Application of the singular perturbation method to a steam power system

Electric Power Systems Research, 8 (1984/85) 219 - 226

219

Application of the Singular Perturbation Method to a Steam Power System D. S. NAIDU* Department of Electrical Engineering, Indian Institute of Technology, Kharagpur 721302 (India) A. K. RAO Department of Electrical Engineering, College of Engineering, Kakinada 533003 (India) (Received August 4, 1984)

SUMMARY A fifth-order steam p o w e r system with reheat unit is considered. The discrete model o f the continuous system is cast in the singularity perturbed form. A perturbation m e t h o d is applied to the discrete model. The important features o f singular perturbations in discrete systems are clearly illustrated. The numerical values represent a typical set o f real data. The results show a new avenue for the application o f the singular perturbation m e t h o d to discrete models o f large.scale p o w e r systems. Key words: steam power system, system dynamics, modelling of large-scale systems, discrete models, order reduction, singular perturbation m e t h o d . 1. INTRODUCTION An electric power system, composed o f sets of p o w e r plants, transmission lines, transformer stations and substations, distribution networks and consumer loads, is a complex dynamic system, characterized by differential equations of high dimensionality and stiffness [1]. The high order is often due to the presence of several small 'parasitic' parameters and the stiffness is caused by the clusters of widely separated eigenvalues of the system. Therefore, there is a strong motivation to study large-scale p o w e r systems via singular perturbation methods which are renowned for their ability to reduce dimensions and remove stiffness. The application of the singular perturbation technique to power *Present address: Flight Dynamics and Control Division, NASA Langley Research Center, Hampton, VA 23665, U.S.A. 0378-7796/85/$3.30

systems described by differential equations has been significant [2 - 12]. There is a growing need for discrete modelling of continuous systems, owing to its suitability for digital simulation [13]. The study of singular perturbations in discrete systems described by difference equations is of relatively recent origin [14 - 25]. In this paper, a steam power system described by a fifth-order differential state space equation is considered. By suitable reindexing of state variables and a discretization process, a discrete model is obtained in the singularly perturbed form [17]. Using the authors' recent results [ 2 1 - 2 4 ] , the singular perturbation method is applied to the discrete model. The results are obtained for a typical set of real numerical data. The salient features of singular perturbations such as order reduction, loss of initial conditions, and boundary layer corrections are clearly exposed. 2. FORMULATION OF THE PROBLEM A steam p o w e r plant consists of an installation for steam generation and a t u r b i n e generator set. The transients of the installation for the production of steam are much slower than the electromechanical transients of the turbine-generator unit; these will be neglected in the derivation of the steam power plant model by assuming that the storage capacity of the installation for steam generation is large enough to be considered as an unlimited source of working fluid (superheated steam) having constant pressure and temperature [ 1 ]. Figure 1 shows the principal configuration of a reheat unit steam plant using a pure tachometric governor with steady state feedback. The h o t steam flows from the boiler via © Elsevier Sequoia/Printed in The Netherlands

220

EMERGENCY STOPVALVE

SPEEDAND LOADCONTROL CROSSOVERPiPNG

I

IF

SrEAM[I

II

TURBINE

TURBINEII

!

FTL--I II

_ !

III

,

A

~

RE~E'ATIN G

----~ ;S~LV~ AET 2 ' 0NDENSER ~

REHEAT

WATER IFEEDWATER INLETPIPING TREATMENT PLANT

FOR

"~"~'~A DD~ • WATER

"

I

I

COND.wATER

Fig. 1. Simplified scheme of reheat

turbine steam power plant.

steam piping and a steam chest and enters the turbine through control valves. The steam expands through a high pressure (HP) turbine, and enters an intermediate pressure (IP) turbine, after passing through the reheat section of the boiler. From the IP turbine, steam flows to a low pressure (LP) turbine via crossover piping, where it expands completely and then enters the condenser. Here it is cooled and converted to water, and then this condensing water is sent to the boiler in a closed cycle. Losses in the cycle are compensated for by additional water entering the plant for water treatment; the plant supplies the feed water to the boiler. The dynamic state equation for the steam power system is given by (Appendix) = Dy + Bu + Eq

