IEEE Transactions on Pover Apparatus and Systems, Vol. PAS-95, no. 1, JanuarylFebruary 1976

INCLUSION OF DC CONVERTER AND TRANSMISSION EQUATIONS DIRECTLY I N A NEWTON POWER FLOW

DUANE A . BBAUNAGEL, MEMBER LEONARD A , KRAFT, " B E R JENEL L. WHYSONG, SENIOR bff3MBEE COMMONWEALTH EDISON COMPANY

CHICAGO, ILLINOIS

ABSTRACT

This paper presents the method used t o include the equations for DC converters and t r ansmiss ion l i nes d i r ec t ly i n a Newton A C system power flow. The DC equations are presented in a per un i t form t h a t i s com- pat ible with the per uni t A C equations. The DC equations can therefore be incorporated directly into the Jacobian of the Newton power flow. The convergence propert ies of the Newton, method are not a l tered. The DC model and equations used make it possible to inves t iga te any number of two terminal DC l i nes or multiple terminal DC networks a s component pa r t s of an A C system.

INTRODUCTION

The Commonwealth Edison Company became ac t ive ly i n t e re s t ed i n HVDC transmission some years ago when it became apparent that such a system might provide an economical answer t o some urban transmission problems which were an t ic ipa ted for the ear ly or middle 19801s. A preliminary economic study with the ASEA Co. of Sweden was suff ic ient ly promising to war- rant a more detailed engineering study. (1)

One purpose of the detailed study will be t o examine how DC l i nes w i l l in te rac t wi th the exis t ing A C system with regard t o normal and emergency power f lows, faul t dut ies and e f f ec t s on system s t a b i l i t y . Another purpose of the detailed study i s t o provide an educa- t ional opportuni ty for some of the engineers who might be involved i n t h e work in the fu- t u r e i f a DC l i ne , were authorized.

Although AC/DC power flow programs were avai lable , i t was the thought that there would be some advantages i n producing our own DC power flow so tha t i t could be t i e d t o our ex is t ing A C power flow in t he most e f f i c i e n t manner. Furthermore, we were i n t e r e s t e d i n a program t h a t would handle DC networks as well a s two terminal l ines because both possibil- i t i e s a r e under consideration.

The conventional Newton power flow, used by Commonwealth Edison, was o r ig ina l ly de- signed to solve only A C networks. (2 , 3, 4 ) Methods have been devised for solving AC sys- tems with a few DC l i nes (5, 6, 7, 8, g ) ,

Paper F 75 425-9, recommended and approved by the IEEE Power System Engineering Committee of the IEEE Power Engineering Society for presentation at the IEEEPES Summer Meeting,& Francisco, Calif., July 20-25,1975. Manuscript submitted August 28,1974; made available for printing April 28,1975.

76

but these methods a re somewhat i ne f f i c i en t because they do not solve the A C and DC sys- tems simultaneously. The usual procedure i s to es t imate the DC parameters, solve the A C system, and then use the A C so lu t ion to readjust the DC arameters. This process cont inues unt i l 'convergence" i s obtained. Each time the DC parameters are modified, an en t i r e A C solut ion must be calculated. The DC modelling and the method used r e s t r i c t the type of DC systems tha t can be studied,

This paper describes a method whereby the necessary DC equations are converted by a per uni t system into equat ions that are compatible w i t h the A C equations. As a re- s u l t of this conversion, the A C and DC equa- t ions may be solved simultaneously, rather than ser ia l ly . It i s p o s s i b l e t o solve multi-terminal DC networks as wel l as two terminal DC lines. Furthermore, it i s a l s o possible to solve unbalanced DC networks,

ASSUMPTIONS

The following basic assumptions have been made in t he de r iva t ion of the equations re resenting the AC-DC converter: (10, 11, 127

1) The A C source delivers a constant voltage of sinusoidal waveform and constant frequency. Three phase voltages and currents are balanced.

2 ) A l l higher harmonics of voltage and current produced by the converter a r e f i l t e r ed ou t and do not appear i n t h e A C system.

3) The transformers associated with the converter have no resis tance and no exci t ing impedance.

4) The valves of the converter are i- deal and have no arc voltage drop.

5 ) The DC voltage and current have no

6 ) In a 6 pulse converter, the valves f i r e a t 600 in te rva ls , and fo r a 12 pulse converter the interval is300.

r ipp le ,

MODEL OF AC-DC CONVERTER

Figure 1 i l lus t ra tes the bas ic conver t - e r model used in t h i s pape r . The A C supply i s assumed t o be a 316 system, however only a s ing le l i ne diagram i s shown. A l l symbols are defined in Appendix I. The area en- closed by the dot ted l ine shows the repre- sentat ion of the conver te r i t se l f . The transformer within the converter i s .repre- sented by an ideal tap-changing transformer with a ser ies reactance. The valves, with

t h e i r 3(6 connections, are represented by the diode symbol. This model i s v a l i d f o r e i t h e r 6 o r 12 pulse converters.

Id EQUATIONS FOR AC-DC CONVERTER

~ ~ / l / l / / ~ / ~ l ~ / " ' / ' / " / ' / ' / ' / ~ / ' / ' / ' / / / / / ' / / /

Fig. 1 Single bridge converter model

For the multiple bridge converter model shown i n Fig. 2 , the transformer reactance, X t , i s assumed t o be the same f o r a l l trans- formers. Tap r a t i o s a r e a l s o assumed t o be equal.

r-----------

/ % / ' / % / I / / / / / / ' / / / ' / / ~ ~ / / / l / f / / / / ' / / / / r

Fig. 2 Multiple bridge converter model

These models may be adapted t o permit the use of three wlfding transformers. A s shown i n Fig. 3, a T" model of transformer i s used. The model of the converter i t s e l f remains the same a s that shown i n Fig. 1. X t i s included within the converter, while X 1 and X2 have t o be placed in the A C system.

/ I / / / / / / / / I / / / / / / / / / / / / / / / I / / / / / / / / / / ~ / / / / / / / / / 1 / / ' / / / / f

Fig. 3 Single bridge converter model w i t h 3 winding transformer

When there are mult iple br idges, then X1 and X2 must be modified. If B equals the number of bridges and the primed quantities indicate values for multi-bridge configura- t ion then ,

sic = B Sac Assuming tha t t he A C voltages remain un- changed, then

The basic converter equat ions that have been developed by others (10, 11, 12) will be used in this paper . A s l i s t e d below, these equat ions a re va l id on ly for a s ingle br idge conver te r and a re no t in per unit form. These equations are based on the previously l i s t ed assumptions and use the symbols de- f ined i n Appendix I. Power leaving a bus i s def ined as posi t ive. Assume T = l .

