404 IEEE Transactions on Energy Conversion, Vol. 3, No. 2, June 1988

REFERENCE FRAME ANALYSIS OF A SLIP ENERGY RECOVERY SYSTEM

P. C. Krause 0. Wasynczuk M. S. Hildebrandt Fellow Member Member

School of Electrical Engineering Purdue University

West Lafayette, IN 47906

Abstract Until the late 1970's, the performance of a slip energy

recovery system was predicted analytically by means of steady-state power balance relationships. Since then, three papers have been published by the same group of authors, wherein reference frame theory is used to derive the equa- tions describing the dynamic behavior of this system. Unfortunately, this analysis is incorrect, yielding erroneous results. It appears t ha t this error has gone undetected. In this paper, reference frame theory is used to establish the equations which can be used to predict adequately the dynamic and steady-state performance of a slip energy recovery system. The steady-state performance a s predicted from this analysis is compared with tha t predicted from a detailed computer simulation. The errors of the previously published approach are discussed and the inaccuracy in predicting the steady-state torque-speed characteristics is illustrated. The stage is now set t o develop linearized, small displacement equations which can be used for control system design.

INTRODUCTION The slip energy recovery system wherein a rectifier and

inverter and placed in the rotor circuits of a n induction machine was developed in the early 1960's [1]-[3]. The ini- tial analytical approach was one wherein power balance relationships were used to predict steady-state operation [ l ] . Apparently, it wasn't until the late 1970s t ha t a n attempt was made, using reference frame theory, t o develop equa- tions which could be used to predict the dynamic and steady-state behavior of the slip energy recovery system. This analytical approach is reported in [4]-[6]. Unfor- tunately, the approach is incorrect, making the usefullness of the resulting equations very questionable.

In this paper, a slip energy recovery system is analyzed using reference frame theory. The system performance predicted by the resulting equations is compared with tha t predicted by a detailed computer representation of the system. The dynamic equations presented herein form the basis from which small displacement or linearized equations can be derived for use in control system design.

SLIP ENERGY RECOVERY SYSTEM ANALYZED

The slip energy recovery system shown in Fig. 1 has been used over the last two decades to achieve subsynchro-

87 SM 614-1 A paper recommended and approved thy the IEEE Rotattng Xachiuery Committee of the IEEE Power Eng inee r ing S o c i e t y €or presentation a t the 'LEEE/PES 1987 Summer Veet ing, San Francisco, C a l i f o r n i a , J u l y 1 2 - 17, 1987. Manuscript s u h m i t t e d .January 3 0 , 1987; made a v a i l a b l e for pr in t ing May 11, 1987.

nous speed control of induction motors. Examples of early work on this system are given by 111-[3]. The rectifier- inverter combination connected to the rotor windings via slip rings provides a means of supplying a voltage in series with the rotor windings. The rectifier operates without phase control while the firing of the inverter is controlled to provide a negative voltage (y negative) which, in effect, shifts the torque versus speed characteristics of the induc- tion motor so tha t zero electromagnetic torque occurs a t subsynchronous speeds. Hence, the induction motor operates a t ' low speeds without the relatively large rotor power loss inherent in conventional operation. Since the inverter voltage is negative, power is extracted from the rotor circuits. Consequently, this system has become known as a slip energy recovery scheme where slip frequency energy is extracted from the rotor circuits and returned to the source. There are numerous design features of this sys- tem such as placing a transformer between the inverter and the source as well as inverter firing schemes which are important from a practical standpoint but will not be con- sidered here since these design variations are of secondary importance to the analysis presented.

METHOD OF ANALYSIS

With the exception of 141-[6], which were written by the same authors, it appears tha t there has been only a n approximate analysis of the slip energy recovery system based upon the power balance relationships. Apparently, the only attempt of a rigorous analysis using reference frame theory is given in 141-[6]. Unfortunately, this deriva- tion of the system voltage equations is incorrect and the resulting voltage equations are invalid. In this section, the derivation of the system voltage equations is set forth.

