New algorithm for D-curve computation

Electric Power Systems Research, 13 (1987) 203 - 208

203

New Algorithm for D-Curve Computation M. NOVOTN~£, D. MUDRON(~I'K and P. M[/LLER Department of Automatic Control Systems of Industrial Processes, Faculty of Electrical Engineering, Slovak Technical University, Bratislava (Czechoslovakia) (Received August 13, 1987)

SUMMARY

A n effective algorithm for the calculation o f stability regions in the plane of two control parameters is proposed. The algorithm is suitable for small computers. In comparison with the classical technique, the derived algorithm requires fewer manual calculations. The program and its short characteristic equation is included. Utilization o f the program is illustrated on an example o f an automatically controlled p o w e r system having a 'forced' excitation system.

1. INTRODUCTION Stability analysis of dynamical systems and synthesis of regulators in special cases can be made effective b y the D-partition technique. This technique is suitable for the case of two real parameters, when the characteristic polynomial is linear with respect to both parameters. When the degree o f the characteristic polynomial is high, the computation of a D curve is very complex and requires many operations with polynomials. Moreover, the form of t h e characteristic polynomial depends on the control loop structure and a special calculation must be performed for each case. In this paper a new algorithm for the c o m p u t a t i o n of D curves is derived, reducing significantly the number of manual calculations. A simple programming language with real arithmetic is used. Special cases are also taken into account. 1.1. Classical D-partition technique Let us suppose the characteristic polynomial to be linear with respect to parameters 0378-7796/87/$3.50

p and q. Hence the form o f the characteristic polynomial is A ( s ) = P(s)p + Q(s}q + H{s)

(1)

Then the stability region in the plane of p , q is defined by the set of pairs (p,q) which are solutions of the equation with complex coefficients A(k +jco) = P ( k + jco)p + Q(k + jco)q + H(k + j w ) = 0

(2)

For the two-parameter case p and q are computed from the system of two real equations Re(A(k + jco)} = P,(k, co)p + Ql(k, co)q + HI(X,CO) = 0

(3) Im{A(k + jco)) = P2(k, co)P + Q2(k, co)q + H2(k, co) = 0 which yields P -

Dv(~, co) D(k, co) '

q

Dq(X, co) D(k, co)

(4)

where Dp(k, co) = Q l(~, co)H2(k, co) -- Q2(k, co)H,(k, co)

(5a)

Dq(k, co) = H , ( k , co)P2(k, co) -- H2(k, co)P,(k, co)

(5b)

D(k, co) = PI(~k, co)Q2(k, co) -- P2(~k,CO)Ql(~k, CO)

(5C)

Thus, for the given degree of stability k0, the D curve is determined by the set of points (P(ko, co), q(ko, co)) for co E (comm, comax). The algorithm for computation of the D regions using the classical technique consists of the following steps. © Elsevie¢ Sequoia/Printed in The Netherlands

204 Step 1. Determine the characteristic polynomial in form (1). Step 2. After substitution s = k + jco compute the coefficients of polynomials Pi, Qi and Hi (i = 1 and 2) in system (3). Step 3. Compute the polynomials Dp(k, co), Dq(k,co) and D(k, co) using (5). Step 4. Compute the particular D-curve point for selecting the values of k0 and COo. It should be noted that only the 4th step in the above-mentioned algorithm must be calculated by using a c o m p u t e r program. 1.2. A m o d i f i e d D-partition technique

The classical algorithm f o r c o m p u t i n g the D curve requires a lot of manual calculation to find the polynomials (5). If the structure of the control loop varies then this algorithm is n o t suitable for practical applications. For such a case we propose a modification of the algorithm in Step 2 using the substitution s = k 0 + c o 0 , where k0 and coo are appropriate given values of k and co respectively. Then all the transfer functions involved in the polynomials P(k, co), Q(k, co) and H(k,co) will be replaced by complex values. Thus the values of polynomials Pi(k0,COo), Qi(ko,co0) and Hi(k0,co0) (i = 1, 2 ) i n system (3) are computed by elementary operations with complex numbers. Evaluation of parameters p and q is made as usual, using eqns. (4) and (5). Now we suppose that the polynomials P(s), Q(s) and H(s) are composed of quadratic polynomials (6)

