FAST FUNCTIONAL MODELLING FOR 18-PULSEAUTOTRANSFORMER RECTIFIER UNITS IN MORE-

ELECTRIC AIRCRAFT

T. Yang, S.V. Bozhko, G.M. Asher

Department of Electrical and Electronic Engineering, University of Nottingham, Nottingham, NG7 2RD, UKE-mail: eexty4@nottingham.ac.uk

Keywords: Multi-pulse Autotransformer Rectifier Unit,Functional model, DQ0 model, More-Electric Aircraft.

Abstract

This paper studies the development of a functional model foran 18-pulse autotransformer rectifier unit (ATRU) in a more-electric aircraft (MEA). The developed model is a non-switching model and is in the DQ0 frame. The three-to-ninephase autotransformer is simplified to a three-to-three phasetransformer and then reduced to a functional model in theDQ0 frame. The rectifier is modelled as a DC transformerwith commutation losses represented by an equivalentresistor. The developed functional model is simulated andcompared with an ABC benchmark model, which is builtfrom the Dymola standard library. The DQ0 model showssignificant improvement in performance under both balancedand unbalanced conditions. Particularly the DQ0 model ismore than 1000 times faster than the ABC benchmark modelunder balanced conditions.

1 Introduction

The more-electric aircraft (MEA) is the developing trend forthe next generation of airplanes. Recent advances in powerelectronics, electrical drives and modern control techniquesmake it possible to replace many functions which areconventionally managed by hydraulic, pneumatic andmechanical power, with electrical power driven devices[1][2]. Alongside this, the significant increase in on-boardelectrical devices results in a new challenge for electricalpower system (EPS) designers in order to ensure theavailability, integrity, stability, and performance of the EPSunder all possible conditions. Within modern systems the EPSarchitecture for MEA involves various topologies, includingAC, AC frequency-wild, DC or hybrid distributions [3].

The development of a new EPS structure for MEA willinvolve extensive simulation studies, under balanced andunbalanced conditions, in order to assess the systemavailability, power quality and transient behaviour. Due to theswitching behaviour of power electronic devices, it is verytime-consuming, sometimes impractical, to simulate suchcomplex EPS with some nonlinear, time-varying models,which take into account the switching behaviour and high

bandwidth transients. Thus, a fast modelling technique isquite necessary for system level studies.

For large-scale EPS simulations, the prime challenge is tobalance the simulation speed and accuracy. This is highlydependent on the task addressed. Models for EPS can becreated within four modelling levels, i.e. from the bottom tothe top, component, behavioural, functional and architecturallevels [4-6]. Each lower level will successively include higherharmonics. The functional level is aimed at addressing low-frequency transient behaviour, where the model is able tohandle dynamic frequencies up to 1/3rd of base grid frequency(i.e. 100-150Hz) with a time waveform accuracy of 95% inrespect of the behavioural model [7]. This paper details thedevelopment of a model targeted the functional level.

Conventional AC-DC converters within aircraft are multi-pulse converters, with each unit consisting of a phase-shiftingautotransformer and two sets of uncontrolled rectifiers. Forthe 12-pulse ATRU, an interphase reactor (IPR) is necessary,which will increase the overall size of the converter. Inaddition, the increased requirement of power quality in theMEA makes it impossible to meet such requirement with a12-pulse ATRU. Moreover, some power qualityspecifications demand that, the distortion of the input linecurrent should be typical of an 18-pulse converter system forloads above 5kw [8]. Several different topologies of 18-pulseATRU have been studied [9-12] and the state-space averagingmethod has been used to study transients and dynamics ofsome particular ATRUs [13][14], however, these models areall for balanced systems.

This paper continues previous studies reported in [15-17], andaims to develop a DQ0 model for one differential-delta type18-pulse ATRU to extend the DQ0 modelling library forMEA. The harmonics of the switching function in the rectifierhave been neglected in the DQ0 model in order to acceleratesimulation speed. Compared with existing models, our modelshows good performance and accuracy under both balancedand unbalanced conditions. In addition, the developed DQ0model is over 1000 times faster than the ABC model underbalanced conditions.

2 ABC model of 18-pulse ATRU

The ABC model of an 18-pulse ATRU, used as a benchmarkmodel, is shown in Fig.1 and is built in Dymola. Fig.2 showsthe coil configuration of the phase shifting delta-differentialautotransformer. The primary windings are Δ-connected and the secondary windings are used to achieve displacement ofvoltages. On the secondary side of the autotransformer, thevoltage phasors are 40o from each other. The three sets ofvoltages produced by the autotransformer are fed directly to

three sets of diode bridges which have their DC outputsconnected in parallel. It is worth noting that, with the selectedphase shifted autotransformer, the three sets of diode bridgesare equal to a nine-phase AC to DC converter and thus thereis no need for the IPR after the autotransformer. The diodebridge is from the standard Dymola library and each diode ismodelled as an ideal diode with a conducting resistance inseries. The design procedure of the autotransformer ispresented in [18] and the parameters are shown in theappendix.

