Matrix properties relating to stability analysis

Electrical Power and Energy Systems 23 (2001) 229±235

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Matrix properties relating to stability analysis U. Di Caprio* ENEL s.p.a. Via A. Volta 1, Cologno Monzese, Italy Received 12 March 1998; received in revised form 7 October 1999; accepted 25 November 1999

Abstract With reference to a multimachine power system are presented properties and conditions to be satis®ed by matrices M, K, D (inertia coef®cients, synchronizing coef®cients and damping coef®cients) in order that the system can be stable. The analysis is carried out with the assumption that the transfer-conductances are negligible while the damping effects (of the ®eld and damper circuits) are taken into account. The formulation is general, i.e. it can be applied to any system with n degrees of freedom, subjected to conservative positional forces and to dissipative forces linearly dependent upon the speed. q 2001 Elsevier Science Ltd. All rights reserved. Keywords: Stability analysis; Transfer-conductance; Damping effects

1. Introduction The equations of a multimachine power system belong to the general class M x 1 f…x† 1 Dx_ ˆ 0

…1†

where x is an n £ 1 state vector, 2f…x† an n £ 1 vector of functions 2fi …x† that represent positional forces, M an n £ n real matrix whose elements represent inertias and D an n £ n real matrix whose elements represent damping coef®cients. We assume throughout that the forces 2fi are conservative: 2fi =2xj ˆ 2fj =xi ;

i; j ˆ 1; 2; ¼; n:

…2†

Linearizing Eq. (1) we get x ˆ 2M 21 K…x 2 x0 † 2 M 21 Dx_ It is shown in Ref. [1] that if matrices M, K, D satisfy the * Tel.: 1 390-2-7224-1; fax: 1 390-2-7224-5465.

D 1 DT

# .0

…3†

…positive 2 definite† then the following function is a Lyapunov function: Zx _ ˆ f…x† dx 1 12 x_ T M x_ 1 12 x_T D…x 2 x0 † V…x; x† x_0

…4†

Here we return on this by clarifying the signi®cance of condition (3).

Consider the 2n £ 2n matrix A " # A11 A12 Aˆ AT12 A22

is symmetric, i.e. K ˆK

DM 21 D

DT M 21 DT

2. A general property of 2n £ 2n matrices

Then the matrix   2f …x0 Equilibrium point† Kˆ 2x x0

T

condition " …DM 21 †K 1 K…DM 21 †T

…5†

where A11 ; A12 ; and A22 are …n £ n† real matrices. Assume that A11 . 0:

…6†

Then A . 0 if T A11 2 A12 A21 22 A12 . 0

Proof.

0142-0615/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S 0142-061 5(00)00006-5

…7†

Assume that conditions (6) and (7) are veri®ed.

230

U. Di Caprio / Electrical Power and Energy Systems 23 (2001) 229±235

Then AˆB

1=2

1=2

B ;

B.0

…8†

Also, T 1=2 1=2 1=2 T ‰xT2 A1=2 22 1 x1 A12 A22 Š‰A22 x2 1 A22 A12 x1 Š

ˆ xT2 A22 x2 1 xT1 A12 A22 AT12 x1 1 xT2 AT12 x1 1 xT1 A12 x2

…9†

where x1 is an n £ 1 vector and x2 an n £ 1 vector. On the other hand the quadratic form

D=2

can be represented in the equivalent way T T T T xT Ax ˆ xT1 ‰A11 2 A12 A21 22 A12 Šx1 1 x1 A12 x2 1 x2 A12 x1 T 1 xT2 A22 x2 1 xT1 …A12 A21 22 A12 †x1

…10†

It follows from (9) and (10) that

1

1

1

DT =2

…positive-definite†

#

M

.0

Condition (18) implies that " # K DT =2 .0 K . 0; Det D=2 M

…18†

…19†

As M . 0; the second of (19) results in

T T xT Ax ˆ xT1 ‰A11 2 A12 A21 22 A12 Šx1 1 x2 A22 x2 1=2 xT2 A12 A1=2 22 Š‰A22 x2

