1454 IEEE TRANSACTIONS ON CIRCUITS AND SYSTFMS, VOL. 36. NO. 11, NOVEMBER 1989

R. M. Biernacki and M. A. Styblinski. “Statistical circuit design with a dynamic constraint approximation scheme.” in Proc. I E E E Inr. S-vnip. Circuits und Svsrenis. San Jose. CA, pp. 976-979, 1986. S. W. Director and G. D. Hachtel, “The simplicial approximation approach to design centering,” I E E E Truns. Circuits s p s r . , vol. CAS-24, pp. 363-372. 1977. R. S. Soin and R. Spence. “Statistical exploration approach to design centering,” Proc. Insr. Elm. Eng., vol. 127, pt. G.. pp. 260-269, 19x0. M. A. Styblinski and A. Ruszczynski, “Stochastic approximation ap- proach to statistical circuit design,” Electron. Letterr. vol. 19, no. 8, pp. 300-302. 19x0. E. Polak and A. L. Sangiovanni-Vincentelli, “Theoretical and computa- tional aspects of the optimal design centering, tolerancing. and tuning

Truns. Circuits Syst.. vol. CAS-26, pp. 795-813. 1979. F. Pinel, “Statistical design centering and tolerancing

using parametric sampling,” I E E E Truns. Circuits Syst., vol. CAS-28, pp. 692-701, 1981. J. W. Bandler and S. H. Chen, “Circuit optimization: the state of the

Truns. Micronuije Theor); Tech.. vol. MTT-36. pp. 424-443. 198X. E. Wehrhahn and R Spence. “The performance of some design center- ing methods,” Proc. I E E E In t . Symp. Circuits and Spstenis. Montreal, Canada, pp. 1424-1438, 1984. J. W. Bandler. S. Daijavad, and Q. J . Zhang. “Exact simulation and sensitivity analysis of multiplexing networks,” IEEE Truns. Microwuile Theory Tech.. vol. MTT34, pp. 93-102. 1986.

Identification Via Fourier Series for a Class of Lumped and Distributed Parameter Systems

B. M. MOHAN AND K. B. DATTA

Abstract -Operational matrix of integration, as well as one shot opera- tional matrix for repeated integration (OSOMRI) is used in this paper to estimate the parameters, and initial and boundary conditions of linear time-invariant (LTI) lumped parameter systems. It is demonstrated that OSOMRI provides better accuracy over conventional operational matrix of integration. Moreover, an algorithm for distributed parameter system identification via Fourier series is also included. Finally, a comparative study of the estimates obtained by the proposed method for both the systems with those available in the literature by other methods is also carried out.

I. INTRODUCTION

Identification of lumped parameter systems (LPS) via Block- Pulse and Walsh functions [2], Chebyshev first and second kinds, Legendre and Jacobi, Laguerre and Hermite ([l] and the refer- ences therein) has already been studied. Similarly, others have used various orthogonal functions [5] apart from Walsh functions [ 3 ] for distributed parameter systems (DPS) identification. Inspite of these large number of contributions, it appears that no con- certed effort has been made to explore the potentiality of Fourier series in system identification which consequently forms the main objective of this paper.

In the context of analysis of linear time-invariant (LTI) sys- tems, Paraskevopoulos et al. [4] have already developed an inte- gration operational matrix for Fourier series. In this paper, a more general operational matrix, with a merit that it can be applied over any arbitrary finite interval of integration, is devel- oped. In order to improve the accuracy of operational matrix for

Manuscript received March 10, 19x7; revised April 23, 19x8 and February 16, 1989. Thia paper was recommended hy Associate Editor R. Liu.

B. M. Mohan was with the Indian Institute of Technology. Kharagpur. India. He is now with the Department of Electrical and Electronics Engineer- ing, Regional Engineering College, Tiruchirapalli 620015, India.

K. B. Datta is with the Department of Electrical Engineenng. Indian Institute of Technology. Kharagpur 721 302, India.

IEEE Log Number 8929761

repeated integration of Fourier basis vector the concept of OSOMRI, originally introduced by Rao and Palanisamy [2] in connection with Walsh functions, is introduced. Essentially based on this OSOMRI, an algorithm for the identification of LTI single-input single-output (SISO) continuous LPS is presented in addition to an algorithm for the identification of first-order continuous DPS in this brief. Numerical examples with detailed comparison of results obtained via proposed Fourier series method and other existing methods are provided.

