fast decoupled load flow the hybrid model

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134 IEEE Transactions on Power Systems, Vol. 3, No. 2. May 1988

Fast Decoupled Load Flow: T h e Hybrid Model

K. Behnam-Guilani I l l i n o i s Power Company 500 S . 27th S t r e e t Decatur, IL 62525

Abstract - This paper descr ibes a load-flow model w h i c m l e a s t f i f t y percent f a s t e r than the t r a d i - t i ona l Fast Decoupled Load Flow model as described by S t o t t and Alsac. The paper i nves t iga t e s the reasons for the speed-up relative t o the t r a d i t i o n a l Fast Decoupled' Load Flow and gives d e t a i l s of the model's performance. The performance of the model is inves- t i g a t e d by examining a number of power systems the l a rges t of w h i c h is a 1435-bus system.

INTRODUCTION

The f irst generation of d i g i t a l computer models f o r solving the load-flow problem was introduced i n the 1950's. In the 1960's, these models were improved i n order t o study l a rge r and more complex systems. In those earlier decades, the load-flow s tud ie s required a s i g n i f i c a n t amount of computer m r y and execution time. These issues were addressed i n the e a r l y 1970's when S t o t t and Alsac introduced the Fast Decoupled Load Flow model [l].

The Fast Decoupled Load Flow (FDLF) model has been accepted by the power industry. This acceptance i s based on the f a c t t h a t t h e Fast Decoupled Load Flow model requires l e s s computer memory and f a r less execution time than the Newton-Raphson model. T h i s model is f a s t , simple and r e l i a b l e . I t uses two f ixed Jacobian-like matrices a s shown below:

p i s vector q is vector v is vector e is vector

B 'AB = AP/V

B'Av = ~ q i v

of a c t i v e power bus in j ec t ions . of r eac t ive power bus in j ec t ions . of bus vol tage magnitudes. of bus phase angles.

B ' i s Jacobian-like matrix ( a c t i v e power). B" i s Jacobian-like a a t r i x ( r eac t ive power).

The Jacobian-like matrices a r e f ixed a s long a s there is no change i n system topology and/or the s t a t u s o f regulated buses. Both matr ices must be updated when system topology changes ( l i n e outages). Changes in the s t a t u s of regulated buses a f f e c t the second matrix only.

Modern system r e l i a b i l i t y s tud ie s a r e based on p r o b a b i l i s t i c methods. These methods d i r e c t l y account for the fact tha t generat ing u n i t s and transmission

This paper was sponsored by the IEEE Power Engineering Society for presentation at the IEEE Power Industry Computer Applica- tion Conference, Montreal, Canada, May 18-21,1987. Manuscript was published in the 1987 PICA Conference Record.

lines a r e forced ou t of service i n a random manner. The process is based on the simulation o f a l a rge number of scenarios each of which represents a snapshot of the system as i t i s exposed t o random occurrences of generation and transmission outages. Each scenario i s then examined by a load-flow model [e]. In other words, modern system r e l i a b i l i t y s tud ie s a r e based on l a rge numbers of load-flow solut ions each of which represents a d i f f e r e n t system i n terms of i t s topology and/or s t a t u s of i t s regulated buses. Thus , the use of the Fast Decoupled Load Flow i n modern system re- l i a b i l i t y s tud ie s r equ i r e s the frequent updating of the two Jacobian-like matr ices . The updating process requires a s i g n i f i c a n t amount of computer time. For t h a t reason, i t was decided t o inves t iga t e a l t e r n a t i v e load-flow models, as described i n the next sect ion. However, before proceeding t o the next s ec t ion , i t would be useful t o review some research which was reported while this p ro jec t was i n progress.

In e a r l y 1986, Chan and Brandwajn 131 introduced two methods f o r updating t r i angu la r f ac to r i zed matr ices such a s B ' and B". These methods e f f e c t i v e l y re,solve the problem created by the need for updating B ( t o r e f l e c t topology change). The second matrix, B", should be updated if any of the following occurs.

1 - There is a topology change 2 - There i s a Q-l imit v io l a t ion 3 - There is a generator outage 4 - There is a newly-turned on generator.

The two methods introduced by Chan and Brandwajn e f f e c t i v e l y resolve the topology change issue. One of the two methods can a l s o be used t o update B" i n order t o account for changes i n the bus s t a t u s ( regulated or nonregulated). This method (known as PR1, f o r Pa r t i a l r e f ac to r i za t ion Method 1 ) is very e f f i c i e n t if there a r e only a few buses t h a t change s t a t u s . For example, a s a load-flow problem i s i t e r a t i v e l y solved, the r eac t ive output of a few p lan t s might v i o l a t e the MVAR l i m i t s of those p l an t s (9 - l imi t v i o l a t i o n ) . In such cases , those few p lan t buses a r e deregulated and PRl i s used t o e f f i c i e n t l y update B". The 9- l imit v io l a t ion issue, i n other words, i s a l s o e f f e c t i v e l y resolved. The e f f i c i ency of P R 1 i s inversely proportional t o the number of bus s t a t u s chanaes. As mentioned e a r l i e r , modern system r e l i a b i l i t y models account f o r random outages of generating units. This means t h a t i n such s t u d i e s , a l a rge number of buses change s t a t u s . Some buses become deregulated because generators connected t o them a r e forced out of service. Many more buses become regulated because their associated generators are turned on i n order t o make up for the l o s s of t h e forced out generators . For t h i s reason, PR1 i s a r a the r i n e f f i c i e n t method f o r updating B". In such cases , B" would have t o be refactor ized.

