ac-dc power flow calc

IEEE TRANSACTIONS ON EDUCATION, VOL. E-27, NO. 1, FEBRUARY 1984

A Simple Approach to the Solution of theac-dc Power Flow Problem

DANIEL J. TYLAVSKY, MEMBER, IEEE

Abstract-The advent of inexpensive and reliable high-power SCR's hasled to the reemergence of dc power systems for industrial and utility applica-tions. Calculations of the voltage profile for these systems require that theac-dc power flow problem be solved. While many texts treat the ac powerflow, none to date have incorporated the ac-dc power flow. Presented in thispaper is an approach for simultaneously solving the ac and dc systemsequations which constitute the ac-dc power flow problem. The approach isvalid for multiterminal HV dc links as well as industrial dc networks. Theuse of Newton's method in this approach allows the ac-dc power flow to bepresented as a generalization of the ac power flow.

INTRODUCTION

T HE number of power electronic applications is growingat a rapid pace due to the developments in solid-state

technology which have made reliable and inexpensive SCR'sreadily available. The theoretical basis for many of theseapplications has been established for years and is covered bymany textbooks in the area [ 1]. Many applications, however,are new, and reliable analytical methods for treating theseproblems have only recently been established. The analysisof converter-fed industrial and utility dc power systems isone such area. Today, industrial dc power systems oftenconsist of interconnected dc grids which are fed throughoutby many, often uncontrolled, converters. These systems aregrowing in size and number. Utility use of HV dc transmis-sion links for connecting remote generation facilities andlong distance power transmission is also increasing. Today,25 HV dc links are in operation throughout the world withmany more in the planning stages. Fig. 1 shows the growthrate of HV dc installations [2]. At this pace, it is essential thatour power engineers be familiar with dc system design andplanning if they are to cope with this trend.One of the principal design tools used in power system

planning is the power flow study. Of the many books thattreat the ac power flow problem [3]-[12], few consider ac-dcsystems at all and none addresses the ac-dc power flowproblem. Of the books which deal with ac-dc systems [ 13]-[16], none treat the ac-dc power flow problem. The reasonsfor this lack of treatment may be that only recently havemethods for treating this problem been proven in practice towork satisfactorily and reliably.Two separate iterative techniques have been used to solve

the nonlinear equations which constitute the ac-dc powerflow problem. The sequential method involves a two-stepprocedure at each iteration [17]-[20]. In the first step, the ac

Manuscript received January 3, 1983; revised August 15, 1983.The author is with the Department of Electrical and Computer Engineer-

ing, Arizona State University, Tempe, AZ 85281.

H

zn

0Ir

950 1960 1970 1980 1990

Fig. 1. Direct current transmission capacity in the United States andworldwide.

power flow problem is solved while the dc system is modeledas a constant load on the ac system. The second step involvessolving the dc system equations while all ac system quantitiesare held constant. The simultaneous method involves solvingthe ac and dc power flow equations simultaneously at eachiteration [21]-[27]. The simultaneous method is attractivebecause of the more elegant way in which the Newton-Raphson method is applied to the problem. In addition,convergence properties of this method are slightly betterthan those of the sequential method. The sequential methodhas the advantage that it can easily be interfaced to anexisting ac power flow program without restructuring the acalgorithm.The goal of this treatment is to use the simultaneous

method to introduce the ac-dc power flow solution proce-dure to the student in such a way that it becomes a logicalextension of the ac power flow solution. It is assumed thatthe student is familiar with the ac per-unit system, theNewton-Raphson (NR) method and the equations whichdescribe the terminal behavior of the three-phase bridgeconverter under normal operating conditions [13].

PER-UNIT SYSTEM

The use of an ac per-unit system reduces all three-phasepower system transformer turns-ratios to 1:1. When bal-anced operation is assumed and single-phase analysis used,the equivalent single-phase I 1: turns-ratio transformer

0018-9359/84/0200-0031$01.00 © 1984 IEEE

31


E 2

(a)

IE(4 Id.|IL± IE,IcosCc '6 XC E2

(b)Fig. 2. Bridge converter: (a) schematic diagram; (b) per-unit equivalent

circuit.

[(E - HARMONIC FILTER

Fig. 3. Generalized ac-dc power system.

model may be replaced by a circuit node. The three-phasebridge converter of Fig. 2(a) may also be viewed as a trans-former which transforms ac to dc. In this figure, the symbol(A) is used to indicate actual rather than per-unit quantitiesand

El = ac side phasor voltageE2 = dc side voltageI, = fundamental-component phasor currentId= direct currentXc= commutation reactance.