11. PRELIMINARIES OF FOURIER SERIES

The Fourier series or Fourier expansion corresponding to a function /( t ) , which satisfies Dirichlet conditions, is approxi- mately given by

n - 1

/ ( I ) = A ) + o ( t ) + c [/ ,+,( t )+h*+W>] = f 7 + ( r ) (1) , = 1

where the Fourier coefficients fo. f , and f,* are

f0 = a / r f / ( t ) dt 1,

f , = 2a/”/( t ) +, ( t ) dt: A* = 2a/”f( t ) +;* ( t ) dt I, t ,

/( t j - 1 , )

A. Operutionul Matrix /or Integrution of Fourier Basis Vector +(t)

Operational matrix for integration of +( r ) , denoted by E , may be obtained from the following steps.

i) ii)

iii)

Integrate every element of + ( t ) with respect to t , Express the result of step (i) in truncated Fourier series, Put the result of step (ii) in a vector-matrix form to obtain

p T ) d T = E + ( i ) (6)

where E is a nonsingular (2n - 1) x (2n - 1) matrix having the form

with

H’ = [ 1, - 1/2,1/3; . . ,( - l ) ” / ( n - l)] ’ OO98-4094/89/11OO-1454$01 .OO 01989 IEEE

I€EE IRANSACrlONS ON CIRCIJITb AND SYSTEM>, VOL 36. NO 11, NOVEMBER 1989 1355

and

Since the first row elements of E are simply the Fourier spectrum of ( t - t , ) , integration becomes more and more accurate as n increases. Inequality sign in ( 6 ) signifies the fact of truncation of Fourier series of ( t - I ,) .

B. OSOMRI of +(t)

For k times integration of +( t ) , (6) becomes

-7

k times

It is equivalent to repeating all three steps in the previous section k times-each time with the same amount of error due to truncation of Fourier series in the development of E . That means thia algebraic approach, even though seems simple, will lead to an accumulation of error at each stage of k stages of integration. Hence, in order to bring down the error in repeated integration we introduce the concept of one shot operational matrix for repeated integration (OSOMRI) here. According to this, we first integrate every element of + ( t ) k times with respect to t and then express the result in truncated Fourier series so that the error of approximation is restricted to one (final) stage of integra- tion only. We show this mathematically as

v

k times

where EA is the intended nonsingular (2n - 1)X(2n -1) OS- OMRI. For instance, if k = 2,3 and n = 2 we have

1/4- 1/2n2 - 1/2n2 1 / 2 ~

-- 3/47? - 1/47' - 1/47'

0

- 1/47 - 3/47>

1 1

1/8- 1/2n2 -- 1/4n2 1/47 - 3/4n3 0 - 3/8n3

- 1/87 t- 3/87' 3/8n3 - 1 / 4 d

1/24 - 1/4n2 1 / 6 ~ - 1/4n3 0 - 3 /8n3

- 1,427 -t 1 / 8 d 3/8a3 - 1/4n2

It is interesting to note that the elements of Eh are recursively related by the following expressions:

form an orthogonal system in the rectangle x, < x < xf, t , < r d t,. Just similar to + / ( t ) and +:(t) in (3), + 1 ( ~ ) , , =0,1;.. and +:(x), 1 =1 ,2 , . . is also a complete orthogonal system of cosine-sine functions in the interval x, < x < x, Uf f ( x. t ) can be expanded in a uniformly convergent double Fourier series, the series is approximately given by

where the double Fourier coefficients are

1456 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. 36, NO. 11, NOVEMBER 1989

with ; = [ A , ] , P = [ L r ] , F=[f , ]and p = [ f r k ] , i=O,l ; . . ,m , = [ I , o;..,o I T

- 1; j = 0,l; . ., n -1; k =1,2;. ., n -1; r=1,2;. ., m -1. - Now, from (7) to (9) it is possible to write 2( I - 1) terms

-- v times p times

= +W( E:)'FE,Y+(t) or

where = +'( x) Ex', FE,"+ ( t 1 (10)

E, - (2m - 1) X ( 2 m - 1) operational matrix for integration

of + ( x )

of + ( t )

E, - (2n - 1) x (2n - 1) operational matrix for integration

and Eh,, and E,, are, respectively, (2m -1)X(2m -1) and (2n - 1) X (2n - 1) OSOMRI's.