_ .0 H‰V…x; x†Š i.e. " K

xT Ax ˆ xT1 A11 x1 1 xT1 A12 x2 1 xT2 AT12 x1 1 xT2 A22 x2

‰xT2 A1=2 22

condition (3) is a suf®cient one for (4) to be a Lyapunov function. As regards the second part, note that, as shown in Ref. [1], _ x† _ is condition (3) simultaneously implicates that V…x; _ is positive- de®nite. negative-de®nite and V…x; x† _ at …x0 ; x_ ˆ 0† is to Consequently the Hessian of V…x; x† satisfy the condition

A1=2 22 A12 x1 Š

…11†

Det…K 2 …1=4†DT M 21 D† . 0

Then, taking into account that:

This then proves that Eqs. (15) and (16) are veri®ed. As regards Eq. (17) note that Eq. (3) results in

T 1=2 1=2 1=2 xT2 A1=2 22 1 x1 A11 A22 ˆ A22 x2 1 A22 A12 x1

K…M 21 D† 1 …M 21 D†T K . 0:

and consequently

Then, as K . 0; it is inferred from a noted Lyapunov's theorem that the linear system

‰xT2 A1=2 22

1

1=2 xT1 A12 A22 Š‰A1=2 22 x2

1

A1=2 22 x2

1

T A1=2 22 A12 x1 Š

.0 …12†

is asymptotically stable and, consequently, that the eigenvalues of …M 21 D† satisfy Eq. (17). A

one infers from Eqs. (11) and (12) that xT Ax . 0

T if ‰A11 2 A12 A21 22 A12 Š . 0

It is then concluded that A . 0:

Corollary 2.

A

Corollary 1. tions

Assume that M, K, D satisfy the two condi-

D 1 DT . 0

…13†

K…M 21 D† 1 ……M 21 D†T K 2 4…DT M 21 DT †…D 1 DT †21  …DM 21 D† . 0

…14†

Then the V-function (4) is a Liapunov function and the Equilibrium point (x0 ; x_ ˆ 0) is asymptotically stable; moreover the following relations turn out veri®ed K.0

…15†

det‰K 2 …1=4†DT M 21 DT Š . 0

…16†

Re {li …M 21 D†} . 0;

…17†

i ˆ 1; 2; ¼n

x_ ˆ M 21 Dx

If

DT ˆ D

…20†

and M 21 D 1 DM 21 . 0

…21†

…K 2 2DM 21 D†M 21 D 1 DM 21 …K 2 2DM 21 D† . 0

…22†

the function (4) is a Lyapunov function. Moreover, K 2 …1=4†DT M 21 D . 0

…23†

li {M 21 D} . 0

…24†

Proof. Assume that conditions (20)±(22) are veri®ed. Then, as D is symmetric, Eq. (22) is equivalent to (14). Also, as M is positive-de®nite then (due to a noted Lyapunov's Theorem) Eq. (21) results in D.0

Proof. The ®rst part is an immediate consequence of the aforesaid property and of the fact that, as shown in Ref. [1],

Consequently Eq. (13) is satis®ed and hence we conclude that the three conditions (20)±(22) are equivalent to the two

U. Di Caprio / Electrical Power and Energy Systems 23 (2001) 229±235

conditions (13) and (14). On the other hand, we know (Corollary 1) that in such a case function (4) is a Lyapunov function. So we can say that conditions (20)±(22) implicate that (4) is a Lyapunov function. Moreover, as D and M are positive-de®nite matrices, then on the one hand

li {M 21 D} . 0;

i ˆ 1; 2; ¼n

…25†

which proves that Eq. (24) is veri®ed, and, on the other hand, …M 21 D† ˆ DM 21

…26†

Lyapunov function. In fact, in a such case " # 6:1875 5:9375 21 21 K…M D† 1 …M D†K ˆ 5:9375 50 " T

