Design of nuclear reactor controlsystems to minimize boiling noise

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Authors Peterson, Loren Rolf, 1938-

Publisher The University of Arizona.

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DESIGN OF NUCLEAR REACTOR CONTROL SYSTEMS TO MINIMIZE BOILING NOISE

by-

Loren Rolf Peterson

A Thesis Submitted to the Faculty of theDEPARTMENT OF NUCLEAR ENGINEERING

In Partial Fulfillment of the Requirements For the Degree ofMASTER OF SCIENCE

In the Graduate CollegeTHE UNIVERSITY OF ARIZONA

1 9 6 L

STATEMENT BY AUTHOR

This thesis has been submitted in partial fulfillment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library.

Brief quotations from this thesis are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author.

SIGNED:

APPROVAL BY THESIS DIRECTORThis thesis has been approved on the date shown below:

/ L. E. WEAVER > ofessor of Nuclear Engineering

Da

ACKNOWLEDGMENTS

The author wishes to express his gratitude to Dr0 Lynn E0 Weaver for his guidance and inspiration during this study« The author also wishes to thank Dr® Hubert B® Streets*, Jr® for his consultations and assistance®

iii

TABLE OF CONTENTSPage

LIST OF ILLUSTRATIONS vlLIST OF TABLES o o e o o e o e o o o » o o e e o o e o o o o VXXX

ABSTRACT o o ® # o o @ » » » o o » » o @ @ ® » o o o o x%Chapter

1 BOILING REACTOR NOISE o 1Introduction . e 6 o o e o o o » e e » c o 6 0 0 0 0 1Mean Square and RMS Values of Noise » „ ® e „ e o 2Wiener Optimization 2Minimization of Boiling Reactor Noise <> <,<,«,<> » h

2 STREETS' METHOD FOR OPTIMUM CONTROL SYSTEM DESIGN . 0 8Streets' Method 8Assumptions and Applicability Requirements <,<,.» 8Form of Solutions ......... . . . . 11Spectral Densities ......... 12Bode Plots ......... 13Rules of Streets' Method litWhite Noise Example 0 . 20 Colored Noise Example 2 itMean Square Values from Bode Plots « « = . . , . « 28Saturation Constraints ,..0 » . 32Saturation Constraint Example 31+

3 APPLICATION OF STREETS' METHOD TO BOILING REACTORNOISE MINIMIZATION . I............................. 1+0

EBWR Experimental Data . « . . * « « « 1+0Reactor Transfer Functions 1+0Boxling Noxse o o o o o o o e o e e o e o o o e o 1+1+Model System Studied 1+7Application of Saturation Constraints 1+9Control System Design e « . « . . » « . o o o > o o 53Wide Bandwidth Systems e o e » < , o e . o » e , « o 56Intermediate Bandwidth Systems • • • > « . . o « . s 60Narrow Bandwidth Systems a e e e o . . « e » . » o 63Study Results 0 o o a o a a a « * © o o « 0 « o o 61+

iv

V

TABLE OF C ONTEMTS— ContinuedPage

Chapterit ANALOG SIMULATION OF INTERMEDIATE BANDWIDTH SYSTEMS » , e 6?

Analog Simulation e e o o * o © e e » e e o o e o o o 6*Frequency Response Measurements . . . . . . . . . . a 69Transient Response Measurements 70Output Noise Reduction o o o « < > » . «....<><>. o 75

5 CONCLUSIONS 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 9 0 0 0 0 75Summary o e o e o e o o o e o e o o c o e o o o o o o 79Stability Considerations e . . , , o , o . o o . . e 80

APPENDIX ANALOG COMPUTER PROGRAM o , o o 8l

LIST OF ILLUSTRATIONSPage

Figure1-1 Recording of EBWR Power @ ........ 31-2 Closed-Loop Compensated Reactor System o o * , * , . * 51-3 Equivalent System Block Diagram o e o . o . o . * * . * , , 71-ll Simplified Equivalent System * 7

2-1 Block Diagram of Standard System o , . . * . . , . , . * 92-2 Bode Plot Amplitude-Frequency Relationships * „ . 152-3 Bode Plot of $ Region o o e o » 6 < , o ® e o o o o .172-lj. ' Ifhite Noise Example Bode Plot * . . . * . . . e a o o < , o 222-5 Solution of White Noise Example • o « » o « o - o « o o « o 232-6 Colored Noise Example Bode Plot * o e o e » o e o o o o © 262-7 Solution of Colored Noise Example 272-8 Approximation of Mean Square Value from Area

Under Bode Plot @ o o o o * o , . . * . . * o e o o e 312-9 Saturation Example Bode Plot ........ 362-10 Solution of Saturation Example o . o o , . , o o e . , o 37

3-1 EBWR Transfer Functions « , e . * e . a . . o » o , , * o @ l&l3-2 Bode Plot of o o o © * © © « 6 © o « « ® « e > « « < ? o © li33-3 EBWR Output Noise Spectrum * .......... U53-ii Bode Plot of ip 23 ^ Gxs xn r3-5 Block Diagram of System Studied 1|83—6 Bode Plots of ^ and ^ © © © © © © © © < » © a © © © © 5lx mv x ma

vi

vii

LIST OF ILLUSTRATIONS— ContinuedPage

Figure3-7 Velocity and Acceleration Requirements 0 0 ® o . 6 o < , o 533-8 Bode Plot to Determine System Mean Square Error „ „ «> o 573-9 Bode Plot for Wide Bandwidth System Design <, <> „ 0 „ » <= 583-10 Bode Plot for Intermediate Bandwidth System Design „ o o 6l3-11 Reduction in RMS Output Noise vs System Bandwidth 66

li-1 Analog Computer Block Diagram 68it-2 Reactor Frequency Response 71ii-3 Compensating Element Frequency Response 72it-lj. Closed-Loop System Frequency Response 73it-5 Power Demand Step Response of Compensated and

Unc ompens ated System O o o o . . < , o o « o e o o o e 7itit-6 Output Noise of Compensated and Uncompensated

System for J = 0»U (71 Mwt) « . o . e « o 76it-7 Output Noise of Compensated and Uncompensated

System for ^ *0«2 o o o o o o . e . o o o o o o o o o 77it-8 Output Noise of Compensated and Uncompensated

System for J = 0 o6 77

A-l Analog Computer Program of Reactor Transfer Function „ o 83A-2 Analog Computer Program of Compensating Element

Transfer Function e . . 83

LIST OF TABLESPage

Table2-1 Definitions of Symbols o o o o o o o e o e o o o o o e 102-2 Butterworth Forms for * * o o * o * ? < > o e o <> o & 182-3 Values of Integrals of the Form s I Z (s)j> o o 293-1 Tabulation of Results a » » o « o , * * , * a o e o o 65

viii

ABSTRACT

The design of control systems to minimize boiling reactor out­put noise is investigated using graphical techniques which give good approximations to Wiener optimum systems. In the design of these sys­tems, velocity and acceleration constraints are considered.

Boiling reactor transfer functions and the spectral density of boiling noise are determined from EBWR data. It is found that the ve­locity and acceleration limitations are directly related to the band- widths of the boiling noise spectral density and of the closed-loop compensated reactor system. These limitations are independent of the reactor transfer function.

System bandwidths greater than give excellent noise reduc­tions, have well-behaved transient responses, and are independent of variations in the reactor transfer function except for possibly a gain constant. However, the velocities and accelerations required of the control system are large,

Bandwidths between the reactor resonance and P/£ give good output noise reductions, have good transient responses, and require mod­erate control system velocities and accelerations. Under the practical assumption that the zero power and full power reactor transfer functions are identical in this region, the compensating element is independent of changes in the reactor transfer function due to changes in power level except for possibly a gain constant.

