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Direct digital control algorithm for low power nuclear reactors Harvey, Geoffrey Alan 1973

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DIRECT DIGITAL CONTROL ALGORITHM FOR LOW POWER NUCLEAR REACTORS by G..A. Harvey B.Sc.(Eng)(Elec) U n i v e r s i t y of P r e t o r i a 1968 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENT FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of E l e c t r i c a l Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA January 1973 In present ing th is thes is in p a r t i a l f u l f i l m e n t o f the requirements fo r an advanced degree at the Un ive rs i t y of B r i t i s h Columbia, I agree that the L ib ra ry sha l l make i t f r e e l y a v a i l a b l e for reference and study. I fu r ther agree that permission for extensive copying of th is thes is for s c h o l a r l y purposes may be granted by the Head of my Department or by h is representa t ives . It is understood that copying or p u b l i c a t i o n of th is thes is f o r f i n a n c i a l gain sha l l not be allowed without my wr i t ten permiss ion . Department The Un ivers i ty o f B r i t i s h Columbia Vancouver 8, Canada ABSTRACT A d i r e c t d i g i t a l c o n t r o l algorithm f o r low power reactors i s proposed using logarithmic power l e v e l as input. The logarithmic power l e v e l s allow the use of f i x e d point arithmic r e s u l t i n g i n f a s t e r c a l c u -l a t i o n speeds than are obtainable with algorithms using f l o a t i n g point arithmetic. A s t a b i l i t y analysis for various sampled data hold types i s shown to have a 25% safety margin. A time optimal control sequence fo r power increases i s derived using switch points. The switch points are determined using simulation techniques, e l i m i n a t i n g the use of complex and approximate c a l c u l a t i o n s . A p r a c t i c a l demand l e v e l c o n t r o l l e r i s developed using machine language programming to minimize the delay from the sampling of the neutron power to the output of c o n t r o l a c t i o n . The c o n t r o l l e r i s tested with d i g i t a l and analog simulations of a thermal reactor showing that a successful, near time-optimal, c o n t r o l algorithm with general applications to low power reactors has been developed. TABLE OF CONTENTS Page ABSTRACT i TABLE OF CONTENTS i i LIST OF ILLUSTRATIONS - i v LIST OF TABLES v i NOMENCLATURE v i i PROGRAMMING NOMENCLATURE i x ACKNOWLEDGEMENT . x i 1. INTRODUCTION 1 2. DIGITAL CONTROL SYSTEMS FOR NUCLEAR REACTORS 4 2.1 Basic D i g i t a l Control System 4 .2.2 Sampled Data Holds 6 2.2.1 Zero-Order Hold 6 2.2.2 First-Order Hold 7 2.2.3 Linea r i z e d Hold 8 2.3 Control Rod Servo System 10 2.4 Reactor Model 11 2.5 Neutron Power Level Measuring C i r c u i t s 11 2.6 O v e r a l l System S t a b i l i t y 13 3. DIRECT DIGITAL CONTROL ALGORITHM 20 3.1 Constraints on Demand Power Level Changes . . . 20 3.2 Summary of. E x i s t i n g Algorithms 21 3.3 Logarithmic D i g i t a l Control Algorithm 24 3.4 Logarithmic D i g i t a l Control Algorithm Demand Power Level Changes 26 3.5 Logarithmic Power Level Measuring C i r c u i t s . . . 27 4. TIME OPTIMAL REACTOR CONTROL . . . 28 4.1 Review of Present L i t e r a t u r e 28 4.2 Time Optimal Power Increases 29 4.3 Time Optimal Power Decreases 35 5. PRACTICAL DEMAND POWER LEVEL CONTROLLER . 38 5.1 Control Computer S p e c i f i c a t i o n s and Programming 38 5.2 Demand Power Level C o n t r o l l e r . 