2D M

21

T 21

D…D 1 D † DM

…indefinite†

K 2 2DM 21 D . 0

We can also see that

K 2 …1=4†DM

21

T

K 2 …1=4†D M

D . 0:

Finally, since D is symmetric, the above condition is equivalent to condition (23). A Numerical example. If " # " 4:5 0:5 0:1 Kˆ ; Mˆ 0:5 5 0 " # 0:1375 0 Dˆ 0 0:12 " 21

D† 1 …M

21

T

D† K ˆ

2D M

21

T

T 21

D …D 1 D † DM

Theorem 1.

0:59375

0:59375

5

Dˆ

69:3

21

"

Dˆ

#

20:226

0:5

0:5

2:915

…indefinite†

;

6:1875

21

0

#

(i) Assume that

DT ˆ D

" T

0

Note that in the case in question (in which D equals 10 times the original value) the linearized system eigenvalues are unstable.

…27†

‰M 21 K 2

then K…M

0:12

Dˆ

259

_ 3. Relation betwween the system eigenvalues and V…x; x†

#

0

21

K…M 21 D† 1 …M 21 D†K 2 2DT M 21 D…D 1 DT †21 DM 21 D " # 359:75 5 ˆ 68:75 219:3

It follows from Eqs. (25) and (26), and from a noted Lyapunov's Theorem, that Eq. (22) results in

which entails that

231

1 4

…M 21 D†2 ŠM 21 D ˆ M 21 D‰KM 21 2

1 4

#

0:2599

0

0

0:12

#

Consequently K…M 21 D† 1 ……M 21 D†T K 2 2DT M 21 D…D 1 DT †21 DM 21 D " # 5:9276 0:59375 ˆ .0 0:59375 4:88 So the system in object satis®es the conditions for (4) to be a Lyapunov function. Moreover, it is evident that …M 21 D† has real positive eigenvalues. Finally " # 4:452 0:5 21 .0 K 2 …1=4†DM D ˆ 0:5 4:75 If, e.g. we multiply D by 10, the function (4) is no longer a

…M 21 D†2 Š

…28†

Then the V-function (4) represents a Lyapunov function and, simultaneously, the linearized system eigenvalues are represented by n complex and conjugate pairs with a negative real part (so that the dynamics of system (1) in proximity of …x0 ; x_ ˆ 0† are represented by n damped oscillatory modes). (ii) The preceding conclusion is valid if DT ± D and the matrices M, K, D satisfy conditions (13) and (14) together with the following additional conditions: DT ‰M 21 K 2

M 21 ‰K 2

1 4

1 4

…M 21 D†2 Š 1 ‰KM 21 2

1 4

…DT M 21 †2 ŠD . 0 …29†

DT M 21 DŠ

…30†

has real eigenvalues M 21 ‰K 2

Proof. T

D ˆ D;

1 4

…M 21 D†2 ŠM 21 D ˆ M 21 D‰KM 21 2

1 4

…M 21 D†2 Š

…31†

(i) The given assumptions entail that M 21 D 1 DM 21 . 0

…32†

232

U. Di Caprio / Electrical Power and Energy Systems 23 (2001) 229±235

…K 2 2DM 21 D†M 21 D 1 DM 21 …K 2 2DM 21 D† . 0 ‰M 21 K 2

1 4

…M 21 D†2 ŠM 21 D ˆ M 21 D‰M 21 K 2

1 4

…33†

…M 21 D†2 Š

…34†

Eqs. (32) and (33) result in V being a Lyapunov function and, moreover, they entail that K 2 …1=4†DT M 21 D . 0 As M

21

…35†

is positive- de®nite, it follows from (35) that

M 21 …K 2

1 4

DM 21 D† . 0

so that the matrix 2M…K…1=4†DM eigenvalues. Consequently we can write

…36† 21

D† has real positive

M 21 K 14 …M 21 D†2 ˆ A2

…37†

with A2 being a convenient real matrix. It is inferred from (35) and (36) that matrix A2 and matrix A1 de®ned by A1 ˆ 2…1=2†M 21 D