Fi(s) = A t s 2 + B i s + Ci

which may be added, subtracted and multiplied. Brackets may also be used. E x a m p l e . The characteristic polynomial according to eqn. (1) is A ( s ) = (p2s 2 + Po)P - - q l s q + h3s s + h2s 2

Re{Fi} = A i ( ~ 0 2 - (.~02) + B l k o + Ci Im{Fi}

=

2Ai~0co 0 + Bico 0

(8)

Now the computation of each pair (p,q) is made by operations with complex numbers. The algorithm for the computation of the D curves will be carried out step by step. Step 1. Generate a set of characteristic polynomial structural parameters. Step 2. Choose the values k0 a~d coo. Step 3. Using the structural data c o m p u t e the values of the polynomials Pi, Qi and Hi (i = 1, 2) (eqn. (3)). Step 4. C o m p u t e the parameters p and q using (4) and (5). Step 5. Execute Step 2 for the next COo or end. In the above algorithm we need to determine a set of structural parameters as follows. 1.3. Specification o f the characteristic p o l y n o m i a l structure

Now we propose one way to specify the structure of the characteristic polynomial. We shall try to make the data input as simple as possible. The basic data are: (i) the number of quadratic polynomials Fi(s) (6) and their indexes; (ii) coefficients Ai, Bi and C i of the polynomials Fi(s); (iii) the value of the parameter X; (iv) the set of values of frequency co, the set to be determined by the values ¢Oram (minimum of co), ¢Omax (maximum of ¢o) and ACO (increase of ¢o). The structure of the characteristic polynomial will be defined using the constants C as follows: C=0:

multiplication

C = -- 1: addition C = --3: subtraction

+ his + ho

The polynomial can be written in the form A ( s ) = Fl(s)p -- F2(s)q + Fs(s)F4(s) + Fs(s)

where F l ( s ) =P2 s2 + Po,

F2(s) = - - q l s ,

F4(8) = has + h2,

Fs(s) = his + ho

Fs(s) = s 2

Hence the substitution s = ko + Jcoo transforms the polynomial Fi(s) into the form F~(k0,coo) = Re{Fi} + j Im(F~} (7) where

C = --4: writing final values of polynomials P, Q andH Further we use the memory index I for controlling the variables X~. The values of the index I are defined as follows: I=0

always at the beginning of the calculation of the polynomial value P(k, co), Q ( k , w ) or H(k, co)

I = I + 1 after each assignment of the value Fi(k, co) to the auxiliary variable Xi

205 I =

1 after executing an operation

I --

Use of parameters C and I is illustrated in the next example. Example. The characteristic polynomial is

A(s) = FI(s) -- F3(s)p + FI(S)F2(s)Fa(s)q + [FI(s) + F2(s)] Fa(s)

+

F4(s)F~.(s)

The structure of the polynomial P(s) is given by the set Cp = (1, 3 , - - 3 , - - 4 ) Thus the computation scheme is I=0 C=I:

I=I+l,

C =3:

I=I+l,

C=--3:

Xi=X,=Fl(ko,¢Oo)

X i = X 2=Fa(ko,coo) R=Xi_I--XI=XI--X2, I=I--1,

C=--4:

Xi=R

P ( k o , W o ) = X i =X1

The polynomial Q(s) is c o m p u t e d using the set Cq = (1, 2, 0, 3, 0 , - - 4 )

2. P R O G R A M F O R C O M P U T A T I O N OF THE D C U R V E S

Now we present the program for computation of the D-curve points. The program consists of three parts (see Appendix A). In the first part, appropriate sets of parameters C are generated. The next part of the program is the computation of the D-curve points for given ~0 and co0. The last part is a subroutine for indication of an infinite point or computation of singular straight-line coefficients.