Induction Machine The voltage equations of the induction machine in the

synchronously rotating reference frame may be expressed [7]

where p is the operator d/dt, w, is the base radian el€ ::I ri-

tal frequency generally selected -to correspond to the rated frequency of the machine, and the slip, s, is expressed

(2) We - wr

we

s = -

where we is the radian electrical frequency of the voltages

0885-8969/88/0600-0$01.0001988 IEEE

405

Source

Induction iar + ' as Motor - Inverter Rectifier Filter

Stator Rotor t Firing Signals

Fig. 1. Slip energy recovery system.

applied to the stator windings and q is the electrical angu- lar speed of the rotor. For the sake of brevity, the zero- variables are not included in (1) since only balanced condi- tions will be considered.

The electromagnetic torque, positive for motor action, may be expressed in per unit as

T, = xM(iG id", - ides i;) (3)

Filter The voltage equation for the filter circuit is

where v, and are the rectifier and inverter voltages, respectively and id, is the dc current.

Converters Neglecting the commutating inductance, the average

output voltage of a full converter with balanced ac voltages may be expressed [7]

vd = vdo cosa (5) where Vd is the output voltage, which is vr in Fig. 1 for the rectifier and vi for the inverter. Also a is zero for the bridge rectifier. The voltage Vdo is

where vp is the peak value of the ac phase voltages (rotor voltages for the rectifier and source voltages for the inverter).

Combined System Equations If the harmonic components of the rotor variables are

neglected then the rotor phase voltages will be sinusoidal. If the q-axis is positioned so tha t it alyays coincides with the maximum or peak positive value of v,, then

v; = 0 ( 8 )

where v; is the peak value of the rotor phase voltages referred to the stator windings by the appropriate turns ratio.

If the commutation of the rectifier is neglected and since a=O, the rectifier voltage may be expressed from (5) as

I 3 6 1 vr = -

7r vm Substituting (7) into (9) yields

(9)

where v: is the rectifier voltage referred to the stator wind- ings by the appropriate turns ratio.

With commutation neglected in the rectifier, the funda- mental currents into the rectifier are in phase with the input voltages. ,Since the q-axis is always positioned a t the peak value of v,,, then ig must be zero. If the losses in the rectifier are neglected then the power on the ac side of the rectifier must equal the power on the dc side. In per unit,

where, as before and hereafter, primed quantities are referred to the stator windings by the appropriate turns ratio. From (10) and (11)

Substituting (10) and (12) into (4) yields

If commutation is neglected in the inverter then from (5)

'e vqr = vm (7)

406

where v,, is the peak value of the stator (source) phase vol- tages. Clearly, if a transformer is connected between the source and inverter, its turns ratio must be appropriately taken into account.

When per unitizing the qre- and dre-variables, fi Vbase is often used, where Vbase is the rms value of the abc- variables [7]. Hence, (1) may be considered expressed in per unit. Substituting (13) into (12) yields

I , x; v; = -R i e - - p i; -vms cos cu

Wb f qr

where R; and X; are defined by comparison with (12).

and set v g and ide, both equal to zero. This yields Let y s now ,go back to (1) and substitute (14) for v$

P x, Wb

s x , - XM -SX:, q P

With i2r equal t o zero, the per unit electromagnetic torque may be expressed

(16) T, = -XM ' e . le 'ds lqr

The q-axis is positioned so that it always coincides with the peak value of var. Hence, for a given machine, the values of v; and v; will be determined by the amplitude of the source voltages and the system operating conditions. Tha t is, v: and Vd; are related to v,, as

v& = (vqt)z + (vd",z (17)

Torque may be related to rotor speed as

where H is the inertia constant in seconds and T, is the load torque expressed in per unit.

ANALYSIS OF STEADY-STATE OPERATION

It is of interest t o determine the steady-state charac- teristics of this slip energy recovery system. For balanced, steady-state operation with harmonics neglected, p = 0 in (15), hence

where uppercase letters are used to denote steady-state quantities. Equations (16) and (17) change only in the use of uppercase notation; tha t is

T, = -XM Id: 1;

v& = (vqE)2 + (vdt)2

(20)

(21)

There are numerous procedures which can be used to manipulate the previous equations to express the torque in

terms of the machine and filter parameters, the phase delay angle, cy, and Vm,. One approach, [8], is to use the last two rows of (19) to first eliminate Iqz and I& whereupon Vq: and Vd', may be expressed in terms of I;, V,, and a. Next, sub- sltitute these expressions for V$ and V$ into (21); solve for I: and substitute it into the expression for torque given by (20). Since Id: is also required in (20) we must again use the third row of (19) in order to express Id: in terms of 1; and

The steady-state torque versus speed characteristics are shown in Fig. 2 for values of phase delay angle, a, rang- ing from 90' t o 150'. The parameters of the slip recovery system are those used in [4]-[6]. The induction motor is a 2-pole, 5 hp, 3-phase, 400 V (line-to-line), 60 Hz machine with the following parameters expressed in per unit where base torque is 9.89 N.m and base impedance is 42.9 R.

cy.

