·"./

IjI.\

CI-IAI'TE.R

NINETHE ENERGY SYSTEM IN STEADY STATE-

THE CONTROL PROBLEM

The two oreceding chapters were devoted to the problems associated with theselection ~f a normal operating state for a power system. In this chapter weconcern ourselves with the problem of keeping the system in this state by meansof continuous automatic closed-loop control.

As the demand deviates from its normal value with an unpredictable smallamount the state of the system will change. The automatic control system mustdetect these changes, and initiate in "real time" a set of countercontrol actionswhich will eliminate as quickly and effectively as possible the state deviations.

9-1 BASIC GENERATOR CONTROL LOOPS

. Figure 9-1 depicts the two major control loops with which most large generatorsare equipped. Th~ automatic voltage regulator (AVR) loop controls the magni-tude of the terminal voltage V. The latter voltage is continuously sensed,re~tified, and smoothed. This de signal, being proportional to I VI, is compared":'lth a de r~ference IV Irer. -r:he resulting "error voltage," after amplification andsignal shaping, serves as th-e Input to the exciter which finally delivers the' voltaVf to the generator field winding. . ge

The automatic load-frequency control (ALFC) loop regulates the me awattoutput and frequency (speed) of the generator •.The loop is not . I g .the c f h AVR . a Sing e one as 111e case 0 t e . A relatively fast primary loop responds t f .

hi h h '. . 0 a requency signalw IC • as we ave noted, IS an indirect measure of me awat .speed governor and the control valves the steam (or hvd g) ft t ~alance. Via t.he

- ro aw IS regulated With

ALFC loops IIVlllo()p

~--------_/'-------------.Secondary IILFC loopr- A- ~

Tie~1ineVOllil!JO unOI

Primary IILFC loop

mixer

\dP,To network\

Figure 9-1 The automatic load-frequency and voltage regulator control loops of a synchronous generator.

-..;-== ... ___

....:I:s"-c>Z

Z...."ooc:~oz

fJ. 'd

SYSTEM IN STEADY STATE-THE C01?TROL PROnLt.~ .THE EN ERGY .\ ~\ ~

F ~ .f'l a... . the me awatt output to relatively fast load fluctuations. -, d.., t:

the intent of matchmg t~ t take place in one to several seconds. By thus. \\By" fast" we :nea.n changes t~ b lance this primary loop performs indirectly a ... .,tending to mamtain a megawa a . , . \coarse speed or fre~Uen~y contra~~tains the fine adjustment of the [requency; and '\

A slower secon ary o~p ~ . har with other pool \also by "reset" action mamtams proper megawatt mterc ange b \members. This -Ioop is insensi tive to rapid load and frequ~ncy cha~ges u tfocuses instead on driftlike changes which take place over pen ods of rmnutes.

9':'1-1 Cross-Coupling Between Control Loops

The AVR and ALFC loops are not in the truest sense noninteracting; cross-coupling does exist and can sometimes be troublesome. There is little if anycoupling from the ALfC loop to the AVR lo,?p, but interactio~ exists in. theopposite direction. We understand this readily by realizing that control actionsin the AVR loop affect the magnitude of the genJrator emf E. As the internal emfdetermines the magnitude of the real power [Eq. 4-22) it is clear that changes inthe AVR loop must-be felt in the ALFC loop. I

However, the AVR loop is much faster than the ALF9 loob and there istherefore a tendency for the AVR dynamics to settle down before they can makethemselves felt in the slower load-frequency conttol channel. In thejdiscussions tofollow we shall first study the two loops independently starting with the AVR loop.Later we shall discuss the effects of cross-coupling.

\I!I.)

9-1-2 Small-Signal Analysis

In analysis of power system dynamics one distinguishes between large-signal andsmall-signal analysis. The former type of analysis is encountered (d1ap. 12) wheneffects of major disturbances are being studied. In such situations generatorvoltages and powers may undergo sudden· changes of magnitudes that mayapproach 100 percent of normal operating values. Usually this type of analysisleads to differential equations of nonlinear type. I

Small-signal analysis is used when variable excursions are relatively small,typically at most a few percent of normal opera ling values. Differential equationsare now mostly linear and the powerful Laplace transform analysis methods canbe employed.

We distinguish between the two analysis types by means of variable symboldesignations. In large-scale analysis symbols like f, 0, and I V I represent actualfrequency, pO;rer angle, ,,:n? ..voltage magnitu<~(: respectively.· (Compare power-fl0w analysis I'P Chap. 7). In small-scale analysis we use the symbols Ilf, Ao, and'·61 Vito mean the deviations of f:equency, angle, and voltage magnitude fronormal operating values. All analysis in this chapterfalls in this latter cacegor;

';j v;- c:(:l.' ~::;. tI:l c(\) '< ~o =..... .-. O· RIC ENERGY SYSTEMS THE'JRY: AN INTRODUCTION.':;;!;; .JI1~ eLecT~ on '

;-/ 9-2 THE AUTOMATIC' VOLTAGE REGULATOR (AVR)/ ' The exciter is the main component in the AVR loop. It delivers the de power to

the generator field. It must have adequate power capacity (in the low megawattrange for large generators) and sufficient speed of response (rise times less than0.1 s~conds). The bas.lc role of the AVR is to .provide constancy of the generatorterminal voltage during normal small and slow changes in the load. However, itis com~on practice to design tile exciter with enough margin to give powerfulboosts in the excitation level also during emergency situations.

·9-~-:n. Exciter Types

There exists a variety of exciter types in use, the more important ones describedin Refs. 1 arid 2. .,

In older' power plants, the exciter consisted of a de generator driven by themain generator shaft. This arrangement required the transfer of the de power tothe generator field via slip rings and brushes.

Modern exciters tend to be of either brushless or static design. A typical.brushless AVR loop is shown in Fig. 9-2, where the exciter consists of an "in-verted" three-phase synchronous generator. The latter has its three-phase arma-ture on the rbtor and its field (Ill the stator. Its ac armature voltage is rectified indiodes mounted on the rotating s.aaft, and then fed directly into the main generatorfield. This deisign obviously eliminates the need for slip rings and brushes.

Comparator Amplifier Exciter Rectifier,...---......, ~,-~

\ ~Wlrel e 8~ ~r-J=hllirt+ 0 . '_ _ + I I Gsit:> VR

. I - I II I I II I I-I I I iI L.r-._j IIl I Gf I J Rotating components .-'-_I

Synchronousgenerator

8

Stability compensators' WI Rectifier and filter

~_---:--<l------=:~IL-~ 1IF'~e 5Jl-2 Brushless AVR loop.

I'I

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I.II,~!II\ .I

THE ENERGY SYSTEM IN STEADY STAlE-THE CONTROL PROULEM 303

In a static AVR-Ioop the excitation power is obtained directly from the genera-tor terminals or from the station service bus. The ac power is rectified in lhy-ristor bridges and fed into the main generator field via slip rings. Static excitersare very fast and contribute to improved "·transient stability" (Chap. 12).

9-2-2 Exciter Modeling

Here follows first a brief description of the AVR loop depicted in Fig. 9-2. (Therole of the" stability compensators" will be discussed later.)

Assume that for some reason the terminal voltage I VI would decrease. Thisimmediately results in an increased "error voltage" e which, in turn, causesincreased values of VR' i., vj',t and if' The d axis generator flux increases as aresult of the boost in if' thus raising the magnitude of the internal generator emfE and terminal voltage V.. Mathematical modeling of the exciter and its controls follows. If for the

moment we disregard the stability compensator (shown dashed in the figure) wehave for the comparator and amplifier respectively

61 Vlref - ~IVI = ~eand

(9-'1 )

where KA is the amplifier gain.Laplace transformation of these two equations yields

~ IV Irer(s) .; ~ I VI (s) = ~e(s)

G ~ ~VR(S) _A - 4e(s) - KA

and (9-2)

where GA is the amplifier transfer function.The last equation implies instantaneous amplifier response. In reality the

amplifier will have a delay that can be represented by a time constant T and itstransfer function will then be of the form A ,

(9-3 )

If R, and L, represent' respectively the resistance and inductance of theexciter field we have for voltage equilibrium in the latter

(9-4 )

t IEEE (see Ref 1) uses the symbol E'd for the field voltage We d~part from thi '.reasons: . IS praxis lor two

(a) Capital letter symbols are reserved for phaso .(b) S b 1M" .' rs,

ym 0 e IS reserved for emf and/or "error volt •.age.

:r.

"rI~

1

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3()~ ~I.EC'TIOC ESEI(t;Y S\'STc~IS THI;llRY: AN INTIWDUCTION

. . d es K armature volts perMeasured across the rnain field the exciter pro u~ )'1ampere of field current i., that is, we have the proportlona ity

t:"vf = K, t:"i,

Upon Laplace transformation of the 1;5.t two equations and elimination oft.ie we obtain the transfer function of the exciter

(9-6)

From these equations and using accepted block symbols we assembledirectly the transfer function model shown in Fig. 9-3. The time constants TAand T, have values in the ranges 0.02-0.10 and 0.5-1.0 seconds respectively.

iNote: Sometimes the exciter block representation is given as

VF

VR K£ + STE

We have chosen the form shown in Fig. 9-3 as it conforms with the generalcon trol system defini tions for" gain" and" time constant.")

9-2-3 Generator Modeling

We need to "close the loop" in Fig. 9-3 by establishing the" missing dynamiclink" between the field voltageu. and the generator terminal voltage IVI. Asthe terminal voltage equals the internal emf minus the voltage drop across the

Comparator~

Amplifier~

Exciter

Figure 9-3 Linear model of t;IC comparator-amplifier-exciter portion of the AY.R loop in Fig. 9-2.

(9-5)

I.

II

\

'1-\'"v'':IV)

'VIVI'ro'~'''' ~

o .-~--:J. ~

'~o....'.i 'PoTHE ENERGY SYSTEMIN STEADY STATE-THE CONTROL ""ell"_" .....~;>,_ 0»

<, ~

\

\

\iI

Exciter field Generator fieldAmplifier

-----, LlIV IKF-1 + sTdo

la) vj~r

uLlWI->

lb)

(c)

Figure 9-4 Closing the AVR loop; (a) i~dividual bloc~ representation; (b) condensed model;(e) closed loop model.

internal impedance it is clear that the relationship between V f and IV I depends onthe generator loading. The simplest possible relationship exists at low orzero load-ing in which case V equals approxir:tately the internal emf E. Kirbhhoff's voltagelaw applied to the field winding yields in this case: I

. d..Ji [ d]I1vf = Rf I1lf + Lff -d (111f)= -L - Rf I1E + Lff - (I1E)t· W f a dt

IThe last step follows from Eq. (4-5). . IAfter Laplace transformation. the last equation yields the field transfer ratio

I

(9-7)

I1E(s) ~ 111 VI (s)I1vAs) = l1iJAs) (9-B)1+ s~o

where

K A wLfo_

F - -fiR~and T;,. was defined by Eq. (4-63). .

We 'can now complete the AVR loop as .hown in Fig. 9-4a. In tws easy-to-. . I

follow steps we condense the diagram down to) the final form shewn' in Fig. 9-4c.

'V;uo'" .~j/

1~ ELECTRIC ENERGY SYSTE~tS THEOR v: AN INTRODUCTION

//// _ Tl]c oven-loop t1'(II.J.~,/C,.[unction (;1s) equals

~

.,;0ot:

_J!.1;ft~~Fjw.--;_"2'i:'102:.";:.'ii:-:;,·£j'S{"_"O=/J.::.~j;_T:'JJ.i5.,?J;;iS!f!./!i~;it!i!!fif![;!.{fi!!ifliff."

»:/

KG(s) = _._ -~~--:::-:-:- __(1 -l- sT,.j)(l + s:z;,)(1 + sT~o) (9-9)

. where the open-loop gain K is defined by

1~£ KKK,.j c F (9-10)

9-2-4 Static Performance of th.: AVR LoopThe AVR loop must

1. regulate the terminal voltage I V I to within required static accuracy limit2. have sufficient speed of response3. be stable

The sranc accuracy requireme-nt can be stated thusly:For a constant (subscript 0) reference input the likewise constant error l1eo

must be less than some specified percentage p of the reference. We thus can writethe static accuracy specification as follows:

, '

l1eo = I1jV Irer. I) - I1jV 10 < 1~ . 111 V Irer. 0 (9-11)

For a consfant input the transfer function is obtained by setting s = O.FromFig. 9-4c we thus obtain .

I G~}l1eo f I1jV Irer. 0 - 11IV 10 = 6.1·V Iref. 0 - 1 + G(O) 6.1 V Irer. 0

1 1- 1 + 0(0) 6.jVlrcr. 0 = 1 -I- K 6.jVlrer. 0 (9-12)i .

Clearly, the. static error decreases with increased loop gain. Substitution of(9~12) into (9-111) yields the minimum gain needed to obtain a specified accuracy:

K> 100 - 1 (9-13)p

For example, if: we specify that the static error should be less than onepercent the open-loop gain must exceed 99.

9-2-5 Dynamic Response of tilt! AVR LoopFrom the theory of linear control :;y:;tCI11S12 it is known that the time responseto. 1V 1(/) of the loop in Fig. 9-4 equals

. . t.1V1(t):o= 2.1-'{t.I~o((s).G(~)(1'1} . (9-14)

I

III

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:'::'."

