Development of an energy function reflecting the transfer conductances for direct stability analysis in power systems

Y.-H. Moon E.-H. Lee T.-H. Roh

Indexing terms: Energy function, Transfer conductances, Direct stability analysis, Power systems

Abstract: Direct stability analysis using an energy function has recently been widely used for angular and voltage stability analysis in electric power systems. Considerable efforts have concentrated on seeking energy functions for lossy power systems, accomplished only for the simple two-bus system. A new approach is presented to derive an energy function by using the complex integral of bus current equations with respect to bus voltages. Under an assumption that all transmission lines have a uniform RIX ratio, it is shown that the energy function can be developed for multimachine power systems with losses. The proposed energy function is tested on sample systems by comparing the time simulation method, which shows the validity of the proposed method.

1 Introduction

Power system stability analysis by time simulation methods requires much computation time as the system size increases. This gives an impetus to apply the direct method by an energy function to practical power sys- tems. A great deal of effort has been devoted to con- structing the energy function or Lyapunov function to reflect the system accurately. However, reflecting trans- fer conductances into the energy function still remains an unsolved problem for multimachine power systems.

Pai et ul. [l] proposed a local energy function for two machine systems with a lossy transmission line. Gudaru [2] attempted to extend this local energy function to multimachine systems. Henner [3, 41 showed later that his local energy function is applicable for only two machine systems. Uemura et al. [5] also derived an energy function by assuming a linear path of integra- tion for the path-dependent terms. Chiang [6] discussed the non-existence of the general energy function for multimachine power systems with losses. On the other hand, the energy function method has also been

0 IEE, 1997 IfiE Proceedings online no. 19971458 Paper first received 25th April 1996 and in revised form 26th March 1997 The authors are with the Department of Electrical Engineering, Yonsei University, Seoul 120-749, Korea

applied to voltage stability analysis, which again emphasises the importance of this subject [7, 81.

This paper presents an energy function to reflect the line resistances exactly in case of the uniform RIX ratio, which can be evaluated as a remarkable theoreti- cal improvement in direct stability analysis. It also presents several new theorems related to the energy integration. These theorems play important roles in proving the seminegativeness of the time derivative of the proposed energy function.

The proposed energy functions can be applied to improve the accuracy in angular stability analysis of a system with a large RIX ratio or voltage stability analy- sis greatly affected by the line resistances.

2 conductances

New energy function reflecting transfer

The power system can be represented by a multibus network with bus admittance matrix Ysus as shown in Fig. 1. Each generator can be considered as a complex injection power supplier.

Fig. 1 Network representation of multibus system

To begin with, we will introduce several new theo- rems related to the integral relationships between the generator input and output. Theorem 2.1: The energy integral of generator output power can be represented by the cross product line integral of current and voltage phasors as follows:

IEE Proc-Cener. Transm. Distrib., Vol. 144, No. 5, September 1997 503

where VGi = ( VDi + jVQi)e-jd2 is the phasor voltage rep- resented on the system reference, IGi = (IDi + jIQi)e-jx’2 is the phasor current represented on the system refer- ence, and PGi + jQGi = VGi ’ IC:. Proofl

= C /vGz Re [ PG% + ~ Q G %

- - c kG,, Im(Ic*;.zdVGz)

VGio VG, 3 VG L

Theorem 2.2. If the statorlsystem transients are negligi- ble for generator i, it holds that

1 2

PmzdSz - -Mzwz - 1 D,w:dt

E,% + c kgot (IdtdEqi - IqidEdz) (2)

The proof is given in the Appendix (Section 7.1). Theorem 2.3:

with Qei = 1m[E~iI&] 5 EqiIrji - EdiIqi

This can be proved in a similar way to Theorem 2.2. This study adopts a new approach to derive an

energy function by using a complex integral of bus cur- rent equations with respect to bus voltages. By virtue of the theorems developed here, this complex phasor integration yields two kinds of energy integral expres- sions for the real and imaginary parts. The latter gives a structure preserving energy function similar to a con- ventional one, and the former a new kind of energy function (second kind energy function). By using these results, we have derived a new energy function which can reflect the line resistances exactly in case of the uni- form R/X ratio. Here, we will present only the final results rather than providing the long derivation proce- dure.

Let’s assume all transmission lines have uniform R I X ratio, i.e.