…38†

satisfy the equation A 2 A1 ˆ A1 A 2

…39†

Eqs. (37)±(39) are equivalent to A 1 Ap ˆ 2M 21 D

…40†

AAp ˆ M 21 K

…41†

(this follows from (13) and (30), and a well noted Lyapunov's stability Theorem) M 21 ‰K 2

1 4

DT M 21 DŠ . 0

…45†

(this follows from (14) and (44)). Then, taking into account Eq. (31) and reusing the line of reasoning in part (i) we conclude that the linearized system eigenvalues are represented by n complex conjugate pairs. A Numerical example With reference to the data of the preceding example we have " # 2:747 0:498 21 21 …K 2 2DM D†M D ˆ 0:6675 4:74 " DM

21

…K 2 2DM

21

D† ˆ

2:747

0:6775

0:498

4:74

…K 2 2DM 21 DM†M 21 D 1 DM 21 …K 2 2DM 21 D† " # 5:494 1:175 ˆ .0 1:175 9:48 Also M

21

M

21

"

…K 2

1 4

D…KM

21

M

21

D†M

21

Dˆ

5

5:52

41:65

"

where A is the n £ n complex matrix de®ned by …42†

and A is the complex conjugate of Ap . In turn (41) and (42) result in det…A 2 lI†…Ap 2 lI† ˆ det…l2 I 1 lM 21 D 1 M 21 K† …43† Since, on the other hand, the linearized system eigenvalues are detemined by the solutions of the equation

2

1 4

M

21

T

D M

21

#

66:875

p

Ap ˆ A1 2 jA2

#

D† ˆ

66:875

5:52

5

41:65

M 21 …K 2 14 DT M 21 D†M 21 D 1 M 21 D…K 2 " # 150 10:52 ˆ .0 10:52 83:33

1 4

#

DT M 21 D†M 21

det‰l2 I 1 lM 21 D 1 M 21 KŠ ˆ 0

Therefore the conditions of Theorem 1 turn out to be veri®ed.

we see that such eigenvalues are also equal to the solutions of

4. Time-varying equivalent system

p

det…A 2 lI†…A 2 lI† ˆ 0 Consequently they are given by n complex and conjugate pairs. Moreover, since V is a Lyapunov function and …x0 ; x_ ˆ 0† is asymptotically stable, they must have a negative real part. (ii) If DT ± D, but M, K, D satisfy conditions (13), (14) and (30)±(32) then V is a Lyapunov function and, in addition: Re{li ‰M 21 …K 2

1 4

DT M 21 D†Š} . 0;

i ˆ 1; 2; ¼; n …44†

We now prove that system (1) is equivalent to a timevarying system which is asymptotically conservative if the conditions for Lyapunov stability are satis®ed. Consider the n £ n symmetric ªfunctional matrixª _ de®ned by the implicit equation M2 …x; x† _ x_ ˆ x_ T D…x…t† 2 x_ 0 † x_ T M2 …x; x…t††

…46†

ªalong the trajectoriesº _ 1 †† ˆ M2 …x…t2 †; x…t _ 2 †† M2 …x…t1 †; x…t

_ 1† ˆ 0 if x…t

U. Di Caprio / Electrical Power and Energy Systems 23 (2001) 229±235

and note that, because of (46) and (47),

Also, set _ ˆ M 1 M2 …x; x†M _ 21 M2 …x; x† _ Meq …x; x† f eq …x† ˆ f…x† 2

1 4

T

D M 1 4

_ ˆ D…I 1 Deq …x; x†

21

…47†

D…x 2 x0 †

…48†

_ M 21 M2 …x; x†

…49†

_ ˆ Deq …x; x† _ 2 DTeq …x; x† _ Geq …x; x†

…50†

(the subscript eq means equivalent). _ is a positive-de®nite matrix whose Then Meq …x; x† elements represent masses: _ .0 Meq …x; x†