I,, = I, - jI,

I n a more convenient form,

The power relat ionships are as fol lows:

In o rde r t o mesh the A C and DC equa- t i o n s d i r e c t l y i n t h e Newton power flow, the converter equations must be expressed i n a per unit system ,that i s compatible with the A C per unit system. (13) Both A C and DC per unit systems are l i s t e d below.

A C PER UNIT SYSTEM = VA3@

Ebase = E1-g

Ibase = VAbase 1 3Ebase Zbase = Ebase / Ibase

DC PER UNIT SYSTEM VAbase = VAac base

Ebase - E1-g = base 'base = / Ebase = 31ac base

'base - Ebase 'base = K2Zac base

-

- / 3

77

Note tha t the cons tan t K may d i f f e r f o r each converter. For DC systems with negative polar i ty , K will a l s o be negative. The DC voltage base must remain unchanged within a DC network, but each separate network may have a different base.

Before putting the converter equations in to pe r wit form, consider the effect of multiple converter bridges and an off-nominal t a p r a t i o on the transformer modelled within the converter. The bridges of a converter a r e connected i n p a r a l l e l on the A C s ide and i n s e r i e s on the DC s ide. If the primed quantit ies indicate values for only one bridge, then for multiple bridges

vd = B V& E = E '

I d = I d Iac = B d c 1

I f E" i s t h e A C voltage on the DC s ide of the transformer, then E" = E/T. To in- c lude the effects of an off-nominal transfor- mer r a t io , subs t i t u t e E/T f o r E i n t he p re - vious equations.

The converter equations in per unit are l i s t e d below, X t i s the per un i t reac tance of the transformer on the A C base. For the remainder of t h i s d i scuss ion , a l l quan t i t i e s are expressed in per uni t .

'aC = 'dld (10) IMPLEMFNTATION OF DC EQUATIONS

Each converter w i l l add three equations and variables to the Jacobian. Mathematical- ly, the converter equations can be arranged to maintain a constant A ) DC v o l t a g e a t Vb, o r B) DC current through the converter, o r C A C wat ts del ivered to the DC system, o r D ] AC vars c'onsumed in the conver te r . The last option has been fully programmed and i s operat ional , but for physical reasons this type of c o n t r o l i s n o t p r a c t i c a l . Hence,

we will not discuss i t f u r t h e r i n t h i s p a p e r . For each type of converter control, there a re two possible modes of control :

a variable , y variable a variable , y variable, while

maintaining a minimum ext inc- t ion angle .

A detai led presentat ion of the equations in- volved can be found i n Appendix 11.

The var ious poss ib i l i t i es of modelling the con t ro l fo r a converter may prove useful to the planner when running power flows, For example, while some converters are nor- mally current controlled, the current set- t i ng i s o f t en ad jus t ed t o ma in ta in a con- s t an t power flow. With these control equa- t ions, the planner can direct ly specify the desired power flow.

Associated with each converter there i s an additional transformer equation. The purpose of the var iable tap t ransformer is t o a l t e r t h e A C voltage going into the valves of the converter . I f th is were not done, there would be no guarantee that there i s a feas ib le in te r face between the A C and DC systems. Changing t h e f i r i n g and com- mutation angle changes both the DC current and voltage, however there a re cons t ra in ts on both of these quant i t ies . By introducing a transformer into the converter model, the A C voltage can be a l te red to a l low the vo l t - age and current constraints to be matched slmultaneously.

DC l i n e s and buses are handled i n the power flow i n much the same manner a s AC quant i t ies , with a few re s t r i c t ions . A DC l ine can have resistance only as a l i n e or shunt parameter. A DC bus has p lus 'o r minus voltage magnitude with no associated phase angle. A DC bus may have only watts as gen- e ra t ion and load, and resis tance only as a bus shunt element. Each DC bus introduces into the Jacobian one equation, C Pdc = 0 , and one variable, vdc.

The neu t r a l bus, Vn, i s handled just l i k e any other DC bus. I f the posi t ive and negative DC systems a re unbalanced, then current will flow i n t h e n e u t r a l system. This current flow w i l l a f fec t the vo l tage of the posi t ive and negative DC systems. The resis tance between neutral buses, whether due t o l i n e o r ground return, i s modelled in the power flow as a l i n e . Any one o r a l l of the neutral buses may be connected d i r ec t ly t o ground. The neutral buses that are so l id ly grounded are designated as system swing buses with a voltage magnitude of zero. The power flow program has the cap- a b i l i t y of handling 25 swing buses, A C and/ o r DC.

When a l l of the conver te rs in a DC ne t - work a re cont ro l l ing the DC cur ren t , the a l - gebraic sum of a l l c u r r e n t s i n t o t h e system must equal zero. If the converters are modelled to con t ro l t he A C watts, then a DC network requires a "Swing" converter f o r the same reasons that the A C -system re- quires a swing generator. One converter must be ava i lab le to supply l ine losses and

78

system wide power mismatches. If there a r e N converters i n a DC system, a t most, N - 1 converters can be control l ing the power. It i s our e x p e r i e n c e t h a t a t l e a s t one DC bus i n each network should be voltage con- t r o l l e d . The function of t h i s bus i s t o define the reference voltage, similar t o t h e fixed voltage of the system swing bus i n t h e A C system.

The cont ro l on each converter can be specified independently of the other conver- t e rs i n t h e DC system. Mathematically, the model of the converter and the associated equations will accommodate any configuration tha t the user des i res to cons ider . However, a s i n a l l a p p l i c a t i o n s of the power flow, i t i s the respons ib i l i ty of the user to def ine what i s reasonable.