The above discussion i l l u s t r a t e s t h a t the methods presented i n [3] e f f e c t i v e l y resolve the updating problem associated w i t h B ' . However, the updating problem associated w i t h B" (when random generator outages a r e modeled) remains unresolved. I t is impor- t a n t t o model random generator outages i n r e l i a b i l i t y s tud ie s . Consequently, i t is important t o resolve the updating problem associated with B". Therefore, i t was decided t o inves t iga t e a1 t e r n a t i v e load-flow models, as

0885-8950/88/05oOa734so1 .WO 1988 IEEE

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described i n the next sect ion.

NODAL ITERATIVE MODEL '

A promising solut ion t o the matrix updating problem appeared t o be the Nodal I t e r a t i v e Model [ 4 j because i t does not use Jacobian o r Jacobian-like matr ices . Instead, i t uses the following r e l a t ionsh ip :

N - n= 1

V is complex vol tage Y i s admittance matrix * is the conjugate function

This model does not use Jacobian-like matr ices , t he re fo re , the matrix updating problem mentioned e a r l i e r would not occur w i t h this model. However, th is model is extremely slow t o converge ( p a r t l y because i t does not use Jacobian-like matr ices) and, t he re fo re , not s u i t a b l e for our purpose.

As a next s t ep , we decoupled the Nodal I t e r a t i v e Model and invest igated i ts performance. That model i s defined by the following re1 at ionships .

N - n = l n m

8, =

bmn vn L n=l nm

njcm v, =

N 1 bmn

n=l n;tm

The above eauat ions were derived from the

( 4 )

(5 )

basic equations f o r ' a c t i v e and r eac t ive power flows in a transmission l i n e . In der iving (4 ) i t was assumed t h a t t h e angular d i f f e rence between any two connected buses is small enough so t h a t the cosine of the angular d i f f e rence could be assumed t o be un i ty and the sine equal t o t h e angular d i f f e rence when the l a t t e r i s described i n radians. No assumptions were made i n der iving (5 ) .

The Decoupled Nodal I t e r a t i v e Model a l s o proved t o be a slow-to-converge model. However, this invest iga- t i o n was not a t o t a l l o s s s ince i t was discovered t h a t the reason f o r this slowness is the a c t i v e network model which is described by (4 ) . In o the r words, the

r eac t ive network model (5 ) has an acceptable convergence c h a r a c t e r i s t i c . T h i s discovery led t o the development of the hybrid model a s discussed i n the next sec t ion .

THE HYBRID MODEL

The purpose of t h i s study was t o resolve the matrix updating problem which was discussed e a r l i e r . As this study was in progress, Chan and Brandwajn introduced two methods t h a t e f f e c t i v e l y resolve the updating problem associated w i t h the a c t i v e network of the Fast Decoupled Load Flow model ( 1 ) . The previous sec t ion discusses a model which does not have the matrix updating problem, but cannot be used because i t s a c t i v e network model ( 4 ) i s slow t o converge. The next logical s t e p i s t o create a hybrid model by combinin the a c t i v e network of the Fast Decoupled Load Flow (17 w i t h the reac t ive network of the Nodal I t e r a t i v e Model (5 ) . The r e s u l t a n t model which i s shown below is the Hybrid version of Fast Decoupled Load Flow Model ( t h e Hybrid model f o r s h o r t ) .

B ' A B = Ap/V (6 )

n+m v, = ( 7 )

N

n = l n m

The Hybrid model has the pos i t i ve a t t r i b u t e s of the FDLF model ( f a s t , simple, and r e l i a b l e ) w i t h the added advantage t h a t i t resolves the matrix updating problem. Further, a s the study r e s u l t s (presented i n the next sec t ion ) show, the Hybrid model is faster than the FOLF model even i f B" is not updated.

RESULTS

The study r e s u l t s a r e given a s p robab i l i t y d i s t r i - butions of the following quot ients .

Number of i t e r a t i o n s of the FDLF model = Number of i t e r a t i o n s of the Hybrid model

CPU time of the FDLF model = CPU time of the Hybrid model

The performance o f the discussed method was evaluated by examining four power systems which a r e defined bel ow.

System Lines Generators 5 1 20 25 5 2 1 2 320 496 33 9 8 3 1,435 2,475 218 45 245 4 20 22 4 2 0

LTC = Load-Tap-Changing Transformer OFF = Fixed-Tap Off-Nominal Turns Rat io Transformer

The l a s t system is a d i f f i cu l t - to - so lve ( i l l - conditioned) system. I t was used t o test the performance of the Hybrid Model w i t h regard t o i l l - conditioned systems (Figure 1 ) .

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I8 t --

6

7

P 17

0 6 L 19 SQ -

I 6

--I2 15 - @- I1 I! 10

I- I I .

