IEEE Transactions on Power Apparatus and Systems, Vol. PAS-100, No. 11 November 1981

A SIMPLE AND EFFICIENT AC-DC LOAD-FLOWMETHOD FOR MULTITERMINAL DC SYSTEMS

H. Fudeh, Student, IEEE C. M. Ong, Senior Member, IEEE

School of Electrical EngineeringPurdue University

West Lafayette, Indiana

Abstract

A new ac-dc Load-flow method for multiterminal dc-ac systems that uses a novel approach to solve theequations of the multiterminal dc system is described.A simpLified version of the ac-dc load-flow methodwith all of the capabilities of estabLished Newton'smethods is shown to have the best qualities of simul-taneous and sequential solution ac-dc methods com-

bined. Next, important features such as discretetaps, distinction between scheduled voltage andscheduled angle controls, fixed taps and extendedphase controls, and current limits are incorporated toenhance the versatility of the method, especially fortransient stability studies. Numerical examples areused to ilLustrate the various features of the method.Its performance is compared with those of an esta-blished Newton's method and a recent method. Themethod is simpLe to program, economical and fast.

INTRODUCTION

Most ac-dc load-flow methods [1-9] can be cLassi-fied broadLy into two main categories: the simultane-ous and the sequential soLution methods. In simul-taneous solution methods, the equations describing thevarious dc system's components - the dc network, thedc terminals and their controls - are incorporatedwith the equations of the ac system; the collectiveset of equations of the ac-dc systems is solved, usu-alLy by a Newton's method. But the sequential solu-

tion methods maintain and solve the equations of thedc system separately from those of the ac system byconsidering the real and reactive powers and the ac

voltages at the converter buses as interface condi-tions that can be iterated upon until these conditionsin both ac and dc solutions match. Often, both dc andac solutions have to be repeated a few times. For ex-

isting ac-dc systems with few dc terminals, the com-puting effort and time for the dc solution are only

small fractions of those for the ac soLution; the ef-ficiencies of sequential solution methods are, there-fore, diluted mainly by the extra effort of having torepeat the ac solution. On the other hand, thesequential solution methods are simpler to program andwill adapt much more easily than simultaneous solutionmethods to future development of ac load-flow tech-nique and to new forms of dc system controls.

Depending on which representation is used [5,2),the total number of variabLes in the equationsdescribing a bipolar terminal can be 11 or 15. Withthese many variables per bipolar terminal, the comput-ing times and storage requirements of conventionalNewton's methods wilL increase rapidly if the numberof terminals in the dc system is increased.

81 sl 301-1 A paper recommended and approved by theIEEE Power System Engineering Committee of the IEEE

Power Engineering Society for presentation at the

IEEE PES Summer Meeting, Portland, Oregon, July 26-31,1981. Manuscirpt submitted August 29, 1980; madeavailable for printing March 23, 1981.

An important part of dc load-flows which has re-ceived considerable attention is the representation or

the implementation of the conditions established bythe controls of the converter and its transformer.Scheduled current, scheduled power, and scheduled voL-

tage with a certain minimum angle (amin for a rectif-

ier or ymin for a rectifier) are the types of convert-er controls that have been considered with transformertaps that are assumed to be continuous.

Very little effort [9] has been made to distinguishbetween scheduled voLtage (constant voltage obtainedwith variable extinction or ignition angle charac-teristics) and scheduled angle (constant extinction orignition angle) controls that are used on the voltagecontrolling terminal. This distinction is not onlynecessary for handling a new power factor control [10]but also essential for representing the differenttransient behaviors of these two controls. It is alsoproper that discrete tap be considered along withthese controls, because the approximation of thediscrete tap by a continuous tap can introduce as muchas 9% error in the control angle alone.

Ac load-flow methods play a useful role in tran-sient stability studies of ac systems. For an ac-dcload-flow method to have the same useful role in tran-sient stability studies of ac-dc systems, it should becapable of handling the conditions of fixed taps andextended phase controLs during a transient: the tapsoften remain unchanged for considerable Length of timeof the transient because of the built-in delays in thetap controLs, but the responses of the phase controlsof the converters are prompt. Furthermore, the dccurrents of the terminals with scheduled power con-trols and the dc current of the voltage controlling,or slack terminal, can change significantly during atransient, the current limits at these terminalsshould be considered. All these conditions of tran-sient operations have received very little attention.

A new ac-dc load-flow method for multiterminal dc-ac systems that uses a novel approach to solve theequations of the multiterminal dc system is described.

