 136.19

-10 " A 0:l 0:2 0:3 0:4 015 016 -5A 0:l 0:2 013 0:4 0:5 0:s time, s time, s

C d Fig. 10 - A

B C

Response to step positwn corrmland Or,= 2arad (no loud) .......... _ _ ~ _ a Rotor position response e,&) b Rotor speed response cy,,@) c Stator phase current i,(t) d Electromagnetic torque re([)

a

, -5 -

0 0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 2.5 3 3.5 time, s time, s

c d Respomes to step loud torque TL = 2 N m uppliedut t = 1.4s

-15' ' ' ' ' ' ' '

Fig. 11 a Rotor position response e&) h Rotor speed response ry,,(t) c Stator phase current iUx(r) d Electromagnetic torque Te(t)

Another important issue of the electrical drives is the capability to reject the effects of load disturbances and shaft moment of inertia variations on system performance. Figs. 1 l a d show the responses when a step load torque T, = 2N.m is suddenly applied at t = 1.4s after running up. With a maximum position change of only about O.OSrad, the rotor position returns to its steady state within 0.35s as shown in Fig. 1 la. This reveals that the performance of the overall system is robust to the load disturbance. The per- formance robustness for the shaft moment of inertia varia- tions is illustrated in Figs. 12~-c with nominal, five times and one-fifth of the nominal moment of inertia. It is seen

from Fig. 12 that the performance of the overall system is also robust to the shaft moment of inertia variations.

E

0:l 0:2 0:3 014 0:5 0:s time, s

C

0 0.1 0.2 0.3 0.4 0.5 0.6 time, s

d Fig. 12 __ nominal moment of inertia

_ _ _ _ U Rotor position response B,>,(r) b Rotor speed response o&(l) c Stator phase current iu,(r) d Electromagnetic torque T,(r)

Respomes ro slufi inoinent of' inertia vciriation

......... five times of nominal moment of inertia one-fifth of nominal moment of inertia

6 Conclusions

Based on the nonlinear model of induction motors, a new inner-loop sliding-mode current control scheme for IM position drives has been presented. The parameters of the position and speed controllers in the outer-loop are taken into consideration in the inner-loop sliding-mode current control. Thanks to the invariant property of SMC, the inner-loop possesses good rotlustness. A rigorous stability verification for the overall sqstem is provided. The pro- posed scheme is carried out by a PC-based controller. The experimental results verify the proposed control scheme. The overall system exhibits robust stability and robust per- formance despite the presence of motor parameters varia- tions and external load disturbances.

7 Acknowledgments

The authors wish to express sincere gratitude to the anony- mous referees for their constructive comments and helpful suggestions which led to substantial improvement of this paper. This work was supported in part by the National Science Council of the Republic of China under project NSC 88-221 3-E-006-095.

References

LIN, J.L., and SHIAU, L.G.: 'PI speed control for an induction motor with inner sliding-mode torque control scheme'. Proceedings of the Automatic Control conferenci: of R.O.C., 1998, pp. 515-519 CHAN, C.C., and WANG, H.Q.: 'New scheme of sliding-mode con- trol for high performance induction motor drives', IEE Proc. Electr. Power Appl., 1996, 143, (31, pp. 177-185 DUNNIGAN, M.W., WADE, S., WILLIAMS, B.W., and W, X.: 'Position control of a vector controlled induction machine using Slo- tine's sliding-mode control', IEE Proc. Electr. Power Appl., 1998, 145, (3). pp. 231-238 SAHIN, C., SABANOVIC, A.. and GOKASAN, M.: 'Robust posi- tion control based on chattering free sliding-modes for induction motors'. Proceedings of IECON'95, 1995, pp. 512-517 JEZERNIK, K., HREN, A., arid DREVERSEK, D.: 'Robust slid- ing-mode continuous control of an IM drive'. Conference Record of IEEEDAS annual meeting, 1995, pp. 335-342 HUNG, J.Y., GAO, W., and HUNG, J.C.: 'Variable structure con- trol: a survey', IEEE Trms.. 1993, IE40, (l), pp. 2-22 SLOTWE, E.J.J., and LI, W.: ,Applied nonlinear control' (Prentice- Hall, Englewood Cliffs, NJ: 1991) JANSEN, P.L., LORENZ, R.D., and NOVOTNY, D.W.: 'Observer- based direct field orientation: analysis and comparison of altemative methods', IEEE Trtms., 1994, IE-30, (4), pp. 945-953

