The Calculation of DC Fault Currents With Contributions From DC Machines and Rectifiers

1 7AN t 1i<ANSAC1TIONS ON INDU)STRY APPlICATIONS. VOL IA-21, NO. 1. JANUJARY/FEBRUARY 1985

The Calculation of DC Fault Currents with Contributionsfrom DC Machines and Rectifiers

DANIEL J. TYILAVSKY, MENMBER, lEhE

Abstract-Since 1970 at least 12 menl have died from underground coalmine fires which were started due to undetected high-resistanice dc faultson trolley systems. A technique is proposed which utilizes Newton'smethod to solve the nonlinear equations which determine such faultcurrents. The formulation is unique in that it is the first known method toaccount for de fault current contribuitions from both rectifiers and dcmotors. Results from the technique can be used to size and coordinatedevices used for system protection. Ilie problems of accurately predictingsuch fault currents is discussed. A sample problem is included as anexample.

l. INTRODUCTIONfEFORE the availability ot iniexpensive and reliable SCR's,

ac/dc power conversion was m)ost ofteni accomplishedthrough the use of miotor-generatoi sets. "[he miethods forcalculating dc short circuit currents for systemiis consisting ofonly motors and generators is straightforward and has beenavailable for some time [I]. Today, combined ac/dc industrialpower systems, such as minle electrical power systemns, are fedby multiple converters and serve imiultiple dc motor loads.Previous methods developed for ttault currenit calculations arenot capable of handling modern systemns for various reaso-ns.Early work on rectifier dc fault curretnts was based on theassumlption that only one rectifier was presenlt, that the tfaultoccurred at the rectifier termiinals, anid that all fault currenttclame only from this rectifier [21, 131. More recent work takesinto accounit dc side resistance and induIctanice but is stilllimited by the one-rectifier, onee-ftault-cnirrenit source assumiip-tion [4]. The reascendance of high-voltage direct current (HVdc) power transmission in the formii of two terminal dc linikshas focused some attention otn two-converter systems. Faultcurrent in these systems is limited by the surge impedance ofthe transmission lines. When the fault occurs at the terminalsof the controlled rectifier, tlhe rectifier acts as a source of faultcurrent for oinly a short time since fast control is used to blockthe rectifier valves quickly [51-171 Hence the techniques usedfor HV dc fault current calculations are limited by a one-rectifier assumption and caninot he used in ain environnmentwhich utilizes multiple uncoontrolle(d rectifiers.The recent deaths of men fron unidergrouLnd coal mine fires

started by utidetezted high-resistance dc faults has promptedmore research in this area [81. Since dc load and fault currentsare of the same order of miagnitude for high-resistance dcfaults, one approach to designinig a miiore reliable dc protection

Paper P1D84-12, approved by the Minling Industry Comimnittee of tihe IEEEIndustry Applications Society for presentation at the 1983 Industry Applica-tions Society Annual Meeting. Mexico City, NMexico, October 3-7, 1984.Manuscript released for publication April 27. 1984.The author is with the Departmllent of Electrical and Computer- Engineering,

College of Engineering. Arizona States University, Temipe, AZ 85287.

system has been more accurate determination of dc systemload and fault currents. Load currents on the dc system aredetermined by means of a multiterminal ac/dc power flowprogram. Such computer programs have been written formultiterminal HV dc links [91--[121. These, however, lack thecapability of including dc motor loads. Recently, three power-flow computer programs have been written which have amitultiterminial and real-power load-modeling capability. Twouse a Gauss-Siedel approach [131, [141 while the other usesthe Newton-Raphson approach [151. Both methods are capa-ble of accurately defining ac/dc system load currents and theprefault voltage profile of the system.A method for accurately determining the dc system fault

current in a multirectifier fed ac/dc power system has onlyrecently become available [141. Since, unlike the ac faultcurrent problem, the defining relationships in an ac/dc powersystem are nonlinear, an iterative approach to the solution ofthese equations is required. The Gauss-Siedel approach isused in [141.The approach for the calculation of dc fault current

proposed in this paper differs from that of [141 in threeimnportant respects. First, the nonlinear equations which definethe ac/dc system under faulted conditions are solved usingNewton's method. Newton's method is known to convergereliably and quickly if a good initial estimate can be made ofthe solution vector and if the nonlinearities in the equations arenot too bad. Seconid, the proposed method is capable ofincorporating fault current contributions from both rectifiersand dc motors. Third, the size of the matrix used in theiterative procedure can be made proportional to the number ofrectifiers in the system. Since the number of calculationsneeded to solve a matrix equation by lower diagonal upper(LDU) decomposition is proportional to the cube of the size ofthe matrix, this greatly cuts down on the computation timerequired.The development required for explaining the proposed

method may be broken into six sections. The first section isused to develop a model for the uncontrolled three-phasebridge rectifier. An uncontrolled rectifier is assumed since thisis typical in most coal mine applications. Next, a brief reviewof ac system three-phase fault analysis is conducted toestablish the notation. In the next section, a model of a dcsystem consisting of only motors and generators is shown toobey the same matrix equation under faulted conditions as theac system. The fourth section is used to show how the ac/dcconverter interface may be modeled in the ac and the dcsystems. The equations describing the combined ac/dc systemunder dc faulted conditions are developed. In the fifth section

0093-9994i85/0100-0170$0l1.00 C 1985 IEEE

'I 70

TYLAVSKY: CALCULATION OF DC FAULT CURRENTS

Xc I = Id

E E

Fig. 1. One-line diagram of rectifier conduction.

