LFC using SMES and EDLC

8/12/2019 LFC using SMES and EDLC

1/66

Adaptive control for improved LFC in an interconnected power system incorporating energy storage devi

Department Of Electrical Engineering Page 1

ABSTRACT

This project presents a method of automatic generation control of power system by including

energy storage devices (SMES & EDL !" This techni#ue is applied to a control system that includes two

areas having reheat and non$reheat steam turbines and connected through power$line" % discrete time

model of this system is developed and simulated by using M%TL%' rograming" %s a conse#uence of

continually load variation) the fre#uency of the power system changes over time" *n conventional) studies)

fre#uency transients are minimi+ed by using conventional integral and proportional controllers aiming of

secondary control of %, and +ero steady$state error is obtained after sufficient delay time" *n this study)the conventional * controller is retained and storage device unit incorporated in both areas" The effect of a

small capacity energy storage system is studied in relation to supplying sudden power re#uirements of real

power load" %ccording to the deviation of the power system energy demand) the storage device releases

the needed energy or absorbs residue energy from power system" The results obtained by using SMES &

EDL devices outperform than those of the conventional control method as settling time and overshoot as"

shown at simulation

The SMES device is controlled by a two #uadrant chopper ( lass D! arrangement) by controlling

duty cycle of the choppers" The EDL device is also controlled by two #uadrant chopper ( lass !

arrangement) by controlling the desired value of the current"

ower system parameters are a function of the operating point" To -eep the system performance near its

optimum) it is desirable to trac- the operating point and use the updated parameters to compute the control

signal"*t may be possible to represent such a system by a linear low order discrete time model with time

varying parameters" % recursive least s#uares(.LS! parameter estimation techni#ue) which has a fast

convergence rate is used in this wor- to estimate these time varying parameters" 'ased on this model

which is trac-s the operating conditions of the system) control is computed using a minimum variance

strategy" Thus) the process is the e#uivalent of a controller with dynamically changing coefficients and is

referred to as the self tuning regulator (ST.!"


2/66



Chapter 1______________________________

Modeling And Control Of

Electric Power Systems


3/66



1. INTRODUCTION________________________________________________% power system must be able to meet reasonable power demands by large and small customers of

domestic) commercial and industrial type" *t must withstand with reasonable security the capricious forces

of nature" *n an age of high energy costs it is called upon to transform the prime energy resources into

electric form with an optimum overall efficiency" The control functions are obviously many and varied"

Some control and decision processes) e/emplified by the optimal utili+ation of the controlled flow of

river systems involve dynamics with month$long time constants" 0ther phenomena) li-e the transients on

the transmission lines following lightning stri-es) run their course in a few milliseconds"

The slower control processes are normally handled by computer$assisted human operators" The

faster control functions are trusted to fully automatic control systems of either open or closed$ loop nature"

% power system involves various control strategies of varying degrees at different stages" %t the

initial stage of a power system (generating unit!) there is a direct control over generation while the load1

demand is unpredictable" Thus there should e/ist a control system) the purpose of which is to continuously

monitor the mismatch between the generation and demand and to adjust the new operating point

accordingly" Such a generation control is performed automatically and is referred to as %utomatic

,eneration ontrol ( AGC !"The first attempt in the area of AGC has been to control the fre#uency of a power

system via the flywheel governor of the synchronous machine" This techni#ue was subse#uently found to be

insufficient) and a supplementary control was included to the governor with the help of a signal directly

proportional to the fre#uency deviation plus its integral" This scheme constitutes the classical approach to the

AGC of power systems" 2ery early wor-s in this important area of AGC have been by 3 ohn4" These wor-s

were based on tie$line bias control strategy" 35ua++a4 illustrated non interactive control considering

i! non interaction between fre#uency and tie$line powers controls and

ii! each control area ta-ing care of its own load variations"

The investigations with large signal dynamics of load 6re#uency onrol ( LFC ! systems were reported by

3%ggarwal4 and 3'ergseth4" The revolutionary optimal control concept for AGC regulator designs of

interconnected power systems was initialed by 3Elgerd4" % techni#ue based on coordinated system$wide

correction of time error and inadvertent interchange was incorporated in an AGC study by 3 ohn4"


4/66



Supplementary controllers were designed to regulate the ACE s to +ero effectively" Later on) energy source

dynamics were incorporated in %, regulator design"

2. AGC INCORPORATING BES, SMES AND EDLC STORAGE DEVICES____onventional AGC system is followed by fre#uency and tie$line power deviations) which are

characteri+ed by larger settling times) higher overshoots etc" Such oscillations can be effectively reduced

and1 or damped by incorporating storage devices" *n this project report) the controllers for such storage

devices have been devised and then the improvement in AGC with such devices have been studied) both

for reheat and non$reheat types of steam turbines in parallel with the bac-lash non$linearity"

Most of the solutions proposed so far for AGC have not been implemented practically due to system

operational constraints associated with thermal power plants" The main reason is the non availability of

re#uired power other than the stored energy in the generator rotors) which can improve the performance of

the system) in the wa-e of sudden increased load demands" % fast$acting BES can effectively dampen

electromechanical oscillations in a power system) because they provide storage capacity in addition to the

-inetic energy of the generator rotor) which can share the sudden changes in the power re#uirement"

The problems li-e low discharge rate) increased time re#uired for power$flow reversal) and the

maintenance re#uirement have led to the evolution ofSMES s for their application as load fre#uency

stabili+ers" The use of ACE for the control ofSMES units substantially reduces the tie$line power deviation)

and the action of SMES is locali+ed with diminished contribution for load disturbances in the other

interconnected area) as compared to using fre#uency deviation as the control signal" *t has been observed that

when theSMES control is adaptive) the performance is almost insensitive to controller gain parameter variation"


5/66



DYNAMIC INCREAMENTAL STATE VARIABLES____ _ ___________*n static operation the angular nominal fre#uency

o = 2 f o

is constant throughout the system) and the individual bus voltages have the form

vi = 2 |V i o| sin ( o t+ i o) (1)

7e identify |Vi| and i as our static state variables" 7hen the system is subject to small dynamic

perturbations) these state variables will undergo small changes8 i"e") we can write

i = i o + i

(2)

|V i| = |V i o| + |V i|

and the bus voltages will therefore be of the form

V i = 2 (|V i o| + |V i|) sin ( o t + i o + i) (3)

The angular velocityi of the ith bus e#uals i = d ( o t + i o + i) = o +d i (4)

dt dt

%nd is no longer constant) since it evidently is characteri+ed by a non$+ero perturbation

i = d i r/s

Dt

or) if e/pressed in cycles1sec)

f i = 1 d i Hz (5)

2 dt


6/66


7/66



Fig 1.1 Functional Diagram Of Governor Control Loop.

"! LIN#AGE MECHANISM

*t consist of two rigid lin-s %' and DE) hinged at points ' and D" The lin- L< provides thefeedbac-" Lin-age arms L= and L> are stiffly coupled) and so are arms L? and L


8/66



1" Directly ) by moving the lin-age point % by 3raise4 or 3lower4 commands of the speed changer"2" Indirectly ) via feedbac-) due to position change of the main piston"

3" Indirectly ) via feedbac-) due to position changes of lin-age point ' resulting from speed changes"

%! SPEED CHANGER The speed changer provides the steady state operating point for the speed governing system" *n

order to change the reference setting (steady state operating point! we give @.aise and lower: commands to

speed changer and this command is translated into opening and closing steam valve (gate! through lin-age

mechanism and hydraulic amplifier"

&. MATHEMATICAL MODEL OF SPEED GOVERNING SYSTEM __________ The model that we develop applies to small deviations around a nominal steady state" 7e

conse#uently assume the following chain of events9

1" The system is initially in a constant steady state) characteri+ed by a constant nominal speed offre#uency fo) a constant prime mover valve setting x E o) and a constant generator output power P G o"

2" 'y means of the speed changer) we command a power increase P c" %s a result of this command) the

lin-age point % moves downward a small distance x A proportional to P c "