…51†

and system (1) is equivalent to the following:

_ x ˆ M x 1 ‰I 1 Meq …x; x†

1 4

M2 M 21 ŠM2 x

ˆ M x 1 ‰I 1

1 4

_ M2 M 21 Š…DT x_ 2 M2 x†

_ 2 …x; x†M _ 2 …x; x†Šx1G _ 21 M _ _ x_ ˆ 0 2‰ M eq …x; x† 1 4

…52†

_ 2 …x; x† _ denotes the time derivative of M2 …x; x† _ along where M the trajectories of system (1). Eq. (52) identi®es a system with n degrees of freedom, which has time-varying masses, and is subjected to three forces: conservation positional forces 2f eq …x†; time-varying _ x_ and time-varying dissigyroscopic forces 2Geq …x…t†; x…t†† _ 2 M 21 M _ 2 x: _ pative forces represented by the vector 2‰ 14 ŠM The dissipative forces are asymptotically equal to zero if the Lyapunov condition for stability is satis®ed. It follows from (46) and (47) that

_ x_ ˆ x_ T M x_ 1 x_ T DT …x 2 x0 † 1 xT Meq …x; x†

1 4

 ‰DT M 21 DŠ…x 2 x0 † _ xŠ _ satis®es the Hence the quadratic form ‰x_ T Meq …x; x† equation _ x_ ˆ aT …x; x†a…x; _ _ x† x_ T Meq …x; x†

f eq …x† ˆ f…x† 2

M 21=2 DT …x 2 x0 †

…along the trajectories of system …1††

_ is a positive-de®nite matrix whose elements Thus Meq …x; x† represent ªmassesº. This being stated, rewrite Eq. (52) in the following way:

1 4

_ 2 M 21 M _ 2 Šx_ 1 Geq …x; x† _ x_ ˆ 0 M

1 4

_ 2 x_ DM 21 M

M x 1 DT x_ 2 M2 x_ 1

…55†

…56†

1 4

M2 M 21 DT x_ 2

1 4

M2 M 21 M2 x_ 1 f…x†

2

1 4

DM 21 M2 x_ 1 M2 x_ 1

1 4

M2 M 21 M2 x_ 1 Dx_

1

1 4

DM 21 M2 x_ 2 DT x_ 2

1 4

M2 M 21 DT x_ ˆ 0

(57)

Eq. (57) is evidently equivalent to (1) and so we ®nally conclude that (51) is equivalent to (1) and vice versa. A _ in (4) satis®es the Corollary 3. The function V…x; x† equation Z _ x_ _ ˆ f Teq …x† dx 1 …1=2†xT Meq …x; x† …58† V…x; x† and represents a Lyapunov funcion with regard to (51) provided that M, K, D satisfy condition (3). Also, the timederivative of V satis®es the equation …59†

Proof. By virtue of Eq. (48) we can derive from Eq. (4) the following relation: Zx _ ˆ f Teq …x† dx 1 12 x_T M x_ V…x; x† x0 …60† 1 4

…x 2 x0 †T DT M 21 D…x 2 x0 † 1

1 4

…x 2 x0 †T DM 21 DT …x 2 x0 † ˆ

1 2

x_ T D…x 2 x0 † ˆ

1 2

xT DT …x 2 x0 †

…53†

1 2

1 4

xT M2 M 21 M2 x_

…61†

x_ T M2 x_

we get V _ ˆ V…x; x†

_ x 1 f eq …x† Meq …x; x† 1 ‰Meq 2

M2 M 21 M2

From Eqs. (46) and (60), that entails that along the trajectories

Consequently _ .0 Meq …x; x†

1 4

Inserting above equations into (47), besides accounting for (49) and (50), and recalling that M is symmetric one obtains from (52)

1 1 2

M2 M 21 M2 ˆ M2 1

Also, because of Eqs. (47) and (49),

with _ ˆ M 1=2 x_ 1 a…x; x†

1 4

Meq 2

_ 2 x_ _ ˆ 2…1=2†xT M V…x; x†

…x 2 x0 †T

…54†

while

d _ xŠ _ 1 f eq …x† ‰M …x; x† dt eq

Proof.