Systems with bandwidths below the reactor resonance give small noise reductions with small velocity and acceleration requirements,

ix

In this bandwidth region, the compensating element transfer function varies with changes of reactor power»

An analog computer study verifies the behavior of the designed control systems*

CHAPTER 1 BOILING REACTOR NOISE

IntroductionThe boiling process occurring in boiling moderator-coolant

nuclear reactors produces variations of reactivity within the reactor core. The vapor bubbles formed by boiling have very poor moderating characteristics compared to the liquid coolant. Since the formation of vapor bubbles by boiling is a random process, the variations in re­activity will also be random. These random variations in reactivity are defined as boiling noise.

At steady state operation, boiling noise produces random var­iations in the neutron density about some average operating level.These variations in neutron density, referred to as reactor output noise, are directly related to the power production of the reactor.

The reactor transfer function, G^(s), relates the Laplace trans form of the input reactivity, I(s), to the Laplace transform of the cor responding reactor output power, P(s). This relationship is given by

P(s) - I(s)Gr(s) . (1-1)

The reactor output noise can thus be related to the boiling noise by the reactor transfer function provided suitable mathematical descrip­tions of the noise can be found.

The magnitude of the output noise fluctuations of a boiling re­actor operating at steady state power becomes greater with respect to

the steady state value at higher power levels« Figure 1-1 shows thevariations in reactor power of the Experimental Boiling Water Reactor(EBWR) at Argonne National Laboratory, Argonne, Illinois, while operat­ing at 100 Mwt.^^ This recording shows that the fluctuations can ex­ceed i 15% of the reactor steady state power level. It also exhibits the random nature of the output noise«

As the noise fluctuations of a boiling reactor become greaterin magnitude with respect to the steady state power level, the reactor becomes less stable. Thus, the output noise is an indication of the stability of a boiling reactor® If the reactor output noise is reduced at high power levels, the reactor will be more stable®

Mean Square and RMS Values of NoiseMean square and root mean square (RMS) values are used to de­

scribe the magnitude of random signals over a long period of time® The RMS value of a signal is merely the square root of its mean square value® In comparison of random signals, the RMS values have more intuitive phys­ical significance since they have the same physical units as the instan­taneous yalues of the signals®

Wiener;OptimizationWiener optimization involves the design of linear time invariant

control systems which minimize the mean square error of the system® Er­ror is defined as the difference between the desired system output and the actual system output® This study considers linear models of boiling reactors with variations of the output around steady state operating levels® The input to the system at steady state operation and the

RE

AC

TOR

P

OW

ER

, M

«

3

M S

110103

100

9 3

9 0

«S -

3 489 3 08 488 8 480IQ- 18 144 T" r i— i i i T i tM S IONM O

1 05

100

9 5

9 0

837 0• 4 • 6 • 88838 SO88SO 82 8 44 4 4 6 4 838 4 0 48

Pig, 1-1 Recording of EBWR Power

desired steady state output are assumed to be a constant which is set equal to zero. Any fluctuations of the output about the zero steady state level are considered to be error.

Minimization of Boiling Reactor NoiseThe object of this study is to design closed-loop linear con­

trol systems to minimize the output noise of boiling reactors. The basic configuration of these systems can be visualized as a unity feedback sys­tem around the reactor and a compensating element which in some manner introduces reactivity into the reactor core to offset the boiling noise, Figure 1-2 is a block diagram in transfer function form of the system under consideration. In the remainder of this thesis, capital letters denote functions of (s) . The symbols of Fig, 1-2 are:

G Reactor Transfer FunctionrGc Compensation Element Transfer FunctionS Laplace Transform of the Power Demand InputN Laplace Transform of Boiling NoiseC Laplace Transform of Power Output

The overall transfer function of the unity feedback system of Fig, 1-2 is given by

G Ow - T t W - • ( 1 - 2 )c r

The reactor power output is

5

PowerDemand

o

BoilingNoiseN Power

Output

Fig. 1-2 Closed-Loop Compensated Reactor System

6or

C - SW + N G (1-W) . (l-u)r

Figure 1-3 is the block diagram of a system mathematically equivalent to that of Fig. 1-2. For S-0 steady state operation Eq<> 1-U reduces to

C - N G (1-W) o (1-5)r

Simplifying Fig. 1-3 for the case where S=0 gives the block diagram of Figo 1-h. This is the basic system used in this study. It is noted that the input to this simplified equivalent system is N G^, the prod­uct of the Laplace transforms of the boiling noise and the reactor transfer function or the Laplace transform of the reactor output noise of the uncompensated reactor. It is noted also that the system output, C, is the system error since for S-0 the desired output is zero.

In this investigation, a method developed by Dr. R. B. Streets to obtain good approximations of Wiener optimum control systems was applied to the system shown in Fig. 1-1*.

7

Fig* 1-3 Equivalent System Block Diagram

Fig. 1-U Simplified Equivalent System

CHAPTER 2STREETS' METHOD FOR DESIGN OF OPTIMUM LINEAR CONTROL SYSTEMS

Streets' MethodStreets' method is a technique which gives good approximations

to the open-loop transfer functions of optimum linear time invarient control systems. This technique involves the use and interpretation of straight line log-log Bode plots. It omits a great deal of the mathe­matical complexity encountered in strictly analytical solutions of Wiener optimization problems. Compared to analytical methods, it also gives more physical insight into the behavior of systems.

Figure 2-1 shows the standard system used in applying Streets' method. Inputs to the system are separated into the input signal and noise. The symbols used in applying this method are listed in Table 2-1. "

Assumptions and Applicability RequirementsThe standard assumptions of Wiener theory are made in the appli­

cation of Streets' method. These assumptions are that random inputs to the system are stationary and ergodic and that the system designed will be a linear time invarient system. System error is defined as the difference between the desired system output and the actual system out­put. The optimum linear system is defined as that system which minimizes the mean square error when random inputs are applied or the integral of the error squared when deterministic inputs are applied.

9

♦ L

Fig. 2-1 Block Diagram of Standard System

10TABLE 2-1

DEFINITIONS OF SYMBOLS

s Input Signaln Noise Signalr System Input Signali Output Signal from Compensation Networkq Input Signal to Saturating Elementc System Output Signale System Error SignalGj. Fixed Plant Transfer FunctionGc Compensation Network Transfer FunctionG Open-loop System Transfer Function G ■ G^G^

n.W Closed-loop System Transfer Function W

s

1 + GD Linear Operator Applied to Compensation Network Output

Signal to Form Input Signal to Saturating Element-L Linear Range of Saturating ElementJ Allowable Maximum Mean Square Value of Saturating

Signal q® Power Spectral Density of Random Signal x1 xxXX+ T

Factored Power Spectral Density of 0 x -L xx2^ Mean Square Value of Signal x

Root Mean Square Value of Signal x m \/cr %

11In Streets’ method it is assumed that the desired output is the

input signal, free of any noise or disturbances that may be present in the system. It is further assumed that the input signal and noise are uncorrelated.

In order to obtain a meaningful solution using Streets' method the RMS error must be finite and less than one-half the RMS value of the input signal. For physical realizability, the fixed plant transfer function, G^, must have a pole-zero excess of one or more. The closed- loop transfer function, W, must have a pole-zero excess greater than or equal to that of G^0

Form of SolutionsStreets' method determines the overall open-loop transfer func­

tion of the optimum linear system of Fig. 2-1,

The open-loop transfer function is related to the closed-loop loop transfer function, W, by the conventional relationship for unity feedback closed-loop systems.