3" The movement of lin-age point % causes small position changes xC and x D of the lin-age points

and D" as oil flows into the hydraulic motor) the steam valve will move the small distance x E ) resulting in

increased turbine tor#ue and) conse#uently ) a power increase P G"

4" The increased power output causes a momentary surplus) or accelerating power in the system" *f thesystem is very large (@infinite:!) the increased generator power will not noticeably affect the speed or

fre#uency will e/perience a slight increase f that will cause the speed and point ' to move downward a

small distance x B proportional to f " The speed governor being fast) we neglect any time delay in it"

onse#uently) we set x B proportional to f "

%ll incremental movements x A. . . x E are assumed positive in directions indicated"

Since all lin-age movements are small) we have the linear relationships

xC = k 1 f k 2 P c


9/66



(6)

x D = k 3 xC + k 4 x E The positive constants k1 and k 2 depend upon the lengths of the lin-age arms 1 and 2 and upon the

proportional constants of the speed changer and the speed governor" The positive constants k 3 and k 4

depend upon the lengths of the lin-age arms 3 and 4"

*f we assume that the oil flow into the hydraulic motor is proportional to position x D of the pilotvalve) we obtain the following relationship for the position of the main piston9

x E= k 5 (- x D) dt (7)

The positive constant k 5 depends upon orifice and cylinder geometries and fluid pressure"

'y ta-ing the Laplace transform of E#ns" (A! and (B!) and eliminating the variables xC and x D,

we obtain the following e#uation9

x E( s) = k 2 k 3 P c( s) k 1 k 3 F ( s) (8) k 4 + s / k 5

7e rewrite E#n" (C! as follows9

X E( s) = K G / (1+sT G) [ P c( s) F ( s) / R ] = G G( s) [ P c( s) F ( s) / R ] (9)

where

R = k 2 /k1 = speed regulationdue to governor action

K G = k 2 k 3 / k 4 = static gain of speed-governing mechanism

T G = 1 / k 4 k 5 = time constant of speed-governing mechanism

GG( s) = K G / (1 + sT G) = transfer function of speed-governing mechanism

T G is a measure o the reaction speed of the mechanism" ormal values are less than = ms"


10/66



'. TURBINE MODEL_____ ________________________________________ 7e are not primarily interested in turbine valve position per se) but rather the resulting generator

power increase P G. The change in valve position) x E , causes an incremental increase in turbine power) P T , which) via the electromechanical interactions within the generator) will result in an increasedgenerator power P G.

This overall mechanism is relatively complicated) particularly if the generator voltage

simultaneously undergoes wild swings due to major networ- disturbances"

*f) as in the present case) we can assume that the voltage level is constant and the tor#ue

variations are of small si+e) then an incremental analysis of the type we performed for the speed governor)

above) will give a relatively simple dynamic relationship between x E and P G. Such an analysis revealsconsiderable differences) not only between steam turbines and hydro turbines) but also between various

types (reheat and non reheat! of steam turbines" *n the crudest model representation we can characteri+e a

non reheat turbine generator with a single gain factor K T and a single time constantT T , and thus write

GT ( s) = P G( s) = K T __ (10) X E( s) 1 + sT T

Typically) the time constantT T lies in the range "> to > s" *n standard bloc-$diagram symbols we

can represent E#ns" (7) and (10) ) as shown in Fig 1.2 , which diagram therefore represents the lineari+ed

model of a nonreheat turbine controller) including the speed governor mechanism"

KT1+ S T T

KG1+ S T G

1R

X _

dP G(s)dX E(s)dP C(S)

dF (S)

+

Fig 1.2 Transfer function representation of power control mechanism of generator


11/66



1. NETWOR# DYNAMIC REPRESENTATION_____ ____________________ The turbine power , P T ) will be used for four different purposes9

1" To supply the demanded @new: load P L"2" To accelerate the turbine$generator) thus increasing the -inetic energy)W kin ) of the unit"3" To increase the powers in outgoing lines) i"e") P l "4" To meet the increase in the @old: load"7e discuss briefly the three last power components"

! #INETIC POWER INCREMENT, (Pkin

The kinetic energy serves as buffer storage" 6or e/ample when a customer suddenly connects a= -7 motor to the system it obviously cannot be met by a corresponding increase in the slow changing

turbine power" *nstead) the generator will supply it by 3borrowing4 from the -inetic energy"

Since the latter varies as the s#uare of the speed this power component can be e/pressed as

follows

P kin = d/dt (W kin ) =d/dt [W kin o ( f / f o) 2] = d/dt [W kin o ( fo+ f / f o) 2] 2W kin o / f o d/dt ( f ) (11)

7here) W kin o represents the -inetic energy as measured at normal speed"

"!. LINE POWER INCREMENT, (P)*+.

P tie, i = P tie, iv

6or simplicity we assume there is only one outgoing line connecting our generatorbus F i) with

another generator bus F j"

P i = |V i| |V j| sin ( i- j)

x ij

P tie,ij |V i| |V j|/ X ij cos ( i - j) ( I - j) (12)


12/66



6rom E#n" (G!) we have

= 2 f dt

we can write

P tie,ij 2 T ij o ( f i dt - f j dt ) (13)

7here the parameter) T ij) the synchronizing coefficient of the line) is defined by

T ij o = |V i| |V j| cos ( i o - j o) (14)

x ij

= T ij,max cos ( i o - j o)

The total incremental tie$line power out of areai , thus ta-es on the final form

P tie,j = 2 T jv o ( f i dt - f v dt ) )summation overv (15)

$! FREQUENCY DEPENDENCY OF OLD- LOAD %s the fre#uency increases) so will the speed of all the motors fed from the bus" %dded speed

means added tor#ue and power" 0ne may e/press this fre#uency dependency of e/isting load by an

empiric parameter) D) having the unit M71 H+" Thus the increase in the 3old4 load e#uals D f "

'y Laplace transforming all the above power components by adding them the dynamic power

balance at the bus reads9

P Gi = P Di + ( 2W kin o / f o) ( s f ) + 2 T ij o ( f - f j) / s + D f (16)

Dividing by P ri) the total megawatt rating of area i) and noting that

H i = W o

kin,i / P ri )above E#n" will become


13/66



P Gi = P Di + ( 2H i /f o) s f + ( 2 T ij o / s ) ( f - f j) + D f (17)

7e can rearrange this e#uation as follows9

f = [ P G - P D ( 2 T ij o / s ) ( f - f j)] K G / (1+sT G ) (18) 7here

K P = 1 / D

(19)

T P = 2H i / Df o

G pi ( s) = K pi / (1 + sT pi) (20)

( 2 T ij o / s ) ( f - f j) = P tie,i

The dynamic fre#uency behavior f ( t) can be obtained by state variable modeling of the a bloc-diagram shown in Fig 1.2 "

X = AX + BP ( I )Y = CX ( II )

7here)

A = -1 / T p K p / T p 0 , B = -Kp / Tp

0 -1 / T T 1 / T T 0

-1 / RT H 0 -1 / T H 0

C = 1 0 0


14/66



KT1+ S TT

KG1+ S TG X

1R

X

b

-1S

+

_ _

+ KP1+ S TP

dP D(s)

dP G(s)dX E(s)dP C(S) dF (S)

dF (S)

1S

X

2 p

i T 1

2 piT1

X

X

dP tie

++

d F

j ( S )

d F v ( S

)

+ +

_

+

_

Fig 1.3 Complete block diagram representation of control area i


15/66



Fig 1.4 shows the response of the single$area system" 6or comparison) we also simulated the

loop response with the inclusion of the time constantsT G and T T . 7e ma-e the following observations in

regard to our results9

1" The overallclosed loop system time constant e#uals only0.393 sec) which is a considerable reductionfrom the value Tp = 20 sec) characteri+ing the plant itself" This speedup is a result of the feedbac-

arrangement of the speed governor" ote that the system can be made still faster by reducing R , that is) by

increasing the static loop gain"

2" .eduction of R also reduces the static fre#uency error"3" *f we performed the above analysis bynot disregarding the time constantsT T and T G, then the responsewould not be purely e/ponential as above" *n Fig 1. 4 we show the difference" ote that the added delays

cause a larger transient fre#uency dip"

4" The speed governor operated in the abovenon controlled mode(i"e") without manipulation of the speedchanger! gives a reasonable performance with a static fre#uency drop of only >"< H+ between +ero and full

Fig 1.4 Dynamic frequency deviation following a step load change.Speed changer position fixed.