2.1. Data structure input This block is established for the automatic generation of structural parameter sets. The data input is given in the form of a string of expressions which are composed of the polynomial names and operation symbols (rows 260 - 290). Example. Correct examples of the structure input are F I F 4 + (F2 + F3)F7

in the steps I=0

( F 1 F 2 F 5 + F14(F7 + F8))F1

C=I:

I=1,

Xi=F,(ko,COo)

(F1 + F 2 ) F 3 + F 6 ( F 5 + F4)

C=2:

I=2,

X~=F:(k0,co0)

C =0:

R =X2XI, I = l , X I = R I = 2 , Xi=Fa(ko,C~o) R=X2X1, I = I , X I = R

After the structure input the C-set elements are generated and stored in the array P (the main block rows 310 - 1030). Auxiliary subroutines are in rows 1 1 0 0 - 1240. In this block the input of coefficients Ai, Bi and Ci of F~ is also performed.

C=3: C =0: C = --4:

Q(~0,co0) = XI

Similarly, for the polynomial H(k, c o ) w e define the set Ch = (1, 2, --1, 3, 0, 4, --1, 2, 0, --4) and the relevant steps are I=0 C = 1:

X1 = Fl(k0,c~0)

C = 2:

X 2 = F2(k0,co0)

C=--I:

X~=X2+XI

C = 3:

X 2 = Fa(X0,co0)

C = O:

X 1 = X2X 1

C = 4: X2 =F4(Xo,¢Oo) C = --1: X 1 = X2 -b X 1

2.2. Computation of the D-curve points This is the main part of the program with the following blocks. (1) Evaluation o f polynomials Ft (6) using (7) and (8) (rows 1700 - 1730). (2) Subroutine for subtraction and addition of quadratic polynomials (rows 1 8 0 0 1900). (3) Subroutine for quadratic polynomial multiplication (rows 2000 - 2030). Note. Variables X l ( I ) and X2(I) represent the real and imaginary parts of variables Xt respectively. The cycle parameter K (row 1530) indicates that for

C = 2: X2 = F2()~0,co0) C=0: XI=X2XI

K = 1:

polynomial P(s) is evaluated

K = 2:

polynomial Q(s) is evaluated

C =--4:

K = 3:

polynomial H(s) is evaluated

H(ko,~o) =XI

206 2.3. S p e c i a l cases T h e special cases are i n d i c a t e d b y D(k0,¢o0) = 0 (3). T h e n if Dp(k0,¢o0) and Dq(k0,¢o0) are n o t equal to zero t h e r e is an infinite p o i n t . I f Dp(k0,¢o0) or Dq(k0,~o0) is e q u a l t o zero t h e n t h e singular straight-line c o e f ficients are evaluated. T h e s e special cases are solved in the last p a r t o f t h e p r o g r a m (rows 2 2 6 0 - 2723).

F o r t h e linear m o d e l , the e q u a t i o n s b e c o m e T~s: A5 = AMT - - AMe

(12)

G ( s ) A E q e + X d ( s ) A i d = AUq + Aiqr

(13)

Xq(s) Aiq = - - A U d - - Aid r

(14)

and t h e b l o c k d i a g r a m f o r the s y s t e m is s h o w n in Fig. 2. A f t e r very t e d i o u s r e a r r a n g e m e n t , t h e characteristic e q u a t i o n b e c o m e s