Two approximations relating W to G which are of basic importance in ap plying this method are

(2-1)

(2-2 )

W 1 for G » 1 (2-3)and

W G for G « 1 (2-1*)

Streets' method can be applied to systems with either determin­istic or random input signals. In this thesis only the random signal case is used.

Spectral DensitiesTo apply this method in the random signal case, each signal

must be described by its power spectral density. A random signal which is stationary and ergodic can be described statistically by its auto­correlation function.

product f(t) f(t +T) where ^ is a time delay.The power spectral density of the signal f(t) is the Fourier

transform of its autocorrelation function,

oo

-00

Conversely, the autocorrelation function can be obtained from the power spectral density by the inverse transformation,

(T ) - E [f(t) f(t +T)] - ^Too — f(t) f(t *'T) dt.T/

(2

-T

In this expression E is the ensemble average of the

(2-6)

13The power spectral density of any random signal can be factored

into functions of (+s) and (-s) such that♦

0 (s) ■ 0 (+s) k (-s) - 0 <j)1 ff ± £. -I f If

(2-8)

In applying Streets' method, the spectral density of each signal under- +

consideration is factored in this manner. The 0 factor of the spec­if

tral density, subsequently referred to as the factored spectral density, is then treated in a manner analogous to a transfer function and plotted on a log-log Bode plot.

Bode PlotsConventional Bode plotting techniques are used. The vertical

scale is plotted in units of log^Q of amplitude M. This scale can readily be converted to decibels amplitude by the relationship

N db - 20 log10 M . (2-9)

The horizontal scale is plotted in units of log^Q of frequency w , where w is in radians per second.

In most cases the amplitude scale is normalized so that the fac­tored spectral densities of the input signal and the noise have a magni­tude of one at their crossover point. The frequency where signal-noisecrossover occurs is eu •c

Whenever the slope of a straight line Bode asymptote is used in this thesis, the slope is assumed to be in units of decades M per decade tu . Thus, a slope of -1 corresponds to a slope of -20 db per decade when the amplitude scale is converted to decibels.

In working with Bode plots the relationship

A ((vj'k - B (W. r k (2-10)a o

is useful in determining the amplitude and frequency relationships be­tween two points on a plot. As shown in Fig, 2-2, A and B are the am­plitudes at frequenciescua and W and k is the slope in decades M per decade w of the straight line joining the two points.

Rules of Streets1 MethodAll problems which can be solved by Streets’ method are basi­

cally input-noise problems. Each problem is broken down into the• - + y 4 • - + -y 4*

consideration of the frequency ranges where 0 , where 0 < (j) , t + ir+ s - ^ n s n

and where 0 ^5 0 .8 n r + I*In the frequency range where d) (b , the optimum open-loop

s In _ +transfer function, G, is equal to or proportional to t . The opti--*-smum closed-loop transfer function in this region is W • 1. Thesecriteria are compatible since they are consistent with the approxima­tion of Eq«, 2-3. Physically, this specification means that in thisfrequency range where t> i) , the optimum realizable system is

s J-none that transmits the dominant signal plus the less significant noisewithout any alteration.

In the frequency range where t d) , the approximation of-in +Eq. 2-U applies and G % W. Here the transmitted noise. Wo) , must be attenuated to have a finite mean square value. This is attained if

/ T +the slope ( W $ ) - -1 . •i n

15

10

A --

decades Mdecade w

u/,LU balo®10 w

Fig* 2-2 Bode Plot Amplitude-Frequency Relationships

A (u/ rk - B (u/. )-ka d

16To assure adequate phase margin for stability, the open-loop

transfer function, G, must have a -1 slope at the crossover frequen­cy. Butterworth breaks are inserted in G in the crossover regionwhere t ^ y . Butterworth breaks in this region give maximum phase **-s -L nmargin with a minimum width band of -1 slope. The form of the Butter­worth formula Gg, describing the open-loop transfer function in the(b ^ <b region, depends upon the slopes of i) and d) at the signal- -L s - n s -Ln +noise crossover frequency, W • Let k be the slope of u) and let

+ c J- sh be the slope of j) at COc as shown in Fig* 2-3. Then refer toTable 2-2 to determine G_ - G in the a) ^ a) region for the combin-s J-nation of k and h under consideration. The complex frequency X usedin Table 2-2 is normalized to the crossover frequency so that

A su> UJ,c cIn summary, the three basic rules of Streets1 method are:

-p + y -f1. For 0 » 0

s n

-- +

GOC ()-L n

W - 1 (2-11)

T +2. For I) « ( )

T +

n

T +G % W slope ( W O ) - -1 (2-12)

•L n

3. For•*> s n

aB slope (G) - -1 at (2-13)UJ according to Table 2-2.

In the design of optimum systems, the $ S5 $ region should•L s nbe considered first to determine Gg. Then G is designed to furthermeet the specifications of the I I and $ $ regions.^s -Ln ^s J-n

1

slope * k

slope * h

clogic w

Fig. 2-3 Bode Plot of Region

18TABLE 2-2

BUTTERWORTH FORMS G_

Slope T + Slope T + Open-loop Transfer Function G g in Region d) £3 a)

L(k)

in(h)

A - — - j —w c

- 1 0

1

1A

(.7 0 7 )+

2 (.5oo)X [1 +

3 (.383)X ( 1 + t 1 + + ( i r k ) 2 J

-2 0 1 + 7 w A 2

1 (.5d°) ( i + 7 ^ 5 ) X ^ 1 z.oocP

2 ( . 2 9 3 ) ( , ♦a z [ 1 * 4 W - * < i A > > 2 J

3( . 191) (1 ♦ 7 ^ )

A (1 + ^ 5) Ji * + (ligg) ]

TABLE 2-2— Continued19

.707

< • « ) [x . i i j g i u . .

(.191)

(.109)

+Region.Table of Butterworth Forms Gn in the

20Examples for both white and colored noise can now be considered.

White noise is defined as a random signal having constant spectral den­sity over all frequencies. Colored noise is a random band limited sig­nal with a spectral density which varies as a function of frequency.

White Noise ExampleAn example with white noise present is first considered.

Given:

L (s) . loU-s2(102 - s2)

and _(s) -

nn ICTIdetermine the optimum open-loop transfer function, G •

Solution:The signal and noise spectral densities are first normalized

in amplitude so that 1 (s) ■ 1 and then are factored yielding1 nn

and

I

; io3

— +0 and Y are then plotted on a Bode plot. Fig, 2-lw It can be -L s -L n

seen from the Bode plot or by applying Eq, 2-10 that the crossover fre­quency is uo - 100. The slope (t ) at tU is -2 . Since G must

C lg Chave a -1 slope at crossover, a Butterworth break must be inserted.

21Referring to Table 2-2 for k - -2 and h - 0, it is seen that

. 1 * mfrmj . ^ ( 1 • 7° . ? . „ . u ,« & >

Gg is also plotted Fig. 2-U.Figure 2-ii shows that f°r 10 < CU <70.7 and then

breaks up to a -1 slope at W * 70.7. The first rule, Eq. 2-11, states that 0oe J * in the J)* region. Since Gg - G coincides withinto the region where G oc * for it is apparent that inthis instance, G « ([) * in the ^ r e g i o n . The optimum G isformed from Gg for tv < 10 by making slope (G) ■ -1 so that G - <[)* atlow frequencies.