(1)

where y , u and q are fifth-order state, control and disturbance vectors and D, B and E are matrices of appropriate dimensions. To apply the singular perturbation method to the system of equation (1), only the state distribution matrix D is of primary importance. After reindexing the state variables and sampling the continuous system, the discrete model is obtained as X(k+

I

1)]---- IAll

hal2 ] Ix(k)]

z(k + 1)J [A21 hA22 ] [z(k)J with initial conditions

x ( k = 0) = x ( 0 ) ;

"~AT~ ' ER

"

z ( k = 0) = z ( 0 )

(2)

(3)

where h is the small positive scalar parameter responsible for singular perturbation and [x'(k), z'(k)] corresponds to the discrete ver-

sion of the state vector y . The above system of equations (2) and (3) is said to be in the singularly perturbed form in the sense that, by suppressing the small parameter h in eqn. (1), the resulting degenerate system

x(°)(k + 1) = Allx(°)(k) z(°)(k + 1) = A21x(°)(k)

(4)

cannot be expected to satisfy all the prescribed initial conditions given by eqn. (3). The degenerate system of equation (4), being a second-order difference equation in x(°)(k), satisfies the corresponding two initial conditions of x(0). That is, x(°)(k

= O) = x ( 0 ) ;

zC°)(k = 0) ¢ z ( 0 )

(5)

This loss of initial conditions of z(0) during the process of degeneration is attributed to the existence of a boundary layer p h e n o m e n o n at the point k = 0 [21].

3. APPLICATION OF THE SINGULAR PERTURBATION METHOD The singular perturbation method, developed by the authors [ 2 1 - 2 3 ] , retains the advantage of the reduction in order associated with the degeneration process and recovers the loss of the initial conditions by incorporating an appropriate correction. As in any perturbation method, the solution is assumed to be in the form of a power series: x ( k } = xC°)(h ) + h x ( ' ) ( k ) + ...

z(k) = z(°)(k) + hz(l)(k) + ...

(6)

221

This is referred to as the outer series since it corresponds to the solution outside the boundary layer. By substitution of eqn. (6) in (2) and collection of coefficients of like powers of h, a set of equations is obtained. The zeroth-order approximation is given by eqn. (4). The first-order approximation is x(1)(k + 1) = AllxO)(k) + A12z(°)(k) z(l)(k + 1) = A21xO)(k) + A22z"°)(k)

(7)

Similar equations can be obtained for secondand higher-order approximations. The initial conditions, z(0), lost in the process of degeneration, are recovered by using the corrections obtained from the transformations v ( k ) = x(k)/h~+l;

w(k) = z(k)/h k

(8)

Using eqn. (8) in (2), the correction system becomes

[""

[,,,,(k + w(k + 1)

J

[A2, A22J[w(k)]

(9)

The correction series is assumed to be v(k) = vt°)(k) + h v ° ) ( k ) + ...

w(k ) = w(°)(k) + hw(~(k ) + ...

(10)

Insertion of eqn. (10) in (9) and collection of coefficients gives, for zeroth-order approximation, 0 = AHv(°)(k) + A12w(°)(k) w(°)(k + 1) = A21v(°)(k) + A22w(°)(k)

(11)

For first-order approximation, v(°)(k + 1) = Allv°)(k) + A12w(l)(k) w ° ) ( k + 1) = A21v(1)(k) + A22w(l)(k)

(12)

Similar equations are obtained for higherorder approximations. The total series solution, composed of the outer series of eqn. (6) and the correction series of eqn. (10) along with the transformations of eqn. (8), is given by x ( k ) = [x(°)(k) + h x ( ' ( k ) + . . . l + h~÷l[v(O)(k) + hv(~>(k) + ...]

z(k) = [z(°>(k) + hz('~(k) +...]