While the ideal ac power transformer is a linear device, theac-to-dc transformation characteristics of a bridge converterare nonlinear. Hence, the effective turns-ratio of the "ac-to-dc transformer" changes as the current through the "ac-to-dctransformer" changes, and no one ac-dc per-unit system canbe chosen which will allow the bridge converter to bereplaced with a circuit node. The selection of an ac-dc per-unit system may therefore be made arbitrarily. For conven-ience it is usually chosen so that per-unit current and powerare equal on both sides of the converter. Use of such an ac-dcper-unit system allows the per-unit equations which describethe terminal relationships of the bridge converter of Fig. 2(a)to be given by

E2=IE, Icosa- Xcr d (1)6

Id = I,I (2)where a is the commutation delay angle.An additional per-unit equation which is useful in ac-dc

system analysis is obtained by equating the real power at theac and dc terminals of the converter. In per-unit, this equa-tion is

IdE2 = |IiJ I|E, cos X (3)

where ¢ = angle by which I, lags E, and cos 0 is the dis-placement factor. Substituting (2) into (3) and cancelling Idfrom both sides gives the result

E2-= II cos 0. (4)For a detailed discussion of this per-unit system and a deri-vation of these equations see the Appendix.An electrical circuit analog for (1) and (2) is conceptually

helpful when using these equations for network analysispurposes. Kirchhoffs voltage law can be used to verify thatFig. 2(b) is the per-unit electrical analog for (1) and (2).

PROBLEM SPECIFICATIONA generalized model for an ac-dc power system is shown

in Fig. 3. Each bridge converter symbolically shown mayrepresent a rectifier or inverter, depending on the controlstrategy. The ideal harmonic filters shown serve to limit the

32

TYLAVSKY: SOLUTION OF THE AC-DC POWER FLOW PROBLEM

commutation reactance to the series reactance measuredbetween the filtered bus and the ac terminals of theconverter.The ac-dc power flow problem may be broken into two

special cases: the utility power flow problem and the indus-trial power flow problem. These cases differ because of thedifferent power system configurations encountered in utilityand industrial networks. A utility ac-dc power system isusually comprised of a large ac network and a multiterminaldc link for high voltage power transmission. Today onlytwo-terminal links are in operation, but true multiterminaloperation can be expected in the near future. The dc portionof the utility dc system is characterized by controlled con-verters which appear as real and reactive power sources andloads to the ac system.

Like a utility ac-dc power system, an industrial ac-dcpower system is comprised of an interconnected ac system.This ac system is most often radial. The dc portion of theindustrial power system differs from a utility system in twoimportant aspects. First, the dc system is usually a grid withmany interconnected, and often uncontrolled, rectifierswhich act as sinks of ac electrical power. Second, the loadson the industrial dc system are often dc motor loads ratherthan inverter loads.

For convenience, let the term "industrial ac-dc powersystem" refer to the case where all rectifiers are uncontrolled.Similarly, "utility ac-dc power system," as used here, willrefer to the case where all converters are controlled and all dcloads are inverter loads.

APPLICATION OF THE NEWTON-RAPHSON METHOD

It is convenient to consider the industrial ac-dc powersystem first since it represents a somewhat simplified case inwhich no converter control is required. The utility case willbe considered second and will be shown to be a generaliza-tion of the first case. In the subsequent analysis it is assumedthat all equations describing the system are expressed usingthe per-unit system developed.

The Industrial Power Flow ProblemThe converters of Fig. 3, as they represent an industrial

ac-dc power system, are uncontrolled. Formulation of thepower flow problem may be accomplished by initially ignor-ing (i.e., removing) the converter connections between the acand dc power system. The bus-power-equilibrium (BPE)equations which are needed to solve the separated ac-systempower flow problem are well known to be

,

Pp-jQp = E Ypq Eq, p = I * N (5)

where

Ypq = Gpq + jBpq (6)

N = number of ac system buses

and Ypq is the (p, q) element of the nodal admittance matrix.The separated dc system is also governed by (5) where all

voltages and power flows are real. The dc system BPE equa-tions are

M

Pt= Et E Gtq Eq,q=N+I

t=N+ 1, * M (7)

where

M - N = number of dc system buses

Pt = dc motor load at bus t

and Gtq is the (t, q) element of the nodal conductance matrix.Notice that for each ac-system bus added, two equations[i.e., the real and imaginary parts of (5)] and two unknowns,1EP4 and Op, are added. Similarly, for each dc system busadded, one real equation (7), and one unknown Et, areadded to list. Application of the polar form of the Newton-Raphson (NR) method to the BPE equations of (5) is wellknown to give a form which may be written symbolically as

,AQ 1 H IN1 AO1[AP= Jv L [AE

where

H Pp - Epl 2 Bpp-QHpq= - =

aoq - Im (Ep Eq Ypq)

-app - + IEpINpq OlEqj Re (Ep Eq Ypq)

lEq|aQ~ S|EpI2 Gpp + PpaOq -Re (Ep* Eq Ypq)

aQ l-IEPI + QP

LP - aEqI -Im (Ep Eq Ypq)

Eq|

(8)

p - q

p# q

(9a)

(9b)

p= q (I Oa)

p = q (lOb)

p = q

p. q

(lla)

(1 Ib)

p= q (12a)

p# q (12b)

and Pp and Qp are the calculated real and reactive bus powersbased on the assumed system voltage profile.

