Static and Dynamic Considerations for a Systematic Model-Reduction Procedure of Large Scale Energy Systems

Copyright © IFAC Large Scale Systems: Theory and Applications 1986. Zurich. Switzerland . 1986

STATIC AND DYNAMIC CONSIDERATIONS FOR A SYSTEMATIC MODEL-REDUCTION PROCEDURE OF LARGE SCALE ENERGY SYSTEMS L. F. Lopez and H. A. Nour Eldin Group of Automatic COl/trol and Technical Cybernetics, University of Wuppertal. Wuppertal, FRG

Abstract. Static and dynamic considerations for a systematic model-reduction procedure of large scale energy-systems are presented. The algebraic network relation is splitted up into a internal and a Ixternal ~ystem (IS and ES) defining the IS as the region of generator nodes or node of interest. Static evaluation factors for the IS are given grouping the ES into 2 areas if influence . Nonlinear or linear dynamical modelling of the first area is only taken. SystematiC order reduction through singular perturbation is reached. due to the choosed "3-Time Scale" structure of the ES-model. Keywords.

Energy Systems;

systematic order reduction;

Singular Perturbation.

1. INTRODUCTION Both . static and dynamic aspects should be taken into account in order to reach a systematiC model-reduction procedure for large scale energy systems. It becomes mandatory if . for example. the analysis of modern decentralized control concepts for a single generator or generator groups in the network is needed and a dynamic equivalent of the connecting network should be choosed. Many dynamic equivalencing techniques have been developed. prevailing : Modal approaches (Undri 11 and Turner. 1971; Undri 11 and co-workers . 1971; Spalding. 1977; Price and co-workers . 1978; Grund. 1978) . Coherency recognition and aggregation (Germond and Podmore . 1978; EPRI. 1977). as well as idendification methods (Price and co-workers . 1975 ; Ibrahim. Mostafa and ElAbi ad . 1976; Yao-nan Yu. EI-Sharkawi and Wvong. 1979; Yao-nan Yu and El-Sharkawi. 1981). Nevertheless open questions about the electrical strongness or coupleness between the system to be studied (internal ~ystem IS) and the rest of the network (Ixternal ~ystem ES) . as well as the degree of the system order for the external machines to be choosed has not completely been discussed. (Lee and Schweppe. 1973) introduced a

static distance measurement through the network admitance matrix and a "deflection" distance measurement assuming classical models of the machines in order to identify circles of influence for the IS. In this paper static weighting factors for the ES-machines are introduced grouping them into 2 regions of influence. Dynamically. only the first influence ~rea (lA) is taken into account. modelling each machine in this area with a set of nonlinear differential equations of the 7~ order in "Eigen-axis" state space description. (Lopez and Nour Eldin. 1984) (Lopez . 1984). The choosed mode I-structure leads to a "3-Time" Scale Interpretation of the coupled ES-machines and defines a suitable form that could be handeled by Singular Perturbation methods (Kokotovic and co-workers. 1976; Chow and co-workers. 1978). Depending on the case of analysis a systematiC order reduction of the ES nonlinear model is reached . Together with the Boundary Layer Systems a full reconstruction of all state variables in different "Time-scales" could be achievied . leading this to different system orders of the ES ·s .

129

L. F. Lopez and H. A. :'>lour Eldin

130

Terminal voltages and currents:

2. STATIC RELATIONS The large power system to be studied is splitted up into a Internal and a External System as shown in Fig. 1 :

I

I I T2 .

I

I

11 -T

= ( I Tl .

VI -T

= ( VT1 • VT2 .

I ITn t ) Terminal-current I VTnt) Terminal-voltage

VI

G,

-

v'G,

v'

-

v'G, ..

G"".

G and V~ are expressed in a synchronously rotating (w O=2.f 0) Q-D reference frame (Anderson and Fouad . 1978) . The admittance matrix yI describes the connections between the IS-machine and the Terminal buses with: 11

I VG G I t VG T= ( VTG) I VTT

Fig.1: Splitted network A subset of ngi-Generators define the dynamical structure of the ES. represent! ng the loads in the system as constant impendances to ground. The interconnection buse s between both systems. a re located at the termed Termi nal buses . midpoint of the original network interconnection branch. dividing the impedance of the branch equally. In the case of one machine as IS. the decomposition is shown in Fig.2:

IS ·Generator Ma t r i x

Dim n 9 i

x

Generator ·Te rminals

Dim n 9 i

x

nt

Di ago n a 1 mat r i x

Dim n t

x

nt

n9 i

The current state space description will be used therefore the relation: I ':GT

=

ZI

I .!.G T

(2.2)

is needed . with I ZGG ZI

( VI )-1

':' 1

I Z TG

Z TT

and n9 i nt

_-----_ -

....