Newton's method is applied to these equations. The finalsection includes a sample problem which provides an exampleof the application of this method. Unless specifically statedotherwise, the equations developed in all sections will use per-unit variables. The ac/dc per-unit system used may be foundelsewhere [15].

II. RECTIFIER MODELMany types of rectifiers may be used to derive dc power

from ac. Three-phase bridge rectifiers are by far the mostcommon type used in coal mine applications and hence will bethe only type considered here. Letting the symbol (^) indicatean actual rather than per-unit quantity, Fig. I shows a one-linediagram of an ac bus connected through an inductive reactanceX; to a bridge rectifier which supplies power to a dc bus. Assuch, the rectifier has three distinct modes of operation, andeach mode is characterized by a different set of equations. Itcan be shown that the per-unit expressions characteristic ofeach mode take on the general form of

E2=a,JE,I +IdRc. (1)The values that constants a, and RC take on are listed in TableI. Inspection of (1) shows that the dc side per-unit equivalentof Fig. I is Fig. 2.

In the normal range of operation (i.e., mode 1), theapproximation

Nf1l6- / I (2)is usually made. The maximum error of (2) associated witheach mode of operation occurs at the operating points on themodal boundaries (i.e., (a, it) = (0, 60°), (30°, 600), and(30°, 120°) where a is the commutation delay angle and it isthe commutation overlap angle). The maximum errors formodes 1, 2, and 3 are 3.16, 4.39, and 10.26 percent,respectively [161, [17]. The 10.26-percent error correspondsto the conditions of a bolted fault at the rectifier terminals. Ifthe dc fault current is limited to 90 percent of the bolted faultvalue, then the error drops to less than six percent. The use of(2) in estimating dc fault currents is conservative since -'/lrIdis always greater than or equal to IILI f. Approximation (2) willbe used with the assumption that this error limit is acceptable.A real power balance at the ac and dc buses of the rectifier

can be used to show that the displacement factor is given, forall modes of operation, by

E2cos +-= . (3)

When E2 is not known, it is possible to find expressions for thedisplacement factor in terms of IE, and I2. These expressionsare different for each mode of operation and are listed in Table1. Also listed are the ranges in cos 4 over which theseexpressions remain valid.

III. AC SYSTEM FAULT ANALYSISConsider a K-machine M-load pure ac-machine system. For

the calculation of fault currents for simultaneous three-phase

TABLE IRECTIFIER MODEL CONSTANTS

Mode a Rc cos ¢ Cos ax)1 1.O 7r/6 Xc - aX X2 0°750

2 cosfsinj1FXc21 3001 1/6 Xc a-I 0.433

3 vr-3 7r/2 xc ^ 1T Xc 12 0.02 JE1f

i=11|I,j/ I -III RC

IEII -ElE (i) aftE| E2

Fig. 2. Equivalent circuit for ac to dc converter.

El(O)

z2-i

Ek(O)

Zk-~~~~~

INTER-CONNECT IONNETWORK

m+11m+2

m+2S

nn

Fig. 3. Equivalent circuit for fault current calculations.

faults on buses m + 1, m + 2, * , n, the equivalent circuitof Fig. 3 may be used. In Fig. 3 each machine is replaced byan ideal EMF source in series with an impedance elementwhich corresponds to either the synchronous, transient orsubtransient reactance. Also, all nonrotating loads are ne-glected since they draw negligible current under faultedconditions. Under faulted conditions, the ac system perform-ance equation is

Eacbus(f)- Eacbus(O) - ZacbusIacbus(F) (4)where

Eacbus(O) vector of prefault ac system bus voltages,Eacbu(p) vector of ac system bus voltages during the fault,Zacbus ac system bus impedance matrix,Iacbus(F) vector of ac system fault currents.

If faults are assumed to occur only on buses m + I through n,then labeling the other ac system buses of interest as 1, 2, - *,m gives the matrix form of the performance equation as

[El )E2(fi - Em(E+ I() ...- En(Pj I

[El(o)E2(o) .. Em(O + 1(0) ... En(o)J T-

Z2,7rz.1z,l,2K7

Zm,lZm+ .li

Zn,"

. .. Zi,mZim+i .

... Z2,mZ2,m + I

Zm,mZm,m + I ...

... Zm+ i,mZm+ Im+ I

..- Zn,mZn,m+ I

Zl,n

Z2,n

ZZm,n

Zn,n

00

01

In

(5)

171

IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. IA-21, NO. 1, JANUARY/FEBRUARY 1985

IV. DC SYSTEM FAULT ANALYSIS

It has been shown that, under faulted conditions, a dcmachine model consists of an ideal EMF source in series withan effective resistance [1]. Thus the equivalent circuit for faultcurrent calculations is also shown in Fig. 3. By analogy withthe ac system just discussed, the dc system performanceequation under faulted conditions is

Edcbus(F) = Edcbus(O)-Rdcbusldcbus() (6)where

Edcbus(O)Edcbus(F)RdcbusIdcbus(F)

vector of prefault dc system bus voltagesvector of dc system bus voltages during the fault,dc system bus resistance matrix,vector of dc system bus fault currents.

