stability of loadflow techniques for distribution

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Stability of loadflow techniques for distributionsystem voltage stability analysis

G.B. Jasmon, PhDL.H.C.C. Lee,BEng

Indexing terms: Stability, Loadflow techniques, Power systems protection

Abstract: Loadflow techniques are widely used in

the planning and daily operation of power systems

including that of the on-line monitoring of dis-

tribution system operation. The performance of

three loadflow techniques is investigated in their

ability to analyse distribution network loadflows.

The criterion for voltage instability is presented

and the ability of the three loadflow techniques to

predict it is discussed.

List of symbols for distflow equations

= net active power injected at busbar i

= net reactive power injected at busbar i

=line impedance in line connecting busbar i

=line resistance in line connecting busbars i

= line reactance in line connecting busbars i

= voltage at busbar i

= real power loss in line connecting busbars i

=reactive power loss in line connecting

= real load at busbar i + 1

= reactive load a t busbar i + 1

and i + 1

and i + 1

and i + 1

and i + 1

busbars i and i + 1

List of symbols for SONR method

p , = net active power injected at busba r i

Qi = net reactive power injected at busbar iEi (or v) = ei + f, = complex voltage at busbar i

J ( x ) = Jacobian evaluated for the initial estimates of

H ( x ) = second derivatives evaluated for the initialestimates x

fis) = vector of scheduled quantities, i.e. scheduledload powers and generator voltages

f i x e ) = vector of estimated values, i.e. the power

equations evaluated for initial voltage estim-

ates

= errors of corrections of unknown variables,

i.e. voltage corrections Ae i; AJ= vector function of corrections in unknown

variables, i.e. the power equations evaluated

for the voltage corrections

X

A x

f i A x )

Paper 8247C (H),irst received 16th January and in revised form 23rdMa y 1991

The authors are with the Faculty of Engineering, University of Malaya,59100 Kuala Lu mp u r , Malaysia

IEE PROCEEDINGS-C, Vol . 138, N o . 6 , N O V E M B E R 1991

P 8 ,Q , = scheduled net active and reactive powers

evaluated with the initial voltages

P o , Qo = active and reactive powers evaluated with theinitial voltages

List of symbols for FDLF method

A P

AQ

HLK

6,jQi

X i jG i j

Bi j

Bs'

=difference between actual and scheduled real

=difference between actual and scheduled reac-

= submatrix of Jacobian matrix

= submatrix of Jacobian matrix

= voltage at busbar i= load angle between busbar i and j

= reactive power flow at busbar i

= reactive between busbar i and j

= real part of the admittance between busbar i

=imaginary part of the admittance between

= submatrix of B matrix

= submatrix of B matrix

power interchange

tive power interchange

and j

busbar i and j

1 Introduction

The operation and planning of power systems depends

heavily on the use of loadflow techniques [3, 51. The

primary objective is to check that any solution offered is

within the permissible (security) operating limits. In real

time operation, any system whose state variables falloutside the permissible limit will be subjected to correc-

tive actions to alleviate the problem. In the event that the

system cannot be corrected, certain drastic actions such

as load shedding or practical shutdown has to be exer-

cised. In such an event some or all of the customers will

not have power supply.

In the real world, power systems are constantly being

threatened by various elements that could cause violation

of the safe or secure limits of operation. On some

occasions, the system may be subjected to such an unex-

pectedly high demand and multiple faults that the

system states can go into extreme conditions. The

occurrence of voltage collapse [6] has been observed and

this problem is being actively investigated with the aim of

determining suitable methods for predicting the onset of

such collapse. The ability to predict this collapse is most

vital as it can save the system from the unwarranted

event of system collapse. It has been shown that the load-

flow technique can be used to determine the voltage

stability limit of power systems [SI. Voltage collapse

occurs when a loadflow solution is unachievable by a

419



loadflow technique. Loadflow techniques [2, 41 are pre-

sented for predicting the voltage collapse point of power

networks.