! 0

T,

--

2Y"' ~

CD

\

\ VT 2 \

\ 1 _

\

\

,

-

"

// _

~

I

I

Vi G

nu mbe r 01

number

01

/

....

VTG VTT

/ /'

I I l: 2 . . I Gj 91J

vi

• it h

vI -G

I I GO i

+

I vG i

I v GO i

+

E v -G (2.1)

'11

-+

J GDi 1 ... . n 9 i

. I J vG Di

l:

z I.. I ITj 1J

(2.3)

I z~ i z .. IJ

i

E E

I ZGG I I ZGT. Z TG

External System E E ZGG Z GT

E v - T

E

[

ZTG Z TT

E ':GT

ZE

E .!.GT

• it h I I Gi

+

The first sumand of equation (2.3) describes the interaction between the IS-generators. the second gives the sum of the injected currents of the terminal buses for the IS.

VI - T

I ':GT

IS-Generator

j= 1

j= 1

/

[

-G

"

I .!.GT

each

nt

n9 i I VG i

I lIS

Internal System

[':' ':'1

for

v"

The matrix relation with already reduced SystemLoads (Kron-reduction) (Anderson. 1978) are defi ned to:

11 -T

Terminal-Buses

The explicity equation Voltage is:

Fig.2: One machine as IS

11 -G

in t ern a I Generators

lE -T

(2.4)

Static and Dynamic Considerations for a Systematic Model

The resulting structure of equation (2.8) combined wi th eQuati on (2.3 ) is shown in Fig.3

with E ( v Gl . v GZ ' [ E ( I G1 . lGZ'

)t vE Gnge I[ )t Gnge

[

I"

and E vG i

[

[

[

[

I Gi

+

I GO i

[

VI Z • E ( Ill' I I Z .

VIn t

[

1~1 I~,,~.

i = 1 •••• n ge )t

[

( V11'

Zm

j vG Di [ j I GO i

+

VGO i

[

131

[

)t

I In t

The impedance matrices E

[

ZGt

[S - Gen.

ZGG (Z[ ) I IG

Dim n g e

[S · Gen. - Terminals

[

Dim n g e

Diagonal mat r i x

ZII

Dim n t

x

nge

x

nt

x nt

with nge

:

Number of [S-generators

The analog relation to equation (2.3) for the ES is : nge l:

nt z

[

gii I Gj

+

j= 1

[

l:

ZIj

E

llj

CZ.5 )

Fig.3: Each Terminal current results from the weighted linear combination of the ES-generators and the IS-generator. The influence of the IS and the ES are reflected therefore at the terminals. The histogram of Fig.4 shows for a decomposed 9 generator system (standard AEP57 nodes), one of the terminals with weighting factors, assumig generator node 57 as the IS:

j= 1 SEN')I9ILITY-HISTOGRAHH FOR IT(1)

[ Z~

Z

i j

RELATIVE WEIGHTS

lii

.. GENERA TOP...

57 2

Again. the first sumand describes the coupling relation between all the ES-generators and the second the linear combination of al I Terminal currents acting on the ES. The linking condition between the ES and the IS is given to

"i j +0

100 ..

50

"'. 2~ •••••••••••••••••••••••••• .. .51 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... ..

.1 0

..................

~

... ..

o

..

12 21 30

.. .. ..

.10 .05 .04 .08 .04 .04

••••••••• ••••• •••• •••••••• •••• ••••

Fig.4 : Histogram v[ -I l[ -I

vI -I 11 -I

(Z.6 )

is the relative weighting factor of the jth ES-generator for the i~ Terminal current:

Xi]

If equations (2.6) . (2.4 ) and (2.1 ) are combined. the Termi nal currents a re obtained:

X

ij

Ix

=1 ZZ i jl

i=l.Z ..... nt

imax

CZ.9 )

j=l.Z .... nge II Zl -G

.!.I

+

cz.n

I[ Zz -G

with:

and X

Z1 Zz

I Z·l S Z IG [ Z-1 S Z IG

Dim n t x nge

With these factors. the ES is ordered in Influence Areas depending on the conditions:

and Zs

I

(lIT

+

E

Ill): Dim n t x nt

Assuming only one generator in the IS. ponent of iT is written to: zli

I IGl

IA1

+

1:

For all x

[ ZZij IGj

two

each com-

nge .