I

r, = 0.058 XM = 2.9 r r ,= 0.072 XI, = 0.10 XI, = 0.10

The filter parameters are I f

rf = 0.02 Xf = 1.0

The harmonics have been neglected along with the commutation of the rectifier and inverter. With the excep- tion of [4]-[6], it appears t ha t only the power balance rela- tionships have been used to establish a n approximate expression for the steady-state electromagnetic torque [ 11.

In order to verify the torque-speed characteristics shown in Fig. 2, a detailed computer simulation of the induction motor was implemented with the switching of the rectifier simulated as an ideal rectifier without commuta- tion. The average electromagnetic torque computed using this type of simulation was found to be identical to tha t depicted in Fig. 2. The effects of commutation of the rectifier will be discussed in a later section.

In Fig. 3, V; is plotted versus rotor speed for rw=90°, 100 and 110'. The voltages VG and Vg are plotted versus speed in Fig. 4 for cy=90, 100 and 110'. Similarly, Id versus speed is shown in Fig. 5; I$ and Id: versus speed in Fig. 6 for these same phase delay angles. Finally, Fig. 7 depicts Idc, the average filter current, versus speed for (1

ranging from 90' t o 150'.

1.80

1.35

0.90

0.45

0 0 0.25 0.50 0.75 1 .oo

Speed, pu

Fig. 2. Steady-state per unit torque versus speed charac- teristics of a slip energy recovery system.

0.28 - 3 a

9, & >

0.14 -

0

Speed, pu

I I

Fig. 3. V: versus rotor speed for steady-state operation;

vd", =o.

3 a >-

Fig.

a 1 .oo

0.67

0.33

0 0 0.33 0.67 1.00

Speed, pu

vqt and vdz versus rotor speed for stea..,r-state operation.

0

-1.34

-2.68

-4.02

Speed, pu

0 0.33 0.67 1.00

Fig. 5. I,$ versus rotor speed for steady-state operation;

I)r = 0.

4.16

2.77 3 a d

1.39

0

407

0 0.33 0.68 1.00 Speed, pu

Fig. 6. 12 and IC versus rotor speed for steady-state operation.

3.64

2.73

1.82

0.91

0 0 0.25 0.53 0.75 1.00

Speed, pu

Fig. 7. ic versus rotor speed for steady-state operat on.

PREVIOUS APPROACH USING

REFERENCE FRAME THEORY

In, [4]-[f], it is Yssumed tha t the q-axis is positioned so tha t v,"r=v,, and v$=O as has been done in the approach presented herein. However, i t is falsely assumed tha t simul- taneously, the q-axis is positioned so tha t v,: is equal to v,, and vg is equal to zero. If, in the equations of transformation, the q-axis is posit,ioned a t the peak value of the fundamental component of var then v: is equal to this peak value and v k is zero. However, in order for v; t o be equal t o the peak value of vas and VdE equal to zero, this same q-axis must also be positioned, in the equations of transformation, a t the peak value of vas. In general, both cannot occur simultaneously. Rather than becoming overly involved in reference frame theory to show this analytically, the fallacy is perhaps a,dequately revealed by letting q = O . In this case, vas and var are of the same frequency and in order ,for the q-axis t o be simultaneously a t the peak of vas and var, these voltages must be in phase. Clearly, this can- not be the case in general. We see from Fig. 4 tha t vd: is zero for only one rotor speed for a given value of a.

408

=I Q

6 P z

1.71 - questioned. In [6], experimental results are included in an effort to give credence to the analytical work. Unfor- tunately, experimental results were shown only for stable conditions. Regions of instability predicted analytically were left unverified.

The development of valid small displacement equations from which operating point stability can be predicted appears to be a formidable task. It certainly is not as clear cut as the approach presented nearly twenty years ago for a drive system with the rectifier and inverter in the stator phases [SI.

