advanced load frequency control

ASW£ i fADVANCED LOAD FREqr

N ^ *'e-W11H, REQUM,~~~~ ~ ~ J. Duncan Glover

4§m<el J-7DectricPower Service CorporationNew York, New York

f<sto~--vA-t $t3fAbstract-The problem of designing a load frequency ntrol law

which reduces transient frequency oscillations (swings) and reduces thenumber of control signals sent to power houses is investigated. A linearmodel of an aren is developed, and a

discrete time, linear-plus-deadband, feedback control law is designed.Feedback variables include cumulative inadvertent interchange, fre-quency deviation, integral of frequency deviation, real power absorbedby loads, and governor-turbine variables. This linear-plus-deadband con-trol is an application of a special case of a more general "set-theoretic"class of control laws. A simulation of two areas with two hydro sourcesis presented. The dynamic response to a step load change is determinedfor the case of no load frequency control, load frequency controlpresently used by power companies, and load frequency control de-signed in this paper.

I. INTRODUCTION flJR¶ PA'il;Present load frequency control (LFC) laws adequately perform the

tasks they were designed to handle. However it is conceivable to use amore sophisticated LFC law (denoted advanced LFC) which can domore. This possibility is investigated here by developing such a controllaw and comparing its performance with LFC presently used (denotedconventional LFC).

Recently there have been other investigations [], [ 2] using mod-ern control theory techniques to develop new LFC laws. Fosha and Elgerd[1] use a state variable model and the state regulator problem of op-tional control theory to develop new feedback control laws. Cavini,Budge and Rasmussen [21 develop a control law which is related toFosha and Elgerd's, but also includes a state estimator.

It is desired that advanced LFC achieves the same objectives asconventional LFC; i.e. the control law should perform the followingfunctions:

1) Each area regulates its own load fluctuations (if possible).2) Each area contributes to the control of system frequency.3) In steady state, frequency and net tie-line power are returned

to schedule in all areas (if all areas can regulate their own load fluctua-tions).

In addition to the objectives of conventional LFC, it is desired thatadvanced LFC:

4) Reduces transient frequency oscillations (swings), without alairge increase in the magnitude and speed of control.

5) Reduces the number of LFC signals sent to power plants,without compromising other LFC objectives.

In order to design advanced LFC laws, a model of an area of aninterconnected power system is developed (Section II). The (state) vari-ables of the model include mechanical variables of sources, electricalfrequency and angle deviations of the sources, load variables, and tie-linevariables. The model is chosen to be linear and time-invariant, and alsocontains uncertainty. The uncertainty is assumed to be "unknown butbounded"; i.e. the uncertainty is contained in a specified bounded set,but is otherwise unknown.

Using the model, a discrete time, "linear-plus-deadband," feedbackcontrol law is designed (Section III). The linear portion of the controllaw is designed (Section 111.2) to reduce swings, and is related to Foshaand Elgerd's design. However, a different model is used here, and a dis-

Paper T72 085-4, recommenided anid approved by the Power System Engineer-ing Committee of the IEEE Power Engineering Society for presentation at the IEEEWinter Meeting, New York, N.Y., January 30-February 4, 1972. Manuscript sub-mitted September 16, 1971; made available for printing November 15, 1971.

HQUENCY CONTROL

Fred C. SchweppeMassachu tts Institute of Technology

Cam ridge, Massachusetts

ete time o-second control period) rather than continuous time LFCis designed here. Thus it is difficult to compare the performance of thetwo control laws.

The deadband portion of the control law is designed (Section 111.3)to reduce the number of LFC signals sent to power plants. The dead-band logic is related to the "Error Adaptive Control Computer" (EACC),which was developed by Ross [31 and is presently used in some LFClaws. The differences between EACC and deadband logic are:

1) EACC logic is based on a probabilistic analysis. Deadbandlogic is based on a magnitude bound analysis.

2) EACC analyzes the area control error (and its integral). Dead-band logic analyzes all of the state variables (or an estimate of them) ofthe area.

It is assumed here that all of the (state) variables of the model are

monitored (perfectly) and are available for control implementation. Ifnot then a dynamic state estimator could be used to estimate the statevariables. The state estimator would precede the LFC (as in Ref. 2).

Simulations of a two area power system with two hydro powersources are presented (Section IV). The dynamic response (frequency,tie-line power, and source mechanical power) to a step load change forthe cases of no LFC, conventional LFC, and advanced LFC is determined.

The linear-plus-deadband control is an application of a special caseof a more general class of control laws designed using set-theoretic tech-niques. Such set-theoretic controls are especially designed to handleproblems with unknown-but-bounded disturbances. Set theoretic con-

trol is discussed in Ref. 4.

II. MODEL

A model of an-area of an interconnected power system is presented.The area consists of a group of power sources, loads, transmission lineswithin the area, and tie-lines which interconnect the area to the system.First each of the components (sources, loads, lines, ties) is modelledseparately; then the component models are connected to form the area

model.

GOVERNOR TURBINE

Fig. 1. Mechanical Dynamics - Hydro Source

1. Power Source

The dynamics of a power source consist of three parts: mechanical,electrical, and electromechanical coupling.