Many loadflow techniques have been developed and

the capability of some modern techniques have overtaken

some of the earlier ones. The earlier methods are inferior

in speed and storage requirements but are useful for illus-

tration of basic principles [3, 51.One important new technique is the second order

Newton-Raphson method (SONR) [2, 71 which has

gained widespread application in many facets of powersystems. In many ways this method is comparable in per-

formance with that of the fast decoupled loadflow

method (FD LF ) [4]. Another technique dedicated to the

analysis of power distribution networks is the method

developed by Baran and W u [I]. The development of

this method is revolutionary in view of its superior fea-

tures for appli cation in radial networks loadflow analysis.

The theory of the SONR and F D L F is briefly pre-

sented and the theory of the third loadflow method, i.e.

distflow method, is presented in detail. The criterion for

voltage collapse will then be presented and the ways in

which the loadflow methods can predict the voltage col-

lapse point are also discussed.

2 AC loadflow techniques

The theoretical basis for the three loadflow methods to

be used for comparison are briefly outlined.

2.1 Second order Newton-Raphson method [2]

The loadflow equations, in rectangular co-ordinates, for

each node in a power network are

P i = CCe i e j G i j- , f j B , + J A G i j + J e j B i j )

Qi = ( e , f j G i j f i f ; B i j - i f j G i j- i e j B i j ) (1 )

where N is the number of nodes in the network. Using

the Taylor series expansion of these equations, the

general form of the expression is

(2)

j = 1

N

j = 1

AS)= i x )+ J ( x )Ax , + H( x) ( Ax i Ax j ) /2

where

and

( A X , A x j )=

A x l A x l

A x l A x 2

Ax, , Axn

It can be shown [2] that the third and higher order terms

of the Taylor series expansion vanishes because the load-

flow equation is quadratic. According to Iwamoto [7],the second order terms of the equation can be simplified,

thus resulting in the form

(3 )

where superscript k indicates the iteration number.

If these variables are replaced with the normal load-

flow variables, the second order loadflow equations are

Axkt ' = J ( x e ) - ' ( y ( s ) A x e ) - AX'))

480

obtained in matrix form as

(4)

where the Jacobian matrix Jo is constructed from

a p a p_ _

J o = [ $ $1In eqn. 4, notice that no voltage-controlled busbar equa-

tions are present because the problem addressed is a dis-

tribution network which has only one voltage-controlled

busbar at the main input node or the reference node.

These equations can easily be modified to take into

account systems with more than one voltage-controlled

busbars. Eqn. 4 forms the basis of the SONR method

used for one method presented.

2.2 Fast decoupled loadflow method [4]Consider the decoupled loadflow equations

A P = H A S

A Q =L [ 7 ]

where the elements of the submatrices H and L can be

expressed as

Hij= K(Gij sin S i j - B , cos S i j ) i # j (7)

(8)

L, = Y(Gi jsin 6 , - B i jcos S i j ) = H , i # (9)

(10)

H . .= - B . . V2 - Q.i =I I I

L . .= - E . ., V ? + Q . i =

Even though the developed loadflow method reduces the

memory storage requirements considerably, it still

requires significant amounts of computational effort.

Therefore, Stot and Alsac [SI have developed the fast

decoupled loadflow method. The basic assumptions used

are the following:

(i) The power systems have high x / r ratios

G i j sin 6, < B i j (1 1)

(ii) The difference between adjacent bus voltage angleis very small

sin 6. .= sin ( S i - . )E 6. - .= 6 . .J - 8 J 'J

cos Si,.= cos ( S i - j ) 1.0 (12)

Q i < E,, V: (13)

CAP] = [ V x B x V][AS] (14)

(iii) Also

Therefore eqns. 5 and 6 can be further approximated as

[AQ] = [ V x B x V ]-"V"lwhere the elements of the matrices B' and B" are the ele-

ments of the matrix - B . Therefore

Bi j = - / X , j i # (16)U

B i i = 1 1 /X . . i = j (17)j = l

E!'.= - E . . (18)

IE E PR O C E E D IN G S-C , V o l . 138.N o . 6, N O V E M B E R 1991



The decoupling process in the fast decoupled loadflow

can be concluded after additional modifications based onthe simplifying assumptions as

A P / V = IT A6 (19)

A Q J V = B” A V (2 0 )

This forms the basis of the fast decoupled loadflow

method used.