K = Rij fX iJ ’da and j (4) Now, let the system energy function be as follows:

1 U = p z w :

z

504

m n m , -

( 5 )

where 1 1

B - zzo -E-- - Xi , and B~~~ - xi, By using the theorems introduced earlier, it can be shown that the energy function in eqn. 5 has the semi- negativeness of its time derivative:

dv - = - C D i w ? 5 0 d t

The proof is given in the Appendix (Section 7.2).

3 generator models

The proposed energy function in eqn. 5 includes many path-dependent integral terms related to generator var- iables. These terms should be removed for the practical application by considering the model of the generator.

3.1 Energy function with classical model of generator In the case where all generators are represented by a classical model, the energy function in eqn. 4 can be further simplified. We assume that all generators are of ideal type with no leakage flux and thus the internal impedances are all zeros. (i) XqL = X d L = X d , = 0. (ii) Pm, = constant. (iii) EGL = constant (let EG, = EqL’). Under these assumptions, all the integral terms related to the generator inteneral voltage vanish to zero. By using these results, the energy function (eqn. 5) can be simplified as follows:

Energy function with consideration of

N r

(7 )

IEE Proc.-Gener. Transm. Distrib., Vol. 144, No. 5. September I997

where Q e 2 = I ~ ( E G & )

N [E;, - E G z 4 cos(St - e,)] =E 2 = 1 X2,(1+K2)

The above energy function satisfies the condition of dvldt s 0 and the convexity in the vicinity of an operat- ing point. This guarantees that the energy function in eqn. 7 can be a local Lyapunov function.

3.2 Energy function with Ed model of generator Consider the derivative of energy integral for a multi- machine system with the Eq' model for the generator. In this example, the generator resistances are neglected in the generator current analysis for simplicity. Then, the Eq' model of the generators can be represented as follows:

dui M.- zz T,; - E ' . I q z 4 i - ( X d ; - XAt)IdiIqi- DW;wi dt

(86)

Idi (E;i - & cos 6 L i ) / x i , (9n) Iy; = V; sin GL;/Xqi (9b)

where S L . - - 6. 2. - 8. - - angEqi -angV,

is the load angle.

tions between Ed/, Eg/ , Edi and EqL: For the above Eq' model, we have the following rela-

EA2 = (Xp - x;$q%

Edi = E d i , + Xi ; Iy ; = Xy;Iq; E,i 1 E ' . q t - X' d i I d t .

(104 ( l o b ) (10c)

To solve the path-dependent integral, the energy func- tion (eqn. 5 ) is examined term by term. Here, it is required to make some appropriate assumptions in order to get a path-independent energy function from the above path-dependent energy integral: (i) All governors and exciters are fixed (Pmi and Eql' are constant). (ii) All loads are constant power loads. Under these assumptions. the following energy function is obtained from eqn. 5 by using theorem 2.3:

1 U = p i u :

1 r 1

m.

where

The above energy function satisfies the property of the semi-negativeness of the time derivative and the con- vexity in the vicinity of the operating point, which are the necessary and sufficient conditions for the local Lyapunov function.

E L 6 $--&4!:: bus

Fig.2 One muchiine infinite bus system

4 Illustrative example: Energy function derivation using classical model

4. I Test for a one-machine infinite bus system The proposed energy function is tested on the one- machine infinite bus system as shown in Fig. 2. By using eqn. 7, the energy function for the above system is:

1 2 1 v 2 E V U = -Mw 2 + (x + g) - cos6

sin S I<E2 K E V

X ( K 2 + 1)' + X ( K 2 + 1) + cos6 - Pm6 + C(VO, Eo, 6,) K 2 V E

+ X ( K 2 + 1) (12)

where K 1 RIX and C(V,, Eo, 6,) is the constant value reflecting the intial state energy. Here, we can let C(V,, E,, 6,) be zero without the loss of generality and the energy function can be rewritten as follows:

u = - M u 2 - BEV cos S + G E V sin 6 - P,b 1 2

(13) B(K2 + 1) (E2 + V2) 3

+ G E ~ ~ + where

1 K 1 1 and B = - - - . G = - - . x (K2+1) x ( K 2 + 1)

The semi-negativeness of the time derivative of the above energy function can be easily shown to be:

~ du(w,S) - - - -+ - -= -Dw d u dw aud6 2 5 0 d t aw dt d6 d t

This energy function can be compared with the follow- ing energy function developed by Chiang [6].

vcon(6,w) = - M u 2 - B E V c o s d + G E V s i n S 1 2 - P,S + G E ~ S (14)

505 IEE Proc.-Gener. Transm Distrib., Vol. 144, No. 5, September 1997

The differences between eqns. 13 and 14 result from the fact that the energy function (eqn. 14) is derived under the assumption that bus voltage is constant. Conse- quently, it is obvious that the energy function proposed here is a more generalised energy function.