This paper is organized into three main sections.First, a simplified version of the ac-dc load-flowmethod with all of the capabilities of establishedNewton's methods is shown to have the best qualitiesof simuLtaneous and sequential solution ac-dc methodscombined. Next, important features such as discretetaps, distinction between scheduled voltage andscheduled angle controls, fixed taps and extendedphase controls, and current limits are incorporated toenhance the versatility of the method, especially fortransient stability studies. FinalLy, numerical exam-pLes are given to illustrate the various features ofthe method. Its performance is compared with those ofan established Newton's method and a recent method.

A SIMPLE VERSION OF THE METHOD

The simplicity and the economical computationaL re-

quirements of the new approach are best ilLustrated bya simplified version of the ac-dc load-flow. Thissimple version, nevertheless, has all of the capabiLi-ties of established load-flow methods. For this sim-plified version of the ac-dc load-flow, the usual as-sumptions of continuous converter transformer tap,

1981 IEEE

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scheduled voLtage control with a certain minimum con-troL angle, and fixed voltage margins at those termi-nals with a scheduled current or power control, as inestablished methods, are made.

Basic Equations

The basic equations describing the converter withits firing angle and tap controls and the dc networkare summarized. The equations and assumptions aresimilar to those given in references [5).

Converter equations

The converter model is based on the relationshipbetween the ripple-free average dc quantities and thefundamental frequency ac quantities.

Based on the per unit system given in Appendix III,the following equations can be written for every con-verter terminal. For the kth converter, its dc vol-tage equation in terms of its tap ak, ac voltage Vk

control angle 0k' commutation resistance Rck, and the

dc current Idk is

(1)Vdk = akVkcos k Rck dk

Its dc power equation is

Pdk =VdkIdkNeglecting the losses in the kth converter and itstransformer and equating the expressions for powers onthe ac side and dc side, the equation obtained for itspower factor angle (* -Ck) is

V=~ ~~Vdk akVkcos(*k3k))

For the simple circuit representation of the con-verter transformer shown in Fig. 1, the equation forthe reactive power flowing from the ac bus into thekth converter terminal is

Qk Pdktan(C1k-k) (4)

But with the more elaborate representation of thetransformer and auxiliary equipment shown in Fig. 7 ofAppendix I, the real and reactive powers on the ac busside of the transformer are no longer given by Eqs.(2) and (4) respectively; they can, however, be deter-mined by the procedure outlined in Appendix I, usingthe known values of Pdk' Qk' Idk and akVk.

6+

++zo+Idk

-0.9.

-- POSITIVEPOLE

Converter controls equations

A practical operating scheme for multiterminal dcsystem using local terminal controls is to have the dcsystem voltage determined at one terminal - the vol-tage controlling terminal. The other terminals areprovided with scheduled power or current settings.

To keep the reactive power consumptions of the con-verters and the losses in the snubber circuits low,the control angles should be small. But to maintainphase control and reliable commutation, a minimum con-trol angle should be maintained. Typical values of

min 0 0the minimum ignition angle a range from 5 to 7and those of the minimum extinction angle range from

150 to 200.In most load-flow methods, the voltage controlLing

terminal that is operating at the scheduled voltagesch

Vd is also assumed to operate with a certain minimum

control angle Omin Thus if the mth terminal is thevoltage controlling terminal, its dc voltage and con-trol angle are

V Vschdm d

and e = 0minm m

(5)

(6)

For the terminal with a scheduled current or powercontrol, it is common practice to coordinate the tapcontrol with the phase control so that the terminalwill operate at some dc voltage below its own minimumignition or extinction angle characteristic to avoidfrequent mode shifts from occurring with normal acvoLtage fluctuations. Typically, a 3% voltage margin

is provided; with the average a or y givenabove, typical values of the control angles a and y

are 150 and 200 respectively for those dc terminalswith a scheduled current or power controL. This typi-cal voltage margin of 3% in practice can be consideredin the load-flow computation by modifying the dc vol-tage equations for such terminals with a coefficientof K = 0.97 [53.

Thus if the kth terminal has a scheduled currentcontrol, its dc current is equal to the scheduledcurrent Idk h that is

dkanis d schv

and its dc voltage equation is

Vdk = Kk [akVkcosOk RCkIdk (8)

Similarly, if the kth terminal has a scheduled powercontrol, its dc power is equal to the scheduled powersch

pd , that is

(9)p pschdk dk

NEGATIVEPOLE

dk

Fig. 1. Equivalent representation of a bipolar station

and its dc voltage equation is also given by Eq. (8).

Dc network equations

The equations for the dc network can be formulatedto suit the procedure that is used to solve them.Since multiterminal dc networks in the near future areunlikely to have greater than 30 buses, the presentchoice is the R Gauss-Seidel method.bus

Although the algorithm is applicable to a generalbipolar network (Appendix II), there is no loss ingenerality by considering a symmetrical m-terminal bi-polar system that can be economically represented byan equivalent m-terminal monopolar system.