41 5.2.1 Fetching of Neutron Power Sample . . . . 42 5.2.2 Erro r C a l c u l a t i o n 43 5.2.3 Output of Control Action 45 5.2.4 Demand Power Level C a l c u l a t i o n 46 5.2.5 New Endpoint and Switch Point C a l c u l a t i o n s 56 i i Page 5 . 2 . 6 General Remarks 59 5 . 3 D i g i t a l Simulation of a Nuclear Reactor 59 5 . 4 Analog Simulation of a Nuclear Reactor 63 5 . 5 Test of D i g i t a l C o n t r o l l e r 64 5 . 5 . 1 C a l c u l a t i o n Time of Control Algorithm . . . 65 5 . 5 . 2 S t a b i l i t y Test of C o n t r o l l e r 65 ~ 5 . 5 . 3 Power Level Increases 66 5 . 5 . 4 Power Level Increases with Noisy Reactor . 72 5 . 5 . 5 Power Level Decreases 72 6. CONCLUSIONS 75 APPENDIX 7 7 A. REACTOR KINETICS EQUATIONS 77 A . l General Reactor K i n e t i c s Equations 77 A.2 L i n e a r i z e d Reactor K i n e t i c s Equations 78 A . 3 Reactor K i n e t i c s Transfer Function ,. . 78 A.4 Thermal Reactor Parameters 78 REFERENCES 80 i i i LIST OF ILLUSTRATIONS Fig . No. Page 1.1 Possible Reactor Control System Using P a r a l l e l Mode Minicomputers 2 2.1.1 Basic Block Diagram of a Continuous Reactor Control System 4 2.1.2 General Continuous Closed Loop Control System . . . . 4 2.1.3 General E r r o r Sampled Closed Loop Control System . . . 5 2.1.4 Basic Sampled Data Control System f o r a Nuclear Reactor 5 2.2.1 Output of Zero-Order Hold Device 6 2.2.2 Output of First-Order Hold Device 7 2.2.3 L i n e a r i z e d Hold Device Sample Points 8 2.5.1 Neutron Power Level Measuring C i r c u i t Schematic (Scaler plus Voltage-to-Frequency Converter) . . . . 12 2.5.2 Neutron Power Level Measuring C i r c u i t Schematic ( F i l t e r plus Multiplexer plus A/D) 12 2.6.1 Root Locus Plot of a Thermal Reactor Sampled Data Control System with Zero-Order Hold 16 2.6.2 Root Locus P l o t of a Thermal Reactor Sampled Data Control System with F i r s t - O r d e r Hold 16 2.6.3 Root Locus P l o t of a Thermal Reactor Sampled Data Control System with L i n e a r i z e d Hold . . . . . . . . 17 2.6.4 Amplitude versus-Frequency for Zero-Order, F i r s t -Order, and Linearized Holds 18 2.6.5 Bode P l o t of Thermal Reactor plus Control Rod Servo System 19 4.2.1 Time Optimal Control Sequence f o r Prompt Reactor . . . 30 4.2.2 Time Optimal Control Sequence with Delayed Neutrons Included 30 4.2.3 Time Optimal Control Switch Point C a l c u l a t i o n . . . . 33 4.3.1 Power Decrease with 100 Second Period Constraint . . . 36 5.2.1 Basic C o n t r o l l e r Flow Diagram 42 i v F i g . No. Page 5.2.2 Merging of Upper and Lower Measuring Ranges 43 5.2.3 Flow Diagram of Neutron Power Fetch 44 5.2.4 Flow Diagram of Erro r C a l c u l a t i o n 45 5.2.5 R e a c t i v i t y Rate Signal Types 46 5.2.6 Flow Diagram of Control Action Output 47 5.2.7 Inverse Period for Log and Linear Constraints 48 5.2.8 Inverse Period f o r Time Optimal Power Increase (Step Increase i n Period) 50 5.2.9 Neutron Power Level Increase with Step Period Change . . 50 5.2.10 Inverse Period for Time Optimal Power Increase (Con-tinuous Increase of Period) 50 5.2.11 Delay i n A t t a i n i n g R with Continuous Period Increase Case 50 5.2.12 Inverse Period for Time Optimal Power Increase (Con-tinuous plus Step Increase of Period) 51 5.2.13 Inverse Period for Power Level Decrease (Continuous Increase of Period) 54 5.2.14 Inverse Period as a Function of Power Level 55 5.2.15 Inverse Period as a Function of Power Level (Linear Constraint) 56 5.2.16 Flow Diagram of Demand C a l c u l a t i o n . . . . . . . . . . 