16/66



load and settling times of the order of G sec (as depicted in Fig 1. 4 !" However) with the e/tremely severe

restrictions we in reality impose on fre#uency constancy) the results are) in fact) entirely unacceptable) we

must do much better"

2. INTEGRAL CONTROLLER_________________ ______________________ *t is necessary to achieve much better fre#uency constancy than is obtained by the speed governor

system itself) as demonstrated above" To accomplish this we must manipulate the speed changer in

accordance with some suitablecontrol strategy.'efore we do so) it is necessary to settle for a set ofcontrol

specifications.

ere follo! some realistic specifications9

1" The control loop must be characteri+ed by a sufficient degree of stability"2" 6ollowing a step load change) the fre#uency error should return tozero.3" The integral of the fre#uency error (this integral has dimension of cycles! should not e/ceed a certainma/imum value"

4" The individual generators of the control area should divide the total load for optimum economy"

'y using the control strategy shown in Fig 1.5 ) we obtain an overall$system that will meet performancespecifications = and > above"

7e let the speed changer be commanded by a signal obtained by first amplifying and then integrating) the

fre#uency error"

P c = -K I f dt (21)

ote the negative polarity of the integral controller" This polarity must be chosen so as to cause a

positive fre#uency error to give rise to a negative) or @decrease:) command" The signal fed into theintegrator is referred to asarea control error(% E!) i"e")

ACE = f (22)


17/66



KT1+ S TT

KG1+ S TG X

1R

X

b

-1S

+

_ _

+ KP1+ S TP

dP D(s)

dP G(s)dX E(s)dP C(S) dF (S)

dF (S)

*ntegral control will give rise to +ero static fre#uency error following astep load change) for the

following physical reason9

%s long as an error remains) the integrator output will increase) causing the speed changer to

move"The integrator output, and thus the speed changer position, attains a constant value only !hen the

fre"uency error has been reduced to zero.

The gain constant K I controls the rate of integration) and thus the speed of response of the loop"

The integration is actually performed in electronic integrators of the same type as found in analog

computers"

Fig 1.5 Integral control of single-area system

(a) (b)

g 1.6 Dynamic frequency response with integral control, following a step load disturbance. The responseis shown for different integral gains. (a) T G = T T = 0, (b) T G =0.08 s, T T = 0.3 s.


18/66



3. E TENSION TO MULTI/AREA- SYSTEMS_______ _________________The loop model in Fig 1.5 is in strictest sense valid for a single generator only" 7e have noted

that the fre#uency dynamics is relatively slow" This tends to ma-e a whole group of generators in

unison) orcoherently ) thus permitting us to represent them all with thesame f " 6or this reason it is

common to let the model in Fig 1.5 represent a whole 3area4 which in practice typically can embrace a

whole power system" *f this 3area4 via tie$lines is connected to neighboring 3areas4 then we tal- about

multi$area dynamics" *n such among all generators that are under control"If each generator in the area

has the same percentage #regulation$ then each generator !ill participate in proportion to its rating.

The secondary ALFC loops in multi$area systems contain control signals) now referred to as 3area

control errors4 ( ACE !) which in addition to fre#uency error) f , also contains the errors in the contractedtie$line powers" % typical such % 6 would be of the form

ACE = P tie + B f (23)

4. THE TWO/AREA SYSTEM________________________________________

Let us turn our attention to a system consisting of two control areas of the type indicated in Fig1.7 ) interconnected via a relatively wea- tie$line" The areas are generally of different si+e and

characteristics"

BLOC# DIAGRAM OF TWO/AREA SYSTEMEach area is characteri+ed by the bloc- diagram shown in Fig 1.3 in terms of its incremental Pf

dynamics" 'y connecting two of these bloc- diagrams together8 we therefore obtain the two$area model

shown in Fig 1.7 .

%n e/planation must be given about the bloc- having the transfer function a 12 . 7e remember that

in E#n" (17) the term P tie,i represented the tie line power out from area i, e/pressed in per%unitmega!atts of the area capacity P ri . 6or the two$area system the tie line power P tie,1 must e#ual thenegative of P tie,2 ifboth are e/pressed in mega!atts. E/pressed mathematically) this means that

P r1 P tie,1 = -P r2 P tie,2 (24)or P tie,2 = a 12 P tie,1


19/66



where a 12 = -P r1 / P r2 (25)

&. STATIC RESPONSE OF UNCONTROLLED TWO/AREA SYSTEM________7e shall first investigate the response of the two$area system under uncontrolled conditions) i"e")

fi/ed speed changer positions9

P c1 = P c2 = 0

7e assume that the loads in each area are suddenly increased by the incremental steps P D& and P D' . 7e shall limit our analysis to finding thestatic changes that result in fre#uency and tie line power" Letus call those changes f stat and P tie1,stat .

Since the incremental increase in generation in this static case is determined by the static loop

gains only) we obtain from Fig 1.7 "

P G1,stat = (-1 / R 1) f stat (26) P G2,stat = (-1 / R 2) f stat


20/66



Fig 1.7 Load frequency control of two-area system


21/66



Since the incremental increase in generation in this static case is determined by the static loop gains only)

we

6rom E#n"(17) we get) upon setting d ( f i) dt = 0 and ma-ing use of E#ns"(24) and (26) )-1/ R 1 f stat - P D1 = D 1 f stat + P tie1,stat

-1/ R 2 f stat - P D2 = D 2 f stat + a 12 P tie2,stat

7e solve for f stat and P tie1,stat and obtain f stat = - P D2 a 12 P D1 Hz

b 2 - a 12 b1 (27)

P tie1,stat = b 1 P D2 b 2 P D1 pu MW b 2 - a 12 b1

where) we have defined the %6. s of each area9

Fig 1.8 Frequency deviation in each area and tie line power swings following a step

load increase of 1 percent in area 1


22/66



b1 =D 1 +1/R 1

(28)

b 2 =d 2 +1/ R 2

E#ns"(27) become particularly simple if we assume identical area parameters8 i"e" )

D1 =D 2 =D

R1 =R 2 =R

b1 =b 2= b

and

a 12 =-1we then get

f stat = - P D2 + P D1 Hz 2b

P tie1,stat = - P tie1,stat = P D2 P D1 pu MW 2

6or e/ample) if a step load change occurs in area >) we get

f stat = - P D2 Hz 2b

P tie1,stat = - P tie1,stat = P D2 pu MW 2

These two last e#uations tell us) in a nutshell) the advantages of pool operation9

1" 6ifty percent of the added load in area > will be supplied by area = via the tie line"2" The fre#uency drop will be only half that which would be e/perienced if the areas were operating alone"


23/66



'. THE CONTROLLED TWO AREA SYSTEMS__________________________The response curves of Fig 1.8 indicate that) as in the single area case) we must add integral

control to our system" Let us state) first) the minimum re#uirements the system should meet"

(uggested control%system specifications9 $ 7e shall re#uire that our system meet the four$point

specifications that we stipulated for the single area system" *n addition) we shall re#uire that the steady$

state tie$line power variation) following a step load change) must be +ero" This re#uirement guarantees that

each area) in steady state) absorbs its own load $$$ the guiding principle in pool operation"

The tie%line bias control strategy 9$ Since we must now use a strategy that will cause both the

fre#uency and tie$line deviations to vanish) we shall) as in the single$area case) adopt integral control) butwith the tie$line deviation added to our area control error 8 i"e") we attempt

ACE 1 = P tie1 +b 1 f 1 ACE 2 = P tie2 +b 2 f 2

The speed changer commands will thus be of the form

P c1 = - K I1 ( P tie1 + b 1 f 1 ) dt (29)