111. IDENTIFICATION OF LUMPED PARAMETER SYSTEMS

Consider a lumped LTI SISO continuous dynamical system modeled by the differential equation

y'"'( t ) + u , l - l y ( ~ l - l ) ( 1) + . . . + a,y( ' ) ( t ) + a,y( t )

= b , , , u ( " ' ) ( t ) + b , , , - l ~ ( " ' - l ) ( t ) + . . . + b , d ' ) ( t ) + b , u ( t ) (11)

in which U( t ) and y ( t ) are, respectively, the input and output of the system, assumed available over an arbitrary active finite interval t E (t,, t f ) ; n is the order of the system, assumed known; an- 1, a, 2 , . . . , a,, b,,, , b,- ,,. . . , bo are the parameters of the system to be identified, and m < n.

Let the initial conditions (IC) of (11) be

} (12) a , = y ( " ( t , ) , i=O,1,2; . . , (n-1)

P, = U(')( t , ) , i = O,1,2 ,..., ( m -1).

Even though a. and Bo are actually known from the output and input records, they will be treated as unknowns along with the other IC's in the identification process as they may not represent true values in noisy situation [8].

Now expand input and output signals in finite Fourier series to have

1-1

u ( t ) = .o%(t) + c [ U,+,(f) + .7+7w] = uT+( l )

A t ) =Yo+o( t )+ c [ Y , + , ( t ) + Y , * + f W ] = Y ' + ( t ) . (14)

(13) / = I

/-1

/ = I

Integrate (11) successively n times with respect to t to get an integral equation, introduce (13) and (14) in the integral equa- tion, make use of property in (7) or (8) and simplify to get

Q P = Y (15)

A =

a2 a3

. . . 1 a n - 1 an-2

0 1 un- l 0 0 1 . . . 0 4

0 0 0 . ' . 1 -0 0 0 . . . 0

. . .

. . . .

and

l o 0

A is always a nonsingular matrix; B does not exist if bo alone is present in (11). Since Q is (21 - 1) X (2n + m + 1) matrix, (21 - 1) must be at least equal to (2n + m + 1) in order to solve the linear algebraic system (15). Least squares estimate of the augmented parameter vector p is given by

B = [ Q ' Q]- ' [Q' Y ] . (17) Thus it is possible to identify all the system parameters. Some- times it is also possible to identify IC's a, from (16) provided B is a null matrix.

Example I: Consider the problem of identification in a second order lag-free model given [8] by

J ( t ) + . , L ( t ) + a o v ( t ) = b , u ( t ) with input-output data on [0,1) simulated with unit step input, zero IC's and a, = 0.3, a. = 0.02, and bo = 1.

The proposed identification algorithm is applied by letting n = 2 and m = 0 . Results due to Ek and E k with k = 2 are shown against the actual results in Table I. It appears from the literature that OSOMRI is so far studied for only Block-Pulse functions (BPF) and Walsh functions (WF). No studies are reported in this direction for polynomial type orthogonal func- tions yet. Hence for comparison sake, results obtained via BPF approach and WF approach are also shown in the same table.

In Table I, p and p denote the actual and estimated aug- mented parameter vectors respectively. Since there are altogether five unknowns to estimate, at least five equations are needed to

IEEE TRANSACTIONS ON CIRCIJITS AND SYSTEMS, VOL. 36, NO. 11, NOVEMBER 1989 1457

TABLE I RESULTS OF EXAMPLE 1

-- A 2

Parameters a a b y(0) y t 0 ) E - lr;l 1 0 0

Actual resu l t 0.3 0.02 1.0 0 . 0 0.0

Fourier I E .3000738 ,0198364 1.0000138 -.0172229 .0001789 2.96692144 aprroach I u i th 1-3 I E .3000738 .Ill98364 1.0000138 ,0027236 ,0001789 7.482401-06

1 2

1 2

1 2 BPP I E .29997 ,0199967 ,9998334 -.0033324 .0009998 1.21331E-05 ap roach I wish 1-5 I E .29995 ,0199953 .9997667 .0000004 ,0009997 1.05635E-06

1 2

1 2 LIF I E ,2999883 ,0199987 .9999349 -.0013019 ,0003906 1.851888-06 ap roach I u l L 1-8 I E .zgggeos .oigggez .9999oeg 0 . 0 .o003906 i . 61251~-07

1 2

solve (15). This is done by choosing I = 3 for proposed approach, I = 5 for BPF approach and I = 8 for WF approach. The main disadvantage with WF approach is that the selection of I depends on the relation I = 2 / where j is a positive integer. The results obtained by the present approach with merely three terms for I seem to be good. Moreover, it is very interesting to note that the use of OSOMRI, i.e., E, in identification process has reduced E

by 39 times without any change in the estimates of system parameters a i , a,, bo. Although there is an improvement in E by using OSOMRI in BPF and WF approaches the estimates of system parameters obtained via E' (not via E,) are closer to the actual parameters.