1.28 - pu

0.86 -

CONCLUSIONS 0.43 -

Reference frame theory has been used to analyze a slip energy recovery system. The steady-state performance of the system as predicted by the resulting equations has been compared with that predicted from a detailed computer

0 simulation of the system. Although it appears tha t the assumptions which have been made in the analysis are ade- quate, in some applications it may be necessary to incor- porate the effects of Commutation of the rectifier and

0 0.25 0.50 0.75 1 .oo Speed, pu

Fig. 8. Steady-state per unit torque versus speed charac- teristics of a slip energy recovery system calculated using the equations from [4]-[6].

There is another oversight in 141-[6]; in particular, i&T- is not restricted to zero even though commutation in the rectifier is neglected. Mindful of these differences, we would not expect the behavior predicted by the equations presented herein to correspond to tha t given in [4]-[6]. This is illustrated very vividly by comparing the torque-speed characteristic given in Fig. 2 with tha t shown in Fig. 8 which was obtained for the same system using the equations derived in 141, 151 or [6].

E F F E C T S O F C O M M U T A T I O N

It was found from a detailed computer simulation of the system that commutation can have a notable effect upon the value of the average electromagnetic torque. In particular, with the rectifier simulated in detail, the torque was found to be somewhat less than tha t predicted when commutation is neglected, especially near stall. With ( Y = 11O0, for example, the stall torque, with commutation included, was approximately 15% less than tha t calculated with commutation neglected. IIowever, this error decreases as speed increases and becomes essentially zero a t no load.

The question arises as t o how the effects of commuta- tion might be included. Clearly, the equations must be modified to include the presence of the commutating induc- tance associated with the converters. However, what value of commutating inductance should be used in (5), particu- larly in the case of the rectifier? Is it the rotor leakage inductance? If so, what voltage should be used for vdo in (5)? These are questions which remain unanswered a t this time.

STABILITY ANALYSIS

Two papers, [4] and 161, deal with the determination of the operating point stability of the slip energy recovery sys- tem. It is now apparent t ha t the results presented and con- clusions which have been drawn from these results must be

inverter. The stage is now set to s ta r t work towards the development of small displacement equations for the slip energy recovery system which can be used for control design purposes.

111

121

PI

141

151

161

171

181

191

R E F E R E N C E S

A. Lavi and R.J. Polge, "Induction Motor Speed Con- trol with Static Inverter in the Rotor,'' IEEE Trans. Power Apparatus and Systems, Vol. 85, January 1966,

W. Shepherd and J. Stanway, "Slip Power Recovery in an Induction Motor by the Use of a Thyristor Inverter," IEEE Trans. Industry and General Applica- tions, Vol. 5, January/February 1969, pp. 74-82. W. Shepherd and A.Q. Khalil, "Capacitive Compensa- tion of Thyristor-Controlled Slip-Energy-Recovery Sys- tem," Proc. IEE., Vol. 117, May 1970. V.N. Mittle, K. Venkatesan, and S.C. Gupta, "Deter- mination of Instability Region for a Static Slip Power Recovery Drive,'' J. Inst. Eng. (India), Vol. 59, October 1978, pp. 59-63. V.N. Mittle, K. Venkatesan, and S.C. Gupta, "Switch- ing Transients in Static Slip-Energy Recovery Drive," IEEE Trans. Power Apparatus and Systems, Vol. 98, July/August 1979, pp, 1315-1320. V.N. Mittle, K. Venkatesan and S.C. Gupta, "Stabil- ity Analysis of a Constant Torque Static Slip-Power- Recovery Drive," IEEE Trans. Industry Applications, Vol. 16, January/February 1980, pp. 119-126. P.C. Krause, "Analysis of Electric Machinery," McGraw Hill Book Co., Inc., 1986. M. S. Hildebrandt, "Reference Frame Theory Applied to the Analysis of a Slip-Recovery System," M.S. Thesis, Purdue University, 1986. T.A. Lip0 and P.C. Krause, "Stability Analysis of a Rectifier-Inverter Induction Motor Drive,'' IEEE Trans. Power Apparatus and System, Vol. 88, January

pp. 76-84.

1969, pp. 55-66.