2.3 Distflow method [ l ]The equations for the loadflow solution as used in the

method by Baran and W u [l ] is derived. Fig. 1 shows a

“Le “1 LP0 Z & 1

Fig. 1 Distribution line

line connecting bus 0 to bus 1 having a line impedance of

r + x .From Fig. 1,it can be shown that

Real power loss:

R = Z CO S tr(P; + Q @ / V ’

= r(P; + Q g ) / V 2

Reactive power loss:

X = Z sin a(Pg + Q @ / V ’

= x ( P i + Q @ / V ’

and,

V : = V ; - (rPo + x Q o )+ (r’ + x’)(Pi + Qg)/Vg (23)

Thus for a line as in Fig. 2 where there is a cumulative

load at the end of the line, i.e. P,, + Q L l

P I = Po- (P; +Q g ) / V 2- L1

Q i = Q o -4% +Qi)/v’ L ~

(24)

(2 5 )

V : = Vg - (rPo+ x Q , ) + (r’ + x 2 ) ( P i+ Q;) / V g (2 6 )

r + j x

“0Pi *IQ1

po+l*Ql

Fig. 2 Example of distribution line

A more general form of eqns. 24-26 , between two

busbars i and i + 1 can therefore be derived [I ] as

(2 7 )

(2 8 )

(29)

P i + , =Pi- i p : + Q:) / V : - P , i + l

Q i + 1 = Q i - Xi(PZ + QZ)/V?- L I + I

V; + = V : - (r iPi x i Q i )

+ ( r: + xZ)(P?+ Q: ) /V:

The proposed method can also be used for loadflow

analysis simply by iterating on the loss terms. The dis-tribution network can be reduced into its single line

equivalent by calculating the equivalent resistance, rer ,and reactance, xeqrrom the total real and reactive losses.

IE E PR O C E E D IN G S-C , V o l . 138,N o. 6, N O V E M B E R 1991

The next iteration uses values of real and reactive

P i +,= ( 2 x 2 P ,- rx Q, + r) / (2(r2+ x2))

power injections by using eqns. 30 and 31

- ( 2 x z P ,- r x Q , + r)’ - (r 2 + x2)

x ( x ’ P , + r 2 Q , - r x P l Q 1 + r P l ) ) 1 ’ 2- 2(r2+ x’)) (30)

Q i + ,= ( 2 r 2 Q ,- rx P l + x)/ (2(r2+ x’))

- ( 2 x 2 P 1- rxQl + r)’ - (r 2 + x’ )

x ( x 2 P l+ r Z Q l- r x P , Q l + r P 1 ) ) 1 1 2

i 2(r2+ x’)) (31)

The values of req and xe q can be obtained by using the

previous power injection of Pi Q i .

(32)

(3 3 )

req = R/(PZ + Q t )

xe s =X / ( P t + Q z )

2.3.1 Loadflow solution algorithm: The fundamental

equations for solving a loadflow problem of a distribu-

tion network using a single-line equivalent has been

derived. The loadflow algorithm using these fundamental

equations can be set up using the following steps:

( a ) Start the initial iteration by using the total real

loads and reactive loads as the initial power injection.

(b) Sum all the real and reactive losses and find theequivalent resistance, re q , and reactance, x e q , for a single

line system from eqns. 32 and 33 .( c )Calculate the new power injection by using eqns. 30

and 31. If P i + l- Pi < e , then go to step (e)where e is a

set tolerance, else go to step (d).

(d) Iterate with new power injection from step (c), then

go to step (b).( e ) Calculate other parameters required, e.g. voltages.

The voltage can be calculated from the losses and power

injections in the individual lines from eqn. 29 .

It has been found that the proposed loadflow requires

fewer iterations than other known techniques.

3 Criterion for stability [8]

3.1 Mathematical formulation of technique3.1.1 Governing equations of a single-line system:Before proceeding into the actual system, we first derive

the equations that characterise the behaviour of a single-

line system. Consider the single line in Fig. 1 which has

the parameters shown in Fig. 3.