(13)

+ hk[w(°)(k) + h w O ) ( k ) + ...] The initial conditions required to solve the various series eqns. (7), (11) and (12) are obtained using the fact t h a t the total series solution of eqn. (13) should satisfy all the given initial conditions of eqn. (3). They are:

w(°)(O) = z(O) - z(°>(O)

x{°>(0) = x ( 0 ) ;

(14) x(,)(0) = -v(O>(0);

w(,~(O) = - z { , > ( O )

4. NUMERICAL RESULTS In order to illustrate the application of the singular perturbation method to the steam power system, the following numerical data are used [ 1 ] : r = 0.05;

kt = 0.95;

Cv = 0.3;

Cs = 0.4

e=1.35;

eT=0.15;

er=l.0;

ep=l.5

Ts = 0.25 s;

Tu = 0.2 s;

T~ = 6.0 s

Tn = 0.50 s;

T = 12 s;

h = 0.25

Using the above numerical data, the various series solutions are obtained and plotted in Figs. 2 - 6 along with their exact solutions for the sake of comparison. It is seen that the degenerate solutions for x l ( k ) and x 2 ( k ) shown in Figs. 2 and 3 retain their initial conditions, whereas the degenerate solutions for z l ( k ) , z 2 ( k ) and zs(k ) shown in Figs. 4, 5 and 6 respectively lose their initial conditions. This situation explains the significance of eqn. (5). The initial conditions are said to have been lost in a narrow region called the boundary layer or layer of rapid transition. This is sometimes referred to as the boundary layer point in the case of discrete systems [14]. The lost initial conditions are recovered by using the correction series as shown by the zeroth-order solutions in Figs. 4, 5 and 6. Further, it is noted that, except for reinstating the initial conditions, the zeroth-order solution is c o m m o n to the degenerate solution. As expected, the first-order solution shows an improvement over the zeroth-order solution. 5. CONCLUSION A reheat unit steam power plant using a pure tachometric governor with steady state feedback has been considered. The dynamic behaviour has been described b y a fifthorder differential state equation. By suitable reindexing of state variables and a discretization process, a singularly perturbed discrete model has been obtained. Using recent results of the authors, the singular perturbation method has been applied to the discrete

222 •

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I

1

I

T

I

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t

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0.8 : : : E x a c t Solution e---.e---e First order Solution ~ = "- Zeroth order ,, ~ Degenerate ,,

1~X ~_~ ~ll

0.6

•. \

_-

_-

SolutionCommon to Oegeoerot

'

0.4

0.2

00 -0.2 -0A 0

i 2

I

I 6

4

L 8

J 10

I 12

1 14

L 16

I 18

I 20

I 22

I

I

l

I

I

=,k

Fig. 2. Exact and a p p r o x i m a t e solutions for xl(k).

0.5

I

0.4

I

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I

[

; ~ Exact Solution e - - - - ~ - - - - t First order Solution

0.2

®

~

=

~

=

Zeroth order ,, Degenerate ,, Solution Common to Degenerate & Zerothorder

0.0

-0.2

I -0.4 -06 J

-0.8 -1.0

0

t 2

I 4

I 6

I 8

I 10

Fig. 3. E x a c t and a p p r o x i m a t e solutions for

I 12 ~ k

I 14

I 16

I 18

[ 20

i 22

24

x2(k).

model. The results have been plotted for a typical set of real numerical data. The distinguishing features of singular perturbations such as order reduction, loss of initial conditions, and boundary layer corrections have been clearly brought out.

It is hoped that the singular perturbation method, with the remedial features o f reduction in order and removal of stiffness, will become a powerful tool in the analysis of large-scale power systems via their discrete models.

223 o.s1-

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i

it

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1

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= "= Exact Solution e---e---e First order So[utiofl ": ~ ~ Zeroth order ,, ~ Degenerate " ~ Solution Common to Degenerate & Zeroth order

0.4~I~ I| 0.2~-~I| -~ I~

l_i.°

....... i

"0'i~'" -- "- ~ -0. 0

2

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l

6

9

10

12

14

16

18

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L- k

Fig.

4. Exact

and

o.3/

approximate

i

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solutions

i

for

i

z](k).

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~°0,

f-° . '

. I~ I 1

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E.act So,u.on" i.,'

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~----~-

-

F,,,torder So,u,~oo --- Zerot~ order ,,

O---o----o

Degenerate

I/I

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-

or .r

0

2

4

6

8

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12

1L

16

18

20

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Fig. 5. Exact and approximate solutions for z2(k ). 0.~

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0.~

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Fig. 6. Exact and approximate solutions for zs(k).