The mechanical dynamics are represented by a linear model. Theinputs of the model are the LFC of the source, u(t), and the change

2095

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(from 60 hertz) in electrical frequency of the source, Afs(t). The outputis the change (from nominal value Pmnom) in mechanical power perunit delivered by the source, APms(t). A model of the mechanical dy-namics of a hydro source [ 5 ] is shown in Figure 1 and is used in simula-tions presented here (Section IV).

The electrical dynamics of the source are modelled by the follow-ing voltage source;

V (t) = ,TCOS eS (t) 1

Qs(t) = WOt 4. bs t)WO = 2 n 60

w (t) = 2 nr f$ =9 (t) = ot + s

AW tJ(t= 2 rr &f (t) = Lo (t) - w° 5-d S(

= dt (t)A s (t) = (t) - bsnom dt s

It is assumed that the three-phase power system is balanced (balancedsources, loads, lines), and only equivalent one-phase models are pre-sented. The magnitude of the sinusoidal voltage source is assumed to beconstant (RMS value V). The phase angle 6s(t) is assumed to be slowlytime varying compared to wot (65s(t) is a narrow band process modulat-ing the phase of the source voltage). A6s(t) is the change (from nominalvalue bsnom) in phase angle of the source. A source reactance is alsomodelled, but is assumed to be "lumped" into the transmission linesystem.

The electromechanical coupling is modelled by Newton's SecondLaw (swing equation).

3. Load

A load is modelled here by representing its voltage and power char-acteristics. It is assumed that the voltage across each load is:

VL (t) = f2 V CoSQL (t)0 (t) = Wot 4- 6L (t)

A6w 1 (t) = WL (t) -w = -t L (t) dt &L(t)dt dt

(3)

The load voltage is assumed to be sinusoidal with constant magni-tude (RMS value V). The phase angle 6L(t) is assumed to be slowly timevarying.

The power absorbed by each load is modelled as:

(4)fAPL (t) = 0

where APL(t) is the change (from nominal value) in electrical powerper unit absorbed by the load. Only step load changes (i.e. Eq. 4) areconsidered here. Also the dependence of load power on frequency (andvoltage) is neglected.

Kirchoff's laws and a knowledge of the transmission line configu-ration of the area can be used to determine the source electrical poweras a function of source phase angles and load powers. Assuming smallangle differences, a linearized relation is:

dtAWt) = _ A (t) + APms tt) - AP(.t )M

(2)

APS(t) is the change (from nominal value) in electrical power perunit delivered by the source. M is the normalized moment of inertia ofthe source, and D is the normalized damping constant.

2. Transmission Line

A transmission line is modelled as a constant reactance. Real andreactive losses on the line are neglected.

SP(t) = ] (t)LARL (t)J

D itsS(t)1 (5)

EL(t)]

APS(t) (an Ns vector) is the vector of (changes in) electrical powers(per unit) delivered by the Ns sources in the area. A6s(t) (Ns vector) isthe vector of (changes in) source angles. APL(t) (NL vector) is the vectorof (changes in) electrical powers (per unit) absorbed by the NL loads inthe area. D = [5s PL] is a constant (Ns by Ns + NL) matrix which isdetermined from the system configuration (of sources, loads, and lines),and Kirchoff's laws.

Fig. 2. Block Diagram of an Area of an Interconnected Power System

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4. Isolated Area Model

A complete model of an isolated area, consisting of Ns powersources, NL loads and NT transmission lines, is obtained by marryingthe component models previously discussed. The result is a linear systemwhich is shown in Figure 2 (disregard tie-line dynamics and "w" terms).It is assumed that the area is not connected to other areas (no tie-lines).The inputs to the area model consist of the vector of electrical loadchanges APL(t), and the vector of LFC signals, u(t).

5. Interconnected Area Model

If the area is connected to a power system via tie-lines, then thearea is affected by electrical power flows on the tie-lines. The powerflow on a transmission line (tie-line or otherwise) is a function of thedifference in phase angles at the two nodes to which the line is con-nected. These two phase angles can be written as a function of loadpowers and source phase angles of the system.

It is assumed, however, that models of only those sources and loadswithin the area are available. For this reason, only an approximate modelof a tie-line power flow can be obtained. It is assumed, then, that a tie-line power flow is a function only of load powers within the area, andpossibly some loads outside the area but electrically "near" the tie-line.

A YXTl (t)ldt LXT ( t) AT

L, (t) = t2(t

X (t5}

XT2 (t)J+ (t) (6)

+ T. "LL(t) (7)

(9)

Eq. 9 defines an ellipsoid 2w, which contains the vector w(t). The posi-tive definite matrix W determines the "size" of the ellipsoid, and is de-termined from the bounds on the uncertain variables. The ellipsoid setis chosen for mathematical convenience.

A model of an area of an interconnected power system, with un-certainty added, is shown in Figure 2. It is a block diagram of a linear,time-invariant model. In equation form, the model can be representedvia:

(10)adt x (t) = A x (t) + 2 O (t) + 5 (t)

x(t) is the "state" vector of the system, which consists of the mechanicalvariables of the sources, frequency and angle deviations of the sources,load variables, and tie-line variables. u(t) is the vector of LFC signals.

Iw(t) is the vector of uncertainties, which is constrained via Eq. 9. Thematrices A, B, and G are determined from Figure 5 and previous discus-sions.