57 5.2.17 Endpoint P r i o r i t y Chain 58 5.3.1 D i g i t a l Simulation of Nuclear Reactorl-Flow Diagram (One Sampling Only) 62 5.4.1 Analog Simulation of Nuclear Reactor . . 64 5.5.1 Time Optimal Power Increase 67 5.5.2 Power Level Increases 70 5.5.3 Power Level Increase with Linear Rate Constraint . . 71 5.5.4 Power Level Increase with Noisy Reactor 73 5.5.5. Power Level Decrease with 100 Second Period . . . . 74 v LIST OF TABLES Table No. Page 2.6.1 Maximum Allowable R e a c t i v i t y Rate R per Unit max -Error to Ensure S t a b i l i t y versus Sampled Period (Thermal Reactor) 18 4.2.1 Time Optimal Switch Points f o r Power Increases . . . . 33 5.1.1 Arithmetic Sub-routine Functions and C a l c u l a t i o n Times 41 5.2.1 Times for Reactor and Demand to Reach Endpoint from Switchpoint 53 5.2.2 T 1 and x' for Simultaneous A r r i v a l of Reactor and m e Demand at Endpoint 54 5.2.3 Parameters for Power Decreases with 100 Second Minimum Period Constraint 55 A.4 Parameters of Delayed Neutron Groups of a Thermal Reactor . . . 79 v i NOMENCLATURE 3 Concentration of delayed neutrons (neutrons/cm ) Er r o r between demanded neutron power and actual neutron power Unit error (E/N) 3 Neutron density (neutrons/cm ) Neutron power l e v e l Demanded neutron power l e v e l at present sampling Demanded neutron power l e v e l at next sampling F i n a l endpoint power l e v e l Predicted neutron power l e v e l at next sampling i f no c o n t r o l action from present to next sampling Neutron power l e v e l at l a s t sampling Neutron power l e v e l at present sampling Switch point power l e v e l Maximum r e a c t i v i t y rate (mk/s) 3 Neutron source (neutrons/cm /sec) Sample period (sec) Fracti o n of delayed neutrons Decay constant of delayed neutrons (sec "*") Mean e f f e c t i v e neutron l i f e t i m e (sec) Deviation of neutron density from steady state R e a c t i v i t y (mk) Error between logarithmic demand power l e v e l and logarithmic neutron power l e v e l Reactor period (sec) Minimum allowable reactor period at endpoint (sec) Minimum allowable reactor period as imposed by l i n e a r rate constraint (sec) v i i T Minimum allowable reactor period (sec) m x Minimum allowable reactor period as imposed by time optimal c o n t r o l constraint (sec) v i i i PROGRAMMING NOMENCLATURE AC Computer accumulator ALPHA Variable f o r merging two measuring ranges BETA(X) Normalized f r a c t i o n of delayed neutrons i n the Xth group (B^/n) BETAT Normalized t o t a l f r a c t i o n of delayed neutrons (3/n) BIAS Bias added to input s i g n a l to obtain f u l l use of A/D range C Constant so that demand does not diverge too f a r from actual neutron power DEAD Erro r deadband DELAY(X) Normalized concentration of neutrons i n delayed group X DERT Inverse reactor period at f i n a l endpoint f o r power decreases ( T / T « E ) DMRT Inverse reactor period at switch point f o r power decreases ( T / T - ) m DRAT Inverse reactor period f o r power decreases ( T / T ) EP Endpoint (Flag) ERRO Er r o r between logarithmic demand power l e v e l and logarithmic neutron power l e v e l FLXD Logarithmic demand power l e v e l of present sampling FLXE Logarithmic f i n a l endpoint power l e v e l FLXL Logarithmic neutron power l e v e l at l a s t sampling FLXP Logarithmic neutron poxver l e v e l at present sampling FLXT Temporary storage for neutron power l e v e l s FUNC(X) Input function for delayed group X i n t e g r a t i o n c a l c u l a t i o n GAIN C o n t r o l l e r gain v a r i a b l e HI Upper end of merging range f o r two measuring ranges