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2/,

224 ACKNOWLEDGEMENTS

The authors are grateful to the Indian Institute of Technology, Kharagpur, and J.N. Technical University, Hyderabad, for providing the necessary facilities to carry out this research work.

NOMENCLATURE

Upper case italic characters represent absolute values and lower case italic characters represent relative per-unit values of variables and coefficients A turbine gate or valve opening Cv fraction of total power generated by high power turbines e coefficient of mutual influence of two variables F system frequency h small positive scalar parameter k proportionality factor connecting control valve position variation and turbine output variation P power output of turbogenerating set r non-dimensional speed-droop coefficient T system acceleration time constant Tn time constant characterizing time delay in intermediate pressure turbine and crossover piping T~ time constant characterizing time delay in high pressure turbine and reheat piping Ts time constant of system pilot valveservomotor-turbine gates Tu time constant of turbine characterizing time delay between control valves and turbine nozzles X servomotor piston motion Y pilot-valve piston motion e participation coefficient of particular generating unit in total system load

REFERENCES 1 M. Caloric, Dynamic state-space models of electric power systems, Ph.D. Thesis, Dept. of Electr. and Mech. Eng., Univ. of Illinois, Urbana, U.S.A., 1971. 2 P. Sannuti and P. V. Kokotovic, Near optimum design of linear systems by singular perturbation method, IEEE Trans., AC-14 (1969) 15 - 22.

3 P. B. Reddy and P. Sannuti, Asymptotic approximation of optimal control applied to a power system problem, Proc. Inst. Electr. Eng., 123 (1976) 371 - 376. 4 P. V. Kokotovic, R. E. O'Malley, Jr. and P. Sannuti, Singular perturbations and order reduction in control theory -- an overview, Automatica, 12 (1976) 123 - 132. 5 J. J. Allemong, A singular perturbation approach to power system dynamics, Ph.D. Thesis, Coordinated Science Lab., Univ. of Illinois, Urbana, U.S.A., 1978. 6 P. K. Rajagopalan and D. S. Naidu, Application of Vasileva's singular perturbation method to problems in large scale power systems, IFAC Symp. on Computer Applications in Large Scale Power Systems, New Delhi, India, Preprints, Vol. 1, Inst. Eng. (India), Calcutta, India, 1979, pp. 41 49. 7 B. Avramovic, P. V. Kokotovic, J. R. Winkelman and J. H. Chow, Area decomposition for electromechanical models of power systems, Automatica, 16 (1980) 637 - 648. 8 M. S. Mahmoud and M. G. Singh, Large Scale Systems Modelling, Pergamon Press, Oxford, U.K., 1981. 9 S. Sastry and P. Varaiya, Coherency for interconnected power systems, IEEE Trans., AC-26 (1981) 218 - 226. 10 J. R. Winkeiman, J. H. Chow, B. G. Bowler, B. Avramovic and P. V. Kokotovic, An analysis of interarea dynamics of multimachine systems, IEEE Trans., PAS-IO0 (1981) 754 - 763. 11 D. S. Naidu and S. Sen, Singular perturbation method for the transient analysis of a transformer, Electr. Power Syst. Res., 5 ( 1 9 8 2 ) 3 0 7 - 313. 12 V. R. Saxena, J. O'Reilly and P. V. Kokotovic, Singular perturbations and time-scale methods in control theory: survey 1976 - 1983, Automatica, 20 (1984) 273 - 293. 13 J. A. Cadzow, Discrete time systems: an introduction with interdisciplinary applications, Prentice Hall, Englewood Cliffs, NJ, U.S.A., 1973. 14 C. Comstock and G. C. Hsiao, Singular perturbations for difference equations, Rocky Mount. J. Math., 6 (1976) 561 - 567. 15 F. C. Hoppenstead and W. L. Miranker, Multitime methods for systems of difference equations, Stud. Appl. Math., 56 (1977) 273 - 289. 16 H. J. Reinhardt, Singular perturbations of difference methods for linear ordinary differential equations, Applicable Anal., 10 (1980) 53 - 70. 17 R. G. Phillips, Reduced order modelling and control of two-time-scale discrete systems, Int. J. Control, 31 (1980} 765 - 780. 18 P. K. Rajagopalan and D. S. Naidu, A singular perturbation method for discrete control systems, Int. J. Control, 32 (1980) 925 - 936. 19 P. K. Rajagopalan and D. S. Naidu, Singular perturbation method for discrete models of continuous systems in optimal control, Proc. Inst. Electr. Eng., Part D, 128 (1981) 142 - 148. 20 G. L. Blankenship, Singularly perturbed difference equations in optimal control problems, IEEE Trans., AC-26 (1981) 911 - 917.