The second rule, Eq. 2-12, states that slope ( W ^ ) ■ -1 andthat G = W in the ^ ^ * region. For white noise the slope

■ 0, and thus slope (W) « slope ) - -1. Since G alreadyhas a -1 slope at crossover from Gg - G, no further changes in the formof G are needed in the $ « $ + region.s n

The optimum open-loop transfer function obtained from Fig. 2-L and plotted Fig. 2-5 is

G . 103 ,.l 10-j L . (2-15). (1 ♦ 5 j)

If a fixed plant were given, the optimum compensation networkis

22

10'

10

10

K f 1 ''

70.710

3,2 1010 10'

Fig. 2-lt White Noise Example Bode Plot

23

10

310

210

10

1

110

-21010'101010

LU

Fig, 2-5 Solution of White Noise Example

Colored Noise Example

Using these same techniques, the more involved case where colored noise is present can be solved in a similar manner.Given:

U9 (1C^) s1*I

(s) -m (U9-S2) ( i c r - s )

and0 (s) ■-L 88 (10-2 - sS)

Solution:Putting these given spectral densities into time constant form

and factoring yields

r . a!--(1 + |) (1 + Jo q)

and

$ 10

^ g and are plotted on a Bode plot Fig. 2-6. Normalization of the Bode plot amplitude scale is not required in this instance since the signal-noise crossover occurs at M - 1.

25The formula for Gg for k • -1 and h * -2 obtained from Table

2-2 is

G. .5oo (2-17)B

Gg is plotted Fig. 2-7. To make Goc J^for $ J a break is in­serted at W - 0.10 to make slope (G) - slope (Jg ) • In the $ +g C 1 +n region where G % W and where slope (Wj)^ - -1, changes in slope (G) are made to offset the changes in slope (£)*). Thus, the slope (G) must in­crease by one at GU - 7 and again at W » 100. The solid lines Fig. 2-7 indicate these modifications to the Bode plot of Gg to obtain the optimum open-loop transfer function.

Once proficiency in applying this method is gained, all plotting can easily be done on a single Bode plot with no confusion. In many cases it is merely necessary to visualize the Bode plot to arrive at a solu­tion for G. Streets has shown that less than 5$ increase in RMS error is caused by using the approximate solutions obtained by this method instead of analytically derived Wiener systems for cases where

G -

(2-18)

26

10

-1 12 1001010

Fig. 2-6 Colored Noise Example Bode Plot

27

Mio 2 --

10

10

10

10

102

I V10

Fig, 2-7 Solution of Colored Noise Example

28

(o)s 10(0)

n

Mean Square Values from Bode PlotsIn order to compare the results of applying this method, the

mean square values of random signals must be obtained. Using Parseval's theorem, the mean square value of x(t) is

oo oo0-^ • ( dt - j X(B) X(-s) ds - l{x(s^ (2-19)

-00 -jOO

Letting

- srfiC(s) Co * Cl8 + D2a2 + ...... ^ - l 8"'1

d + d-,5 + dns^ + d sno 1 2 n

can be expressed as a function of X^(s) alone. The values of ■ I^Xn(s)| for the commonly used cases of n-1, 2, and 3 are tab­

ulated Table 2-3. If cases for U n ^ 10 cannot be avoided, consult Newton, Gould, and K a i s e r . C a s e s for orders n ^ 3 become very complicated and should be avoided if possible. This can be done by proper interpretation of the Bode plot of the signal factored spectral density, .

The area under ^*on a Bode plot is equal to the mean square value of x(t). Thus,

29TABLE 2-3

VALUES OF INTEGRALS OF THE FORM - I {x(s)}00 , jOO

< r i ’ j X2 (t) dt - 27j j Xn(s) Xn (-s) ds - l ( x n (s)j-00 -JOO

n Xn (3) I (xn (s)}

c c 2o odo + dls 2 dodl

C + C s o 1d + d_s + d0s o l d.

Cl?do * Co2d2 2 dodld2

C + C_s + c0s o 1 c C2do V (Cr 2CoC2 )dod 3 + W 32 3d + d-s + d0s + d^s o 1 d j 2 d d~(-d d_ + o 3 o 3 dld2 )

In visualizing the area under a curve on a log-log Bode plot,it must be remembered that the geometrical area to the upper right isthe most heavily weighted numerically. This allows modification ofJ)* at lower values of frequency and magnitude to reduce the order of

2- (7^ with little loss of accuracy in determining the mean square value of x(t)0

For example, consider the factored spectral density

T +I - - - - - 5- - - - - —-X (1 ♦ 8) (1 + 5J5_) ! * B ♦ JL s2

plotted in solid lines Fig. 2-8.2The exact solution for (j- is

Using the form I | (s obtained from Table 2-3, the exact mean square value is

* ' i ' a . # ! ® - • * ■ t o -6

Little accuracy is lost by making the lower order approximation indicated by the dashed line. Fig. 2-8, giving

31

M

10- -

10

10

1010‘ 10101010

Fig. 2-8 Approximation of Mean Square Value from Area Under Bode Plot

In applying Streets’ method, it is frequently convenient to use such approximations to determine the mean square value of signals.This is particularly true in cases where saturation constraints are present.

Saturation ConstraintsIn real control systems, the linear response region is limited

in some physical manner. To take this into account, saturation con­straints are placed on the output of the compensation network of the standard system under consideration. Fig. 2-1. The mean square valueof a function q(t) of the compensator network output, i(t), is lim-

2ited to a constant value, . The function q(t) is visualized asImaxthe input to a linear element that saturates at i L , The value ofJmo~ is chosen according to the desired probability that q(t) will qmaxexceed i L .

For a Gaussian distributed signal with zero mean, choosing gives a probability of saturation. For the satur­

ation probability is 1.2U%.^^Referring to Fig. 2-1, the value of the linear operator D, op­

erating on i(t) to form q(t), depends upon the physical significance of i(t). If the output of the compensation network corresponds to posi­tion, then D*1 is used to impose position constraints, D-s for

2velocity constraints, and D»=s for acceleration constraints.

33Again referring to Fig. 2-1, it is seen that

iY +

W ()X r (2-21,)

and

w TD0 - Do) ■ D — x™ • (2-2$)

lq Xi uf

Let Y +D 0 .. r

K - (2-26)

If has a numerator of greater order than that of its denominator,

slope (J ) — 0 as W — ► oo

mthen

and the mean square value becomes infinite. In order to make the mean square value of q have a finite value, W must insert a break at some

dw£frequency b to make slope ( - slope (- — - -1 for W > b . Thechoice of break frequency, b, determines the mean square value of q

2and thus applies a constraint in order to satisfy <7 • J. Since Wis the closed-loop transfer function, the constraint determines the bandwidth b of the system or, conversely, the bandwidth b deter­mines the mean square value of the input to the saturating element of the system.

In the case where the noise input to the system is - 0 and thus , the quantity <[) * can be treated like a noise signal.Here, j) * and $)* are plotted on a Bode plot with their crossover at the

bandwidth frequency, b. The magnitude scale of the plot is normalized so that M-l at the crossover point. The problem is then handled as a colored noise problem. The case with $ * “ 0 will be the only satur­ation case applicable to this thesis since the equivalent reactor sys­tem under consideration is that of Fig. 1-U.

Using Streets' method for saturation constraints, problems constraining n different signals are treated as n separate prob­lems. One constraint will be found to be critical and the remaining constraints may be neglected. A basic requirement for use of this method in the saturation constraint case is

(0)-crtor- < 00 • ( 2 - 2 7 )

Saturation Constraint ExampleA simple example of a saturation constraint problem with

$ * - 0 is now considered. x nGiven:

$» * * < 1 * #>> 1 * ft

2D - s , and b ■ 200 radians per second .

