The effect of regulating elements on the dynamic behaviour of power systems

'l']lJi; ON

OJi' FOWK£I

f'o:c

Doctor of

irl

of

ERRATA FeB.

"The Effect of Regulating Elements on the ]):'l!H1ulic

Behaviour of ~Jower ;)Jstems."

pa e

12.

equation

,2.1)

Xl f_l,

x

_l

,x

₂' •••••• ,x_n

etc •

.

pa e

43.

e'luation

(3.16)

= ( g' + g' ) (

1

+

h)3/2 _

n-'

o ~o

a e

49.

eCluation

D.46)

left hand side siould be e

a e

70.

second last li~e should read

efficient ( zero for constant tor~ue and -1 for constant power ),

fi

.17

Base 0.16

fi

.26

Omit G = 1.0

a e

166.

fifth line

Haireici par meter table

from caption

Tb Hydraulic toru~ue for part load e,,,-uili briulfl oi)er~tion with

rated speed and head.

I wish

thEmk

lTIy behalf'@

M:r

to

Woodward t.heir

Dr Butn

to Ike

ana. 1+ t r,j. s

Mr Millard and

at

the New Zealand

[ am support~

I thank the

staff of

the

Mobil m;sisbJ:tl.ce in Lterg o.f

:Ln

Perrfold

in 811 sta

fl

heir on

Genera1 of the New Ze

conneC

I

to

fat' vEtlu.able ,i:i~seussions of'

the Direntors of the

fo:1:' their Ba:LnhE~r

J NrrTID])lJC'L' ION

1

1

1. h Init:i.al C;rmsideY's.tions

1

G

'i of Povre:c

1 2 ration of

1 ~ ;( _}")OW6:('

. .1

'1 <

2

261 Introch).r.:tion 2 Definition::!

2. 1 De tion of

2"

5

TVfO theo1"crnG of

2. ~['he f,lJbov Jviethoil for the Construotion of

?" 1 The

trun

of t.ho [:~e:c:Les Solution 2 m.3 j Solution of the 1/ecurrenoe

'l'he lvT atrix

[

G)

]

1

2 'J'ests fen' the cia1

2

.3

He

(

,,12) and

(

~?2 )

D:tsc;l:tssion of ng

'['uT'bine Col.LiJllfl

5

'l'he Gate Position v8..tion

~) 1 The Mechanical Inertia

')" ~'he '[1urb:i.ne

:3

I) 'rb.8 Governor

Pilot j\ctuato:e

of

'I Calculation of the from

r~ 8:blJ,lcrJdon

CI1I\.Tjfl:iH II

1 tion

2 }i.:valuation of the lviode] of ConCllti

2", 'I'he Niethod of 'i\!valuation 'Che Determtnation of

~

c ' 1 '['he of

'1"

3

2 A Correct

Ivlet.bo(1 of

JL:f'f'ect of' 1.0813 on the

P:r.nct:i .. caJ l~ esul t of i~)e r; Method

I~f:fcct oflJermanent and Dea.d Band on ;imal1

7 _I

t

th

9

Cain

V£llidi fOl~ 011"""

I~etwo:ck

the tho Ceneral Tcrrn:Lnal

Network

for :3ffiall about a

f33,turatton F'aetrn'

Gnt ;:~ollJ --'cion.s

f'jolutiofl of the I~incar:i

model

? Prog:c8lTi DPIi:HT

6

TTl tT'ollLlct~!.o~n

1 1

the

1 q:: funetions

6

L

Lion of tbo

6

1!~

Differential

on LOE'cl

6 GoveTDDl'

F'urwLi o:nal 6 3 I) Value s of 'I.'

:r

6 '['he

:'3 on Rotor

'J1he

V{or.k on Rotor

6

The Genera.tion of

6

He Ck,jn

6. 5

'i:'ransient

G 2

Vol

0t at Pilot

F'unetiom.:

}(O to:c

fi'uneti ons in terms of

'rhe Effect ShtJ.nt Load.s on

on Hotol'

tb

")Note

Zubov

Lo

wit], Vol

TOII

{\I'PB:NDIX

ial

{'PP1%!DJX TIll:l

Genera1

ForTII

j!I!:li'J':nli:NC1')S

'!'

of ;"3I;andard of te Srn(lll

Connection :('01' ~~olutioIJ

fOT

~l

1

?07

\T()clc 1

21

220

W J. fa.

,v.

o.

led

'\Ira.

w

N T

rp

.g n

g

'2~

11.

e

G·enerator Piela. li'lux

Genera1;m' d.-axis armatu:re flux

Generator a;cmature fhu

Generator d-axis arnortisseur flux Generator

Field

Botor

bus

Rated

Hatecl

8Inorbiss6u.r flux

from rotor to

of

)

of set

p®u" turbine Instantaneous

p.u

deviation

unle

of infinite

(

or

Normalized p"Ue instantaneous pilot actm).tor

~Per uni1~ d('lvia:tion as indicated on a ze:co-1;o-uni indieator

Normalized p .. u. act.uator

p.u" (leviai:ion in head due

water

Instantaneo'Uf'\ rotor relative to refex'cmce fixed in. stator frame

Phase t;eI'm:i.ncl1

Phase tenrlirk'll cUr':ren1~f'l

d and te:cnrLnal

y Illfin:L vol

k Saturation

Gate for part load

Baae as indicated by

lORd

indicator full load

~l'GN '" 'l'utbine

B.AS1:: Base*'fGN A

D eoefficient

Nordnal load turbine coefficient

D

gen

H

Nominal damping coefficient of

intercon:nected electrical loacl.

Inertia Constant

Moment Inertia

~- 2fT time consiJa:n1;

'I' Inertia Time Constan1;

(

.,.

.,

fr'

'1 + ()

Dashpot Helaxation ~.'ime Constant

'l'ernporary droop for a 1 p"Ue

gate hioH

for a 1 p

tion

Pilot valve·

Ga:Ln

~~:ime Com-:d:;~)nt

ma.ch:lnc and.

:x; / I

q

[

C

_J

r

]

[

T

_]

r _W

_]

L

[

]

[

V]

[

AI]

[

.11 I

J

of

gap

subtrarF2 .. Len.t reactanee

OJ;. lj'ield time constant

F'ielcl resistance

i\mortisseur oircuit reSif,\tancel:; Arrnr:d:;ure cireui t seri8s rCt3is tance

p u" resistarlce balanced transmission

line

p. u~ reactarll~e of transmission

llne

~L'otal line

or

Sb.Uf.tt resisi;i ve load at

rPotal p.u" , ox' twice

shunt

reaetive load at bus

P.u" resistEUlce of transformer

p. U. :r:eE).ctance of' transformer

Terrnlnal network

Rows

Co18

2 2

2

6

6

2

1

'I 1

[

PI:]

[A

[

]

[B]

[ dJ

[

]

[PJ

[ GJ

[oj

V(~ctor

Matrix fm:' linear of

CA]

['3. matrix

Vector of coefficients of

Vector of c:oefficients of

Vecd;or of coefficients of <p

of

in

Zubov

matrix of coe:t'fieients of' second order form [x][ (}][ xT]

Symmetric

matrix of

quadratic form [ ] [

c

J [

xT]

Differentia,l Operator Perturbation

C l·l1\pr[,I~ll 1

,['his thesis describes the analY~iis of the dynamic of a hydro electric generating 13et und.or conditions of balanced three phase ope:cation. 'l'he hydTo generating set, or system~under conxideration ba,] been limLted to ~) single synchronous gene:catur, itn control devices a slln.nt 108e1 8t its terminaJ s and a balanced tl-rcee l)hase t:cal1Ennission 1 ioe terminating at an infinite bus. J.iLi.g. 10 'J if, an overall block diagram of

'1'he system as ~;lhown in 1 1 l S a good representation of an isola.ted

povreT' sta.t50n supplying power to a large centraJ s.Y;,tem via a h:Lgh vol taf~e

tran;:lInission line, sj nee the shunt loao at the sending end of the line c~m be used to represent line cha:cging capacitEtYlce as well aB the local load.