_V

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If the buses are numbered so that the mth terminalis the voltage controlling terminal and its networkterminal is also the reference bus for the Rbus' the

voltage equations for the dc network of the equivalentm-terminal monopolar system can be written as

m-1vdk k Id + Vd, k = 1, ..., (m-1) (10)ik =1 di dm

where rki 's are elements of the dc network's Rbus with

the terminaL of the mth terminal as itsreference.

Note that Vdm is the dc voltage at the terminal of the

voltage controlling terminal; Vdm is equal to the

scheduled voltage Vsch of that terminal.

Method of Solution

The method of solution is simple but different fromthose of previous ac-dc load-flow methods.

First, the voltage equations in Eq. (10) are solved

by using Vdm = Vsch I = Isch for the terminals withdm d di di

scheduled current settings, and Idi = Pdch/Vdi for

those with scheduled power settings in the Gauss-Seidel iterative procedure of the Vd's that starts

s chwith all Vd's initially set equal to Vd . This dc

network solution establishes the values of dc current,dc voltage, and dc power at every dc terminal of them-terminal network.

Next, the product of the tap a and the ac voltageV, or aV, at every terminal is determined individual-ly: For the voltage controlling terminal, this isdetermined by substituting its values of Vd and 0 from

Eqs. (5) and (6) and its value of Id from the dc net-

work soLution into Eq. (1). For the other terminalswith scheduled current or power controls, their aV'sare determined individually by substituting theirvalues of Vd and Id from the dc network solution and

the value of K into Eq. (8). Note that at this point,the V's at the converter buses are still unknown quan-tities.

With the knowledge of the aV's and the Vd's, the

power factor angLes (*-U)'s can now be determined fromEq. (3).

With the simple transformer representation shown inFig. 1, the real and reactive powers flowing from theac bus to the converter terminal are given directly byEqs. (2) and (4). But with the more elaboratetransformer representation of Fig. 7 in Appendix I,the real and reactive powers flowing from the ac bushave to be determined iteratively using the knownvalues of Pd' Q, Id, and aV at that converter termi-

nal. In any case, the real and reactive powers flow-ing from the ac buses into every converter terminal,or the whole dc system, can now be obtained.

Knowing the real and reactive powers fLowing fromthe ac buses into the dc system complete the descrip-tion of all the real and reactive loads on the ac sys-tem; the ac load-flow can be determined.

The ac load-flow provides the values of ac voLtagesat those ac buses connected to the dc system; knowingthese ac voltages V's, together with the aV's obtainedpreviously, the tap a of every converter transformercan be determined. And if these taps are within theirupper and lower limits, the complete ac-dc solution isobtained. However, if any of these taps exceeds its

limits, the dc system's voltage at the voltage con-trolling terminal is normally rescheduled and thewhole procedure has to be repeated.

A way of rescheduling the dc system's voLtage hasbeen described in reference [5). Briefly, the tapwhich exceeds its limit by the largest amount is firstidentified, say ak. If ak is greater than amax, the

scheduled dc voltage at the voltage controlling termi-a

nal is decreased by the ratio max And if a is lessak k

than amin, the scheduled dc voltage is increased by

athe ratio min

ak

Upper and lower tap limits can be separately ex-ceeded at different terminals, for example, when thereis an unusually steep voltage gradient in the dc net-work. However, this condition is not common in normalsteady-state operations and has been excluded in cer-tain methods by stopping the computation whenever sucha condition is detected.

A flowchart of the simple version of the ac-dcload-flow method for ac-dc systems with multiterminaldc networks is given in Fig. 2.

From this description of the method of solution,many desirable features of the method become apparent:

(1) The new ac-dc method can adopt any ac load-flowmethod for the ac solution.

(2) It is an economical method; both dc and ac solu-tions need only be computed once to obtain theac-dc solution with taps that are within theirlimits.

(3) Its storage requirement for the dc solution ismainly for the Rbus Gauss-Seidel dc network solu-

tion; this is minimal compared to that for theJacobian in most Newton's method.

(4) It is a simple approach to understand and to pro-gram.

2 1. ENTER WITH DATA AND SET INITIALCONDITIONS

2. SOLVE DC NETWORK'S EQUATIONS FOR V 's$3 I's AND P 's d

d d3. DETERMINE THE aV PRODUCTS AND POWER

FACTOR ANGLES (X - E) S4 8 L 4e DETERMINE REAL AND REACTIVE POWER

LOADINGS OF DC SYSTEM AT AC BUSES5, AC LOAD-FLOW

5 6. DETERMINE TAPS a's FROM AC VOLTAGES V'sAND aV PRODUCTS

7. TAP LIMITS EXCEEDED?6 8. RESCHEDULE DC VOLTAGE

9 . PRINT RESULTYES

Fig. 2. Flowchart for the simple version of the ac-dcload- flow

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Capabilities of the simple version of the method

In spite of its simplicity, the simpLe version ofthe method has many capabilities:

It wilL handle general bipoLar dc systems with sym-metrical or asymmetricaL network configurations.