Mad 1 zzd ) j=l.Z ..... nge

i j ma x

Dim n t x n g i

ij >

i

E

1.2 ..... nt 1.2 ..... nge

=

(2.10 )

CZ.8 )

j= 1

IA2

For all x

ij

<

E

i

= 1. 2 .....

nt 1.2 ..... nge

E :::

0.1

132

L. F. Lapez and H. A. Naur Eldin

Graphically:

--

Mechanical part : IA-2

....... , "-

" ",,

,,

(3.2)

\

d 6 ( I ) I d I = Ul 0

\

S(

I)

with the current state vector in the Eigen-axiscoordinates i (I) = Z- 1 I ( I ) 0.3) ; Z: real Eigenveclor malrix

def i ni ng i (t) to : Fig.S

ES - grouping

i (I

)

elf id I D iq

TO)I

( -if ' did iOO xqi q iOO)1

0.4 )

3. DYNAMIC RELATIONS A

,E.f'

!?N

and Me are given as

The full dynamic representation of each single generator is given through a set of nonlinear differential equations of the 7th order (Anderson and Fouad,198O). In (Lopez and Nour Eldin, 1984) a novel state space description of the machine in "Eigen-axis" coordinates has been described. With this formulation a systematical order reduction procedure of the machine model using the singular pertubation methods could be reached. The dynamical structure of the model is given in Fig. 6:

Fig.6 Synchronous machine coordinates

in

Eigen-axis

The electrical and the mechanical system of the machine defines a "Three-Time"-Scale subsystem structure - "Very fast", 2nd. order "Equivalent dqsystem" wi th 60 Hz - TI«l 1st. order "fast Equivalent Dampers" (msec) - One 1st. order "slow Equivalent Field" an a 2nd order slow mechanical system Its nonlinear model formulation is :

0 . 5)

Z

-1

-1-

Le

£.f

Z -1 L -1i, (t)

e -N

(3.6)

The matrices Le' R(9(t)), Mand the vectors ~f and ~(t) are given in the Appendix A. The "Eigen-axis" coordinates representation has a physical meaning and describes clearly the different speed component of the electrical part of the machine . The dynamical equations for the IS and the ES will now be given. For conviniency , the state vector of the IS is fo rmu Iated in the physical variables (equation (3.4) and the ES in the form (3.3) . Assuming one machine in the IS a systematical order reduction procedure for the nonlinear model will be given. Internal System: Electrical part

and ( 0

Electrical part d~(I) / dl

= A i(1) - £'fu f - £'N(I)

0.1)

vTd ' vTq are obtained from the network relation (2.3) transformed to the d-q axis:

Static and Dynamic Considerations for a Systematic Model

133

Mechanical part nt

(3.16 )

•E

using the general relation:

Wo

6 jR (t)

.

E

(Sj(t) -

sR(t))

i=1,2, ... ,nge (3.10)

IOD exp(j6 )

I = I, U

Mechanical part

Combining equations (3.12) and (3.15) the full description of the electrical part of the coupled ES generators in Eigen-axi s coordi nates is obtained: d~~(t)/dt =

For the consideration of the ES the IS machine Torque angle 6R will define the reference angle:

A(t)

!~(t)

-

!:~c

-

!:~c(t)

(3.17)

A(t) is defined as:

ji=j

(blockwise)

External System:

=

A( t)

(3.18 )

ilj -wo z- ~ L - ~ /

Ki j ( 6 i j ) pt Zq d j

Electrical part and [

Cl

(3.12 )

K .• lJ

j [

"Yj

E Jr+x(6jj)/xdj

J

x- r

E (6 .. )/x 1J dj

i=1,2, ... ,nge

with

E

xijsjn(6~-6~)

r .. COS(6 E-6 E) i j 1J

J + (6 jj ) x r

(3.19 )

(t\ =1)

-

W

E t oP Ki i P ) zE1

) J _ (6 jj x r

=

E

x .. 1J

COS(6~-6~)

(3.13)

1

J

-

rjjSin(6~-6~) (3.20 )

(L [. )-1 'E

!:N i ( t )

e1

Now, reordering equation (3.17) according to the state vector i~ definition: and , E !:Ni (t)

(0

".[:::::

r

E

rii /

K~ . 11

E

0.14) E

xdi -

E

"Y i xi j / x d j

E Cl

i

E "Yj

E

Xii /

E

rji/

':, 1 E

x . q1

The ES generator voltages V~di and V~qdi are obtained transforming relation (2.5) to the qdreference frame using (3.10) for the currents and taking into accout, that the IS-machine is reference.

the very different transients of the coupled electrical parts of the machines are identified in a "Three-Time Scale" structure 1. Level: coupled equivalent d-q machine subsystems 2. Level: coupled equivalent machine Dampers 3. Level: coupled equivalent machine Field subsystems Analog to the single machine case (Lopez and Nour Eldin,1984) each "Time-Scale could be written separately and depending on the analysis case a model reduction is systematically achieved leading to different model orders for the ES. The description remains nonlinear. 4. CONCLUSION This paper introduced static weighting factors in