THE ENERGY SYSTEM IN STEADY STATI'.--TIIE CUNTROL I'IWIlLEM 3{)7

Mathematically, the- response depends upon the eigenvalues or closcd-toonpoles, which are obtained from the characteristic equation

G(s) + I = 0 (9-15)

In the AVR loop (Fig. 9-4) the open-loop transfer function G(s) is of thirdorder and we thus obtain three eigenvalues, Sl' S2. and S). If these are distinctand real the transient response components are of the form

(9-10)

If two of the eigenvalues. for example S2 and SJ. are conjugate complex.a ±jw. then the transient component will contain an oscillatory term of theform

Ale'" sin (wt + [3) (9-17 )

For the AVR loop to be stable the transient components must vanish withtime. We thus must require that all three eigenvalues be located in the left-hand s

. plane. For the loop to possess good tracking ability the transients must not onlyvanish but they must do so speedily. As the real parts of the eigenvalues deter-mine the rapidity of the exponential decay, a high-speed loop must possesseigenvalues located well to the left in the s plane.

The amplitude factors AI' A2, and AJ in formulas (9-16) and (9-17) expressthe relative size of the transient terms. If one or several of the terms are relativelylarge then the corresponding eigenvalues are said to be dominant. Generally thecloser the eigenvalue is 'located to the jw axis the more dominant it becomes,

9-2-6 AVR Root Loci

The location in the s plane of the eigenvalues depends upon the open-loop gainK and the three time constants TA, I:, and Tdo' Of these parameters only theloop gain can be considered adjustable. It is interesting to study how the magni-tude of this gain affects thes-plane locations of the three eigenvalues-and thusthe transient response.

A root locus plot yields valuable information in this regard. Figure 9-5 de-picts the root-locus picture for the AVR loop. There are three loci, each startingfrom an open-loop pole (marked ), The latter are located at s = - l/T.~, - liToand -l/T~o respectively. For low values of the loop gain the eigenvalues

, (marked 0) are located close to the open-loop poles. their positions marked a.Because T~o is comparatively large the pole - IIT~o is very close to origin. Thedo~inating eigenvalues 52 is thus small, resulting in a very slowly decaying expon-ential term, givmg the loop an unacceptably sluggish response. The gain Kwould not satisfy the inequality {9-13} thus rendering an inaccurate static re-sponse as' well.

By incre.as!ng the loop gai.n th~ eigenvalue 52 travels to the left and the loopresponse ,quIckens ..At a certain gam setting-the eigenvalues S3 arid S2 .. collide,"Further increase III the loop gain results in S3 and S2 becoming conjugate

._------

, , ., . r'tS THFORV: AS INTRODUCTI0NEI.Ee'T!It(-" FSER(,\ ~\Sr", .

. \ compl:Xls plane_j

t. . '"1'(') ,/<('

c

a_ ..J,C:-: __ ...;_''!.. .!J.·~-*------l!H)--f--{}-X;:-t---------.,_ ~~I

Pole at1s = --

TA

<,'Critical gain

crossing

Pole at1s = --

Te

Figure 9-5 Closed-loop rOOI loci for AYR control loop in Fig. 9-4.

complex. This dominant eigenvalue pair (b) makes the loop oscillatory, withpoor damping. If the gain is increased further the eigenvalues wander into theright-hand s plane (c). The loop now becomes unstable.

9-2-7 Stability Compensation

High loop gain is needed for static accuracy but this causes undesirable dynamicresponse, possibly instability. By adding series and/or feedback stability compen-sation (Fig. 9-2) to the AVR lo"op, this conflicting situation can be resolved.

As the stability problems emanate from the three cascaded time constantsthe compensation networks typically will contain some form of phase advance-ment. Consider for example the addition of a series phase lead compensator,having the transfer function

G,=l+sYc

The open-loop transfer function will now contain a zero

. K(1 + s7;)G(s) = (1 + sTA)(l + s'4)(l + sT,;o) (9-18 )

The added network will not affect the static loop gain K, and thus the staticaccuracy. The dynamic response. characteristics will change to the better. Con-sider for example the case when we would tune the compensator time constant

",.. !.

IN STEADY STATE-THE CONIROLTHE ENERGY SYSTEM .

. T The oDen-loop transfer functionT to equal the exciter time constant e'. •

. cthea equal

K 1(9-19)G(s) = (1 + sTA)(l + sTdo)

sated system are depicted inThe root loci (there are only two) of the con:pen 1 el'genvalues the dornin-

. )·U It in negative rea '.Fig. 9-6. Low loop gam (a sn resu. s term Increasing loop gam (b)ant one of which, S2' yields a sluggIsh respo.nse f the oscillatory term will,results in oscillatory response. The ~ampmg °h in the uncompensated., . n a- was t e case 1however, not decrease with mcreasmg gal :>

system. . ..' k ensation is ·employed.In reality a combination of senes and feedbac cornp

(Compare Prob. 9-5.)

y, withIto the

9-2-8 Effects of Generator LoadingThe above analysis of the AVR loop was based upon the assumption (Eq. 9-8)that E ::::IV I which holds true only for a lightly loaded generator. HO\;r. doesincreased loading affect the above analysis? It should be noted that the ability ofthe AVR loop to maintain a well regulated terminal voltage under widely shift-ing load conditions ("robustness") is an absolute prerequisite. . '

Added load does not change the basic features of the A VR loop as depictedin Fig. 9-4 . It will, however, affect the values of both the gain factor KF and thefield time constant. High loading will make the generator work at higher mag-netic saturation levels. This means smaller changes in IE I for incremental in-creases in if' translating into a reduction of KF• The field time constant will

"likewise decrease as generator loading means closing the armature current paths.