Fig. 3 Single line system

From Fig. 1 , the real and reactive power equations

have been derived in Reference 1 as

P = r (P2 + Q 2 ) / V 2+ P I

Q = x ( P 2+ Q 2 ) / V 2+ Q 1

(34)

(35)

48 1



From eqns. 1 and 2, the (P’ + Q 2 ) / V 2 terms can be

eliminated thus obtaining

x(P - Pi )= r( Q - Q i ) (36)

By rearranging eqn. 6, and eliminating Q in eqn. 4, a

quadratic equation in terms of P is obtained

(r’ + x’)P’ - 2 x 2 P 1- rxQl + r) P

+ ( x ’ P , + r’Q1 - r x P l Q l + r P l )= 0 (37)

The voltage at the sending end is the reference voltageand its magnitude is kept constant, and in this case

V’ = 1 p.u.

Hence, from eqn. 37

P = ( 2 x 2 P 1- rxQ, + r ) / ( 2 ( r2+ x’))- ( 2 x z P 1- r x Q l + r)’ - (r 2 + x’)

x ( x ’ P , + r Z Q 1- r x P l Q l + rPl))li’

x ( 2 ( rZ+ x’) ) (38)

Similarly for reactive power Q , because of symmetry of

equations, the reactive power equation can be derived as

Q = 2 r Z Q 1- rxP l + x) / (2(r2+ x’))- ( 2 x 2 P 1- rxQl + r)’ - (rZ+ x’ )

x ( x ’ P , + r’Q, - r x P l Q l + rP1)) ’ / ’

x (2(r2+ 2)) (39)

The above equations are quadratic in form and for P and

Q to have real roots

( 2 x 2 P 1- rxQl + r)’ - (r 2 + x’)

x ( X’ P , + r 2 Q 1- r x P l Q l + r P l ) > 0

which on simplification can be reduced to

4( ( xP l - Q1)’ + x Q , + r P l ) -= 1 (40)

3.1.2 Reduction of real network to single line equiva-lent: A given power distribution network can be reduced

to a single-line equivalent. From Reference 1, the real and

reactive power flows in any line are given by

P i + = P i - , (P?+ Q?)/VZ- l i + (41)

Q i + i = Q i - APZ +QWZ

- , i + i (42)

R i = r i p ? + QZ)/VZ

X i = x,(P?+ QZ)/V?

The real and reactive loss terms are

(43)

(44)

Using eqn. 42, the ratio of real losses between line i and

preceeding line i + 1 can be computed as

R i+ i /Ri = (ri+ (P?+ + Q?+ iW?+)

x ( rdp? + Q ? ) / V ? )

x (rXPZ +QZ)Xv?/v?+i) (45)

= (ri+ (P?+ + Q?+1))

By considering the current flow in the line

(P? + QZYVZ = (Pi+ + PI)’ + Q i + 1 + Ql)’)/VZ+

which gives

VZ/vZ+i =U? + QZ)/( (Pi+i+ Pi)’ + ( Q i + i + Q i ) ’ )

(46)

482

From eqns. 45 and 46

R i + , / R i= ( r i+ l / r iXP Z +l+ Q f + J

Similarly for reactive losses

X i + i / X i = ( x i + i / x i X P? + i+ Q?+ i )

For a given distribution network

(49)

(50)

From eqns. 47 and 48, t can be seen that the losses in the

distribution network are ratios of the losses in the first

line of the network.

Hence

P = r,(P’ + Q 2 )+1 l i

Q = xeq(PZ+ Q’) + Q i i

where re q and xe q is the equivalent resistance and reac-

tance in the single line. Hence the real distribution

network consisting of many lines has been reduced into a

system with only one line.

3.1.3Voltage collapse: By using the single line method

for reducing a distribution network, the occurrence of

voltage collapse can be studied easily as every line in thenetwork does not have to be considered.

Recall from eqn. 40, for P and Q to have real roots

L = 4( ( xP l - e l ) ’ + XQ l + r P l )

Hence for the reduced network

where L < 1.0.

L = 4 ( ( x e q P i - r e q Q i ) ’ + x e q Q i + r e q P i ) (53)

If the network is loaded beyond this critical limit, the

power becomes imaginary and it is at this point that

voltage collapses.