4.2 Test for a 3-bus sample system The proposed energy function is tested for the follow- ing 3-bus sample system, where all generators are mod- elled by the classical model. We also adopt the assumptions of constant voltages at all generator buses and constant powers in all loads. For simplicity, the generator saliency is also neglected. The data .of the above sample system are given in Tables 1 and 2. The system state before faults is given in Table 3.

Table 1: Synchronous machine data for 3-bus system

Direct axis

reactance, xb

Inertia constant, Bus code, P transient

1 160 0.1

2 3 0.3

Table 2: Line admittances and line charging for 3-bus system

Half line Bus code, Admittance, P q

Line no. charging, YP4 YbqI2

1 1-2 0.37 -j5.88 j0.15

2 1-3 0.74-j11.77 j0.07

3 2-3 0.58 -j9.17 j0.04

Table 3: Base load flow results for 3-bus system ~ ~

Generator Load Buscode, Bus

P "P MW MVARs MW MVARs

1 1.04LO" 212.14 93.26 0.0 0.0

2 1.02L3.0" 100.0 70.0 50.0 20.0

3 0.93L8.12" 0.0 0.0 250.0 150.0

The energy function for the above system is given by eqn. 11. The path-dependent integration should be cal- culated along the system trajectory. However, the sys- tem trajectory cannot be obtained before solving the system differential equations, which is against the objective of direct stability analysis. In this study, an approximate integration method is introduced by eval- uating the path-dependent integral term along the straight line path instead of the system trajectory (see Appendix (Section 7.3)). The simulation results are compared with those of the time simulation method.

Table 4: Stability analysis for 3-bus system

The critical clearing time has been obtained by the energy function method and by the conventional time simulation, and both results are compared. In order to analyse the system stability, the system equilibrium points should be calculated by solving the partial deriv- ative equation of the energy function, which is just the same as the power flow equations. By solving the power flow equations, stable and unstable equilibrium points can be obtained. In order to obtain unstable equilibrium points, we adopt the minimisation tech- nique by considering the problem of min lfg2 + f,"]. The system stability can be determined by comparing the lowest UEP energy with the system energy just after fault clearing. Stability analysis is performed with vari- ous RIX ratios for the sample system.

The above results show that the proposed energy function yields the results closer to the time simulation results by considering the effects of transfer conduct- ances. In case of K = 0, the energy function is just the same as the conventional one. When K = 0.063, 0.105, the proposed method gives the reduced critical clearing time by 0.5-1.0 cycle as compared with a conventional direct method, which is an expected accuracy improve- ment from considering the effects of transfer conduct- ances. This accuracy improvement in the critical clearing time may prevent serious misleading in the power system operation by the conventional energy function method. The proposed energy functions can be applied in many fields such as stability analysis of a system with a large RIX ratio or voltage stability analy- sis greatly affected by the line resistances.

5 Conclusions

A great deal of effort has been devoted to constructing the energy function or Lyapunov function to reflect the system transfer conductances, and the importance of this subject has further increased since the energy func- tion method has been applied to voltage stability. This paper shows that an energy function reflecting the transfer conductances can be developed for a multibus system with uniform RIX ratio for all transmission lines, which can be evaluated as a theoretical improve- ment in direct stability analysis. The proposed energy function provides an approximated direct Lyapunov approach to improve the accuracy of stability analysis for a multibus system with high RIX ratio, and is espe- cially applicable to voltage stability analysis which is greatly affected by the line resistances. The validity of the proposed energy function was verified through the sample test.