In this thesis a general analY:3is of this system "vil1 be produced and

a rneu.ns of reducing this general analysis daVin to any of the common ~lpeciaJ co.so:::, such as a voltage regulated ~')ynchronmls machine connected to an infinite bus, vrlll be providr-;d.

1 • 2 THE: APPROACH TO 'I'HB HI.Ol:jU~\fl

'J'he problem of obtaining information on. the pe.rformEUl.Ce of the generating set for all load.ing conditioIlf3 hDB been approached from two separate but

interrelated a~;pects.

1 Tbe development of improved c1:Lf'ferential equ.ations for the gcnt)1'.'3 gove:cnor and hydra.fJ.lic tm~bine; with the aim of a :cepresentation suitable for the

con.t:col theo:eye

2 1'be stabili ano of the gener"Jting set the

p;en,coTaJ method knovin. an th,," "Second Method of' Liapounov, 11

T'h.e math.ematie;clJ model of the goveJ'nor,~tuJ;binc i~\ based on the

seTT/Nt:

WATE£ ~ESSV/t'G

SYSrE,/Vt

C eNE..eATO,.e

.Lne-rr/O'

e'.I""£Y

C' 0/1 V~ ""~/(;'n

rV~"//V$

Ene"..,?y

Con v.

vo",TACE

A!.G<: V.L.ATo;€

TA;:4NS-,;r=o~E..e

t-...,."...,.,. ...

.8A~A/VCe.P

r;4!A.lVs ~/ $,$/<

HI--+-.iI ...

.5/?V'/VT L OAJt'

Y'-.1:..1 A/ E C' k' A...-eCZ,/f/ ~

A4

/'n,,#a./1~e

)

ved from ( )

reel for cUli.trol tuclies but it will be sklovrn heTt~in that [3ssumptioI1E.; in the c1e:l'i of

in

tbis

thesis

t'o:[' 8.11 COTldt

BeCc;lJ)Be there

eyc' accu:racy of governor mo(lel~;

to suitable test

'I'he mathemE,tical

it is clLrected eguations in the form

f. l

rather than a sei; of o:perational p:co

x:-e8u1t

tion

)

or fins'] generator differential equation:,:) are fltlX orLly, iDBtead of in 8.

~lonlC fluxes ancl some Cu.cTents Bfl 1;I.:3e<'1 i n

~L'he machine model :Lncludes

amortisi3eUl' circuits, 9.rmature resistance 'fhe complete lW3.thAlllAtical model of the of such complexity that i t cannot be handled

computers at the l:Lne8rised

~d;ion::; of the

the

ion of

of doubtful aC10Ctel which

of the ff1':)del c1eri ved. in

is unu.su81 in

set of differential

( 1 . 0 "I '\ _ /

in terms of the machine

of some

the

set is nonJ.incar cmd

on t.he or

the: thiB linear

J

1'0

! S seCD:rld due to ZuboV',wJd.ch

Ot

OF CDM

vrhich

th(~

I

1J~M 1

re s triction.

th("~;A

possible to solve

'1

KLltt:) ~ Gill

were

IUM 1620

on [I hinc"

nvtchLnc

th(" fiela

and V'lhile bo.ndle

of the nonlinear set e

i~his to be of

Thus, while nOllline of r,he rl.s~_;~

;:~ tu,(t:L{]~:;, _to ·he :cc

Ll

?,1!

JON

i diJi~ed fou

Ird);coduc"i;ion

conside:CDtion tri.c gove:r."nO:CE, ond the

and the

the 1

to

the

in

.iUJ __ material on ~jn(J test

in

1 e 6 INI'J'BI, CONSI

1 1

'l'be

of

in 'i

Tn

fau.} hod

tr~ln.sient

'f'he

of control

the main work OIl.

of

(

) a rrurtibr';:r'

Whi}e the voI

on inte:ccoymcct;eCl power Vlere first studied j,n America were mad.e in

), artd SI, 'te"l'rl ( J . , .. ~ J11.::1de contrlbutioxlf.l to the of

in incU vidual effects of water

turbine c::ha:CE1.cterif:\tics and interconnection

S11cee ss fu.} to tIle transient

under ault conditions of multi machine pow

of governors 8x!d h:::l.ve been made Aldred and. flhackshaft Th.c 8t.;:,1;e of

at

is known and it is these cont:rol element[; do

te and trl).1]sient of power whole. t in

calculate curves for tr1'lxJ.sient GOfH1Ltioml on m:ultim<';lchine power

or , but at it is not econom:i.cnJ to

includ.e detailed.

oontrol s'1:;u.clies of'

the varicH) s control SlLClt c:ls 'r'(lO+ ]" "'. .J N ()C'll co ~T I:> ('1 ~

7)

• , the II.outh

Hu.rwitz test ("113)

(1':)),

(6)

'I'lle

have the

HlclClIJ .. nAG o.n a

Cl)Jfin~on II 1:[1 oJ'de:!:' to !lnde:t:'c3t~)nd tbH oJx;ration of a

in some

,")c'l;iO[l of' the governors 01.i sucb, 8

When a govern.or acts on an isol"ded prime mDver and load such 8,S an c.LTlsyncbronisml hydro genera.ting set it acts simply i'tS a speed :cegillating

~,1.1.~)'.1~ L . , • . _ ,al.,J, ,-L·.'t~.· ·,r.·)~l·!llc , w . J _ I.. .. r11(~/-p·.r,.'.') ·rr.~,l.,,~.t,; l-.·Ol~a.~C,! • ~., . . . 1.1. ,."I.1, J,. '(-~lP L\. ,,' crJnp L " ) , L > (E'l~n~l~l'ro-L) -'~'.",d,- -'~O~ Q-rec' "'fj,~ "'" ~l'(' v 1 ~

governor::: assurne the Cil1Dl roles of speed and tOJ'qnl'l ree;ulators0

best illustrated by conside'cing a single gellDrating set which is connected to a hypothetical infinite machine runn:ing at consta.nt. frequencyo

//')creQ.f//')9 IV.h .

.spiff? lff?eI Selt/ng

li\OVeT

i

e

tJI.H iG f'o:eced to

nolon(j

I t L:~ will 'lOW

:in y!II]

d

)

frOIrl the

Lhat

ef:fect b:cium lOio,d Il'hus the governor has become

f'n:(' the

G()nnect·-:;d",

(;(;(1 HlB .. ebine wi{;h

to

of' (J, power

.'1 v(role th(' fbet th.s.t i t i~~

'OlD.y be cOl1f';irJered as a.lJ.nit the

and so the

to

the dLWJ rolf'!