6

9

Figure 1. The Ill-Conditioned System

16 -

15 - 14 -

13 - 12 - 11 - 10 - 9 -

8 -

7 -

6 -

5 - 4 -

3 -

2 - 1 -

The 20-Bus System

probability (%I

0 7

The study r e s u l t s f o r the 20-bus system are shown ‘ i n F igure 2. The r e s u l t s are i n the form o f the proba- b i l i t y d i s t r i b u t i o n o f the r-values. As F igure 2 shows, the average r -va lue i s approximately 1.5 which means t h a t the FDLF model requi red 1.5 times as many i t e r a t i o n s as the Hybr id model. It can be estimated t h a t the average s-value f o r t h i s system i s approximately 1.3 s ince one i t e r a t i o n o f the Hybr id model takes approximately f i f t e e n percent more t ime than one i t e r a t i o n o f t h e FDLF model.

15 -

14 - 13 -

12 -

11 - 10 - 9 -

8 -

7 -

6 -

5 -

4 -

3 -

2 -

1 -

I probability (%I

16 1

0.9 1.0 1.1 1 2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 22 2.3 2.4

Figure 2. Probability Distribution for the 20-Bus System

The 320-Bus System

Figure 3 shows the study r e s u l t s f o r the 320-bus system. This system i s a reduced model which represents downstate X l l i n o i s and the surrounding regions. The study r e s u l t s f o r t h i s system i n d i c a t e t h a t the Hybr id model converges approximately f i f t y percent

faster than the t r a d i t i o n a l Fast Decoupled Load Flow model ( the average r -va lue i s 1.5).

r-value - 0.9 1.1 1.3 1 5 1.7 1.9 2 1 2.3


The 1435-Bus System

Figure 4 shows the p r o b a b i l i t y d i s t r i b u t i o n o f t h e s-values f o r the 1435-bus system. As F igure 4 i n d i - cates, the Hybr id model i s approximately f i f t y percent f a s t e r than the FDLF model.

11 -, ;b { probability (%I

o f s-value -

1 1.5 2 25


As discussed e a r l i e r , the purpose o f t h i s study was t o develop a load- f low model t h a t mi t iga tes the mat r ix updating problem. That i s t o say, a load- f low model t h a t would be f a s t e r than the FDLF model because i t would n o t requ i re mat r ix updating. As i t turned out, the developed model ( the Hybrid model) i s f a s t e r than the FDLF model even when mat r ix updating i s n o t needed (F igure 4). If matr ix r e f a c t o r i z a t i o n i s required, the Hybr id model becomes approximately one hundred percent f a s t e r than the FOLF model, as shown i n F igure 5. That f i g u r e shows the p r o b a b i l i t y d i s t r i b u - t i o n o f t h e s-values f o r the 1435-bus system when B" i s re fac tor ized . B" i s r e f a c t o r i z e d because, as mentioned e a r l i e r , p a r t i a l updating techniques are i n e f f i c i e n t when a la rge number o f buses change regu la t ion s tatus.

13 - probability (%)

B' is refactorized

l2 1 11 -

10 - 9 -

0 -

7 -

6 -

5 -

4-

3 -

2 -

1 -

0 7 T

s-value - 1.2 1.6 20 24 20

Figure 5. Probability Distributioin for the 1435-Bus System

Figure 5 represents a bimodal d i s t r i b u t i o n . The f i r s t d i s t r i b u t i o n i s centered a t 1.5 and the second a t 2.3. There are two p r o b a b i l i t y d i s t r i b u t i o n s because there &re two types o f load- f low runs ( t h a t produced Figure 5). The f i r s t type used a f l a t s t a r t whereas the second type used a basecase f o r the i n i t i a l condi- t ions. A load-f low run t h a t uses a f l a t s t a r t gener- a l l y requi res more s o l u t i o n t ime than cne t h a t uses a basecase. Since the r e f a c t o r i z a t i o n t ime f o r B" i s f ixed, i t s impact on the p r o b a b i l i t y d i s t r i b u t i o n of the s-values i s t o d i v i d e i t i n t o two par ts . The d i s t r i b u t i o n on the l e f t represents the load- f low runs t h a t were based on a f l a t s t a r t and the d i s t r i b u t i o n on the r i g h t represents the load-f lows t h a t were based on a basecase.

Special Cases

It i s t r a d i t i o n a l t o examine the robustness o f a load-f low model by apply ing i t t o a hard-to-solve ( i l l - condit ioned) system. The il 1-ccndi t ioned system used i n t h i s study had a predominantly r a d i a l topology (weakly-interconnected) and a t y p i c a l c i r c u i t parameters (see APPENDIX A.). Furthermore, i t s generation and load busses were no t dispersed. This combination

737

r e s u l t s i n a hard-to-solve system. F igure 6 shows the p r o b a b i l i t y d i s t r i b u t i o n o f the r-values f o r the ill- condi t ioned system. As F igure 6 shows, the Hybr id model performed a t l e a s t as w e l l as the FDLF model.

26 1 probability (%)

r-value - 0.7 04 1.1 1.3 1.5 1.7 1.9 21

Figure 6. Probability Distribution for the Ill-Conditioned System

DISCUSSION

The presented data i n d i c a t e t h a t the Hybr id model i s approximately f i f t y percent f a s t e r than the FDLF model. When the Hybr id model i s used i n modern re - l i a b i l i t y studies, where generator outages are randomly modeled, the speed-up i s c lose t o one hundred percent. We have i d e n t i f i e d f o u r reasons f o r the speed-up. Those reasons are g iven below.