LAMDA(X) Decay constant of delayed group X LIND Linear demand power l e v e l i x LINP Linear neutron power l e v e l LNRT Inverse period for l i n e a r rate constraint LOW Lower end of merging range f o r two measuring ranges MAXE Error required to give maximum output s i g n a l NRATE Inverse reactor period ( d i g i t a l simulation) PERD Inverse reactor period ( c o n t r o l l e r ) REACT T o t a l r e a c t i v i t y RRATE R e a c t i v i t y rate SCALE 1 Variable f o r c a l i b r a t i o n of A/D #1 sample SCALE 2 Variable for c a l i b r a t i o n of A/D #2 sample SWLD Logarithmic switch point for l i n e a r range SWSD Logarithmic switch point f o r time optimal c o n t r o l range (power decreases) SWST Logarithmic switch point f o r time optimal c o n t r o l range (power increases) TEMP 1 to 4 Temporary storage f o r d i g i t a l simulation TEM 1 to 3 Temporary storage f o r c o n t r o l l e r UERT Inverse reactor period at f i n a l endpoint f o r power increases ( T / T E « ) UMRT Inverse reactor period at switch point f o r power increases ( T / T ' m ) URAT Inverse reactor period f o r power increases (T/T ) © Exclusive OR function x ACKNOWLEDGEMENT The author wishes to express h i s gratitude to Professors F.K. Bowers and A. Soudack f o r t h e i r supervision of t h i s research. Sincere appreciation i s due to Dr. A.J.A. Roux, President, and Mr. W.K.H.A. Weidemann, Director: Instrumentation, of the South A f r i c a n Atomic Energy Board for making this study opportunity a v a i l a b l e . Thanks are due to the author's wife, Fiona, for her understanding and f o r typing the d r a f t and also to Miss Norma Duggan for preparing the f i n a l manuscript. x i 1. INTRODUCTION U n t i l recently, the r o l e of the d i g i t a l computer i n nuclear reactor systems has been that of a supervisory and data->collection nature"*" S t r i c t safety regulations r e s u l t e d i n conventional methods being used for reactor control due to the low r e l i a b i l i t y , slow speed and tremen-dous expense of early computers. Advances i n computer technology have removed these objections. With reactor systems becoming l a r g e r and more complex, i t i s advantageous that d i g i t a l computers be used i n the c o n t r o l of nuclear reactors. At f i r s t , computers were used only f o r i n d i v i d u a l tasks such as f u e l l i n g machine con t r o l and f a i l e d f u e l detection"''. For the actual c o n t r o l of nuclear reactors, dual computer systems have been used; one operational and the other i n a "watchdog" and "backup" mode. With the a d d i t i o n of more duties, such as load matching, turbine c o n t r o l , s p a t i a l control and automatic f u e l l i n g , the s i z e of the computers has entered the medium range. Each of these duties i s normally handled by separate design groups. The co-ordination of these groups i n the devel-1 opment of a s i n g l e operating system i s extremely d i f f i c u l t . Recent developments make the use of several small minicomputers economically feasible"'". Each computer i s assigned i t s own s p e c i f i c task. An added advantage i s that each design group can develop and com-mission i t s own separate system, without too much dependence on the other groups. Figure (1.1) shows a pos s i b l e system of minicomputers working i n a p a r a l l e l mode. A l l the computers are l i n k e d together by a bus system as w e l l as to the common mass storage u n i t s . Transfer of data to or from these mass storage units i s processed by a s i n g l e c o n t r o l computer to f a c i l i t a t e f i l e orientated transfers without r e p e t i t i o n of 2 software d r i v e r s . Mass Storage Units Input/Output Channels D i s c Drum 1 Output Multiplexers A/D P Tape Input/Output Watchdog Mass Computer ! Computer Storage ! Control Minicomputers Demand Power Controller Sp a t i a l Controller T r Turbine Control F u e l l i n g Machine Control Mini-computers Additional Input/Output Channels. F i g . 