Fig.1: Configuration of an 18-pulse autotransformer rectifier unit

Figure 3 shows a diagram of the input voltage phasors, Vabc,in the primary windings and the displaced output voltagephasors in the secondary windings, i.e. Vabc1, Vabc2 and Vabc3.The three sets of three-phase secondary voltages are shiftedby 0, +40o and -40o respectively with respect to the primaryvoltages. The magnitude of secondary voltages is 91.3% ofthe original input voltage.

3. DQ0 model of 18-pulse ATRU

The development of the DQ0 model for the 18-pulse ATRUinvolves two steps.

1. Reduce the three-nine phase system to a three-threephase system. Both systems are in the ABC frame.

2. Transform the three-three phase system to the DQ0frame.

The first step is possible due to the symmetry of the 18-pulseATRU. The second step is achieved by simply multiplyingthe three-phase variables derived from step one by the ABC-DQ0 transformation matrix. As can be seen in Fig.4, theoriginal 18-pulse ATRU is now reduced into two parts: theAC-AC part and the AC-DC part. The coil resistance andleakage inductance of the transformer are represented by Req

Fig.2: Configuration of the differential-deltaautotransformer

Fig.3: Voltage phasor diagram of the delta-typeautotransformer for 18-pulse operation

Vc2

Vc1

40o

Vc3

Vb

Vb2

Vb1

40o

Vb3

and Leq respectively. The development of each part of themodel is detailed in the following two sections.

eqR

eqR

eqR

eqL

eqL

eqL

Fig.4: The equivalent representation of 18-pulse ATRU

3.1 Autotransformer

In the autotransformer shown in Fig.1, the AC powertransferred from the primary side is shared between threechannels on the secondary side, carrying ia1,b1,c1, ia2,b2,c2 andia3,b3,c3 respectively. Assuming the power is equally shared inthe three channels, the symmetry of the 18-pusle ATRUmakes it possible to use only one channel to represent thewhole ATRU system. In this paper, the channel carrying iabc3

is selected and the relationship between variables in theprimary windings and secondary windings is derived asdetailed below.

The relationship between voltages can be derived accordingto the vector diagram in Fig.3.

c

b

a

c

b

a

v

v

v

kkkk

kkkk

kkkk

v

v

v

31

333

3331

3

33331

1221

2112

2211

3

3

3 (1)

where k1 and k2 are the length of windings 1abc and 8abc inFig.2 respectively. Assuming that the power is shared equallybetween the three channels on the secondary side, andneglecting the magnetizing current in the autotransformer, therelationship between the currents on the primary andsecondary windings can be derived as follows:

3

3

3

1212

2121

2121

31

333

331

33

33331

3

c

b

a

c

b

a

i

i

i

kkkk

kkkk

kkkk

i

i

i (2)

The coefficient ‘3’ is due to the assumption that the threechannels on the secondary side of the autotransformer sharethe power equally.

Using the DQ0 transformation

abcsdq fKf 0(3)

With

1/21/21/2

)3/2sin()3/2sin(sin

)3/2cos()3/2cos(cos

3

2

sK

Where θ is the synchronous angle and θ =ωt in the matrix Ks.

Multiplying Ks on both sides of equations (1) and (2) andrearranging the equations yields:

003

3

3

100

00.6994020.586850

00.5868500.699402

V

V

V

V

V

V

q

d

q

d(4)

03

3

3

0100

00.6994020.586850

00.586850-0.699402

3

I

I

I

I

I

I

q

d

q

d(5)

3.2 Rectifier

Due to the switching behaviour of the diode bridge, the ABCmodel requires abundant simulation time and storage.Therefore, in order to accelerate the simulation, a non-switching functional model will now be developed.

The non-switching model for 6-pulse rectifier has been wellstudied in [19][20] using an equivalent time-varianttransformer. The DQ0 model for the AC-DC part of an 18-pulse ATRU will be developed in the following sections.

3.2.1 Voltage relation

In the 6-pulse diode bridge, the commutation voltage drop canbe represented by an equivalent resistor located on the DCside [21]. The same idea has been used for the modelling ofthe 18-pulse ATRU. The commutation effect associated withthe leakage inductance is represented by a resistor, ru, and thediode bridge itself, thus it can be treated as ideal.