Special consideration must be g iven to the s ta r t ing va lues ass igned to the DC para- meters when f i r s t beginning the i terative processes of the Newton method. When using previously solved values, no problems a r i s e . However, when not using previously solved values, DC buses in the pos i t ive por t ions of the DC system should have s t a r t i n g v o l t - ages of +1 pu, and those in the nega t ive portions -1 pu. Neutral buses that are not grounded should be assigned voltages close t o zero. Star t ing values of 2 .O5 have yielded s a t i s f a c t o r y r e s u l t s . The transformer tap r a t i o , T, should be s e t t o 1. The f i r i n g and commutation angles should be given typical

Eac'l.012 er3.928. Idct4.9

I

Converter Parameters

For a l l converters: K = 2 2 . 7 E

X t = .01 (P.U.) B = 2

CONTROL MODE 1 ( a var iable , y variable) For individual converters :

Control Control Control Control Type Value Type Value

21 Idc 4.900 c2 Vdc * 971 c3 Idc -4.900 c4 vdc - -971 C5 Pac 4.901 c6 Idc -4.900

DC Line Parameters Resistance (P .u . )

L1 = ,0002 L4 = ,002 ~6 = . O O ~ ~2 = ,0018 L3 = .002

L5 = .002

Per Unit Base Quant i t ies

Table I. Input Data

79

values, remembering that the former may be e i the r pos i t i ve or negative.

COMPUTATIONAL RESULTS

The DC equations have been ful ly pro- grammed in to the Commonwealth Edison power flow program (PEPCE). Converters and DC l i n e s of various configuration have been add- ed t o a 500 bus A C model, Tolerances of l e s s than 5 MW and 5 WARS mismatch per bus have been easily obtained. It has been found t h a t the addition of the DC equat ions direct ly in- to the Jacobian of the- Newton power flow d i d not increase the number of i t e r a t i o n s needed f o r convergence of the A C system alone.

For i l lustrat ion purposes only, the sys- tem i n Fig. 4 i s o f f e r e d a s a n example of the methods described in this paper. Table I contains the input values. It should be noted t h a t t h i s i s a mult iple l ink DC network t i e d i n t o a large A C system.

A diagram of the Jacobian representing the portion of the network shown in F ig . 4 i s presented in Appendix 111.

CONCLUSIONS

A suitable choice of a per uni t system for the DC components permits including the DC equations with the AC equat ions in the solut ion of a Netwon power f low.

The modelling of DC elements may include multiple two terminal l ines or a DC network a s a par t of the A C network. The model cor- rectly represents both the posit ive and nega- t ive po les and separate neutral conductor for a l l DC l i nes .

There i s no degradation of the conver- gence propert ies of the Newton method a s t he solutions of the DC and A C equations are ob- tained simultaneously rather than sequential- l Y *

Some addi t iona l . work must be done. Minor refinements ' involve inclusion of con- verter transformer resistance and converter a r c drop. Considerably more work Will be re- quired to incorpmate the DC converter model and the associated equations developed in th i s paper in to the fau l t s tudy and t rans ien t s t a b i l i t y programs.

1.

2.

3.

REFERENCES

J. L. Whys;ong, B. Skoglund, and R. A . Naata, Comparison of High Capacity A C and DC underground Transmission for Com-

W . F. Tinney and C . E. Hart, "Power Flow Solution by Newton's Method", PAS-Vol. 86, pp. 1449-1460, November 1967

J. P. Bri t ton, "Improved Area In te r - changt Control for Newton's Method Load Flows , PAS-Vol. 88, pp. 1577-1581, October 1969

4.

5.

6.

7 .

8.

9.

J. P. Brit ton, "Improved Load Flow Per- formance Through a More General Equation Form", PAS-Vol. 90, pp. 109;116, January/ February 1971

G. Hingorani and J. D. Mountford, SJiimulation of m c systems i n A.C,., Load Flow Analssis b s Dig i ta l ComDuters

J. J. Vithayathil, "Digital Simulation of DC systems f o r LoadllFlow and Trans- i en t S t ab i l i t y S tud ie s , unpublished notes

A . Gavrilovic and D. G. Taylor, "The Calculation of the Regulation Character- i s t i c s of D-C Transmission Systems", PAS-Vol. 83, pp. 215-223, March 1964

G. D. Brever, J. G. Luini, and D. C. Young, "Studies of Large AC/DC Systems on the Digi ta l Computer" PAS-Vol. 85, pp. 1107-1116, November 1966

W. L. Powell, unpublished notes describ- ing Bonneville Power's implementation of DC i n t o t h e i r power flow

10. C . Adamson and N. Hingorani, High Volt- age .Direct Current Power Transmission, London, Garraway Limited, lgbo

1 2 . L. Neiman, S . Glinternik, A . Emel'yanow, and V. Novitskii , D - C Transmission i n Power Systems, t ranslated from Russian bv I s r a e l Program f o r Sc ien t i f ic Trans- - v - lat ions, Jerusalem, 1967.

13. A . G. Phadke and J. H. Hylow, "Unbalanc- ed Converter Operation , PAS-Vol. 85, pp. 233-239, March 1966

APPENDIX I: LIST OF SYMBOLS

'dc ,'de 9 Idc Vd

I d vb Vn

X t T K

B a

Complex A C power, voltage and current A C voltage magnitude, 1-g, peak of Vac, A C voltage magnitude, 1-1, rms of Vac, Active and react ive components of A C current f lowing into con- ver te r , rms DC power, voltage and current DC voltage developed across con- ve r t e r DC current f lowing in converter DC cvoltage a t "high" side of converter DC vo l tage a t "neut ra l" s ide of converter Transformer reactance Transformer tap ratio Ratio of A C and DC per uni t voltage a t t h e DC s ide of the transformer within the convert- e r Number of br idges in converter Firing angle of converter

ao

Y Commutation angle of converter 6 0 Minimm ext inct ion angle Bac j A C bus Bdc j DC bus pb DC power going i n t o c o n v e r t e r a t

non-neutral DC bus Pn DC power going in to conver te r a t

neut ra l DC bus * complex conjugate

APPENDIX 11: DC EQUATIONS I N THE JACOBIAN

When t h e f i r i n g and commutation angles of the conver te r a re var ied to . maintain a constant DC current , or AC Watts, equa- t i o n s i n a d d i t i o n t o t h e normal bus power equations are introduced into the Jacobian. Although the equations used will depend on the type of control, the same types of equa- t i ons w i l l be used f o r each type of control. The equations themselves and the 'appropriate pa r t i a l s a r e g iven below.

1) Converter maintaining constant DC current .

Using equation 1

The var iable associated with this equat ion i s the f i r ing angle , a. The par t ia l s requi red are given below.

Section 4 of Appendix I1 presents the equa- t i o n used t o govern the relationship between t h e f i r i n g and commutation angles a t t h i s converter.

2 ) Converter maintaining constant A C watts

Define the following variables :

X = cos(2a) - COS[Z(U+Y)] B = sin(2a) - sin[2(u+y)]

Using the var iables just def ined and equa- t i o n 6

The var iable associated with this equat ion i s the f i r ing angle , U . The p a r t i a l s r e - quired are given below.