For asymmetricaL network configurations, the net-work equations for both positive and negative poles(an example of the positive pole equations is given inAppendix II) can be solved simultaneously by the RbusGauss-Seidel iterative method using the scheduledvalues of the converter terminals in both poles.

Back to back 'systems, and also pole-parallelingoperations, with zero resistance network connectionsbetween adjacent terminals present no problem to theRbus formulation used.

Voltage control at a remote dc bus, or line voltagedrop bias, can simply be accommodated by rearrangingthe dc' network equations so that the specified voltageVdv at the remote bus appears on the right hand side

of the network equations. This is accomplished by ex-pressing Vdv in terms of Vdm using Eq. (8) and then

substituting back to eliminate Vdm; Eq. (8) becomes

m-1Vdk = 2 (rk - r.)Id + V, k=1,v ... mdk i=1 i V i d k*v

(11)

SPECIAL FEATURES ADDED

Features such as discrete tap, distinction betweenscheduled voltage and angle controls, fixed taps andextended phase controls, and converter current limitsthat are important for transient stability studies ofac-dc systems can all be easily incorporated with theefficient method of solution above to form a trulyversatile ac-dc load-flow method.

The numerous but simple steps taken to incorporatethese features are best described by flowcharts.First, a simplified flowchart of the ac-dc load-flowis given in Fig. 3. A number of blocks containing

W1

3o51 ENTER2. DC LOAD-FLOW

YES 3, AC LOAD-FLOW<4 ) 4. UPPER TAP LIMIT EXCEEDED?

5. RESCHEDULE DC VOLTAGEtYES 6. CONTINUOUS TAP?

7. PRINT RESULT\6 L 8. FIXED TAP?

9. DETERMINE NEAREST TAP POSITIONS ANDANO cos' s ASSUMING FIXED acose PRODUCTS

<8 > 10. DETERMINE cosO's USING UPDATED AC VOLTAGES11. UPDATE O CONSUMPTIONS12. SCHEDULED ANGLE CONTROL AND DISCRETE TAP?13. DETERMINE OPEN-CIRCUIT VOLTAGE OF TERMINAL

10 | JWITH SCHEDULED ANGLE CONTROL USING THEUPDATED AC VOLTAGES

14. AC LOAD-FLOWl11 1 15. CHECK CONVERGENCE OF AC VOLTAGES OF AC

ESBUSES WITH DC TERMINALS

sYES ______16. PRINT RESULT

minor calculations have been added to the simple ver-sion of the ac-dc load-flow that is given in Fig. 2.

Limits on the dc currents of the terminals can beconveniently applied during the Gauss-Seidel iterativesolution of the dc network equations.

For scheduled angLe with discrete tap and fixed tapwith extended phase controL, iterations between dc andac soLutions are required. But these are specialfeatures which neither 'simuLtaneous nor sequentialsoLution Newton's methods has been shown to be capableof providing. Besides, the numericaL examples willlater show that for conditions requiring thesefeatures the numbers of iterations between ac and dcsolutions of the method are about that taken by an es-tablished Newton's method to handle conditions thatthe simple version of this method can solve with justone dc soLution and one ac solution.

The main steps in the dc load-flow are shown inFig. 4. Many of these steps involve very little com-puting effort.

Because the terminal with a scheduled angle controland discrete tap can have only discrete step changesin its open-circuit voltage, its condition is betterhandled by manipulating directly with its open-circuit

voltage es (or aVcossch) than with its terminal vol-tage Vd. Thus for the terminal with a scheduled anglecontrol, an internal bus with open-circuit voltage esis defined, and at the same time its commutatingresistance is incorporated into the Rbus of the dc

network.

1

NO

I. ENTER|4 2. FIRST DC LOAD-FLOW?

3. READ DATA AND SET INITIAL CONDITIONS

YES 4. FORM R-BUS MATRIX{5x--- A{6> 5. FIXED TAP?