If current is injected at some dc system buses during the fault,then this current will have an effect on the dc system voltagesopposite to that of the fault current. The performance equationunder the combined current injection and fault currentconditions is, by the assumption of linearity of the dc system,

Edcbus(n = Edcbus(O)- RdcbusIdcbus(n

+ RdcbusIdcbus()- Edcbus(o)- Rdcbus(Idcbus(F- 1) (7)

where

Idcbus(G) vector of injected dc system bus currents'dcbus(F- ) combined vector of injected and fault currents in

the dc system.

Let the dc buses be numbered n + 1, ,p,p + 1, , u -

1, u, u + 1, * , q. Further, let current injections occur atbuses n + I through p while bus u is the faulted bus. Clearly,many of the bus currents will be zero. Specifically, thefollowing bus current vector is zero:

[Ip+, * , I-l IU+ 1 T Iq]T= . (8)

With this in mind, the matrix form of the dc performanceequation is

[En+ I(F) ... Ep(Ep+I,*Egj. *@.. Eq(F)lI[En+ I(0) ... Ep(O)Ep+ I(0) ... Euo) ... Eq(O)]-

Rn+ 1,n+ 1

... Rpp,p + I

* * Rp+1,pRp+ I,p+IRp,n + I

Rp+ l,n+lI

Rq,n+ 1 ... Rq,pRq,p+I

... Rp,q

... Rp+I,q

* Ru,q

Rq,q... Rq,u

7n\+i n+2 /7p

DC POWER SYSTEM u

El HARMONIC FILTER R f

Fig. 4. Generalized ac/dc power system.

V. COMBINED AC-DC SYSTEM FAULT ANALYSIS

Consider the generalized ac-dc power system shown in Fig.4. The branches between ac buses m + 1, m + 2, * * *, n, anddc buses n + 1, n + 2, * * *, p represent power rectificationvia uncontrolled three-phase bridge rectifiers. The idealharmonic filters at the ac buses serve to limit the commutationreactance to that of the converter transformer leakage reac-tance plus any reactance of the lines connecting the ac bus tothe converter transformer. Commutation resistance is assumedto be negligible. This assumption is often acceptable and, inany case, will yield conservative fault current results.

If the rectifier branches are removed from Fig. 4, thenclearly fault analysis of each independent system may proceedas described in the previous sections. The Zacbus and Rdcbu,matrices used in the independent fault analysis would reflectthe network impedances contained entirely within the respec-tive boundaries of the ac and dc systems as shown in thisfigure. If these branches are present, then any current whichflows through the fault resistance at bus u acts as a load on theac system as well as any dc machines capable of feeding thefault. Using the ac and dc network models developed in theprevious sections, along with the rectifier model previouslydeveloped, an equivalent circuit for the ac-dc system underfaulted conditions is shown in Fig. 5. Note that the numberingof the ac and dc system rectifier buses corresponds to thenumbering used in Sections III and IV for ac system faultedbuses and dc system current injection buses, respectively.Thus (5) and (9), when taken together, also describe thecombined ac-dc system under faulted conditions. Taking aquick tally of the number of equations and unknowns to insuresolvability shows that (5) contributes 2n - m complexunknowns (i.e., n complex Eacbus(I; values and n - m nonzerovalues of Iacbus(F)) or 4n - 2m real unknowns. However, (5)only contributes n complex equations or 2n real equations.Similarly, (9) contributes q - n unknown real voltages and p- n + 1 unknown real currents. Realizing that, to each acrectifier connection there exists a dc rectifier connection, thetotal number of rectifiers is given by

(9) n-m=p-n.

171

Reli

... Rp,u

... Rp+ I,u

Ru,n + I ..*. Ru,

Xl_jn+, - - - _IPo ... I ... ol T. (10)


El(O) Ep .+ i(°)

Fig. 5. Equivalent circuit for dc fault current calculations with ac rectifier contribution.

Hence (9) contributes q - m + 1 unknowns but adds only q- n equations. Since m is less than n, (5) and (9) have moreunknowns than variables. Subtracting real equations from realunknowns leaves a deficiency of 3(n - m) + 1 equations.The additional constraining relationships which are needed arethose that relate the ac and dc terminal behavior of therectifier. These relationships are defined by (I)-(3) as

En+, =am+jjEm+jj -In+,Rcn+l

Ep =a.IE.1and

(1 1)

In+l = I'm+11

Ip = |Inh (12)and

En+1=IEm+iI cos Om+i

Ep =E,E cos On, (13)

respectively. Note that (11)-(13) each consists of n - mequations. Including these into the system of descriptiveequations reduces the deficiency of equations to one. The lastequation needed relates the dc fault current to the voltage at thedc bus as

Eu(, = IfRg- (14)

Equations (5), (9), and (11)-(14) constitute 4n - 3m + q +1 equations in as many unknowns. At this point any solutiontechnique capable of solving simultaneous nonlinear equationsmay be used. In the next section the application of Newton'smethod to this problem is discussed.