Similarly, application of either form of the NR method tothe BPE of (7) gives a form which may be written symboli-cally as

[,AP] = [N][,AE]where

MtU == (Et Gt1 +Et

aEuEu Gtu

(13)

t=u

t = u

(14a)

(14b)

and P1 is the calculated real bus power based on the assumedvoltage profile. Note that matrix equations (13) and (8) maybe combined while still retaining the symbolic form of (8).This type of equation will be termed the bus-power-mismatch (BPM) equation.The definitions of (9)-(12) and (14) show that the coeffi-

cient matrix of form (8) is simply the Jacobian of the ac anddc BPE equations with respect to the selected bus variables(i.e., El and 0 for polar form and Re{E}, ImnEl for rectangu-

33


lar form). A prerequisite for a unique solution of (28) is thatthe Jacobian be square. A square BPM equation is guaran-teed by the correspondence of unknowns and equations.Thus for the case of the two independent ac and dc systems,the BPM equation for ac busp connected to ac bus q and dcbus t connected to dc bus u is given by

A PpA QPA Pt

-

. 1

Hpp Npp 0

Jpp Lpp 0

0 0 Nit

AGp

A!EpA Ei

A0q

Al Eql

AEU

Hpq Npq... Jpq Lpq

0 0

0

0

Ntu

(15)

where it has been assumed without loss of generality thatbus t is ordered directly after bus p in construction of theJacobian.Now consider the effects of including the bridge converter

connections between the ac and dc systems. It may beassumed that there are any number of such connections.Rather than consider all interconnections, however, it is bestto consider one such connection since the process is identicalfor a11 connections.Assume that a rectifier connects ac bus p to dc bus t. The

J3PE equations which characterize these connected busesmust be modified to account for power flow between the acand dc systetns. Thus, for buses p and t the BPE equationsare

N

Pp -jQp = Fp* L Ypq Eq + IEpl Idp COS Opq=I

M

Pt = Et E GtqEq-E, Idp, (17)q - N + I

where

Idp = magnitude of the ac per-unit fundamental-component current flowing from ac busp to therectifier

= dc per-unit current (Id) flowing from the rectifierto bus t

cos fp = displacement factor associated with ac busvoltage Ep and ac fundamental-component cur-rent Idp.

Notice that one of the three terminal relationships previouslyderived, namely (2), has been used in (16). This allows theterms associated with the real and reactive power deliveredto the converter to be expressed as a function of the dcper-unit current rather than the magnitude of the ac per-unitfundamental-component current. Notice also that the varia-bles Id and 0 are assumed to be associated with ac bus p in(16) and (17). This assignment is arbitrary.The added real terms in (16) and (17) account for real

power flow from ac bus p to dc bus t. The imaginary termaccounts for reactive power flow from the ac bus to therectifier. From its ac terminals the rectifier looks like alagging power factor load.The modification of the BPE equations requires that the

partial derivative entries in the Jacobian be reevaluated. If() is used to indicate the entries which need to be modifiedin (15), then the modified entries are given by

app _Npp Epl lEpl Gp + 11+ Idp COS k,p

(18)= Npp + Idp COS 'p

PP- QIlPWEpI BPP + IEl+ Ipsin OpaEEpl p ±Idsi

-= LPP + Idp sin 4p-apt ~~Pt

Nt =- = Et G1t +--IpAE, E,~£= Ntt - Idp-

(19)

(20)

In addition to the modification of existing Jacobianentries, applying Taylor's theorem to (16) and (17) showsthat additional terms, which correspond to the addedunknowns Id and 4, must be accounted for in the BPMequation. These termns, which result after truncating higherorder terms in the Taylor series expansion, are shown in(2 1)-(23).

a P 0AiP =* +Aldp +-

ap

aIdp a3p

A Q = *- + ApIdp + M A$ +

AP, --- +-AIdp + * * * .aldp

(21)

(22)

(23)

(16) Thus, these additional terms along with the corresponding

34

-jlEpl Idp .in Op,


variable increments must be incorporated into the Jacobianand column vector of variable increments, respectively.Notice that this will lead to a nonsquare Jacobian with anonunique solution. This, however, should have been ex-pected since new variables have been added by the modifiedBPE equations [i.e., Idp and 'p in (16) and ( 17)] while addingno new constraints. In general, a unique solution of the BPMequation may be obtained by adding additional independentconstraining relationships between these added variables. Inpractice, the input-output characteristics of the converterconstrain the independent variation of Id and 0. This con-straining relationship must be accounted for if a realisticsolution of the ac-dc power flow problem is to be obtained.The terminal relationships necessary are taken from (1) and(4) as