1.1 Possible Reactor Control System Using P a r a l l e l Mode Mini-Computers. In Chapter 2, a basic e r r o r sampled data c o n t r o l system i s de-veloped and the t r a n s f e r functions of the various system components are 2 given. Based on a study by Marciniak the o v e r a l l system s t a b i l i t y i s analysed using various sampled data holds and sample periods. A review 2 3 of e x i s t i n g d i g i t a l c o n t r o l algorithms i s given i n Chapter 3 ' . An a l -gorithm r e q u i r i n g a logarithmic neutron power l e v e l as input i s developed, r e s u l t i n g i n a much f a s t e r and simpler d i g i t a l c o n t r o l l e r . The use of logarithmic power l e v e l s allows the use of f i x e d point arithmetic which i s much f a s t e r than f l o a t i n g point a r i t h m e t i c . Using the r e s u l t s of pre-2 A vious studies on the time optimal c o n t r o l of nuclear reactors ' , a time optimal c o n t r o l sequence using switch points i s developed i n Chapter 4 f o r power l e v e l changes. The switch points are determined using simulation techniques. In Chapter 5, a p r a c t i c a l demand power l e v e l 3 c o n t r o l l e r i s developed using machine language programming. The per-formance of the c o n t r o l l e r i s tested using d i g i t a l and analog simulations of a thermal reactor. The s t a b i l i t y a n a l y s i s of Chapter 2 i s shown to have a 25% safety margin and power l e v e l changes were e f f e c t i v e l y c a r r i e d out, maintaining the reactor within the safety constraints, with l i t t l e overshoot of the f i n a l end point power l e v e l . 4 2. DIGITAL CONTROL SYSTEMS FOR NUCLEAR REACTORS A b a s i c e r r o r sampled closed loop c o n t r o l system i s presented i n this s ection for various types of hold u n i t s . The o v e r a l l system tr a n s f e r functions are derived, followed by a s t a b i l i t y a n a l y s i s f o r low-power or zero-power reactors using a l i n e a r i z e d point k i n e t i c s model^ 2.1 Basic D i g i t a l Control System A b a s i c nuclear reactor continuous control system i s shown i n figure (2.1.1). The input to the system i s a demand power l e v e l as w e l l as a constraint on the minimum allowable reactor period. (The reactor period i s defined as the time necessary f o r the, power l e v e l to change by a f a c t o r "e", the n a t u r a l logarithm base). These two inputs are combined with the measured reactor power l e v e l to generate an e r r o r Demand Power Level r r o r 'B(s) Control Rod Drive R e a c t i v i t y Keactor Neutron Power Level Measuring C i r c u i t s C(s) F i g . 2.1.1 Basic Block Diagram of a Continuous Reactor Control System G(s) C(s) H(s) F i g . 