VI. APPLICATION OF NEWTON'S METHODThe application of Newton's method to a set of simultaneous

equations results in two linearized matrix/vector equations of

the form

AR = JAX

X=X+AXwhere

(15)

AR vector of residual mismatches between the knownequation values and those found for the solutionestimate X,

I Jacobian of the system of equations with respect to thesystem variables evaluated at the solution estimate,

I AX variable increment vector needed to correct thesolution estimate X of the linearized problem,

X better estimate of the solution vector.

The application of this method to the problem at handrequires that certain decisions be made. First, polar orrectangular form of the matrix equations may be used. Thisdecision will impact the form of the Jacobian and thereforewill impact the number of computations needed to invert theJacobian. Second, it is possible to incorporate all of (5), (9),and (1 )-(14) into the form of (15), or it is possible to select asubset of these variables and hence reduce the size of theJacobian. The two extremes of the many possible ways toattack this problem are discussed.

Method 1: Full Equation Set Solution

One approach to the problem is to linearize (5), (9), and(11)-(14) in the form of (15) and solve for all of the busvoltages and rectifier currents in the system. The dimension ofthe Jacobian used in this solution process can be somewhatreduced if use is made of the fact that many of the impressedbus currents are known to be zero. If (5) and (9) are partitionedto use this fact, they take on the following form, respectively,

Em()

Em + I (F)

El(o)

Em(O)

Em+ 1(0)

En(o)

11 12Zacbus Zacbus

21 22Zacbus Zacbus

0

0

Im+ I

In

(16)

i

173

-IpRcp,


1p(0)

Ep_ I(O)

Eu-(o)

Eq(O)

En+ (o)

Ep(o)

Eupo

11 12Rdebus 'Rdcbus

13Rdcbus

I

21 22 23Rdcbus Rdcbus IRdcbus

l l~~~~~~~~~~~~~~~~~~~~

31Rdcbus

32 7:Rdcbu.

33Rdcbus

0

00

0

- In + I

-Ip

If

(17)

Eliminating the portions of the Zacbus and Rdcb,S matrices whichare multiplied by the zero current entries in (16) and (17) andcombining the results while rearranging gives

Ei(o)

Em(O)

Eu.+ l(o)

Eq(O)

Em+ 1(0)

IEn(c)

En+ 1(0)

Ep_ I(o)

Eu(o)_

12Zacbus 0

12 13° dcbus, Rdcbus

I

22Zacbus: 0 0

22 :230 :I Rdcbus Rdcbus

_ . .~~~~~~~~~~~~~~~~~~~~~~~~

32 :33Rdcbus: Rdcbus I

Im+ I

In

In+1

Ip

_ If

Written as a residual, (19) may be expressed as

Ri= - EbUS(F) + EbUS(O)- ZbusIbus(F) (20)Recognizing that if (20) is kept in rectangular form, it is alinear equation and may be linearized in the form of (15) as

[AR, ARmIARp±I ARI. 1UARu+ I ARqIARm+ 1

... ARn ARn +.1.. ARPJIARu]"T

l- -01

1

12

ACBUS

0

22

ACBUS

_ I

0

R12

DCBUS

0

22MDCBUS

- - r

0

13RDCBUS

0

23DCBUS

R 33RDCBUS

_In__-4 0

* _ _

I- -

I

.

I-_- -- 1L4-Ix[AE, ... AEm IAEp+ I ... AEu - AEu +1 ... AEqIAEm + I

* AEnlAEn+ ... AEp I AEuIAIm+1

AInIAI AI* I (21)

Equation (21) may be written symbolically as

(22)

where

[AR1j vector of linear residual mismatches,AET(P) vector of bus voltage increments associated with

buses which have (no) rectifier connections (i.e.,transfer (pure) ac and dc system buses),

ITF vector of bus current increments associated withcurrents at the transfer and faulted buses.

If (-) is used to indicate the variable estimates at the

beginning of each iteration, then the linear residual mismatchvector is given by

ARlin= Ebus(F) EbUS(O) + Zbusibus(F). (23)

For convenience, let (18) be written as

Ebus(F)= Ebus(o) Zbuslbus(F).

(18) As mentioned in the previous section, there are 3(n - m) + 1

more unknowns than equations in (18). The equations to beadded, (11) and (13), use variables which correspond to ac buscurrent and voltage magnitudes. Since rectangular form was

(19) assumed in the construction of (21), these quantities must be

. (1)

Eq(,f)

En+ l(F)

Ep(F)

Eu(F)

0 1 i. I. I I

II11 0.*.