4 Performance test results between three

loadflow methods

The three loadflow methods have been tested on several

distribution systems. The loadflow data and results for

the 32-node test system from Reference 1 are given asTables la and lb . The following comparisons can be

made :

4.1 Speed of convergenceBoth the SONR and distflow methods require about the

same number of iterations to converge to a tolerance of

O.OOO1 on the real and reactive power accuracy, but

FDLF requires about three times more. The relative

speed is given in Table 2.The distflow method does not require any matrix

inversion and as a result the computation time is con-

siderably less.

4.2 Accuracy of solutionThe accuracy of the methods depend on the tolerance

specified and for all the results presented here a tolerance

of O.OOO1 on the power is used. All the results are exactly

the same and exact (depending on the tolerance).

4. 3 Storage requirementsIt is obvious that SONR requires more storage than dist-

flow because the Jacobian matrix has to be formulated

IE E PR O C E E D IN G S-C , Vol .138, No . 6, N O V E M B E R 1991



Table l a :Test system

Branch Receiving Sending P, (kW) 0, kVar) Branch parameters

number node nodeR (D.u.) X (D.u.) SusceDtance (D.u.)

1 0 12 1 2

3 2 3

4 3 4

5 4 5

6 5 6

7 6 7

8 7 89 8 9

10 9 10

1 1 10 1 112 1 1 12

13 12 13

14 13 14

15 14 15

16 15 16

17 16 17

18 1 18

19 18 19

20 19 20

21 20 21

22 2 22

23 22 23

24 23 24

25 5 25

26 25 26

27 26 27

28 27 28

29 28 29

30 29 3031 30 31

32 31 32

Table 16: Loadflow results

Branch Receiving Sendingnumber node node

1 0 12 1 2

3 2 3

4 3 4

5 4 5

6 5 6

7 6 7

8 7 8

9 8 9

10 9 10

1 1 10 1 112

1 112

13 12 13

14 13 14

15 14 15

16 15 16

17 16 17

18 1 18

19 18 19

20 19 20

21 20 21

22 2 22

23 22 23

24 23 24

25 5 25

26 25 26

27 26 27

28 27 28

29 28 29

30 29 30

31 30 31

32 31 32

100

90

120

60

60

200

200

16060

45

60

60

120

60

60

60

90

90

90

90

90

90

420

420

60

60

60

120

100

15021 0

100

1318

750

479

660

630

1049

1425

1106602

298

895

961

599

621

947

936

371

970

1099

658

431

776

1053

673

495

772

963

684

732

909408

261

0.0058

0.0308

0.0228

0.0238

0.051

0.01 7

0.1 68

0.06430.0651

0.01 23

0.0234

0.091

0.0338

0.0369

0.0466

0.0804

0.0457

0.01 2

0.0939

0.0255

0.0442

0.0282

0.0560

0.0559

0.01 7

0.01 7

0.0661

0.0502

0.031

0.06080.01 94

0.021 3

0.0029

0.01 7

0.01 6

0.01 1

0.0441

0.0386

0.0771

0.04620.0462

0.0041

0.0077

0.0721

0.0445

0.0328

0.0340

0.1 074

0.0358

0.0098

0.0846

0.0298

0.0585

0.01 2

0.0442

0.0437

0.0065

0.0090

0.0583

0.0437

0.0161

0.06010.0226

0.0331

0.0052

0.0277

0.0206

0.021 4

0.0460

0.01 5

0.0961

0.05780.0586

0.01 0

0.021

0.0824

0.0304

0.0332

0.041

0.0724

0.041

0.0092

0.0845

0.0230

0.0398

0.0253

0.0504

0.0503

0.01 4

0.01 0

0.0595

0.0452

0.0285

0.05470.01 74

0.01 91

Sending node Power loss

p, (P.U.)

0.01 00.0090

0.01 0

0.0060

0.0060

0.0200

0.0200

0.01 0

0.0060

0.0045

0.0060

0.00600.01 20

0.0060

0.0060

0.0060

0.0090

0.0090

0.0090

0.0090

0.0090

0.0090

0.0420

0.0420

0.0060

0.0060

0.0060

0.01 0

0.01 0

0.01 0

0.0210

0.01 0

Q, (P.U.)