6. SCHEDULED VOLTAGE CONTROL?\ -NO 7. DETERMINE

YES 8. SCHEDULED ANGLE CONTROL?R< 9 10 7 9. IF DISCRETE TAP, SET ITS OPEN-CIRCUIT

e TO CORRESPOND TO THE SCHEDULED ANGLE

AND TAP10. ADD R OF SCHEDULED ANGLE CONTROL TERMINAL

I l ~~~~~~~~~~TOR-BUS11. UPDATE DC VOLTAGES12. DETERMINE DC CURRENITS OF SCHEDULED POWER12 lCONTROL TERMINALS, SET TO MAXIMUM VALUES

IF THESE LIMITS ARE EXCEEDED/13 - 13. CHECK CONVERGENCE OF DC VOLTAGES

NO 14. DETERMINE CURRENT, POWER AND VOLTAGE OFSLACK TERMINAL

15. SCHEDULED VOLTASE CONTROL?14 16. FIXED TAP?

17. SET V = Om AND CALCULATE aV OF BUS ATNO NO 5 mn

1/ 19 21 SLACK TERMINAL18. DETERMINE 0 FROM DC VOLTAGE EQUATION

NO 19. FIORD TAP?e10 20 20. DETERMINE 0 FROM cos = S/

21 - sch s oH5 SaV17

, , l L22. FIXED TAP?| 1fiM 1 23. DETERMINE cosO FROM DC VOLTAGE EQUATION WITH

KNOWN DC CURRENT, VOLTAGE AND aV PRODUCTYES 24. DETERMINE aV AND cos9 FOR NON-SLACK TERMINAL

<22 25. DETERMINE P AND Q LOADINGS ON AC SYSTEM26. RETURN

Fig. 3. Flowchart for the ac-dc load-flow Fig. 4. FLowchart for the dc Load-fLow

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TEST RESULTS

In t-his section, numerical examples of two samplesystems solved by the ac-dc load-flow method that hasall of the features mentioned above are given. Theac-dc load-flow program that is used has all of thefeatures mentioned above. The fast decoupled acload-flow method of Stott and Alsac C133 has been usedto solve the ac load-flows in all these numerical ex-amples.

Table 1 gives a summary of the test cases of thetwo systems solved by the new ac-dc load-flow method.It gives the types and scheduled values of the con-trols on the converter terminals, the size of the tapsteps, and the current limits.

Bd Bdc Bd

B~ ~ ~ ~ ~ LL

Edl p 4 C L JD B

ac)~~~~~~~~~~~~~~~~~~~~~~~~~~~~a~

d 8

B r 5 Bdc9 Bac2~~~. ac5

System A shown in Fig. 5 is the asymmetrical bipo-lar system of Braunagel, Kraft and Whysong; the dataof this system is given in reference C6]. System Bshown in Fig. 6 is an ac-dc system similar to thatused in reference E8], it is based on the AEP 14 busac test system C14]. The ac interconnections betweenbuses 2 and 5, 2 and 4, 4 and 5 have been replaced bya 3-terminal dc system using the same ac lines. Atthe three converter terminals the reactive power re-

quirements are compensated locally so that the reac-tive powers flowing from the ac buses are about thosewith the ac connections before.

Fig. 5. Sample bipolar system Fig. 6. Sample ac-dc system

Table 1. A summary of test cases

Isysteml types of converter control iterationspecial /

case Cl C2 C3 C4 C5 C6 constraints timetsec)

A II(sch)IV(sch)IItsch)IV(sch)IP(sch)lI(sch)I continuous I"2"dc4.90 .980 4.90 .980 4.90 1-4.90 TAP

I 1 1.001

A |I(sch)l II(sch)IV(sch)lP(sch)ll(sch)l continuous I"2"dc4.90 4.90 .975 4.90 1-4.90 TAP,remote

2 IV(sch)l controL of 1.001.980 C10's voltage

A II(sch)|Y(sch)tI(sch)IA(sch)|P(sch)II(sch)I discrete |"2"dc4.90 .980 4.90 .980 4.90 1-4.90 TAP

3 I I step=.015 1.002

A II(sch)It(sch)II(sch)1A(sch)IP(sch)I(sch)I discrete 1"2"dc4.90 18.0 4.90 18.0 4.90 1-4.90 TAP

4 I step=.015 1.002

A II(sch)| V I1(sch)I A IP(sch)tI(sch)l fixed I"2"dc4.90 1.8855 4.90 1.8912 4.90 1-4.9 TAP

5 1(max)=5.0 1.002

B IP(sch)IP(sch)IV(sch)l continuous I"2"dc,(16A,12V)acI.976451.211351.9800 TAP 1.001 + .26

I1 1 I

B IP(sch)IP(sch)| I continuous I"2"dc,(160,12V)ac1.976451.211351 TAP,remote 1.001 + .26

2 IV(sch)l control of.980 IC2's voltage

B IP(sch)|P(sch)I|(sch)l discrete 1"2"dc,(16A,12f)ac,(1tl1V)ac|.976451.211351.9800 TAP 1.001 + .26 +.03

3 I step=.015

B IP(sch)IP(sch)10(sch)l discrete I"2"dc,(160,12f)ac,(10,2V)ac,"2"dc,(1f,1V)acI|1.976451.211351 18.0 I TAP 1.001 + .26 + .04 + .001+ .03

4 step=.015

|8 IP(sch)IP(sch)l A fixed 1"2"dc,t(16,12V)ac,t(1,2V)ac,"2"dc,t(1,1V)acl1.976451.211351.8751 TAP 1.001 + .26 + .04 + .001+ .03

5 I I1(max)=.80

The number in quotes are the Gauss-Seidel iterationsfor the dc network solution in block 2 of Fig. 2 or bLocks11, 12 and 13 of Fig. 4.

