Partial pole placement of H, based PSS design using numerator-denominator perturbation representation

C.Y.Chung, C.T.Tse, A.K.David and A.B.Rad

Abstract: A new H, PSS design method which uses the numerator-denominator perturbation representation and includes the partial pole placement technique and a new weighting function selection method is proposed. This overcomes certain conventional H, PSS design algorithm limitations. A sixth-order machine model is used to increase the accuracy of selected weighting functions. A robust PSS has been successfully designed for single and two-machine systems by treating the highly nonlinear characteristic of the power system as model 'uncertainty'. The design is verified to have better performance for a wide range of system operating conditions when compared with the conventional PSS designs.

1 Introduction

Power system stabilisers (PSS) are employed to provide damping of rotor-angle electromechanical oscillations (EMO) [l]. In recent years, various control methods have been applied to PSS tuning among which are optimal con- trol [2] and combined method of model frequency, root locus and sensitivity techniques [3]. These studies have con- centrated on a single nominal operating point without con- sideration of the robustness of the power system and cannot guarantee stability of a highly nonlinear system such as an electric power system. Although some robust PSS designs are based on online tuning techniques such as adaptive [4] and neural network [5], power utilities still pre- fer the fmed-structure and fixed-parameter (FSFP) control- lers due to limited confidence in online tuning schemes [6]. A robust FSFP controller to enhance system damping over a wide range of operating conditions is desirable.

The main advantage of H, optimisation methods is that the model uncertainties can be accounted for at the design stage. H, algorithm has been used to design PSS control- lers [&8] and thyristor controlled series compensator auxil- iary controllers [9]. However, these are based on the conventional H, algorithm and direct application of this method to PSS design has certain difficulties: (i) certain uncertainty modelling restriction, (ii) unobservable modes due to pole-zero cancellation, (iii) over design and hence performance degradation of the controller. [9] has proposed a technique to overcome limitations of (ii) but the controller will have a pair of weakly damped poles [7]. Limitations (i) and (ii) have been solved in [7] but the controller becomes suboptimal. Limitation (iii) is solved in [6, 81, however, the uncertainty cannot be handled in the

0 IEE, 2001 IEE Proceedings online no. 20010696 DOL 10.1049/ipgtd:20010696 Paper fmt received 22nd July 1998 and in revised form 28th January 1999 The authors are with the Department of Electrical Engineering, Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

design stage [6] and the resulting nonminimum phase con- troller is unacceptable for practical applications because of stability problems. Moreover, several internal states of the system need to be used in the design stage which makes it difficult for practical implementation [8].

Since there is no satisfactory method to solve these limi- tations simultaneously, this paper will introduce a com- bined technique of numerator-denominator perturbation modelling, partial pole placement and a new weighing func- tion selection method.

I

Because the generator phase characteristic is vital for the selected weighting function [lo], the widely used version of the Heffron-Phillips model is not adequate. A sixth-order machine extension which can be applied to both single machne and multimachine environment is used. The per- formance of the proposed PSS is examined and compared with the conventional PSS (CPSS) which uses pole place- ment technique [l I].

IEE Proc-Gener. Trnnstn. Dhirih., Vol. 148. No. 5 , Sepfenlher 2001 413

2 System modelling

2. I Sixth-order generator model In Park's two-axis machine model, machine behaviour can be described by a set of sixth-order equations in the machine dq-frame so as to include amortisseurs effects. Under small perturbations, these equations are linearised (Appendix 9.1) and expressed in the convenient block for- mat which is easily associated with the block model of con- trol equipment such as the governor-turbine system (GOV), excitation system (EXC) and PSS (Fig. I). Thus, a self-con- tained individual machine representation with five input/ output variables: AId, AIq, A & AVq and A6 is formed.

Fig. 2 System conjigumtion a n d p h ~ o r diagrum

2.2 Single machine infinite bus model From Fig. 2, using the infinite bus as reference, the machine terminal voltage equations are:

V, =I IdXe + I, Re + Vo COS S (1)

and for small perturbations, take the form:

+ V, ( X , sin S - Re cos S) Ad] (3)

1 Xz + .RZ

AI, = -- [R,AV, - xeav, + v,(R, sin S + X, cos S)AS]

(4) The block diagram (Fig. 3) has the same five extemal con- nections as Fig. 1 and can be readily interconnected. The reference frame is stationary, that is AWref = 0.