Rip = Et- Epi cos 4.p (24)

R2p= E, -|EP cos a, + (rr/6)Xcp Idp (25)

where a = 0 for an uncontrolled rectifier and Xcp is thecommutation reactance associated with the rectifier trans-former at ac busp. Equations RI and R2 are termed residualequations because when all of the variables are known, eachequals zero. Like the BPE equations, R1 and R2 are non-linear equations. The handiest way to simultaneously solvethe BPE and residual equations is to use the NR method onthem.Expanding R1 and R2 about an assumed solution point in

a Taylor series and truncating higher order terms gives

AR, = O- R1 = dE+IOF+ + AIP(26)

OR2 O~R2 OR2A R2 =-R2 AF + ZAIEp1 + - AhIdp.aEt aIEplj aIdp(27)

Incorporating the modifications of the original Jacobianentries along with the addition of new variables gives therevised bus power and residual mismatch (BPRM) equationas

where

APP = - = EpI cos 'paIdp

Bpp =-O EP= Idp sin pIdp

Dpp= =- IFp Idp cos 'pdopDP= OR=E1|dPCS-

FPP = aFI =- Cos (APOREp,

KPP = Rp-EpI sin kpdEp

Mp=alEpl=-Rpp = =_(7r/ 6) XcpkIdp

Fp,- O, -1OR2,,

Mpt- -1OFt

A,,p F=a Et.aIdp

(29)

(30)

(31)

(32)

(33)

(34)

(35)

(36)

(37)

(38)

(39)

The residual portion of the BPRM equation is usuallyincluded as the last rows of the Jacobian to preserve sparsityand minimize fill in. For the sake of compactness, this is notthe case shown in (28).Once the BPRM equation is defined by applying the

method presented to every bus in the system, the solution ofthe power flow problem is straightforward. An initial esti-mate of the system unknowns is used to evaluate the Jaco-

Hpp Npp App Bpp O

Jpp Lpp Cpp Dpp O

O Fpp 0 Kpp Fp,t0 Mpp Rpp 0 Mp,0 0 A,,p 0 Nt,

Hpq Npq

Jpq Lpq0 0

0 0

0 0

0

0

0

,Aop

A Idp

ASEt

Aop,AlEql

AEu: _

(28)

35


bian entries and mismatches. Next, the variable incrementsare obtained using the desired solution procedure. Theseincrements are added to the initial estimates of the systemunknowns to obtain better estimates. The procedure con-tinues until the mismatches are sufficiently small.

Normally, the initial estimate consists of choosing all volt-age magnitudes and angles as 1.0 and 0.0, respectively. Agood starting point for 0 is 0.0. A good starting point for Iddepends on the specific industrial power system being stud-ied. Without previous experience, any small value for Id(i.e.,0.1-0.5) may be assumed and convergence will usually beobtained.The selection of system variables used in the Taylor series

expansion of the BPE and residual equations is somewhatarbitrary. For instance, an equally valid choice of variables isId and cos ¢.

The Utility Power Flow ProblemThe use of converters as terminals of a multiterminal dc

link requires that all converters be controlled via the delayangle a. Four types of converter control are common in HVdc systems. The extinction advance angle, dc voltage, dccurrent, or dc power may be scheduled at each dc converter.In practice, one dc bus is under constant extinction advanceangle or dc voltage control while others are either current orpower controlled.From an analytical point of view, the difference between

the utility and industrial power flow problem is that a is avariable which must be identified in the solution process.Since an additional variable is to be added to the system ofunknowns, either an existing variable must be eliminated oran additional constraining relationship added. The emphasisin the rest of this section is on showing how this may beaccomplished.

Before considering how a is identified for each type of

(25) will improve the linearity of the expansion. Using cos aas a variable, the truncated Taylor series expansion of (25)which is necessary for the case of controlled converters isgiven by

aR2p 3R2p _____EAR2p-= - AE + d A|E+ P A (cos a,)aE, Al EP 8(cos a,)

+PA Idp. (40)

Comparing (40) to (27) will show two important differencesin the expansion of R2p in the utility power flow problem.First, the partial of R2p with respect to |EP| will have adifferent form. This term corresponds to the Mpp entry in theJacobian of (28) and is given by

Mpp = alEp - cos at. (41)

The second difference is that an additional term which cor-responds to the variable cos a appears in (40) but not (27).This is the additional variable which must be identified in thesolution process. The way in which this may be handled foreach type of control is discussed in the remainder of thissection.

Extinction Advance Angle Control: When a converter isoperated as an inverter with a constant extinction advanceangle, a is fixed. Therefore, A(cos a,) = 0 in (40) and neednot be included as a variable in the BPRM equation. Thus,the BPRM equation which is characteristic of this type ofcontrol is given by (28) with Mpp defined by (41).