As mentioned before, the three sets of diode bridges are equalto a nine-phase AC to DC converter. Under balancedconditions, the conducting period for each diode leg is π/9, thus the relationship between the DC and AC side voltages ofthe rectifier can be derived as follows:

)9

sin(18

)()cos(9

9/

9/

imimdc VtdtVv

(6)

Where vdc is the DC side voltage and Vim is the magnitude ofthe phase voltage at the AC terminals of the rectifier. Thespace vector of vim can be defined as:

3/4,

3/2,,

3

2 jcim

jbimaimim evevvv

(7)

With the DQ0 transformation in (3), the magnitude of thevoltage phasor Vim can be calculated from the d- and q-axiscomponents of correspondent vector vim

2,

2, qimdimim vvV (8)

Where vim,d and vim,q are the d and q-axis components of

vector vim respectively and, under balanced conditions, thesetwo variables will be DC-like. Substituting (8) into (6) gives:

2,

2,)

9sin(

18qimdimdc vvv

(9)

The analysis of commutation loss for an 18-pulse rectifier canstart from the study of a 6-pulse rectifier. For a 6-pulse diodebridge, the commutation occurs every π/3 period and the voltage drop ∆Vd due to commutation is:

dcsd iLV

3 (10)

Where Ls is the inductance on the front end of the dioderectifier and idc is the DC-link current. Similarly, for the 18-

pulse rectifier, since the commutation occurs every π/9, the voltage drop due to that can be expressed as:

dcsdcs

d iLiL

V

9

9/ (11)

Combining equation (9) and (11), the DC link voltage can bewritten as:

dcsqimdimdc iLvvv

9)

9sin(

18 2,

2, (12)

3.2.2 Current relation

For each diode in the three sets of rectifiers, within the 18-pulse ATRU, the conducting period is 2/9π. The switching function of Sa3, Sb3 and Sc3 is shown in Fig.5 and thefundamental components of the switching function can begiven below:

)cos()9/sin(4

3 tsa

(13)

)3/2cos()9/sin(4

3

tsb

(14)

)3/2cos()9/sin(4

3

tsc

(15)

Fig.5: Switching function of the diode bridge on channel 3 ofthe ATRU

The relationship between the currents at the AC terminals andthe DC terminals of the rectifier is:

dc

T

cba

T

cba iSSSiii 333333 ,,,, (16)

Substituting (13)-(15) to equation (16) and using the DQ0transformation yields:

dcii

)9/sin(43 (17)

Whereหi3หis the magnitude of the current vector for ia3,b3,c3

and 2,3

2,33 qd iii

.

As for the diode bridge, the fundamental components ofvoltage and current on the AC side are in phase [16]. Thus thephase angle of the AC current can be given from the voltagevector.

)/(tan ,,1

3 dimqimi vv (18)

Hence, the currents on the d and q axes are:

)cos( 33,3 id ii

(19)

)sin( 33,3 iq ii

(20)

Equations (17)-(20) give the relationship between the currentson the AC and DC sides of the rectifier.

3.2.3 Leq and Req

The parameters Leq and Req can be calculated as below:

2N

LLL p

seq (21)

2N

RRR p

seq (22)

Where, Lp and Ls are the primary and secondary leakageinductances respectively, Rp and Rs are the primary andsecondary winding resistances respectively and N is the turnsratio between the primary and secondary windings.

4 Model validation

The effectiveness of the developed DQ0 functional model ofthe ATRU is verified by comparison with the ABCbenchmark model in both balanced and unbalancedconditions. Simulations have been performed on a Pentium(R)4 CPU/3.40GHz/3.00GB of RAM using Modelica/Dymolav.7.4 software. The Radau IIa algorithm has been chosen inthe solver and the tolerance has been set at 1e-4. As aquantitative evaluation of the computation efficiency of thesemodelling techniques, the CPU time has been compared.Meanwhile, the evaluation of the accuracy is performed bycomparing the plots of the system quantities in figures.

The developed model has been tested under both balancedand unbalanced conditions. In order to examine the dynamicsof the system under balanced conditions, a resistive step loadis applied on the DC-side. Under unbalanced conditions, aline-to-line fault is also applied within the system. The faultinjector in the DQ0 frame was developed in [14] and will notbe detailed here.

The simulation scheme is shown in Fig.1 with the loadresistor changed at t=0.02s and line-to-line fault implementedat t=0.05s. Parameters for the system are shown in Appendix.The DQ0 model for the generator can be found in [22] andwill not be shown here.

Under balanced conditions, the resistive load is changed att=0.02s by closing a switch on the DC side. The transients ofthe system are compared in Fig.6 and Fig.7. Fig.6 shows thephase currents flowing into the autotransformer within theATRU, with the load resistance changed at t=0.02s. As can beseen, the results from the DQ0 model exactly reflect thefundamental component of those from the ABC model. TheDC-link voltage comparison is shown in Fig.7. It is obviousthat the DC voltage from the DQ0 model well reflects theaverage of the result from the ABC model.

Fig.6: Comparison of the phase currents flowing into theATRU between the ABC and DQ0 models, with load changedat t=0.02s.