2.3 M ultimach ine network representation A multimachine version is established by linking together the five external connection variables for all machines in the following form [12]:

[ U i q l = [Yml [AV,,] + [III[ASl (5)

where [ y,] and [a are derived from the steady-state oper- ating condition of the system. For a two-machine system, the full matrix equation is:

Y11 Y12 Y13 Y l 4 ] [AVdi] Y21 Y22 Y 2 3 Y24 AV,, Y31 Y32 Y33 Y34 AVQ

4 2 Y41 Y42 Y43 Y44 AV,, K11 K12

K41 K42 (6)

The machines are together as shown in Fig. 4 where only the connections to AId, are shown for simplicity.

mochine 2 I I machine 1

3

Additive and multiplicative uncertainty representations are often used to model perturbed plant in the H, approach. These perturbed plant (P(s) models with, W&) = 0 for the additive and W2(s) = 0 for the multiplicative cases, are rep- resented by

H, mixed sensitivity optimisation problem

p ( s ) = (1 + a M ( s ) W 3 ( s ) ) pO(s) + AA(s)WZ(s) (7)

where ilAA(s)llm 5 1, IIAM(s)ll, 5 1, W2(s) and W3(s) represent the upper bound of the allowable additive and multiplica- tive perturbations and Po@) is the nominal plant.

From the mixed sensitivity optimisation formulation, the optimal H, controller which minimises the effects of distur- bance on the plant output and guarantees the robustness of the plant can be obtained by minimising J:

J = 1 1 WZ(S)R(S) :E:: Ilm (8)

where the weighting functions W,(s), W,(s), W&) are used to define the acceptable magnitude of output error in the presence of disturbances and the maximum perturbation of the nominal plant of additive and multiplicative type, R(s) = K(s)S(s) where S(s) is the sensitivity function of the closed-loop system, K(s) is the controller and T(s) = P&)K(s)S(s). This is shown in Fig. 5 where w is a distur- bance signal.

The process of J minimisation can be achieved by solving two Riccati equations [ 13, 141. If the minimal value of J I 1 the obtained controller can stabilise all perturbed plant rep- resented by eqn. 7.

However, the conventional H, PSS design based on direct application of this method has the following limita- tions:

TEE Proc -Gener Trunwn Dlyttrb , Vol 148, No 5, September 2001 414

2 2 2 3 +

Kls) Pg(s)

Fig. 5 method

ilie augmented plmit wifli controller for the conventioturl H, design

(i) Restrictive uncertainty modelling under stableiunstable transition: A&, AM(s), W2(s) and W3(s) in eqn. 7 are assumed to be stable. Therefore, the perturbed plant will have the same number of the right-half plane (RHP) poles as the nominal plant. In other words, these models are unable to represent uncertainty when a nominal stable (unstable) plant becomes unstable (stable) after being perturbed [15]. How- ever, stable poles of a power system can become unstable after being perturbed especially for the oscillatory mode (local or interarea mode). (ii) Unobservable modes due to pole-zero cancellation: The closed-loop system poles of the designed controller and the nominal plant will include both the poles (stable) and the mirrored left-half plane (LHP) poles (unstable) of the nominal plant. If the nominal plant contains poorly damped modes, the poorly damped modes will be ‘can- celled’ by the zeros of the controller and become unobserv- able for the specified output. Consequently, improvements, if any, due to the controller cannot be recognised. Since the electromechanical mode of the power system is always close to the imaginary axis, this pole-zero cancellation phe- nomenon is a serious limitation for PSS design. (iii) Overiunder consideration of the uncertaintyithe special specification of the controller: The main objective of a robust PSS is to damp oscillatory modes in different system operating points, that is the main source of the uncertainty arises from the change of system operating points. However, in conventional H, designs, the uncertainty model described by eqn. 7 will cover a wide range of unrealistic plants which is impossible to be obtained by varying the system parameter of the model mentioned in Section 2. Here, the mixed sensitivity optimi- sation method is a compromise between the robustness and the performance deterioration. Stabilising these unrealistic cases degrades the performance level of the realistic cases. Furthermore, the conventional H, method may not be able to deal with some practical constraints. For instance, gain restriction at hgh frequency for a A!2-input PSS cannot be considered since the torsional mode is not detected in the model.