When n povfer

machine tn

gc)\n~rr]orB :in

~tn con~;jd.e rnean~" thl:J.t th~~ 'w.hole of i its own si

since

the

£1,0 Ire l'no r 5

In

to

settixlg of' sorne

of the ::W"S\;cr<!

But

wben n.ny indiVidual mac.hLne of a 'is eOTlsiderecl pf 1;11e

controls the

the overall conLrolled tb!"; eol1ecti ve

di 1'd:r:ibution of' the l;r,erlf3T'at

eontrol1ec1

the

between the no lOftel

al governors

~'hc c011t1'o1 of oveJ'.9.11 one rn'~'c rd.ne or,

t~hc ','(,8.porw i bili lTIt3c:bine

to

manner te) coni;:co], f'rcquency i t :l:,

tIle

can be msae

l[le no 10::,;.<1

C!)~Cr8ct Whr;";n OIle

E18 a 11

to

G8V

of tb.e

I i

f)-f' _thi

of' (;h(';

II

in the

nontrol machines so

rated

dealt with

det:D:i.l in referl:lnce (

power' foetor of will 'be; shifted in. in exci t"tion EUId in the le direction 17oJ.

in power

the

P + j

v

i13 8

:in power

'1 @ .An indud;i vo

a power

A

puwer

it

in 8. network

whLch Cl)rrent throv

of this (lcmventhm

"l,o a

1;1:'01

deereasi3 in. exe.ite:c

'I/,e vars

the e of to

aT G()wluctor

excitation

knoYvn dj fficlJI

The CBXl be

tr<J.I1srnission line with i line can

JHuch

the

reactive 1'13 of .normal inductive load~ ThU.i3 the

pmve:c with lower excita.tion 'l'he combination of

excitation a medhun will re,;\ulL

rotor wlL1.ch in 13 reduced stab:ili

'T'he of the of underexcited of

vol '[.or:::: h::Js recleved. o. de;:)] .. of attention in

(9)

_.

and Jt ",till a

continue to rise.

rrtle

of' a as

excited anCi this

1 ~

Pr6blems in the transient

decade before the eot 0:[' control

came tb.i s cLee adf)

were J:18J1dlec1

po\<;rex'

co:Llective 'l'he tcadi

of some OJ~ all of the

the

thiB thesi;:; is sui t.ible for unde}~~ reeeives clt ten,tiOl1

of power

ana

to "be known under

a short circuit

'Phe Fiest the

) > Jiovrever

differential

'l'hi is II stabil in in definj.

j

of

1;io(1 of stf:ibili :in

the seeond s is in,solu.b1e

the

;tin the II

diffe:centiaJ. fOT sirruJIDtion "in the InD.C b:1ne "

(-l) (16) ( fo:crnatiorJ

for mu.ltimachine fox),

(

in tbe indu it woul(1

8.ppear that before

But

control thex'6 no diG tinction bel/Heen !1

stal)j l:i stclbLl:Lty " Both terms refer to

mathematical

condl. tion of II in definition 2 of section

'rhe corrcrol s t1J.dies of Concordia.

(6)

(

) AId.red and

(

tan

(18)

ton

(1

been concernecl In this the

but

line

ntion

11(~nco

it

II II

_to

i of 'I;he Jines

Problems power were rneaJts

}:Tt!'CO(~8Siorl W:1.S

'I'he common ut3e of'hi

g;ovex·t1.0r have now m;>,de :L t t.o conf:l;iJ:le:c pov/er tra.!1sient

in their control

e.ilTIl).lation mnl timachine power been 3.chi

ar.ld several indirect control e

to

indtviduaJ. in the simula l~ion. of power

limited. of computers wliicn are

c1vailable.

'['hi thesi tb.e

tho '.1. Zubov to element of

IN'l'HODfSC'J:ION

'rhe basic noulinea.),,:, oontrol 8.:ce

de:f:'initie>ns and which are necessary for discussion of

method of the r::rtatemtmts of

defied tions and a statement of the Zubov and

and

to

studjes~

The conventional control based on

methods

deal with

b.igh or(ler linear differential

contra.st nonlinear

control

deals with sets of

differential

ca.rl 'be

[

]

[ f

that

the nut!iber [ ] an(l [

by

de

of the fornl

.

.

@ ,

•

•

.,

in

matrix

form as

t)]

t;hia thesis the

t)

t)

t)

( '1 )

used

f[' bll[J

in

tif.;,Iteci eorrtrol

Ji'or the purpose of'

[

fo:rm. for som"e

and

the

equa't;iO(lS

( x

J

--

[Al[

1

as

J

1- _,

r

r(

J

(2" )

that (201) beeOlYlBS

in a more

pr:l.rts from the

r

A

1

is a sqUEL1'e rr!4'Jtrix of com::tant 8.nd

the

t'

_{' i}

now

desoribed

8.l,\(1

terms of seeond and

the;

02.,2)

[xl

order.

become

[AJ [x]

i t is a

straightforward

lineal"

(

to order

linea.r fliffer'en:!;i,al equation with constant coefficierd;s in the form ( the r<'lmainder of this

[fA ]

will be

to

ffi8EUl the

matrix (

linear a:od nonlinear control

terms of the

planelt _{where the magnitude of a variable is}

and its time derivative is plotted

on

the

other

axir:l For

a linear

or nonlinear system the

caube extended to

a the eqlJ.ations (2.L,~) in the form

(

time

of the form of the

:HoweveI'

)

we deal not with

one

the x.

l

space but ea.c:h

term this n-8p8.ce the l!~:ta.te }1'urther clisous::.don of the and state space is

) and

that

Each

Hovtever we note one

of the n-dimensional state or sp ac f) assoeiatect

an n-dimensional determines a state of of that

2.

When a is released from 8.

(Xi (0)

j)

in s tate space i t vdJ.l move a8 by (2,,2) The

of

) )

the

at successive instants after release will follow a curve in stat~; space which is described. by the co-ordinates (xi

(t ) ,

(t) ) Sttch a curve in :3tate spl'l.c:e is called a

An

pass through any

(10 not have a

of trajectories is that

of a state space" Any where

with the result that more than one

o e ,

2)

would have to pass through is a in state Rpace.

Such a

We must now define in the

the behaviour of a state where

to an

of the

of d,ynamical B.nalysis "mel disoussion. when its state of is

It mllst be noted. that the behaviollJ'." a

released from a

that the stat€"} variables a:1:'e not

from an.

when

i t

(2. )

on ( ).

that CO!lcli i;ioH

In

behaviour of the

space it released from different

from

be assumed have <been

such the

the

(

(202)

th(~n beoome

[

]

[

-

0

(

~['he (2,,6) _~~<=;:;,

if

there exists for

8..>0 cL {) :;. OJ such that the

[ I

[

[

t,

) I

J

<[

e:: ] for t ~

r/C

2 /

a

S//4I1/L/TY

def

of

s

2@1a, ~~.'Ib and 2 onto a

for stable unstable and stable

It' the i) includes the of' the state space

to

defirLi titUlS 1 and

It :1ho111d 'be different clEl.sses of

and

Such fA si tua.tion.