A discrete time version of Eq. 10 is:

( 1)

where the state x(je) is described at discrete times je, j = 0, 1, 2,. . at in-tervals of e seconds. The matrices ox, 13, and -y are given by:

Eqs. 6 and 7 represent a second order linear system. The input of thesystem is APL(t), and the output is APT(t), the net change (from nomi-nally scheduled value) in tie-line power flow out of the area. The con-stant matrices A B H and HT2 are determined from a knowledgeof the system configuration and tie-line oscillations (period and timeconstant).

A model of an area of an interconnected power system, includingtie-lines, is shown in Figure 2 (disregard "w" terms).

6. Uncertainty

Many assumptions and approximations are made to obtain thelinear model of Figure 2. In order to account for modelling errorscaused by these assumptions and approximations, the following varia-bles are augmented:

t P-L (W-, API, (t) + RitL (t)(E(t) A ES (t) + MS (t)8

T t ST (t)E MT (t) (8)-u (t) (t) u(t)

0e-A(r -r)- JyA(e %dr G

= 7eA(d.1. G0_(12)

The bound on the discrete-time uncertainty vector, yzje), is repre-sented as in Eq. 9, with "t" replaced by "je".

7. Discussion

Figure 2 contains Ns sources. A source could represent one gene-rator, one plant, or a group of "closely coupled" plants. Also the NTtransmission lines and NL loads could represent all the lines and loads inthe area, or a reduced set (e.g. just the EHV System, with the lowervoltage system "lumped" into the loads). Thus the model is flexible,allowing for as much (little) complexity as desired.

The variables of the model are written as changes from nominalvalues. The nominal values can be determined from a "load flow"equation.

The purpose of the model is to develop advanced LFC laws. Firstthe model is used to design control laws. Then it is used in simulationsto analyze control law performance.

III. CONTROLA vector of "uncertain" variables, wL(t), is added to the vector of

load powers. wL(t) can be viewed as a small perturbation of APL(t) in 1. Descriptionorder to account for the uncertain nature (inexact modelling) of theloads. Similarly "uncertain" vectors ws(t) and wT(t) are added to the This section discusses advanced LFC. Only the basic LFC signals,source and tie-line power vectors. An "uncertain" vector yu(t) is added which are assumed to be dispatched every two seconds, are considered.to the vector of LFC signals in order to account for the inexact model- Economic dispatch and adjustments of frequency schedule and net tie-ling of the mechanical dynamics of the sources. line power flow schedule (to control system time deviation and cumula-

It is assumed that the uncertain variables are bounded or con- tive inadvertent interchange) are not considered. These can be viewed as

strained in magnitude by the following equation: refinements of the basic LFC which operate more slowly (minutes versus

2097

x (t) - x -1JalW- r(-) .. 13I I

I

(t) (t) )., (t)-WS 311.i 11.w (t) = [ MI, (t)

X( i e + e ) =-ccx( i 6) + Au( i C-) + VW( j G)V .1


seconds).The following control law is used to achieve the control objectives.

ii(t)=~ ~ ~ E if x (je) is not contained in 9231(t) = s(je--i) otherwise

je Lt - (j+1) e (13)

e= 2.0 seconds

Eq. 13 is a discrete time, "linear-plus-deadband," feedback control law.Every two seconds LFC is either updated by an amount proportional tothe change in state variables or left unchanged. K is a feedback gain ma-trix which linearly relates the state vector to the LFC vector. Q2 is a"deadband set" which specifies a region in state space. The LFC is leftunchanged if the state vector is contained in O.

2. Feedback Gain Matrix

The feedback gain matrix is based on the discrete-time model, Eq.11, and is determined from the state regulator problem of optimal con-trol theory [6].

A quadratic cost function, J, is specified.

= z ', (j) Q X (ji)+ u'(ja)Ru(Je)J-O

(14)

Q is a positive semi-definite, symmetric matrix which is chosen to pena-lize the (squared) magnitude of source frequency deviations. R is a posi-tive definite, symmetric matrix which is chosen to penalize the (squared)magnitude of control signals.

The control law which minimizes the cost J, subject to the model,Eq. ll,is

1U (QE) = Lz (ii ) (15)

where K is the steady state solution to a discrete time Ricatti equation[6]. The uncertainty vector, w(je), is neglected in the design of K.

After K is determined from Eq. 16, it is modified to include a feed-back of cumulative inadvertent interchange, in order to regulate inadvert-ent interchange (so that the area regulates its own load fluctuations).

3. Deadband Set

The deadband set Q2 is based on the discrete time model, Eq. 11,and some set theory. Since the model is driven by the uncertaintyvector, wvOe), a steady state is never achieved. However, a "steady stateregion" can be defined. For a given load change, APL, the state will,after a transient period, settle into a steady state set (if the area and itsmodel are stable), which is denoted by Qss

The model, Eq. 11, is rewritten.

l(je+e) = M1;X1(j ) *-&1J- L +*2 ia(j)4+iE )(16)

The load vector, APL, is written as an input in Eq. 16 (rather thanbeing included in the state). xl(je) is a reduced state vector (loads notincluded) and the matrices cl 1, 131,12 and 71 are determined from Eq. 1.

Assume linear LFC is used, i.e.