Proposed Proposed Proposed energy function energy function energy function

Conventional energy function ( K = 0) ( K = 0.063) ( K = 0.105)

vuep (energy at UEP 5.7277 5.7277 5.6378 5.6352

CCT (critical clearing time) by 0.275 (s) 0.275 (s) 0.267 ( s ) 0.265 (s) energy function method

v,p(energy at CCT) 5.7491 5.7491 5.6170 5.6093

Energy and CCT at critical clearing 5.7488 at t=0.271 5.7488 at t=0.271 5.6470 at t=0.268 5.6391 at t=0.268 by time simulation point

506 IEE Proc-Gener. Transm. Distrib., Vol. 144, No. 5, September 1997

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References

PAI, M.A., and MURTHY, P.G.: ‘On Lyapunov functions for power systems with transfer conductances’, ZEEE Trans. Autom.

GUDARU, U.: ‘A general Lyapunov function for multimachine power systems with transfer conductances’, Znt. J. Control, 1975,

HENNER, V.E.: ‘Comments on “On Lyapunov function for power systems with transfer conductances’, IEEE Trans. Autom. Control, 1974, AC-19, (5) , pp. 621-623 HENNER, V.E.: ‘Comment on “A general Lyapunov function for multimachine power systems with transfer conductances’, Int. J. Control, 1976, 23, (l), pp. 143 UEMURA, K., MATUSKI, J., YAMADA, J., and TSUJI, T.: ‘Approximation of energy functions in transient stability analysis of power systems’, Electr. Eng. Jpn., 1972, 92, (6), pp. 96-100 CHIANG, H.D.: ‘Study of the existence of energy functions for power system with losses’, ZEEE Trans. Circuits Syst., 1989,

DEMARCO, C.L., and OVERBYE, T.J.: ‘An energy based secu- rity measure for assessing vulnerability to voltage collapse’, ZEEE Trans. Power Syst., 1990, PWRS-5, (2), pp. 419426 OVERBYE, T.J., and DEMARCO, C.L.: ‘Improved technique for power system voltage stability assessment using energy meth- ods’, ZEEE Trans. Power Syst., 1991, 6, (4), pp. 1446-1453 PAI, M.A., and VARWANDKAR. S.D.: ‘On the inclusion of transfer conductances in Lyapunov function for multimachine power systems’, IEEE Trans. Autom. Control, 1977, AC-22, (6),

Control, 1973, AC-18, (2), pp. 181-183

21, (2), pp. 333-343

CAS-36, pp. 1423-1429

pp. 983-985

Appendix

7.1 Proof of Theorem 2.2 The full model of generator gives the following termi- nal voltage equations:

1 d@, = Vo --

us d t where U+ = U, + U and U, is the rated angular velocity in [pu].

By adding eqn. 15 and j x eqn. 16, we obtain 1 d w+ .

- - (@d + j Q q ) = - - J ( Q ’ d + j Q q ) + ( V d +jVq) w, d t W .9

(18) With the use of phasor representations, the above equation can be rewritten as

For generators comected to a power system, the electri- cal dynamics of the statorinetwork can be assumed in general to be very fast, so that all flux linkages change instantaneously during transients. With the use of this assumption, eqns. 15-17 can be rewritten as

W+ (20) -_ j!Pg + 1<=0

WS

The system-referenced terminal voltage has the follow- ing relation:

(21) VG, z - - V gz , e j ( L - r / 2 )

The substitution of eqn. 20 into eqn. 21 gives

By using eqn. 22 and transforming I,, into the genera-

IEE Proc -Gen?r Transm Distrzb , Vol 144, No 5, Septembev 1997

tor-referenced current, we have the following equation: VG z

c LGto Im(I&dVGz)

( 2 3 ) On the other hand, the generator induced voltages are given by

The electric output of the generator is also given by U+

p e z = EqzIqz f EdaIdz = - ( I q t @ d z - I d z q q z ) ws

Consequently, the substitution of the above equations into eqn. 23 yields

% z

c LGT0 lm(l&dVGz)

= c 1;; PezdJz + c Lr:: (IqzdEdz - IdzdEqz)

By substituting the relationship P,, = P,, - M,w, - D , y into this equation, one can easily obtain eqn. 2.