3. NUMERICAL EXAMPLE

P(s)k 2 + Q(s)Toc + H(s) = 0

C o n s i d e r t h e voltage c o n t r o l o f a singlem a c h i n e s y s t e m . T h e task is to calculate t h e region o f stability in t h e p l a n e o f t w o p a r a m eters Toc and k2 f o r a s i n g l e - m a c h i n e - i n f i n i t e bus s y s t e m (Fig. 1) w h o s e e q u a t i o n s are given b y the swing e q u a t i o n and P a r k ' s e q u a t i o n s :

where

Tjs2~

(9)

M T --M e

=

G(s)Eae

+

Xd(S)i d

=

(10)

Uq + iqr

-- (F1F6 + F7)F12F8 + (F1F9 + F10)FllF5F12 H(s) = (F1F2 + F3)F4F5

-- (FIF6 + F7)F8 + (F1F9 + F10)FllF5 w h e r e F I are the p o l y n o m i a l s (6)

where

F l = s2 + Os + O

Ts+ l =

Q(s) = ( F 1 F 2 + F 3 ) F 1 2 F 4 F 5

(11)

X a ( S ) i q = - - U d -- id r

G(s)

P(s) = ( F 1 F 2 + F 3 ) F 1 2 F 8

2 r/~rP qn S Zdo-Ldo + (Tld + Tdo)S + 1

F 2 = - - 0 . 0 0 0 94s2 - - 0 . 6 7 0 12s + 0 F3 = - - 0 . 2 1 9 96s2 - - 1 3 . 2 0 2 3 s + 1 . 3 1 4 42 F 4 = 0s 2 + 0 . 3 2 1 s + 1 F5 = 0s 2 + 0.16s + 4 . 9 9 3 24

Yli' Y12

F6 = 0s 2 + 7 × 10-Ss + 0.037 68 F7 = 0s 2 + 2 . 8 8 × 10-3s + 1 . 6 2 2 37 F 8 = 0s: + 0s + 1 0 0 . 0 1 8 F 9 = 0s ~ + 8.3 × 1 0 - % + 0 . 1 9 9 44 F 1 0 = 0s 2 + 0 . 0 1 5 78s - - 1 . 3 1 4 42 F l l = 0s 2 - 0 . 3 1 5 0 6 6 s + 1 . 6 5 4

Fig. I. Single-machine system.

F 1 2 = 0s 2 + s + 0

AEqe

-AU G

A f t e r t h e c o m p u t a t i o n m a k i n g use o f t h e d e v e l o p e d p r o g r a m , we o b t a i n , w i t h o u t d i f f i c u l t y , t h e results w h i c h are p r e s e n t e d in A p p e n d i x B. 4. CONCLUSIONS

AEq k6

1+sk5Te

J

!

Fig. 2. Block diagram for the single-machine system.

T h e a u t o m a t i c v o l t a g e c o n t r o l l e r design o f a s y n c h r o n o u s g e n e r a t o r c a n be e f f e c t i v e l y d o n e using t h e D - p a r t i t i o n t e c h n i q u e .

207 The classical D-partition procedure requires many manual computations. In this paper a new algorithm and a program for computation of the D-curve points is presented. The main advantages of the presented program are the following: (1) The data preparation and input are very simple. (2) The program is suitable for small (personal) computers. (3) Changing of the control loop structure and parameters is feasible simply by changing the input data. (4) The proposed program is universal and independent of the control loop structure.

REFERENCES V. A. Venikov, Transient Phenomena in ElecPower Systems, Izdatelstvo Veeshaja Shkola, Moscow, 1970. 2 S. B. Pandey and V. V. Chalam, Dynamic stability investigations -- effect of damping, saturation, governor action and stabilizing link, Electr. Power Syst. Res., 2 (1979) 155 - 164. 3 S. B. Pandey and V. V. Chalam, Dynamic stability investigations -- on obtaining a c o m m o n region of stability, Electr. Power Syst. Res., 2 (1979) 21 - 26. 4 J. Kulh~ny and D. Mudron~fk, Ur~ovanie oblastf stability m e t 6 d o u D-rozkladu bez vlTpoStu koeficientov charakteristickej rovnice, Elektrotech. Cas., 9 (1981) 666 - 675. 1