2Determine W, G, G , and g— •* ' c q

Solution:First Eqs. 2-2U and 2-2$ are used to determine

35

and

0

-p +0-Lq

Thus, from Eq, 2-26

W

.33

1 + .33

y +0-I- m 7 3

Then, and are plotted on a Bode plot Fig. 2-9 so that their cross­over is at M-l and b-200 radians per second. It is seen from on this plot that W must break downward at frequency b with a -2 slope in order for slope (j) ) ■ slope (wj* ) * -1 for ut b . As in the col­ored noise case, W-l for w b . Thus, the form of W is determined to be

W - ------tr-J---- n— (2-28)

which is plotted Fig. 2-10. For the optimum closed-loop system, criti- , cal damping with J - *707 is usually considered to be the best system giving

W (2-29)d + '/ ? ' 3 + sL)200 U0000;

36

10

10

10

10

10

20021 10'1010

Fig, 2-9 Saturation Example Bode Plot

37

M

2 ..10

10

10

200 U0010

h32icr1010

Fig. 2-10 Solution of Saturation Example

The open-loop transfer function G can be obtained from W byformula

0 . -yJi-g- . (2-30)

However, the calculations usually become very involved in non-trivial cases• At this point, the problem is treated as a colored noise prob­lem to determine G •

The Butterworth form in the region obtained fromTable 2-2 for k--2 and h-1 is

.500( 1 + £00(200) ) 2(10)1* ( 1 + 100 )b " — r — — — 5— 7 ~ — r;;.-------------- (2-31)

^ U (1 + 2T20oy) s(1 + E o o )

The $ 4 » $ 4 and $ 4« $ 4 regions are treated in the same manner x s x m x s -L mas was the colored noise case giving

1260 (1 + rS-)0 - ---- — ------- • (2-32)

(i + 3 5 ) (i + 3 5 ) (1 + Coo)

The optimum transfer function G is plotted Fig* 2-10.Since G-G^G^ , the optimum compensation network is

Q 1260 (1 + I5 o ) . (2-33)

° " + + ^The mean square value of the saturating signal is obtained by

plotting J 4 Fig. 2-9 and then taking the area under the curve. The -2 slope for tv < .33 can be neglected.

39Thus,

2 ^ I f (5xlO~3) 3 j . 2g (10)~6

q I 1 + + (2io)J / 2(1) (2&) (n 5 ^ 0 )

CTq2 ~ 70.7 (2'31*)These examples illustrate all of the techniques of Streets'

method used in the next chapter.

CHAPTER 3APPLICATION OF STREETS» METHOD TO BOILING

REACTOR NOISE MINIMIZATION

EBWR Experimental DataSince experimental data from EBWR are readily available in lit­

erature and since the behavior of that reactor is typical of boiling reactorsj, EBWR data are used in this study. The techniques developed here can be applied to similar data from any boiling reactor.

Reactor Transfer FunctionsThe Bode amplitude and phase plots of Fig. 3-1 show the reactor

transfer functions of EBWR at various power l e v e l s . T h e s e transfer functions were obtained by rod oscillator frequency response measure­ments using cross-correlation noise rejection techniques. Because of uncertainty of oscillator rod worth in the presence of coolant boiling, the amplitudes of these transfer functions were normalized to that of the zero power transfer function by a best fit at the three highest frequencies measured. This type of normalization is valid provided feedback within the reactor becomes negligible at high frequencies.Experimental studies of EBWR internal feedback functions by DeShong and

(5 )Lipinski indicate this assumption is valid.The phase shift of minimum phase systems is uniquely determined

by the amplitude characteristics of the system. Comparison of the am­plitude and phase shift plots of Fig. 3-1 show that the reactor is a

Ul

1" l \ l m r i j —■ | i - t r r n j - i - i - i i j n r | —

-X . 60 M w - ^ A - — 71 Mw

_ ZERO POWER 2 0 M w ^ X / U — 40MW _

io m w^ X j L V

-

— —

____ 1__1_L_I J l u i 1 - J 1 i 1 1.1.11.. 1 l_ l.l_ l_ lix l---__ i__1_1-1-1 l. ii

8071 Mw

6 0 MW

4 0 MW

20 MW

£ 20 10 MWX</>

I20 ZERO POWER

4 0

100100.10.01CJ, ro d /s e c

Fig. 3-1 EBWR Transfer Functions

An *= peak value of sinusoidal component of neutron flux Anc « average neutron flux

k = reactivity f3 5 delayed-neutron fraction

minimum phase system. Thus, Bode amplitude plots are sufficient for accurate analysis of this system.

Figure 3-2 is a straight line Bode plot of the reactor trans­fer function based on the 60 and 71 Mwt curves of Fig. 3-1. The reac­tor transfer function equation determined from this straight line plot is

0 . -33 (1 * Tig) (1 * 3 ) (1 * IT^ (1 * . (3.1)

(1 + (1 + ^ (1 * t § * (1 * i#o)

The final downward break at 120 radians per second in Fig. 3-2 and Eq. 3-1 is based on the assumption that the zero power and the full power reactor transfer functions become identical in the high frequency region. The high frequency downward break of the zero power reactor transfer function occurs at w * , where ft is the delayed neutronfraction and £ is the neutron generation time.^^ The quantity 0 is a constant for each individual reactor, depending on the fuel, mod­erator, and absorbers present in the reactor,, For EBWR, |3 — 0.007 and£ ^ 6 (10)*" , making the &/£ break occur at L0 — 120 radians per

(7)second.The dominant change in the reactor transfer functions at the

different power levels shown in Fig. 3-1 is a resonance at approxi­mately 10 radians per second. This resonance is represented in the transfer function of Eq. 3-1 by the second order denominator term,(1 + 21-2- ♦ ^2— ). In this expression the height of the resonance depends upon the value of the damping ratio, J . Resonance occurs for J 0.707. As the damping ratio decreases below this value, the

1*3

10 - -

1 “

•1..10

1201.810

32■12 1010101010

Fig. 3-2 Bode Plot of

height of the resonance increases, the system transient response becomes more oscillatory, and the system becomes less stable.

The transfer function G^, Eq. 3-1, is used as the fixed plant for the application of Streets1 method to the problem of boiling reac­tor noise minimization. This transfer function characterizes the re­actor at high output power levels. The specific power level represented is determined by the value of the damping ratio that is used.

Boiling NoiseThe power density spectrum of EBWR output power noise is shown

Fig. 3-3 .^ This plot shows the spectrum to be relatively flat to cu - 0.33 radians per second. It then drops off with a slope of roughly 100 (Mwt) per decade cu at the higher frequencies. Taking the square root of the amplitude scale which corresponds to factoring the power density spectrum gives a high frequency slope of 10 Mwt per decade uj 0

Recalling the relationship between boiling noise and output noise, it is possible to determine the boiling noise spectral density using this plot and the known reactor transfer function. Using factored spectral densities to describe the noise signals. Fig. 3-3 is a plot of

T * 1r + r +

0 « () G - <)r-Lp JLn J- s

Since the break at u; = 0.33 does not occur in G^, it must beassociated with the boiling noise signal. Other investigation has

(7)shown that the boiling noise has a band limited form. The boiling noise of EBWR before modification for 100 Mwt operation was found to

10

10

1001 101.01rREOUENCV. nd/eec

Fig. 3-3 E3WR Output Noise Spectrum

1

33

10

210

1.8310.22 1010 10'1

Fig. 3-U Bode Plot of • 5 n Gr

U6be white up to a break point near W = 0.3 cycles per second ^ 1*8 radians per second. Its power density spectrum then has a -1 slope for w > 0.3 cycles per second. The spectrum shown in Fig. 3-3 indicates that after modification of EBWR for 100 Mwt operation, this break point frequency is approximately 0.33 radians per second. Thus, the boiling noise fac­tored spectral density used in this report is the band limited function

t — . (3-3)1 * 733

This function is normalized to unity amplitude in the white portion of the noise spectrum since the boiling noise is the basic input to the system. A plot of the uncompensated reactor output noise asymptotes obtained by substituting the values 0^ of Eq. 3-1 and J * of Eq. 3-3 into Eq. 3-2 is shown in Fig. 3-ii« Comparison of Figs. 3-3 and 3-U shows that the two curves are basically identical, especially in the high frequency range.