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In Table 1, and also in Table 4, the numbers ofiterations in the dc and ac solutions are given along

with the recorded computing times taken on a CDC 6500computer. The numbers enclosed within the double quo-

tation marks are the Gauss-Seidel iterations for thedc network solution. The notations of reference C13]have been used to present the recorded numbers ofsolutions for EAe) and [AV] in each ac load-flow.

With continuous taps, the method took just one dcsoLution and one ac solution to solve the ac-dc sys-tem. If the taps were discrete, onLy a second dcsolution was required. However, for scheduled angle

control with discrete taps, and also fixed taps, notonly a second dc solution but also a third ac solutionwas required.

The resuLts of the tests on each of the two samplesystems are given separately in TabLes 2 and 3. Theasterisk placed after a value of power, voltage or

current indicates the type and the scheduled vaLue ofthe converter control used.

In the comparative study, method A is the method ofReeve, Fahmy and Stott E5), which is an establishedNewton's method with many of the capabilities men-

tioned earlier. Method B is the sequential method ofOng and Hanzei-nejad E93, which has all of the capa-bilities of the new method. The convergence toler-ances used in methods A and B were 0.001pu for theresiduals of dc and ac quantities in the ac and dcLoad-flows and 0.01 pu for the residuaLs of the reaLand reactive powers that were exchanged between ac anddc load-fLows. In aLL the tests, the convergence

tolerances used in the new method were 0.00001 pu forthe Gauss-Seidel dc network soLution and 0.001 pu forthe ac Load-fLow.

TabLe 2. NumericaL results on sample bipolar systems.

case Iconvertericonvertericonverterl converter tap controL neutralnumber voLtage I current Load setting angle voltage

Vd Id P Q Ti |(degrees)l Vn

C1 I .9900 4.900 *| 4.851 1.580 1.0289 15.605 0C5 .9891 4.954 I 4.900 * 1.600 1.0353 15.604 0C6 .9695 -4.900 o| -4.751 2.015 1.0521 21.126 .0098

(1) C2 .9800 o| -4.954 -4.855 1.618 1.0498 16.000*1 .0001I C3 .9897 4.900 of 4.850 1.580 1.0285 15.605 0

C4 .9800 of -4.900 -4.802 1.598 1.0433 16.000*1 .0001

ci I .98000 *v 4.900 *o 4.802 1.566 1.0186 15.604 0Cc5C .95110 5.152 I 4.900 * 1.616 I .9965 I 15.560 0C6 .93150 --4.900 of -4.564 1.943 I 1.0113 21.124 .0098

(2) C2 .98930 -5.152 I -5.097 1.705 1.0602 16.000*1 .0005C3 .98430 4.900 vf 4.823 1.572 1.0230 15.604 0 IC4 .97500 of -4.900 I -4.777 1.591 1.0380 16.000*1 .0005

C1 .9900 4.900 *o 4.851 1.598 |* 1.0300 15.829 0C5 .9891 4.954 4.900 * 1.749 f* 1.0450 17.412 0C6 .9695 I -4.900 o| -4.751 2.112 |* 1.0600 22.206 .0098

(3) I C2 .9800 of -4.954 I -4.855 1.769 f* 1.0600 17.825 .0001C3 .9897 4.900 of 4.850 1.603 fo 1.0300 15.892 0C4 .9800 of -4.900 I -4.802 1.624 fo 1.0450 I 16.327 .0001

C1 I .9332 f 4.900 *o 4.573 1.717 |* 0.9850 18.364 0C5 .9329 I 5.252 I 4.900 * 1.738 f* 0.9850 I 16.994 0 IC6 I .9133 --4.900 of -4.475 2.007 f* 1.0000 222.293 .0098

(4) C2 .9221 -5.252 f -4.843 1.804 1* 1.0000 18.000*1 .0007C3 .9378 I 4.900 o 4.595 1.656 f* 0.9850 17.508 0C4 .9287 f -4.900 f -4.551 1.680 f* 1.0000 18.000*| .0007

C1 .8957 4.900 *o 4.389 2.308 a 1.0000 26.138 0C5 .8949 f* 5.000 f 4.474 2.287 0 1.0000 25.382 0C6 .8753 I -4.900 of -4.289 2.380 0 1.0000 27.472 f .0098

(5) f C2 .8855 -5.000 f -4.427 2.146 f* 1.0000 24.049 .0003C3 .9008 4.900 of 4.414 2.259 |* 1.0000 25.467 0C4 .8912 -4.900 -4.367 2.112 fo 1.0000 24.049 .0003

The comparison of performance given in Table 4shows the merits of the new method. The new methodrequired only one dc solution and one ac solution, theother methods required more for the same ac-dc solu-

tion. The new method is also a faster method than the

others.