- b Pac = - 2 C B -- ba

bPac 2 5 S l ~ 2 ( U + Y ) ] by

5 h - P 2 bT T

Section 4 of Appendix I1 presents the equa- t i on u sed t o govern the relationship between t h e f i r i n g and commutation angles a t t h i s converter.

3) Converter maintaining constant DC

The implementation of t h i s type of control d i f f e r s somewhat from the previous two des- c r ibed i n s ec t ions 1 and 2 of Appendix 11. Normally, an equation i s w r i t t e n f o r each DC bus t h a t sums the power around i t , c P d c f o . The var iable associated with t h i s equation i s the voltage, vdc. when the converter con- t r o l s t h e DC voltage, vb i s no longer varia- ble. Since the power around th i s bus must s t i l l sum to zero, the power equation re- mains the same but the var iable associated with i t now becomes the f i r ing angle of the converter, The p a r t i a l s f o r t h i s equation are descr ibed fur ther in Sect ions 6 and 7 of Appendix 11. Section 4 of Appendix I1 presents the equa- t i o n used t o govern the relationship between t h e f i r i n g and commutation angles a t t h i s converter.

voltage, vb

4 ) Relationship between the firing and commutation angles of a converter.

Whenever the f i r i ng ang le i s changed, the commutation angle will a l s o change. Neither angle i s independent of the other. Since the number of variables must equal the number of equations, an equation must be obtained to govern the relationship between the f i r i ng and commutation angle,

Using equations 2 and 3,

Vb = m [coe(a) + cos(a+Y)] + v, 2nKT

a t a = O

v;, = - 2nKT [l + cos(yo)] + vn Pa, = A o r

81

where

The par t ia l s a re g iven below.

= - sin(a) - sin(a+y) 6a = - sin(a+y)

b y

The equation f o r I d may a l s o be used t o obtain a re la t ionship between the f i r i ng and commutation a n g l e . I f t h i s i s done, the following equation i s obtained:

COS(^) - cos(a+y) + COS(^,) - 1 I 0 where

6 TX& COS(Y0) = - KE

Notice tha t t he pa r t i a l s of this equat ion with respect to a, y di f fe r on ly by con- s t a n t from the respect ive par t ia ls of I d . This means tha t i n t he t r i angu la r i za t ion of the Jacobian, one diagonal term will become zero and the standard algorithm will not work. Since the usage of the voltage equa- t ion to der ive the re la t ionship between the two angles d i d no t y ie ld th i s d i sas t rous result, it was chosen. Upon convergence, both methods w i l l give the same re su l t .

5 ) Transformer equation associated wi th

+l

each converter.

The equat ion tha t i s used i s ob ta ined from equations 2 and 3.

s = S l E coos(a) + oos(a+y)] + vn - vb 0 2nKT

The var iable associated with th i s equa t ion i s the t ap ra t io , T, of the converter t ransfor- mer. The par t ia ls are given below.

6 ) Power equations for converter DC terminal buses,

An equation i s normally written for each DC bus tha t sums the power around that bus. The var iable associated with t h i s e q u a t i o n i s

normally vdc. When a DC bus i s connected t o another DC bus, the power equation for the DC l i n e and the appropriate par t ia ls are presented i n Section 7 of Appendix 11. Equa- t i ons a r e a l so needed to ca l cu la t e t he power flowing into the converter from the DC bus. % i s given i n ~ q . 8. using equations 2 , 3 , 8, 9, 10 it i s e a s i l y shown tha t

Pb = -Pac - 'n The p a r t i a l s f o r Pac are given in Sect ion 2 of Appendix 11. The p a r t i a l s f o r Pn a r e given below. The p a r t i a l s f o r % can be obtained by combining t h e p a r t i a l s for Pa, and Pn.

7) Power equation for DC l i n e s

Note that i f DC buses are assigned a magni- tude and zero angle and DC l i n e s a reactance of zero and a line charging value of zero, then the standard A C l i n e power equations w i l l compress to those given above.

8 ) Power equations for converter A C terminal buses.

Two equations are normally writ ten for each A C bus that sum the watts and vars around that bus. The variables associated with these equations are normally the voltage magnitude and angle. Equations a re needed to ca l cu la t e t he power flowing into the converter from the A C terminal bus. The equations for paC are g iven in Sec t ion 2 of Appendix 11. The equation for &ac i s given below. Note t h a t Pac and kc a r e not dependent on the angle of the A C bus.

Using the var iables def ined in Sect ion 2 of Appendix I1 and Equation 7

Q,, = (2Y + 3

82

The necessary partials are given below,

- - 2 F X bQao ba

9 ) Converter control while maintain- ing minimum extinction angle,

When operating at minimum extinction angle, the following relationship must be main- tained :

c5

c5 C1 c2 c3 c4 c5 c5

3ac~

3 a c ~

%c3

3x4 3ac5 Bdcl Bdc2 3dc5 Bdc6 3dc8 3dc9 adc 1 1

Id .

LP

At the time of this writing, the best way to maintain converter control and simultaneous- ly maintain a minimum extinction angle has not been determined. Two possible methods are currently under study:

a) Set up the converter to maintain a desired voltage, and dynamically (between iterations) change the voltage. This will force T, a ? y to vary and perhaps the minlmum extinction angle can be obtained. Even if this method works, it is neither desirable nor computation- ally efficient.

b) Include specifically this relation- ship into the equation associated with the' transformer ratio. One possibility is

h a A 7

A a AY A a AY A a A Y

A a A Y A U AY A T A T A T A T A T A T A b A E/E A 6 A E/E

A E/E A b

A b A E/E A b 0 v/v m rn m m

A v/v

A V / v

- Fig. 5 Jacobian of Sample AC-DC System

a3

If th is op t ion works, and there is no apparent reason why it will not, then th i s will be a very desirable and e f f i c i e n t method.