The band limited spectral density of the boiling noise is as­sumed to be relatively constant at all power levels for constant pres­sure operation of the reactor. The results obtained using Streets’ method in this particular problem are not very sensitive to small var­iations in the boiling noise bandwidth.

A numerical value of the mean square value of the boiling noise of Eq. 3-3 is calculated for use in later comparisons. Using Eq. 2-20 without any approximations gives

^ " I - °-167 •755

hi

(3-U)

Taking the square root of this value to determine the RMS value of the normalized boiling noise gives

CT^ - O.I4I . (3-5)

The mean square and RMS value of the uncompensated reactor out­put noise are calculated utilizing the Bode plot of Fig. 3-U* Taking the approximate area under the ^ n J curve gives

.01*5" T o

- 1

0.01*0 + 0.010 - 0.003

^ or 0.01*7 . (3-6)

The RMS value of the uncompensated reactor output noise, therefore, is

(j— — 0.22 . (3-7)

Model System StudiedComparing the simplified reactor system, Fig. l-U, and the

standard system for application of Streets' method, Fig. 2-1, shows that the two systems are identical. Fig. 3-5 shows the system and the parameters used in this study. The fixed plant is the reactor trans­fer function, G^, of Eq. 3-1 • The factored spectral density of the

Fig. 3-5 Block Diagram of System Studied

input to the system is1*9

T +0-*-s

(3-8)1 + .33

The factored spectral density of the compensating element output is

T +0i

T +0 W -L-n

W . (3-9)1 + 7 3 5

Application of Saturation ConstraintsIn the design of physically realizable systems, either satur­

ation constraints or required linear operating ranges of system com­ponents must be considered. To do this, the amplitude of the compen­sating element output is visualized as the position of a control rod inserting reactivity into the reactor core to offset the boiling noise.Compensation reactivity position, velocity, and acceleration are then

2associated with values of D*l, D-s, and D-s , respectively.For velocity constraints, the input to the saturating element

is— — * — A s WY + y + Y *0 x s 0 - s 0-Lqv -1 i J-n

(3-10)1 + 735

T +and <) is 3-mv

T T +0 ■ s 03-mv J-n

(3-111 + 733

For the case of acceleration constraint50

and

Y +0■i-qa

Y +0-Lma

2 I + 2s () - s 0 WJ-i

s2 r1 + .33

s2W (3-12)1 + .33

(3-13

It can be seen from these equations that the saturation con­straints of the reactor system are independent of the reactor trans­fer function. The saturation constraints are directly dependent upon the bandwidths of the closed-loop system and the band limited boiling noise.

A Bode plot of 3) + and <j) + is constructed. Fig. 3-6. From thisr mv maplot it can be seen that W must break downward at some bandwidth fre­quency, b, with at least a -1 slope to impose a velocity constraint on the system, and it must break with at least a -2 slope to impose an ac­celeration constraint. For bandwidths b^>0033 radians per second, the lower order approximations of J)q introduced in Chapter 2 can be used to determine the mean square values of the saturating signals.

These approximations yield

&~qv.33 0.10 b

1 + §- 0.05 b (3-H)

for the velocity constraint case and

51

ma10

qv

2 1 210 10 10 10

Fig, 3-6 Hode Plots of j>+ and-Lmv ma

52

for the acceleration constraint case.These values are normalized to the mean square value of the

boiling noise, converted to RMS values, and plotted as a function ofthe closed-loop system bandwidth in Fig, 3-7, This plot and Eqs, 3-lhand 3-15 show that the acceleration constraint will be the dominant con-

2straint in most problems, since the mean square acceleration, ' , isqaproportional to the bandwidth cubed. Since some maximum acceleration limit will be present in any real system, the final slope of the closed- loop transfer function, W, must be a -2 slope regardless of the pres­ence of a velocity constraint introducing a -1 slope at a lower frequency.

The systems studied are designed for acceleration constraints imposed at various closed-loop system bandwidths. Fig, 3-7 is then interpreted as the required RMS values of compensation reactivity velo­city and acceleration, in units of the RMS value of the boiling noise.

These general quantities can be related to physical quantities of a hypothetical system in the following manner. Assume that a con­trol rod having a total reactivity in a rod travel of one foot is the output of the compensating element. The limits of rod travel set

P m± L - * o For 1.2k% probability of saturation with a Gaussian in­put signal, the maximum mean square value of compensation reactivity must be

53

M

10

10

10 10'

System Bandwidth (radians/sec)

Fig* 3-7 Velocity and Acceleration Requirements

<7^ ' J - (2 3 ) ’ (3 -^ * (3-l6)max

If the maximum mean square value of the compensation reactivity is set equal to the mean square value of boiling noise reactivity, then

- cr/ - 9 (3-17)MTiax

or in RMS values

P T" T * (3*18)^max

Assuming a system bandwidth, b-300 radians per second. Fig. 3-7 gives

qV-^2 - 9.5 (>19)CTnand

- qa3- ° - 21*00 . (>20) cr5

Substituting Eq. 3-18 into Eq. 3-19, the required RMS value of compen­sation reactivity velocity for this hypothetical system is

qv3009.5 PT p ,— g • 1.9 'T/sec - 1.9 ft/sec (3-21)

and the RMS compensation acceleration required is

5521(00/° P

^ ■ g--- - U8 0 IT/sec - U8 0 ft/sec . (3-22)qa300

This example indicates that large compensation acceleration capabilities are required for systems with bandwidths b » $/& • Nar­rower system bandwidths will obviously decrease the required compen­sating element acceleration capabilities to more easily realizable values•

Control System DesignControl systems are designed for various bandwidth regions with

reference to the characteristics of the reactor transfer function. Streets’ method for acceleration constraints is applied to the reactor transfer function and band limited reactivity noise developed earlier in this chapter. The desired closed-loop transfer function in all cases is

W « ------------- g— • (3-23sl + v£j> +

b b2

The output error of the closed-loop compensated system isgiven by

T+ -r +0 - 0 Gr (1 - W) . (3-2M-L e -L n

Since the form of W is known, Eq. 3-23, the quantity (l - W) as a function of system bandwidth is

56

(3-25

In order to determine the mean square and RMS error, the factored error spectral density

gions that are of interest in control system design. The region be-

moderate bandwidths. The region below the resonance gives systems with relatively small bandwidths. These three regions are referred to, re­spectively, as the wide bandwidth, the intermediate bandwidth, and the narrow bandwidth regionsv

Wide Bandwidth Systems

shown Fig. 3-9# The open-loop transfer function obtained from this

eT +0

is plotted. Fig. 3-8, for various system bandwidths.Referring to Fig. 3-8, there are three distinct frequency re

yond 0/g gives closed-loop systems with large bandwidths. The region between the resonance and $/1 results in closed-loop systems with

The Bode plot for a bandwidth of b«300 radians per second is

plot is(3-27)

57

30b

300b

300 '

21-2 10101010 uu

Fig. 3-8 Bode Plot to Determine System Mean Square Error

58

--10

--10

10 - -

/ - -10

10

--10ma

10

--10

300

10'101010

Fig, 3-9 Bode Plot for Wide Bandwidth System Design

Dividing this function by the reactor transfer function gives the com­pensating element transfer function

G 2 m 1050 (1 * l f o ) . (3-28)

C3°0 " Gr ' (1 + 33) (1 + 555)

The mean square and RMS error for this system is calculated using the Bode plot of Fig* 3-8 and the area approximations introduced in Chapter2. This calculation gives

C7-2 - I < ----------- :--- r- - 1.06 (10)'1* (3-29)e300 d + m )

and the RMS error is

cr~ ^ 0.010 . (3-30)8 300

Normalizing this value to the RMS value of the uncompensated reactor * output noise gives the reduction of RMS output noise obtained by using the closed-loop compensated reactor system. In this case for 300 rad­ians per second bandwidth, the calculated RMS output noise reduction is

6'e'— 22<L ^ 0.05 . (3-3i)

0~T)

Looking at the Bode plot. Fig. 3-8, and the form of Eqs. 3-27 and 3-28, it is apparent that 0^ is independent of G^, except for possi­bly a gain constant, for bandwidths b » ®/£ • Equation 3-31

60indicates that the reduction in RMS output noise for wide bandwidth systems is excellent. However, the high reactivity accelerations re­quired for bandwidths of this order. Fig. 3-7 and Eqs. 3-20 and 3-22, make physical realizability of such systems quite difficult.