1 1

I.I0S__ 1 _

II 1. '

EI(F)

Em(F)

Ep+ 1(F)

EU_ I(F)Eu+ 1(F

Eq(fl

Em+ 1(F)

IL

174

-4

AE-Vol p

[ARlin]- Zbus0 I: T:i[TF


derived through the use of additional residuals. Letting

Imx = Re [Im]

,my = Im [VmlEmx = Re [Em]

Emy = Im [Em], (24)

then the nonlinear residuals needed to constrain the systemequations at the (m + 1) ac rectifier bus are

Rq+ = -In+ 2+Im+lx2+Im+ly2

Rq+2 - Em+ 12+Em+lx2Em+ly2Rq+3 -En+l+ |Em+1 cos m+lI

(25)

(26)

values. When the solution estimates are accurate, each of theresiduals given by (25)-(30) should be ideally zero.The last residual needed corresponds to the fault constraint

at the faulted dc bus. It is given by

Rq+6(m-n)+ I -Eu(F)l+ IfRg (36)

where Rg is the resistance of the fault. The linearized form ofthis residual is

ARq+6(m-n)+ 1= [-1 Rg] [ jIu]

with

(37)

(38)(27) ARq + 6(m - n) + 1 = Eu-IfRq,Rq+4=-En+, +am+IlEm+l1-In++Rc(n+l) (28)

Rq+5-sIm+Ix+In+I cOs (km+I-km+i) (29)

Rq+6=-Em+,x+ jEm+Il cos i,m+1. (30)

In the construction of these residuals the ac voltage at bus m +1 is assumed to have a phase angle of {m + 1 and the ac rectifiercurrent at this bus is assumed to lag the voltage by 4m±+ 1. It iseasily verified that this set of residuals is not unique in kind ornumber. For example, it is possible to rearrange (25) andsubstitute it into (28) and (29), thus eliminating I,+ l. Further,Rq+ 5 may be replaced by

Rq+5=-Im+ly+In+l sin (ikm+,-m+,). (31)

Thus many ways exist to formulate the residuals. Usingresiduals (25)-(30), their linearized form is

[ARq+l ARq+2 ARq+3 ARq+4 ARq+5 ARq+6] T=

If (22), (32), and (37) are combined, then the general form ofthe residual mismatch equation is

Zbus

B

F

0

C

0

AE-

AET

AITF

AU

I16Rlin 0 I

ARnln 0 A

ARF L.D E

(39)

where

ARl,, vector of nonlinear residual mismatches,ARF = ARq+ 6(m - n) + 1, residual mismatch of the faulted

bus constraint,AU vector of added variable increments, and

000

21m + Ix

2Im + ly- 2In + I

00

- 0

2Em+ ix2Em + Iy

0000

-2IEm+ll00

00- I000

Cos Om+ I

e0

00- I00

-Rc(n+ I)am + I00

000-10b0c

-c

-100000

Cos I

0d

X [AEm+ix AEm+ ly AEn+1 AIm+ix AIm+Iy AIn+I

AIEm+jt A4Om+j AO6m+lT (32)

where

b = cos (Om+,I-¢m+,1) (33)c=In+l sin ({m+I-4m+,) (34)

d=I-Em+,I sin Vm+I (35a)

e=-IEm+iI sin Om+,. (35b)

The residual mismatches AR are given by the negated valuesof (25)-(30) when all variables are replaced by their estimated

T AU=[AIEm+lI A4om+l A1'm+l ... AIEIAn Ain']T (40)

The form of the A, B, C, D, E, and F matrices is easilydetermined using (32) and (37).The nice feature of using this method is that when

convergence is attained, all of the bus voltages are known aswell as the rectifier currents and fault currents. Further, ifcomplex variables are used, the Zbus block of the coefficientmatrix in (39) may be constructed using well-known tech-niques. The drawback of this method is that the size of thecoefficient matrix that needs to be inverted at each iteration isproportional to the number of ac system buses plus the numberof dc system buses plus six times the number of rectifiers inthe system. For a four bus ac/6 bus dc power system with tworectifiers this would require a coefficient matrix which is 27real elements square.

Method 2: Minimum Equation Set SolutionThe objective of reducing the coefficient matrix of (15) is to

minimize the computation time of each iteration. To see howthis may be done, consider (16) and (17). The rectifiercurrents Im+ through In have the effect of reducing the ac

.I

L

175

1EF7E TRANSACTIONS OsN INDUSTRY A.PPLU'(- A FIONS., ViM.-. -) l. ;IANUARY FEBRUARY 9s.s

system bWs voltages and increasing the dc system bus voltages.These currents are independent variables while the voltagesare dependent variables. Clearly, any choice of currents willsatisfy (16) and (17). The unique solution to the dc faultproblem is achieved when the currents are choosen so that theac and dc system rectifier bus voltages obey the rectifierconstraining relationships. Thus only rectifier bus voltagesneed to be included in the solution procedure since any choiceof system currents will satisfy the performance equations forthe nonrectifier bus voltages of (16) and (17). If only thesevariables are included in the solution procedure, (15) and (16)can be reduced to the following form:

Elti+ I( Em+l(o) lF 1 Fin I

J-[£, ~ ( O ,J a Jb[s

1+I (F) En 1(0)

EP(O)

Eu(o)

22 23- Rdcbus, Rdcbus

32-RdCbUsR. u

tIn +I

LInf

If

(41)

. (42)

i44) to simplifx' the notation. Sinmllariv. the residuals associ-ated with (42) Lire

Rkz=-EA. + Eko±+ RAldI- E RkArlIr -(I- mn),Ir - n -

(45)

RU- + EU + Eu(o) + RuIlI1f- I RiirlIr- (rl - tl1- -- rz 4

(46)

where Rkp is the (k, p) entry in the Rbus matrix of (42). Theresiduals associated with the rectifier constraining relation-ships are

RkI- EA + (n -n),f+ EAI cos 4k, k-=n+ 1, in+2, , n

Rk2 =-Ek+ n +attE)+klA-1- k+nnRck+(n-rin),. k=rn+1, rn+2, -., n.