(namicompen-:d.rstantsvance-nsator,

'b t

1 1TA 0 a - Tdo a

~~~------~----~D-~~4---- --------~

(9-18)

.staticCon-

~s~antb t

Figure' 9-G Root-loci for zero-compensated loop.

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~. ~O ELECTRIC ENERGY SYSTEMS THEORY: AN INTRODUCTION

/7

//r

~1~.,

i~."11;.,~j,'I'&\i~~\:.i •: }I

~!: t

I:~-,

I'.~\;'1 ~'Il( .~\11 I,:'

Ii! \.,?iJ,l '~1{l'~ \ 'I

'fi'.1fi I;~

~ t The term" Automatic Generation Control ~ (AGe) is also commonly used.I ! :t The terminology used agrees wit~ the standard 'accepted by the industry. AIl systc fI course. not identical. We chose a representative example for demonstration. ms arc, 0

~'J;,)",;?-.=~~.. ..,..:=_. .. ... -

This circumstance permits tlu: formation of U transien t " stator current the'. f hi h . ld S, e.exIstence 0 W IC yie s a lower U effective" field inductance. The rotor-field time

cons tan t, wyer load can be sh a wn to 'equal

I X' X .

\

T - d + exl I

d, load - X + X . Tdo s (9-20)_ d ext

where X ext i~ the exterior reactance viewed from generator terminals. Comparealso Eq. (4-64).

II

I9-3 AUTOMATIC LOAD ..FREQUENCY CONTROLOF SINGILE-AREA SYSTEMS .

1The basic role of ALFCt is to maintain desired megawatt output of a generatorunit and assist in controlling the frequency of the larger interconnection. TheALFC also Helps to keep the net interchange of power between pool members atp~ed~termin9d. values. Con~~\ should. be appl~ed in such a fashion that high!ydiffering response characteristics of units of vanous types (hydro, nuclear, fossil,I . I .etc.) are recognized. Also, unnecessary power output changes should be kept to aminimum in-lorder to reduce wear of control valves.

·The AL~C loop will maintai~ con~ol only during nor~al (small and slow)changes in load and frequency. I~ IS typically unable to provide adequate controlduring emergency Situations, when large megawatt imbalances occur. Then moredrastic" emergency controls" must be applied {Chap. 12).

In the present section we shall first study ALFC as it applies to a singlegenerator supplying power to a local service area. We shall later extend thestudy to embrace several generators allpart of a single" control area."

9-3-1. Speed-Governing SystemThe real power in a power system is being controlled by controlling the drivingtorques of the individual turbines of the system. Figure 9-7 shows in a highlyschernatical fashion the operating features of such a speed-governing system.tThis system constitutes the" I?rim(}ry" control loop in Fig. 9-1.

By controlling the position, measured by the coordinate XE, of the controlvalve (or gate, in the case of a hydroturbine), we can exert control over the flowof high-pressure steam (or water) through the turbine.

A downward small movement of the linkage point E increases the steam (orwater) flow by a small amount which, if measured in valve power, represents a

THE ENERGY SYSTEM IN STEADY STATE-THE CONTROL PROn'LE,\.1 311

Steam

1

.IPv

High- ~pressure oil

Speed governorHydraulic amplifier

Figure 9-7 Simplified functional diag~am of the primary ALFC loop.

megawatt increment 6oPv. This flow increase translates into a turbine powerincrement 60PT in the turbine (not shown in the figure).

Very large mechanical forces are needed to position the main valve (or gate)against the high steam (or water) pressure, and these forces are obtained viaseveral stages of hydraulic amplifiers. In our simplified version we show onlyone stage. 'rhe input to this amplifier is the position XD of the pilot valve. Theoutput is the position XE of the main piston. Because the high-pressure hydraulicfluid exerts only a slight differential force on the pilot valve, the forceamplification isvery great.

The position of the pilot valve can be affected via the linkage system in threeways:

1. Directly, by the speed changer. A small downward movement of the linkagepoint A corresponds to an increase 6.P ref in the reference power setting.

~. Indirectly, via feedback, due to position changes of the main piston.3. Indirectly, via feedback. due to position changes of linkage point B resulting

from speed changes. .

It should prove a useful' exercise for the reader to find, qualitatioely, theworkings of the mechanism. For example, give a "raise" command to the speedchanger and prove that this indeed results in an increase in turbine outpwt.Prove also..that a sm~""cinr"" U/;1I o-;,,~ th~ ~~~~ ~I'r __ •

. Of'Y' AN INTRODUCTlON3:\'2 .t,(_EC'FRIIf ENERGY SYSTEMS THE' .

(ilr(>s~ntly we shall give a qual1cicaciue description of the mecha~is_m' . ints.Il<lthe analysis to follow, incremental movements .of the five linkage po

II .. , E in Fig. 9-7 are of particular interest. In reality these movements aremeasured in millimeters but in cur analysis we shall rather express them aspower increments expressed in megawatts or per-unit megawatts as the case maybe. The movements are assumed positive in the directions of the arrows. The

. . h (1. Thegovernor output command 6.Py is measured by the positron c ange Xc'governor has two inputs:

I. Changes 6.P ref in the reference power setting2. Changes tl/ in the speed of frequency of the generator, as measured by 6.xB

An increase in tlPg results from an increase in 6.Pref and a decrease .in 6.f.Wethus can write for small increments

1Mg = sr.; - Ii 6./ MW (9-21)

The constant R has dimension hertz per megawatt, and is referred to asregulation or droop. (For numerical values see Example 9-2 below.) Laplacetransformation of Eq. (9-21) yields

16.Pg(s) = 6.Pref(s) - Ii 6./(s) (9~22)

Using well-known block diagram symbols we have represented the governoras shown in Fig. 9-8.

9-3-2 Hydraulic Valve Actuator

The input position 6.xD of the valve actuator increases as a result of an increasedcommand 6.Pg but decreases due to increased valve output, 6.Pv. Equal in-creases in both 6.Pg and 6.Pv should result in 6.xD = O. We can thus write

tuD = 6.Pg - 6.Pv MW . (9-23)

For small changes IlxD the oil flow into the hydraulic motor is proportionalto position 6.xD of the pilot valve. Thus we obtain the following relationship forthe position of the main piston:

6.Pv = kH f 6.xD dt (9-24 )

The positive constant kH depends upon orifice and cylinder geometries andft:blid pressure.

Upon Laplace t.ransformation of the last two. equations and upon elimina-lioF} of 6.XD w~ obtain the actuator transfer function

"'-'--,. __, '---~~--__..'------~~-----~(2~.y!'~:;-a

"

::t;;;'~jJ Z~~==-e5 -t'ci-~ arouzri Q.! s,1t.~ b;'-~:-2.'l:t1:-{zl-;e 2.....~to.:- j;_as ceen represented by me transfer- function _

G;![s):n :~~ 9-8.

9-3-3 Turbtne-Cenerator Responsefu LO::::;:-a] S:~~': s-.z;r~ 2I.Q ....~ Ete- ",-ec1l~aiSm described in Sec, 4-9 the turbine,!:-07~l';:.rP '7 k~ ';n!~c:e 'him &e e!~tromecil2illca1 air-gap Dower P G resultin a~ zero a:;;:;e~efo~ 2":"'" a consrant speed or frequency. P~bations sr; andLPG b i::zs~ FD7,-::;:S 71'iTI ~t me above balance. If the difference power,!l:P;- - b.P /G, :s :;::-OS:"i3·.-e:::e nrr~me generator unit will accelerate; if negative it;:;~'5;Je:.~%a:-;..

(9-25)

71:1:: ;:;,.;r;;rcff.,-epo,;=:- mcre-m-.~t M, de~ds entirely upon ·the valve power~~ !::'Pr at.d !te zesponse t:..hzr-ac!eristics o~!he turbine, Different turbine<!)l~ -i~} n~..c.el;,1:: :this regard. It .is pOSS11>leto express the turbine dynamics in~~ cf:a t1rb:.r:~trn!!L-::(e:- ftmcuon

b. AF~' D "''-:;:b~ye Gf:T.--eo Gy for the most common turbine types. A~.t<tailil.e;flt;r;-Be~?at S1P..am3J'POirE has the simples, transfer function, 'consisting ofa .~~ mre cc;t~.?iPt, i.e...

iI ~bEe1'R'I~ ENERGY SYSiFEMS TI-!EORY: AN INTRODUCTION' .

. o'tm~r,ltlrbine types have more complex t~ansfer functions In I··,' ,. . . genera it ca·r;}!D,e S'al,d t'h"ll[ the tU,rhine response is slow with response times . d' 1,,' nd measure' In severase(\:Qn 's. . .

" The, gen,er.atoi". P?wet increment I:lP G depends entirely upon the changes C1PvlIil the l?ad Pj;) being fed from the generator, Via the transient mechanism dis-~ussed 10 ~ec. 4-13 the genei'ato~ always adjusts its ,output so as to meet tne

e.man?, c~anges. !1P D '. These adjustments are essentially "instantaneous," eer-tainly m c9mpanson With the slow changes in Pr, and we can therefore set

\ !1PG = !1PD (9-28)In vie~ of Eqs. (9-26) al1ci (9-28) we can thus symbolize the turbine-genera-

tor dynamic response as shown in Fig. 9-8.

9-3-4 Static Performance ~1Speed Governori

At this stage in our analysis the control loop in Fig. 9-8 is "open." We cannevertheless obtain some interesting information about the static performance ofthe speed governor. The relationship between static signals (subscript "0") isobtained by letting s --+ O. As GH(O) = Gr(O) = 1 we obtain directly.from Fig. 9-8

1APr, 0 = !J.Pref, 0 - R !1Jo (9-29)

We consider three cases.

Case A The generator is synchronized to a network of. very large size, solarge in fact, that its frequency will be essentially independent of any changes inthe power output of this individual generator (" infinite" network).

Since we thus have '

6.1=0'we obtain from Eq. (9-29)

!J.PT,o = !J.Pref, 0 (~-30)

For a generator operated at a lforced) constant speed, we thus have a directproportionality between the 'turbine power and reference power setting.

Example 9-1 A lOO-MW generator is operated onto an .. infinite" network. Hew would YO\lmake this generator increase its turbine power by 5 MW?

SeW'IJ:0N Simply by giving a "raise" signal of $ MW to the speed-changer motor,

Case B Now we consider the network "finite," i.e., its frequeney is variable.W'e do, however, keep bae speed-changer setting constant, that is, Mrcf = O.

FFOIn Bq. (9-19) we obtai;l'l ,'j, 1 .:; 4FT, 0 == - if. A/o (9-31)

.. '.:.

THE ENERGY SYSTEM IN STEADY STATE-THE CONTROL PROULHI 315

For a COl1stant setting of the speed chanqer the static increase ;11 turbine !'(JI\'('I'

output is directiy proportional to the static [requency ·drop.This result points out the physical significance of the feedback parameter R.

We remember that the physical unit for R is hertz per megawatt. III practice. thepower is measured in per units, and in this case the unit for R will be hertz perunit megawatt. If the frequency drop is likewise measured in. per units of normalfrequency (= 60 Hz), and if the power is measured in per-unit megawatts. thenthe unit for R will be in per units also.

Example 9-2 Consider the tOO-MW generator in (he previous example II ha, a re!?lIIalit)11parameter R of four percent (or 0.04 pu). By how much will the turbine power increase if thefrequency drops by 0.1 Hz with the reference unchanged?

SOLUTION A 0.04-pu regulation parameter means thai the turbine power will increase I I'll. ortOO MW for a 0.04-pu, or 2.4-Hz drop in frequency. Thus we have

2.4R = 100 = 0.024 HzjMW

According to Eq. (9-3l) for a frequency change of 6.[= -0.1 Hz the turbine power will thusexperience a static change of

I6.Pr.o=-0.024 x(-0.1}=4.17MW

Case C In the general case, changes may occur in both reference setting andfrequency in J.,hich case the more general relationship (9-29) applies.

In a frequency-generation graph Eq. (9-29) represents a family of slopinglines as depicted in Fig. 9-9, each line corresponding to a specific referencepower setting.

Percent iof ratedfrequency Speed changer set to give rated

frequency at 100 percent Of ratedoutput

---I>Percent ofrated output98

96, Speed changer set. to give ratedfrequency at 50 percent of ratedoutput94.

Figure 9-9 Static speed-power response of speed governor (Graphs correspond to R = 0.04 pu.)

___ --:' ~ ...:t>::I._~_"'" .---:- -----.-

RY' AN INTRODL;CTION".11'0 El.ECTRIIC EN:ERGY SYSTEMS THEO . .

.. MW enerator in the previ0us example. If the frequencyExample 9-3 Consider again the 100- g . h ed by how much should thedrops by 0.1 Hz but the turbine power must remain unc ang ,

referense setting be changed?

SOLUTIO:--: As t>.P T. 0 must be zero we obtain from Eq. (9-29)

D.P =.!.. t>.f, = -'-- x (-0.1) = -4.17 MW·,r.o R 0 0.024

Rcsul«: We mUSI command the speed changer 10 "Iower" by 4.17 MW.

Th . t' ngs are 50 andExample 9-4 Two generators arc supplying power to a system. err ra I ,

500 M W respectively, The frequency is 60 Hz and each generato( is hulf-Ioaded. The systemload increases by 110 MW and as a result the frequency drops to 59.5 Hz. What. must rheindividual regulations be if the two generators should increase their turbine powers In propor-tion to their ratings?

SOLUTION The IWO generators should pick up 10 and 100 MW respectively. From formula(9-31) we thus compute their R values:

-0.5R = - -- = 0.05 Hz/MW

10

-O.SR = - -- = 0.005 Hz/MW

100

(smaller unit)

(larger unit) ,

If we express the regulation in per-unit hertz per per-unit megawatt, the R figure will be0.0417 pu (or 4.17 percent) for both units. .

... ~.~-

This result thus teaches us that generators working in parallel on the sa~enetwork ought to have the same regulation (expressed in per unit of their ownrating) in order to share load chanqes in proportion to size.

9-3-5 Closing the ALFC LoopWe observed earlier that the loop in Fig. 9-8 is "open.'; We proceed now to "close"it; i.e., we seek to obtain a mathematical link between !::"PT and!::"'f.As our genera-tor is supplying power to a conglomeration of, loads in its service area, it' isnecessary in our following analysis to make reasonable assumptions about the"I umped " area behavior. We make these assumptions:

1. The system is originally running in its normal state wijh complete powerbalance, that is, Pg = P~ + losses. The frequency is .at normal value [". Allrotating equipment represents a total kinetic energy of Wkin= H1in

fTIegawatt-seconds.2, By' connecting additional load objects to the system the load demand in-

creases by !::"P D which we shall refer to as "new'; load. (If load objects arediscennected thea tJ.PD < 0, that is, the "new" load is negative.] The genera-tor immediately increases its output !::"'Pa to match the..new load, that is,!!.PG = !::"PD,

~. .;:;:co k<~, zs:

'~ ~, ~,I..;

THE ENERGY SYSTEM IN STE.\I)Y STATE-THE CONTRO n"9-. L PROl\L~ ~ .

o~ \;\