P c2 = - K I2 ( P tie2 + b 2 f 2 ) dt (30)

The constants K l1 and K I2 are integrator gains) and the constants b1 and b 2 are the fre"uency bias

parameters" The minus signs must be included since each area shouldincrease its generation ifeither its

fre#uency error f i or its tie line power increment P tie,i is negative"

STATIC SYSTEM RESPONSEThe chosen strategy will eliminate the steady state fre#uencyand tie line deviations for the

following reason9

6ollowing a step load change in either area) a new static e#uilibrium)if such e"uilibrium e)ists,can

be achieved only after the speed changer commands P c1 and P c2 have reached constant values" 'utthis evidently re#uires that both integrands in E#n"(29) be +ero8 i"e")

P tie1,stat + b 1 f stat = 0 (31)


24/66



P tie2,stat + b 2 f stat = 0 (32)

*n view of E#n"(24) ) these conditions can be metonlyif

f stat = P tie1,stat = P tie2,stat = 0 (33)

DYNAMIC SYSTEM RESPONSEThe added integrators have) contributed to the oscillatory nature of the system" The situation is

depicted in the simulated response graphs in Fig 1.9 " These recordings were obtained for the same

system data as in Fig 1.7 "

The graphs of Fig 1.9 depict the transient response for a step load change (in area =!" %ll graphs

correspond to stable parameter combinations) but we have demonstrated the effect of varying the bias

parameter b. 'oth areas are characteri+ed by identical parameters" The graphs of Fig 1.9a correspond to

b = 0" The system now properly controls the tie line power) but the fre#uency will have a static error"

The graphs of Fig 1.9b relate to the other e/treme case) with b = " ( ote that the product K I b isfinite) which means that K I = 0 !" The control actions are now insensitive to tie line power errors) and as a

result we left with a static power error) but the fre#uency is properly controlled"

The graphs of Fig 1.9c show an intermediate case" either the fre#uency nor the tie$line power

will have any static error" This is the practically important case"


25/66



Fig 1.9 Frequency and tie-line power deviations in a controlled two-area system following a step load disturbance of 1 percent in area 1 (only area 1 frequency is shown)The three cases correspond to the following widely different bias settings:(a) b = 0, K I = 1 (b) b = , K I b =0.425, (c) b=0.425, K I =1.


26/66



Chapter 2_________________________________

AGC Using Intelligent Control

Techniques In Power System

With EDLC.


27/66



1. INTRODUCTION

The automatic generation control ( AGC ! problem) being the instantaneous mismatch

between supply and demand of real power in power system) can also be effectively reduced by the

addition of Electric Double Layer apacitor (EDLC ! unit" 6ast$acting energy storage systems can

effectively damp electrochemical oscillations in a power systems) because they provide storage capacity in

addition to the -inetic energy of the generator rotors which can sudden changes in power re#uirement" To

store the energy re#uired for AGC applications batteries are used) however) they have some

disadvantages"

;ltra aps electrochemical double layer capacitors are new energy storage devices that close

the gap between aluminum electrolytic capacitors and batteries in terms of power and energy density" They

have two outstanding features9 their energy density is appro/imately = times higher than that of

conventional capacitors and power density is appro/imately = times higher than those of the batteries"

urrently ;ltra aps come with ) A ) => ) =C ) >B ) ?A and G 6" They can be

combined to power modules by series or parallel connection) available either in open or closed design"

;ltra aps are typically preferred in industrial) automotive and drive system applications as well as in power

#uality or ; S systems"

Supercapacitors are well suited to replace batteries" This is because at the moment theirscale is comparable to that of batteries) from small ones used in cellular phones to large ones that can

be found in cars" Even though supercapacitors have a lower energy density compared to batteries) they

avoid many of the batteryIs disadvantages"

'atteries have a limited number of charge1discharge cycles and ta-e time to charge and

discharge because the process involves chemical reactions with non$instantaneous rates" These chemical

reactions have parasitic thermal release that causes the battery to heat up" 'atteries have a limited life

cycle with a degrading performance and acidic batteries are ha+ardous to the environment"

Supercapacitors can be charged and discharged almost an unlimited number of times" They

can discharge in matters of milliseconds and are capable of producing enormous currents" Hence they are

very useful in load leveling applications and fields where a sudden boost of power is needed in a fraction of

a second" They do not release any thermal heat during discharge"


28/66



Supercapacitors have a very long lifetime) which reduces maintenance costs" They do

not release any ha+ardous substances that can damage the environment" Their performance does not

degrade with time" Supercapacitors are e/tremely safe for storage as they are easily discharged" They

have low internal resistances) even if many of them are coupled together"

Even though they have a lower energy density) are bul-ier and heavier than an

e#uivalent battery) they have already replaced batteries in many applications due to their readiness in

releasing power"

%S% has a research project to use super$capacitors in an electric bus called 3The

Hybrid Electric Transit 'us4" The energy used to start the engine and accelerate the bus is regenerated from

bra-ing" During test runs) a bus loaded with ? super$capacitors) each of them weighing ?>-g and releasing

energy of G -J at > 2 managed to run for < miles" T0K0T% has developed a diesel engine using the

same technology and is claimed to use just >"B liters of fuel per = -m"

Such devices if used with a Stand %lone 7ind ower lant) can effectively damp out the

electromechanical oscillations caused by varying wind speed and will thus improve the power #uality"


29/66



2. DOUBLE/LAYER ULTRACAPACITORS_____________________________0rdinary capacitors store energy in the dielectric material at a value of1/2 CV 2, where 'C'

is its capacitance (farads! andV (volts! is the voltage across its terminals" The ma/imum voltage of

a regular capacitor is dependent on the brea-down characteristics of the dielectric material" The

charge Q (coulombs! stored in the capacitor is given byQ = CV . The capacitance of the dielectric

capacitor depends on the dielectric constant( ) and the thic-ness (d) of the dielectric material plus

its geometric area"

C= A/d

During the past few years) electric double$layer capacitors with very large capacitance values havebeen developed" Those capacitors are fre#uently called Supercapacitors ) Ultracapacitors , or

Electrochemical Capacitors "*t has two electrodes immersed in an electrolyte with a separator between

them" The electrodes are fabricated from porous high$surface$area material that has pores of diameter in

the nanometer range which gives the materiala very large active internal surface) in the order of =

m>"g$=" harge is stored in the micropores at or near the interface between the solid electrode material and

the electrolyte" The charge and energy stored are given by the same e/pressions as those for an ordinary

capacitor) but the capacitance depends on comple/ phenomena that occur in the micropores of the

electrode"

Fig shows the construction details of a double$layer ultracapacitor" The capacitor contains two

particulate$carbon electrodes formed on conductive$polymer films" %n ion$conductive membrane separates

the two electrodes) and a potassium hydro/ide electrolyte permeates the capacitor" The micropores in the

carbon particles result in an enormous surface area and yield e/tremely high capacitance values which

conventional capacitors cannot attain" Energy is stored in the double$layer capacitor as charge separation

in the double layer formed at the interface between the solid electrode material surface and the li#uid

electrolyte in the micropores of the electrodes" The ions displaced in forming the double layers in the

pores) are transferred between the electrodes by diffusion through the electrolyte" The separator

prevents electrical contact between the two electrodes" *t is very thin) with high electrical resistance) but

ion$permeable) which allows ionic charge transfer" olymer or paper separators can be used with

organic electrolytes and ceramic or glass fiber separators are often used with a#ueous electrolytes


30/66



The energy and charge stored in the electrochemical capacitor are given by the same

e#uations as for ordinary capacitors" However) the capacitance is dependent primarily on the

characteristics of the electrode material (i"e") surface area and pore si+e distribution!" The specific

capacitance of an electrode material depends on the effective dielectric constant of the electrolyte and

the thic-ness of the double layer formed at the interface"These are very comple) phenomena that are

not fully understood " The thic-ness of the double layer is very small (a fraction of a nanometer in li#uid

electrolytes!) which results in a high value for the specific capacitance"

The performance of electrochemical capacitors depends on the specific capacitance (61g or