233

Zx x0

f Teq …x† dx 1

1 2

_ x_ x_T Meq …x; x†

…62†

This proves the ®rst part of the statement. Also, from (51)

234

U. Di Caprio / Electrical Power and Energy Systems 23 (2001) 229±235

Since

Table 1 Time behaviour of the equivalent inertia constant t

d…t†

d_ …t†

0 1 2 3 4 5 6 7 8 9 10

0.2 0.696 0.615 0.5878 0.549 0.5372 0.53 0.527 0.525 0.5218 0.519

0 0.00013628 0.0136 0.025 0.017 0.0118 0.0077 0.0036 0.0026 0.00176 0.0006

M2

Meq

1 174.54 0.96 0.3564 0.21 0.04 0.125 0.1527 0.10576 2 0.09 2 0.084

1 1.314 1.1875 0.4679 0.319 0.275 0.23468 0.2616 0.2097 0.02 0.025

_ x† _ ˆ f Teq …x†x_ 1 x_T Meq x 1 V…x;

1 2

_ eq x† _ x_ T M

ˆ f Teq …x†x_ 2 x_T …Meq x_ 1 f eq …x† 1 1 2

1 4

_ 2 M 21 1 M2 x_ M

_ eq x_ x_ T M

d ‰M …t†d_ Š 1 feq …d† 2 Pm 2 dt eq

1 4

_ 22 M d_ ˆ 0 M

with Meq …t† ˆ M 1 M2 …t† 1

_ 22 1 M 4 M

M2 …t† ˆ D‰d…t† 2 d0 Š=d_ …t†; M2 …t1 † ˆ M2 …t2 †

feq …d† ˆ Pe …d† 2

D2 …d 2 d0 † 4M

In the ®rst approximation d…t† and d_ …t† can be determined from the linearized system equation; hence

d…t† ˆ ke2at cos…v t 1 w†

_ eq x_T ‰ 1 …M _ 2 †Šx_ _ 2 M 21 M2 1 M2 M 21 M ˆ 2 x_ M 8 1 2

both conditions turn out satis®ed. Also, system (63) is equivalent to

if d…t1 † ˆ 0

and (62) one gets

_ x† _ 1 1 Geq …x; x†

KM 21 D 1 M 21 DK ˆ 3:8563

D ˆ 0:1375;

T

d_ …t† ˆ ke2at ‰2a cos…vt 1 w† 2 v sin t 1 wŠ†

It follows from the above equation and from (47) that _ 2 x_ _ x† _ ˆ 2 12 x_T M V…x;

where a ˆ 0:625; v ˆ 6:31 …for D ˆ 0:1375†: k; w are real constants to be determined from the initial conditions. For example, if

5. A numerical example referring to EPS

d…0† ˆ 0:2;

we have k ˆ 1:195; w ˆ 3:043

Consider the scalar system M d ˆ Pm 2 Pe …d† 2 Dd_

d_ …0† ˆ 0

…63†

d…t† ˆ 1:195e20:625t cos…6:31t 1 3:043†

where M ˆ 0:10968; Pe …d† ˆ 5 sin d; D ˆ 0:1375; Pm 2:5: The point …d0 ˆ 0:5236; d_ ˆ 0† is an Equilibrium point. Linearizing Eq. (63) one gets

d_ …t† ˆ 1:195e20:625t ‰20:625 cos…6:31t 1 3:043†

d_ ˆ 2M 21 K…d 2 d0 † 2 M 21 Dd_

M2 …t† ˆ 0:1375‰d…t† 2 0:5236Š=d_ …t†

ˆ 239:478…d 2 d0 † 2 0:1375d_

…64†

with

2 6:31 sin…6:31t 1 3:043†Š

Meq …t† ˆ 0:10968 1 M2 …t† 1

_ 22 1 M 4 0:10968

In Table 1 are reported the values of d…t†; d_ …t†; M2 …t†; Meq …t† in the interval (0±10 s).