3, "There will now be a power imbalance ill the ~rea that equals 6.PT - ~t ~megawatts. As a result the speed or frequency will change. This change will b(ilssumed uniform throughout the area. As the kinetic energy is proportional t~ ,the square of the speed we can thus write for the area kinetic energy

. ( J ) 2Wkin = ~in JO MWs (9-32)

4. The" old" area load has' a frequency dependency ap olaJ (compare Example3-6) that we can lump into one area para neter

IvI:W/Hz (9-33 )

Area power balance requires that the increase in turbine power equals the,sum of "old ,; and "new" load changes plus the: rate of change of kinetic energy. Wecan thus express this area power balance as follows:

(9-34 )

Asf=fO + /)"f

and as I1f is small relative tor we can write Eq. (9-32)

(fO + I1f)2 [, 2!if (N)2] ( 6.f)~in = ~jn fO = ~jn 1 +: fO - + fO ;::::;~In 1 +: 2 fO

(9-35)

By substitution of Eq. (9-35) into Eq. (9-34) the power-balance equationtakes on the form

(9-36)

By dividing this equation by the generator rating P, and by introducing the "per-unit inertia constant

H~ ~jnPr

MWs/MW (or s) (9-37)

it takes OF! the form

2H dI1PT - sr; = fO dt (~f) + D ~f puMW

The ~P's are mow measured in .per unit (on base Pr) aHO D in per-unitmegawatt per hertz.

The H para~eter h~s the advant~~~ ~ver ~ln that it is essentially iadepen.deat ef system S'1~e.TYPlcalll values 'he 'Ill the f~nge 2-8 seconds. ,

Il '.:'

pu MW (9-39)

M 'The eq1uation is now written in terms of AI/ I", that is, the per-unit frequencyany ana ysts prefer this.] .

Laplace transformation of Eq. (9-3&) yields

?HAPr(s) - .1PD{s) = =Os lJ.f{s) + D lJ.f(s). I (9-40)

which equation can be written in the form

(9-41 )

where, for brevity, the following new parameters have been introduced:

G ( ) A Kpp s - 1 + sT.

p

A 2HI;, - IOD

K ~ _!_p D

(9-42)

s (9-43)

Hz/pu MW (9-44)

Equation (9-41) represents the" missing' link " in the control loop of Fig. 9-8.By adding a summing ju~ction. and a transfer block we close the loop asdepicted in Fig. 9-10. (Disregard at this time the dashed portion of the diagrammarked" secondary loop.")

Secondary ALFC loop

Primary ALFC loop

\! Ir-------------r------------------.

IIIII'III ...·----.,LlPr.'(s)_I I. 'K, 1_+.....--1. --~I S I

df.(s) L.._-' __ ~

Power systemI

/1 ' lFiin!lr~9-10 Closing the ALFC 100H.

~,:',

THE CONTROL PROaLEI>! 319OY STATE- "

THE ENERGY SYSTEMIN STEA

'-40)

9-3-6 Concept of " Control Area" .' lor feed inglion of a single genera

Our above theory was based upon the ~ssur:nP I the striClest sense the theorypower to a local service area-a rare sltuatl~n. a~ arallel-working gcnerato~Sdoes not apply to the more common case 0 m YtPms normally control their

st power sys e I loopsserving a larger area. However, mo . lier the individual contra .. . F easons mentiOned ear I . . ortaIH---t hegenerators In Unison. or r d his is qUite Imp

have the same regulation parameters. If also-an t 11 onse characteristics thenindividual generator turbines tend to ha.ve t~e s;~~r;:gresent the whole systemit is possible to let the control loop In Fig ..which then would be referred to as a control area.

39)

1Cy.

r a conlrol area having theExample 9-5 Determine the primary ALFC loop parameters or

following data:

1-41)Total rated area capacity P, = 2000 MWNormal operating load P~ = 1000 MWInertia constant H = 5.0 s )

= 2.40 Hz/pu MW (all area generalClr'Regulation R

)-42) . r aning that the" old ..We shall assume that the load-frequency dependency IS Inear, me

load would increase one percent for onc percent frequency increase.

i-43)SOLUTION The latter load assumption yields D:

iJP~ 10. MW/HD = - = - = 16.67 zof 0.609-44}

~.9-8.)p as.gram

or in per units of area capacity

D = 16.67 = 8.33 x 10-3 pu MW/Hz2000

We then get from Eqs. (9-43) and (9-44)

2 x 5.0Tp= 60 x 8.33 X 10-3 = 20 s

1tc, = D = 120 Hz/pu MW

9-3-7 Static Response of Primary ALFC Loop

Having closed the primary ALFC loop we wish now to-study both its static anddynamic features. In this section we shall limit attention to the static character-istics. One of the basic objectives' of the loop is to maintain constant frequency inspite of changing loads. How accurately does the loop maintain the frequency?

The primary ALFC loop in Fig. 9-10 has one output 6.J and two inputs6.P rer and IlP D' From the block diagram we obtain by inspection'

{[6.Prer - ~ N ]GHGr - MD}Gp = N (9-45 )

.r ' . u t (A P - 0) we thus haveFcii.r a (:,OI1SlUllt relenmce inp D. ,d-

Gp 6P (s)6./(s) = - 1 + (I/R)GpGHGT D

For a step load change of constant magnitude 6.Pv = M we haveM

6.Po(s)=- sdil bta i from Eq. (9-46) the staticUsing the final-value theorem, we rea I Y 0 tarn

frequency drop

(9-46)

6.j~ = lim [s 6./(s)] = Kp M _ _ M1 + Kp/R I - D + l/R

Hz (9-47)

We introduce here the so-called area frequency response charactertistic(AFRC) j3. defined as

D. 1 pu MW/Hz (9-48)P=D+-R

and obtain, then,

6.fo =M

Hz (9-49)f3

Example 9-6 Find the static frequency drop for the 2-GW system iri previous example follow-ing a one percent load increase. that is. 6.PD = M = 20 MW = 0.01 pu MW.

SoLUTION For the above numerical data we get

, If3 = 8.33 X 10-3 + - = 0.425 pu MW/Hz

2.40

0.01.'. 6./0 = - 0.425 = -0.0235 Hz

or 0.04 percent of normal frequency.

Example 9-7 What would the frequency drop in the previous example have been if the speed-governor loop were nonexistent or open? ,

50LtJT'I0N Opening the loop is tantamount to setting R = co. We now would have

P = D = 8.33 x 10- 3

.and

6.( = -~ ~-v o 0.00833 - 1.20, Hz

or fw.o: percent of normal value.

'.' .:

=:

"., ,.~

THE ENERGY SYSTEM IN STEADY STATE-THE CONTROL PROfiLGM 321

ic

9-3-8 Dynamic Response of ALFC Loop

The static response of the ALFC loop yield ~d. important information aboutfrequency accuracy. The dynamic response of tile: loop will inform about" track-ing" ability and stability of .t,~e loop.

Finding the dynamic response (for a step load) is quite straightforward.Equation (9-46) upon inverse Laplace transformation yields an expression for!::"f(c). However, as GH, GT, and Gp contain at least one time constant each, thedenominator will be of third order, resulting in unwieldy algebra.

We can simplify the analysis considera oly by making the reasonableassumption that the action of the speed governor plus the turbine generator is.. instantaneous" compared with the rest of till: power system. The latter, asdemonstrated in Example 9-5 has a time constant of 20 s, and since the othertwo time constants are of the order of 1 s, we will perform, an approximateanalysis by setting TH = TT = O.

From Eq. (9-46) we then get

7)

til-

48).s.:

!::,.f(s) ;:::; _ 1 + sTp M1 K s

1 +- --p-

R. 1 + a,

=-MR:K~p(~_- Rl+Kp)s+~

. p

If we make useof the previous numerical values, we obtain \

(9-50)49)

tow-

t::,.j(s) ;:::; -0.0235(~ 1_)s s + 2.55.

The approximate time response is therefore purely exponential.t::,.j(t);:::; -0.0235(1 - e-2.S5,) Hz (9-51 )

peed-

Figure 9-11 shows an analog computer recording. of this response. Forcomparison, we also simulated the loop response with the inclusion of. the timeconstants TH and TT' We make the following observations in regard to ourresults: .'

1. The overall closed-loop system time constant iequals only .1/2.55 = 0.393 s,which is a considerable reduction from the value T" = 20 s, characterizing the" plant" itself. This speedup is a result of the feedback arrangement of thespeed governor. Note that the system can be made-still faster by reducing R,that is, by increasing the static loop gain.

2. Reduction of R aliso reduces the static frequency error.3. If we performed the above analysis by not disregarding the turbine response,

then the response would net be purely exponential as above. :I.o,'Fig. 9-1i ,we

Jfo = -0.0235 Hz

-0.01

-0.02

Response if THand T T are not neglected (nonreheatturbine assumed)T T =, 0.3 sTH ,~, 80 ms

Figure 9-11 Dynamic response of the primary ALFC loop to a step-load increase.

show the difference. Note that the added delays cause a larger transientfrequency dip. Why?

4. The speed governor gives a reasonable performance with a static frequency.drop of only 2.4 Hz between zero and full load and settling time of the orderof 3 s (as depicted ill Fig. 9-11). However, with the extremely severe restric-tions we in reality impose on frequency constancy (see Sec. 9-3-10), theresults are, in fact, entirely u.nac.ceptable. We must do much better.

5. The foregoing analysis may not have given the reader a full physical under-standing of the load frequency mechanism of the' single-area system. Since

.such an understanding is invaluable to appreciate fully so-called bias controlin multiarea systems (see Sec. 9-3-10), we shall attempt now to shed somelight on the physical mechanism.

9-3-9 Physical Inrerpretation of Results

When the load suddenly increased by 1 percent (= 20 MW), where did thispower come from? Certainly, it must hav~ come from somewhere, as can becertified by the customer who threw the switch and expected and got instantan-eously the demanded 20 MW. """'-.'.

In the milliseconds following the closure of the switch, the frequency has notchanged a measurable amount, and therefore no power increase has had time todevelop in the turbine (where the steam valve has not yet moved)... . In those first instants the total additionalload-demand 20 MW is obtainedfromthe stored kinetic energy, which therefore will decrease at an initial rate of 20 M W.The kinetic energy is released. by speed reduction. Since the speed is dropping(and from. Eq. (9-51) .we note that the initial deceleration is 0.0235 x 2.55 =Q.06 Hzjs), the steam valve is opened up; due to the mechanism' described ea~lierand thllis meains increased generator output. Also, and this, is important to real'iz-e',

se1-

~tLO

m'lIV,:ng=er,ze,

THE ENERGY SYSTEM IN STl:,\I)Y Sl'ATIi,-THll CONTIlOL PROBLEM 323

since the speed is now dropping, the, "old" 10tHI llOOO rvlW in this case)decre~ses at the rate of D = 1000/60 = 16,67 rvtW/Hz,

SInce the appearance 011 the scene of-thts H I'c/t;oscd" power //IeallS that lesspower l1eeds to be qenerated. we carr in effect consider it to /)(1 (l direct cOlllributiollto the new load demand, '

- In conclusion,' as the speed is dropping, the demand increase of 20 MW isthus made up or three components:

l. "Borrowed" ·kinetic energy from the rotating system machines2. Increased generation3. "Released" old customer load

Initially, the last two components are zero, but as the speed is dropping,they will account for an increased contribution. Consequenlly. the kinetic energywill lie consumed at a decreased rate and this is confirmed in Fig, 9-1 L which,shows that the deceleration decreases as time goes on,

Eventually (theoretically, after infinite time), the speed will level off at a newconstant lower value. At this time the kinetic energy is constant (at a lowervalue), and the 20-MW load increase is therefore made up of the last twocomponents only. It is interesting to see in what ratio the last two componentscontribute.

The static speed drop being 0.02.3.5Hz, we can compute these contributionsas follows:

1. The generator regulation is 4.00 percent, or 2.<.l.()Hz/pu MW,'and the increasein generation .after the new steady state is reached will be

0~~5 = 0.0098 pu MW

= 0.0098 x 2000 = 19.6 MW

2. Since the" old" load decreases at the rate of 16.67 MW 1Hz, the speed drop of0'.0235 Hz will," release"

0.0235 x 16.67 = 0.4 MW

These two components evidently add up to 20 MW. Note that the lar tibuti b c. is f Lt gescontn ution, y tar, IS rom new generation.' ,

9-3-10 The Secondary (" Reset") ALFC Loop

.It is necessary toachieve n:uch better frequency constancy than is obtained bthe speed-governor system Itself, as demonstrated above T '. ymust manipulate: the speed changer in aCGordance " 0 acco~phsh this westrateqy, Before we do so, it is necessary to settl r WIth some suitable control

, e lor a set of control snecifir.nt.i(')P1'<

324 ELECT~IC ENERGY SYSTEMS THEORY; AN INTRODUCTION

the stringency of which will in the end determine the sophistication of the

proposed control method.

A Suggested control system specifications We are presently discussing ~ singblel-

. . .f . .r h a system are considera yarea system, and the con trol spec! catI~ns lor suesimpler than those imposed upon a multiple-area system (power pool).

Here follow some realistic specifications:

1. The control loop must be characterized by a sufficient degree of sta~ilitYT' hi2. Following a step load change, the frequency error should retur~ to ,zero. IS

is referred to as isochronous control. The magnitude of the transient frequencydeviation should be minimized. (This magnitude depends, of course, upon themagnitude of the load change.) ,~

3. The integral of the frequency error should be minimized. 14. The individual generators of the control area should divide the total load for

optimum economy (Chap. 8).

Let us comment on each of these requirements:

1. Stability is always a problem in closed-loop control. The tighter the errorspecifications, the greater the risk-that the proposed loop win turn unstable.

2. The need for frequency constancy was discussed in Chap. 7. Isochronous(iso = equal, chronos = time) control gives assurance that the synchronousclocks run on time, but not without error (see point 3).

3. Isochronous control guarantees that the static frequency error following astep load change will vanish. No control system can, however, eliminate thetransient frequency error (see Fig. 9-12 for a demonstration of this point.)The time error of synchronous clocks is proportional to the integral ofthe frequency error. This integral has the dimension of" cycles." If we dividethe integral by I" we obtain an expression for time error. !:it expressed inseconds, i.e.,

1 r!:it = fO fo !:if dt s (9-52)

The particular control strategy we shall propose below does not have theability to reduce the error integral automatically to arbitrarily small values.During the periods of heavy load, the system frequency will have a tendencyto be below 60.00 Hz, during periods of the light load above.

For example, ass~me that the frequency averages 59.99 Hz during a pro-longed heavy-load period. In one hour we would now accumulate a time error of

1 3600 36!:it = 60 fo (-0.01) dt = - 60 = -0.60 s/h