APPENDIX I11

Figure 5 shows a diagram of the Jacobian f o r t h e AC-DC system i n Figure 4. For pur-

AP andaQ represent the wat t and var m i s - match a t a bus. The equations for R and S are given in Sections 4 and 5 of Appendix 11. Those positions occupied by . a n X in - d ica te that ;the appropr i a t e pa r t i a l must be calculated and placed i n t h a t l o c a t i o n . The converter or bus that the equat ion i s writ- t e n f o r i s i n d i c a t e d on t h e l e f t hand s ide of the char t , and the var iables on the r igh t hand side correspond to the same converter or bus.

poses Of this . diagram Vac = 1EI-&, V = Vdc,

Discussion

Carl E. Gmad, k e y H. Happ, and Ray V. Pohl (General Electric Company, Schenectady, N.Y.): 'Ihe authors have made a major contribu- tion to the solution of system power f low by formulating the solution of general AC/Dc networks in the form of the Newton power flow. Our review has resulted in the following comments and questions.

up the two terminal DC line between converters C3 and C4 of Figure 4, As an independent check on the validity of the solution we have set

setting Vn to zero at both converters. To achieve consistent results we had to resolve a few differences in conventions. For three-phase power systems the normal base voltage is the line-to-line voltage rather than the line-to-neutral voltage used for single phase AC systems. 'Ihe con- ventional defintion of angles (Reference 11) is given in Figure 1 with parenthetical values as used in this paper.

i i

Fig. 1. Relationships for Angles of Rectifier (left) and Inverter (right) Valve Currents

(130.5)

I - 1.012 '37*3

Using the equations of Reference 8 for a two terminal Dc line, our results are given in Figure 2 in parentheses where different from those in the paper. It should be noted that our solution requires specification of 7, the inverter margin angle, and the inverter power and voltage. The LTC transformer adjusts the commutation voltage to achieve the required inverter conditions. The rectifier conditions are calculated from inverter conditions and the IR drop with adjustments of its commutation voltage to amve at the required delay angle, a.

correct results for a R / h and aR/&y, appears to warrant some The derivation in appendix 114, although yielding essentially

clarification. The DC voltages for a = a and a = 0 cannot be equated unless the effective commutation voltage E/T is allowed to vary (since the commutation angle is a dependent variable). This implies that lower- ing u to zero will increase the DC voltage unless the tap ratio is used to reduce the effective commutation voltage and thereby the DC voltage to maintainVb=Vb. ' IhusforVb=Vb

E r {cos a + cos (a + y) 1 = 11 + cos y 1 E' and

R = $cos a + cos (a + y))- 7 {1 + cos y 1- 0 E E' T The partials are now

Would the authors please comment on the assumptions used for their derivation of R(u,y).

able to specify a desired inverter margin angle and adjusting the effec- We agree with the authors comments on the desirability for W i g

tive commutation voltage E/T to achieve the desired inverter DC voltage

those calculated in the sample system (Figure 4), typically about 18 vb. We would recommend operation with margin angles b a t e r than

degrees rather than 13.2 degrees.

G. T. Heydt (Purdue University, West LaFayette, Indiana): The

increasingly moving in favor of direct current (dc) transmission over economics of bulk power transmission by underground conductors is

conventional alternating current circuits. The dc links augment the alternating current network, and electric power flow studies must be expanded to include the high voltage dc converter and the dc network. This paper presents the details of the method to handle dc transmission in electric power flow study techniques. Apparently the authors have had considerable experience in the area, and I would like to take the opportunity to raise a few points and inquire their opinion on certain technical matters.

The fixing angle a is generally considered a valve control parameter, but the commutation angle 7 is not usually regarded as an authentic control parameter. The latter is the case since 7 is largely determined by the converter transformer reactance which is clearly not variable. It is

Manuscript received July 28, 1975.

( 136.9)

'37*0 0.990

+ Q = 13.2 (12. I ) I - T= 0.993 T = 0.977 I c3 - I =4

480.6 sso.'s 475.8 477.3 (480.41 (480.61 (475.6)

- vb=o.981 Vb'0.971

(0.975) (0.966)

Fig. 2. Comparative Solution Using Reference 8. (Values in parentheses where different)

Manuscript received August 25,1975.

a4

true that a and y are interrelated in a complex fashion (Equs. (1 -1 0) in the paper), and y may be made to vary by varying anode voltage or other valve parameters. Nonetheless, y is not generally a control parameter and this quantity does not always appear in the post- multiplying vector to the Jacobian (Figure 5 of the paper). In order to minimize the required reactive power (Q) to be dispatched at converter busses, the angle 6,

6=Ct+y

must be minimized. Under this condition, y appears in the solution

vector. This point needs to be clarified since control to effect minimum vector. Under other control methods, y may or may not appear in the

6 is not the exclusive form of converter control. On a different topic, the several converter transformers associated

with a converter may not have tap changers which are tied together (Le., the tap setting on each transformer may be different). How can this feature be incorporated?

In the example shown in Figure 4, it is not clear whether the five ac busses are interconnected by adjacent ac transmission. If this is the case, then the ac network will contain a single swing bus. If the ac busses are not interconnected to each other by ac transmission, then there will be multiple swing busses - one bus per ac island. Each isolated ac system is unable to transmit phase angle information to other isolated ac islands; therefore each ac system will have its own reference bus.

separate ac systems (eg.: New Zealand, Northern Manitoba). In these Ihis point is raised because often times dc is used to connect two

cases, multiple swing busses will typically occur in the ac network. On the same topic, in work with Sheble reported in [A], we did not use a dc swing bus in the dc network. Sheble and Heydt did not investigate multitenninal dc networks, but in the conventional two terminal

voltages are determined by Equations (1-10) [B] . With this in view, the balanced bipolar case, or the two terminal monopolar case, the dc bus

dc bus voltages are determined by converter firing angles, controls, and ac system voltages and angles. Why is a reference bus or swing bus required in the dc network?.I believe that bus Bdc4 is actually a voltage controlled bus (V=O), not a swing bus.

Finally, a question on line filters may be of interest. Should line filters be treated as constant impedance loads?

Messrs. Braunagel, Kraft and Whysong have done an excellent job in presenting a timely topic. Their conclusions agree with our fmdings that ac and dc busses should be intermixed in the Jacobian matrix (rather than iterating alternately between ac and dc solutions) [A]. The authors and their colleagues at the Commonwealth Edison Co. deserve recognition for their forward approach to system design and analysis.