2.1.2 General Continuous Closed Loop Control System. s i g n a l which drives the c o n t r o l rods. S i m p l i f i c a t i o n gives the general feedback co n t r o l system i n figure (2.1.2) with the o v e r a l l t r a n s f e r function C(s) =  R(s) 1 + G(s) H(s) G(s) (2.1.1) 5 where G(.s) = G c(s) G r(s) = feed-forward tran s f e r function (2.1.2) H(s) = F(s) = feedback trans f e r function (2.1.3) G (s) = c o n t r o l rod t r a n s f e r function (2.1.4) c G^(s) = reactor transfer function (2.1.5) The most s u i t a b l e sampled data c o n t r o l system to use for reactor c o n t r o l i s the e r r o r sampled closed loop system given i n f i g u r e (2.1.3). Using the z-transform notation, the o v e r a l l t r a n s f e r function i s : ( s ) G(s) 0 ( B ) H(a) F i g . 2.1.3 General Er r o r Sampled Closed Loop Control System K d ( s ) G h(s) G 0(s) G r Cs) F i g . 2.1.4 Basic Sampled Data Control System for a Nuclear • Reactor G z(z) R (z) 1 + G H (z) zs z z (2.1.6) which i s of s i m i l a r form to that of the continuous case (see footnote). Figure (2.1.4) gives the basic sampled data c o n t r o l system f o r a reactor where G ^ ( s ) i s the t r a n s f e r function of the hold device following the Note: Throughout this thesis the z-transform notation i s the same as -used i n previous d i g i t a l reactor c o n t r o l studies^. 6 sampler, G (s) the tr a n s f e r function f o r the con t r o l rods, G (s) the c r tra n s f e r function of the reactor and F(s) the tr a n s f e r function of the neutron power l e v e l measuring c i r c u i t s . The tr a n s f e r functions of each of the system blocks w i l l be analysed i n sections 2.2 to 2.5, followed by a s t a b i l i t y analysis of the o v e r a l l system using various types of hold. 2.2 Sampled Data Holds i s used to drive the con t r o l rods at the required v e l o c i t y . A pure impulse s i g n a l i s unsuitable for t h i s task due to i t being p r a c t i c a l l y unrealizable and the sampled s i g n a l i s passed through some hold device that performs the function of reproducing the sampled s i g n a l u n t i l the next sampling. Two basic holds are the zero order and f i r s t order holds. Holds of greater order are to be avoided, not only due to the d i f f i c u l t i e s of p h y s i c a l r e a l i z a t i o n but also because of the delays they may i n t r o -duce i n t o the system. 2.2.1 Zero Order Hold In the sampled data reactor c o n t r o l system, the sampled e r r o r (n-1)T n T (n+1)T (n+2)T (n+5)T Time F i g . 2.2.1 Output of Zero-Order Hold Device. 7 I f the sampled s i g n a l i s held u n t i l the next sample such that N(t) = f(nT) nT < t £ (n+l)T (2.2.1) as i n f i g u r e (2.2.1), the device i s c a l l e d a zero order hold and has the t r a n s f e r function H q ( S ) = (l-exp(-Ts))/s .In the case of the sampled data reactor c o n t r o l system E(t) = N d(nT) - N(nT), nT < t * (n+l)T where = demand neutron power l e v e l (reference) N = measured neutron power l e v e l T = sample period E = error 2.2.2 F i r s t Order Hold «(t) (2.2.2) (2.2.3) (n-1)T nT (n+1)T (n+2)T (n+3)0? Time F i g . 2.2.2 Output of F i r s t - O r d e r Hold Device I f the l a s t two samplings are used to c a l c u l a t e the slope of the s i g n a l such that f(nT) - f [ ( n - l ) T j N(t) = f(nT) + -(t-nT), nT < t ,< (n+l)T (2.2.4) 8 as i n f i g u r e (2.2.2), the device i s c a l l e d a f i r s t order hold and has the t r a n s f e r function 2 H ( S) - (1+Ts) (l-exp(-Ts))-I T S In the case of the sampled data reactor c o n t r o l system E ( t ) = E(nT) + E(nT) - E((n-1)T) (2.2.5) nT < t £ (n+l)T where E(nT) = N d(nT) - N(nT) and E, and N are as before. 2.2.3 L i n e a r i z e d Hold (2.2.6) (2.2.7) (n-1)T nT (n+1)T T i m e F i g . 