The customary way to handle such a slowly accumulating time error (inthe U nited States) is as follows. .

1",o~~

.).

1"l-1E ENERGY SYSTEM. IN STEJ.!}Y STATE-THE CONTROL ph '':_".(...OB~

f (t), H:j2 3 4 -

Without inte ~I'al control

(a)

-0.Q1

Without intec.r sl control

(b)

Figure 9-12 D~amic re~ponse of the ALFC loop including the reset action of the secondary loop.

The error is measured by comparing the "system time" with that of theNational Bureau of Standards. When an error of ± 3 s has accumulated (and,as exemplified, this may normally take many hours), the speed changer isoffset intentionally an amount of, say,.O.02 Hz for a sufficiently long time forthe time error to reduce to zero.

In the case of a power pool, one usually designates a pool member tokeep track of the time error and inform the other pool members regularly ofthe need to reduce' the error by a unison effort.

4. The first three specifications' are taken care of by a control system (see below)with a response time of the order of a few seconds or half a minute. Whenthese three control requirements are met, one attends to the fourth economic'requirement. This is usually being done by a slower economic dispatch con-'trol scheme, having response times of the order of a minute or longer.

. (Sec. 9-3-11.)

B. Integral control By using the control strategy shown dashed in Fig. ~-10, weobtain an overall system that will meet performance specifications 1 to 2 above.We have.added to the primary ALFC loop.in Fig. 9~1O.s.()~called integral control':i.e., we let the speed changer be commanded by a signal obtained by .first. am-plifying and then integrating the frequeacy error; i.e.,

- '.

tlPref ~ ., K[ J t:.fdt (9053)

- ~:?

.:.~~

$.>..~

t!7/

~.'>j

8sff erNfiJ,(£JY SYSTBMS TI-mOR Y: AN INTRODUCTION

, ,'the [flni,tfor K_I is "per-unit megawatt per hertz and second."Fef exaI?ple, If t~e frequency drops by 1Hz (t:.J == -1 then . ,

salIs fer an Increase In power with the "call" . .) the Integrator-K[ pu MW/s. Note the negilti~e polarit of cthe .ll1creasmg at the init!al rate. ofmust be chosen so as to cause " iti ~ Integral controller. This polarityor "decrease," com~a~d ~ha P?SI ive re~uency er~or to give rise to a negative,area control error (ACE) '. e signal fed Into the Integrator is referred to as, I.e.,

Cgain,

i.e. if

, ' ACE ~ N (9-54)I dI:nhtcgral control will give rise to zero static frequency error following a stepoa c ange, for the following physical reason.

As long as an error remains, the integrator output will increase, causing thes~ed ch~nger to move. The integrator output, and thus the speed-changer posi-tion, attains a constant value Dilly when the frequency error has been reduced tozero.

The gain constant KI controls the rate of integration, and thus the speed ofresponse of the loop. The integration is mostly performed in electronic integra-tors of the same type as found in analog computers.

then'

whenconjutherel

n

C Analysis of Loop Respons-e Here follows an analysis of'the proposed system,subject to a step load change. To avoid cumbersome numerical analysis, we shallas before neglect the turbine dynamics. In addition, we also make the assump-tion that the speed-changer action is instantaneous. This is not perfectly correct,since the device is electromechanical and will therefore have a nonzero responsetime. These approximations will make possible relatively simple analysis withoutdistorting the essential features of the response. It is also worth mentioning thatthe errors we thus introduce into our analysis affect only the transient, not thestatic, response, (This was already demonstrated in Fig. 9-11.),

From Eq, (9-53).we get, upon Laplace. transformation,, K

llP,ef(s) = - _!_ 6./(s) (9-55)s

then'

wheret

corre

After making use of the block diagram in Fig. 9-10 and Eq, (9-55) we get, aftersome algebra,

we ilSeC0,!

simucup",Figtl

~

(9~56) l.U01We obtain the time response !lJ(t) upon inverse transformation of this

ex,pFessien. Since the response depends upon the poles of Eq. (9-56)we must first, turn 011f' attention to the second-order-denominator polynomial, whJ.s:hcan be

w;nlttca

CI

U:

Fe

i+,Kpl'R +'K/K~ ,= (s + 1+ Kp/R)'2 + K/ Kp _ (1 + Kp/R) 2

T" Tp' 27;, Tp 2Tp'(9-57)

2. PI.fe

s.

)r

)f~y

is

~)

p

e

o

'f

\iI

THE ENERGY SYSTEM IN STEADY STATE-THE CONTROl. rRonLE~1 327

. Clearly, the nature of the poles depends upon the magni tude of the in tcgr a Igam K/. If

i.e., if

1 ( K)2 (9-58)K/ > 4~Kp 1 + RP~ K,. eril

then we can write the denominator polynomial in the form

(s + Q:)2 + w2

where Ct. and w2 are both positive and real. The expression ~f t,f(s) now has aconjugate-complex pole pair in the stable s plane, and the time response t,f(t}therefbre contains damped oscillatory terms of the types

e-ar sin wt and e-·r cos (VI

If, on the contrary,

(" subcritical " gain setting) (9-59)'

then we can write the polynomial in the form

(s + P,)(s + /32)

where /31 and P2 are both positive and real.Equation (9-56) now has a real pole pair in the stable s plane, and the

corresponding nonoscillatory time response t,f(t) contains terms of the types

and

In either of the above two cases, t,f(t) will thus approach zero, proving tha twe indeed have bolh stable and isochronous frequency control. Our first andsecond system requirements are thus met. In Fig. 9-12 we depict the actualsimulated time responses for different values of the gain setting K/. The family ofcurves in Fig. 9-12a neglect. turbine dynamics, i.e., the case we just analyzed.Figure 9-12b includes the effect of turbine and hydraulic motors.

We make the following general comments about integral type control:

1. If we use subcritical gain settings we obtain sluggish nonoscillatory responseof the control loop, This means that the integral of 6.f(t), and thus the timeerror, will be relatively large. In a practical situation thi~ setting is most oftenused. The advantage is that the generator now will not unnecessarily" chase ,.rapid load fluctuations, causing equipment wear.

2. A careful study of the response graphs in Fig. 9-12 r~veals the followingfeatures:

As the sudden 'load increase sets in, the frequency starts falling off at tilesame exponential rate as for the system of Fig. 9-11. During these first instants

3./.X EI.ECTRIC tl" EIH.;\' SYSTEMS runon Y: AN II"TI((lDl)C'I'lON

rhc inicnral controller has not yet had time to IJO j:nto t~Cli()ll,.an,d the :r;l:.m~. d . d by the 1)!'!'I'II'll'y (\1 1':'(, loop which VIC (!Jf;r;WWC( PIC-rcspOflse IS elermtne u , J..... ,

viously in great detail. After a certain lime (lh,c short~r the y/(J~,_til: h,I;~)_1t;:integral gain Kil. the integral controller comes Into acuon and eventually 11ft.,the frequency back to its original value. . . .' .. ,It

3. The reader must realize that in order to keep the analysis (jf thCnyW tc1h"ra1JJcontroller simple. we have made several simplifying assumptions. v: C 13 a

summarize these:l d (I l th "y v/cten. The hydraulic and turbine dynamics were neg cctc iu ie

included in the simulated graphs in Fig. 9-12/J).1>. The speed-changer response was assu rued ins tan tuneous.c. All nonlinearities in the equipment, such as dead zone, ctc., have been

neglected.d. It has been tacitly assumed that the turbine can change its torque as fast as

It is commanded by the speed changer to do so. In reality, there is apractical limit to the rate, expressed in megawatts per second, at which asteam generator can pick up load. We have neglected this rate limitationduring the few seconds we are considering.

e. We have assumed that the ACE is available as a continuous signal. Inreality, the measurement of the frequency deviation t..j takes place discon-tinuously in sampled-data fashion. If the sampling rate is relatively high(compared with the fastest changes in the response of Fig. 9-12), then theabove analysis gives good results.

9-3-11 Economic Dispatch Control

The integral controller just described results in a system that meets the first threeof the specified control requirements. The fourth requirement, i.e., that pertain-ing to economic dispatch, can be met only by application of the optimal dispatcheql:lations,\ODE's (Eq. 8-15). .'

The primary ALFC loop makes the initial coarse readjustment of frequency.By its actions the various generators in the control area track the shifting loadand share it in proportion to their size. The speed of response is limited only bythe natural time lags of the turbine and the system itself. Depending uponturbine type the primary loop responds in 2 to 20 seconds, typically.

The secondary ALFC loop takes over the fine adjustment of the frequencyby resetting, through integral action, the frequency error to zero ..This loop isconsiderably slower and goes into action only when the primary loop has doneits job. Response time may be, of the order of one minute.

B-oth of the above loops base their simple control decisions upon thefrequency.error that can be measured locally at the power stations. They canthus be implemented locally in the plants.

Economic dispatch control can be viewed as an additional tertiary control1·0OY.p. As the control decisions in this loop are based upon the solutions of theOPE's it is necessary to incorporate a digital computer as part of this control)'c:m,p.Typiea,Hy this computer is located in an .. energy control center II which is

'V 'Q .W] ~

~ .Q, § '0 :9.-r$ '~,.~:::.$s /j <~, tty s:

f,' .:l'"'.... v~ !::. 'J)" ~ o· ...

~ ~;§,:,.q_j$~ <,

- ~ ~CtJ'tJOo~~'?,~ s# /\~ q;t; ,,0 ~I ~ ~ ~ ~.:l' ~ J5 ~~~~ \ 9:,. h o·e, ~ f) .

THE ENERGY SYSTEM IN STE/my STATE-THE CONTROL PROBt :" ~ ~ ~ ~".; 0 O~ (

. . I . .~~ I t':§ $ (;~ttrlinked to the various power plants via communication channels (micro~,~~ & G.& ::S;s c.-

telephone, etc). Periodically, e.g., every five minutes, the com~uter is providtf ~ f:1'-l ~ ~ "q;'with the megawatt settings in the power plants, These settings are compare\~ W ~ ~ §,.'f ~with the optimal settings derived from a run of the type of "optimal dispatch\~ S:"," ~ ,'tJ <$"

I • ~ \",. '- ..

program" discussed in Chap. 8,.lf the ~ctual settings are ofT~i:O~ theoptimal \ R' ~ ~ ~ (values the computer sends back instructions to the plants to readjust the mega- ~ {J.g ~t::;- !..'"

. .. I fl. h d ,~i/r" ~. ~watt outputs accordingly. This readjustment takes ...p ace, 0 course, via t e spee \ 1f ~ 0 ~changer. ' '~if. ~

& ~~ ~

~

\ierfts

re

:n

9-4 ALFC OF MULTI-CONTROL-AHEA SYSTEMS(POOL OPERATION)

is'aa

From a practical point of view, the problems of frequency control of inter-connected areas, or power pools, are more important than those of isolated areas.Practically, all .power systems today are tie-l together with neighboring areas,and the problem of load-frequency control becomes a joint undertaking. Closelyassociated is the problem of controlling the power flows on the interties.

Many advantages can be derived from 1'001 operation, and they can all besummarized in two words, mutual assistance We should at the outset state thefollowing operating principles, which are basic:

on

n1-

he 1. Under normal operating conditions each pool member or control area should

strive to carry its own load, except such scheduled portions of the othermembers' loads as have been mutually agreed upon.

2. Each control area must agree 'upon adopting regulating and control strategiesand equipment that are mutually beneficial under both normal and abnormalsituations. The advantages of belonging tl) a pool are particularly evidentunder emergency conditions.

Let us consider some of the advantages referred to in point 2.First, consider the effect of size. We have: seen in the preceding examples in

this chapter how the frequency, following.a sudden load change, will sag as aresult of the fact that, during the first moments following the load increase, the·needed energy is being "borrowed" from the kinetic energy of the system.Clearly, the larger the system, the more kinetic energy it possesses, and thereforethe. more energy can be temporarily borrowed for a 'given speed drop. Forexample, the large interconnected group that covers the central and eastern partsof the United States and some Canadian provinces can absorb a sudden loadchange of about 3000 MW and experience a frequency deviation of less than0.1 Hz.

A small system of, say, 1000-MW capacity, if it suddenly loses 300-MWcapacity (which represents 30 percent), .and if operating alone, may be 1R diretrouble. Its frequency will undergo an extensive drop, and chances are that thetransient power angle swings that would ensue (Chap. 12) W0u!cl tear the whelesystem apart, Fesulting in a complete blackout. . .

~ - -~. (0 o '\ - . --':~~.u~an i'1.lea""'L«;.,--,-_. • __--, -uu mrroduction~'" \1 ..-\<fl ~ ~ \\ WhlCh IS tj..~~ ~

• ~. ~ 'if ',... c; ~ ~ Step 3 r- - _--=-- __ -~~ Jf ~ ~ ? () ~ /'" '~~~<fl()<;- 0 .1/

'" '" (.> 0~, --z_ - w/{p (.> 0 .... ":l e- •..-l ,> •,... 0 ~ :;;.S', '2..---'- ." I-

, (0) ;::::l ,i' ·v-.... 0; ~ -<"'~Q.-~~ ~ r" / RGY SYSTEMS Ti-IEORY; AN INTRODUCTION~--?,-''''o -0 ~, \\ ~ / .. ic £N£... e- <;::. ". ,..- o,6c1f1.rr> ~. 8 B ~ <ft'~. t$)pi'. I _'s. ~ ~ ~ g ~ ~ / same system were part of a pool of say 100 000 MW .;. ~/~ ~ ~ ~.,~/ l('l~J-MW-generation f~lure. would repres~nt ~nly ~ 03 perce catpalcltY'Tthe

(l ~ .... ?' ~ ~./ J1lf: - Id b" d '1 d. ' n oss. he<fl ~ d .... ("> C<;/JP ency WOU e save, an support power to the extent 0[300 MW .--0 00 ';.> o o, frcqu 1 fl . he cri '. would%.. ~ e-?' jflS!antaneous y. 0V: Into t e crippled area via the tie-lines to carry its load until

normal generation IS restored.

System size also reduces the need for reserve power among' the pool mem-bers. A system that op~r~tes alone as a "single area" must provide power tohandle not only the anticipated peaks, but also the sudden requirements due toequipment failures. The reserves are classified on the basis of their readiness." Spinning" reserves can be called upon for instantaneous (at least as fast as thecontrollers can react) assistance. They are made up of fully operating but onlypartially loaded units. Hydroelectric-, diesel-, and gas-turbine-operated units canbe called upon at a Jew minutes' notice. Steam reserves range all the-way from"low bank" (i.e., idling at pressure and heat levels below that required forservice) to "cold .reserves " (operative, but not in operation): .