trical

APPENDIX A 10 REM D-PARTITION 20 ~IM P(500),A(99),B(??),C(9?),F(??) 30 "Fx="LP:~ 100 OPEN T~ FOR OUTPUT AS FILE #3 110 PRINT #3:\PRINT #3:\PRINT ¢3: 120 PRINI #3:' . . . . . . . . . . . . . . . . . . . . . . . . SAS ' ; 150 PRINT ¢3:~D-PARIITION'; 170 PRINT #3:TAB(36);DAT~; 180 PRINT #3:TAB(45);' ..................... 190 PRINT #3:\PRINT #3: 200 PRINT ~D-PARTITION" 201 PRINT 260 PRINT 'P(S)='; 261 INPUT' PI~\L=LEN(Plx) 262 IF SEG~(PI~,L,L)()'/' THEN 264 263 P~=P~&SEOx(PIx,I,L-I)\O0TO 261 264 P~=P~&PIx 270 PRINT 'O(S)='; 27]. INPUT O1~\L=LEN(QIw) 272 IF SEG~(QI~,L,L)()'/~ THEN 274 273 OK=Q~&SEG~(O1~,1,L-1)\GO TO 271 274 O~=O~&QI~ 280 PRINT 'H(S)='; 281 INPUT HI~\L=LEN(HIK) 282 IF SEO~(HI~,L,L)()'/' THEN 284 283 Hx=H~&SEG~(HI~I,L-1)\GO TO 281 284 H~=H~&HI~ 290 PRINT

,

300 REM STRUCTURE COEFICIENTS C COMPUTATION 310 LET R~=P~&'.'&Ow&'.'&H~&'.' 320 FOR I=1 TO LEN(RK) 330 LET Cw=SEG~(R~yI,I) 340 IF C~=' ' THEN 1020 350 IF C~='F' THEN 500 360 IF Cx=')' THEN 800 370 LET 0=0 380 IF Cx='w' THEN 600 390 IF Cx='÷' THEN 700 400 IF C~='-' THEN 700 410 IF C~='( ' THEN 1000 420 IF C~='. ' THEN 900 430 t.ET Q=I 440 LET N~=N~&C~ 450 O0 10 1020 500 IF Q=O THEN 1020 510 LET C~= ' * ' 600 GOSUB 1110 610 IF Z=O THEN 1000 620 IF Z~(Z)(>'*' THEN 1000 630 GOSUD 1210 640 GD TO 610 700 GOSUB 1110 710 IF Z=O THEN I000 720 IF Z~(Z)='(' THEN 1000 730 GOSUB 1210 740 00 TO 710 800 GOSUB 1110 810 IF Z~(Z)=~(' THEN 850 820 GOSUB 1210 830 GO TO 810 850 LET Z=Z-1 860 80 TO 1020 900 OOSUB 1110 910 IF Z=O THEN 950 920 GOSUB 1210 930 GO TO 910 940 LET P=P+I 950 LET P=P+I 960 LET P(P)=-4 970 GU TO 1020 1000 LET Z=Z+I 1010 LET Z~(Z)=Cw 1020 NEXT I 1030 80 TO 1300 1100 REM INPUT OF COEFICIENTS A,B,C 1110 IF N~= ' ' THEN 1200 1120 LET P=P+I 1130 LET P(P)=VAL(NK) 1135 LET N~='~ 1140 FOR J=l TO F 1150 IF P(P)=F(J) THEN 1200 1160 NEXT J 1170 F=F+I 1171F(F)=P(P) 1180 PRINT 'F';STRx(F(F));':' 1181 PRINT ,CHR~(26);'A='; 1182 INPUT A(F(F)) 1183 PRINT ,,CHRX(26);'B=~; 1184 INPUT B(F(F)) 1185 PRINT ,,,CHR~(26);'C='; 1186 INPUT C(F'(F)) 1200 RETURN 1205 REM PARAMETER C 1210 LET P=P+I 1220 LET P(P)=42'-ASC(Z~(Z)) 1230 LET Z=Z-1 1240 RETURN 1300 PRINT # 3 : ' * * * * * * * ~ * D - PARTITION , , * * * * * * * * * * * * w " 1302 PRINT # 3 : ' - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1304 PRINT #3: 1306 PRINT #3:'CHARACTERISTIC EQUATION:~ 1310 PRINT #3: 1320 PRINT #3:'P~(~;P~;')+Q*(';Q~;')+~;H~;'=O" 1330 PRINT #3: 1335 PRINT # 3 : , ' A : ' , ' B : ' , ' C : ' 1340 N=O 1350 M=100\S=O 1360 FOR J=l TO F 1361 IF' F(J)>=M THEN 1364 1362 IF F(J)(=N THEN 1364 1363 M = F ( J ) \ S = I 1364 NEXT J 1365 IF S:O THEN 1390 1370 PRINT #3:'F';STR~(M);':';