61cm?! of the electrode material and the ionic conductivity of the electrolyte used in the device" The

specific capacitance of a particular electrode material depends on whether the material is used in the

positive or negative electrode of the device and whether the electrolyte is a#ueous or organic" Most

carbon materials e/hibit higher specific capacitance9 in the range BG to =BG 61g for a#ueous electrolytes

and < to = 61g for organic electrolytes) which allow much more ions" %lthough a lower specific

capacitance is achieved with organic electrolytes) they have the advantage of higher operating


31/66


32/66



?" EDLC SYSTEM____________ ____________________________________

The bloc- diagram for two$area system developed in chapter > can be put in the state variable modelas follows9

X ! AX " #$ %

& ! CX

7ith EDLC unit included as shown inFig this model will change to

X ! AX " #0$ %' $ sc ! & ! CX

7here X ) $ % are state) disturbance vectors respectively)$ sc is the EDLC power vector8# and A areconstant matrices associated with the above vectors" A is the state matri/" Here)

-KI1S

_ _Kp1__ 1+STp1

__ 1 _ 1+STt1

-KI2S

__ 1 __ 1+STt2

___ 1 ___ 1+sTg1

_1_ R2

___1___ 1+STg1

__ Kp1__ 1+STp1

X

X

X2piT 12S

X

X

a 12

1R1

X

X

b2

b1

a 12

Governer Turbine

dP L1(s)

dP L2(s)

dP t ie

+

+

+

+

++ +

+

_

_

_

_

_

_

dF 1(s)

dF 2(s)

+

_

x 1(s) x 2(s)

x 5(s) x 6(s) x 7(s)

x 8(s)

x 9(s)

x 4(s)

x 3(s)

EDLC

EDLC X

X

Fig 3.2 Two- area interconnected power system with EDLC.


33/66



1 T 1 # 1 T 1 /# 1 T 1

0 -1/Tt1 1/Tt1 0 0 0 0 0 0

-1/(R1*Tg1) 0 -1/Tg1 1/Tg1 0 0 0 0 0

-b1*Ki1 0 0 0 0 0 0 0 -Ki1

0 0 0 0 -1/Tp2 Kp2/Tp2 0 0 -a12*Kp2/Tp2

A= 0 0 0 0 0 -1/Tt2 1/Tt2 0 0

0 0 0 0 -1/(R2*Tg2) 0 -1/Tg2 1/Tg2 0

0 0 0 0 -b2*Ki2 0 0 0 -a12*Ki2;

2*pi*T12 0 0 0 -2*pi*T12 0 0 0 0

-K p1 /T p1 0 P d = p d1

x1 0 0 p d2

x2 0 0

x3 0 0

X = x4 B= 0 -K p2 /T p2 and, Psc = p sc1

x5 0 0 p sc2

x6 0 0 x7 0 0

x8 0 0

x9


34/66



1 0 0 0 0 0 0 0 0

C = 0 0 0 0 1 0 0 0 0

0 0 0 0 0 0 0 0 1

1. CONFIGURATION OF EDLC UNIT_______________________________0nce the rated voltage in the super$capacitor is reached) the EDLC unit is ready for automatic

generation or load control" The change in ACE is sensed and used to control the power released or

absorbed by the EDLC unit"The EDLC unit is connected with the ?$phase % supply by means of the circuit configuration shown in

System voltage is reduced by means of transformer which is then converted to constant D voltage using

Si/ ulse 7M onverter involving *,'T:s"

Fig 3.3 Configuration of EDLC in Power System.


35/66



Fig 3.4 Two Quadrant Chopper Arrangement for Control of EDLC Unit.

7hen there is surplus power in the system) S .= and D= will operate in order to store the surplus

energy in the super$capacitor" The waveforms pertaining to such a condition are shown in Fig 3.5 " During

one period (T !) average power (Vd*I des avg ! is stored in the super$capacitor" onse#uently when the

system is deficient of power)S .> and D> will operate and re#uired energy is released by the super$

capacitor" The waveforms pertaining to such a condition are shown in Fig 3.6 " During one period (T !)

average power (Vd*I des avg ! is released by the super$capacitor" ;nder normal conditions) when there is

e#uilibrium between the generation and demand) neither any S . nor any Diode will conduct" This will lead

the super$capacitor to remain in floating mode" To serve the purpose) the super$capacitor is employed with

the two #uadrant chopper arrangement as shown in Fig 3.4.


36/66



SCR 1 D1 SCR 1 SCR 1D1 D1

T

I min

I max

I s c

I d e s

_ Ides

S C D e v

i c e s

O N

(a)

(b)

(c)

Fig 3.5 (a) Switching Sequence of Chopper (b) Super-Capacitor Current Waveform

(c) Current Waveform Supplied by the system.


37/66



SCR 2 D2

T

I min

I max I s

c

I d e s

_ Ides

D2 D2 D2 SCR 2 SCR 2

S C D e v

i c e s

O N

(a)

(b)

(c)

Fig 3.6 (a) Switching Sequence of Chopper (b) Super-Capacitor Current Waveform

(c) Current Waveform Supplied to the system.

The wor-ing of the said controller is based on the 6u++y Logic" The 6u++y Logic ta-es two inputs

(%rea ontrol Error @ ACE : and the deviation in EDLC voltage!"6or proper functioning of the super$capacitor8 it should neither be discharged below some minimum value nor should be charged above some

ma/imum value" %s such when there is surplus of power in the system and if super$capacitor is charged to

nominal value) the 6u++y Logic should fire the command to chopper to absorb ma/imum power from the

system else if when super$capacitor is charged to ma/imum value) it should fire the command so that

super$capacitor remains in floating mode"


38/66



Chapter (_________________________________

AGC Using Intelligent Control

Techniques In Power System

With SMES


39/66



INTRODUCTION

The automatic generation control (AGC) problem, which is the major requirement in parallel

operation of several interconnected systems, is one of very important subjects in power system studies

!n this study, power systems with two areas connected through tie"lines are considered The

perturbation of frequencies at the areas and resulting tie"line power flows arise due to unpredictable

load variations that cause mismatch between the generated and demanded powers The objective of

AGC is to minimi#e the transient deviations and to provide #ero steady state errors of these variables in

a very short time although unpredictable load variations present !n literature, for AGC, some controlstrategies based on classical control theory have been proposed $nfortunately, because of operating

point continuously changes depending on demand of consumers, the selected fi%ed controller can be

unsuitable at other operating points Therefore, many of controllers with variable structure are

proposed in literature

!n the AGC problem, the instantaneous mismatch between supply and demand of real power in

power system can be reduced by the addition of fast acting superconducting magnetic energy storage

(&' &) unit &' & is a device for storing and instantaneously discharging large quantities of power !t

stores energy in the magnetic field created by the flow of direct current in a superconducting coil which

has been cryogenically cooled to a temperature below its superconducting critical temperature

%s an energy storage device) SMES is a relatively simple concept" *t stores electric

energy in the magnetic field generated by D current flowing through a coiled wire" *f

the coil were wound using a conventional wire such as copper) the magnetic energy

would be dissipated as heat due to the wireIs resistance to the flow of current" However)

if the wire is superconducting (no resistance!) then energy can be stored in a

persistentII mode) virtually indefinitely) until re#uired" Superconductors have +eroresistance to D electrical current at low temperatures so that ohmic heat dissipation is

eliminated) hence the refrigerator is needed in the SMES to cool the coil" *n %

applications) there are still electrical losses) but these can be minimised through

appropriate wire architecture and device design" 6or both D and % applications)

energy savings will be significant" The current carrying capacity of the wire is dependent

on temperature and the local magnetic field" The optimal operating temperature for


40/66



most of the devices will be G $BB "

% typical SMES system includes three parts9 superconductingcoil) power conditioning

system and cryogenically cooled refrigerator" 0nce the superconducting coil is charged)

the current will not decay and the magnetic energy can be stored indefinitely"

The stored energy can be released bac- to the networ- by discharging the coil" The