Since peak demands occur at various hours of th~ day In vanous areas, theratio between peak and average load for a large pool IS smaller than that of theindividual systems. It is obvious, therefore, that all pool members can benefitfrom a reduced need of reserve capacity by a scheduled arrangement of energyinterchange.

9-4-1 The Two-Area System.' 7' . troduce the reader to the fundamental problems

It was possible m Chap. to ill (on of an n-bus system by discussing, first, theassociated with the ~tatlC"Qo:p'~ra1 th two-bus system. We shall find it equallysimplest of all multlbus SYSb,mcts,t e the load frequency dynamics of the n-area

- d ce the rca er 0instructive to mtro u d . mics of the two-area system,system by studying first the ynatt ti on to a system consisting of t,_vocontrol

. Let us therefore turn OU~. a .en 19_io interconnected via a relatively weakdeJled U1 Fig. . ,- of the type mo f diff rent size and charactenstlcs.areas rail)' 0 lue . h . ed b. li The areas are gene, . that a control area ISc aracterize ytIe- me, . d the 'lSsumptlon . th

W had earlier ma e • This' tantamount to saying that e areae h uj,hout. IS - d h-frequency t fO: '. th . gle-area case we coul t us representthe same "'t- n o n, In e sin- -, 1"" fiend" or: . ;; ro J t,'. . ble Af..In the present case we assumenetwor ( IS 0- • b the Single varia . h " k "

fre uency deviatlODS .. Y, "Having interconnected them WI~ .a \~'ca-the q . dividuaJly stl,oqg.. t' that the frequency deviations 111 theeach area m. rhe assump IOn '.tie-line therefo~ leads us t(:d' by twO variables Ail and !J..i2 respectively.

. 'be representt,two areas can

J• . the Tne-JLiII,e .'9-4-2 M[9de.11!1g h t'e-line follows from equation (3-49), i.e.,. h (I\'I'(:r on tel_ I operatIOn t e.p ,IN norma I0f II ~ I sin (c5~ _ c5~) (9-60)

P~:: =, . X

, NTReJL PReJB,LEM '3;31THE ENERG¥ SYSTEM 'IN STEAIDY STATE-T'HE CO, ,

. , 1 The order. ... so - 0 J/ d V respective y, .Wl'l~,ne (;/1 and 02 are the angles of end voltages 1 an 2 "in direction(§)'fr tlii'e sum scripts indicates that the tie-line power is defined positive

1 :-' 2: . " , " , -1' e ower changes with. For small deviations In [he angles 01 and O2 the tie In p.the amoant

6P!2 ~ 10? 1111 I cos (c5? _ (5~)(tlc51 - l'lc)l). MW ,(9-61)X

, 'rr "f synchronous machinesAnalogous to the concept of "electric su nes,s, 0 • ffici t " of a line(compare Eq. (4-33)) we now de~ne the "synchronizing coe cien

-r<l_A 1V? 11111 0) MW/rad (9-62)1 - "'" -'---'-':::-::-- cos (c5? _ t5 2\)(

The tie-line po~er devia~n then takes on the form

I.5.P12 = TO(nc5!_ nol) MW

d the reference angle nc5 by theThe frequency deviation nJ is relate to

(9-63 )

formula1 dId61=- -W+6c5)=- -d (/lc5)

211: dt 211: tHz (9-64)

or inverselyr

110 = 2n: J 61de rad

By expressing tie-line power deviations in terms of Co.! -rather than Co.fJ wethus get

MW (9-66)

Laplaee transformation of the,l~st formula yields

211:YO .APl~(S) =:=-' s-(Ml(s) _ Co.!2(S)) (9-67)

Rel\)'r-esenting this equation in terms of block diagram symbols yields the'<i!lia:gram in Fi:g, 9-13,

"(9,..4;-30Bl~k fi'iagram ltepresentafion of TWO-Area System

. ~~~w a-Q.estae tie.o;lineF'0:w.er enter our area mOdels? Clearly we m~st g0 baek t'€l. '~UF :~:~JJ1¥wmw.er-lDailall€e elq,Ua'ti0Fi ~9-40); In writing such a'foleql1ati011 f0. area 1""we: iN,ilis;t '9Iea,iIy ad~ i~Mi)12 to t,lireright-Ran€! side, SiFFlila,dy, fer aeea :2., We liFloUM

leit~&t, Sr;la(7raot AP~~ 0F qdd AP2 t 1 whene :P21 r.elDFese,ll'~s the t'ie-lilae p.Gwe-r j,t\}'; ...!d1!ntet!i'0,t!\~,9; it. Wre Aa,ve;JF'l!»ssesa-Fe aegfeetem.( .

~' ,

,~~ I = :-Mi"u . MW

I

If I! !v

,.:IP1?-:--p...----~,....... .._-.

FI1:ur~ 9-13 Linear representation ortic·lill~ ..

We shall find it convenient when we work with multiarea systems always toconsider the tie-line power positive in the direction out from the area in question.We remember that the sum of the right-hand power terms in Eq, (9-40) isrepresented by the right summing junction of Fig. 9-10. If we thus observe theabove tie-line sign rule we can interconnect two single-area diagrams by meansof the tie-line symbol in Fig. 9:13. The end result is the block diagram depictedin Fig. 9-14. In reading this diagram the following clarifying poin ts are in order:

.:1f 1 (s)+--

r - -- - - -----~------------------.. Ir....Ll18; 1L~J+i

..J.. r--l/'L')---+ K/, + L\ .,---,--l>-LS--l>-cr I __.J IIP+ 1 rof.'

L1P'2(Slj I ACE, = LlP'2 + 8,.1(,I .dP,2Is)I 04-~-----------------------I

r..LlI -/ IL.,-J

.IP () 1 I ACE2 = LlP21 + 82M2- 21 S ~ 1 I

Ll"P+J.. r---,' rof.2/"",'\~__j' K/2 ~ "'"I i- f---,I- - L..'"1" ../ L _ _!_J -I- •+1 -

r..L-,I 82 ILTJ

\ ...,-.t --- ----O-------- ...:..:...____J

i

i

Figure 9-14. Linear model or two-area system.

.i\.

\

,

l~.I!

\u~ .........-~ • 0~t ~~? ~ ,THE ENERGY SYSTEM IN STEADY STATE-THE CONTROL PROBu:.~~' e;<>t$' '?\,~'&...,~ (5

i \. ~ ~ (A~ '!:J'<.J (• • \1) i$ V ~O ....,~

1: Th~ dashed portions should be disregarded for the .trrne being, (~~ r! ,,0 '0.-"; eoSec. 9-4-7.) ! ~. ~ iP;' ;S>"'<J~

'2. We r~rne~ber that .the powers i~ the single-area diagram (Fig. 9-1~) welrf~vc.,.l~f:'~Oexpressed in per unit of area rating. The parameters R, ~, arid H, likewise -.,Vl s; ~were based on the same base power. When two or several areas, generally of ;It"different ratings, are involved, we must refer all powers and/parameters to the ~one chosen base power. I

3. The added blocks of transfer function -1 follow from Eq. {9-68}.II

9-4-4 Mechanical Analog of Two-Area System IBefore we proceed with an analysis of the two-area system" we shall present,without proof, a mechanical system that is a perfect analog of our electricaltwo-area system. Consider the train in Fig. 9-15. If this traiq is traveling at aconstant speed VO and ,suddenly is subjec t to incremental load changes, thevelocity will change. The two car-assemblies will experience the velocity pertur-bations t..Vl and t..V2, respectively, and the t.ie-spring power will change with theamount t..P'ic' By writing the incremental dynamic equations for this system,one can readily confirm that they are of the identical form as our electricalsystem equations. This means that the systems are analogs of each other. Thefollowing variables correspond to each other :

t..Vl ~ !jfl

AV2 (-~ af2

Tie-spring power H tie-line power

The train analog can thus be used as an aid in achieving a better feel forthe behavior of the electric system.

[.

!:" "i'Ii

!

9-4-5 Static Response of Two-Area System

We shall first investigate the response of the two-area system with fixed speedchanger positions; i.e.,

Analog of area 1 Analog of area 2Relatively soft "tie-spring"

... ,::: .•:if(.,'··:; analog .of weak tie-line ..:..-:;;i,){;

~~.;"' 1 ~ \ ";~:::;o.2 .. I . .

. ~~ . :,;,: ~(~~~.~~~~~~. ~"~ ~ ~~~

.,0+.::1".1 . "O"'~U2 ---

Figuie 9-15 A mechaniealanaleg of two-area system.

',-.'0 iB _ o j;'I:v"-- 9 -~~ §. ~ /al~' ~ ~ /"J~ "....p g- ::/< 6N£KGY SYSTEMS THE9RY: AN INTRODUCTION~ ~ :::, ~.It':[(._'

'~~' C"JQ /''',(1) • /

~. ~ (//~~;e assl:H11C that the loads in each area are suddenly increased by the constant~ @ '::;~JcreIl1ental steps tl.PD1 = MI and 6PD2 = M2• We shall presently limit our

/,(' IlrlaJysis to finding the static changes that result in frequency and tie-line power.,/ Let us call those changest 6Jo and 6P 12. 0 respectively.

Since the incremental increase in turbine dynamics in this static case isdetermined by the static loop gainsj we obtain from Fig. 9-14

(9-69)1

DPn.o= - R2 t.Jo

By adding the powers at the summing junctions we obtain

(9-70)

1- Rz

~fo - Mz = Dz ~fo - ~PI2.0

We solve for ~fo and ~Pu. ()and obtain

!'J.fo = _ Ml + Ml131 + 132

131M 2 - 132M I

!'J.P12.o=-!'J.Pu.o= /31+/32

where, in analogy with Eq. (9·-48), we have defined the AFRC's of each area:

Hz

(9-71 :

pu MW

Co 1/31=DI+RI

~ 1132 - D2 + Rl

Equations (9-71) become particularly simple if we assume identical area par-ameters; i.e.,

(9-72)

Di=D2=D

RI = R2 = R

/31=fJ2=/3

.t In ~tead~"~tate the frequency dr~'ps in the two areas will be equal (compare graphs in Fig. 9-1'6').,~''Obtained Ib~letting s .... o.

," " J

IIIIIlI

IIJ

I,

'!

THE ENERGY SYSTEM IN STEADY STATE-THE CONTROL PROULDI 335

We then gel

No=(9-73 )

Hz

M2-M16.Pt2.0 = -6.P21•0 = 2

For example, if a step load change occurs only in area 2. we gel

pu MW

Hz(9-74)

pu MW

These two last equations tell us, in a nutshell, the advantages of pooloperation:

1. Fifty percent of the added load in area 2 will be supplied by area 1 via thetie-line.

2. The frequency drop will be only half that which would be experienced if theareas were operating alone (compare Eq. (9-49)).

Example 9-$ A 2-GW control area (I) is interconnected with a 10-GW area (2). The 2-GWarea has the system parameters given in Example 9-5, i.e..

R = 2~40Hz/pu MW

D = 8.33 X 10-1 pu MW/Hz

Area 2 has the same parameters, bur in rerms of the 1O-G W base.A 20-MW load increase takes place in area 1. Find static frequency drop and tic-line

power change ..,

!IIII

II

III

SoLUTION We will choose the generator capacity of area I, 2000 MW, as our power base. InEJtample 9-6 we had already computed PI = 0.425 Hz/pu MW in terms of this base. p, has thesame numerical value based on I()"GW. Based on 2-GW it must be five times larger numerically,i.e.

p, = SPI ;". 2.125 Hz/pu MW

Also20

MI = 2000 =O.QI pu MW

Equations (9-71) yic:ld the static deviations

0.01No = - 0.425 + 2125 = 0.00392 Hz

tlP _ - 2.125 x om12.0 - 0.425 + 2.125 = -0.00833 pu MW (or -16.7 MW)

. Note that the frequency drop is now only one sixth of that experienced by area I runningalone (co~pare Example 9-6). Note also that this "frequency suppcrt " is accemplished by anadded delivery of 16.7 MW from the larger area.

........v LI.I.\.. I r..1\.. r.."'CIU ....11 .y't~Il:MS !"I-IEORY: AN lNTRODUCTION

9-4-6 Dynamic Response of Two-Area System

Even with the v€ry simple turbine model that we have used, the two-area systemin Fig, 9-14 is of seventh order. It would therefore be meaningless, in our text, toattempt a direct analytic approach for finding the dynamic response of :hesystem. We perform instead an approximate analysis based upon the followingassumptions:

I, Consider the case of two equal areas,} Consider the turbine controller fast relative to the inertia part of the systems.

i.e .. we set Gil = GT = I.3. Neglect the system damping, This means that we assume the load not (0 vary

wi til frequency; i.e.. we set D 1 = D 2 = 0, This means, in accordance wi thEqs. (9-42) to (9-44), that Gpi (s) = Gp2(S) -+ fO js2H.

Under these highly simplifying assumptions we can readily derive the follow-ing expression for the tie-line power from Fig. 9-14

nfoyo llPD2(S) - llPD1(S)llPi2(S) = ~ S2 + (f0j2RH)s + 2nfoYOjH (9-75)

This expression tells us several important facts:

1. The denominator being of the form

S2 + 20:s + w2 = (s'+ 0:)2 + w2 - 0:2

where 0: and w2 are both positive; we know that the system is stabieresv:damped.

2, Following a disturbance, the system will oscillate at· the damped angularfrequency

(9-76)

3. The system damping is strongly, dependent upon the 0: parameter. Since fOand H are essentially constants, the damping will be a function of the R par-ameters, Low R values will give strong damping; high R values, weak damping,The system will perform undamped oscillations of frequency Wo = w if R = co,tha t is, if the speed governor is nonexistent.

4, The fact that the system _is inherently oscillatory could have been immediatelypredicted from the mechanical analogy in Fig. 9-15.

Example 9-9 Consider two equal areas, each having the parameters

R = 3,0 Hzjpu MW

If = 5 s

I" = 60 Hz

THE ENERGY SYSTEM IN STEAfilY Sl'ATE~TME GGNT;''''' . '\. "''''''' 'P:R0BLEM ~

:, "\

I'"\ .

I:~. ,

TH1 = TH2 = 0 R 1 = R 2 = 2.4 Hz/pu MW ITn = Tn = 0 VI = 02 = 8.33 x 10-3 pu MWrZ

Figure 9-16 Dynamic response of two-area system sublect to a step-load increase in area 2.. r I

I

From Eq. (9-76) we thus get

COo = .j75.3To - 1.0 r/s

Assume the tie-line has a capacity of 0.1 pu (10 p er cent of area capacity) and is operatingat a power angle of 45°. From Eq. (9-62) we thus have

T" = 0.1 cos 45° = (·.0707

The oscillating frequency is thus

COo = .j75.3 x 0.0107 :- 1.0 = 2.1 rls (or 0.3314 approx.)

The graphs in Fig. 9-16 depict the tie-line power a id frequency oscillations (in both areas)for a step load increase in area 2. (The dashed pc rtions of the graphs are explained in·Sec. 9-4-8.)

9-4-7 Tie-Line Bias Control

The response curves in Fig. 9-16 indicate clearly Ililat some form of reset .i:n'tegralcontrol must be ·added to the two-area system, The persistent static 6req}oHemcy.error is intolerable for the same reason as was the case in the siagle-area (4Se.AIs@, a persistent static error in tie-line power flow=-se-ealled "inaQ'ver~e:m't. h Iexchange "-would mean that one area wOli,}ld ave to llUp,p<l>rt t1<!,eether on as-t~acl'Y-sta'te basis, A basic glJi&ing principle in [.'0,,)[ '(f)peratJi"e.n nlu§.t beor:h.qt eael:area" in normal steady staoe, absorbs its o~n l(i)tUl·(,~.ee Jll.• 32~). . '.

Vari0.l.'lS met1t0d:s of reset integrai contrel have -over t1t<?years .l:>een: trieo6i , 0,lllltc.,......,airlQJ',·a'ba>Ad@uecl-fQf FFlu>].ti@l1~a s~steflilS. If'I@t e~~Iijflli, ·iIi ·<i>u,t::tw.Q~af.~a

~,S'1!em',we. ca.u'ld eoneejve of the a(r3ilil~eFQetit tbat at~a. -1 ·:be i'~~n$i.Il:[i'le· f€lF

I ", "

<.. >

!"it':;

';;jIttTl

oez.,j;0'®r

r"e"quehCY reset and area 2 take care of the tie-lineU' power. We would thusa-rr~nge for the following area control errors: .

ACE1 ~ Afl(9-77)

h These AC:E's would be: fed via slow integrators on to the respective speedc angers. This arrangement would work-but not too good. Actually in theearly days of pool opers.tion one area was designated to reset the system'frequenc,~ and the others would be responsible for zeroing their own" net inter-changes (see also Sec. 9.4-9). The problem with this arrangement proved to betha.t the central frequency controlling station tended to regulate for everybodytrying to absorb everybod y else's errors and offsets. As a result it would swingwildly ,between its generating limits. '

