dynamic and steady state modelling of brushless dbim

8/3/2019 Dynamic and Steady State Modelling of Brushless DBIM

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Dynam ic and Steady State Modelling of BrushlessDoubly Fed Induction MachinesG . Boardman, J. G. Zhu, andQ.P. HaFaculty of Engineering, University of Technology, Sydney

P.O. Box 123 Broadway NS W 2007 AustraliaEmail: [email protected] This paper addresses issues of the modelling ofthe doubly fed twin stator induction machine, using spacephasors. Dynamic and steady state models are includedboth in the voltage and current controiled modes.Keywords-doubly fed twin stator induction machine,dynamic and steady state modelling, voltage andcurrent controlled modes, speed control.Nomenclature

A .iILNPP93RTy .VZR8

f 'i

T

w

Main VariablesFrequency (Hz)Instantaneous current (A)r ms current (A)Imaginary operatorInductance (H)Mechanical speed (rpm)Differentiation with respect to timeNumber o f pairs of polesReal part of com plex quantityResistance (Q)Instantaneous torque (Nm)Steady state torque (Nm)Instantaneous voltage (V)R M S oltage (V)Angular velocity (r ads )Impedance (Q)Flux linkage (Wb)Angle of alignment of rotor 'a' phases

B.dDeImMnP4r

C

CC

QSVCz

Subscript and Superscript VariablesControl machineRotor direct axisCurrent controlledStator direct axisElectricalLeakageMechanicalMutualNaturalPower machineRotor quadrature axisStator quadrature axisRotorStatorVoltage controlledComplex conjugate

Bold tower case variable denotes instantaneous spaceBold upper case variable denotes rms spacehasor.

phasor. Power winding refers to the stator winding ofthe power machine and control winding refers to thestator winding of the control machine.I. INTRODUCTION

About 80% of all electrical motors produced areinduction motors, and over 90% of the world's power ofelectric drive is provided by induction motors. Manyapplications require a variable speed drive to operateefficiently, often over a narrow speed range. Suchapplications include fans for ventilation and air-conditioning, and water pumps, where typically a speedm g e of 75% to 100% is required.The doubly fed, twin stator, induction machine(DFTSIM) is being investigated as a variable speed drive[1,2]. The DFTSIM being studied consists of two,nominally identical, wound rotor induction machines,shown schem atically in Fig. 1. A number of studies havebeen conducted on the performance modelling of thebrushless doubly fed machine (BDFM) [3-61, which ishnctionally equivalent to the DFTS IM. Perm anentconnections can be made between the two rotor windings,under which conditions the D FTSIM is brushless.In this paper, a positive control winding frequencydenotes a balanced three phase set which produces amagnetic field that rotates in the same direction as thatproduced by the power winding voltages. A negativefrequency denotes a voltage set that produces rotation inthe opp osite direction.The DFTSIM can operate in the synchronous imode,in which there is a single frequency of current in therotor and the rotor speed is a simple function of the statorsupply frequencies and numbers of pole pairs, as follows:

The natural, or synchronous, speed, N,,, occurs with dcapplied to the control winding.Power machine Control machine

--Utility Variablesupply: voltage.Fixed variablevoltagendrequencycUlJPIYrequencyFig. 1. Arrangement of DFTSIM

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This paper revises dynamic and steady state modelsof the DFTSIM, both in the voltage and currentcontrolled modes, in terms of space phasors and makescomparison with the m odel for the BDFM .11. MODELLING

A . Gen eral Assumptions(a) Balanced three phase windings are distributed toproduce sinusoidal space variation of flux density;(b) Only the fundamental components of voltage andcurrent are considered;(c) The magn etic circuits are linear, i.e. the effects ofsaturation and hysteresis are neglected; and(d) Zero sequence quantities are not present.B. Dynamic Voltage Equations

The DFTSIM is modelled as two cascaded woundrotor induction machines that have, in general, differentnumbers of pole pairs. The stators are fed fromindependent supplies. The rotors share a common shaftwith their windings connected in reverse sequence toproduce contra rotating magnetic fields. To analyse theDFTSIM, a power invariant, per p hase, direct quadratureaxes (dq ) ransformation is applied in the rotor referenceframe. In this analysis, the rotor a phase is assumed tobe aligned with the direct (d) axis and the quadrature ( q )axis leads the d axis by d 2 .The two-axis theory of the three phase induction motoris well developed and is used as the starting point for thedevelopment of the dynamic equations of the DFTSIM.The dynamic equation of the induction motor, in therotor reference frame, is written as [7]:

The following assumptions are made:

[.I = [zl[ilwhere

and the space phasors are

5, =2 ~ C i pM iz i , ). (4)2In the space phasor the operatorj represents the n/2

Power machine rotor Control machine rotorFv c

icd

Fig. 2. dq connections for any valid rotorinterconnections and with non-aligned rotor a phasesspatial displacement between the d and q axes.To produ ce the contra rotating fields on the rotor, therotor windings must be connected in reverse sequence.There are six different ways to connect the rotorwindings that praduce contra rotating fields on the rotor.Each of the six ways introduces aphas e shift of mI3 intothe control winding space phasors. The phase shift is thespatial orientation of the control machine rotor fieldwhen the power machine rotor field is coincident withthe d axis. A further phase shift may be introduced if thetwo rotors are coupled with their U phases out ofalignment by an angle 8. This is depicted in Fig. 2.For the case shown in Fig. 2, with p=0, the equationsrelating the currents and voltages in the rotors are

The rotor space phasors arei,, = i p d + i w ,i, = i c d + J Z c 9= i p d - j i p q ,

V p r =vpd +PP q. . (7)

vcr = vcd + vc 9 = -(vpd - Vw ) . (9)In this case, the control machine space phasors arerelated to the power m achine space phasors as

.*i,, = i P r , v, =-vPr *In general, Q, #O, and (1 0) becomes

* . I i, = ipreJQ,v, = - vp , eJp ,where

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Fig.3. Per phase equivalent circuit; voltage controlled

whereR, = R , +RCr nd Lr = L , +L,, .

Equation (1 3) is the general equation for the DFTSIMwith any valid rotor interconnection, with an arbitraryangle of rotor alignment and for any input voltage orcurrent. The per phase equivalent circuit, based on (13),is shown in Fig.3, in which 4 and &are the flux linkagespace phasors.The total electromagnetic torque is

As an example of the application of the dP factor,consider the case, where the value of the integer n in (12)is n=3 and S O . This corresponds to the rotor fields beingcoincident on the negative d axis. The resultant equationcorresponds with the model obtained for the BDFM inC. Qnamic Current Equations

In current controlled mode, the control windingcurrent, i,, is imposed by the controller. This results inthe second row of (13) being eliminated, since it is nolonger the governing equation for the circuit. Theresultant equation is

r41.

"I,D. teady State VoltageEquations

In the steady state, the DFTSIM is assumecl to besynchronized, in which the rotor carries only a singlefrequency of current. Each machine acts as an inductionmachine with the angular frequency of rotor current inboth ma_+ines given by0, =uvn Pa, . (16)The slips are defined as

s, = m p -ppw* , a n d s = % (17)Using r m s values (1 3) becomes, at the power windingfrequencyrvvc1 = r ~ , c 1 [ ~ v c l ~where

The per phase equivalent circuit, based on (15), is shownin Fig,4(a).- 4.14 -


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"P

VP

(a) (c)Fig.4. Per ph ase equivalent circuits, (a) dynam ic, currentcontrolled, (b) steady state, voltage controlled, and(c) steady state, current controlledEquation (18) can be represented by the circuit shown inFig.4(b).

Equation (14) gives the instantaneous torque. In termsof rms values, the total torque, T,, is

T h e first term is the torque developed in the powermachine and the second is the torque developed in thecontrol machine.E. Steady State Current Equations

In steady state, (1 5) becomes[Vccl =[ZccI[ ICCI,where

Equation (20) can be represented by the equivalentcircuit show n in Fig.4(c).111. SYSTEM SIMULATIONA, Machine Parameters

The-m achine parameters used in the simulations

were those of a laboratory DFTSIM set consisting of two,nominally identical, wound rotor induction machines,A/Y onnected, 6 pole, 1.5 kW, 1 5 Nm, 100 V, 50 Hz,14.8 A. The parameters of the machines, calculated at 50Hz, are given: ri=0.627 $2, xi=1.20 R, ,=18.0 R,r2=1.29 R, x2=1.73 0, tator to rotor turns ratio=1.06,and J=0.31 kg m2. In all simulations and experimentsrated voltage and frequency were applied to the powerwinding, unless otherwise indicated.B, Closed loo p, voltage controlled. performance

A closed loop, proportional plus integral (PI),controller was implemented in which the controlledvariable was the control winding voltage. The controlschem e is illustrated in Fig.5.Utilityontrol angle* omputationer