SMES recharges within minutes and can repeat the charge1discharge se#uence

thousands of times without any degradation of the magnetic coil" .echarge time can be

accelerated to meet specific re#uirements) depending on system capacity"

SMES is devoted to improvingpower #uality due to the energy re#uirements of

refrigeration and the high cost ofsuperconducting wire which causes it to be used forshort duration energy storage"*f SMES were to be used forutilities it would be a diurnal

storage device) charged frombaseload power at night and meeting pea- loads during

the day"

O + )*56 57 6 SMES 86*)The basic operation of a an SMES unit is very simple" The transmission voltage

(from the % networ-! is first stepped down from a few hundred -2 to several

hundred volts using a step$down transformer" This is then converted into D which is

fed into the superconducting coil" Hence when the power flows from the system to the

coil) the D voltage will charge up the superconducting coil and the energy is stored in

the coil" The ma/imum energy stored depends on the design of the device"

7hen the % networ-s re#uires a power boost) the coil discharges and acts as a

source of energy"The D voltage is converted bac- into % voltage through the

converter" Thus the power conditioning system uses aninverter 1rectifier to

transform alternating current(% ! power to direct current or convert D bac- to %

power" The inverter1rectifier accounts for about > ?N energy loss in each direction"SMES loses the least amount ofelectricity in the energy storage process compared to

other methods of storing energy" SMES are highly efficient at storing electricity (greater

than OBN efficiency!) and provide both real and reactive power" Thus the addition of a

small capacity SMES unit to a power system significantly improves transients of

fre#uency and tie$line power deviations against to small load disturbances" 7hile there

is a sudden rise in the demand of load) the stored energy is almost released through the


41/66



power conversion system to the power system as alternative current" %s the governor

and other controllers start wor-ing to set condition) the coil changes bac- to its initial

value of current" Similar action ta-es place during sudden release of loads" *n this case)

the coil immediately gets charged towards its full value) thus absorbs some portions of

the e/cess energy in the system) and as the system returns to its steady state) the

e/cess energy absorbed is released and the coil current attains its normal value" During

normal operation) the stored energy is ta-en about > $? MJ because of factors of

superconductor stability) mechanical forces and fatigue" So) SMES unit is forced its

lower and upper current limits" Therefore SMES units are separated from the system if

the inductor current reaches its limits and SMES units are again connected only whenthe control signal changes its sign" 7henever SMES unit is disconnected) it is noticed

that the power system is e#uivalent to the one without SMES unit"

Types of power system models

The AGC problem has been dealt with e%tensively for more than three decades The major past

of the wor reported so far has been performed by considering lineari#ed models of two*multi area

power systems +ater on, the effect of GRC was included in these types of studies, considering both

continuous and discrete power system models !ncorporating the dynamics of the energy source in AGC

regulator design, -watny. have proposed an optimal trac ing approach to AGC , considering load to be

the output of the dynamic system

The small signal analysis is justified for studying the system response for small perturbations /owever,

the implementation of AGC strategy based on a lineari#ed model on an essentially nonlinear system does

not necessarily ensure the stability of the system Considerable attention has been paid by researchers

to consider the system nonlinearities +ater, also the destabili#ing effect of governor dead"band non linearity

on conventional AGC system was demonstrated !t is shown that the governor dead band nonlinearity tends to

produce the oscillations in the area frequency and tie"line power transient response


42/66



Control Techniques

The pioneering wor by a number of control engineers, namely 0ode and 1yquist has

established lin s between the frequency response of a control system and its closed"loop transient

performance in the time domain The investigations carried out using classical control approaches reveal

that it will result in relatively large overshoots and transient frequency deviation 'oreover, the settling

time of the system frequency deviation is comparatively long and is of the order of 23"43 s

The AGC regulator design techniques using modern optimal control theory enable the power

engineers to design an optimal control system with respect to given performance criterion 5osha and

lgerd were the first to present their pioneering wor on optimal AGC regulator design using this concept

A two"area interconnected power system consisting of two identical power plants of non"reheat thermal

turbines was considered for investigations

The feasibility of an optimal AGC scheme requires the availability of all state variables for feedbac

/owever, these efforts seem unrealistic, since it is difficult to achieve this Then, the problem is to

reconstruct the unavailable states from the available outputs and controls using an observer Considering

state reconstruction, many significant contributions have been made 6ue to practical limitations in the

implementation of regulators based on feedbac of all stale variables, suboptimal AGC regulator designs

were considered

Discrete data model for AGC

The bloc diagram for two"area system developed in chapter 2 can be put in the state

variable model as follows7

X` = AX + BP d Y = CX

8ith SMES unit included as shown in Fig 2.1 this model will change to

X` = AX + B (P d + P sm)

Y = CX

8here X , P d are state, disturbance vectors respectively, P sm is the SMES power vector9 B and A are


43/66



constant matrices associated with the above vectors A is the state matri% /ere,

Fig 2.1 Two- area i !er"o e"!ed #ower s$s!em wi!% SMES.

-1&T #1 ' #1 &T #1 ( ( ( ( ( ( -' #1 &T #1

( -1&T T1 1&T T1 ( ( ( ( ( (

-1&R1T )1 ( -1&T )1 1&T )1 ( ( ( ( (

-*1' 1 ( ( ( ( ( ( ( -' 1

A = ( ( ( ( -1&T #2 ' #2 &T #2 ( ( -a 12 ' #2 &T #2

( ( ( ( ( -1&T T2 1&T T2 ( (

( ( ( ( -1&R 2T )2 ( -1&T )2 1&T )2 (

( ( ( ( -* 2' 2 ( ( ( -a 12 ' 2

2 T o ( ( ( -2 T o ( ( ( (


44/66



, 1 ' #1 &T #1 ( P d= #d1

, 2 ( ( # d2

, ( (

, / ( (

X = , 0 B = ( ' #2 &T #2 a d P sm = #sm1

, ( ( # sm2

, ( (

, 3 ( (

, 4 ( (

1 ( ( ( ( ( ( ( (

C = ( ( ( ( 1 ( ( ( (

( ( ( ( ( ( ( ( 1


45/66



SMES S9:)+;The schematic diagram in Fig 2.2 shows the configuration of a thyristor

controlled SMES unit The SMES unit contains 6C superconducting coil and

converter which are connected by :"; * :": transformer The converter consists

of a si%"pulse P5M rectifier*inverter using insulated"gate"bipolar"transistors

( GBT6s) interfacing with the AC power system and 6C"6C chopper interfacing

with the SMES coil The P5M converter and 6C"6C chopper are lin ed by a 6C"

lin capacitor The GBT is a new power switching device, which is basically a

hybrid ') due to a reverse

biased p ? " n" junction Therefore, in voltage"fed converter in application an anti"

parallel diode is connected e%ternally

Configuration of &' & in @ower &ystem

The energy stored in the &' & coil in Fig 2.2 at any instant is

5 7 = 7 sm 2 & 2

where,

7 = inductance of SMES coil

sm = direct current in SMES


46/66



D2

D1S1

S 2

3 PhaseAC System

SMES Coil

+ _

Ism

Vsm

Fig 2.2 Co 8ig9ra!io o8 SMES : i! Power S$s!em.