L. F. Lopez and H. A. Nour Eldin

134

order to evaluate the relative influence of ESgenerators to a defined IS-Area. For the analysis only the 1st. lA could be taken. A fully dynamic modeled ES has also been given based an the Eigen-axis coordinates description of each machine. A "Three-Time Scale" structure was found, making a systematic model reduction procedure possible leading to several reduced order models for the ES. 5. ACKNOWLEDGMENT The research work for this paper was supported by the German Society for the Encouregement of Scientific Research (DFG) under No 133/1-4. The authors are grateful to the DFG. 6. REFERENCES Anderson, P.M. and Fouad (1978). Power System Control and Stability, 2nd. ed. . IOWA State Uni vers i ty Press. Iowa. Chow. J.H .. Allemong. J.J. and Kokotovic P.V. (1978). Singular Pertubation analysis with sustained high-frequency oscillations. Automatica. 14. pp.271-279. EPRI (1977). Development of Dynamic Equivalents for Transient Stability Studies. Final Report EPRI . Project RP 763. April. Germond. A.J. and Podmore. R. (1978). Dynamic Agregation of Generator Unit Models. IEEE Transact. on PAS. Vol PAS-97. No. 4. July/Aug. 1060-1068. (1978). Power System Modal Grund. G. E. Identification from Large Scale Simulation Using Model Adaptive Reference Control. IEEE Trans. ~ PAS. May/June. pp. 780-788. Kokotovic . P.V . • O'Malley Jr .. R.E. and Sannuti.P. (1976). Singular Pertubations and order reduction in control theory - An Overview. Automati~. 12. pp. 123-132. Lee. S.T.Y. and Schweppe . F.C. (1973). Distance Measures and Coherency Recognition for Transient Stability Equivalents. IEEE Trans. on PAS. pp. 1550-1557. Sept/Oct. Lopez. L. F. and Nou rEI din. H. A. (1984). Mode ling and Model-Reduction of the Synchronous Machine through Singular Pertubation. Proceedings of the IX. IFAC-World-Congress. Paper 09.4/B-5. Budapest. July 2-6. Lopez. L.F. (1985). Netzdezentralisierung durch Modellreduktion. Restnetz- und BeoabachterAnsatz fOr nicht-lineare Netzprozesse. Final Research Report. DFG-133/1-4. (in German)

Price. W.W .. Ewart, D.N . . Gulachenski. E.D. and Si I va. R. F. . (1975). Dynamic Equivalents from On-line Measurements. IEEE- Trans. - -on-PAS. July/Aug. pp. 1349-1357. Spalding. B.D. and Goudi. D.B. (1977). The use of dynamic equivalents in the analysis of Power System Steady State Stability. IFAC-Symposium 1977. Melbourne. 21-25 February. Undrill. J.M .. Casazza. J.A .. Gulachensky. E.M .• and L.K. Kirchmayer (1971). Electromechanical Equivalents for Use in Power System Stability Studies. IEEE Trans. on PAS. Sept/Oct. pp. 2060-2071 . (1971) . J . M. and Turner. A.E. Undri 11. Construction of Power System Electromechanical Equivalents by Modal Analysis. IEEE Trans on PAS. Vol PAS-90. Sept/Oct. 2049-2059. Yao-nan Vu. EI-Sharkawi. M.A .. and M.D. Wvong. (1979). Estimation of Unknown Large Power System Dynamics. IEEE Trans. on PAS. Jun/Feb. pp. 279-289. Appendix A The matrices of the synchronous machine are: L

e

." 1

(s ( t»

R

Rqd

R r

Oiag(- ( 1- ~O )/ lf

-ClwO/Td

-'( Wo I I q

-(1- ~f )/ IO

- 1I 1 ) 0

and: ( 1- ~ 0 ) Cl

Led

Cl Cl

(1 - cr f 0 )

le q

( 1- ~ f )

Cl

0 R dq

Cl 0

"-·''"1 .[: [-: ~

'(

-'(

Rqd

:1

1

The given vectors and Parameters are : ~f

( -(1-

IN

(0

Cl

( 1-

~D )/ lf

ClW 0 U d

0

~ f) ( 1-

0 0 O O)t '(W 0 U q ~d )( 1- ~ O)

Id : xd / r ; Iq : x q/ r ;

0) t ;

A(t ) :

The electrical Torque is defined to: M ( t): 1/2 i ( t ) t

e

with M

-

M

i(t )

-

'(

: ( 1-

1 • set)

cr q )