Voltage or

InverterControl

itio1PIControllex

Fig.5 Space phasor conti61 schemeThe system response was better than in open loop butless than optimal. Fig.6 shows the system response to acommanded speed of 400 rpm, corresponding to acontrol winding fiequency of -10 Hz. The response wasstable but slow. In the open loop it was not possible tosynchronize the machine at -10 Hz. After two secondsthe speed was stable but greater than comman ded. Theproportional and integral gains having been set to closeto the limits of stability. Over time, this error wasremoved and the speed settled at 400 rpm but the speedof response was slow.In the open loop, the system was uhstable even for acontrol winding frequency of 5 Hz. In the closed loopthe system was stable at 5 Hz, ut slow to respond. Fig.7shows the response to a commanded speed of 600 rpm,corresponding to a control winding frequency of -10 Hz.Again, the error after 2 s was quite pronounced but wasforced to zero over time.The results obtained for other speeds were varied.The proportional and integral gains that could be usedeffectively, varied with speed and as a consequence thesystem was, at times, unstable or, at best, very slow to-respond. The greater the variation from the natural speed,the lower the value ofgain that could be used.-4r --


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C.The reduced order of the current controlled model in

(15) suggested it could be more stable than the voltagecontrolled model. This proved to be the case. A PIcontroller was implemented, in which the controlledvariable was the control winding current. The controlscheme is the same as that illustrated in Fig.5, in whichthe voltage source inverter was replaced by a cumentsource inverter. Figs.8 and 9 show the system response! ocommanded speeds of 400 and 600 rpm respectively. Inboth cases it was noted the response was faster and was,in general, more stable than in the voltage controlledmode. In both cases the maximum value of gain thatcould be used was increased. It can be seen that thespeed error in both cases is much reduced compared tothe voltage controlled case.This schem e was also prone to instability, caused bythe choice of the proportional and integral gains, as i s tobe expected with this type of con troller.

IV. CONCLUSIONSDynamic and steady state models of the DFTSIM, inboth voltage and current controlled modes, have beendeveloped. The models are valid for any pole paircombination, any possible interconnection of rotorwindings and with arbitrary angle of alignment of thetwo rotors.Two schemes, both involving PI controllers, weresimulated. The first involved an ideal voltage sourceinverter while the second involved an ideal currentcontrolled inverter, connected to the co ntrol winding.

Closed loop, current control, performance

-8W.d

-1000 0 1 0 . 0 0 . I ,.I 4 . I1 I ITh. 11

. t o . . . . . . . . .

Fig.6. Simulation of closed loop response,voltage controlled, reference speed = 400 rpm

do0

.om0 0.1 0.. 0 1 0.. I 1 1 t.4 1. 1.1 1TLn. 11

. 2 0 1 . . . . . . . , IFig.7. Simulation of closed loop response,voltage controlled, reference speed = 600 rpm

lo*-, .ZOO

, 40 00 0.2 0.4 0.8 0.0 1 1.2 1.4 1.8 (. I 14 0 .T l i i s1

Fig.8. Simulation of closed loop response,current controlled, reference speed = 400 rpm100. , , , , , , , . , . 00

REFERENCESN. hilakapati, V. S . Ramsden, V. Ramaswamy, andJ. G. Zhu, Investigation of doubly fed twin slatorinduction motor as a variable speed drive, Proc. Inf.Con$ P ower Elec tronics Drives an d Energ y Systemsfo r Industrial Growth PEDES98, pp. 160-165.N. hilakapati,V. S. Ramsden, V. Ramaswamy, andJ. G. Zhu, Comparison of closed-loop speed controlschemes fo r a doubly fed twin stator induction motordrive, Proc. Znf. Power Electron. Motion ControlCo nj IPEMCtOOO,pp. 786-791 .R. Li, A. Wallace and R. SpCe, Dynamic sirnuhitionof brushless doubly fed machines, IEEE Tram:. onEnergy Conversion, Vol. 6, No. 3, 1991, pp. 445-452.R. Li, A. W allace, R. SpCe, and Y. Wang, TWO-axis model development of cage-rotor brushlessdoubly fed machines, ZEEE Trans. on EnergyConversion, Vol. 6 , No. 3, 1991, pp . 453-460.R. Li, R. Spk, A. Wallace and G. C. Alexander,Synchronous drive performance of brus hlessdoubly-fed motors, IEEE Trans. on Indusf?yApplications, Vol. 30, N o. 4, JulylAVg. 1994, pp.963-970.

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dynamic and steady state modelling of brushless dbim

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