IV. IDENTIFICATION OF DISTRIBUTED PARAMETER SYSTEMS

Consider a LTI SISO DPS modeled by the following differen- tial equation:

ay (x7 ') +a,- ay(x ' ') + aoy( x, t ) = b,u( x , t ) (18) ax at

in which u(x, I) and y(x, t ) are respectively the system input and output, both assumed available over a region defined by x E

(x, , xf) and t E ( t , , t,); and ul, a,,, bo are the system parameters to be estimated.

Let y ( x, t , ) and y ( x,, t ) be the initial and boundary conditions (IC and BC) of (18). Even though IC and BC are known from the output record y(x, t ) in practice, they will be treated as un- knowns in the identification process for the reason explained in the preceding section.

Now expand u(x, t ) , y(x, t ) , IC and BC in truncated double Fourier series as follows:

4 x , t ) = 4J7( x ) U+( t )

Y ( x, t ) = +7( x) Y+( t )

Y(X,t,) = / ( X I

(19)

(20)

/ - I

r = l

Y(X,,t) k -1

(22)

TABLE I1 RESULTS OF EXAMPLE 2

c a al bo Parameters

Actual 129 ----____-__ 12 1 .0

Four i e r s e r i e s e s t ima tes 2 .0000104 1.00003 14 1.000006Y

WF e s t h e t e s 2 .0 0.99999'39 3.99'39999

LaP es t ima tes 2 .o 1 .o : .o

___-____________-__-____________________-------------------- C P 1 e s t ima tes 2 .o 0 -9999909 0.9999959

where I < m , k < n; U and Y are just similar to F in Section 11-C; and A , , is a (2m- l )x (2n- l ) matrix having ( i , j) th element unity and all other elements zeros.

Integrate (18) once with respect to x and once uith respect to t to get an integral equation, introduce (19)-(22) in the integral equation, make use of (10) with p = v = 1 and manipulate to arrive at the following equation:

and vec( .) is the vector valued function of matrix (.), defined for a p X q matrix A with columns U , , U , , ' . . , a q as

Least squares solution of (23) is the desired parameter vector which contains all the system parameters explicitly and the spectra of a , f (x) and g ( t ) . If a, f 0, f ( x ) can also be deter- mined from (24). Notice that Q in (23) is a (2m -1). (2n - 1 ) ~ (21 + 2k + 1) matrix. In order to apply least-squares technique: (i) columns of Q must be linearly independent and ,i:ii) (2m - 1)(2n

Example 2: It is required to identify the system described by (18) whose input and output are, respectively, given by u(x, t ) =

t + 2x + xt and y ( x, t ) = xt with zero IC and BC. By assuming that u(x, t ) and y ( x , t ) are available in the

region defined by x, t E [0, l), the proposed identification scheme is applied with m = n = 2. Estimates of the parameters are as shown in Table 11. For comparison sake, other existing methods such as the WF approach, Chebyshev first kind polynomial (CP1) approach, and Laguerre polynomial (Lap) approach are also applied with m = n = 2 and the parameters estim.ited.

In Table 11, LaP estimates seem to be exactly the same as the actual parameters. This is due to the fact that the Laguerre representation of input and output signals is exact and all the estimates of Laguerre integration operational matrix are zeroes and ones. Since the estimates of proposed approach are found to be slightly inferior to that of other approaches, Fourier series approach need not be regarded as the inefficient approach. Superiority of chosen orthogonal function approach depends on

- 1) > (21 + 2k + 1).

the nature of signals to be dealt with as certain class of orthogo- nal functions fit certain signals more accurately than others.

Section I1 presents a concise description of structure preserving models. Sections I11 and IV present the results on equilibria for the various cases of special damping properties. Section V looks at the special case of a two generator system. REFERENCES

C. C. Liu and Y. P. Shih, “System analysis, parameter estimation and optimum regulator design of linear systems via Jacobi series,” Int . J . Conrr., vol. 42, pp. 211-224, July 1985. G. P. Rao, Piecewise Constant Orthogonal Functions and their Application f o S.vsfems und Conrrol. (LNCIS 55). Berlin, Germany: Springer-Verlag, 1983, chap. 10. P. N. Paraskevopoulos and A. C. Bounas, “Distribution parameter sys- tem identification via Walsh functions,” Int. J . Sysf. Sci., vol. 9, pp. 75-83, Sept. 1978. P. N. Paraskevopoulos, P. D. Spans, and S. G. Mouroutsos, “The Fourier series operational matrix of integration,” Int. J. Syst. Sci., vol. 16, pp. 171-176, Feb. 1985. T. T. Lee and Y . F. Chang, “Analysis and identification of linear distributed systems via double general orthogonal polynomials,” I n f . J . Contr.. vol. 44, pp. 395-405, Aug. 1986.