CONCLUSIONS

This paper has described a simple method for solv-

ing the equations of the multiterminal dc system in an

ac-dc load-flow.

The simplified version of the ac-dc load-flowmethod which has all of the capabilities of esta-bLished Newton's methods is shown to have the bestqualities of simultaneous and sequentiaL solution ac-

dc methods combined: It is an economical method; bothdc and ac solutions need only be computed once to ob-tain the ac-dc solution with taps that are withintheir limits. It can adopt any ac load-flow methodfor the ac solution. Furthermore, its storage re-

quirement for the dc solution is minimal compared tothat for the Jacobian in conventional Newton'smethods.

The versatility of the ac-dc method is enhanced bythe incorporation of features that are important fortransient stability studies, such as discrete tap,distinction between scheduled voltage and scheduledangle controls, and converter current limits. Itsversatility is illustrated by the variety of dc con-

trols and system conditions of the numerical examples

that have been solved by the method.

The comparison of performance with those of an es-

tablished Newton's method and another recent methodshowed that the new method is economical and fast.

Table 3. Numerical results on sample ac-dc system.

case fconverterlconverterlconverterl converter tap control ACnumber voLtage current load setting angle voltage

Vd Id P Q Ti |(degrees)lmagnitudel

I C1 .99778 f .9786 .97645 * .4245 1.0411 l 15.381 1.045(1) C2 .98465 f .2146 .21135 * .0678 f 1.0402 15.614 .9941

C3 .98000 *1-1.1933 1-1.1694 .4975 1.0405 16.000 *| 1.024

C1 .99319 f .9831 .97645 * .4257 f 1.0368 15.378 1.045(2) C2 .98000 o| .2157 .21135 * .0679 1.0354 15.614 .9941

C3 .97533 -1.199 1-1.1692 .4987 1.0360 16.000 *l 1.024

C1 .9978 f .9786 .97645 * .4343 1.0450 16.133 1.045(3) C2 .9846 f .2146 f .21135 * .7072 f 1.0450 I 16.436 I .9936

C3 .9800 *f -1.193 1-1.1694 .5067 1.0450 16.562 1.023

C .9230 1.0579 .97645 * .4820 0.9850 18.102 1.045I (4) C2 .9090 .2326 .21135 * .7485 0.9700 17.204 I .9931

| C3 .9038 1-1.2905 1-1.1663 .5602 0.9850 18.000 *| 1.017

C .8898 |* .8000 .71185 .4384 f* 1.0000 26.927 1.045(5) C2 .8793 .2404 .21135 * .1067 l* 1.0000 25.121 .9850

C3 .8751 1-1.0404 1-.91039 .5287 |* 1.0000 25.519 1.012

Table 4. Comparative performance on a CDC 6500 computer.

Isysteml method A method B newcase method

I A iteration 2dc 2dc I"2"dc. I.-.-.

___________ lI- _I_ ---------------| 1 | time(sec) | .85 .01 1.001. ______- -.-------------------------------------- --------------------------------------------_

B iteration ld|c,(160,12V)ac,ldc,(10,2V)ac,ldcI 2dc,(160,12V)ac,3dcl"2"dc,(160,12V)acIIl ---- ---- ----- ---- ---- ---- ----I.-------- - ----------------

1 time(sec) 1.095+ .26 +.095 + .04 +.0951.005+ .26 +.008 1.001 + .26I- --_I-.-- -- -- -- -- --- - --

4395

ACKNOWLEDGEMENT

This work was sponsored by the Electric Energy Sys-tems Division of the U.S. Department of Energy underthe contract ET-78-S-01-3396.

REFERENCES

1. M. M. El-Marsafawy and R. M. Mathur, "A new, fasttechnique for load-flow solution of integratedmulti-terminal dc/ac systems," IEEE Trans. PowerApparatus and Systems, Vol. PAS-99, No. 1, 1980,pp. 246-253.

2. J. Arrillaga, B. J. Harker, and P. Bodger, "Fastdecoupled load flow algorithms for ac-dc sys-tems," IEEE Paper A78 555-5, Presented at the PESSummer Meeting, Los Angeles, CA, July 16-21,1978.

3. J. Arrillaga and P. Bodger, "AC-DC Load-flow withreaListic representation of the converter plant,"Proc. IEE, Vol. 125, No. 1, 1978, pp. 41-46.