2.2.3 L i n e a r i z e d Hold Device Sample Points 3 Cohn developed a hold that has p a r t i c u l a r bearing on nuclear reactors such that E'(t) = N d[(n+1)T] - N'[(n+1)T], nT < t s> (n+l)T (2.2. where Nj[(n+1)T] = = demand neutron power at next sampling and N'[(n+1)T] = = predicted neutron power at next sampling i f no c o n t r o l action i s taken from the present to the next sampling. 9 Let = Nl(n-1).T] = neutron power l e v e l at l a s t sampling Np = N[nT] = neutron power l e v e l at present sampling and T = acutal reactor period. (See figu r e 2 . 2 . 3 ) . 3. Now N f = N p exp ( T / T ^ ( 2 . 2 . 9 ) where the reactor period T , i s the inverse of the logarithmic slope of the neutron power l e v e l such that 1/T = ( l n N - l n N J / T ( 2 . 2 . 1 0 ) a p 1 Therefore N f = N p exp [T(ln N p - l n N^/T] ( 2 . 2 . 1 1 ) = N V ( 2 . 2 . 1 2 ) p 1 and E'(t) = N^ - ( N ^ / N ^ , nT < t $ (n+l)T ( 2 . 2 . 1 3 ) The t r a n s f e r function of t h i s hold can be obtained i n the following manner: Assuming that N p and N^ deviate only s l i g h t l y from the demand N^ and that t i s the present time and T the sample period, N = N(t) = N' (1+Y ) , |Y I « 1 ( 2 . 2 . 1 4 ) p d p 1 p 1 and N 1 = N(t-T) = N^ ( 1 + Y ^ , \Y±\ « 1 ( 2 . 2 . 1 5 ) S u b s t i t u t i o n i n t o equation ( 2 . 2 . 1 3 ) and neglecting high order terms of Y and Y . gives P 1 . E'(t) = [2Y - Y J N ' ( 2 . 2 . 1 6 ) p 1 J d But from equations ( 2 . 2 . 1 4 ) and ( 2 . 2 . 1 5 ) Y = (N(t) - N')/N' ( 2 . 2 . 1 7 ) p d d and Y , = (N(t-T) - N')/N' ( 2 . 2 . 1 8 ) 1 d d therefore E'(t) = 2[N(t) - N^] - [N(t-T) - N'] ( 2 . 2 . 1 9 ) d d As E(t) = N(t) - N^ , ( 2 . 2 . 2 0 ) therefore E(t-T) = N(t-T) - N' ( 2 . 2 . 2 1 ) d 10 and E*(t) « 2E(t) - E(t-T) (2.2.22) As the output of an erro r sampler i s given by E*(t) = I E(t) 6[t-nT] (2.2.23) n=0 and s u b s t i t u t i n g for equation (2.2.22) and taking the Laplace transform gives E'*(s) = 2E(0) ( 1 " e x p ( " T s ) ) + [2E(T) - E(0)] exp(-Ts) s (l-exp(-Ts)) + [ 2 E ( 2 T ) _ E ( T ) j exp(-2Ts) (l-exp(-Ts)) s which may be s i m p l i f i e d to E'*( S) = 1 " 6 X P ( " T s ) [2 - exp(-Ts)] E*(s) (2.2.25) Thus the tr a n s f e r function of the l i n e a r i z e d hold i s H 1(s) = [2-exp(-Ts)] X " e x p ( " T s ) (2.2.26) The t r a n s f e r functions for the three types of hold w i l l be used i n the o v e r a l l system s t a b i l i t y analysis to determine which hold gives the best performance. 2.3 Control Rod Servo System The simplest transfer function f o r the c o n t r o l rod servo system i s G (s) = R/s (2.3.1) c where R i s the r e a c t i v i t y rate per unit e r r o r input. A time constant should also be included i n the t r a n s f e r function, however, due to the complexity of the o v e r a l l system gain, i t i s neglected. In p r a c t i c e , there i s also a l i m i t placed on the maximum r e a c t i v i t y rate which 11 2 constrains the maximum system gain. This, too w i l l be ignored . 2.4 Reactor Model Examination of the d i f f e r e n t i a l equations f o r the reactor k i n e t i c s shows a reactor to be h i g h l y non-linear. (See Appendix A). Schultz^ developed a l i n e a r i z e d t r a n s f e r function of a reactor model about a steady state power l e v e l , incorporating a l l s i x groups of 2 4 delayed neutrons. Marciniak and Lipinsky have derived a time inde-pendent, l i n e a r , monoenergetic, one-delayed-neutron-group k i n e t i c s trans-f e r function given as follows: r ( s ) e ^N(s) = s + X V S ; N o6k(s) £sls + X + B/A] U . 