F i r s t , the r e a c t i v e network o f the Pybr id model (7 ) requ i res l e s s CPU t ime than t h a t o f the FDLF model (2). More s p e c i f i c a l l y , the r e a c t i v e network o f the Hybrid model requi res approximately one t h i r d o f the CPU t ime requi red by the r e a c t i v e network o f the FDLF model (see APPENDIX B f o r f u r t h e r in format ion) .

Second, because o f the above, one can a f f o r d t o execute ( 7 ) more o f t e n than (6). This approach can be advantageous espec ia l l y when LTC adjustments are made. The procedure i s described beTow.

FORTRAN Statement Descri p t i on

CALL SOLVE solve (7) CALL LTC make LTC adjustments CALL SOLVE solve (7)

compute accelerat ing fac to rs

CALL AF

Subroutine AF computes accelerat ing fac to rs f o r chang- i n g the tap se t t ings o f the LTC's, as shown below.

138

F i s the acce le ra t ing f a c t o r VS i s the s e t vo l tage va lue f o r the LTC Vk i s the LTC vol tage a t i t e r a t i o n k

The tap se t t ings f o r the LTC's i s computed i n the fo l low ing manner.

tk - tk-l = -F*(Vk - VS)

The above equation i s s i m i l a r t o the r e l a t i o n s h i p given i n [l] w i t h the exception t h a t [l] f i x e s the value o f F a t u n i t y . The procedure o u t l i n e d above a l lows F t o vary and thus accelerates the convergence process.

Third, the Fast Decoupled Load Flow model can have many i t e r a t i v e schemes. Some examples are:

Scheme Descr ip t ion

i - (l0,lV) ii - (19,2V) iii - (20,lV) S t o t t and Alsac inves t iga ted these schemes and se lected the f i r s t scheme as the most e f f i c i e n t [l]. Instead o f being bound by a predetermined (user-defined) scheme such as i, ii , o r iii, the Hybr id model i s designed t o dynamical ly s e l e c t the scheme which i s most e f f i c i e n t f o r the case t h a t i t i s so lv ing. The se lec t ion tech- nique i s based on the observat ion t h a t as the i t e r a t i v e s o l u t i o n progresses, t h e two networks ( a c t i v e and reac t ive) become truly decoupled. That i s t o say, f u r t h e r v a r i a t i o n s i n vo l tage magnitudes would have no impact on t h e phase angle ca lcu la t ions and v i c e versa. A t t h a t stage, the convergence t e s t o f the two networks can a lso be decoupled. I n other words, when one of the two networks converges, i t i s excluded from f u t u r e i t e r a t i o n s .

The preceding d iscuss ion describes three o f the f o u r f a c t o r s which make the Hybr id model f a s t e r than the FDLF model. The speed-up a t t r i b u t a b l e t o those f a c t o r s i s approximately f i f t y percent. The f o u r t h f a c t o r makes the l a r g e s t c o n t r i b u t i o n t o the speed-up. It increases the speed-up t o approximately one hundred percent. This happens when random generator outages are taken i n t o account. Which means t h a t B" needs t o be re fac tor ized . Since the Hybr id model does n o t use B", i t becomes approximately tw ice as f a s t as t h e FDLF model.

Th is l a s t p o i n t i s r a t h e r s i g n i f i c a n t . A la rge p o r t i o n o f load- f low stud ies are performed i n r e l i a - b i l i t y studies. There i s a d e f i n i t e need t o represent random generator outages i n such studies. The savings represented by the Hybr id model can, therefore, be subs tan t ia l .

(1) and (2) are solved a t the same ra te . (2) i s solved twice as o f t e n as (1). (1) i s solved twice as o f t e n (2).

CONCLUSIONS -~ The Hybrid Fast Decoupled Load Flow model i s a t

l e a s t f i f t y percent f a s t e r than the t r a d i t i o n a l Fast Decoupled Load Flow model. A usefu l fea ture o f t h i s model i s t h a t i t does no t use the mat r ix E " , thus add i t iona l speed-up i s obtained when B" needs t o be re fac tor ized . For t h a t reason, i t i s espec ia l l y s u i t - ab le f o r modern system r e l i a b i l i t y s tud ies. This model has been incorporated i n t o a p r o b a b i l i s t i c cos t /benef i t model. The r e s u l t s have been p o s i t i v e .

REFERENCES -- [l] B. S t o t t and 0. Alsac, "Fast Decoupled Load Flow."

IEEE Transactions on Power Apparatus and Systems, pp. 859-869, May 1974,.,

[2] K. Behnarn-Guilani, A Monte Carlo Method for Simultaneous Import Capab i l i t y Planning." Pro- ceedings o f 13th Inter-RAM, pp. 130-134, June 1986.

[3] S . K. Chan and V. Brandwajn, " P a r t i a l Mat r i x Refac tor iza t i on .I' I EEE Transactions on Power Apparatus and Systems, pp.193-200, February 1986.