REFERENCES

G. Sheble, G. Heydt, Power How Studies for Systems with HVDC Transmission Proc. Power Industry Computer Applica- tions Conference - New Orleans, La., May, 1975. L. Neiman, S. Glintemik, A. Emelyanow, and V. Novitskii,

Israel Program for Scientific Translations, Jerusalem, 1967. D-C Transmission in Power Systems, translated from Russian by

C. V. Thio (Manitoba Hydro, Winnipeg, Manitoba, Canada): At Manitoba Hydro we have been using load flow programs incorporating DC representation for some time since the early installation stages of

subroutine where the DC is solved sequentially or serially to the AC as Nelson River bipole-one. Our present load flow program utilizes a DC

described in the paper as the usual solution method.

include the DC representation and we therefore find this paper of.value We are presently also developing a Newton power flow program to

and interest. In the development of the basic converter equations the parameters

voltage. In dealing with the DC power equations I prefer to relate all Em, E, /El , EL%, y d Vac all appear to refer basically to the same

equations to the parameter E, unless special conveniences dictate other-

voltage referred to the valve side of the converter transformer. The wise. E is taken as the rms, line-tdine, magnitude of the commutating

nominal value can also be chosen as the AC base voltage. If my inter- pretation of the per unit system in the paper is correct the resulting base quantities would be:

VAdc base = v%c base = Edc base Idc base

Edc base = &c base

Idc base = f l %c base Iac base = E

K%c base K Iac base

zdc base - - K2Eac base fl Iac base = K2Zac base

and voltage equations would be multiplied by fl. The power equations In the per unit equations (1 ) to (IO) the corresponding current

remain the same but the complex power equation (4) could be divided by the transformer tap ratio T.

calculated from the basic DC equations are all referred to the com- The complex AC power and power factor or displacement factor

mutating point as shown in Figure 1. In this case X, is the total commutating reactance to the commutating bus with sinusoidal voltage. In the simple case of a two winding transformer this will usually be the leakage reactance of the converter transformer. The DC equations then include the reactive drop across the transformer and in effect they need not be a part of the AC solution except for the tap ratio.

condenser is connected to the tertiary the solution becomes more For a three winding transformer, for example, if a synchronous

complex because the condenser and tertiary reactances form a part of the commutating reactance. Also, if the transformer is placed in the AC system as depicted in Figure 3 of the paper it must be ensured that the reactive drop is not accounted for twice - once in the AC solution, and once more in the DC equations. The interface between the DC and AC solutions is then somewhat confusing. Can the authors comment on how the program handles this situation or if they foresee any difficulties in this respect with the simultaneous type solution?

Xc= 10 A ON VALVE SI DE 230/127kV vd=150kV

+pV= 2.7p.u. Id 1800amps 9 . Z 1.059p.u. s= 18

Pd = 270 m w

h= 0.94p.u.

SYNCHRONOUS CONDENSER

Fig. 3. Simple practical example of DC power flow

n X, I

Fig. 1. Basic Diagram for Converter Equations

Manuscript received July 14,1975.

a5

EV

Xvalve - pv f-, Q

tert iary

T S Y N C H R O N O U S C O N D E N S E R

N

Fig. 2. Example of AC and DC solution interface

established at the valve terminals on the DC side of the commutating A convenient interface between the DC and AC solutions can be

reactance as depicted in Figure 1. The complex power and line-tdine voltage magnitude at the valve terminals then completely defines the DC system for the AC solution as shown in Figure 2 as an example.

To facilitate this solution the following DC equations are established further to those given in the paper:

1 cos $ = -

2 [cosa+cos (a+?)] = - (displacement factor)

3 d 7 E

E, = I E COS $ f j (E sin $ - f i X, IIacl)l + rectifier - inverter Using the base quantities given above the corresponding per unit

equations are :

cos $ = - - K T n vdp.u. w Ep.u.

Qvp.u. = vdp.u. kl p.u. tan $ - IIacp.u.l & p a . 2

The per unit equations were applied to the simple practical example shown in Figure 3. The base quantities are:

VAbase = 100 MVA; Eac base = 127 kV; E& b e = 450kV

K = 450/127

&p.u, = l o x 100/1272 = 0.062

the paper is indeed quite convenient. The example exercise clarifies that the per unit system chosen in

assumptions relative to the power factor of the converters and the AC The equations given above and those in the paper contain

current magnitude. The AC current magnitude corresponds to the case of zero commutation angle and leads to an approximate fundamental power factor sometimes referred to as the displacement factor (references 10 and 11). For the exact fundamental power factor con- sidering the fuing and commutation angles the AC current magnitude should be multiplied by a factor which for most practical cases is very close tounity. It is our experience that the approximate equations are sufficiently accurate for power flow calcualtions and in any case they give a pessimistic var requirement for the converters.

J. Reeve, C. Fahmy and, B. Stott (University of Waterloo, Ontario,

load flow programs which ammodate a multiterminal dc network, we Canada): Being actively engaged in formulating comprehensive ac/dc

have particular interest in the paper. It is evident that there is not a unique method of either interfacing the ac and dc system analytically or incorporating the control and operating constraints imposed on the dc system.

For multiterminal HVdc system representation, the extent to which the possible steadystate control strategies [ 141 must be accomodated is crucial, and this aspect is the least developed in the paper. The difficulties in implementing preferred inverter modes are

in Fig. 4 purports to represent voltage control in Converters 2 and 4. At described in Appendix 11, section 9, and yet the sample study illustrated

the same time, the respective delay and extinction angles are approxi- mately the same as for converter 6 on current control. This is unreason- able for practical purposes. (The same illustration, unless we mis- understand it, appears to provide a power flow of 9.8 p.u. in dc line 6 between 2 dc buses having the same voltage to ground).

a well established philosophy for a parallel multiterminal system [ 15 1 . The concept of one dc terminal providing voltage control is part of

provision must be made for modifying the control strategy between In general, the transformer tap positions must be constrained and

iterations if a tap limit is reached. There is no indication in the paper that these features are included in the authors program. If so, how are they computationally realized? Can the authors clarify the criteria they have used for tap-changing at both voltage or current bower) controlling terminals, in relation to preferred system operation as opposed to providing a feasible interface between the ac and dc systems?

The direct inclusion of the dc system into a combined Newton ac/dc load flow solution has certain attractions, as in the case of simpler controlled transmission devices, and is consistent with previous directions of load flow program development at Commonwealth Edison [3] , [4] . The combined Newton ac/dc solution was successfully applied

ing nonstandard control schemes and fully representing control action for two-terminal dc links in [ 161 , having greater flexibility for specify-

when the various operating limits are reached. The main price paid for these additional features was extra complicated sparsity programming to cater for structural changes in the Jacobian matrix, which can include zero-valued diagonal elements with certain control modes.

Our experience with two-terminal and multiterminal solutions, including nonstandard controls and onerous operating conditions, supports the authors fmdings that the Newton process converges reliably and rapidly in this application. In spite of this, we seriously

Manuscript received August 15, 1975.