208 1371 PRINT # 3 : , \ I F A(M)=O THEN 1372 \PRINT ¢3:A(M); 1372 PRINT ¢ 3 : , \ I F B(M)~O THEN 1373 \\PRINT #3:B(M); 1373 PRINT ¢ 3 : , \ I F C(M)=O THEN 1374 \PRINT #3:C(M); 1374 PRIN'[ #3: 1375 N=M 1380 GO TO 1350 1390 PRINT #3: 1400 PRINT #3:"OMEGA:"," P:"," G:',"SGN(DET)" 1410 PRINT #3: 1420 CLOSE ¢3 1430 PRIN[ °DE~REE OF STABILITY";\INPUT L9 1490 PRINT 1500 PRINT 'OMEGA MINIM:'; 1501 INPUT Ox 1502 IF O~='' THEN 9900 1503 OI=VAL(O~) 1504 PRINT 'OMEGA MAXIM='; 1505 INPUT 02 1506 PRINT 'DELTA OMEGA='; 1507 INPUT O3 1508 OPEN "LP:" FOR OUTPUT AS FILE ¢3 1510 FOR 0=01 TO 02 STEP 03 1520 LET L=O 1530 FOR K=I lO 3 1540 LET I=O 1541LEI XI(O)=O 1542 LEt X2(O)=O 1550 LET L=L+I 1560 LET C=P(L) 1570 IF C=O THEN 2000 1580 IF C=-1 THEN 1900 1590 IF C=-3 THEN 1800 1600 IF C=-4 THEN 2130 1700 REM COMPUTATION OF F(I) 1710 LET R=~-(O~2)wA(C)+C(C) 1715 R=R+LV*(A(C)wL9+B(C)) 1720 LEt G=O*B(C) 1725 G=G+2*Lg*A(C)*O 1730 LET l=I+l 1740 GO TO 2100 1800 REM SUBTRACTION i~10 LE1 X I ( I ) = - X I ( I ) 1820 LE[ X2(I)=-X2(I) 1900 REM ADDITION 1910 LEI R=XI(1)+XI(I-I) 1920 LET G=X2(1)+X2(I-1) 1930 LET I=I-1 1940 GO TO 2100 2000 REM flULTIPLICATION 2010 LET R=XI(1)*XI(I.-I)-X2(I)*X2(I-1) 2020 LET G=XI(I)*X2(I-I)+X2(I)*XI(I-1) 2030 LET I=I-1 2040 GO TO 2100 2100 LET X I ( I ) = R 2110 kEl X2(1)=G 2120 GO TO 1550 2130 LEI U(K)=XI(1) 2140 LET V(K)=X2(I) 2150 NEXT K 2200 LET W=U(1)*V(2)-U(2)*V(1) 2210 LET WI=-U(3)*V(2)+U(2)*V(3) 2220 LEI W2=-U(1)*V(3)+U(3)*V(1) 2230 IF W=O THEN 2260 2240 PRINT #3:0,WI/W,W2/W,SGN(W) 2250 GO IO 2800 2260 IF WI=O THEN 2300 2270 IF W2=O 1HEN 2300 2280 PRINT #3:0,"INFINITE POINT" 2290 GO TO 2800 2300 IF U(1)<>O THEN 2400 2310 IF V(1)()0 THEN 2500 2320 IF V(2)()0 THEN 2600 2330 IF U(2)<)O THEN 2700 2340 PRINT ¢3:0,'******w','*******',0 2350 GO "l'O2800 2400 E I = - U ( 2 ) / U ( 1 ) \ E 2 = - U ( 3 ) / U ( 1 ) 2420 GO TO 2520 2500 E 1 = - V ( 2 ) / V ( 1 ) k E 2 = - V ( 3 ) / V ( 1 )