(17)u.(jc)= iX(jE) = 1ZI(j ) + K2 AFLL

where K is determined from Eq. 15. Using linear control, the model, Eq.16 becomes

(18)xV(jj -+E) olxl( je)+ -l'L + -Wl j )

11 - oQ - "9I IA,

J31 = J1 +- .82K2

(19)

(20)

Assuming Eq. 18 defines a stable system (eigenvalues of oil lessthan one in magnitude) Qss is given by the solution to the following"set equation" [4].

A

ss-~SS=C f2 ss 0SO,1 AL 0 0 31 <l (21)

where a&l Qss denotes the set Qss translated by &1.I denotes vector sum;

I 31 APL I denotes the set consisting of the poillt (vector) 1 1 APL; and

11l2W denotes the set Qw translated by yl.In order to reduce the control action a reasonable approach is to

stop updating LFC when the state is contained in Qss The LFC will not"track" the noise w(je) in this case. Thus Eq. 13 is used for LFC with

a= 9ss,

Unfortunately Qss given by Eq. 21, is not an ellipsoid, even thoughQw is. For computational simplicity, an ellipsoid Qss3 which is an innerbound to Q2ss' is determined.

ni ss -{i:[z1 - sSA LS jjzi-'.s > }= Eq. -kI[I =13-12 M 1 A

rSS= +Ct rS5;l 1

(22)

A A

2ss is an ellipsoid with center css and matrix rss 2ss is contained in

Ss,The linear-plus-deadband control law is

z;(j) =t( i) if [Z1 C)_( 0I

Di)-csSst[v)si>(i otherwise (23)

The deadband region can be tuned to a particular area by adjustingPss. Each adjustment provides a new degree of freedom, and a changein performance. The use of the ellipsoid set, Qss results in a more con-servative control law (LFC signals are dispatched more often). It isassumed that the noise vector, w(je), is magnitude bounded for eachtime interval (Eq. 9, with "t" replaced by "je", describes a discrete timewhite noise process). Many other types of noise (periodic, exponential,frequency banded) can be handled by including noise dynamics in themodel.

4. Role of Set Theory

Eq. 21 is a set equation involving mathematical concepts whichare probably not familiar to most readers, and the derivation of Eq. 22is a nontrivial exercise. Explanations are not given here because theywould greatly expand the paper and are not essential to the basic con-cept of the linear-plus-deadband control of Eq. 23.

Even though the details underlying Eq. 21 and Eq. 23 are omitted,the application of control design using set-theoretic techniques is a majorcontribution of this paper. Details are presented in Ref. 4.

IV. SIMULATION

1. System Description

A power system consisting of two interconnected areas with twosources is investigated (Figure 3). There are four nodes (five including

2098


"area control error", and is dispatched every two seconds. The "frequencybias coefficient" is chosen to be equal to the "area frequency response

characteristic" of the Northwest U.S., and an optimum gain is chosento obtain "best" conventional LFC performance.

YA

SiS

iYD - YB

{PS2(t) ' PL2(t)

Fig. 3. Simulation Configuration - Two Areas, Two Sources

ground). Power sources are connected to two nodes, and loads are con-

nected to two nodes. The nodes are connected via four transmissionlines.

That portion above (below) the dotted line in Figure 3 is viewed as

area one (two). The two areas are interconnected via two tie-lines.

Each power source is considered to be a hyc'tro source, with me-

chanical dynamics as in Figure 1. The loads, transmission lines, andsource electrical dynamics are modelled as in Section IL Constants are

listed in Table 1. A "complete" model of the two sources, two loads,and four lines is used for simulations. Only an area model (plus a modelof the load outside the area) is used to design the area LFC.

The following control laws are simulated.1) No LFC

u U 0 (24)2

2) Conventional LFCConventional LFC in each area is proportional to the integral of the

TABLE 11. Discrete Time Model of Area One& =

.962 1.11 .

-005 - 0 09 4

-*05 -, 53 -

81. 7 565. E8-.2 614. i

0 00 iI 0

0,0 0

A. =

.-qq 1..99

.4c)

.13-,1583.211.90000

.0005.019

.005 .030

.0001-030-.64 -59.-1.01-58..o46 -.12-.35 -1.7

0 1.00 0

-.001-.001-.003.9482.06

0000

0 .00050 .0001o, ooo40 -9-071 -. 65h0 -ig460 - .40 00 0

.019

.030

.0 0-52..104t.8.

0018.243

01.0

0-1 I.627 55.6 -147. .0 0 0

A(jf-+ F-) a-x(je) +.A-U(ifi)X xl x2 x3 x4 x5 x6 x7 x8 X91X1,X2,x3 governor turbine variables of

area oneAPMsi.

= IpMsl - PMslnoml =-71X1+4-3x2-4.25x3x4 SiX5 slx6,x7 - variables of tie-line power model

APT =,3.5 x6 -.18 x8 -t- .4 5 xg

x8 APLlX9 '4PL2

3) Advanced LFC (Eqs. 13, 23)The, gain matrix, Ki, is designed as in Section 111.2. A discrete time

model of Atea one, including the tie-line power flow and load in Areatwo, is given in Table 2, along with a physical interpretation of thestate variables.