On the Equilibria of Power Systems with Nonlinear Loads

DAVID J. HILL

Ahrracr -This paper collects together several useful facts on the equi- libria of structure preserving models used in energy function analysis of power systems. The equilibria of interest reflect the bus power imbalance after a fault and depend on the nature of generator and load damping.

Key worth -Power systems, state models, stability, equilibria.

I. INTRODUCTION

Current progress in the development of energy functions for structure preserving models of power systems could benefit from a clarification of certain issues in the equilibria of the models. Structure preserving models overcome several disadvantages of the usual impedance load reduced network model [1]-[lo]. In particular, they allow well-defined energy functions for a large class of nonlinear load models and promise to facilitate study of network control devices such as SVC’s and HVDC links on large disturbance stability [ll]. This paper collects together some use- ful facts on the equilibria of these models.

The importance of a clear understanding of the equilibria in models used for energy function analysis was promoted by Ribbens-Pavella [12], [13]. Amongst further results on this issue, we are influenced by the work of Willems [13], [14] and Bergen and Gross [15]. These early studies used the impedance load models and focused on the roles of assumptions about generator damping and the sum of injected bus real powers. For structure preserving models, the role of load damping is also important in determining system dimension and equilibria. In fact, loads with- out damping contribute algebraic equations and normal form representations can only be guaranteed locally. Thus the differ- ences with impedance load reduced network models are substan- tial enough to warrant a fresh analysis.

11. STRUCTURE PRESERVING MODELS OF POWER SYSTEMS

We consider a network of rn generators and no buses con- nected by transmission lines. There are no - m load buses, i.e. buses which have loads, but no generation. Following [l], we augment the network with fictitious buses which represent the internal generation voltages E,. These are connected to the gener- ator buses in the transmission network by transient reactances. All the transmission lines are modeled as series reactances. (Hence, all the buses in the augmented network model are interconnected by a network of series reactances.) There are n = rn + no buses in the augmented network. It is convenient to number these as follows. The load buses are numbered 1,2; . ., no - rn, the rn generator buses are numbered no - m + 1; . ., no. Each fictitious bus is numbered i + rn where i is the bus number of the corresponding generator bus to which it is connected.

Let the complex voltage at each bus be the (time-varying) phasor y=ly.;ILS, where 8, is the bus phase angle. The bus frequency deviation is given by U, = 8,. For generator fictitious buses, this describes the instantaneous departure of the rotor frequency from a synchronous reference. Using classical genera- tor models, we assume [y l=lE,- , , , l are constant for i = n o + 1,. . ., n . We could equally well use the fast exciter model in [16] and constrain the terminal voltages to be constant. We will assume in the equations that there is no infinite bus. Minor modification allows for one. Let I VI = [ 15 I,I& 1,. . . , I v., I]‘; this represents the vector of magnitudes of the physical network (generator terminal and load bus) voltages. It is convenient to define a vector of angle differences for later stability considera- tions. Taking the nth bus as the reference, we use the internodal angles

Then let

Since the transmission lines are assumed to be lossless, the bus admittance matrix Y is purely imaginary, with elements y, = jB,,. At each bus, real and reactive power is exchanged between some of the generators, loads and/or transmission lines. Let Ph, and Qh, denote the total real and reactive powers leaving the i th bus via transmission lines. Then

Manuscript received September 22, 1987; revised November 22, 1988. This work was supported by the Australian Research Grants Scheme and by the Electrical Research Board. This paper was recommended by Associate Editor M. Ilic.

The author is with the Department of Electrical Engineering and Computer Science, University of Newcastle, New South Wales, 2308 Australia.

IEEE Log Number 8929762.

In interpreting the dependence of Ph,, Qh, on \VI, we assume the substitution I I = I E, - I, i = no + 1,. . . , n has been made. Also we note a, = 0.

Now consider the modeling of loads. Denote the real and reactive power demands at the ith bus by Pd, and Q d , , respec- tively. In general, these powers are functions of voltage IYI and frequency U,. For the derivation of energy functions, the follow-

1458 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. 36, NO. 11, NOVEMBER 1989

0098-4094/89/1100-1458$01 .OO 01989 IEEE