IEE Proc.-Elecrr. Pover Appl., Vol. 148, No. 1. .Januur?; 2001 14

9 IWASAKI, M., and MATSUI, N.: ‘Robust speed control of IM with torque feedfonvard control’, IEEE Trcms., 1993, IE40, (6), pp. 553- 560

10 GAO, W., and HUNG, J.C.: ‘Variable structure control of nonlinear systems: a new approach‘, IEEE Trans., 1993, IEE-40. (l), pp. 45-55

11 DESOER, C.A., and VIDYASAGAR, M.: ‘Feedback systems: input- output properties’ (Academic Press, New York, 1975)

12 KHALIL, H.K.: ‘Nonlinear systems’ (Macmillan, New York: 1992)

9 Appendix

9. I Proof of theorem I Consider a scalar Lyapunov function candidate

1 2 2

It follows directly from eqn. 21 that the time derivative of V(s) along the system trajectories is given by

V ( s ) = L T s = -(s? + S;)

V ( S ) = sTS zr sT ( F + HITL + H 2 6 v e f

I -7?zIS2J - Jhs; Hence s,(t) + 0 and s2(t) + 0 in a finite time [lo].

9.3 Proof of lemma 2 The homogeneous linear system z,en + en = 0 is globally exponentially stable and the force input s1 -+ EL,([O, -)) owing to sl(t) + 0 in a finite time by lemma 1. Based on these observations the rotor flux tracking error ek(t) is bounded and eA(t) + 0 as t -+ 00 ([I 11, theorem b on p. 59).

9.4 Proof of theorem 2 Define yl( t ) := lo’ e,(z)dz, y2(t) := j l ( t ) = edt) and y3(t) := ekt) then the state-space representation of eqns. 28 and 29 can be expressed as

Y = AY+ B(t)Y+ C(t) + < with state vector Y := bl y2 y31T, where

0 01

1 0 ~ ( t ) := 1{~10,~f(t) - + ( e x ( t ) + ~ Y ) s , ( t ) [ e v e , ( t )

and constant vector ( := [0 (ndJ)T, OlT, Since the matrix A is nonsingular with Kpl > 0, Kp2 > 0 and K,. > 0, the last equation can be rewritten as

Y= A ( Y+ A- E ) + B( t ) ( Y+ A- c) + C( t ) - ~ ( t ) A - I E Now define F := Y + A-’( then

- Y = AY+ B(t)Y+ D(t)

with D(t) := C(t) - B(t)A-’(. The result in lemma 2 en(t) + 0 as t -+ 00 leads to B(t) + 0 as t + -.

Assume that the reference position input O r e , t ) is differ- entiable and O,.,ht) + 0 as t + -. Together with the fact that s2(t) -+ 0 in a finite time, shown in lemma 1 and en(t) is bounded, shown in lemma 2, it follows that Cyt) + 0 as t + 00, and thus D(t) + 0 as t + -.

With the help of the fact that 1imtjm B(t) = 0 and the matrix A is stable if the inequality in eqn. 30 is satisfied, the origin P = 0 is a globally exponent@lly stab& equilibrium point of the unforced system F = A Y + B(t)Y ([12], exam- ple 4.10 on p. 199). Moreover, based on the fact lim, +,-

D(t) = 0, we conclude that 1imL+,= Y (t) = 0 for the differen- tial equation B = AB + B(t)Y + ~ ( t ) (1121, lemma 4.9 on p. 208). This implies that

[TI lim Y(t) = -Ap1< = t+w

since P = Y + A-’(. Based on Y = b, y2 y3IT = [I; e,(z)dz eJt) e(@]T, we get

Ji% lt e,(r)dr = - T L

K22

lim e,(t) = 0 t+cc

and lim ee(t) = 0 t-+m

This completes the proof.

15 IEE Prw-Electr. Power Appl , Vol. 148, No. I , Jiinunry 2001