Voltage Control: Under voltage control, Et is fixed.Thus, AEt = 0 and need not be included in the BPRMequation. This eliminates one variable as required. Inclusionof A (cos a) as a variable from (40) and elimination of A E, asa variable gives the BPRM equation from (28) as

-~~~~Hpp Npp App Bpp O

Jpp Lpp Cpp Dpp 0

0 Fpp 0 Kpp 0

0 Mpp Rpp 0 Spt

0 0 Atp 0 0

control, it is important to determine if the addition of aaffects the portion of the BPRM equations already devel-oped. Inspection indicates that the variable a is found only in(25). Since cos a is the functional form in which a appears,use of cos a rather than a as a variable in the expansion of

where

Spct= o ) =- lEpl

and the other variables are defined by previous results.

,A PpA QOPAR1pA R2pA P,

= I . . .

Hpq Npq

Jpq Lpq

0 0

0 0

0 0

0 0

0 0

0 0

Ntu 0

Aop'Al EpA Idp

A(cos a,)

Aoq

Al EqI

A E,A(cos a,)

(42)

(43)

.a

L

36


Current Control: Under dc current control Id is fixed.Thus, AId= 0 and need not be included in the BPRM equa-tion. The inclusion of A (cos a) by (40) and elimination ofAId allows the revised BPRM equation to be written as

whereT- =-E_T

aIdp

A PpAQPA RipA R2pAAP

0pp Npp Bpp 0 O

Jpp Lpp Dpp 0 0= I . . . 0

0

0

Fpp Kpp Flp, 0

Mpp 0 Mpt SPtO O N,, O

where all variables in the Jacobian are defined by previousresults.

Power Control: Under dc power control Pdt = El Idpiscontrolled to be constant at a scheduled value Pst. Since A Pdis not already a variable, it cannot be eliminated from theBPRM equation. The Jacobian can be made square after theaddition of A (cos a) by adding an additional residual equa-tion which incorporates no new variables. This additionresidual is

R3p= Pt- Et Idp. (45)

A truncated Taylor series expansion of this residual gives

AR3 3p= AEt + AIdp. (46)oE, aIdp

Incorporation of this residual in to thegives

BPRM equation

Hpq Npq

Jpq Lpq0

0

0

0

0

0

0

0

0

00 0

0 0

Ni, 0

7

AOpAI Epl,AL,AsEt

A(cos a,)

.AOqAEq

AEuA(cos au)

-..

(44)

UP, =a

- -Idp.aEt (49)

The preceding discussion on the form that the BPRMequation takes under dc voltage, current, or power control,focused mainly on the changes in the BPRM equation.Comparing (28), (42), (44), and (47) shows that regardless ofthe control strategy, or lack thereof, the BPRM equation isvery similar in form. It should be noted that thep-q and t-ucoupling entries in the Jacobian matrix do not change as thecontrol of the rectifier changes.

Solution of the BPRM equation requires an initial esti-mate of the system state. This is somewhat easier for theutility ac-dc power flow problem than for the industrialproblem. Generally assuming all bus voltage magnitudes

Hpp Npp App Bpp 0 0

Jpp Lpp Cpp Dpp 0 0

0 Fpp 0 Kpp Fpt 0

0 Mpp Rpp 0 Mpt Spt0 0 Tpp 0 Upt 0

0 0 A,p 0 N,, 0

(48)

A PpA QP

A R2pA R3pAPt

= ...

Hpq Npq

Jpq Lpq

0 00.0

0 0

0 0

0 0

0 0

0 00 0

0 0

Ntu 0

Aop

A Idp

AL,

A Eq

ALqAsEq

AEu

A(cos au)

(47)

37


and bus voltage angles as 1.0 and 0.0, respectively, is a goodinitial estimate. Inverters typically operate with a constantextinction angle of yo -15 (i.e., a- 1650 ), while rectifiersunder current or power control operate with ao 150.Under voltage control, rectifiers operate with amin- 50. Byknowing the scheduled bus power a good initial currentestimate may be obtained. In addition, 0 may be assumed tobe equal to 0.0.