[4] A. F. Glimn and 6. bl; Stagg, "Automatic Calcula- t i o n s o f Load Flows. A I E E Transactions on PAS, pp. 817-825, October 1957.

APPENDIX A

The fo l lowing tab les f u r t h e r de f ine the ill- condi t ioned system.

Bus

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

19 20

-

ia

TABLE I

Bus Data For the I l l -Cond i t ioned System

MW-Generation MW-Load MVAR-Load

0 150 30 The Swing Bus 0 0

0 0 0 0 380 60 0 0 0

100 0 0 0 20 0

100 0 0 0 0 0 0 50 10 0 0 0 0 0 0 0 0 0 0 0 10 0 0 0 0 1c 0

100 0 0 0 10 20 0 0 0 0 10 0

TABLE I1

L ine Data For The I l l -Cond i t ioned System

From 3 & & Chy;r-;g R;;i;y

1 2 0.50 5.00 6 250 2 3 0.11 1.52 30 350 2 4 5.00 10.00 50 200 2 15 2.00 4.00 20 300 3 5 20.00 20.00 0 500 4 17 3.00 4.00 20 400 5 14 30.00 40.00 0 400 6 9 0.10 3.00 30 300 6 14 0.10 1.00 0 600 7 12 0.50 5.00 10 200 a 13 0.50 6.00 6 200 8 15 0.15 1.50 0 400 9 10 0.50 5.00 10 200 10 11 0.00 30.00 0 200 11 12 2.00 40.00 6 250 11 13 0.00 '15.00 0 209 15 19 5.00 10.00 0 150 16 19 60.00 80.00 0 600 16 20 0.00 5.00 0 300 17 19 0.6'2 8.00 10 250 17 20 60.00 60.00 0 250 18 20 0.50 5.00 20 300

. *

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APPENDIX E?

This appendix contains add i t iona l in fo rmat ion about the r e a c t i v e network of the FDLF model which i s defined by (2 ) and t h a t o f the Hybrid model which i s def ined by (7) . As mentioned e a r l i e r , the Hybr id model i s f a s t e r than the EDLF model p a r t l y because (7) requ i res about one t h i r d of the CPU t ime requi red by (2) . An explanat ion f o r the d i f fe rence i n CPU t ime requirements i s given below.

The two r e a c t i v e networks are s i m i l a r i n the sense t h a t i n both cases one has t o loop over sets o f compar- ab le computations. The number o f times t h a t one loops over the se t o f computations l a r g e l y defines the CPU t ime requirements. L e t parameters N2 and N7 show the number o f t ines t h a t one loops over the computations i n (2 ) and (7), respect ive ly . The fo l low ing t a b l e shows N2 and N7 f o r the th ree systems discussed e a r l i e r .

TABLE I

Data on N2 and N7

Buses Lines N2 N7 N2 f N7 - - 20 25 78 35 2.2

3 20 496 2,192 837 2.6 1,435 2,475 14,316 3,977 3.6

N2 i s def ined by the number o f nonzero elements i n B". N7 i s def ined by the number o f l i n e s connected t o nonregulated buses.

As the t a b l e shows, N2 i s about three times l a r g e r than N7 and ( l a r g e l y ) for t h a t reason, (7) requ i res much less CPU t ime than (2) . The value o f N2 depends on the optimal order ing r o u t i n e which i s used before B" i s pu t i n i t s f i n a l ( t r iangu lar ized) form. We used an optimal order ing rou t ine which was developed a t the Bonnevi l le Power Admin is t ra t ion. I t i s known as BPA Scheme 2. The subroutines which perform the t r i a n g u l a r - i z a t i o n and the forward-backward s u b s t i t u t i o n were developed by researchers a t The Un ivers i ty o f Texas a t Ar l ington. Add i t iona l in fo rmat ion on the s t ruc tu res o f these models i s shown below. It i s noted t h a t (7) i s executed more o f t e n than ( 2 ) as shown below, and as discussed e a r l i e r . That i s one reason f o r the f a s t e r r a t e o f convergence f o r the Hybr id model. Other reasons are the method used f o r ad jus t ing the tap se t t ings f o r LTC (discussed e a r l i e r ) and (poss ib ly) t h e fac t t h a t (7) i s der ived w i thout making any assumptions regard ing l i n e conductances , vol tage magnitudes , and angular d i f ferences. I n d e r i v i n g (2) f o r the FDLF model, OR the o ther hand, l i n e conductances are ignored, vo l tage magnitudes are assumed t o be one per u n i t , and angular d i f ferences are assumed t o be small.

Reactive Network Chart fo r the FDLF Model

CALL QE CALL AIQOFF Make adjustments f o r f i xed- tap off-nominal transformers. CALL AIQLTC CALL DELTA CALL SOLVE Solve (2) CALL LTC Make LTC adjustments. CALL TEST Perform convergence t e s t .

Compute r e a c t i v e power i n j e c t i o n s fo r each bus.

Make adjustments f o r load-tap-changing transformers. F ind d e l t a q, the e r r o r i n r e a c t i v e power i n j e c t i o n s .