I'eeJ~izable i"ltates of a the

are said to

'I'hw'l a

ma;v be different, be stable in the

buJi:; not stable in the

I3hovm in 2" .3

//\/

T,#'E LARCE A/Vb

SrAL7/L BUT /VoT ASY/"1'P;'V/7C

not of

IIfhole fox~

the

If

of the

ste:ble in then is to

is ensIJ.:cec1 in if aJ.l the

of charaeteris tic have

Houth ~ Hurwitz test p:roves linear

but a

can in B. small

the

st'lmi=·defiai te i'unctions furlctions. These classes of

functions are now defined~

A

v

of sp;;\ce

the same

for

[ x

J

\'Ii ttLin and vl::l.nishes

_LX]

₌₌

For

)

2

i.s ve defini te and.

v (

)

= ( 2 2

is

fUYl.ction V is nD.l1cd

[

]

Before

function we observe that a

V

(x

1

to the

'rhen

the set of

n

Z

w

n

.z

cOD.sidered 'Inth a

( 6)

:md the

defini

te funct.ion the Ji~ule:J":i.an tlerivative of V with

f. ( ~

Now if wi thin some clomain D:~

) V(x

1, @ •

)

is pos ..

definite

(ii) is neg. definite

then V(

for the

2 .. 2 ..

5

One of the most

means of and means that fo:cms

the

1113.sis for the

to nonlineEtl' control

methodr~ of this

the

Method of II _{'rhis second metJ')oc1}

_the

or more of number of theoremf;'

were

We no'll wi

thou 1;

, the tVIO most

of' these o1;her theorems of aTe

'T'he

thero

for the

of t3table:

there funct;ion

that

I~uleri.an de:civative the

the definition

7

it is evident; sufficient

for of a domain D of'the

that

function l'mmd i.'()l'

'l'he of theorems of and of

from tbei'J':' result,.~ are too numerous to

)

(

)whioh discuss corl'trol e

of the 11,11ethoo,1I

'rID:

ZUBOV lvfi';trJ:"lOD ]j'OH Tffi~ GONSTHUG'l'ION OIil LIAPOUNOV li'UNCTIONH

The second method of

the and the of

pui, it is "t:ll unfort-u.nate charactc:cil3i;ic of

app:roach that met;hods 'bafJed on tbe main theorems are often

on One

of' the second

in

f. ( ]~

f'

It ~LG sbown

cp _cP

_(Xi'

in 6 of vre novJ

form

( .)

_l,t', ~7

( 2.

)

n wllf'Jr(; 1;hG . are

l

, It are

fTahn (

.

,

form

fo:rmc~d

::;1;,"1111e"

a.s:"nJmed. that the; f".

1,

'J' and. so one

L.

the linea.r ternl;,:

ElJ'J.O. Vogt

(1.

18) that the solu.tion function fo~(' the.:

( ') 7)

0(;-,0 J'., =-.,ll q(' u ) . l'n" E:~

(j

+ _OX

n

)

o

x

n

...

) n ){

)

x ) i~ a po~;d, ti ve definite OJ,:"

11 t.ive

rl'b.c clo:::ed form SOll1tion of

(2.9)

i:3 often but it :LS pos::abL3 to obtain a sol:u.tion to this equation in the form. of a ~ler:u~['l

u

where

sums u of U l\1 +

III ie' ,~

the

+

an mth m:,der ser:L8S

(2. 'I 0)

+

+ II Tn

homogeneous

by U slJ.eh.

III

form tb.at

U

tTl

in

the

x.

.

~l Vi e denote tb.c p:':iJ:·tial

(2.1'1 )

'l'he series solLltion of (2<o~)j is generated, by Lwing equation (2@ '12)

to Corm ,and. then

d

+ f

n

where l'

[

CC)[lSi:3tf3

;j c:dl

o • +

f +

;jth

ar,ld 13180 snown that space where th.e fU,nGtion u valid 8;:: identi ty holdr~e

(0)

)

)

t

t) )

The dorllr:un of

tho function

domain st[~hili of' tb,e

and Vogt shows that

r

i.

)

]

o

~u(

It sh.oulc1 be noted that this do;:m noi;

any satisfied because

!)ecorne infitute for f'i,n:Lte values of 1ihe

( "I _! \

0

+

]

(20

the doma.in

i'u.rK:tion the

f:l, apouno" fLlTlctJon, is the Tbcormn I [

or

t;})j,,[;l dOlfl21.tn

for

':Che theorems on the domai.n of valiclLty of the (2" 1 0) to the :;:lolJltion of the Zu:bov

is and Vogt@

i8 [lot here taxt the mOBt

tbe

't'hen

fini

w:L the

dorm:lin of'

'J1he

.

-)

calculated and. rmlal1est values

the domatn is bO!lnded.

(

(Zubov

(1-17)

p. (Th.eol'Grn 'I

)

'be the minimum value of

the

Therl

the surfaee U

m

of

for

(m)

C:

1

on the

hs.nd of' the recrtrrence of seoo:r:ul CJJla rntrl in

'"

y x.~ 11

!.J 0 • •

~)

of

ca:"Lctl.lated,

(

Gontl)ined in the

(2 .. -1;;-) B.nd 'l ) are

the

where

y + (;

,'"

~

•

"h 11 -, m and

the coefficients r

h

are the

coefficients

the ind:i vidual

each of wh,1.oh ha.'3 a

of ~the Y:J

c;

V

IJ1he lef't hand of (2@ 12) and (2.15) are

,'c~econd. and mth whicl:.l. the

'I'herefore the sollltion of eqrw.

Goeffic forms

the coofficient:')

[B][d]

J

the r" BJ _L

in the

(ii) [

y

(iii)

k is

k

(n(n

1 )

x y

1

unlc!1.own

terms in the (4*~) as

+

2)

combinations

v

v

m

(n

+

m-1)

(

] is the of coeff'icients = cp or

k

1;0 liUbov (ref page 6~) the determinants of the

[

from ( -12) and (20 1 ) are different from

unigae solu,tioDJ;; f'or the coefficients of the u always

m

of the solution of the recurrence

(2",1 ) for a second order There

is:-The

2 eli x'l

2

wruch are, in

1

Therefore

(2

+ (

+

(2

~

-1)

with the linear

[ J;

[AJ[ ]

(

17)

(x

1l' + L) (

b

+

[f(x)]

(

)

fOJ'JllBtionthe

[:s]

MacF'arltwe ( ) haf3 alterrwtiv('l

It that thEl z,e of th~; mt:ttrix

Inl

increase::;\

very

as nand m are increased since xmmber crr terms in the 81lBwer

cal(mlate~ct froIlt ( 19 c1etermin8.tio:n

for '10 x 10 rna:trix while i;he determina.L.i.on

ot~ _{for a four variable rna.trix}

_x

often

the second.

method

of'

result, of to a

.A second 0 rder Gart be in

v

][ x]

(2021 )

Ivhere [

G] is

a matrix" For a

[ G] is a

definite matrix (

)

..