4 Proposed H, PSS design methodology

The limitations discussed above can be solved by the numerator-denominator perturbation representation (Sec- tion 4. l), the partial pole placement technique (Section 4.2), and a new weighting function selection method (Section 4.3). A single-input single-output model is used for easy explanation but can be extended to multiinput multioutput systems.

IEE ProcGener . Trrrnsni. Distrih., Vol. 148, No. 5, September 2001

4. I Numerator-denominator perturbation modelling A plant with the transfer function (P(s)) can be written in fractional form as:

(9)

The numerator-denominator perturbation model can be represented as:

P ( s ) = No(s) + M ( S ) A N ( S ) W Z ( S )

Do(s) + M(S)AD(S)Wl(S)

where No(s) and Do($) are the numerator and the denomi- nator of the nominal plant, M(s)AN(s)W2(s) and M(s)A,(s) W,(s) are the perturbation of the numerator and the denominator. M(s) W2(s) and M(s) Wl(s) represent the largest possible perturbations of the numerator and denom- inator so the magnitude of AN($) and AD($) will not be greater than one.

Define V(s) = Do-’(s)M(s) where V(s) is used as partial pole placement (Section 4.2), then the following inequalities can be obtained.

N ( s ) - No(s) - - V ( S ) A N ( S ) W Z ( S ) V(S)WZ(S) NO (SI PO ( S ) P O ( S )

(11) V(s), Wl(s) and W2(s) also represent the weighting func- tions of the augmented plant in Fig. 6 [16]. Therefore, based on this perturbation representation, the optimal H, controller can be obtained by minimising J:

Therefore, it can be observed that nominal plant (Po($)) and perturbed plant (P(s)) in eqn. 10 do not need to have the same number of RHP poles. Hence, the limitation (i) is overcome.

Fig.6 tnethod

The uugmxted p h t with controller for the proposed H , design

4.2 Partial pole placement technique Based on the solution to the mixed sensitivity problem in eqn. 12, the following equalising property can be obtained:

I Wl ( j w ) S ( j w ) V ( j w ) 1 2 + iW2 ( j w ) R ( j w ) V ( j u ) l 2 = x2 (13)

where il is a nonnegative constant [17], implying that

Wl(S)WI ( - s ) S ( s ) S ( - s ) V ( s ) V ( - s ) +W2 ( s ) W2 (- s) h’(s) K ( - s ) S (s) s (- s) V ( s ) V ( -s )

= A2 (14) The plant P, the controller K, the weighting functions W,,

415

W, and V are written in rational forms as

where all numerators and denominators polynomials are in s-domain.

The sensitivity function S can be expressed as S = DXI DCl where D,, = D.Y + N Y is the closed-loop characteristic polynomial of the kedback system.

By substituting S into eqn. 14,

D"DM"M(X^'XA;"AI BYB2 +Y"YATAaB," B1)

0;; Dc$-EB,"B1B,"B2 = X2 (15)

where if F(s) is any polynomial function, F- is defined as F-(s) = F(-s)

Because the righi: hand side in eqn. 15 is constant, all factors in the numerator of the rational function on the left will cancel the corresponding factors in the denominator. Without loss of generality, M which has LHP roots only cancels a factor in Dc1 assuming no cancellations between M-M and E-EBI-BIB2-B2. So, choosing M is equivalent to reassigning the open-loop poles (the roots of D) to the locations of the roots of A4 and this is the concept of par- tial pole-placement [16]. In particular, if V is not used and there is no cancellations between D-D and B1-B1B2-B2, D-D must be cancelled by a factor of Dcl-DC1. Therefore, the closed-loop poles (the roots of D,,) will include the sta- ble roots of D and the mirrored LHP poles of the unstable roots of D. It has explained why the open-loop poorly damped modes will reappear in the closed-loop system.