Fig.7: Comparison of the DC-link voltage between the ABCand DQ0 models, with load changed at t=0.02s.

Under unbalanced conditions, the phase A to phase B line-to-line fault has been implemented in the system at t=0.05sbetween the generator and ATRU, as shown in Fig.1. Fig.8shows the phase current flowing into the AC terminals of theATRU and Fig.9 shows the DC-link voltage. It can be seenthat the results from the DQ0 and the ABC models are wellmatched.

Fig.8: Comparison of the phase currents flowing into theATRU between ABC and DQ0 models, with a line-to-linefault occurring at t=0.05s.

Fig.9: Comparison of the DC-link voltage between the ABCand DQ0 models, with a line-to-line fault occurring att=0.05s.

Table 1 shows the CPU time required by different modelsduring both balanced and unbalanced conditions. It can beseen that before the fault occurs, the DQ0 model is more than1000 times faster than ABC model. This is because underbalanced conditions , all the variables in the DQ0 model areDC-like, which allows the computer to use larger simulationsteps. However, after the fault occurs, the simulation speed ofDQ0 model decreases sharply. This is due to the 2nd

harmonics included in the DQ0 model under unbalancedconditions. Table 1 and Fig.10 show a comparison of the CPUtime required by the two models. Please note that the CPUtime required within the initialization period (t=0) is notconsidered.

Scenario ABC DQ0Before fault (0s-0.05s) 59.57s 0.047sAfter fault (0.05s-0.1s) 20.72s 289.7s

Table 1: CPU time comparison between the ABC and DQ0models

Fig.10: CPU time comparison between the ABC and DQ0models, with a step load change at t=0.02s and a line-to-linefault implemented at t=0.05s.

0.015 0.02 0.025 0.03-200

0

200

Ia(A

)

0.015 0.02 0.025 0.03-200

0

200

Time(s)

Ic(A

)

ABC model DQ0 model

0.015 0.02 0.025 0.03-200

0

200

Ib(A

)

0.019 0.0195 0.02 0.0205 0.021 0.0215 0.022 0.0225 0.023500

510

520

530

540

550

560

570

580

590

Time(s)

Vd

c(V

ol)

ABC model DQ0 model

0.045 0.05 0.055 0.06

-200

0

200

Ia(A

)

0.045 0.05 0.055 0.06

-200

0

200

400

600

Time(s)

Ic(A

)

ABC model DQ0 model

0.045 0.05 0.055 0.06

-200

0

200

Ib(A

)

0.046 0.048 0.05 0.052 0.054 0.056 0.058 0.06-500

0

500

1000

1500

2000

2500

Time(s)

Vd

c(V

ol)

ABC model DQ0 model

0 0.01 0.02 0.03 0.04 0.050

20

40

60

80ABC DQ0

0 0.02 0.04 0.06 0.08 0.10

100

200

300

Time(s)

CP

UT

ime

Con

sum

ed(s

)

Load change

Fault

5. Conclusion

The DQ0 model developed in this paper has been shown to behighly effective under both balanced and unbalancedconditions. The dynamics of the system are retained well inthe developed DQ0 model. Under balanced conditions, thedeveloped model is more than 1000 times faster than thebenchmark ABC model. In addition, the developed model canbe easily connected to other functional models developedfrom previous work. The interface between the DQ0 modeland the ABC model can also be conveniently developedthrough a DQ0-ABC transformation matrix in Dymola. Underunbalanced conditions, the simulation speed of the DQ0model decreases sharply, which is due to the doublefrequency in both d and q axes. The dynamic phasortechnique is a potential method which can handle anunbalanced system effectively and a dynamic phasor modelfor the 18-pulse ATRU in the ABC and DQ0 frames is underdevelopment.

Acknowledgements

The author gratefully acknowledges the EU FP7 funding viathe Clean Sky JTI – Systems for green Operations ITD. Theauthor also would like to thank Mr Christopher Hill for thegrammar corrections of this paper.

References

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Appendix

ATRU parameters: Prate=150Kw, V=230Vol, Rp=0.160pu,Lp=0.0202pu, Ls=5.91e-4pu, Rp=0.1596pu, Rs=0.02pu,Winding length: l1abc=0.3471 (pu), l2abc=l3abc=0.5189pu,l4abc=0.2969pu, l5abc=l7abc=0.0502, l6abc=l8abc=0.4133pu

Synchronous machine generator: Rs=0.0044Ω, Lls=1.989e-5H,Lmd=2.201e-4H, Lmq=1.618e-4H, Rf=0.0689Ω, Llf=3.283e-5H,Rkd=0.0142Ω, Llkd=3.408e-5H, Rkq=0.0031Ω, Llkq=1.443e-4H,f=400Hz.

Connection cable: R=0.01 Ω, L=2e-6H.

DC-link side: CF1=CF2=45μF, R1=10Ω, RL=12.5Ω