TABLE III. Cost Function and Feedback Matrices

TABLE 1. Simulation Constants

Governor Turbine Constants

Al=-A2= 1.20 - servo motor gainKHI''=Kli2-- 1.50 - lever gainTR1=TR2= 7.00 seconds - dashpot timeconstantD'Rl=DR2= .430 dashpot compensationDP1=DP2= .05 permanent speed droopPB1=PB2= 826 megawatts- maximum powerT'Wl=TW2= .860 seconds- turbine time constant

Swiyig Equation Constants

Ml=M2= .0177 - normalized moment of inertiaDl=DCO-= .000177 normali-zed damping constant

Transmission Line Admittances

YA=YC= 7.65 per unit (80 miles)YB=YD=3.82 per unit (160 miles)

.Base .u-antitiesSB77: 700 megawatts - system power baseV = 500 kilovolts - system voltage base

2 (line-line,RMS)y = SBAV ) z-.0028 rnhos

Machine constarits ai-e based ori measurementstaken at the Dalles Dam ( Columbia river)and on erigineering judgment. Constants arechosen to represent ten 70-megawattgeDerators. Coristants labelled .1, (2) refer.to area 1 (2)"O q55 = .10.1 qii = 0 otherwise

k;

El- =[-.068 --.310 -.090 -.00004- 0 0.065 o4o -.ool463

Icumulative inadvertantint.erchange feedback gain

The. cost function matrices g and R, and Ki (modified) are listedin Table 3. Q and R were adjusted in a "trial and error" technique byobserving the system response. It was desired to minimize swings withouta large incre'ase in the magnitude of control (the mechanical responseof the, sources should stay in linear region).

2099

Q, = lqj.jl;q44' =1.0

6,R .= 7xlO


.962-.05-.0581.783.2

00

1.11-.01-. 53565.614.00

.398

.125-.1583.211.900

.001 0_.001 0-.003 0.948 02.06 10 00 0

TABLE IV Deadband Terms

o10- 0]

-6~F 71 41 10 8

1.41 0 l.xlOo 0 6.7x10-29

CSS 0ap

Ll F(di agr) l.8x10-487-487- -s S 0.50 .6j-28.6 28.6ILP l.xlO0

TABLE V. Discrete Time Model of Area One

0 -.000610 - .0009O -.0009.008 1.92.oo8 1.93.9998 1.967-.0001 .967

1.9911.99g2a =.627

=59.61-147.

10

-xI.( e i- E ) = otl xl(i) +.gsul(Je);LlJ+II \(iF) 6L2F

xll,x12,x13 - governor turbine vari ab:

xi4 =AUt1 = 21Tf1&-frequency deviation

x15 =A s1 angle deviati-on

xl = (AP +24o.xl +. L2,45P6 = - T 7 18L1 -. 5PL2

x17 = OlPTcAt - cumulative inadverl; nt.rhnnA

.030

.049

.0o49-95.5-95.7-.356-. A15

.0171

.027

.026-51.9-52.41.898.89a

les

/120tant

sources one and two is -.0640 radians.At time zero a step load change of .05 per unit power occurs in

area one (APLl(t) = .05, t > 0). It is assumed that the discrete timecontrol laws are zero for the first two seconds after the disturbance.

The model (Eq. 10) is simulated on an IBM 360/65 digital compu-ter, using the "Continuous System Modelling Program" (CSMP) sub-system [9]. A random number generator is used to model the load pertur-bations, wLl(je) and wL2(je). Two cases are considered, IwLi(je) <10-4 and IwLiOe) < 10-3. Plots of frequency deviation, the envelopeof tie-line power flow, and mechanical power are shown in Figures 4-8.

1.04 PMI.=PM.

102 /

1.01i.1 2.0 40 6.0 8-01.00o

0.99 T- SECENVELOPEAPTIE I

20O 40 60 so.00 I-.Ol T-SEC

-.-C3

.6 -AFI-CPS

.4-

20 40 60 80

I I I^ - 1

V-2

-.4

-6

The deadband logic of Section 111.3 is used to determine css and'ss (Table 4). A different model of the tie-line power flow, which in-cludes the cumulative inadvertent interchange, is used, resulting in anarea model shown in Table 5. Uncertainty is included only in the loadsfor this design and simulation. Also, rather than use the full deadbandmatrix, Pss only the diagonal elements of the matrix are used. Thisreduces the computational effort (the matrix inversion, I7l in Eq. 23,becomes trivial). The effect of this approximation is not analyzed.

4) Continuous Time Advanced LFC

(25)(t)] = K (t)

u2 (t)

The feedback gain matrix is designed via the continuous time stateregulator problem [8].

It is assumed that before time zero the system is in steady state.Each source is generating 1.0 per unit power. The load in area one (two)is absorbing .864 (1.136) per unit power. The net tie-line power flowingfrom area one into area two is on schedule at .136 per unit. Also thefrequency is sixty hertz (in both areas), and the angle difference between

.4.4

V T-SEC

Fig. 4. Dynamic Response -No LFC

I\FCPS

02.0A _~ 80 100 120 140

-. 40 -60-'V *4 T-SEC

Fig. 5. Dynamic Response - Conventional LFC

2. Discussion

For no LFC (Figure 4), frequency deviation in area one oscillatesand, in steady state, is -.063 hertz. The frequency deviation in area

2100

LI

I I I 1 U Z.L- IL;Lltt 1 Ir, IZ


two is approximately the same as in area one. In steady state, the net tie-line power flow is off schedule by -.025 per unit. The mechanical pow-ers follow frequency and, in steady state, each source picks up one-half of the unscheduled load change.