Reactive Network Chart f o r the Hybr id Model

CALL RECALL CALL PIQOFF CALL AIQLTC CALL SOLVE CALL LTC CALL RECALL CALL AIQOFF CALL AIQLTC CALL SOLVE CALL AF CALL RECALL CALL PIQOFF CALL AIQLTC CALL QE CALL DELTA CALL P.V CALL TEST

Recal l defined r e a c t i v e power i n j e c t i o n s f o r each bus. Make adjustments f o r f ixed-tap off-nominal transformers. Make adjustments f o r load-tap-changing transformers. Solve (7 ) Make LTC adjustments. Recal l def ined r e a c t i v e power i n j e c t i o n s f o r each bus. Make adjustments f o r f i xed- tap off-nominal transformers. Make adjustments f o r load-tap-changing transformers. Solve (7 ) Compute LTC tap-se t t ing acce le ra t ing factors . Recal l def ined r e a c t i v e power i n j e c t i o n s f o r each bus. Make adjustments f o r f ixed- tap off-nominal transformers. Make adjustments f o r load-tap-changing transformers. Compute r e a c t i v e power i n j e c t i o n s f o r each bus. F ind d e l t a q, the e r r o r i n reac t ive power in jec t ions . Adjust computed voltages t o r e f l e c t d e l t a q. Perform convergence t e s t .

740

Discussion

B. Stott and 0. Alsac (Power Computer Applications, Mesa, AZ): The main idea in the new Hybrid method is attractively simple to grasp and apply, and its reported performance is impressive. If the method were to work as claimed on a sufficiently wide range of power systems, it would have considerable. potential for practical application. The possibilities include moderate-accuracy solutions in on-line ac contingency analysis, representing generator var l i t s and perhaps utilizing concentric relaxation [Al.

We have devoted some effort to analyzing and testing the method. The paper has not been too helpful in this exercise. Its performance results consist almost entirely of timing comparisons between the Hybrid and fast decoupled methods. Critical details of the respective methods tested, including typical numbers of active and reactive iterations and convergence criteria, are not mentioned at all.

Speed comparisons are sensitive to many factors, particularly implementation efficiencies. A cornerstone of the new method‘s claimed attractive- ness is that a Hybrid reactive iteration is about three times as fast as a fast decoupled one. Our own multiplication-addition counts and timing tests show this ratio instead to be around 1.5. Therefore our suspicion, reinforced by several misinterpretations in the paper, is that a very poor quality fast dewupled version was used. Notwithstanding, there can st i l l be useful areas of application for the Hybrid method, provided that it converges well.

In an attempt to assess this convergence, we have performed a series of tests on the method without control adjustments, on a number of large “normal” North American systems. We tried different schemes for automatically determining the number of reactive solutions at each step, and Gauss-Seidel acceleration with experimentally optirmzed factors. Solutions were made from both flat and good voltage starts, and converged to 1 MW/ Mvar mismatch accuracy at each bus.

The fast decoupled method took from two to five P and Q iterations on the problems. The Hybrid method also converged in every case. On particularly benign problems, it took around twice these numbers of iterations. However, on most it took much more, with correspondingly greater overall computing times. The paper’s 20-bus system did appear to be somewhat illconditioned to the Hybrid method, but not to the fast decoupled, which solved it with 1 pu generator voltages in four iterations.

We did not perform tests with control adjustments, since their details tend to obscure the basic properties of the methods. This may already have happened in the paper’s comparisons. Perhaps the fastdemupled convergence was badly retarded by the LTC and var limit logic used? In a matrix- based approach like the fast decoupled, relatively sophisticated techniques are used to minimize the number of extra iterations required for adjustments (see for example PI) .

In some applications, the main advantage of the new method over more traditional decoupled versions-the absence of a factorized reactive matrix-will become a distinct drawback. This applies to areas such as security optimization, where approximate network sensitivities are calculated from the B” or similar decoupled matrix factors. On a more general note, an important principle highlighted by the paper

is that the coupling from active to reactive power is strong, while the reverse is weak. This has also recently been exploited in the CRIC formulation [C], which could easily be incorporated into’the Hybrid method. A related issue is sensitivity to X/R ratios, where there might be some advantage.

Overall, however, the Hybrid version’s practical usefulness seems to hinge mainly on whether each Gauss-SeideJ reactive iteration has a sufficiently competitive voltage convergence rate. Our present limited results are not too encouraging, but we look forward to the results of testing by others. It would be very helpful if the author could explain in his closure more precisely the details of his version and its performance.

.

References

J. Zaborsky, K. W. Whang, and K. Prasad, “Fast contingency evaluation using concentric relaxation,” IEEE Trans. Power App. Syst., vol. PAS-99, pp. 28-36, JadFeb. 1980. S. K. Chang and V. Brandwajn, “Adjusted Solutions in Fast Decoupled Load Flow,” Proc. IEEE PICA Conference, pp. 347- 353, May 1987. J. Carpentier, “CRIC, a new active reactive coupling process in load flows, optimal load flows and system control,” Proc. IFAC Symposium on Power Systems and Power Plant Control, pp. 65- 70, Bcijing, Aug. 1986.