We now which re~ml t3 :f'roHl

derivative of the

[xTJ[(}J[x]

toa

where [ C

J

form

[ xJ[

T a defirrite matrix@

(

_H

][ A ] ~

[ ]

(

)

(

)

A.

l

of the Ai' of [.II ] is zero. It is not nGCef3SE1J:~y for [ C] or [ G

J

to be positi vo defin:i:te for thi~1 unique solution to exist.

when [ C

J

is t1H-': mat,rix [ (}] o..eterminecl as the sOlU:t.i011 of' (2022) i:01 pCH'litive definite if and only if all the of [ .A ] have

real parts"

Under these conclLtions the guad:r:-atic form v [ XII'l [C'

1 [ xl·;

c·,

oJ

e"ply·~

. : :f' _ , _~ d:.1.:~ ,-,,==:'Cd .. ~,

a Liapounov function for the system ( ), 3Jld f\ince it is positive definite:

with a negative definite derivative over the whole state space the existence of such a

[G]

matrix proves global asymptotie stcibili ty of the linear system

[x

l

0:

[A

J

[xJ.

rrhe traditional teehnigue for proving t.he stability of linear differentia~1 equa.tions has been the Routh Hurwitz test -which ensures that a real polynomial has roots with negative real parts (~ ..

3).

The use of the Eou.th Hunvi tz, tei~t

the characteristic equation to be rectuced to the form of a polynomial, and while this is always possible the algehrB. frequently becomes impractieaT.

However there are two methods of p:coving thE:) asymptotic stabili of linear cLifferential equations with the form

l x

r 0 ]

~

[.AJ[X]

which do not require the algellraic production of tb.o characteristic equation. The first method is to use Bocher i s F'orml~lae ((

69· ),

p8.ge 61) to determine

the coefficients of tb.e characteristic equation arid then to a}Jply theH.outh Hurwi tz criteria in the usual rnannero Tbis method removes the need

for

algebraic manipulation but i t reqtlires the produ.ction of' the matriees

2 n

[AJ, [AJ , ••

@ • ,

[.AJ

@

'I'he 8oeo.nc1 metboc1 of ensurinp: that (?®4.) has a stable eqv·i. .. l:Lb:dunJ is to 'l'heCl if

r

t. G

l

.

is positive defini te the a.,sYlrrptotic stability of

(2.L})

i~1 p~coved aa in 1\ ' . - «"-; < G · · · . - 0 c" ·f'P.· -' .. -1- .,,' -t-'" J 3 _ , [' C' ] J._

.. ' Dcces,_,cu,Y arlu . . _~LL _~ <lc:teD, CO.l1CLl. ,I. on 1. 01 . A _ l.')

definite (51) i:::: tlw.t all its principal minor determixk'3Jd;s have values

>

_n

(

i3lib:cou tineo described in

2"

Consider

the

( ) for the C8.Be of a second order

The

r

a

1

r

g11

G~'11 ₂₁

8'12 [3,22 g21

g2:)

[g11

2J

_r

1 8

12

)

!

°11

,,]

°2:

g21

g22

13

22

NoV! indi viclual equ8.tions

for the

of [

]

we

si.nce ')

~

1 g11 + + g11 8.11 + g'l ~? 1

1 "":-

g11

8.12 + g'l ~l.

2 + 8.12 + &;22

since we can vlr:ite these a8 a matrix

1

J...lut we C8Jl alfJ()wri te the form [ ] [

c] [ ]

in the 2

(

equivalence Gan be extended to and so i t can be

that tbe '17) deveLoped from ( 1 )

'rherefc.re

t

follows that

f (

]

(

) i~; i(lprltical to the solution of

(

c~ ' ) )

. 2 .• 12

[

])

(0 )

'rhus progr'am 1 ) in the ('orrn

2

1 , n

to the ~301ution. of (2 e 2? ) PU

c

'l'he ( 1 ) with (:2022)

of

the

ZUllOV'

8.nd to form the means of (

defin:i te the of th'" recurrence e

is vaJ.id; beCBm:lG the

must have the form

former1 by the linear

the

(

R.H.S.

veetor

[A ] [

x]

and the s tabili ty of' thi~l

be

in order to

matrix routine for

de fi ni te <p function defini te <p fUnction cen

teness of

[G]

to use a (p f'Lmction which a desired functional

so that a1; least

be considered@ Then the to forrmllate the [ G ] one ar'bitrary

u;:> for the

matrix~ and jj' ] is posi ti ve definite the resl),lts of section 2" can be invokec1 to determine tb,e dom~d.n -of' of each fune

the Zubov

It should be noted that when the

(p

function is only positive semi·~ <lefini te the described by ) must be observe<l whf:m

theorems

we notice from (2 .. 13) and ( 1Lf.) that for the linea.!.'

u _

Hence from (

)

[

J[

_{] -}

Q • ,

) dt

(

)

rp

[

J[C][x]

it bas inclicatect K31man

(

tho

to the J:/DU ·1:;11 t.hen

and. Be:ct:cam -I:;

do not of

valid 0

I t front ( ) th.a the Routh

assoei.a.ted with i,he functional

"" / n

J

(

:6 \ J dt

t::.::o i;:;;:~i

[

J[l~]

since

(2e

) shows that

n

Z However wri

been to find any detailed discm.8sion of the

between the Houth Hurw:i.t~ test and (2@22) than in (

)

in La ~ialle and.

r~ef8hEltz ).

(

)

)

It

is now

clear

the tWO ( 22) and the

) ha~; two the Houth Hurlvi tz

manual

solution of (

to form the characteristic

program or else while

) (leD in the first of the Zubov

series const:ruct:Lon

While it hafJ been Elhown in this thElt tho Routh Hurwitz

cri terion is assoc:Lated with an measure there

is

choice of the quadra1;ic in this In contrast the

associated. with (2@ 22) under the control

of the user by virtue of (2,,2J

+),

(20

)

_and (2.26).

i3UM.NIARY

:ccGorcled tl:I.O Zubov

to used fl.S n fnethod control,

location

In stc'lte

if

( iv) 'I'b.c of a index for

':celnerrts Cf1n be

(i)

Obt"'Lin Et set of d::LCferential

in

the

form

(2~7)

b;y

Ilbout the equili'brium

( 7

:2I.nd express them in the

r ® ]

form: x

"" [AJ[x]

) Use the result of section the matrix [ G

J.

if

[G]

is definite the linearised system is asymptotically and as shown Kalman. and Bertram

9,

ref'

(55)) Jehe

nm.l~

linear

is

s'(;able

(l,n linearised of the

:t\xe small perturbations.,

Thus

(

Use the recurrence p1'oeeclure with ) and. ( suffioient nuniber of' term,,; of tl:.!6

) Use theorems III or TV of sec t:.ion 2@

3

e 2 to

'l'hen, wi "chin this

U

_m an to tho

1 ]]\!~lRODUCfI'ION

II.S recordet),

by Almeras and

work of the e:bove autho:cs hut thif'. tb.osis vtill demonst:cF!.te a f)asic

in the The

r'l()r']]'nn,,"~ _l __ ~1-.Lvt;"'" ... 1, C ()rl •. _-'" eli' ~ tJ..i., 4orl'" _ .-~) l"_"~~,~_ ~ 1"1'" ~ ~'.< Y'<"c'e""l"''''r' ~,v v __ t e~OlYl'" at"-c·!~t:' /",;, .),."I.:J '(; tl'-<-u 1 4.(~ '111 ( t;bc intelltion of

no

model has

to be inclttded power where both and. s!l1aJ.l

loading mLlst be corwiderec1"