86

question whether the combined Newton ac/dc approach is the most

Whenever the number of load flow iterations is not minimal, there is promising for more detailed modeling of the dc system and its controls.

scope for conducting separate alternate solutions of the ac and dc systems without penalizing the total computing time, compared with ordinary ac solutions. Good interface techniques are available to promote rapid and reliable mutual convergence of the ac and dc equations. In the Newton load flow solution of practical large power systems, limits on various quantities will always tend to make the total number of iterations significantly more than the 3 - 5 often quoted. developed independently and with more advanced features whose

In the separatesolution approach, the dc program can be

representation would be detrimental to the simplicity and efficiency of the papers combined scheme. As an add on facility, such a dc program requires no essential modification to any existing ac program.

the use of the fast decoupled method, for instance, offers substantial Thus the choice of ac load flow method is not restricted to Newtons -

overall savings [ 171, [ 181. It would be very helpful to know the the typical number of iterations for large power systems with generator VAR limits, LTC transformer limits, etc. using the authors load flow program.

The dc system equations in the paper could have been formulated in alternative ways. The use of overlap angte as a variable, extensive use

avoided in the interests of analytical simplicity and computational of trigonometric functions, and nonlinear dc network equations can be

efficiency.

shunt capacitance aspects of W d c conversion on the converter ac bus? Finally, can we assume that the authors have accomodated the

REFERENCES

K.W. Kanngiesser, J.P. Bowles, A Ekstrom, J. Reeve and E. Rumpf, HVdc multiterminal systems, CIGRE, 1448, Paris, August 1974. J. Reeve and J. Carr, Review of techniques for HVdc multi- terminal system, IEE Int. Conf. Publication, HV AC/DC Transmission, London, Dec. 1973.

Thesis, University of Manchester, 1971. B. Stott, Load flows for aC and integrated ac/dc systems, Fh.D.

Power App. Syst., vol. PAS-93, pp. 859-867, May/June 1974. B. Stott and 0. Alsac, Fast decoupled loadflow, IEEE Trans.

Masiello and B.F. Wollenberg, Comments on Review of load A.M. Sasson, W. Snyder and M. Flam, and separately R.D.

flow methods, IEEE Roc., pp. 712-714,April 1975.

Ku and D. R. Nevius (Public Service Electric & Gas Comuanv Newark, N.J.): The authors should be commended for their innovkve

in the AC load flow program. Since flows on point-to-point DC ties are approach to incorporating a more exact simulation of DC transmission

normally fixed for specified steadystate system conditions, there does not appear to be an overwhelming need for using such a sophisticated program in comparison with the application of the simple block-loading method of representing DC ties in AC load flow. However, if the evalua- tion of a DC network is required, a more sophisticated simulation of AC and DC flow interactions appears desirable.

For transient stability and short circuit studies, a more rigorous representation of DC transmission would be highly desirable for point- to-point DC ties as well as DC networks. We assume that in order to incorporate a more accurate simulation of DC transmission in studying

provide such a feature in the load flow program as the first step. We transient and dynamic system conditions, it would be necessary to

would like to encourage the authors to continue their successful efforts and expand their work into the area of incorporating the DC simulation in the transient stability and short circuit programs. Perhaps it would also be desirable to incorporate the automatic DC control functions with respect to the specified AC requirements in both the load flow and transientstability programs.

The authors indicate that the inclusion of DC equations in the Newton Power flow does not increase the number of iterations needed

time required for each iteration? The authors comments wohd be for the convergence. If so, would there be a significant increase in the

appreciated.

Manuscriut received A w t 4.1975.

H.L. Forgey and J.D. Osborn (Consumers Power Company, Jackson, Michigan): The authors are to be commended for incorporating the method of representing HVDC systems described in this technical paper into the large modem load flow program which has been developed in their Company. Representation of HVDC systems instead of a limited number of DC tie lines in working load flow programs is of great value because of the economic alternative offered to the future expansion of power systems through the use of HVDC facilities.

wealth Edison in this effort, a HVDC tie line representation for the load We have recently developed, and received help from Common-

flow program which is presently being used by our Company. We have added to this tie line representation the ability to represent converter transformer resistances which the authors indicate are yet to be added to the representation discussed in this paper.

planned HVDC system would be connected to AC EHV systems (for From a system planning viewpoint, it is very possible that any

example, 345,500 and 765 kV) made up of transmission lines which are of significant length. Since the present practice is to non-transpose transmission lines in these EHV AC systems, it is possible that significant voltage and power unbdances between phases would occur. Presently, a considerable amount of investigation is being carried out on &@tal three phase representations of AC systems in order to adequately study these unbalances. It would be appreciated if the authors would give their opinion on the need to include this DC system representation in a three phase load flow program in order to adequately simulate expected system performance.

W o n is planning additional work for future incorporation of this As noted in the conclusions, it appears that Commonwealth

HvDC system representation into other programs in use in their Company as well as adding refmements such as converter transformer resistance and converter arc voltage drop. The indication of plans for these future developments is encouraging and we would like to ask the authors to briefly discuss an expected time schedule for the inclusion of these developments into the various programs mentioned and to state any Plans for the preparation of companion papers describing this additional work.

Manuscript received July 28,1975.

L. Carfsson (ASEA Ludvika, Sweden): This is a very interesting paper, which presents an elegant method of including HVDC transmission in a load-flow program. The authors have used a sound approach in making a simultaneous solution of the AC and the DC equations.

of treating multi-terminal HVDC schemes with arbitrary network The most interesting aspect of the program seems to be its ability

topology. To the discussors knowledge this is the first time that such a feature has been described.

conceming a similar inclusion of multi-terminal HVDCscheme models It would be very interesting to know if any work is going on

into a transient stability program.

Manuscript received July 28,1975.

D.A. Braunagel, L.A. Kraft and J.L. Whysong: The information and questions presented by the dicusson are very welcome additions to the paper, and are much appreciated by the authors. Our closing remarks will generally follow the same order as the items are mentioned in the discussions.

We agree that there should be some standarized notation for

notation available. We have kept identical notation for both rectifiers converters. Unfortunately, the notation in Reference 11 is not the only

and inverters to eliminate the problems associated with maintaining and programming two sets of equations.

to-ground voltage. However, to conform to common usage, the voltage In our equations the AC base voltage was assumed to be the line-

ratio, K, is input using the line-to-line AC voltage. The program then immediately multiplies K by fl voltage can be maintained while holding fixed the inverter extinction

In our scheme, only the inverter current, power, or terminal DC

(margin) angle. If the terminal DC voltage is held constant by the inverter, then the desired power or current must be specified by the

Manuscript received September 18, 1975.