0.1318

0.0750

0.0479

0.0660

0.0630

0.1 049

0.1425

0.1 106

0.0602

0.0298

0.0895

0.09610.0599

0.0621

0.0947

0.0936

0.0371

0.0970

0.1 99

0.0658

0.0431

0.0776

0.1 53

0.0673

0.0495

0.0772

0.0963

0.0684

0.0732

0.0909

0.0408

0.0261

IV I

0.9967

0.9824

0.9747

0.9670

0.9477

0.9436

0.9279

0.9203

0.9144

0.91 35

0.91 20

0.90580.9035

0.9020

0.9007

0.8986

0.8980

0.9962

0.9926

0.991

0.991

0.9790

0.9728

0.9695

0.9459

0.9436

0.9344

0.9280

0.9251

0.91 8

0.91 77

0.91 71

P

0.3859

0.3354

0.2273

0.21 3

0.2091

0.1006

0.0791

0.0625

0.0561

0.051

0.0454

0.03920.0271

0.021

0.01 0

0.0090

-0.0000

0.0271

0.01 0

0.0090

-0.0000

0.0846

0.0421

-0.0000

0.0820

0.0758

0.0691

0.0566

0.0464

0.0310

0.0100

-0.0000

0.2401

0.21 3

0.1 25

0.1 85

0.1 629

0.0533

0.0421

0.0297

0.0275

0.0244

0.0209

0.01 720.0091

0.0081

0.0060

0.0040

-0.0000

0.0121

0.0080

0.0040

-0.0000

0.0305

0.0201

-0.0000

0.0737

0.0635

0.0579

0.0525

0.0504

0.0200

0.01 0

-0.0000

(kW) (kVar)

0.0015 0.0008

0.0053 0.0027

0.0022 0.0011

0.0020 0.0010

0.0042 0.0036

0.0002 0.0008

0.0016 0.0011

0.0006 0.0004

0.0004 0.0003

0.0001 0.0000

0.0001 0.0000

0.0003 0.00020.0001 0.0001

0.0000 0.0000

0.0000 0.0000

0.0000 0.0000

0.0000 0.0000

o.oo00 0.0000

0.0001 0.0001

0.0000 0.0000

0.0000 0.0000

0.0003 0.0002

0.0005 O.OOO4

0.0001 0.0001

0.0002 0.0001

0.0002 0.0001

0.0007 0.0006

0.0005 0.0004

0.0002 0.0001

0.0003 0.0003

0.0000 o.oo000.0000 0.0000

Table 2 : Speed comparison

SONR: 1.5

Distflow: 1

and subsequently inverted for SONR. The distflowmethod does not require the formation of a Jacobianmatrix or any matrix inversion. In fact, this is the greatestdisadvantageof SONR particularly because the JacobianDLF: 3

IE E PR O C E E D IN G S-C, V o l .138, N o . 6, N O V E M B E R 1991 483



matrix for radial networks are usually sparse. Both

SONR and FDLF requires more storage as given in

Table 3.

Table 3: t o r a g e comparison

SONR: 8Distflow: 1

FDLF: 4

4.4 Effect of loading up io voltage collapse pointAll three loadflows have been tested to see whether they

fail at the voltage collapse point. From the test run, it

was found that all the three methods gave the same

results, i.e. they can produce a solution for any condition

up to the point when the stability factor approaches close.

to unity.

5 Conclusion

The performance of three AC loadflow techniques has

been compared with respect to their ability to solve dis-

tribution network voltage stability problems. The dist-

flow method has been demonstrated to be the most

suitable method in view of its speed and storage as well

48 4

as its ability to produce a loadflow solution close to the

voltage collapse point.

6 References

1 BARAN, M.E., and W U, F.F.: ‘Network reconfiguration in distribu-tion systems for loss reduction and load balancing’, IEEE Trans. ,1989, PWRD-4, pp . 140- 1407

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IE E PR O C E E D IN G S-C , Vol. 138,N o . 6 , N O V E M B E R 1991 I-

stability of loadflow techniques for distribution

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