225

problems in discrete systems, Int. J. Control, 36 (1982) 7 7 - 94. 24 A. K. Rao and D. S. Naidu, Singular perturbation method applied to open-loop discrete optimal control problem, Optim. Control: Appl. Methods, 3 (1982) 121 - 131. 25 M. S. Mahmoud, Order reduction and control of discrete systems, Proc. Inst. Electr. Eng., Part D, 129 (1982) 129 - 135.

21 D. S. Naidu and A. K. Rao, A singular perturbation method for initial value problems with inputs in discrete control systems, Int. J. Control, 33 (1981) 953 - 965. 22 A. K. Rao and: D. S, Naidu# Singularly perturbed boundary value problems in discrete systems, Int. J. Control, 34 (1981) 1163 - 1173. 23 D. S. Naidu, and A. K. Rao, Singular perturbation methods for a class of initial and boundary value

APPENDIX

The d y n a m i c equations for ~he steam p o w e r plant [ 1 ] (a) Governor equation A j = (--r Aa + Afro -- A f ) I T s

(A-l)

(b ) Steam turbine equations [~, = (k Aa - - A p , ) / T ~

(A-2)

~2 = (k~ Ap, -- A p 2 ) l T~ ~3 = (AP2 -- AP3)IT~ (c) S y s t e m inertia equation A]; = [erCv Apl + er(1 --C~)Cs Ap2 + er(1 - - Cv)(1 -- Cs) A p 3 - - e A f - - A p L ] / T

(A-3)

The a b o v e eqns. ( A - l ) , (A-2) and (A-3) are r e p r e s e n t e d in the state variable f o r m given b y eqn. (1). T h e n Aa ~Pl

y =

Ap2

u = Afro

q = Ap L

(A-4)

hp, ~f

---r/T, klTu 0

D =

0

0

0

--IlT,-

--llTu lIT,

0 --IlT~

0 0

0 0

--llTn

0

0

0

llTn

0

erC~lT

dsslT

1/T 7 0 B =

0

0

i0

E =

0 0 0 0 --1/7

ds41T

(A-5)

--elT

ds3 = er(1 - - Cv)C. d54 = er(1 - - C,,)(1 - - C.)

(A-6)

A f t e r r e i n d e x i n g the state variables y and discretizing with a sampling interval o f 0.8 s f o r a set o f real n u m e r i c a l data, t h e discrete m o d e l is given b y [ 1 7 ]

226

X l ( k + 1) x2(k + 1) z , ( k + 1) z2(k + 1)

z3(k + 1)

0.9014 --0.0196 --0.0071 --0.7500 --0.3060

0.1179 0.8743 0.7342 --0.0557 --0.0169

0.2100h 0.0000h 0.8070h --0.1280h --0.0440h

0.0668h 0.0842h 0.1000h 0.1174h 0.0520h 0.0843h 0.7743h --0.0563h 0.5711h 0.0529h

This is in the form given by eqn. (2), where [0.9014 A11 = --0.0196

0.11791 0.87431

--0.0071

0.7342

A21 =

[0.2100 A12= [0.0000

--0.7500 --0.0557

A22 =

--0.3060 --0.0169

0.0668 0.1000

0.8070

0.0520

0.084~

0.1280

0.7743

--0.05631

0.0440

0.5711

0.0529J

I

'(k)l

x(h)

=

xl(k)l x2(k)]

z(k) =

Ii

z2(k) I 3(k)J

0.0842] 0.1174J

h = 0.25

x,(k) x2(k) z,(k) z2(h)

y3(kL

(A-7)