As a result of the original work by Cohn" a control standard has developedthat has been adopted by 100st operating systems. The control strategy is termed"tie-line bias coritrol " and is based upon the principle that all operating poolmembers must contribute jr.eir share to frequency control in addition Cotakinq careof their own net interchanq.:

9-4-8 Tie-Line Bias Con trol of Two-Area SystemIn applying this reset control method to our two-area system we would ,add thedashed loops shown in Fi:~. 9-14. Th~ c~ntrol error for each area consists of alinear combination of frequency and tie-line error:

A CE I ~ ilP i2 + B 1 6fl

ACE2 ~ ilP21 + B26f2

The speed-changer commands will thus be of the form

ilP,,,r, 1 = -K/1 I (ilP12 + BINI) dt

(9-78)

(9-79)

ilP,ur, 2 = -KI2 f (LlP21 + B2Llf2) dt

Th' It nts K and tc., are integrator gains, and the.con~tants s, and B2 aree cons a." r 1 . ' . b . 1 d d .thefreql ency bias parameters. The mmu~ SIgnS must e inc u e: sl~ce.each areashould ~nc~ease its. generation if either ItS frequency error or ItS tie-line powerincrement IS negative.

I, ,9-4-9 Static System Response ''The chbsen strategy will eliminate the steady-state frequency a~d tie-line devia-tions rdr the following reasons. '

Foliiowing a step load change in either area, a new static equiiibriarn, if such

IiI

I1,

I~

I

THE ENERGY SYSTEM IN STEAI)Y STATE--THE CONTROl. !"WIlLEM ),39

. I . id changer commandsall equilibrium exists, can be achieved only alter I ic spec - b: I . te zrands in

. . I . cs thai ot 1 111 eg ,.have reached COl1stanl values. But this evident y rcqurrcs uia- -

Eq. (9-79) be zero; i.e.,

eyg

dd:JI-e

6P'2.0 + B, 6/0 = 0

6P21.0 + B2 6/0 =·0

In view of Eq. (9-68). these conditions can be met ollly if

6/0 = 6P'Z.0 = 6PzI.o = II

B d B 'allies l n fact, 0/1(:' oflheNote that this result is independent of the ,(Ill 2 (,;. II t) b d lVe still have a gllarantee 10bias parameters (but 110t both can e zero, an .

Eq. (9-8I~ is satisfied. t has beenThe question what "best" value to choose for the B para me ers d

hotly debated. Cohn has shown that choosing B = f3 (i.e., the A ~R C) pro uc~s. t d system The Irltegralor gamsatisfactory over-all performance of the mterconnec eo svsvern. I

constants K" and KI2 are not critical-but they must be chosen 'small :1~OUg 1not to stimulate the area generators to "chase" load offsets of short duration.

The actual effect on the frequency and tie-line power graphs of the. addedtie-line bias control is shown in Fig. 9-16 (dashed parts of graphs). Following theimmediate excursions which are entirely determined by the pnmary sp~ed-governor loops of each area, the secondary integrator loops of each area go mtoaction and reset both the frequency and tie-line power back to original values.

(9-XO)

(Y-X I)

lea

8) 9-4-10 Tie-Line Bias Control of Multiarea Systems

In reality a control area is interconnected not with one tie-line to one neighbor-ing area but with several tie-lines to neighboring control areas, all part of theoverall power pool. Consider the ith control area. Its net interchange equals thesum of the megawatts on all m outgoing tie-lines. As the area control error ACE;ought to be reflective of the total exchange of power it should thus be chosen ofthe form

19) I

iI

·1

ire:1

'ea Ifer!

m

ACE, = I ilPll + BI 6;;j=1

Typically, the reset control is implemented by sampled-data techniques. Atsampling intervals of, say, one second, all tie-line power data are fed into thecentral energy control center where they arc added and compared withpredetermined contracted interchange megawatts. In this way is obtained thesum-error of Eq. (9-82). This error is added to the biased frequency error andthe ACE results. The ACE is communicated with all area generators' that areparticipating in the secondary ALFC If optimum dispatch is employed, a tertiaryslower "OD loop" is added of the type discussed in Sec. 9-3-11.

'ia-

sch

(9-82)

34(} FII·CTRW F:-.iFRGY SYSTIiMS THEORY: AN INTRODUCTION

',i

9-5 .. STEADY-STATE" INSTABILITIES

The control methods outlined above-or variants thereof-are incorporated asstandard in most of the interconnected power systems around the world. Theoperating experience has been good-exceptionally good in view. of the vastgeographical spread and different generator mix characterizing many of thesvstcms.. However. on occasion. u power system may experience stability problems

usuull y in the f0f111of self-excited low-frequency oscillations. Power engineersarc rcfcrr ing to these as "steady-state" instabilities as they occur, mostly unex-pected, as the system is running in what appears as a smooth steady state.ligure 9-17 depicts ,: graph of the frequency of the peninsular Florida grid. It isoperating normally at 60 Hz with a slight unavoidable" noise" (which is alwayscharacteristic of the ALFC loop action) when, gradually, an oscillation starts tobuild of frequency typically less than 1 Hz. The oscillation is detected not only inthe system frequency but also in the tie-line which will experience pow.er "slosh-ings " of increasing magnitude. This type of instability is actually caused bycross-coupling between the AVR and ALFC loops. The phenomenon has beendetected in many parts of the world. It is directly associated with the so-callednatural oscillatory modes of the network and it is therefore fitting that we startthis section with a discussion of the latter.

f. HzjI

60.2

Normal operation Oscillatory buildup •

60.1

600~

I10

I20

I30

I4.0

I50

I60

Figure 9-17 Self-excited oscillations in inter-connected power system.

:jlifi

.~

THE ENERGY SYSTEM IN STEADY ST,\TI:-THI! CO~TI1.0L PROBLE~\ -

Lower natural mode

"-,\

----»5

/I

I

II

/

,,,

Higher natural mode

,te)

. I '11 tory system' (a) identical masses andshaft elasticities; (b) lower naturaFigure 9-1S Torslona OSCIa .mode; (c) higher natural mode.

9-5-1 Natural Torsional Oscillatory, Mode. in a Power System

In Fig. 9-18a are shown three rotational masses interconnected with elasticshafts. The masses can perform torsional dyn arnics, describable in terms of thethree angular coordinates 151> 152, and 153,

The three masses possess two torsional na :Llral modes, each characterized bythe natural (or eigen) frequencies 11 and 12 respe ctively.

If we assume the masses to have equal inertia and the shafts equal elasticitycoefficients the natural modes become particu.arly simple. In the" lower mode»(Fig. 9-18b) masses 1 and 3 swing against each other with mass 2 nonparticipat-ing. In the" higher mode" masses,.1 and 3' swj~i; against mass 2, (Fig. 9-18c). '

The natural modes can be excited in oneol' three ways: ' ,

rrr.- ,;' ~~..

.~

_' _ -::_•. ...,~ S·

':n~'~:f1G):p'~rchoice of initial position setti.:'; • I... Id" 11.., t so 0 mgs. For exa"'" I 'r'"'~.?·te'flil IS ~~e so 1, a o 1 = - f>3 and f>o == 0, '. . "'~~' I InItially the

•.<, !il:J!JQr} release: 2 , It WIll oscIlJat€ In the lower modei: 2. By some oscillatory Source that is " '"

will arise if the SOUlce frequency pu~pmg. the system. Natural Qscillatiofls~'. Sudden major fault ..induced to ,col~clbels with one of the natural frequ€l'Icies

rque irn a ances. .

. The natural oscillations can takRig. 9-18, or around a reference l e place around a fixed reference, as incase is analogous to what h sy~tem rotating at a constant speed. The latterhere the enerator+ r a~pens I? a power system. The individual masses are"shaft" g h I('t.or~ (I~c)udmg the fixed turbine rotors). The elastic

s zle t e transnussior, lines of the grid. The position coordinates are thepower ang es 15 [compare Example 9-9).

If the system conla ins N generators then there exists, in the strictest sense,N - 1 natur~l modes. However, many subgroups of generators, particularlythose belon~m~ ~o th.I,~ same power stations, are swinging as coherent groupsra ther than individually. Their close proximity makes their connecting "electricshafts" very stiff. These natural frequencies typically lie in the range 0.1-5 Hz.When one of them IS b eing excited the power system is said to perform inter-machine oscillations. Every individual generator rotor including its directlycoupled turbine rotors ,:and excitor rotor, if present) participate in the oscilla-tions as one unit. The o scillations shown in Fig. 9-17 are of this type.

For a turbogenera tor unit of very large rating the shaft is very long andgenerator rotor, exciter rotor, and the often three to four different turbinerotorst constitute a rotational system which by itself possesses maybe half adozen natural modes. 'I he eigenfrequencies of these modes typically lie in thefreq)J~ncy range .20-50 Hz. ?ne of these mo~es can on occ~sio~ ~lso b~ excited,partidul'arly when the.f.:lfc~nc network con-tams very ~ong lines WIth series capa-citors! Such an instalinitt:1 IS referred to as subharmonic resonance.Itl is important to realize that both the low-frequency" interrnachine " andthe high-frequency "subharm.oni~" oscillations ~re po~rly .dampe~. This meansthat a\ relatively weak" pumpmg source can excite oscillations which may grew

. to amplitudes of destructive magnitudes causing the" elastic shafts" to break .. . tnl the" subharmonic" phenomenon the elastic shaft consists of the genera-

.1>0r'shrf,t itself. In a elasslc accident this shaft broke causing lOO-millioN-dollar

da'ma9~' 'Itl:\the "iNtefmachine" phenomenon th~ "~lasti~ shafts". are the transmissionlk.J.esof,the ·~itl. They "~r.eak" when the tle~lme power swings become S0 largethat '1'h6p.e:we.l1ang.le 0 g'1e.s beyond. 90 electrical degrees cal!sing loss of synchro-

~""In,ism a~fd"island" feFIDit10n.:J; .

, I." . d·i. t't, ,r.l.i:g!\, low" and' rntermec I~t.!pressure sec rons.~: WOfn, periinsill,ar' flor.ida· e~pe.r.ienCes the s~ihgs. depicted in Fi.g. ~-17, and if the oscillation

'lfi.oi'I{luB'Gannet DC steppe<\; the .e.nd.result usually IS a, ~reakup of-the tie-hiles .to -Georgia, The wJiele.~tlinSU'11 gnq ',wHl th"eFetip-en;Am isolated, One talks- In sueh cases .apont "'int~rar.ca" esciJ.Iat,ion's.

, ~..

..

" "

.9...5QpIt i1SySIselsgenThibenthe

det

Jger

1~ pel~ (E!j foli

1.

2.

p~be

", ,I,', ,

l

THE ENERGY SYSTEM IN STEADY STATE-THE CONTROL PIWBLI;M 343

s

9-5-2 Natural Mode of a Single GeneratorOperating onto Infinite Bus .

It is instructive to study the factors that determine the natural modes in a powersystem. Because the interrnachine oscillation is the most common one we shallselect it for our study. As our" textbook case" we shall choose the simplest of allge~er.ator configurations-a single machine operating onto an .. infinite" bus.This IS analogous to the case of two rotational masses (Fig. 9-18) one of whichbeing of infinite size. With N = 2 the system possesses only nllC' natural mode.the characteristics of which we now explore.

The single-generator case was extensively studied in Sec. 4-12 where wederived static characteristics. In Sec. 4-13 we derived formulas for the cransienrgenerator power which will now prove useful.

In the study of .. area" frequency dynamics in Sec. 9-3-5 we derived thepertinent differential equation (Eq. 9-38) from a .•power-balance" expression(Eq. 9-34). We use the same approach now but need to slightly modify it in thefollowing regard:

1

rece

ysc

y1. We study the power balance of one rotating unit-not an entire "lumped"

area.2. As we shall later use our derived equations for large-scale dynamics (Chap. 12)

we do not limit.our derivations to incremental excursions.

As the differences between the turbine power PT and the generator buspower P G is used for increasing the kinetic energy of the unit we obtain the powerbalance equation .

) .

dPT - PG = di(Wkin)

Note that the last term in Eq, (9-34) is nonexistent here, as we assume thatno local load is served from the generator.r

Following the same steps as in Sec. 9-3-5 we transform Eq. (9-83) Co

MW (9-83 )

2H dfPT-PG=- -fO dt

pu MW (9-84)

We shall find it more useful to let power angle, lJ, rather than frequency.j, beour independent variable. In ·view of Eq. (9-65) we thus get

pu MW (9-85)

t Lale~ w~ shall add a term corresponding to the (relatively small) power loss in the generatordamper winding (Sec. 9-5-3).

SM' ELECTRIC E1'l'ERGY SYSTEMS' THEORY: AN INTRODUCTION

This is the famous genera tor swing equation (GSE) that plays an extremelyirnportant role in all types of studies of power-system dynamic phenomena, Ifwewant to use the GSE to study incremental dynamics around an operatingpoint(P~, Pg, bO) then we readily prove it to be of the form

. H d2

I1P T - I1P G = 1[/0 dt2 (116) (9-86)

This is the form we find useful in the present case.

Example 9-10 Write the GSE in form (9-86) for study of incremental dynamics of the 15-MVAgenerator being operated as specified in Example 4-3. The generator has H = 5 s.]" = 60 Hz.Find natural frequency.

SoLUTION W~ approach the problem in the following steps:

Seep 1 Find the steady-state operating point. This was done in Example 4-3 for the power levelP~ = P~ = 0.667 pu, an excitation level EO = 1.22 pu and with the network voltage held atI v:, I = 1.0 pu. It was found that the steady-state power angle equals

15~ = 25.7° electrical degrees

Seep 2 Find the transient generator power valid for transient deviations around the aboveoperating point.

This was done in Example 4-13. The transient generator power was foundthe form (4-83).

Seep 3 Find 6P G'

For small angular deviations around 15,,_= 15~ we can write

to be of

(9-87)

(9-88)

or

pu MW (9-89)

where S' is the" transient stiffness" of the generator. Compare also Eq. (4-33). The-powercomponent (Eq. 9-89) is referred to as synchronizing power.

From Eq. (4-83) we obtain

(dP ) 0 :

S· = dO; = 2.398 cos o~- 2 x 0.478 cos 2J~

= 2.161 - 0.596 = 1.565 pu .MWjelcctric rad . (9-90)Step 4 Write the incremental GSE. We set for simplicity t!.Pr = O. Substitution of (9-89) into

(9-86) yields

(9-91)or shorter

(9·92),

.'0

I

~THE ENERGY SYSTEM IN STEADY STATE-THE CONTROL PROBLEM:'

where

Step 5 Solve the GSE.The solution of the second-order linear, constant-parameter differential equation (9-92) isof the form