(47)

(48)The residual associated with the fault constraint at the faultedbus is given by

R,3=- Eu + fR,. (49)

The linearized form of these equations may be written as

Note that (12) has been used in constructing (41) and (42). Ifpolar form of these equations is used, then the only additionalresiduals needed are ( 11), (13), and (36). Counting realunknowns and equations shows that there are 5(n - m) + 2of each.The use of polar rather than rectangular form of the ac

performance equations allows the reduction in the number ofresiduals per rectifier from six to two. A computational priceis paid for this reduction. Since (41) is not a linear equation inmagnitudes and angles, construction of the Jacobian entrieswhich correspond to increments on these variables will requireextra computation time. In addition the linearity feature whichis important to the convergence of Newton's method issacrificed when polar form is used.

If superscripts of x and y are used to indicate real andimaginary components, then the residuals associated with (41)may be written

n

R x- -Ek cos O4k+Ek(o) COS Vk(O)- , lZkPIiIPtX= n +

COS (Okp +±lk-kk), k-m + 1, m + 2, * n*,n (43)

Rky= -Ek sin 4k +Ek(o, sin bk(O) E IZkplIlpIXp=tn + I

sin (Okp+4'p- p), k=m+1, m+2, nn (44)

where

lZkpl magnitude of the (k, p) entry in Zacbus,4'kp angle of the (k, p) entry in Zacbus.

Note that the (F) subscripts have been dropped from (43) and

[ARtt? + 1' AR,tr 1' ?,,AR,-1'' Rn +I

*. ARi RU AR,?,,,' AR11+12+* ARn I ARn2 ARI31=[J1X

[L/ Ernj+11 AJIt71+11 Aotwl5 n1+1

... AlEXI AlI,, A4',l A)n AE,,+41* AEp AEuAI '. (50)

The entries needed to construct the Jacobian are merely thepartials of the residual equations with respect to the incrementvariables. These partials are listed in Appendix I. The size ofthe Jacobian is 5(n - m) + 2. For a four bus ac/6 bus dcpower system with two rectifiers, this means a 12-row matrix.If LDU decomposition is used at each iteration, the minimumset will require only about 8.5 percent as much computation asthe full equation set. The total reduction in computation is notquite as good as this since once convergence of the unknowncurrent vector is obtained (5) and (9) must be used if theunknown system bus voltages are desired.

VII. SAMPLE PROBLEMFig. 6 is a per unit impedance schematic of a mine electrical

power system. All impedances are in per unit using a 1500-kVA 7.2-kV ac system base. The 333-kVA induction machinedraws a 200-kW, 0.6 lagging power factor load and is suppliedthrough a 7200:480-V 750-kVA transformer. The rectifiersare supplied by 750-kW transformers which step the voltagedown to 200 V. The dc system operates at a nominal systemvalue of 300 V dc. Each dc motor draws a 200-kW rated load.Bus 1 is the utility connection point.

Normally, the first step in finding the fault currents in the

1 '76

. k n + I , tl 4- -2, - -, p


20014 5 + j .0505

SLACK BUS

V=1.0+jO.O [

7.433 .216

-6

0.133

JRI

3 4.0058 +j002 .2214 +i 1910

0.133 +jO.177

J 0 161

R2

9.

-10

0 13 3

j O161

Fig. 6. Per unit schematic of mine electrical power system.

system is to find the prefault voltage profile through a loadflow study. To avoid unnecessary complexity, it will beassumed that this profile is flat at a value of 1.0 /0 ° per unitfor the ac system and 1.0 per unit for the dc system. The nextstep is the construction of the Zacbus and RdCbus matrices. Toaccomplish this, it is necessary that the per-unit impedances/resistances of the ac/dc machines be known. A typical per-unitreactance of 0.15 on the machines base is used for theinduction motor, and a typical per-unit resistance of 0.10 isused for the dc machines on their bases [1], [18]. Transform-ing these to the base in use gives jO.6757 pu and 0.75 pu,respectively, for these machines. Using these data and thosefrom Fig. 6 allows the bus impedance and resistance matricesto be given by