Adam Semlyen (University of Toronto, Toronto, ON, Canada): I would like to congratulate the author for his very interesting paper. In my opinion, it represents an important advance in the. sequence of practical power flow computation methods: Gauss-Seidel, Newton-Raphson, and the Fast Decoupled Load Flow. The new hybrid method brings the Gauss-Seidel approach again in the forefront due to the fact that, in decoupled form, the Q-V part is not afflicted by the notoriously slow convergence which characterizes the Gauss-Seidel power flow in its classical complex form. The author has demonstrated this by numerous examples.

I would like to suggest the following explanation of the fact that the reactive part of the Gauss-Seidel equations (6) and (7) converges much faster than the active part. Intuitively, this is due to the fact that the voltages at many buses (the voltage controlled buses) are kept constant. Formally, the argument for better convergence is based on the improved diagonal dominance of the resultant system matrix. To see this, we rewrite equation (6) in the form

n # m n # m

In equation (a) the right hand side is constant during the iterations. If we now subtract from (a) its expression after convergence, we obtain

n+m n # m

where E represents the error at a given iteration step. Diagonal dominance can be assessed directly from equation (b) by

comparing the coefficient of e,,, with the sum of the absolute values of the coefficients of e,. Two factors are of significance here: the fact that the number of coefficients bmn is reduced since the generator buses are not included in the summation, and the effect of the load. The first effect enhances the diagonal dominance, while a reactive load reduces it. These are the theoretical results. Do the experimental (i.e., computational) results confirm that a larger proportion of generator buses tends to improve the convergence, while a larger reactive load tends to slow it down? The author’s comments on these matters would be much appreciated.

S. K. Chang and V. Brandwajn (Systems Control, Inc., Palo Alto, CA): For more than a decade, the Fast Decoupled Load Flow (FDLF) has been acknowledged in the power industry as an extremely efficient and reasonably reliable load flow method. Therefore, any new method that has the potential to be at least 50 percent faster than the FDLF deserves to be investigated closely. An experimental code employing the proposed hybrid model was written by the discussers and tested on several systems ranging from 30 to 1600 buses. Based upon the test results, we would l i e to offer the following comments:

1) The proposed hybrid method is faster than the conventional Gauss- Seidel (Glimn and Stagg) method.

2) Test results indicate that the hybrid model indeed converges faster than the FDLF, if the convergence test is based on voltage changes, only. At the solution point, however, Mvar mismatches at some buses can be unac- ceptably large, up to a few hundred Mvar’s with a voltage tolerance of O.ooO1 per unit, and the voltages themselves can be quite different from those calculated by the Newton-Raphson method or FDLF. An improved solution can be obtained by adding an inner iteration loop for equation (7). Additional improvements are possible, for example, by an appropriate ordering of the calculations for equation (7). However, a significant additional computational burden will result.

3) It is well known that the convergence of the nodal iterative method depends on the use of acceleration factors. Our experience has shown that the hybrid method does not alleviate this problem. Consequently, the convergence characteristics of the hybrid method are system dependent, which could affect detrimentally the usefulness of the method.

4) The efficiency of the partial refactorization methods depends to a great degree on the particular implementation and data structures. In our experience, the CPU time for the update of the matrix B ” can vary as much as 1 :4 depending on the particular implementation.

5 ) In view of points 2) through 4), we feel the hybrid method may not be as efficient and reliable as the paper would indicate.

We would appreciate the author’s comments on our discussion as well as some additional implementation de#aik.

F- L. AhrJo (The University of Wisconsin-Madison, WI): TLh k a impaM paper, a final verdict on which will have to d c r b -= . by orhers. However, several ideas that have been

CLt ommunity for some years lend this paper credence. m d rrrStbc potential merits of this paper in the light of some

m-by the author in an attempt tosbed light idothe merits d B a - gofthemethod. Tbt maS d hitations of the fast deunpkd u&od am well known.

0 0 brlacc. ooc m y say that, when applicrble, t& fasl cksuupled method is pabpr tbt fastest known reliable MPM for solving the power flow problem. Tbt nmrber of itedoos reqtitd by the fast decoupled method is macorks. of sysian site. The Gauss-Seidel method, on

rhis -=-, ha bgn discredited for many years as a usful techniqrw for the sohaioo of the power flow problem in large sysram. Tbc masm for this, as mrrectly perceived by the author, is that axrcctiom proppgae slowly through the network, with the net result that the oumbcr of

&-two key issues of the paper are that the amount of computation for each iteration of the fast decoupled power flow grows with system size, and that the convergence of the Q-V portion of the Gauss-Seidel method is quite good and dependent of network size. Let us examine both of these ideas.

The first idea is justified empirically by the author in the paper. It could have also been justified simply by citing [All. In this reference it is established that, even though the ratio of number of branches to number of buses is independent of system size, the amount of computation required for factorization and repeat solution grows faster than linearly (but less than quadratically) with system size. Furthermore, the rate of growth is influenced by the sparsity preserving ordering scheme used. For typical power system matrices using Tinney Scheme 2 ordering, the growth of computation for factorization of typical sparse matrices seems to be, according to [All, of order n1.4, and the growth of computation for the repeat solution step (a most important quantity in the fast decoupled methods) grows as n1.2. Experimentation subsequent to [All has indicated that these numbers are a little too large for actual power system sparse matrices, but that the fundamental idea remains valid. Thus, this is in independent verification.of the observation in the paper that the solution for V from the B” matrix will grow with system size.