Irhe inteni;:i.on of' tbis to governor

which includ.es 8.11 the major nonlineer:itieE. of the wh:ich :l3 still able to be

of' the form

[ J

=

f'or control stuClieSe

set of

) 1

Section

3

2 diflcusses the

but

l,u.led in the governor

the gove:cnor moclc;;)l

and section contr::.:iJJs the derivations

DIi:lGUSSION GE' GOVE:'RNING SYS7.'EM CO'WJIUNE:N'l'S

1

we consiclen.' reaction t.urbines i.n

It

haf;l Ii ttle

the

and

where

l}

(,l

Y

H

on tho that

lived to have any effect em the response of'

~,

:::

a,llows the tur'bine

turbine

turtJine flow (c f s )

of' water

(Tb/ft

j

heac1

(feet)

be based.

the

the

detailed infocmation is ,?nra:Llable from the turbine man,ufacture:CI'I

on the vcfri8tion of'l wi t11 va:riatiol1 in the of a

lIm/ever since our so mmry ef'fecd:;f3 such

on the water and

friction not consider the the

'but we the effec't of variation in T) in the turbine

effects"

.AI though such curves are seld.om used as a basis for turbine

because of

the

difficulties

of' . tbe

charac teris tic~~ tUTbines in our :range of' interest form

shown in

₁ *

fal1B off

mre:e 'wide ranges of _t:ion" In 'the tU:I'hlne

fan off

to

wi 1 of

as defined

It can l)e seen that the

the servomotor

ZncreQS'/nY'

C;o/~

t:joe/?/rJS

eed

of the turbine iH ba.sed

such

that it nm'.1t be as I t

account for the non]j~nea:dties and first order vaI'iati0l1~1

in turbine _"

The water collUlU1 in the preasure of a station has fini te

of veIoel ty of' this colU.mn is obtained at the expense of head V3T'iaticms tbe

since both the fJtruebl'I:'C and the water are elastin any

8udd,en

in the

pressure will be the wt;l,tel'

w!:wewhieh will the well knownw8:ve

) 0 {\ more modern ;:;olul:.ion of

(3

?)

has bee n m~l.de by

Donelson (

15)

e 'I'hi,~

J)9

to

i'OJ"m of the well kOlJVm

~tn be slJcce::;~lful in the :resporu:~e

of Et preSi..,ure

( 5).

'I'he,;;e complete solutions of the pre:3Sl1re are essentiaJ, where Dt;anding waves set up reflections of pressure surge:::; in the penstock can EJ,pproe.ch re::.~onaJlce yv:Lth the oDcillation::1 0[' the mech8Id.cal governing

I3Te not neces:~8:ry when euch. a conctLtion is not to occur.

~I'hus, beCaUE)e of the high Vfllod. of of tfl.e prer'3sure ~'ur.!?:e[) in pem3tock,'3, the problem of resonance J~n

penstocks I~'or mO[Ji; cOflve:ntional sta.tion,o;; vvi 1;h reasonably 8110:Ct pcn8todc (less thl3Xt about jO() feet) it :u:; permissible to

the effects of' compressibility in the preSSll"re system anr1 derive ~) lra.n::d:'e:c function based on the :'Lnerti:::l of the water coluIlLn Thifl watel' inertia t:c~:msi'cr flll1ction :u-; wel]~ known, 8.wI one form of i t 1_5

'r

VI d

where ,'\H

(tm)

(

tl1.rbine head v.<3x'iatiorl

gate tion va:riation.

In terms of frequency response tbiD tranp,fer function ilJ an adegu.::d;e repre[lentation of mOBt typical penstock fwst:erlh'3,. except at high frequencies where standing wave effects become a.pparlfmt@ lnoquent ,justification hAS

been provided by Oldenbul'ger

C:5:i)

for u:::in.g

(j.

3)

in most stuclie[l instead of lwing a complete soJ:lltion of thep:cemm.:ce including elastic ef':('ect[:;.

~L'bus for power stuilies wbere the higb f:ceqLtency ef'fectf.~ a]:'("; f-lUb~;taut:Lal.l.y integrated out by the inertia. of the

., oy vm.l:er ' :Ulert:ta. • , . aeco:ccl"l. " . ng '(:0 , ( -.~ ". '-, _. -.. ... ) ,''' ... l1 .. (~l 'r-," . _Leo .I .• n ..• +; _v

studies of the

su.ch which can

in the

(

the governor the appears be two

(

device with a. finite inertia.

) A source of active

which

Cl'he eiTeci; of the rota:~ionC\l inertL'1 of' the is described

power

ng

the IiInertia

ex.e:t'ted at the tu~('bine sba:Ct

eonsists of friction and and of the air

'l\he and in lOt small

in power t;ern studielc~ ar(~ ei theY' vlith the

eoefficiellt~ 'L'he i::1:Lr gap

ire; a concU tions on

8],ld of and

Of course a full cr:msicleration of a full

the

g£lp but until rlOW

omi tted from atuc1ies

5

of this 1;h0si8 conf:lid(lI'S the accurate

of the air gap in limi tations of tho

used for tlds thesis hHve the execution of Ci.n,y numerical

stlHlies with t'his

It ha.s lmen GOIDJnOn

the

lmwhine to the

l~

'e

0,)) +- D n

i~ gen

(

d.eviatiol1 :(n

'l'hif3 ir\ the

the most of governor

load

ZCl'O tlleS8 C(clS8S the appropria'ce

c

6'1'

In

for. needed 'becaU1::le only off line

condi tions are cons:Ldered, but the purpose the

and accurate a.ssessment of the instar.Ltaneouf>

gap torque

The universal prc\c t:i co is to

the

:3ervon01;ors wb:Lch t:U~E:l in turn Gontrolled

or electric pi.lot aetLlatol'o although it h.as never control the rate of change of

actuator 8.11(1 servovalve has been

of is whether or not the

:tn the derivation of the governor'

of

of the pilot [1C II.e<~d not be i t eOllld e well be

in

::uw

other

~p:Llot=,

The

(

)

)

canbt:')

the reason

ca.sc WQu.ld wh:tch l,S

eoncl:udeCl. tha.t the actlv;,tOI' not

between this

Ctetuatol' is two

sleeve self neutrali

as tlJat used

the Woodward

With this of the

servovalve

will not follow the

but the will be so small as to ·be ThuEl the

actuator / f;crvomotor d.e t ermined

t:he of form

shown

£>MAX

U/VO/<!!/7 If

eecr" yet//'? ~ C;

FIe;

'1'.ho

of actua:t:Ol'" It will be

th1:::

been

found

the G.

discussion t follows tt1;;.l.t reasonHble and.

of the servoV8~ve

.".,rhe:ce z

=-

actua andii'( z) is defined 2

t.1:1e raJ.lge constraints on. thE)

and the preaonce of'

huf'f'ers will

'Gl(

.l' z,~

( 6)

of the function

J"(

z,

g)

d.escribecl in tletail in

~'0 close Lhis f,8C note that ba .. sed OIl

r1.etailed of the Woodward governor

(38)

it is

governors bV most other m8rlU~

facturers" governors are

to

siIlulate the of governor and so too Garl be treated

the of this ttteBi8e

It :w COIOIaon

arrange t.he inctlcator to )cion of the

servomotor a, fraotion of the stroke from eIoned

to

However at. no 108,('1 the will rlOt be

eloseCl and may not be

(

the

(leviation in tlem be 11erivecl subtraction of ir!.d:Lc~a.tor

Cll1(l definitiow>:

'/.'he turbine for rated f:LrH.1 hea.d is rolated to

in t11e :cangc from no load to full loade

in ideal hEwed on th(~ change from no load. to full lOG.d being

ch:3.nge in indicExtcd opening based on the ChEHlgf3 from fnll

clOBed to full opened being 1 p.u.

no load indicated gate position full load ind:iG<:lted gate

'l'GI\j ~ 1 turbine

'rhe interr'elatiorlship of the se qU.ant:Ltie[! is shown in fig.