PEDAGOGIC CONSIDERATIONSThe material contained in this paper is predicated on a

good understanding by the student on the NR ac power flowformulation. The extension then of ac power flow equationsto cover the general dc power flow, as presented here, isstraightforward and follows naturally. The coupling of thesetwo systems of equations is usually the critical point in thedevelopment. It is important to emphasize that the con-straints which couple the behavior of ac and dc converterbuses must be taken into account if a realistic solution of theac-dc power flow problem is to be achieved. To show thatthese constraints are the converter terminal relationshipsrequires only a verbal argument. Consider the followingargument in which real power control is assumed at theconverter. The BPE equations of (16) and (17) guaranteeconservation of power at the ac and dc rectifier buses. If (24)is multiplied by Idp, then it represents a real power equilib-rium equation. This equation shows that since no real poweris consumed by the converter, the power delivered to theconverter equals that converted. This set of equations takentogether guarantees power conservation and the fact thatreal power delivered by one system equals that consumed bythe other. Equation (25) guarantees that this power balanceoccurs for voltages at the ac and dc rectifier buses which areconsistent with the converter terminal relationships. Now,for a given real ac power transfer the dc power is constrainedand the voltage at which it is delivered is constrained; hence,the direct current is constrained. Since the per-unit magni-tudes of the dc and ac currents are identical, the ac currentmagnitude is constrained. This fact along with the known acreal power transfer and ac-dc voltage relationship gives thephase angle of the current relative to the voltage. Hence, fora given ac voltage and power transfer all equations aresatisfied and only one solution is possible. The exact acconverter bus voltage is, of course, a function of the systemadmittances and system loads. Similar arguments can bemade for converters at which other modes of control areused.The next critical point in the development is the inclusion

of the nonlinear terminal relationships in the NR powerflow. Students familiar with the ac NR power flow solutionoften forget its origin. Thus, it is important to quickly reviewthe fact that it is a general method for solving nonlinearequations. Once this is reviewed, the inclusion of the con-verter residual equations via proper partial derivative entriesin the coefficient matrix falls out directly.

If the approach described is used, then the extension fromthe uncontrolled case to the controlled rectifier case via themethod described is relatively easy since it only requires thatone extra variable be accounted for in the Taylor series

expansion of (25). The controlled case is somewhat morecomplex than presented here since tap-changing trans-formers are used at the ac terminals of the converter. Controlof the transformer taps is used to maintain the ac terminalvoltage while allowing the firing angle to remain withinspecified limits. It is felt that at the introductory stage, theinclusion oftap-changing transformers adds an unnecessaryprofusion of "extra" variables and obscures the underlyingsimplicity of this approach. Once this material is understood,then other complexities, such as tap-changing transformers,can be added without undue confusion.Two assignments are usually made which help reinforce

the material presented. In the first assignment, the studentsare asked to come up with other sets of residuals which maybe used in place of (24) and (25). Since a hint is usuallyneeded, an example is usually given. Equation (25) is solvedfor Idp, and this result is substituted into (16) and (17) toeliminate one residual and one variable. It is quite surprisinghow many different residual sets can be found which can beused in the ac-dc power flow solution formulation.

In the second assignment, the students are asked to derivethe BPRM equation which corresponds to the residual andBPE equations obtained in the first assignment. In addition,at rather than cos °It is suggested as the variable to be used inthe Taylor series expansion. This assignment reinforces themethods used in constructing the Jacobian of the system ofequations and shows that the system variables are notunique.

CONCLUSIONS

The approach presented has the advantage of starting withthe ac power flow problem which is understood well by thestudent. Three simple steps are then taken to solve whatappears to be a more complex problem. The first stepinvolves showing that the form of the dc BPE equation isvery similar to, and in fact can be derived from, the ac BPEequation. In the second step, the converter terminal relation-ships are introduced and a verbal argument presented toshow why these equations are necessary and sufficient forobtaining a power flow solution. The third step involvesapplying the Newton-Raphson method to solve these equa-tions, a method which is well understood by the student.Taken one at a time the steps are extremely simple and leadto a solution of the more complex ac-dc power flowproblem.

APPENDIXTHE AC-DC PER-UNIT SYSTEM

The usual ac-system base quantities chosen in an ac per-unit system are

VAac BASE = VA 3 Phase

Vac-BASE - VLL

ac VAac-BASEac-BASE =/3 V\3 ac-BASE

Vac-BASE ( ac-BASE)2\ZA T\ ac-BASE VAac-BASE

(A.1)

(A.2)

(A.3)

(A.4)

38


where VLL represents the ac line-to-line voltage and thbol VA3phase represents the three-phase voltamperechosen. Denoting quantities with (A) as actualvalues, the current and voltage transformations ass(with a per-phase equivalent of a three-phase transforr

A A

El _ 2N1 N2A A

N II =N2 I2.

Implicit in the development of these equations is the ation that the turns-ratio is unaffected by the magnitudapplied transformer voltages and currents (i.e., the ation of linearity). The per-unit equivalents of (A. 5) anmay be written as

E=E2

11 -12.