8 7

rectifier. This differs somewhat from the General Electric AC/DC program.

We are very pleased that the solution by the General Electric

the same results. However, the voltage developed across Q is -.974 AC/DC program of part of our sample system (Fig. 4) yielded essentially

believe that the solved tap ratios at C3 and C4, and the commutation (pu), rather than -.966. From the discussers comments, we are led to

angle at C3, did not differ significantly from the values presented in Fig. 4.

The method of obtaining the equation governing the relationship between the firing and commutation angle (Appendix 11-4) is a common used mathematical technique to defme a relationship between two variables when there is only one relevant equation. This technique is presented in Reference 10, p. 30. However, for the reasons given in Appendix 11-4, we chose touse thevoltage equation. Upon convergence, using either the voltage or current equation will result in identical relationships between the firing and commutation angles. The method presented by Grund, Happ, and Pohl requires the calculation of E , T, cos (70) and Sy/Ga, and we are not sure how this would be accomplish- ed. Since the equation defining R is an attempt to specify the relation- ship between the two angles, the partial derivatives of this equation should not contain a 6y/Ga term.

it is nevertheless a variable. Since the commutation angle has a definite While the commutation angle is not an actual control parameter,

relationship to the firing angle, which does change, the two angles will always appear in the vector post-multiplying the Jacobian.

The multi-bridge model used cannot account for the situation where the transformers in a multi-bridge configuration might have different tap settin@, If it were necessary to study this situation, each bridge could be modelled separately with the appropriate connection of the DC terminals.

The five AC buses shown in our test system are all part of a large network requiring only one AC swing bus. However, the program is capable of considering several AC islands, providing each has its own swing bus. DC buses 4 and 10 are swing buses with voltage magnitudes of zero. At least one bus in each separate neutral network should be grounded. The voltage at a swing bus is a defined rather than a con- trolled quantity. Since the voltage at a ground bus is a defined quantity and since there is no control by either the AC or DC systems to hold the voltage of a ground bus to the prescribed value, the labelling of these ground buses as swing buses with V = 0 is a convenient method of maintaining a reference neutral voltage.

We do not have any experience with line filters and do not feel that we can make any judgements regarding their modelling in a power flow.

In the model we use, the AC/DC interface occurs at the converter AC terminal bus. Mr. Thio assumes his interface to be at the valve side

as a base the line-to-line AC voltage. We do not agree with his assertion of the converter transformer. Moreover, in his per unit system he chose

that ourequations contain assumptions regarding the AC current magni- tude and the power factor of the converter. Mr. Thios equations, how- ever, do _make these assumptions.

from that of the Westinghouse AC/DC program. Both models represent Our modelling of three winding converter transformers differs

the high side and tertiary winding in the AC system. However, Westinghouse has the tap changer on the high side, whereas we have it on the low side. Our arrangement does introduce some error because the tap is actually on the high side, but we believe that this error will be small compared to those introduced by some of the otherassumptions associated with the converter model. In order to adjust the tap in the solution process, Westinghouse must include it in the AC/DC interface. Thus using a Thevenin equivalent of the high side and tertiary windings, a new commutating voltage and reactance is calculated. Since our scheme has the tap changer on the low side winding, this extra computa- tion is not required. We are aware of the fact that the impedance of the low side winding in the T model of a three winding transformer is often very close to zero. At the present time we dont know how this problem can be resolved.

capable of representing the various steady state control strategies. At the We agree that -a converter model for use in a power flow must be

time the paper was written, the methodology for maintaining a minimum extinction angle at an inverter was not fully established. The scheme proposed in Appendix 11-9-b was eventually adopted, with one minor modification. Because of the small values involved when angles are expressed in radians, the program would converge without achiev- ing the required accuracy. Therefore, the equation in Appendix 11-9-b was changed as follows:

For inverters maintaining a minimum extinction angle, the equations in Apkndix 11-5 should be modified appropriately.

only because they are printed to three significant figures. More In Fig. 4, the voltage at DC buses 8 and 2 appear to be the same

accurately, the voltage difference between buses 8 and 2 is .00020, resulting in a flow on Lg of 9.8025 MW.

We could devise no general and workable algorithm for changing control Our program allows for many types of control for each converter.

strategies that would be appropriate for DC networks when some converter parameter limits are reached in the iteration process. When firing or commutation angie limits are reached, they are merely reset at the limit and processing continues. There are no tap limits for the transformer. With the voltage, power and current effectively specified for each converter, the program will determine the required transformer ratio. If the tap differs by more than 5% from unity, the user is warned appropriately. We believe that an unsatisfactory solution is better than none. We do not modify any user data to amve at a preferred system operation.

T h e model and methods presented in the paper will never result in zero elements on the main diagonal. The diagonal elements for the five AC converter terminal buses do contain some zeroes (Fig. 5 ) . How- ever, since these buses are connected to other AC buses, these diagonal terms will be added to other non-zero diagonal terms.

Our Ac power flow converges in 3-1 0 iterations, with an average of

cluding a test model of nearly 10,000 buses. The AC/JX test system 5-6. These numbers are typical of all production and test models, in-

used for this paper contained about 600 buses and converged to a tolerance of .5 MW and 5 MVAR in 4 iterations.

each iteration resulting from the inclusion of DC into the AC model. We can not determine precisely the additional time required for

Our sample DC system had a total of 17 converters and DC buses. The increase in time required is no more than would be normally required by the addition of 17 AC buses to the same model.

We agree that the converter equations could have been formulated differently, However, we did not value analytical simplicity and computational efficiency enough to eliminate trigonometric functions and non-linear network equations. It appeared that some severe and unnecessary assumptions would be necessary if this were done.

is connected to the converter terminal AC bus. Any shunt capacitance associated with the filtering of harmonics

and currents. If this is not true, then a three phase representation would We have assumed that the entire AC system has balanced voltages

be required of both the AC system and the converters. Depending on the

in the converter model, the additional complexity of the converter purpose of the study and in view of some of the other assumptions made

model and the equations may not be justified.

These refinements could be include later. The expansion of our model We did not consider converter transformer resistance and arc drop.

to include dynamic and transient effects will require substantial effort in both the engineering and mathematics involved. Regretfully, the publication of this paper completes this phase of our work on the DC Power Flow Program. We do not anticipate doing any work on a DC Fault Study or Transient Stability program in the near future.

88