2520 PRINT #3:0,"81NGULAR LINE P = ";STR~(EI);"*O"; 2521 IF E2(O lHEN 2523 2522 PRINT # 3 : ' + ' ; 2523 PRINT #3:E2 2530 GO TO 2320 2600 EI=-V(1)/V(2)\E2=-V(3)/V(2) 2620 GO TO 2720 2700 EI=-U(1)/U(2)\E2=-U(3)/U(2) 2720 PRINT #3:0,'SINGULAR LINE G = ";STR~(EI);"*P"; 2721 IF E2(O THEN 2723 2722 PRIN'[ # 3 : ' + ' ; 2723 PRINT ¢3:E2 2800 NEXT 0 2810 PRINT ¢3: 2820 CLOSE #3 2830 GO TO 1490 9900 OPEN "LP:" FOR OUTPUT AS FILE ¢3 9901 PRINT #3: 9910 FOR I=i TO 70\PRINT ¢3:'-';\NEXT I 9920 PRINT ¢3:\PRINT #3:\PRINT ¢3: 9930 CLOSE #3 9999 END

APPENDIX B ****~**~**w*** D - PARTITION ********************* .............................................................. CHARACTERISTIC EQUATION: F'*((FIF2+F3)*FI2F8)~Q*((FIF2+F3)*FI2F4F5-(FIF6+F7)~FI2FS+ (FIFg+F10)~PIIF5FI2)+(FIF2+F3)*F4FS-(FIF6*F7)*FS+ (FIFg+FIO)*FIIF5=O

FI: F2: F3: F4: F5: F6: F7: F8: F9: FIO: FII: FI2: OMEGA~ .2 .4 .6 .8 i 1.2 1.4 1.6 1.8 2 2.2 2~4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5

A: B: C: 1 . - 9 . 4 0 0 0 0 E - 0-.67012 4 -.21996 -13.2023 1~31442 .321 I .16 4.99324 7.00000E-05 .03768 2.88000E-03 I~62237 100.018 8.30000E-04 .19944 .01578 -i.31442 -.315066 1.654 1 P: 3.04315 .762801 .3407 .193192 .125175 .0885127 .0667186 .0529101 .0438079 .0376438 .0334247 .0304536 .0280371 .025816 .0226551 .0179634 .0121423 7.45715E-03 7.19744E-03 .0164082 .068367 -.156431 -.0567035 -.0403441 -.0334233

Q: 2.0597 .256634 -.081927 -.206137 .-.27023 -.312521 -.346431 -.377874 -.409721 -.444452 -.482097 -.522049 -.557766 -.586092 -~578546 -.508438 -.36095 -,165139 .0269051 .20564 .55451 -.615868 -.0723637 .0136724 .0436471

SGN(DET) -I -I -i -I -1 -i -I -I -I -i -I -I -I -1 -I -i -I -i -1 --i -'1 1 1 I 1