7.2 Proof of semi-negativeness of energy function (eqn. 5) By differentiating both sides of eqns. 2 and 3 with respect to time t , the following can be obtained:

dfl, Q G ~ dT/, d t V, d t PGz- + --

By using the differential chain rule, we can get the fol- lowing time derivatives of the energy function (eqn. 5):

507

( 2 6 ) Substituting eqns. 24 and 25 into eqn. 26, the first and the fifth terms can be replaced. Eqn. 26 can then be simplified to

dv - = - C D ~ w : dt

2

r

(27) On the other hand, Ysus = [G + jB] and K is defined

as K = R i X . The admittance G and B can be repre- sented by

G -L(-) K ” - X,, K 2 +1 B -L(-) -1

23 - X,, K 2 + 1 1

X,, K 2 + 1 3 3 -- ( G,, = C(-G, , ) =

We also have the following relations: G,, = -KB,, G,, == -KB,,

= - ( K 2 + l )B%, (29) 1

BaJO - Xt3

Here, we can investigate eqn. 27 term by term. The coefficient of d0,ldt in eqn. 27 is given by

PG, - PL, - K(QG, - QL,) + BzJoVz& sin Q,, 3+

= pG% - P L z

- (KG,, KV, sin Q,, - KB,, KV, cos Q,, ) J

+ -B,~ (K’ + I)T/,v, sin Q,, I f ,

= PG, - PL, + K B , , ~ ~

- x ( K G t , & V , sinQ,, - KB,,&& COSQ,,)

3 f t

+ - B , ~ (K’ + I ) V ~ V , sin o,, I#%

= PG, - PL% - G,,K2

- C(G,, C O S Q , ~ + E,, sinQ,,)xV, 3#,

= o (30) The final result of the above equation is just the load flow equation for the real powers, which makes the coefficient zero.

In the same way, the coefficient of dVJdt in eqn. 27 can be written as follows:

QG, - Q L , + K(PG, - PL,)

- E,,, K2 - B,, 0 K V, COS Q,, ,#,

= QG, - Q L ~

+ ~ ( K G , , COSO,,V,V, + KB,, s i n ~ , , x & )

- B,,, K2 -

= QG, - Q L ,

3

B,, I4 V, cos e,, 3 5 %

+ ~ ( - K ~ B , , cos^,,^,^, - G,, sinQ,,xx)

- C ( - ( K 2 + l)B,,K& COSO,,)

3

3

= QG% - Q L ~ + C ( - G , , sinO,,V,&) J

= o (31) The final result of the above equation is just the same as the reactive power balance equation, which also makes the coefficient zero. Therefore, dvldt can be rep- resented as follows by substituting eqns. 30 and 31 into eqn. 27:

__ dv = - C D , W ; 5 0 d t

2

This shows the semi-negativeness of dvldt.

7.3 Approximation of energy function (eqn. 11) The path-dependent integrals included in eqn. 11 should be evaluated along the system trajectory. How- ever, the system trajectory cannot be obtained before solving the system differential equations, which is against the objective of the direct stability analysis. We need to introduce an approximation method to evalu- ate it without the knowledge of system trajectory. Therefore, this study adopts an approximated integral by evaluating the integral along a straight line connect- ing two points rather than the solution trajectory. The straight will be denoted by C . A point (a, v) on the straight line F can be represented by the use of param- eter h such that

508 IEE Proc.-Gener. Transm. Distrib., Vol. 144, No. 5, September I997

where yo = 0 - 0, y v = v- v,

E [0,11 From the above equation, the differentials of e,, and V j are as follows:

vi e, = 8,; + yo; . x e3 = Oog + Yeg . x V, = VO, + ?vi . X q = voj + YVJ . x

ddi = ~ 0 i . dX d8, = yo.7 . dX d K = yr/; * dX dV, = y v j . dX (34)

If the initial and the terminal points are given, we can approximately evaluate the system energy between them by integrating it along the straight line C.

The third term in eqn. 11 can be evaluated through the following approximated calculation:

(35)

(Note that yo[ = e, - e,, = AO, and yv, = V, - V,, = A K ) . In a similar way. the other path-dependent terms can

be approximately evaluated as follows:

KE;, N - (cos b L , - cos SOL2) (37) -xiL A6LZ

By substituting the above equations into the energy function (eqn. 1 I), we can get the approximated energy function in the case of the generator saliency neglected.

IEE Proc-Gener. Transm. Distrib., Vol. 144, No. 5, Septenzbes 1997