Zacbus2 [0.04789 /89.60903 0.04771 /87.99604 L_0.03605 /102.3360

Rdcbus-S [0.40496 0.205766 0.20576 0.324547 0.16564 0.261278 0.08522 0.134429 0.06965 0.1098610 L0.05408 0.08529

that neglecting this component in fault studies will lead to largeinaccuracies in the computed fault current. While the currentslisted in Table II correspond to bolted faults, the techniquedescribed is capable of modeling resistance faults and can beused to study current patterns in ac/dc systems under highresistance fault situations.The convergence properties of the algorithm are dependent

on the initial estimate of the solution. The computer runswhich generated the data of Table I were all started with a flatvoltage and current start of 0.5 pu with all bus voltage anglesat zero and all rectifiers current angles at 700. Convergencewithin a tolerance of 0.0001 was obtained in five to 11iterations with a mean value of 7.3 iterations for these runs.The computer algorithm discussed is capable of generating

0.04771 /87.99600.05005 /81.79200.03780 /96.1070

0.16564 0.085220.26127 0.134420.38421 0.197680.19768 0.324480.16156 0.265190.12543 0.20589

Using these matrices and Method II of the previous section,bolted faults were simulated at each of the dc system buses.Table II lists the fault current values for each of these faultsalong with the fault current contribution from each machineand rectifier. For faults close to machines, it can be seen thatdc machines contribute significantly to the fault current and

0.036050.037800.23827

0.069650.109860.161560.265190.393260.30532

/102.3360/96.1070/60.771

0.054080.085290.125430.205890.305320.40476 1

(51)

(52)

accurate values for the dc fault current within the limitationimposed by the approximation of (2). There are, however,other problems which limit the degree of accuracy to which thesystem can be modeled. First, the ideal filters assumed at theac terminals of each rectifier are not ideal in reality. This hasthe effect of making the effective commutation reactance

177


TABLE IIFAULT CURRENT CONTRIBUTIONS

FauiltedBus

PaultCurren 5 6 7 8 9 10Source _,, __..__

ml 1.3333 0.8453 0.5748 0.1409 0.1622 0.0566

M2 0.4470 1.3333 0.9067 0.2222 0.2560 0.0893

RI 0.9544 2.7285 5.1239 1.3805 0.4044 0.1075

M3 0.0563 0.1985 0.5868 1.3333 0.8513 0.2776

R2 0.0513 0.0967 0.4088 2.5815 5.0769 2.6031

M4 0.0263 0.1083 0.2976 0.3744 1.0352 1.3333

TOTAL 2.8686 5.3106 7.8986 6.0327 7.7859 4.4675

larger than the leakage reactance of the transformer. The exactvalue of the commutation impedance depends on the type offilter present, harmonic-impedance characteristics of the acsystem and the number of rectifiers which share this system.Second, filters are often not present on the ac system. Thus theeffect of a common commutation impedance must be takeninto account when an effective commutation impedance is tobe calculated. Analytical results for common commutationimpedances are only available for shared inductive andresistive lines under a restricted set of operating conditions[151, [19]. Thus the accuracy of any technique for predictingdc fault currents depends on the accuracy of the effectivecommutation impedance which is supplied in the model.

V1II. CONCLUSIONA range of equation sets is shown to be capable of providing

sufficient constraints to solve the dc fault current problems.Two of tlese, the full set and minimum set of constraintequations, are presented. The residual equations included ineach set are shown to depend upon whether the polar orrectangular form of the system performance equation is used.While any method for solving simultaneous nonlinear equa-tions can be used on these equation sets, Newton's method is

selected and applied to both sets of equations using polar andrectangular form. The form of the Jacobian is provided foreach. Finally, a sample problem is discussed in sufficientdetail to allow the fault current results generated to beduplicated. The convergence properties of the algorithm areacceptable when only a poor initial guess is possible. Theconvergence behavior improves as the quality of the initialguess improves. The results of the sample problem shows thatthe fault current contribution from dc machines is significant.Neglecting these currents can lead to significant inaccuraciesin the estimation of the fault current.The general approach taken shows that many equation sets

can be used to define the dc fault current. The minimumequation set provided represents an improvement over previ-ously published work in several respects. First, it is the onlyknown approach capable of calculating fault current contribu-tions from multiple dc machines and multiple rectifiers.Second, the number of nonlinear equations that need to besolved simultaneous at each iteration is proportional to thenumber of rectifiers in the system.

APPENDIX IJACOBIAN SPECIFICATION

The partials needed in constructing the Jacobian are

aRkx -cos kt'k t=k, k=m

alETl . O, otherwise

3Rky (-sin ilk, t=k, k=m

a3E,l 0, otherwise

aRkx - - IZk,I COS (Oki + C/ )-

aRky { -IZkIl sin (Oki +4Of-i),allil 0,

(53)

(54)

k, t=m+l, ..., notherwise

(55)k, t=m+l, ..., notherwise

(56)

aRkx Ek sin 1k+IZkkIkl sin (6kk+ 4kk k)y

a-t i|Zkl|if sin (Okt+ ,-4,t) I

aRky --E cos xlk -IZkkIIII COS (Okk+tk -k),d -IZk, II,} cOs (Okt + AIt-f ),

aRkx -|Zk,I IJII sin (Oki+rXI-4),

aRk Y IZkIIlA COS (OAkt+AI'-Ot),(01 ot°aRk ( - 1

alERl O,

aRk |Rku,-iti= -Rkts

t=k=ml , - - - notk,k=mhe,ws notherwise

t=k=m+l, --- nt:*k, k=mtl, notherwise

t, k=m+l, ..., notherwise

t, kwm+l, notherwise

t=k=n+1, .**, As uotherwise

t=f, k=n+l, *.-, p, uSot+(n-n)wkins, , p, u, t=n+e *.. notherwise

(57)