For conventional LFC (Figure 5) the frequency oscillations arecentered about sixty hertz and in steady state, frequency is returned toschedule. The oscillations are not reduced, compared to conventionalLFC. In steady state, the net tie-line power flow is returned to schedule,and source one picks up all of the unscheduled load chalge.

1.05

1.04

1.03

1.02

1.01

- .01

- .02

- .03

schedule. Frequency oscillations are reduced by about ten to thirty per-cent after the first swing.

After four and one half minutes, the state enters the deadbandregions, and control action is reduced. As shown in Figure 7, controlaction is eliminated for wliw(je) < 10-4 and reduced by about fiftypercent for IwLi(je) < 10-3.

1.05

1.04

1.03

1.02

1.01

\ PM2- *

PM1 x

20 40 6o 8so

T-SEC

ENVELOPE&PTIEI

20 40 60 80

T-SEC

PM2-*

PM1 -x

20 -xf

40 60

1.99 - T- SEC

ENVELOPE&PTIEI

20 40 60 80.00 i

-.01 T-SEC

-.02-

- .03

- . 6 FI CPS

.4-

o {\ ,:>> 40 60 sov

-.2

-.4

-.6T-SEC

-20

T-SEC

-.81

Fig. 8. Dynamic Response - Advanced LFC, Continuous Time-.81~

Fig. 6. Dynamic Response - Advanced LFC

For advanced LFC (Figure 6), the state is outside the deadbandregion during the first four and one half minutes, and linear control isapplied. Control action in response to the disturbance is not delayed. Insteady state, both frequency and net tie-line power flow are returned to

For continuous time advance LFC (Figure 8), frequency oscilla-tions are reduced by about forty percent during the first swing and anorder of magnitude thereafter. Also the period of the frequency oscilla-tions is about one-half that of conventional LFC, and the sources respondabout twice as fast as conventional LFC. Further reduction in frequencyoscillations could be obtained (with a "larger" feedback gain matrix),but the sources would respond at a correspondingly faster rate.

Log 10 Icli -cSSO2 rss

* - Iwi (je)l =

X Iwi Cje)l=I

Fig. 7. Deadband Logic

V. CONCLUSIONS

Based on the simulations presented in Section IV, the author makesthe following conclusions.

1) A major reduction in swings can not be obtained with ad-vanced LFC signals which are sent with a period of two (or more) sec-

onds. Swings are reduced (by ten to thirty per cent after the first swing),but not substantially.

10-4 2) A major reduction in swings can be obtained by decreasingthe advanced LFC period. This reduction is limited by the rate at which

T- SEC power sources can change their generation levels.3) Advanced LFC is a promising approach to reducing the num-

-3*50 ber of LFC signals which are sent to power sources. The control law isflexible, versatile, and easily implementable, provided the appropriatevariables are monitored and are available for control use. Also, control

M action in response to large disturbances is not delayed by the deadbandlogic.

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41

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Advanced LFC laws are based on a dynamic model of an area of aninterconnected power system. Consequently, advanced LFC performanceis critically dependent on the validity and accuracy of the model. Notuntil adequate models of power system dynamics and an on-line dy-namic state estimator are available, will advanced LFC be feasible.

The linear-plus-deadband control presented here is only a specialcase of a more general theory. In Ref. 4 different but related designsthat could be applied to LFC are discussed. The incorporation of adynamic state estimator based on unknown but bounded uncertainty[10] is also discussed. The general concept of set theoretic control canalso be applied to other power system problems, such as voltage regula-tion, load shedding, etc.

ACKNOWLEDGEMENT

This work was supported by the Bonneville Power Administration,Portland, Oregon. The authors wish to thank Messrs. Arden Benson,Arthur Brooks, and William Hauf for their criticism and support.

REFERENCES

[1 C. E. Fosha, Jr. and 0. I. Elgerd, "The Megawatt-Frequency Con-trol Problem: A New Approach via Optimal Control Theory", IEEETrans. Pas 1, Vol. PAS 89, pp. 563-578, April, 1970.

[2] R. K. Cavin, III, M. C. Budge, Jr. and P. Rasmussen, "An OptimalLinear Systems Approach to Load-Frequency Control" IEEE Trans.Winter Power Meeting, January, 1971.

[3] C. W. Ross, "Error Adaptive Control Computer for InterconnectedPower Systems", IEEE Trans. Pas, Vol. PAS 85, No. 7, pp. 742-749,July, 1966.

[4] J. D. Glover and F. C. Schweppe, "Control of Linear DynamicSystems with Set Constrained Disturbances", IEEE Trans. AC, Vol.AC-16, No. 5, October, 1971.

[5] D. G. Ramey and J. W. Skoogland, "Detailed Hydro-GovernorRepresentation for System Stability Studies", IEEE Trans. Pas,Vol. PAS 89, pp. 106-112, January, 1970.

[6] R. E. Larson, "Survey of Dynamic Programming ComputationalProcedures", IEEE Trans. AC, Vol. AC-12, No. 6, pp. 767-774,December, 1967.

[7] A. R. Benson, Control of Generation in the U. S. Columbia RiverSystem, Bonneville Power Administration, Portland, Oregon, 1966.