K ˆ 5 cos d0 ˆ 4:33 The V-function is represented by

6. The signi®cance of matrix D in EPS

V…d; d_ † ˆ 5…cos d0 2 cos d† 2 2:5…d 2 d0 † 1 0:06875…d 2 d0 †d_ 1 0:0548d_ 2

…65†

The two following conditions are suf®cient for V to be a Lyapunov function 2D . 0 KM 21 D 1 …M 21 D†K . 0

The analysis of transient stability of a multimachine electric power system is successfully carried out making reference to a second-order model for each machine. The introduction of the dissipative force 2Dd_ allows us to take into account the damping effects, upon the electromechanic oscillations, of the ®eld circuit and of the damper windings of the machines, plus the effect of possible stabilizing devices (PSS).

U. Di Caprio / Electrical Power and Energy Systems 23 (2001) 229±235

A concrete way of determining D is the following. Assume that the effective oscillations, as determined, e.g. from ®eld tests or from a multimachine model in which each machine is represented by a convenient ®fth-order model, are ªdamped oscillationsº identi®ed by n pairs of complex conjugate eigenvalues:

li ˆ 2ai 1 jvi ;

i ˆ 1; 2; ¼; n

ai ; and where  vi are positive real constants such that 6i ˆ p ai = a2i 1 v2i equals the ªexperimentalº damping factor of the i-th oscillation (according to Control Theory). Assuming that v2i ˆ li {M 21 K} the problem is to determine matrix D so that, in the simpli®ed EPS model, the electromechanical oscillations have damping factors equal to the assigned ones. The solution can be found determining D from 3 2 2a1 7 6 7 6 2a 2 7 6 7 6 7 21 6 21 ´ M D ˆ G6 7G G 7 6 7 6 7 6 ´ 5 4 2an where G is the matrix of the eigenvectors of M 21 K : M 21 K ˆ GLG21 Then the characteristic equation det‰l2 I 1 lM 21 DM 21 KŠ ˆ 0 becomes equivalent to 2 2 2a1 6 6 6 6 2 a2 6 6 6 6 6 6 2 det6l I 1 l6 6 6 6 6 6 6 4 4

3 7 7 7 7 7 7 7 7 7 5

´ ´ 2a n

2 6 6 6 6 6 16 6 6 6 4

33

v21

77 77 77 77 77 77 ˆ 0 77 77 77 55

v22 ´ ´

v2n

235

yielding the desired solution

li ˆ a 1 jvi The following example illustrates this. Assume that " # " # 4:5 0 0:1097 0 Kˆ ; Mˆ 0 0:55 0 0:12 Then M

21

"

Kˆ

41:02 4:558 4:145 41:65

v21 ˆ 36:96; " Gˆ

#

v22 ˆ 45:705

1

1

20:989

1:027

#

If the assigned damping factors are 61 ˆ 0:1; 62 ˆ 0:1; then constants a1 and a2 turn out determined from q q a1 = a21 1 v21 ˆ 0:1; a2 = a22 1 v22 ˆ 0:1 yielding a1 ˆ 0:611; a2 ˆ 0:689: Matrix D is given by " " # 0:071 0:611 0 21 D ˆ MG G ˆ 20:0051 0 0:689

20:0437

#

0:078

One can check that M, K, D satisfy the stability condition (3). 7. Conclusion We have shown general matrix relations that allow to express, in closed form, the conditions for the stability in a large multimachine power system. The results are framed in the general context of the theory illustrated in Ref. [1]. References [1] Di Caprio U. The effect of friction forces upon transient stability. Int J EPES (in press).