Intermediate Bandwidth SystemsUsing Fig0 3-10, the open-loop transfer function for a band­

width of 1*0 radians per second is calculated to be

(3-32)and the compensation transfer function is found to be

G (3-33)CU° ( 1 * 7 3 3 ) ( 1 * 5 ^ 3 )

The mean square and RMS error are

anda— - 0.053 % 0

(3-35)

This gives an RMS output noise reduction of

61

10

--1

--10io - -

10

1010ma /

--102__10

103__10

•101010'101010

Fig. 3-10 Bode Plot for Intermediate Bandwidth System Design

62

^ h O =a 0,2h » (3-36)

The required RMS compensation velocity and acceleration obtained from Fig. 3-7 or Eqs. 3-lli and 3-15 are

3.U (3-37)

and

^~qa^ - 115 . (3-38)

(2~n

Using the hypothetical compensating element control rod me­chanism described previously, page 52, for a system having bandwidth b «U0 radians per second, the required RMS compensation velocity and acceleration are

3.U m n- 0.7 'T/sec - 0.7 ft/sec (3-39)

and

0 Y 1 T - 23 fr/sec2 - 23 ft/aec2 . (3-1*0)qal*0 5

It is seen from Eq. 3-33 that the ft fa, term, (1 + ), of Galso appears in Gc^ . This zero in cancels the pole ofG^ to give the optimum open-loop transfer function G ^ of Eq. 3-32«, Considering Fig. 3-10, it can be seen that this cancellation occurs for

63any system bandwidth in the flat response range of between the reson­ance and $ fa. • Since 0/^ is a constant for all reactor power levels, the compensating element transfer function is still basically independent of variations in the reactor transfer function due to power level changes. The compensating element gain constant may vary a small amount as a function of the reactor power level. From the standpoint of required compensation acceleration, Eqs. 3-38 and 3-UO and Fig. 3-7 show that physical realizability of intermediate bandwidth systems is much easier than realization of wide bandwidth systems. Equation 3-36 indi­cates that the reductions of output noise achieved with intermediate bandwidth systems are quite significant.

Narrow Bandwidth SystemsIf the bandwidth of the compensated system is decreased below

the reactor resonance, the compensation element must have terms to can­cel the poles and zeroes of the reactor transfer function occurring above the bandwidth break frequency. This is illustrated by the com­pensating element transfer function obtained for b * 3 radians per second.

0 , . ^ (1 A > . (3.U13 (i ♦ « ) (i +i^» ♦ =L) (i . s )7 5 3 ; v x * - j - * T , u - i p

Since the resonance and also the lower frequency breaks of the reactor transfer function vary with power level, narrow bandwidth system com­pensation networks are not independent of reactor power level.

While the acceleration requirements are much smaller for narrow bandwidth systems, the reduction in output noise is less significant*In the case of b ■ 3 radians per second, the reduction of RMS output noise is

~ 0 .U8 „ (3-1*2)Cp

The compensating element dependence upon the reactor power level and the less significant reductions in output noise for narrow system band- widths greatly reduce the practicality of such systems*

Study ResultsTable 3-1 compares the compensating element transfer functions

and corresponding values of , and for se-e 0n Onlected closed-loop system bandwidths. The approximate reduction in out­put noise as a function of system bandwidth is shown in Fig*,3-11*

To further investigate the possibility of using intermediate bandwidth systems, an analog simulation study was made of a control sys­tem designed to operate in this range*

TABLE 3-1 COMPARISON OF STUDY RESULTS

b 2 dTqv -(TH

CTqa Gc

300 1.06 (10 )"1* .05 9.5 2U001050 (i ♦ ^

d + 7 3 3 ) d + ^ 5 )

100 7.9 (10)-1* .13 5.3 U60

uo 2.8 (10)"3 .21* 3oli 11585 (1 + jfg)

(1 > -33) (1 + ?6-;6)

10 7.1* (10)'3 .39 1.7 Ikek

3 1.1 (10)*2 .1*8 .95 2.U5.31* d + s + Y#o) (1 + ifo5

(1 + ^ ( 1 + 4 3 + V (1 + b :)o5 2.1* (10)'2 .73

oVA

66

1.00

75--

50--

0.0210'1.0 10

System Bandwidth (radians/sec)

Fig. 3-11 Reduction in RMS Output Noise vs System Bandwidth

CHAPTER kANALOG SIMULATION OF INTERMEDIATE BANDWIDTH SYSTEMS

Analog Computer SimulationA simulation study of an Intermediate.bandwidth reactor control

system was performed on an analog computer* The analog computer block diagram is shown in Fig* U-l. More detailed diagrams of the computer program used appear in the Appendix* The boiling noise signal was ob­tained from a 0-20 Kc white noise generator and then band limited to 0 , 3 3 radians per second.

The closed-loop compensated reactor system and an uncompensatedreactor were simulated simultaneously in order to compare their re­sponses to the same inputs. It was necessary to modify slightly the reactor transfer function to do this simultaneous simulation with the number of computer amplifiers available. The reactor transfer function used was

This modified reactor transfer function is identical to that derived in Chapter 3 in the region of the resonance and at higher frequencies. In the analysis of intermediate bandwidth systems, the resonance is the most important region of the reactor transfer function because of its proximity to bandwidth break frequency and because its magnitude

Gr.33 (1 + (1 -

^ 2( W )

67

--1Timer

GeneratorNoise

LimiterBand

Fig, U-l Analog Computer Block Diagram

varies with changes of power level• Variations in reactor power level are simulated by changing the damping ratio, J , in Eq. L-l.

Frequency Response MeasurementsThe frequency response of each component was measured to verify

that the simulation was accurate. A low frequency signal generator pro­vided the excitation signal for these measurements•

measured frequency response are shown in Fig. b-2. The damping ratio was varied to simulate various reactor power levels. Comparing this plot with Fig. 3-1 shows that a damping ratio J * O.U corresponds to 71 Mwt operation. Rough interpolation indicates J - 0.5 corresponds to approximately 50 Mwt and J ■ 0 . 2 corresponds to approximately 1 0 0

From Fig. U-2 it can be seen that the resonance of the simu­lated reactor extends out to W - 30 radians per second. To make the closed-loop system bandwidth greater than the resonance frequencies, a closed-loop system bandwidth b ■ 6 0 radians per second was chosen for the simulation study. The transfer function of the compensation ele­ment for 6 0 radians per second bandwidth is

The straight line asymptotes and the measured frequency response for this system are shown in Fig9 U-3* The frequency responses of Figs.

The reactor transfer function straight line asymptotes and the

Mwt.