The &' & control strategy requires that7

1 5or a positive AC i e , decrease in load, the &' & device should absorb the surplus power from the

system, thus eeping the frequency constant

2 5or a negative AC , i e , increase in load, the &' & device should deliver power to the system, thus

eeping the frequency constant

3 5or #ero AC , i e , normal system operation, the &' & should not e%change any power !n other

words it should isolate itself from the system

These three conditions are e%actly met by using a two quadrant chopper arrangement to control

the &' & device

ics

Vdc

S1

D2

+

D1

S2

Fig 2.3 Chopper arrangement for control of SMES unit


47/66



Fig 2. shows the control scheme S1 and S2 are two chopper switches ; 1 and ; 2 are two diodes

The duty cycle of the two choppers is ept same Therefore, if T is the chopper period and T o is the time

for which the chopper remains in on state, then

6uty cycle, ; "%o# = T o & T

and ( < ; "%o# < 1

6epending upon the value of ; "%o#, three regions of operation can be identified for the chopper

arrangement in Fig 2. vi#

a) ( < ; "%o# < (.0

5or this condition, there is net power flow into the system This situation is suited for increase in

load demand

b) ; "%o# = (.0

5or this condition, there is no net power e%change with the system This situation is suited when

there is no change in load

c) (.0 < ; "%o# < 1

5or this condition, there will be a net power flow into the &' & device and &' & will be charged

This situation is suitable when there is surplus power in the system and it is required to dissipate this

power

The charging, floating or discharging mode of &' & is decided by the ind of disturbance and will

thus operate in (a), (b) or (c) region of operation respectively 5urther since the load changes are not

fi%ed in practical power systems, thus we can vary the value of ; "%o# in a particular given region in order

to meet the desired requirement


48/66



Advantages over other energy storage methods

There are several reasons for using superconducting magnetic energy storage instead

of other energy storage methods The most important advantage of &' & is that the

time delay during charge and discharge is quite short @ower is available almost

instantaneously and very high power output can be provided for a brief period of time


49/66



Chapter )_____________________________

Adaptive Control


50/66



*n recent years) interest in adaptive control systems has increased rapidly along with interest and progress

in robotics and other control fields" The term adaptive system has a variety of specific meanings) but it

usually implies that the system is capable of accommodating un$predictable environmental changes)

whether these changes arise within the system or e/ternal to it" This concept has a great deal of appeal to

the system:s designer since a highly adaptive system) besides accommodating environmental changes)

would also accommodate moderate engineering design errors or uncertainties and would compensate for

the failure of minor system components) thereby increasing system reliability"

INTRODUCTION

*n everyday language) Pto adaptP means to change a behavior to conform to new circumstances" *ntuitively

an adaptive controller is thus a controller that can modify its behavior in response to changes in the

dynamics of the process and the character of the disturbances" Since ordinary feedbac- also attempts to

reduce the effects of disturbances and plant uncertainty) the #uestion of the difference between the

feedbac- control and adaptive control immediately arises"

*n most feedbac- control systems) small deviation in parameter values from their design values will not

cause any problem in the normal operation of the system) provided these parameters are inside the loop" *f

plant parameters vary widely according to environmental changes) however) then the control system maye/hibit satisfactory response for one environmental condition but may fail to provide satisfactory

performance under other conditions" *n certain cases) large variations of plant parameters may even cause

instability"

*n the simplest analysis) one may consider different sets of values of the plant parameters" *t is then

desirable to design a control system that wor-s well for all sets" %s soon as this demand is formulated) the

strict optimal control problem loses its importance" 'y as-ing for good performance over a range) we have

to abandon the best performance for one parameter set"

*f the plant transfer function or plant state e#uation can be identified continuously) then we can compensate

for variations in the transfer function or state e#uation of the plant simply by varying adjustable parameters

of the controller and thereby obtain satisfactory system performance continuously under various

environmental conditions" Such an adaptive approach is #uite useful to cope with a problem where the

plant is normally e/posed to varying environments so that plant parameters change from time to time"

DE6* *T*0


51/66


52/66



%D% T*2E 0 T.0L .0 ED;.E

%daptive controller may consist of the following three functions9

=" *dentification of dynamic characteristics of the plant"

>" Decision ma-ing based on the identification of the plant"

?" Modification or actuation based on the decision made"

% bloc- diagram representation of an %daptive control system is shown in the figure below" *n this system

after the identification of the plant) a decision is made based in the findings as to how to modify the

actuating signal "Since the plant is identified within the system itself) adjustment of the parameters is a

closed loop operation"

6ig

Q *DE T*6* %T*0 06 THE DK %M* H%. TE.*ST* S 06 THE L% T

The dynamic characteristics of the plant must be measured and identified continuously or at least very

fre#uently" This should be accomplished without affecting the normal operation of the system" To identify

the characteristics of the system) we must perform a test and analy+e the results" (6or a control system)

this entails imposing a control signal on the plant and analy+ing system response!"*dentification may bemade from normal operating data of the plant or by use of test signals) such as sinusoidal ones of small

amplitude or various stochastic signals of small amplitude" *n practice) no direct application of step or

impulse inputs can be made" ormal inputs and system noise should not disturb the test" However )

identification with normal inputs is only possible when they have ade#uate signal characteristics

('andwidth) amplitude) and so on! for proper identification"


53/66



*dentification must not ta-e too long since if it does) further variations of the plant parameter may occur"

*dentification time should be sufficiently short compared with the rate of environmental changes" 7ith time

for identification limited) it is usually impossible to identify the plant completely8 the best one can e/pect is

only partial identification"

The difficulty of ma-ing a realistic identification will depend on how much information about the plant is

re#uired and on the amount of prior -nowledge of plant"

DE *S*0 M% * , '%SED 0 THE *DE T*6* %T*0 06 THE L% T

Decision here refers to one made on the basis of plant characteristics that have identified"

0nce the plant has been identified it is compared with the optimal characteristics ( or optimal performance!)

and then the decision must be made as to how the adjustable parameters (controller characteristics! shouldbe varied to maintain optimal performance" The decision is accomplished by the computer"

M0D*6* %T*0 '%SED 0 THE DE *S*0 M%DE

Modification refers to the change of control signals according to the results of the identification and

decision" *n most schemes) the decision and modification are conceptually a single operation with the

modification consisting of a means of mechani+ing the transformation of a decision output signal into the

control signal (the input to the plant!"

The control signal) or the input signal to the plant) can be modified in two ways" The first approach is to

adjust the controller parameters to compensate for changes in the plant dynamics" This is called controller

parameter modification"The second approach is to synthesi+e the optimal control signal) based on plant

function or plant state e#uation) desired transient response" This is called control signal synthesis"

The choice between controller parameter modification and control$signal synthesis is primarily a hardware

decision since the two approaches are conceptually e#uivalent" 7here reliability is very important) as in

aerospace applications) the use of parameter change adaptation is often favored over the use of control$

signal synthesis" This is because the system can operate even after the failure of the adaptive loop if thecontrol signal is not entirely dependent on the adaptive portion of the system"

%D% T* 2E S HEMES

*n the following section we describe three types of adaptive systems9 gain scheduling) model$reference

adaptive control and self$tuning regulators"


54/66



,%* S HED;L* ,

*n many cases it is possible to find measurable variables that correlate well with changes in process

dynamics" These variables can then be used to change the controller parameters" This approach is called

gain scheduling because the scheme was originally used to measure the gain and the change) that is)

schedule) it controller to compensate for changes in the process gain" % bloc- diagram of a system with

gain scheduling is shown" The system can be viewed as having two loops" There is an inner loop

composed of the process and the controller and an outer loop that adjusts the controller parameters on the

basis of the operating conditions" ,ain scheduling can be regarded as a mapping from process parameters

to controller parameters it can be implemented as a function or a table loo-up"

The concept of gain scheduling originated in connection with the development of flight control systems" *nthis application the Mach number and the altitude are measured by air data sensors and used as

scheduling variables" *n the process control the production rate can often be chosen as a scheduling

variable) since time constants and time delays are often inversely proportional to production rate" ,ain

scheduling is thus a very useful techni#ue for reducing the of parameter variations" Historically) it has been

a matter of controversy weather gain scheduling should be considered an adaptive system or not" *f we use

the informal definition) that an adaptive system is a controller with adjustable parameters and an

adjustment mechanism) it is clearly adaptive"

6ig

M0DEL$.E6E.E E %D% T*2E SKSTEMS

The model$reference adaptive system was originally proposed to solve a problem in which the performance

specifications are given in terms of a reference model" This model tells how the process output ideally

should respond to the command signal" % bloc- diagram of the system is shown" The controller can be

thought of as consisting of two loops" The inner loop is an ordinary feedbac- loop composed of the


55/66



process and the controller" The outer loop adjusts the controller parameters in such a way that the error)

which is the difference between process output @y: and model output @ym: is small" The M.%S was originally

introduced for flight control" *n this case the reference model describes the desired response of the aircraft

to joystic- motions"

The -ey problem with M.%S is to determine the adjustment mechanism so that a stable system) which

brings the error to +ero) is obtained" This problem is non$trivial" The following parameter adjustment

mechanism) called the M*T rule) was used in the original M.%S9

= ee

dt d

*n this e#uation) e R y$ym denotes the model error and is a controller parameter"

The #uantity

e is the sensitivity derivative of the error with respect to " The parameter determines

the adaptation rate" *n practice it is necessary to ma-e appro/imations to obtain the sensitivity derivative"

The M*T rule has been regarded as gradient scheme to minimi+e the s#uared error e>

SEL6$T; * , .E,;L%T0.S (ST.!