3.:;

<>

I

In this

of section

me charlicel

'{' I

"

or 'I: OJ Wo (;J

All'

"

'-,-~

W

the deftnition of

Ii

therefore there

lS

Iw

o

V.A.

suhsti

T (JJ

D~ 0

2H w

o

f' our gover{lOr l'he derivation8 of known but

in this the

electrical)

inertia conntB.nt

(

~n becomes

0J

o

Now the rotational time

defined as

by the

to

(

2 m

(

on

7)

b'Ltt in of

'r

can

rn

W

o

whenc:e.

have

I

o

'I'

But

Tw()

is the rated power

per urrLt

.

n

'1' HI

so that we have

2H

the equ.ation of motion

Wo

W

o a

has the dimens:lcms sec -1 or ( 3 .' ga.ve

~~Jj

_•

(J

~~

w 0

V 1\0

p",U"

to the r'at",d volt amps in tho

(

)

per lHUt that

I t tbat

NoV! the substitution of (

( 1

C))

(

2H 9

n

V

<:>

T "

-~ n

m

(3

11)

of th.c d.Lf'finult:LefJ that carl a:cise 138 8.

of units u~jed tha lH1it

nett in p~u~ of

constant in seconds

in poU. per sec of

n.eglect:3

The is based on the

P

K

H 8n(1 (~

'"

K'

G ( 1 )

}1'rom 12) caD write

TN

;;;;; K G II 1 )

where

'r

:::: rated

N rated

~

raterl

H rated

Now consider a

('I'

\ b + 6

'r)

'I'N( 1 +

di

headw

6N)

position from zeri) at no load to the valu,!,) and head.,.

K (C~

."

6G)( o

3

K G

n2

1 )

l~ o

/1" 'I' /\

) (1

H18ke

and above

Now the defirtitions of per un:LI~ variables

h , .. P0U. head va:ciation

:3

,2

We

write ideal base

Tbe turbine then becomes

+

Now notice that the .. ~~~.~,"-... ) in the inlwrent

coefficient

cannot be neglected or held constant bees.use

we

~N 'be small and so

it

i.s clea~(' that

coefficient; must be includecL in the tln:;bine 'l'he tel'm indicates th.at

have

some

T

(

3 2

) is

(

( '1 )

( 16)

varLlb1e

of the turbine wi.thout

'I'his variable damping

not tbat and

vari8b1e

increaBes with incre

to

that

This term does not

variat:ions t,urLdne and

effe:cts

vH~,:iEJ . .tion

well

f:)S

with

te:r:'lll Dn in h

flow head ik~viation

by ( 16

.~ but the reduction the

fOrTlI would have been rather

D ) r 1

£1' \

beoome

of'

( 1

dQviB.tion we incluJle tha.t the inereased

If

have been nee(:)seary

the

( 1

To e.xpre3s () 17) and 18) in terms of the incticated. tion Vfe first note that G 'is meaf:mrecl f'rom the no load

f:eom the clm~ed 'Jlhe restt.lt at' which is show.n

g'

(3

1 )

since we wish to use the

of the

indicator in

of

(

)

H!U.st

1 ) and

11)

)

+

(

)

It be claimed that ( 21) and ( 22)

a reasoneiblE;

atioX1 on

the

of linear'

account for the

and

acourate enough for electric: power

studieso

D

"

3

;;;; TGN

(g

+

2

follows

~

D*n

We conside:r a simple

of

II aXld

cross section

at V under ratccl 's second

g

>tAHA

where ¥

of water

g

acceleration

by the rated veloci ty and h(~a,d

g

d

h

there :l.S

)

dis~

(

)

(1

+

the <l:ee not mathematical

for t.he COIlJli tions vJh;Lchbhe

the exiEltenc:e value

of order terms in

)to

we lineari('lG

(3.

) to

AV

V g

1 'h

"2

be

the veloc:itieG

small '['hus

(j.27) 1;hen

elds

where to of of be rated '1' VI dh

IN

gIl h

to the time for the from

head., l"or L, and with

J. a :.eest pres~~ure rated system we

For the case of a preasure of exaot

but a suita:ble

Note

(

) since based on a

based, on from

(

)

watcx' column from intake

a+ .,'

full

by the

made up of a numbel' of' each

3

)

sectiOl)' t:rH~

the form ( )

no't; g n

zel"O

s :i.::~ basecl used

:is a

of the

link between

the lower the

of:' the be wi

of' the servovo_lvf:l

'1'he lever ratios ::;hm'm in the been chosen as combiflation the !y:!qui:ced :cel:'Hl1

t.

Different

the

result in (Efferent relating lever ratiosbui'.

would

11'1113 e, y and

n

°t

y

(

)

'I'ben") a:ce also the equations

e

f z

whence z e

(

)

n y

(

t

Next

definr:; temporaxy droop limd droop 8S th.e values

to maintain wh.en g ~ 1 @

in t~he valu,e of

f:rom the corlstraint he loeked

K I

}t

.£I/s~/qc<!:P'1

(/r.{t)

=

Q~-SC'//ENA

P/A

Next K

substi

or

'l'hus

y, and SJ.,Ilce y

temporary

o

thai;

n

g

foY'

o

1

)

contributed the feedbac.k

trte

is

the value of

Now let tbe relax so that the

and let K take a non zero v3~ue"

K

~[lhe _{is the}

there the

- · - - K

K

t

It HI here tha.'t, the

1 p.u" :in

Now consider the J0_{0:r the}

'I'

Then

values 0

in

(3

)

wrlich we call 0 •

p

Op are defined rlcrl; for :in indicated

for

k

1'heref'ore from t.hree eql)

']'

_:v

so

rc (')

1 +

r

) g

)

NoV! di1:'ferent:Late ( 1+1) w. t. time

1 +

6

a:nd substitute ( 1 ..

5)

to

y between this

)

s the governor

1

•

n

(1

.

g

e :::

(

)

~rhe governor of the servo~

va.Lve

equation ( 6)

g __

[11 (g,

z)

_One

6)

_{is that}

for smftll

where the vcloci

and lLmits arc not

eneountereCi.