[8] M. Athans and P. L. Falb, Optimal Control, McGraw Hill BookCompany, New York, 1966.

[91 IBM, Subsystem/360 Continuous System Modelling Program (360A-CX-I6X) User's Manual, IBM Rept. H20-0367-2, New York, 1966.

[10] F. C. Schweppe, "Recursive State Estimation: Unknown ButBounded Errors and System Inputs", IEEE Trans. AC, Vol. AC-1 3,pp. 22-28, February, 1968.

[11] N. Cohn, Control of Generation and Power on InterconnectedSystems, John Wiley and Sons, Inc., New York, 1966.

Discussion

J. A. Hetrick (Ohio Edison Company, Akron, Ohio 44308): The authorsare to be commended on their efforts to develop a more sophisticatedcontrol law for reducing power system transient frequency oscillations.The addition of a deadband control plus the investigation of "discretetime" load frequency control are contributions to previous studies onthis subject. Unfortunately, anyone responsible for the operation of apower system must look into the future and ask: "Is this concept appli-cable to the real world?" "Will it achieve better control?'" and "Howmuch will it cost?" Of course, the answer to these and similar questionsmust await the arrival of an actual, field tested, control system embody-ing the control theory presented in this article. The authors do, fortu-nately, indicate an awareness of the problems associated with adaptingsuch a control scheme to the real world.

Referring to the work reported in this paper, the discusser wouldlike to comment on the term "area" as defined by the authors. Mostpresent day control areas are defined by corporate system boundariesor by power pool agreements. For this reason, little thought is given to

Manuscript received February 9, 1972.

purposely selecting a control area on the basis of the electrical parame-ters of the system. Separate control areas which are very closely "cou-pled" by strong interconnections are becoming more frequent as individ-ual companies undertake joint ventures in constructing new generat-ing facilities. In the control model presented by the authors, is the'4area" considered to be comprised of closely coupled generation andloads and is the "area" assumed to be controlled by a single area con-troller? How would the author's model be affected if the separate con-trol areas were closely coupled?

Consistent with preceding discussion, it would appear that as inter-connections between neighboring utilities are strengthened, the conceptof a control area defined by company boundaries would invalidate theassumption that the power transferred to an area is consumed within thearea. Have the authors investigated the effect of the approximate modelfor APT on the tie-line power flow and transient frequency oscillations?

With regard to the conclusions of this paper, the discusser has twoquestions:

1. Would not a reduction of advanced LFC period increase thenumber of control signals sent to the power house?

2. Do the results expressed in this paper (Items 1 and 2) applyonly to a system with hydro generation? Have the authorsconducted any studies with non-hydro sources?

The authors are again to be commended for their work and encour-aged to pursue further the application of modern control theory to thecontrol and operation of a power system.

Charles W. Ross (Leeds & Northrup Company, North Wales, Pa. 19454):This discussion deals primarily with the itemized difference betweenthe EACC and deadband logic given in the Introduction of the paper.

The function of the EACC is to analyze disturbances (determi-nistic as well as probabilistic) in the system and to modify the con-troller and economic considerations based on the urgency with whichcontrol action is- required. The net effect of the disturbances in the sys-tem, including the dynamic response of the system to the disturbances,is reflected in the area control error ACE. The EACC takes advantage ofthe fact that even random (white noise) disturbances are colored (corre-lated) by the system. The effectiveness of the EACC results from utiliza-tion of the joint probability characteristics of the observed disturbances,i.e., analyze the steady -state value, the expected magnitudes, the ex-pected times above the expected magnitudes, and the direction of thechange in ACE. This is basicatly a pattern recognition approach which isapplicable to .deterministic as well as probabilistic disturbances.

The principle of joint probabilities is believed to give the fastestknown method for analyzing disturbances and assigning a probabilityweighting to their importance from the control point of view. TheEACC may also be made adaptive to deal with the various classes ofdisturbances which may be encountered in an electric power system.This is accomplished simply by retuning the EACC from fast respond-ing disturbance state detectors as discussed in Reference (1).

Perhaps the basic difference between EACC control (Reference(1)) and the "advanced control" with deadband discussed in this paperis that-the EACC deals basically with "load disturbance states"-the"advanced control" deals primarily with process states.

If pure deadbands are assigned to prevent control of random proc-ess noise the steady state error may be substantial. If deadbands areassigned based on uncertainties in the model for estimates states, it isdoubtful that these states would be active during control, i.e., inside ofthe generation rate limiting band for the dispatch units. Perhaps theauthors could comment on the effects of these deadbands and genera-tion rate limiting as well as how their optimum control law deals withthese nonlinearities.

It is not clear how the authors deal with the problem of steadystate errors. Does the control scheme have reset action? If so, whichstate or states are reset? In conventional control only the area controlerror is driven to zero. If the authors are counting on the inadvertentinterchange in the equation for K1 (Table III) for reset action-this willresult in unilateral interchange correction. Finally, can the reset and deadzones in the control scheme lead to reset-dead zone cycling?

As the authors discuss, the process states are ill-defined and variableand the degree of controllability is limited by equipment constraints.Therefore, the question is raised-why attempt to use linear-optimalcontrol laws over the limited and uncertain range?