(9-94)

The integration constants A and", depend upon the initial conditions, i.e., upon howthe oscillations are started,

In Our numerical case

J11 X 60 x 1.565WI = 5 = 7.68 rad/s

corresponding to a .. natural" frequency of 1.22 Hz,

It is interesring to note that increased inertia lowers the natural frequency.Increased stiffness (caused by lower generator and line reactances) will increasethe synchronizing power and thus the frequency.

9-5-3 Effect of Damper WindingI .

Equation (9-94) implies that the inertial oscillations are undamped. In actuality,the oscillations in all probability would be damped and would ~ctually vanishafter some time. The system evidently possesses positive damping.' This dampingoriginates mainly in the rotor "damper" winding (Sec. 4-1).

As the rotor performs incremental dynamics relative to the synchronouslyrunning "infinite" system its damper windings will no longe~ be stationaryrelative to the armature reaction flux wave. The relative rnotidn' will in factinduce currents in the damper bars and according to Lenz' lawl these 'current~will have a direction such as to try to stop the relative motion. .

The damper winding will in fact dissipate power and the dissipation rate is. ,.almost proportional to the relative velocity, d(~o)/dt. The presence of. thedamper winding thus .adds a positive term on the right-hand sid~ of the powerbalance equation (9-85). The incremental form of the GSE will change from(9-B2) to I

I

(9-95)

The general solution of this equation is

!iON = Ae-br sin (co; r + if;) (9-96)

(9-n)where

Typically, COl p b, and therefore

f:l",

e-<;j

!2'l7 ENERGY SYSTEI~'S THEORY: AN INTRODUCTION

:>;j cr;</c~P

6riN<I--

Stiffness2

d .di (tlbN) 66N---l> 1 --l>

S

3 4Natural dampi~g

d .5 iF (dr)N)

'-q-

9 AVB

10

IFigure 9-1~ A linearized dynamic model of the single generator-Infinite bus system including theAVR coupl,ing effects.

I

In conclusion:' Under the influence of the damper winding, the generator is'" oscillat6rilY 'stable," meaning that if subject to any small disturbance it will returnto its stebdy-state equilibrium by a damped' oscillation.

This i"natural" damping of the system is further enhanced by the resistancesof the network, which we have neglected in our analysis.

It is instructive to vishalize the rotor dynamics by means of the boldfaceportiont of the block dia~am in Fig. 9-19. The blocks 1 through 4, plus thesumming junction Ll, model the basic undamped." inertia H dynamics ofEq. (9-92). The damper loop (block 5) takes into effect the natural damping ofthe system, as expressed by Eq. (9-95).

I9-5-4 Extension to Large-Scale System

It is possible to extend the above theory to the N-generator case. An "inertial"angular. reference is chosen (Chap. 12) relative to which all N machine rotorsare measured. A swing equation is then written for every generator. These GSE'swill be mathematically coupled via the line powers. ,

t The remaining part of the-diagram will be commented on in Sec. 9-5-5.

POIt

1. 1(

2. {

3. 1

9-5·1ft!sta1:Cle:ismEq.In tosei.e.,pre

tivewil

~ th~Ii~ tiv\\ an~~ in~

~r, enI~

gi~~)

Ii\

9-Ir.SI.

a:Qi

f,f

IS

's

, III performing a ULlJ0y nf (Iii; int;(!;rm;nlaI dynamic:; around some operatingpoint one can cOllcludt; Llla~:

1. The response cClIwil;tl! flat of fine radian natural frequency (1)1' as in the abovecase, but N - I natural Ircqucncien (/)1> .,,' (I)/:_!,

2. Under the influence or damper windiJl&l and line resistances all N - I naturalmodes arc positively damped.

3. The system would thus: at; "o;;cillat0rily stable."

9-5-5 Negative Damping

If the" natural" tendency for the Interconnected machines is to be " oscilla~orilystable" how then can phenomena of the type depicted in Fig. 9-17 be explained?Clearly, our above analysis must have neglected some important system ~echan-ism that will be present in real life. To understand this let us return for a minute toEq. (9-95). Assume tha.t somehow the damping coefficient b becomes negative.In that case the factor e"!" will qrow with time rather than decay, and the inertialoscillations will be of the type in Fig. 9-17. Furthermore, the system will self-excite.i.e., the oscillations will be initiated by any infinitesimal disturbance that is alwayspresent.

The damper winding and line resistances will, of course, never yield a nega-tive b. It is entirely possible, however, that signals emanating from other sourceswill enter the loop at the summlnq junction 2:1 in Fig. ~-19. These signals may, likethe damper signal, be proportional to d(Llojjdt. If they, like the latter, are posi-tive they will further help damp the inertial oscillations. If, on the contrary, the vare negative they will reduce the natural damping, possibly to the paine of eliminat-ing it altogether. .

It. is instructive t(: sh?w ho"~ sue? signals may be generated and how theyenter into the dynamic prcture. r or trns reason we will in the next few sectio sgiv~ a qualitative, step-by-step presentation of these added effects. n

9-5-6 Effect of Changing £'

In deriving the formula (9-159) for LlP G we made use of t .sion that was derived from Eq, (4-77). It is im or" a ransl.ent power expres-assumed IVII and IE'I tho h p .lant to realize that we tacitl»o )(1 constant t us rnakinq P fi . .angle ONonly, i.e. G a unction of the power

(9-98 )This Yi.clded the C'xpre;wion (9-89).In reality the emf £' will chance and thi

• • r., IS .means that Eq. (9-9~) changes to

Po = PG(O/l,. E')(9-99)

I I I" 1'\'\ l .. ' 1''''\.1 \ ., 1 .... 11:.'1:'1 IIII:UK'I . ,\:" I;'\; 11«(.JLHJCTIO:"i

Thus we have for the incremental transient power output

(9-100)

or shorter(9-101)sr; = S' t:.b,.,.+ K, t:.E'

where K I has the unit" per-unit megawatt per per-unit kilovolt."The added term will clearly be represented in our block diagram in Fig. 9-19

as rhc new input via block 6.

9-5-7 Factors Causing Changes in E'

Why will E' change? There are two reasons:

I. Manipulation (via AVR) of the field voltage "r2. Changes in relative rotor position b,.,. .

An increase in vf will clearly tend to ill crease E'. Not so clearly, an increaseill rot or angle tends to decrease E'. The latter statement is, however, easily con-firmed from Fig. 4-25. As 0 increases, the voltage X~ I d and thus I d will increase.But since Id is demagnetizing (Fig. 4-25b) the end result is a decrease in the emf.

The relationship between increments of t:.oN, tlvf, and t:.E' will thus read

t:.E' = K2tlvf - K3M,v (9-102)

The transfer blocks 8 and 9 plus the summing juncton 2::2 account for theeffects stated in Eq. (9-102). Equation (9-102) expresses the steady-state changesin E resulting from the changes in t:.vf and t:.oN.', ;

Dynamical/y, the change in E' can take place only after S» f and t:.o,~. haveovercome the field winding time constant, the value of which (under load) wasgiven by formula (9-20). The added delay transfer block 7 takes this field delayinto consideration.

9-5-8 Inclusion of the A V 1{ Loop

At this stage the increment t:.v f in Eq. (9-102) must be accounted for. Return-ing to the A VR block diagram in Fig. 9-4 we obtain AUf directly as the outputfrom the exciter. This results in the addition of block 10 plus the third summing'junction 2::3 to the growing diagram of Fig. 9-19. .

We finally are left with only t:.1 VI, the incremental change in the generatorterminal voltage. t:.1 V I consists (see Fig. 4-25) of the two components t:. V andtl Vd• The former increases as E' increases, the latter as {J increases. Thus we \avethe incremental relationship:

(9-103 )

(

2

3

4.

5.

6.

l)

. _,..".w_-" ....~~<- ----~

. 1THE ENERGY SYSTEM IN STEADY STATE-THE CONTROL PROBLEM 349 - .

In Fig. 9-19, we symbolize this equation with the summing junction 2:...andthe transfer blocks 11 and 12. The block diagram is now complete with alLsignalbranches, except .6.1 V Iref' fully accounted for.

£ )

9-5-9 Discussion of Results

The boldface portion of the diagram in Fig. 9-19 is readily understood. Underthe influence of the synchronizing power the ~enerator rotor when disturbedtends to perform undamped natural oscillations. much like the spring-masssystem of an auto when hitting a bump. The damping power emanating from thedamper winding serves as the system "shock absorber." Note that the synchro-nizing power is in negative phase with the swing amplitude .6.bN, whereas thedamping power is 90° out of phase.

What can we make out of the lower portion of the diagram in Fig. 9··19?Without getting involved _in analytical details we summarize these

observations:

1. The swing amplitude /:1tJN is fed back via a dual-path system, i.e., directly viablocks 6, 7, and 9 and indirectly via the AVR. This feedback can result in" parasitic" power components in phase with either the synchronizing poweror the damping power. The actual phase is determined by the frequency-dependent blocks 7 and 10.

2. DeMello and' Concordia analyzed the actual nature of these parasitic powercomponents in a classic papers and contributed quantitative data. Anderssonand Fouad" applied Routh's stability criterion to find conditions for stability.The resulting formulas are too complex tobe included here.

3. The parameters S', K!, ... , [(5 are partial-derivative expressions (compareEq. (9-100)) and are therefore depending UPOIl the loading. DeM~llo and Con-cordia made the following comment: "It is important-to recognize that ...the parameters change with loading. Since they change- in 'rather complexmanner, it is difficult to reach general conclusions based on parameter valuesfor one operating point only." I - \ .

4. In some systems the combination of long lines, fast AVR's, and heavy loadingcan result in a situation where the parasitic feedback power arinuls the. nat-ural damping thus making the system "'oscillatorily Jnstable" asdemonstrated in Fig. 9-17. I

5. It should be realized that Fig. 9-19 in its complexity models only a singlegenerator operating onto an infinite network. The results obtained from thisdiagram either by analysis or simulation are useful but may nbt be directlyapplicable to the conditions existing in an interconnected netw9rk.

A recent EPRI sponsored project? has resulted in a computer programfor simulation of intermachine oscillations in large systems. I '.

6. When .a system experiences self-excited oscillations addition of so-called

., j'W: EI:;ECTRIC ENERGY SYSTI!MS.THEORV: AN INTRODUCTION

"powt'!r system stabiliz ers " (PSS) will quench the oscillations.V" A PSS in-jects stabilizing "supp: ernentary " signals to the summing junction L3 inrig. 9-19. These signah typically are derived from the generator shaft speedand shaped in special networks.!? See also Prob. 9-14.

I9-6 AGe DESIGN L~;XNG KALMAN METHODS

Th~.rJder must realize by now that design of generator controllers as we knowthem tbday has followed "classical" lines. Root-locus methods, block-diagramrepresentation, intuition, trial and error, empiricism are but a few of the"classi~al" design tools'. The high degree of reliability and general excellentfunctioning of today's power systems is proof of the soundness of design. In viewof this fact is there really any room for improvements? ,

Tpb phenomenon of intermachine oscillations discussed in the previous sec-tion ddmonstrates that our present power systems are not immune to instabili-ties. Itlalso accentuates an inherent weakness of classical design-its inabilityfully tOI.anticipate and adequately to cope with all dynamic features in systems ofhigh dimensionality. '

A inodem gigawatt generator with its multistage reheat turbine, including itsALFCjand AVR controllers, is characterized byan impressive complexity. Whenall its nonnegligible dynamics are taken into account, including cross-coupling

I I 'between control channels, the overall dynamic model may be of 20th order.Th~s "dimensionality barrier" can be nicely overcome by means of

computer-aided optimal control design methods originated by Kalman.Uv'? Yu;Fosha, and Elgerd13

•14 ,made ear~y attempts to apply these methods to power

systems. A recent surveyl~ classifies the many contributions that have laterfollowed. "Optimal 'conho1" covers today a widening spectrum of computer-

, Ioriented techniques of which the so-called optimum linear regulator (OLR) designhas proven particularly useful.

Compactly stated, the OLR design results in a controller that minimizes bothtransient variable excursions and control efforts. In practical power systems terms

, tbis means optimally damped oscillations with minimum wear and tear of COR-

trol valves. OLR design proceeds as follows:

Stefl 1 Casting, the system dynamic model in state variable form and introductionof appropriate central forces ' , '

, Step 2 Choosing: ai:! }ntegi'al-squared-error control index, the minimization ofwhich is the control goal

Step 3 Hnding, the structure of the optimal controller that wiU minimize thechosen control index

" We clemonstr~te brie~y ~e use of QLR clesi~ by applying ',i,t to the siagle-. '~X'ea ;ALF-,C sy,stem depicted in Fig'. ~-lQ. We propose to fiAIil a eon;tF@i felice I\(

v,"ii~IIrl~t1

THE ENERGY SYSTEM IN STEADY STATE-THr: CONTROL I'ROIlI.EM 351

Figure 9-20 OLR control of primary ALFC loop.

(depicted in Fig. 9-20) which in an "optima}" manner steers or controls thesystem. We define "optimal" later; presently we develop an appropriate modelfor the system.

~-6-1 Putting the Dynamic Model in State Variable FormOLR design is based entirely upon the availability of a dynamic system modelin so-called state-variable form. We demonstrate how this form can be obtainedfrom the block diagram model in Fig. 9-20.

For.this system we have the transform equations:

LlPv(s) = _1_ (u(s) - _!_ Llf(S))1+ STH R

ILlPT(s) = -1 -- LlPv(s)+ sTT

KLlf(s) = -1 PT (LlPT(s) - LlPD(s))+ s P

Or, if expressed in the time domain, -

d 1tlPv + TH -d (IlPv) = u - - III

t R

dIlPT + TT dt (LlPT) = sr;

dLlf + Tp dt (Llf) = Kp IlPT - Kp IlP D

(9-104)

(9-105)

At this juncture we introduce the three state variables Xl, X2, and X3 forming thestate vector

[XI] [LlPvlx = ~: £. ~;T

We also define the disturbance force:

p £ tlPD

.-"fIf

.sng

:r;rr-.n

:h~sn-

.0f

ble

!e-.U

(9-106)

(9-107)