The adaptive scheme discussed so far are called direct methods) because the adjustment rules tell directly

how the controller parameters should be updated" % different scheme is obtained if the estimates of the

process parameters are updated and the controller parameters are obtained from the solution of a design

problem using the estimated parameters" % bloc- diagram of such a system is shown below" The adaptive


56/66



controller can be thought of as being composed of two loops" The inner loop consists of the process and an

ordinary feedbac- controller" The parameters of the controller are adjusted by the outer loop) which is

composed of a recursive parameter estimator and a design calculation" *t is sometimes not possible to

estimate the process parameters without introducing probing control signals or perturbations" otice that

the system may be viewed as an automation of process modeling and design) in which the process model

and the control design are updated at each sampling period" % controller of this construction is called a self$

tuning regulator (S*.! to emphasi+e that the controller automatically tunes its parameters to obtain the

desired properties of the closed loop system"

The bloc- labeled P ontroller designP in 6igure represents an on$line solution to a design problem for a

system with -nown parameters" This is the underlying design problem" Such a problem can be associatedwith most adaptive control schemes) but it is often given indirectly" To evaluate adaptive schemes) it is

often useful to find the underlying design problem) because it will give the characteristics of the system

under the ideal conditions when the parameters are -nown e/actly"

The ST. scheme is very fle/ible with respect to the choice of the underlying design and estimation

methods" Many different combinations have been e/plored" The controller parameters are updated

indirectly via the design calculations in the self$tuner shown in 6igure" *t is sometimes possible to re$

parameteri+e the process so that the model can be e/pressed in$terms of the controller parameters" This


57/66


58/66



Chapter *_____________________________

System

Identification & Control.


59/66



*n the fields of physical science and engineering we are often interested in the development of a

mathematical model of some physical phenomena in order to analytical predictions about the behavior of

the system" *n control application we are often interested in modeling a physical plant) which we wish to

control in order to predict the effect of control efforts and disturbances on that plant"

0ften) the system model can be obtained by application of physical principle to the region of space

designated as the system and obtaining the governing dynamic e#uations" 0ften these e#uations result

from force) mass" Energy) or momentum balances or the governing principles of electromechanical

systems(i"e) ewtonIs) irchhoffIs) Len+Is and 6aradays laws!" Sometimes the model cannot be obtained

from physical arguments because of un-nown chemical reactions) un-nown boundary conditions on thephysical processes) or the e/treme comple/ity of the process" *n these cases we must resort to the

e/perimental method to develop a system model" This method is applied to the system where in

measurements of system stimuli and responses are made and the dynamic nature of the system is

deducted from the relations between responses and stimuli" This process) -nown as system identification

or characteri+ation) is the epitome of the scientific method" There are many techni#ues for accomplishing

this process and only two are given here8 however) it is interesting to note that the popular techni#ues are

(=! random or pseudorandom e/citation8 (>! impulse e/citation8 (?! step e/citation8 (

with a sweep through the fre#uencies"

LE%ST$S5;%.ES TE H *5;E

7e will be interested in the identification of systems with constant parameters which will form the

parameter vector n (?">"=!

where di is a p$vector of data corresponding to measurement yi) and < is a p$vector of parameters whichwe wish to find based on the data yi) and diT"Let us define our estimate of the parameter vector as=whichmay differ from the actual parameter values of< " The set of e#uations then are

yiRdiT= e i iR=)> n (?">">!where ei is the error induced because we have employed the estimate of">! can be written as the vector e#uation

9>D = ?+ (?">"?!


60/66



where Dis the n / p data matri/ with an i th row of%*T and 9 is commonly called nU = measurement vectorand + is an n / = vector of errors" There are many practical problems that we can put into the form of(?">"?!"

The techni#ue of least s#uares is not restricted to models that are linear in the independent variable) but

the models must be linear in the parameters" olynomial curve fits are common) but note that these fits are

linear in the coefficients which are the parameters" Similarly) 6ourier series representations are linear in the

parameters) which in that case are the un-nown 6ourier coefficients"

Let us now go to our general discussion of least s#uares" The model of the estimator is

9>D< (?">"

while in reality9 > D=? + (?">"G!

where the +vector represents the errors in the estimates of the points yi "7e want to minimi+e the sum ofthe s#uares of the errors or minimi+e the scalar function

@> +T+ > +* 2 (?">"A!7e see that the error vector is the difference of9and the predicted value of9 (namelyD=!) or

J> 09/D=!T09/D=! (?">"B!ow to minimi+e J with respect to= we differentiate the function J with respect to the vector= and we get

d dJ >09/D=!T0/D!?0/D!T09/D=!R (?">"C!

These two terms are the same) and hence it is sufficient that one of them be +ero and hence DT D=> DT 9 (?">"O!

(?">"O!

This set of e#uations is often referred to as thenormal e"uations) which must be solved for the parameter

estimate vector=" 0ne way would be to invert theDTD to give=R0DT D!/1DT9 (?">"= !

(?">"= !

The matri/ DTD is p / p and must be inverted" *f p Q n) then the ran- ofDTDis less than p and it will besingular and hence not invertible"

learly ) the number of data points (n! ta-en should be larger than the number of parameters to be

estimated (p!" 7hen dealing with a large number of data points possible that numerical problems will arise


61/66



in the inversion re#uired in (?">"= !" *t might be easier to solve (?">"O! by an iterative techni#ue such as

rela/ation"

Transfer 6unction Estimation ;sing Least S#uares

7e sha*l assume that we have a discrete$data system and that we have measured stimulus and response

se#uences u(-! and y(-!) respectively" 7e must assume a priori a form of the +$domain transfer function" *t

is sometimes not an easy tas-) so one must assume several forms for the transfer function and see which

one gives the minimum aggregate error" %n e/ample will best illustrate this techni#ue) ithen can easily be

applied to other transfer functions"

%ssume that the process involved is governed by the transfer function

, (+! R )( )( zU zY R ( bn$= +n$= """" b= + b !1(+n$an$= +n$=$"""" $ a= + a ! (?"?"=!

The e#uivalent difference e#uation is

y(-! Ran$= y(- $=! " " " a y (- $ n! bn$ u(- $=! b u(- $ n! (?"?">!

7e shall assume that y(-! and u(-! are +ero for negative indices -"

Let us define thek th data vector to be

d(-!RVy(-$=! y(-$>!"""y(- $ n!u(-$=!"""u(-Wn$=!XT (?"?"?!

and the parameter vector to be

< >Van$=a o bn$=b oXT (?"?"

y(-!R%T (-! < (?"?"G!*f an estimate of< ) say =) is used) there will be an error in the prediction of y(-!) or)

y(-!R%T (-! = e(-! (?"?"A!*f we startk at n and end it at * ) the resulting set of e#uations is

y(n!R%T (n! = e(n!

y( !R%T ( ! = e( ! (?"?"B!

Define the vector of output data9( ! as9( ! R Vy(n! " " " y( !XT (?"?"C!

Define the data matri/ as

D( ! R V%T(n! " " "%T ( !XT (?"?"O!


62/66



%lso define the error vector

+( ! R Ve(n! " " " e( !XT (?"?"= !*f we intend to estimate the parameter vector) the system of e#uations (?"?"B! can be written as

9(n! RD(n!= +( ! (?"?"==!*f we want to estimate the parameter vector at time*T ) we want to minimi

Documents

LFC using SMES and EDLC