4.3 Selection of weighing functions In general, the weighting functions are used to guide the H, design method to produce a robust controller that meets the specified performance. Limitation (iii) will be improved by using a new method of selecting the weighting functions Wl(s) and W2(s). , I ollothqr I ,

contributions

-Y- I APep

A E pss L, GEP (s 1 I - I P S S ( S )

Fig. 7 Connection ofCEP(s) Wld PSS witli slulft .-peed input

4.3. I Handling the uncertainty in correct direc- tion: W2(s) is used for robustness targeting and it also acts as a penalty factor of the controller. A high gain W2(s) results in low PSS gain. A suitable selection of the phase of W,(s) wiU guide the controller to handle the realistic plant and the uncertainty in correct direction. The transfer func- tion from PSS output ( U p s s ) to the component of the electrical torque (AI%,,) which can be controlled via excita- tion modulation is defined as GEP(s) in Fig. 7 [3]. To make Ape, in phase with Am, the PSS should provide a phase- lead so that it can compensate the phase-lag of GEP(s) to ensure positive damping torque. The phase of GEP(s) can be obtained by the method discussed in [3]. Since the elec-

416

tromechanical mode frequency ranges from 0.7 to 2Hz, W,(s) should provide sufficient phase-lag to penalise the PSS so that the final PSS design has sufficient phase-lead in thls frequency range. However, if the PSS phase-lead is larger than the GEP(s) phase-lag, it will produce, in addi- tion to a damping component of torque, a negative syn- chronising torque component [lo]. To eliminate this desynchronising effect, an undercompensation of about IO" will be provided in this frequency range. The phase of W2(s) is hence determined.

4.3.2 Handling the size of the uncertainty: W,(S) and the gain of W2(s) are for robust stability. They should be chosen such that eqn. 11 is fulfilled to bound the uncer- tainty.

4.3.3 Handling the special specification: To pre- vent the interaction with the torsional mode, the high fre- quency (say above 7Hz) gain of the PSS should be limited therefore the high frequency gain of W,(s) should be adjusted if necessary.

Because of the selected weighing function already guides the controller to the correct direction, the designed control- ler may be unable to make the unrealistic plant of eqn. I O stable and it may have completely different dynamic char- acteristic with the realistic plant. This is the main reason why the penalty factor concept is used in [8] and the obtained minimal value of J is much larger than unity which does not fulfil the requirement of the Small Gain Theorem [18], but the system is still stable and provides good performance. The requirement of minimal value of J less than unity is a sufficient condition only.

5 H, based PSS design

An H, PSS is designed for a single machine system in Sec- tion 5.1 and two-machine system in Section 5.2.

/-- CPSS

a 0 50, \

o- 40 .- 6 30

y 20

5 0 -L 10

-10 0

D

Fig. 8 U X, = 0.3 b X, = 0.5

Dumping ratio of the electr~~iwchicuI nio&

5.7 Single machine infinite busbar system

5.7.7 Sixth-order machine system: The block dia- gram of Fig. l and the blocks of GOV and EXC in Appendix 9.2 with the data provided in Appendix 9.3 are

IEE pro^ -Gener TranJm DrJtrib Vol 148, N o 5. Septeiiiber 2001

used to form the state equation and determine the eigenval- ues ( A = okjw) . System performance can be assessed using the damping ratio defined by -o/d(d + d). Figs. 8a, 8b and 9 show the darnping ratio for a wide operating condi- tions obtained by varying P, Q and X,. The overall per- formance of the original system (i.e. without PSS) is poor, especially when the X, is high and the load is heavy. A robust PSS (with shaft speed AQ as input signal) is there- fore introduced to improve damping.

50 3- LO]

Fig.9 Dmping rutw of the electroinecluuiical mode X, = 0.7

5. ‘1.2 Nominal power plant transfer function: The nominal operating point is taken as P = 0.8pu, Q = 0.2pu, and Xe = 0.5pu and other operating points are regarded as perturbations of the nominal system. The transfer function of the nominal plant (from Mpss to Am has been obtained as:

-1320(s2 + 21.9s + 122.45)(s + 2)s (s2 + 0.24s + 50)(s2 + 22.89s + 136.28) P ( s ) =

1 (s + 44.82)(s + 22.13)(s + 3.6)(s + 1.8) X

(16) The eigenvalues of the nominal system are 4 . 1 2 f 7.07j, -11.44 k 2.31j, -44.82, -22.13, -3.6 and -1.8. By the pole- zero cancellation, the plant can be reduced to P(s) in eqn. 17. The frequency response of the original plant and the reduced plant are similar as shown in Fig. 10.