GcOi-2)

70lt- 2 and h-3 show that the simulations of the reactor and of the compen­sation element were accurate«,

The frequency response of the closed-loop compensated reactor system was then measured„ The straight line asymptotes of the desired closed-loop transfer function and the frequency response of the actual system are shown in Fige U-U» The slight peaking in the closed-loop frequency response shown in Fig, I1-I4. can be attributed to the contribu­tion of the reactor resonance in the crossover region where the approx­imation W = 1 for G » 1 is not completely valid. While this closed-loop system is not the ideal critically damped system that can be achieved with larger bandwidths, its behavior is satisfactory. It is the optimum realizable system for this given bandwidth.

Transient Response MeasurementsThe transient response of the compensated and uncompensated

reactor to a power demand step is shown in Fig. It-5 for J = O.lt (71 Mwt). The upper trace of this figure is the compensated reactor response and the lower trace is the uncompensated response. The time scale is one centimeter per second. This comparison shows that the step response of the compensated reactor is much better than that of the uncompensated reactor. The compensated 71 Mwt reactor has less than 20% overshoot and a settling time of approximately 0 . 5 seconds compared to almost k.00% overshoot and 1 . 5 seconds settling time for the uncompensated reactor.

J - 0.2

1202.2521 10'10110

Fig. U-2 Reactor Frequency Response CU

72

1 0

1 0

10

8$ 1210

.22 1 10'101010

Fig. U-3 Compensating Element Frequency Response

i c f 1 - -

- 1 10 10'

Fig. U-U Closed-Loop System Frequency Response- 4U>

Power Demand Step Response of Compensated and Uncompensated System

75Output Noise Reduction

Figure 1 -6 is a recording of the compensated and uncompensated output noise response to the same band limited boiling noise input.The measured reduction in RMS output noise for a system with 60 radians per second bandwidth and 71 Mwt reactor power ( J - .Ol*) is

- 0.22 . (ti-3)Cp

This compares favorably with the predicted value of 0.20 obtained from Fig, 3-11. The output noise reduction was also measured for other damping ratios. The measured reductions in output noise range from

■ 0.19 for f - 0 . 2 (U-U)els'

to

' 0 .2 U for J ■ 0 , 8 • (U-5)or

This shows that at higher power levels, where the uncompensated reactor output noise fluctuations are of greater amplitude, the noise reduction produced by the optimum closed-loop system is greater than that at­tained at low power levels. Figures U-7 and U- 8 compare the output noise response of the compensated and uncompensated system for the cases when J ■ 0 , 2 and J • 0 .6 , respectively.

The RMS values of the compensated system output noise were meas­ured for various reactor transfer function damping ratios. The RMS val­ues of the output noise were constant within ± 1 0 # of the mean value

76

Fig. U- 6 Output Noise of Compensated and Uncompensated System for J - O.lj (71 tort)

Fig. b-7 Output Noise of Compensated and Uncompensated System for J = 0 . 2

Fig. U- 8 Output Noise of Compensated and Uncompensated System for y = 0 . 6

of RMS output noise« This figure is based upon twenty-five samples measured at each of the following damping ratios: 0*2, Ooitj, 0«6, andO080 These measurements verified that the compensation element trans­fer functions are independent of the reactor transfer function in the intermediate bandwidth range*

CHAPTER 5 CONCLUSIONS

SummaryOptimum linear time-invariant control systems that minimize

boiling reactor noise can be obtained by applying the design tech­niques developed by Dr, R, B, Streets, If the reactor transfer func­tion and the spectral density of the boiling noise are known# this method gives the compensating element transfer function for the Wiener optimum control system.

For the unity feedback closed-loop system formulated in Chap­ters 1 and 3 # the compensation reactivity velocity and acceleration requirements are independent of the reactor transfer function. They are directly dependent upon the bandwidths of the closed-loop system and the band limited boiling noise.

The most easily realizable systems with good output noise re­duction performance have bandwidths in the intermediate frequency range between the reactor resonance and , The design of suchsystems is based on the assumption that the zero power and the full power reactor transfer functions are identical at high frequencies.The compensating element transfer function of intermediate bandwidth systems is independent of reactor power level. The transient response of these systems is good.

The greatest reductions in output noise can be obtained from systems with bandwidths greater than , The compensating element

79

transfer function of these wide bandwidth systems is completely inde­pendent of the reactor transfer function» The transient response also approaches the ideal critically damped response as the system bandwidth is increasedo Howevers the large compensating element accelerations re quired for wide bandwidth systems make physical realization of such systems difficult0

Systems with bandwidths below the reactor resonance cannot be made independent of the reactor transfer function as it varies for dif­ferent power levels„ Use of such systems does not appear practical ex­cept in the case where operation at a single power level is anticipated The reductions in output noise become less significant as the system bandwidth is decreased into this range0 Narrow bandwidth systems do offer the advantage of requiring small compensating element velocity and acceleration capabilities„

Stability ConsiderationsIn applying Streets' method to reactor control system design,

it should be noted that system stability is insured by the specifica­tions used only in the linear operating range about a steady state power level„ Non-linear analysis techniques, such as describing func­tions, must be used to determine system stability in the statistically small proportion of the operating time when the linear operating range is exceededo

APPENDIX

81ANALOG COMPUTER PROGRAM

Time Scale2 seconds computer time ■ 1 second reactor operating time

Frequency Scales - 2. p - or p - 0 . 5 s

Reactor Transfer Function

G • 3 3 ( 1 * ( 1 4

(1 * W + iB o 1 ( 1 + i f o )

.33 (i + T-&?) (i ♦ f)‘ iTTi TgnrT^

Kr(l + TlP) (1 + Tjp)(1 + 2*T2p + T22)(l + T^p)

Compensating Element Transfer Function (b - 60 radians/a

83

10

Fig, A-l Analog Computer Program of Reactor Transfer Function

Amp. 10

Fig, A-2 Analog Computer Program of Compensating Element Transfer Function

sitANALOG COMPUTER POTENTIOMETER SETTINGS FOR FIGS. A-l AND A-2

Potentiometer PotentiometerValue

PotentiometerSetting

ReactorKr T110 t3 .710

10 T. .113

D

E

100 T,

10 T,

.222

.858

G

100 T,

T3 ~ Ta

.600

.8 8 k

Compensating Element H Kc T 6

t5 T? .IliO

S10 + i 26_

100i i . J_

100 T, " 100 T_ 10

.165

.700

.276

85

REFERENCES

lo Ee A 0 WimunQj, M 0 Patrick, ¥„ C, Lip inski, and He Iskenderian,"Performance Characteristics of EB>JR From 0-100 M#t," ANL- 6775, Argonne National Laboratory (1963)=

2e R<, B<, Streets, Jra, "Integration of Analytical and ConventionalDesign Techniques for Optimum Control Systems," Doctoral Disser­tation, University of Arizona (1963)®

3o Go Co Newton, Jr*, Lo A 0 Gould, J» F. Kaiser, Analytical Designof Linear Feedback Controls, pp« 366-381, James Wiley & Sons, Inc., New York, 1957®

lie Go A» Korn and T„ M. Korn, Mathematical Handbook for Scientistsand Engineers, p« 900, McGraw-Hill Book Co®, Inc., New York,196lo

5«> Jo A® DeShong and W„ C 0 Lipinski, "Analyses of ExperimentalPower-Reactivity Feedback Transfer Functions for a Natural Cir­culation Boiling Water Reactor," ANL-5850, Argonne National Laboratory (1958)®

6 0 L« E® Weaver, System Analysis of Nuclear Reactor Dynamics» pp.95-98, Rowman and Littlefield, Inc., New York, 19WI

7o Jo A. Thie, Reactor Noise, pp0 186-188, 227-231, Rowman and Lit­tlefield, Inco, New York, 1963o