Equations (A.7) and (A.8) show that the advantage othe ac per-unit system over using actual system value'the fact that the per-unit system scales all quantitiesphasor currents and voltages on both sides of theformer are the same in magnitude and phase. Tktransformer turns-ratios are scaled so that each is effe1: 1. For the purposes of calculating system voltagcurrents under normal operating conditions, the 1: 1former may be replaced by a circuit node.Now consider the three-phase full-wave bridge re

shown in Fig. 2(a), as a network element that, as

shown, has some similarities to a transformer. The terelationships for this circuit in actual units are

A A 3v',/ A 3 AA

Edc E2 IE(LL)I cos aC- - Xc Id

III Id.7r

Note that (A. IO) is only an approximate relationshipis accurate to within 0. 8 percent under normal conditthird equation which is also useful in defining the behzthe bridge converter comes from equating the real pcthe ac and dc sides of the converter. This expression

A A A A

3 Ei(LL)I I,I cos ) E212.

Substituting (A. 10) into (A. 1 1) gives

3 -x~f6 A A AA

EI(LL)I Id cos 2= £212.7r

Cancelling Id from both sides yieldsA 3v ICA£2 cos(L)CO .

Wr

ratio is fixed (i.e., a linear transformation), the voltage turns-ratio varies as 4 varies. Thus, the voltage turns-ratio repre-sents a nonlinear transformation and no one per-unit systemcan be chosen which will allow Fig. 2(a) to be reduced to acircuit node.

If (A.9) and (A.6) are compared, then the "ac-to-dc trans-former" appears as a lossy transformer with voltage turns-ratio 1: 7r/(3 \/2cos a) and dc side resistance 3/wr Xc. It mustbe emphasized that this dc side resistance is a fictitiousresistance which absorbs no real power. This resistance issimply the circuit analog for the voltage reducing role playedby the commutation reactance. When viewed in this way, thevoltage turns-ratio is linear but variable since the firing anglemay be changed. Hence, no fixed per-unit system can be usedto reduce Fig. 2(a) to a circuit node and series resistance.Since no per-unit system is capable of achieving the desiredreduction in circuit complexity, the dc system per-unit basequantities may be chosen arbitrarily provided they are inter-nally consistent. A convenient choice which permits theper-unit current and real power to be equal on both sides ofthe converter is

Pdc-BASE - VAac-BASE

Idc-BASE - vr/ \f6 Iac-BASE.

(A.14)

(A.15)

Equation (A. 14) permits per-unit real power on both sides ofthe rectifier to be compared directly since the base scalingfactors are identical. Comparing (A. 15) and (A. 10) alsoshows that a 1.0 per-unit ac current equals a 1.0 per-unit dccurrent. For example, if

ac-S=IE = 1.0 LIac-BASE

tA.) then

(A.10 Id =Id -AE ir XIf6aC-BASE

(A.1l)

(A.16)

(A. 17)

Selection of Pdc-BASE and Idc-BASE automatically determinesVdc-BASE and Rdc-BASE as

VPdc-BASE VAac-BASE

Vdc-BASE /'dc-BASE ir /'f61ac-BASE

= Vac-BASE (A. 18)7r

Vac-BASE

Vdc-BASE rV

(A. 12) Rdc-BASE =

Idc-BASE 7rIac-BASE

-\/6

(A.13)

If the bridge rectifier is viewed as an "ac-to-dc trans-former," then comparison of (A. 10) and (A.6) shows that thetransformer has a 1: \/6/ r turns-ratio with a complete lossofphase information. If(A. 13) and (A. 5) are compared, thenthe voltage turns-ratio appears to be 1: or/ (3 V6cos 4) with a

complete loss of phase information. While the current turns-

182 Zac-BASE.

7r(A.19)

Use of this per-unit system is best exemplified by finding theper-unit equivalent of (A.9). The first step is to divide bothsides of (A.9) by Vdc-BASE to get

--E s- 3V\IEl(LL)I (3/w) Xc2IdVdc-BASE 7r ( Vdc-BASE) Vdc-BASE

39

I


Substituting (A. 18) and (A. 19) into (A.20) gives

E3 \f2E2 = -if

3 A

- XC Id3 25 Rdc-BASE Idc-BAS

Vac-BASE7r

3 ^-xc

= lEil cos a - Id 182 Zc-BASE

7r

= lEIl cos a - - Xc Id.6

Note that in going from step I to step 2 in (A.21assumed that the Xc per-unit value will be expresseithe ac-base impedance rather than the dc-base resiThe choice of which base quantity to use in this case isarbitrary. Similar operations on (A. 10) and (A. 13) g

|I,| = IdE2 = |E,| cos ¢.

Equations (A.21)-(A.23) are the per-unit equationscharacterize the bridge converter. Kirchhoffs voltacan be used to verify that the electrical analog for pequations (A.21) and (A.22) is Fig. 2(b).

REFERENCES[I] Task Force on Power System Textbooks, "Electric powe

textbooks," IEEE Trans. Power App. Syst., vol. PAS-100, p4262, Sept. 1981.

[2] "1981-1985 research and development program plan-Prolscription," EPRI, Palo Alto, CA, Special Rep. P-1726-SR,

[3] 0. 1. Elgerd, Electric Energy Systems Theory: An IntroductioYork: McGraw-Hill, 1971.