C~ (

)

(

the governor model

3mr).11 condi tions. When the studies

Ge ( .11'S)

It ll111st be remembe:eed that the levex' ratios used. in the s of the

to the schematic where ELll the

or per the actual construction the levers

the schematic may have different or

not even 'be levers at al]~~ could be cams or the

well lcnown English Eaectric governor or circu,its the eleci;:t'o~ governors"

occur in the same ratios appea,r in

may not but the comlJined result of

both alld lever ratios vJi11 result in

the

schema'hie of although the

1+ or in a direct 'rhus it C8Xl be seen that of the actuator may be small in te:cms of

um

ts it is neither small nor in the based upon no:rmalized

Tt

is to consider the effect of

the actua.tor has infjnite and negligible

The condition for infinite pilot actuator gain is z ::::

o.

( :,,, ) in the new form

°t

.n y

1 g

aXld a11bst:i:\;ution inl;o

(3

) giveEl

!'lb ove which

that

( ) f its

,= fj'

r

(

~')

o

t

c1efinc8 the

hi

o

th

and gl

Subs1:i

wh,ich if1

)

'I'

(

(

0 ) ~ 'if

P'

to

use~3 the

in loael be

~l'GN

these into (

)

e

n n

the

gOV'I;;:rnor

it

can be observed

(

)

"

_('i

°

P'

<:'

0 ) ~

roO ( '1

for

1 p .. Uo

g I of' thi~~ thesis"

Then

.

n + n

of

that

)

tbe

)

lJ

Of)

)

thai;

i l l load and, that

reference

))

a direct

'l'h .. e

condit.ion:;; If terJi; :cesults are a;vailc;J,le the val:lw

by

the

of the indica'l;o:c 3i'te!~

(U:;"(.urbances about gi'V'Eln equilibrivm lo"tding

in the ~J~bsence of test data. the vcllue of Base call be cD,Tcul[xted from the fox"nrula

Ba.se N.I •• indj~cai;ion +- ( ':J())

tb,e 'I'G.N*Ba.i3e appearr': :Ln the fin£:tl -Glll"bine equation

all COlI!:pu.terprogr8Jfla u~{e the nl.odif'ied pa:rameter, BAtm:, defined 'by

~L'GN >I< Base

It follows that

indication) + p.u" eqllilibrium load"

c~.

)

~rhe results of :,lecticm constitute a governing model.

Ine:rtia .11

ter h

~:'urbine D*n

IJ1nrbirle D

Governor

E'

(g,

z)

'1 ]}\['I'HODUC'I'ION

In t})is eha.pter the gover'ning moo.el developed in chapte:c j i:') ai.~'JCUSE!ed B.nd eva.luated. by reference to a class of step respon,'3e tests B.nd model flimu1ation of these tests~

~Cbe lJlOi'3t important question regarding the usefLJ,lne~:s of the governing

model wa}';: the matter of the deterllLination of certain pa:r'ameters, and a ser:ies of tests I'laS performed to p:r:'ovide a ba[lif3 fo:!:' the inve:::l'tigation of this

question" Since any testing technique in power system engineering must involve an absolu.te assurance that the contirmi ty of supply aXlc1 immediate availability of machinery will not be af'fected this chapter does not separate theoretie2.1 and practical. consi(J,eration:::;.

Cbronological1y the eX}lerimenta1 results l;;iven at the end of this cha:pter were obtained first 0.110. the analytical studio::;. of secttons L1

-.4

mld. 1+. 6 were made to explaj.n anomalous results. However the chapi:.er is

developed on the basis of the correct technique ·'i'ihich vms produced all a result of the explana.tion of the an.om,s1lies 0

[I. 2gXPEHI}VIEN~'l\L FNJiLUATION OJ!' ~L'1J1~ MOD)1;L

~'he ultimate proof of any engineering model is the complete of calculated and test results for all rCEJ.lizablc combinatiorm of circumstances" 'l'bLw complete proof' of the governing model would, ~requ.ire a range of tests cove:cing all possible operating concliticms of a hydro set, including some condi tions involving imrtability@ It is est3enl;iaJ" however, tbr! .. t; te;:d:.s on an operational power system be restricted to conditions v,hLcb do not affect

the

security of

have

t

tlle that.

while load do

(;mto the their exccu.tion

~'be off tes ts 8.1:"e rna~ae lilni

g;overno:c error in the ve.,ri;iblcB n,

line

In

the load tests

the

the to produce 1ni tia]. error in the

conoition in both tiests thbt of the no load

ngwi Base =:

The fao1; that test d<>.ta Gould

for

one value of' a

b

wld.le on

13ase :is limi ta.tion of the work 1x'.cor(ted in this but

the exeGut:ton of on-load tests wew oonsidered to be the scope

of the between the Emo the N • .z.~~. D.

'1'he methorl of eval'Ll.at::i.on of the moclel is to corrrp1::1re the measured

response from the small ",mel load ection tests vvl th the trar~sients

vue".',,"'U by model simulation of these tests.

If

the

of the

set and governor were knOl'ln thi direct

check on the of the lIk'lde in the dertv.f;ltion of the modele HOVlever not all of the governors CDJl be deduced from the

data of a the

of the pilot~a(Jtuator/servovalve and the dl'l1nping coeffici.ent

Dnorn cannot; be determined. ~'herefore values for these

mUfd; be trial and erI'or to model s:i .. lTIula.tions whioh

best

to

the test

This of the two mDdel :recluces

the tmt f:lince

sarne values of'

to

rpWQ other the and

Go:rJfJ"tant flUJ.s'C be determined test to dete:r:m:Lne

t:he exeetltion of reco:cded :in no

this

is determined, 'by a

of

Vlhen IS rnethod for' used wi thout mocllfic8,tion i t was found that no could be obtained between sinulation and

test

This lead to the of error in the method of

nn",,,v, which is in the next section~

1

L. M. Hovey, (52), ( ) has d,escribed a tes 1; and c:alcula. tion to the

value the

the

a correction to

for 1 p.n.

1\)1:' a '1 p. u.

the

WE; consici.er

Consi,der now governor

As

locked

so that ( LI)) becomes

(\

(z

Kg)

n ~ +

and to

~ (0)) g

(

Novi 's '\;eI31; to d.etennine

of

test

the

governor" s method of

a.efirled i l l this

lood, but :in the the initial d.etermination of the

for the condition of

section 3

h

there is

.v

°t

g

)

,

when z 0

we

~cespOllse on

(

)

of of a

insta.ntllneous value

g s rnethotl o:f G£1J based on

three

(

)

dev:i.a.tion in needed to

(c) In the closed dashpot test the effect of' water' inex'tia. to

the

deV:LEttiOl1 from i tS.Llyi.

of the

to

(

)

where

p;a.te

with

(

readi ng of

at

0:(' :('i:(,8t

It became

simulated

from to obtain bet,ween test and

response tntnsient::1 thai;

t\

determined

than

true value of 0t@

The

errot'G were eonsig thaI),

of and so not 'Lb.e

r:.ource of' the Neither is

(b)

s.ince

magni

reduce the value

In in

12 deyjat:ion

COflBtarrt of

metbod

1"01'

that

(l,IJI 1;

:in the assumed

consideration of

(

t]:le case of

reCTJ

te~ems

cond:t tions and we

the turbine

J\ll.l where B

is m8.de

.h

(

)

then, theWEl.ter inertia e:ff'eet in the t:ruc no load. te[d. co.ndition

mUCfl amaLler :vL is in full load corlditi on considered

and tbe

ftY'st l.nvD]id~

frorn