- 1320s P’(s) = (9 + 0.249 + 50) (s + 44.82) (s + 2 2.13) (s + 3.6)

(17)

.- frequency, r a d i a n s l s

0 ci, 4 -200

2 -LOO a

-6OOL I

10-1 100 IO’ 102 f requency , r a d i a n l s

f i e uency res o m of the original plmt P(s)-(soiicl line) und Fi . I O re#ced pkwt g(sj-(stur L e )

5.7.3 Selection of weighting functions: V(s) is used for partial pole placement as mentioned in Section 4.2. Choosing 30% as the damping ratio of the electromechani-

IEE Proc.-Genu. T n m m Distrib., Vol. 148, No. 5, Septeinber 2001

cal mode, the open loop system poles (4.12 ? 7.07j) should be relocated to -2.22 k 7.07J. This can be acheved by selecting V as

(18) (s2 + 4.44s + 54.91)

( s2 + 0.24s + 50) V ( s ) =

Based on Fig. 11, W,(s) has to be a double phase-lag trans- fer function for sufficient compensation. However, to avoid torsional interaction and to fulfil the inequality of eqn. 11 mentioned before, the weighting functions W, (s), and W,(s) can be obtained from the method discussed in Section 4.3.

s + 400 s+4

W I ( S ) = 0.01-

W2(s) = 0.1 (19) (0.015s + 1)’ (0.15s + 1)2

-70 - ci, 4- -80- a, m

Q 2 -90 -

-lool -110

-1201 1

0.6 0.8 1.0 1.2 1.L 1.6 1.8 2.0 frequency, Hz

P h e chzructeristic ofGEP(s)-(solicl line) d W,(s)-(star line) Fig. 11

5.7.4 Controller reduction: The H, algorithm and the selected weighting function give a tenth-order controller (K(s). Applying the Optimal Hankel norm approximation [19], this high order controller is then reduced to a third- order approximation. In normal practice, a washout stage is included to suppress steady-state voltage offsets. A large washout time constant of 7 seconds is selected to ensure that the stabilising signals in the interested frequency range (0.7-2Hz) have negligible changes. Finally, the proposed robust controller (K’(s)) is described by:

i s K’(s) = -8.258-

(1 + 7s ) (1 + O.O853s)(l + 0.3023s)

(1 + 0.0382s + 0.0005396~~) (1 + 0.0106s) X

(20) The frequency response of K’(s) is shown in Fig. 12 where the phase deviation at very low frequency is due to the phase-lead property of the washout stage. (The transfer function of PSS will be -K’(s)) The closed-loop system poles of the power system and the proposed controller are -2.02 f 7.04j, -11.57 f 2.23J, -17.08 f 15.14J, 48.42 f 27.64j, 4.144, -1.75, -3.66 and -96.68. The electromechan- ical mode is relocated from -0.12 k 7.07j to -2.02 f 7.04j. The difference between the desired assigned location (-2.22 f 7.07j) and actual assigned location (-2.02 f 7.04j) is due to the approximations made in the plant model and the controller design.

417

103,

I

10-1 100 IO ' 102 frequency, r a d i o n d s

-50,-

2 -150 * x I( x x a

- 2 0 0 1 I

10-1 100 101 102 frequency, rad ions/s

Fig. 12 proposed controller K (s)-(stur line)

Frequency response ofthe iiigii order controller K(s)-(solidline) d

5.1.5 PSS performance comparison: The robustness of this PSS is compared with a conventional PSS (CPSS) designed using pole: assignment (see Appendix 9.4). The performance of these two PSSs for different operating con- ditions is shown in Fig. 9. It is observed that both PSSs will enhance the system damping, in particular at heavy load when the original s,ystem is poorly damped or unstable. (The effect is not so obvious at light load when the system is already quite stable). At low X,, the performance of the CPSS vanes with (! changes, whilst it is quite consistent with the proposed PSS. (Thus, the performance of the CPSS is better than the proposed PSS for some operating conditions.) The advantage of the proposed PSS becomes obvious at more critical conditions. For example, at heavy loading conditions in the vicinity of Prated = O.Spu, Qrated = 0.6pu for X, = 0.7~11, the damping ratio for the CPSS case cannot reach the so-called acceptable level of 10% [20], whilst that for the proposed PSS it is almost 2O'Yo. Indeed, the system damping ratio with the proposed PSS is quite consistent: well above 20% for most operating points.