4. J. Arrillaga and P. Bodger, "Integration of HVDClinks with fast-decoupled load-flow solutions,"Proc. IEE, Vol. 124, No. 5, 1977, pp. 463-468.

5. J. Reeve, G. Fahmy and B. Stott, "Versatile loadflow method for multiterminal HVDC systems," IEEETrans. Power Apparatus and Systems, VoL. PAS-96,No. 3, 1977, pp. 925-933.

6. D. A. Braunagel, L. A. Kraft and J. L. Whysong,"Inclusion of dc converter and transmission equa-tions directly in a Newton power flow," IEEETrans. Power Apparatus and Systems, Vol. PAS, No.1, 1976, pp. 76-88.

7. G. B. ShebLe and G. T. Heydt, "Power flow studiesfor systems with HVDC transmission," Proc. PowerIndustry Computer Applications, 1975, pp.223-228.

8. H. Sato and J. ArrilLaga, "Improved load-fLowtechniques for integrated ac-dc systems," Proc.of IEE, Vol. 116, No. 4, 1969, pp. 525-532.

9. C. M. Ong and A. Hamzei-nejad, "A general-purposemultiterminal dc load-flow," IEEE 1981 WinterMeeting, Paper 81 WM 0207

10. F. Nishimura, A. Watanabe, N. Fujii and F. Ogata,"Constant power factor control system for hvdctransmission," IEEE Trans. Power Apparatus andSystems, Vol. PAS-95, No. 6, pp. 1845-1853.

11. E. W. Kimbark, Direct current transmission, Vol.I 1, John WiLey & Sons, Inc., New York, 1971.12. E. UhLmann, Power transmission by direct current,

Springer-Verlag, Berlin Heidelberg, New York1975.

13. B. Stott and 0. Alsac, "Fast decoupled loadflow," IEEE Trans. Power Apparatus and Systems,Vol. PAS-93, May/June 1974, pp. 859-869.

14. IEEE Computer Applications Sub-Committee StandardTest System, American Electric Power Service Cor-poration, 1962.

APPENDIX I

THREE-WINDING TRANSFORMER REPRESENTATION

Many 3-winding converter transformers with auxili-ary equipment can be represented by the equivalentcircuit given in Fig. 7 [see discussion of [6]].

I :a X.. aV p

xs

P Q'

Vt

Xf

Fig. 7. Equivalent representation of a 3-windingtransformer with auxiliary equipment.

To determine the real and reactive powers fLowingfrom the ac bus into transformer, a simple load-flowof the T network has to be solved using bus 1 as thereference bus with a voltage magnitude aV, bus 3 as aP , IVtI bus, and bus 2 as a Pd' Q' bus. At bus 2 the

real power Pd is the dc power obtained from the dc

network solution and reactive power Q' is calculatedfrom the expression

(12)Q' = Pd tan (4-O) - (d)_2>

APPENDIX II

GENERAL BIPOLAR NETWORK EQUATIONS

In a general bipolar system, the dc network confi-guration, and also the number of converters, of thepositive pole can be different from those of the nega-tive poLe. At a terminal the dc currents of the twopoLes need not be the same, in fact, one of thesecurrents may even be zero. For such asymmetricaLoperation, the equation for the positive pole dc ter-minal voltage at the kth terminaL of the general m-terminal bipolar system is

+ in-1 + + - +Vd = - r(I -I ) +dk X= k~idi rgk dk dk dm

gm Cdm dm (13)

where rki are the eLements of the positive pole net-

work Rbus matrix with the mth terminal, the

voltage controlLing, chosen as reference.

The dc currents flowing into the positive pole dc net-work should sum to zero. Thus,

m

E Idi =0i =1

(14)

Similar equations can be written for the negativepole dc terminal voltage and currents.

4396

APPENDIX III

List of symbolsV/ - ac bus voltage and phase angleI/_ - r.m.s alternating current and phase angle

a - transformer tap ratioa - control angle, ignition or extinction anglee - open-circuit voltage, Vacose

Vd - direct voLtage

Id- direct current

r.. - (i,j)th element of the resistance bus ma-1J

trixrg - resistance of ground connection

X - reactanceRc - equivalent commutating resistance

nb - number of series-connected bridges in a

terminalsuperscripts: + for positive poLe quantities

- for negative pole quantitiesmax for maximum valuemin for minimum valuesch for scheduled value

Per unit systemA common base power Pbase

dc systems.

is chosen for both ac and

Vac base = VMline to line, rms value).

abase,v3 Vac base

Zac base = Vacbasevac base

Zhoe =K

ac base ac bac base

Choose Vdc base = K Vac base

3V4where K = ir nb

then Idc base K ac basez K~~2Zdc base =K Zac base

If Pbase is on one converter transformer rating and Xc

is the per unit commutating reactance, the equivalentcommutating resistance in per unit is given by

R Xc 6nb c