[4] C. A. Gross, Power System Analysis. New York: Wiley, I[5] W. D. Stevenson, Elements of Power System Analysis. N(

McGraw-Hill, 1975.[6] B. M. Weedy, Electric Power Systems. New York: Wiley,[7] A. Fouad and P. Anderson, Power System Control and.'

Ames, IA: Iowa State Univ. Press, 1977.[8] R. L. Sullivan, Power System Planning. New York: McGi

1976.[9] H. Brown, Solution of Large Networks by Matrix Method

York: Wiley, 1975.[10] R. B. Shipley, Introduction to Matrices and Power System

York: Wiley, 1976.[11] G. W. Stagg and A. H. El-Abiad, Computer Methods in Powe

Analysis. New York: McGraw-Hill, 1968.[12] R. A. Hore, Advanced Studies in Electrical Power System

London: Chapman and Hall, 1966.

[13] E. W. Kimbark, Direct Current Transmission, vol. 1. New York:Wiley, 1971.

[14] E. Uhlmann, Power Transmission by Direct Current. Berlin:Springer-Verlag, 1975.

[15] B. J. Cory, High Voltage Direct Current Converters and Systems.E London: Macdonald, 1965.

[16] C. Adamson and N. G. Hingorani, High Voltage Direct CurrentPower Transmission. London: Garraway, 1960.

[17] H. Sato and J. Arrillaga, "Improved load-flow techniques for inte-grated ac-dc systems," Proc. IEE, vol. 116, pp. 525-532, Apr. 1969.

[18] R. Miki and K. Uemura, "A new efficient method for calculating loadflow of AC/ DC systems," Elec. Eng. Japan, vol. 95, no. 1, pp. 81-87,1975.

[19] J. Reeve, G. Fahmy, and B. Stott, "Versatile load flow method formulti-terminal HVdc systems," IEEE Trans. Power App. Syst., vol.PAS-96, pp. 925-933, May-June 1977.

(A.21) [20] C. M. Ong and A. Hamzei-nejad, "A general-purpose multi-terminaldc load-flow," IEEE Trans. Power App. Syst., vol. PAS-100, pp.3166-3174, July 1981.

1), it S [21] B. Stott, "Load flows for ac and integrated ac-dc systems," Ph.D.J using dissertation, Univ. Manchester, Manchester, England, 1971.

stance. [22] D. A. Braunagel, L. A. Kraft, and J. L. Whysong, "Inclusion of dcconverter and transmission equations directly in a Newton power

purely flow," IEEE Trans. Power App. Syst., vol. PAS-95, pp. 76-88, Jan.-ive Feb. 1976.

[23] G. B. Sheble and G. T. Heydt, "Power flow studies for systems with(A.22) HVdc transmission," in Conf Rec. IEEE Power Industry Comput.

Applic. Conf, 1975, pp. 223-228.(A.23) [24] J. Arrillaga and P. Bodger, "Integration of HVdc links with fast-

decoupled load-flow Solutions," Proc. Inst. Elec. Eng., vol. 124, pp.which 463-468, May 1977.

dge law [25] J. Arrillaga, B. J. Harker, and P. Bodger, "Fast decoupled load flowalgorithms for ac/dc systems," in Conf: Rec. IEEE PES Summerer-unit Meeting, July 1978, Paper A-78-555-5.

[26] M. M. El-Marsafawy and R. M. Mathur, "A new fast technique forload flow solution of integrated multi-terminal dc/ ac systems," IEEETrans. Power App. Syst., vol. PAS-99, pp. 246-255, Jan.-Feb. 1980.

[27] D. J. Tylavsky and F. C. Trutt, "Inclusion of a resistance-inductance-

r system fed bridge rectifier bus in a Newton-Raphson load flow," in Conf. Rec.in425e- IEEE-IAS Annu. Meeting, Oct. 1982, pp. 415-420.

gram de-1981.

n. New

1979.ew York:

1972.gtability.raw-Hill,

is. New

is. New

er System

Design.

Daniel J. Tylavsky (S'77-M'77-S'80-M'82) wasborn in Pittsburgh, PA, on August 24, 1952. Hereceived the B.S. degree in engineering science in1974, the M.S.E.E. degree in 1978, and the Ph.D.degree in electrical engineering in 1982, all fromPennsylvania State University, State College, PA.

In 1974 he was with Basic Technology Inc.,Pittsburgh, PA, as a member of the Thermal andMechanical Stress Analysis Group. In 1978 hejoined the faculty of Penn State as an Instructor ofElectrical Engineering. In 1982 he assumed his

present position as Assistant Professor of Engineering in the Department ofElectrical and Computer Engineering at Arizona State University, Tempe,AZ.

Dr. Tylavsky is a member of Phi Eta Sigma, Eta Kappa Nu, Tau Beta Pi,and Phi Kappa Phi.

40

ac-dc power flow calc

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