load 2

Fig. 13 sysrein configuintion

t i m e , h Fig. 14 Duilv loud d e " d curves

5.2 Two machine system A two-machme and double circuit system is shown in Fig. 13. It is assumed that the daily MW demand of Load 1 and 2 vary according to Fig. 14 and the MVar demands are charactised by Q , = (1 + 2*P1)/4 and Q2 = 1 + 2*P2)/5 [21]. The sharing of generation is allocated such that the tieline flow is 300M W, 250MW and 200MW during the peak, off-peak and light load of the dominant load 2

418

respectively. Eigenvalues are computed and the damping ratio of the critical mode (interarea) is shown in Fig. 15. It is observed the damping ratio of the original system is low, and is even negative at noontime.

30r

4

.- = I5t 10 a

u o b s b I ,

.= 0 2 1 6 8 I= 11 16 18 20 22 24 t ime, h

- J L

Fi . 15 -8- without Pss -A- CPSS -.- proposed PSS

Dady curve of tiic clanping rutio

To improve system damping, the robust PSS to be installed at GI is designed by using techniques of the weighting function selection and controller reduction (simi- lar to the single-machine case) where the plant condition at zero hour will be used as the nominal condition and the conditions at other hours are regarded as perturbations of the nominal plant. Again, the effectiveness of the two aforesaid PSSs is compared. It is noticed that although both PSSs enhance the system damping throughout the whole day, the damping ratio of the CPSS could not attain the 20% level when the load becomes heavy. It is also found that the performance of the proposed PSS is particu- larly effective under poorly damped conditions when the damping ratio can be maintained above 20%. This dia- gram, of course, is unable to represent the other advantages (ij-(iii) discussed in Section 3.

6 Conclusions

A new H, design methodology is introduced in the present study which is able to solve certain limitations of existing H, PSS design techniques. The robust PSS is successfully designed by treating the highly nonlinear characteristic of the power system as model uncertainty. System damping with the proposed controller is much enhanced for a wide range of operating conditions. It is superior to the conven- tional PSS in terms of the robustness of the closed-loop sys- tem on the aspect of model uncertainties. This methodology can guarantee the stability of the closed-loop system with some predefined uncertainties such as load var- iation and tieline flow variation. The design procedure has been discussed in detail with a single machine case, and similar technique is also illustrated in a two-machine sys- tem. Further work will be concentrated to deal with the PSS design in multimachine system.

7 Acknowledgment

The authors gratefully acknowledge Hong Kong Polytech- nic University support for t h s work through research grants A/C 350l519.

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Appendices

AEpss T

& AVref

Fig. 16 G O v a n c l E X C ~ y ~ t ~ n

9.3 System data Machine data (1000MVA base) Rated power = SOOMW, Rated power factor = 0.8,

0.45, Xq” = 0.2

Re = 0 GOV and EXC data: K, = 50, T, = 0.03, Kg = 20, Tg = 0.5

9.4 Pole assignment technique for CPSS [I 11 For the notation of Section 4, the closed-loop characteristic polynomial of the feedback system can be represented by

R, = 0, Xd = 1.95, Xg’ = 0.3, X / = 0.2, X, = 1.9, Xy‘ =

Tdu’ = 7.4, TdON = 0.03, Tqi = 0.4, Tqon = 0.07, M = 7.6,

DCl(S) = X ( s ) D ( s ) + Y ( s ) N ( s ) (22) 9, I Machine equations Under small perturbations, the 6th-order machine equa- tions in Park’s dq-frame [22] are:

AE; = [AE; + ( X i - X:) AI,] / (1 + ST;:) AE: = [AEL - (XA - X:) AId] /(I + sTl0)

AEL = [ (X, - Xi) AI,] / (1 + ST&)

BY assi@ing the desired closed-looP Pole to form Dch) and equating the coefficients of equal power in both sides of eqn. 22, the transfer function X(s), Y(s), and the controller K(s) (= Y(s)/X(s)) can be obtained. For the CPSS design in Section 5.1, the reduced plant in eqn. 17 is used and a fourth-order K(s) is obtained. Of course, similar to H, PSS case, an additional washout stage is needed.

IEE Proc.-Gener. Tronsni. Distrib., VOI. 148, No. 5, Septeniber 2001 419