(58)

(59)

(60)

(61)

(62)

178

i+l, ---, n

+1, --., n


aRk ( COS Okg t=k=m+l, , n= -1, t=k+(n+m), k=m+l, , n

l *l 0, otherwise

aRkI_ Ek, t=k=m+l, , nd+,

=

0, otherwise

aRk2alE,l

ak(from Table I) t=k=m+l, ..., n= - 1, t=k+n-m, k=m+l, '-, n

0 O, otherwise

aRk2 Rc,,dll,l=t 0,

t=k+n-m, k=m+l, * *, notherwise

iR"31 (67)

ME,dR 3 (68)dIf

Note that aRk2/1alEt is not strictly correct as listed here for allmodes of operation. For mode 2, the functional form of ak

from Table I should be substituted into (48) before thederivatives are taken. This form, however, is used in thecomputer program with the justification that ak is only one

constant in the Jacobian which is used to predict the magnitudeand direction of the increment vector.

REFERENCES[1] W. R. Crites and A. G. Darling, "Short circuit calculating procedures

for d-c systems with motors and generators," AIEE Trans. PowerApp. Syst., vol. PAS 73, pp.816825, Aug. 1954.

[2] C. C. Herskind and H. L. Kellogg, "Rectifier fault currents," AIEETrans., vol. 64, pp. 145-150, Mar. 1945.

[3] C. C. Herskind, A. Schnidt Jr., and C. E. Rettig, "Rectifier faultcurrent-n," AIEE Trans., vol. 68, pp. 243-252, 1949.

[4] J. L. Paine and R. A. Hamnlton, "Determination of dc bus faultcurrents for thyristor converters," IEEE Trans. Ind. Appl., vol. IA-8, May/June, 1972.

[5] J. Reeve and S. C. Kapoor, "Analysis of transient short-circuitcurrents in HVdc power systems," IEEE Trans. PowerApp. System,vol. PAS-90, pp. 1174-1182, May/June 1971.

[6] J. Reeve and S. C. Kapoor, "Dynamic fault analysis for HVDCsystems with ac system representation," IEEE Trans. Power App.Syst., vol. PAS-91, pp. 688-696, Mar./Apr. 1972.

[7] H. A. Peterson, A. G. Phadke, and D. K. Reitan, "Transients inEHVDC power systems: Part I-Rectifier fault currents," IEEETrans. Power App. Syst., vol. PAS-88, pp. 981-989, July 1969.

[8] J. F. Burr, "Solid state overcurrent relays for the protection of trolleydistribution systems in underground coal mines," in Conf. Rec. 1976IAS Annu. Meeting, 1976, pp. 77-84.

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

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

[11] C. M. Ong and A. Hamzei-nejad, "A general purpose multiterminal dcload flow," IEEE Trans. Power App. Syst., vol. PAS-100, pp.3166-3174, July 1981.

[121 H. Fudeh and C. M. Ong., "A simple and efficient ac-dc load flowmethod for multiterminal dc systems," IEEE Trans. Power App.Syst., vol. PAS-100, pp. 4389-4396, Nov. 1981.

[13] S. N. Talukdar and k. L. Koo, "The analysis of electrified groundtransportation networks," IEEE Trans. PowerApp. Syst., vol. PAS-96, pp. 240-247, Jan./Feb. 1977.

[14] M. M. Hassan and E. K. Stanek, "Analysis techniques in ac/dc powersystems," IEEE Trans. Ind. Appl., vol. IA-17, 5, pp. 473-480,Sept./Oct. 1981.

[15] D. J. Tylavsky and F. C. Trutt, "The Newton-Raphson load flowapplied to ac/dc systems with commutation impedance," IEEE Trans.Ind. Appl., pp. 940-948, Nov./Dec. 1983.

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

[17] D. J. Tylavsky and F. C. Trutt, "Terminal behaviour of theuncontrolled R-L fed 3-phase bridge rectifier," Proc. Inst. Elec. Eng.,vol. 192, Pt. B, pp. 337-343, Nov. 1982.

[18] P. M. Anderson, Analysis of Faulted Power Systems. Ames, IA:Iowa State Univ. Press, 1973.

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

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 fromThe Pennsylvania State University, University

In 1974 he joined Basic Technology, Inc., Pitts-.Xi burgh, PA, working in the thermal and mechanical

stress analysis group. After receiving the M.S.E.E.,he joined the faculty at Penn State as an Instructor of

Electrical Engineering. He retained that position until 1980 at which time hereturned to full-time graduate study with the aid of an RCA Fellowship. In1982 he assumed his present position as Assistant Professor of Engineering inthe Electrical and Computer Engineering Department at Arizona StateUniversity, Tempe.

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

179

The Calculation of DC Fault Currents With Contributions From DC Machines and Rectifiers

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