The second key idea in the paper is that the Q- V portion of the Gauss- Seidel method actually works quite well. Recent evidence by others [A2], including unpublished evidence by this discusser, lends credence to the idea that for most systems Q-V effects are limited to a small portion of the system. In situations where this is true, the fact that changes propagate slowly with the Gauss-Seidel method would not be as important, because these changes would not normally propagate very far, thus lending additional credence to the author’s contention.

Under what conditions is this last assumption true? The reason Q-V disturbances do not propagate far is that most systems have voltage support buses distributed throughout the system. Generally, a change in voltage will propagate in any direction only as far as the nearest PV bus not at its limit. This observation immediately implies a possible limitation of the author’s method to systems with reasonably adequate voltage support. Under conditions of system stress, where many PV buses may reach a limit, this discusser would expect the author’s method effectiveness to suffer. This is n0t.a discredit to the worthiness of the method, merely an attempt to understand its potential limitations. Does the author have any comments on this observation?

increases drastically with system size.

References

[All F. L. Alvarado, “Computational Complexity in Power Systems,” IEEE Trans. Power App. Sysf., August 1976, pp. 1028-1037.

[A21 M. Lauby, T. Mikolinnas, and N. Reppen, “Contingency Selection of Branch Outages Causing Voltage Problems,” IEEE Trans. Power App. Syst., Vol. PAS-102, December 1983, pp. 3899-3904.

K. Behnam-Guilani:

I thank the discussors for their in- sighfull comments. At the PICA meetinq, A. Semlyen and I--and later, F. L. Alvarado and I--privately discussed the advantages ana disadvantages of the Hybrid model. 1 believe

741

those discussions are worth repeating.

The basic advantage of the Hybrid moeel can be traced to the decoupling of the Nodal Iterative moeel ( 3 ) . The decoupling cf ( 3 ) made it possible to see that the slowness of ( 3 ) is due to its active netvork mo2el ( 4 ) . On the other hand, its reactive network nodel ( 5 ) displayed good convergence characteristics. This is due to the presence of regulated buses which serve as anchors embedded within the state vector, thus, greatly localizing the Q-V relationship. The above explanation can be used to clarify sone of the comments made by the discussors. Further comments are offered below.

1.

7 I.

F . L. Alvarado

I agree with the discussor’s observation that in the absence of regulated buses, the performance of the Hybrid model will suffer. The Hybrid model, therefore, should not be used for examining--for example--a distribution system or any system that has very few regulated buses.

B. Stott and 0. Alsac

The question of the quality of the FDLF model (which was used as a benchmark) is highly relevant. I have, therefore, provided the discussors with a copy of the benchmark (the FDLF model) so that they could test its quality. We hope to report our findings in the near future.

The discussors have reported that their version of the Hybrid model took around twice the number of iterations (relative to the FDLF model) to converge. During private discussions, I have received similar comments from other iesearchers. The discrepancy, I believe, is due to the difference in the methods used to adjust the computed voltages to reflect the error in reactive power injection. The computed voltages are adjusted in two steps. First, the error in reactive power injection is divided by the diaqonal element to form the preliminary adjustment. Second, the preliminary adjustment is modified to reflect the preliminary adjustments of the neighbor- ing buses, as shown below.

J: buses that are connected to bus I I,: line number

DV1: preliminary voltage adjustment DV2: secondary voltage adjustment

€3: ljne susceptance D: the diagonal element of bus I

TEMP = 0.0 DO 10 I1 = Il,I2

L = LAT,(II) TEYP = TEMP + DVl (J) x D ( I , )

DV2(I) = DVl(1) + TFVP/E(I)

J = I n A B ( 1 1 )

1 0 CONTINUF

v(r) = V(I) + DV~(I)

The above computations make the voltage adjustments quasi-simultanecus.

?mP

142

It is noted that the simultaneity of adjustment is not as important in reactive network computations as it is in active network computations because of the localized nature of the Q-V relation. The last point is confirmed by F. L. Alvarado's discussion.

3 . A. Semlyen

The mathematical interpretation, provided by the discussor, offers an interesting explanation of the Hybrid model's characteristics. I agree with the discussor that the presence of generator buses (regulated buses) improves the performance of the Hybrid model. The discussor's second point refers to the possible relation between the magnitude of reactive loads and the rate of convergence. I have not noticed a significant correlation between the magnitude of the reactive loads and the rate of convergence.

4 . S . K. Chang and V. Brandwajn

Both the FDLF model and the Hybrid

model used the same convergence criteri- on, i.e., error in the active and reactive power injections. Therefore, it is not possible to have the large mismatches (at any bus) that the discussers have experienced.

The Hybrid model has been used to analyze a number of power systems. It has consistently been faster than the FDLF model. It, therefore, is unlikely that the Hybrid model is a system- dependent model.

The absence of the matrix B" in the Hybrid model would be an advantage only if a large number of buses change regulation status. In such cases, the matrix B" would be literally refactorized. When the above is not the case, the advantage of the Hybrid model (relative to the FDLF model) would be reduced.

I appreciate the comments and the test results provided by the discussors and look forward to similar future exchanges.

fast decoupled load flow the hybrid model

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