REFERENCE

[1] C. W. Ross and T. A. Green, "Dynamic Performance Evaluation ofComputer-Controlled Electric Power System," presented at IEEESummer Power Meeting, Portland, Oregon, July, 1971. (71TP593)

Manuscript received February 17, 1972.

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J. Duncan Glover and Fred C. Schweppe: We thank Messrs. Ross andHetrick for their valuable discussions. Dr. Ross's comments are ad-dressed first.

The function of the deadband portion of advanced LFC is thesame as that of EACC; i.e. to reduce control action without compro-mising other LFC objectives. It is uneconomical to continually changegeneration levels, and it may be futile to attempt to track rapidly vary-ing loads and random disturbances.

EACC [3] deals basically with the area control error (a weightedsum of frequency and net tie-line power fl6w deviations), which Dr. Rossrefers to as "load disturbance states." Deadband logic deals with allavailable information, including frequency deviation, net tie-line powerflow deviation, plant variables, and load variables.

The basic concept of deadband logic is to determine a (multidimen-sional) region. If the control information (above variables) lies withinthe region, control action is eliminated. The construction of the regionis based on a knowledge of the dynamics of the area, which includesgeneration response (rates, delays, limits), load, tie-line, and noise (ran-domness) characteristics.

EACC uses a probabilistic approach towards the random disturb-ances. Deadband logic uses a magnitude bound approach; i.e. insteadof considering averages or expected values, deadband logic considersbounds. Both approaches can take advantage of time correlation (color-ing) by using "noise dynamics" (e.g. exponential filters; periodic filters,etc.).

Since both cumulative inadvertent interchange and time deviationare considered in the deadband logic (they are two dimensions of themultidimensional region), steady state errors will not occur.

In conventional LFC, the area control error is reset. Similarly inthe linear portion of advanced LFC, frequency and net tie-line errorsare reset. Proper adjustment of the reset gains results in mutual, ratherthan unilateral, interchange correction. Reset-deadband zones could leadto reset-deadband cycling. However, the deadband region (and thefeedback gains) can be tuned or adjusted to prevent cycling. Tuning canalso be used to adjust for nonlinearities (e.g. source limits and dead-bands). Also advanced LFC could be made adaptive (this is not donehere).

The reason for considering advanced LFC, which uses more infor-mation and a knowledge of the dynanmics of the area, is simply to utilizea better control law which improves system performance and efficiency(see Introduction).

Mr. Hetrick suggests the possibility of choosing a control area onthe basis of the electrical parameters of the system. We feel that LFCshould occur within the areas defined by corporate boundaries (or powerpools) where it can be coordinated with other controls (e.g., economicdispatch) and planning decisions (which affect future LFC and the integ-rity of the area).

Manuscript received April 24, 1972.

The area model presented (Figure 2) represents Ns sources and NLloads, which may (or may not) be closely coupled. The degree of cou-pling depends on the specific characteristics of the area (transmissionline strength, size and dynamics of sources and loads) and is specified inthe area model by the values of the D matrix, and the source and loadparameters (e.g. inertia constants, time constants). The area is assumedto be controlled by a single area controller. However, the area is affectedby other areas (neighbors) of the system through its tie-lines (Figure 2).The degree of coupling of the area to its neighbors is represented in themodel via (part of) the D matrix.

It is not assumed that power transferred to an area is consumedwithin that area. On the other hand, LFC is (partly) designed to accom-plish this. LFC causes each area to regulate its own load changes bymaintaining net tie-line power flow on schedule (assuming that enoughsources in each area are on automatic control and can respond fastenough to meet area load fluctuations). Scheduled changes in powertransferred from one area. to another are accomplished via (mutual)changing of tie-line schedules.

However, during transient periods, all areas in the system respondto disturbances. Following a disturbance (e.g. a load change) all areasfirst act to reduce frequency changes (generator shaft accelerations) viasource governing action. Second, LFC causes all areas to aid in restoringsystem frequency to schedule. Third, as system frequency returns toschedule, LFC causes the area in which the disturbance occurred tocorrect its own load change (other areas revert to pre-disturbance condi-tions). As interconnection strength between neighboring utilities is streng-thened, all areas contribute more to system frequency regulation.

The approximate model of tie-line power flow is used only to designadvanced LFC laws. For example, the two areas simulated here (Figure3) are closely coupled (each area consists of one source and one load).The LFC law in each area is based on a model of the area source, load,and approximate tie-line model. The performance of the LFC laws(Figs. 4-8), however, is determined from "exact" simulations of bothareas (no approximate tie-line model used).

A reduction of the advanced LFC period will increase the numberof signals sent to power houses. Although the deadband logic is designedto reduce control action, during a sustained load change more signalsare sent with a smaller period. We are opposed to continuous time LFCfor this (and other) reasons.

The ideas of the paper (modelling and control) apply to a generalpower system. The conclusions are based only on simulations of ahydro system. We feel, however, that similar results would be obtainedfor typical existing systems. In a mixed-type system (e.g. AEP) oftenlarge fossil fuel units are base loaded and pumped storage hydro unitsare used for a major portion of regulation duties. Also the longer delays(time constants) inherent in larger fossil fuel units reduces their regulat-ing effectiveness (especially with regard to "fringe" control). Thus thedeadband logic is a promising approach to reducing control action andincreasing the efficiency of (hydro, fossil fuel, nuclear, etc.) powerplants and the system.

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advanced load frequency control

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