4 . ± ; where X = average decay constant B = t o t a l e f f e c t i v e delayed neutron f r a c t i o n I = prompt neutron l i f e t i m e N q = neutron density about which the system i s l i n e a r 6k = e f f e c t i v e r e a c t i v i t y 6N = deviation of neutron density from N . o This t r a n s f e r function w i l l be used f o r the reactor model i n the s t a b i l i t y analysis given i n s e c t i o n 2.6. 2.5 Neutron Power Level Measuring C i r c u i t s Due to reactor noise, the input to the computer must have some smoothing. The method used on the ZPR-9 fa s t c r i t i c a l reactor at the Argonne National Laboratory i s very applicable for the d i g i t a l moni-3 t o r i n g o f the neutron power l e v e l . An ion chamber i s used to measure the neutron f l u x and the output of the ion chamber a m p l i f i e r i s used to drive a voltage-to-frequency converter. The output of the voltage-to-frequency converter i s fed i n t o a counter or s c a l e r which i s read and then reset every T seconds. The counter acts as. an i n t e g r a t o r 12 smoothing the input to the computer. However, i n most systems, computer inputs are multiplexed to measure other v a r i a b l e s and not only the neutron power. In this case, the output of the ion chamber a m p l i f i e r must be s u i t a b l y f i l t e r e d , depending on the sample period. Taking the case of the counter, the average of a s i g n a l f ( t ) over a period T i s p(t) = i C p(t') dt' t-T (2.5.1) Taking the Laplace Transform gives v<*\ - p ( g ) P(s) exp(-Ts)  n S } Ts " Ts (2.5.2) Therefore the Transfer Function i s F(s) = 1 - T ^ ( - T 8 > (2.5.3) Schematic diagrams of the two possible neutron power l e v e l measuring c i r c u i t s are given i n figures (2.5.1) and (2.5.2). Ion Chamber 0 Amplifier Voltace to Frequency Converter Counter To Computer Input Bus • Control from Computer F i g . 2.5.1 Neutron Power Level Measuring C i r c u i t Schematic (Scaler plus Voltage-to-Frequency Converter) l Ion Chamber Ion Chamber Amplifier Suitable F i l t e r M u l t i -plexer A/D Converter To Computer Input Bus Control from Computer F i g . 2.5.2 Neutron Power Level Measuring C i r c u i t Schematic ( F i l t e r plus Multiplexer plus A/D) 13 2.6 O v e r a l l System S t a b i l i t y A b asic sampled data c o n t r o l system for a nuclear reactor was given i n fig u r e (2.1.4) and the o v e r a l l t r a n s f e r function by equation (2.1.4), where G z(z) - Z [ G h ( s ) G c ( s ) G r ( s ) ] ' (2.6.1) G zH z(z) = Z [ G h ( s ) G c ( s ) G r ( s ) F ( s ) ] (2.6.2) and Z[G(s)] = G (z) (2.6.3) i s the z transform of G(s). The tr a n s f e r functions f o r the i n d i v i d u a l system blocks have been derived i n the previous sub-sections. The t r a n s f e r function f o r o the system with zero-order hold i s : SN (z) Z[H (s)G (s)G (s)] z o c r (2.6.4) N N (z) 1+Z[H (s)G (s)G (s)F(s)] o d z l o c . r 3^TKz(az 2+bz + c) 6A' 4TJlz(z - l) 2(z-Y)+K(dz 3+ez 2 T-fz+g) (2.6.5) 1 = X + (6/A) (2.6.5a) K R/£ (2.6.5b) Y = exp(-X'T) (2.6.5c) a = XX , 2T 2Ji+2BTX ,+23(2+Y)-6B (2.6.5d) b = XX'2T2SL(1-Y)-2BTX' (1+y)-2B(l+2y)+6B (2.6.5e) c = -XX | 2T 2£Y+2BTX'Y+2BY-2B (2.6.5f) d = XX , 3T 3£+3BT 2X'-6BTX i-6B(3+Y)+248 (2.6.5g) e = XX' 3T 3£(4-Y)-3X , 2T 2BY+6BTX'(2+Y)+18B(1+Y)-36B (2.6.5h) f = XX' 3T 3£(1-4Y)-3X , 2T 2B-6BTX' ( l+2y)-66(1+3Y)+24B (2.6.5i) g = -XX'3T3X,Y+3X'2T2BY+6BTX'Y+6BY-6B (2.6.5J) 14 where 2 For the f i r s t - o r d e r hold,, the tr a n s f e r function i s : 6 N z ( z ) = . •4X,TK(hz4+kz3+mz2+j>z)  N o N d z ( z ) 2 4 A , 6 T 2 i t z 2 ( z - l ) 2 ( z - Y ) + K ( q z 4 +rz 3+uz 2+vz-fv) (2.6.6) X' = X + (B/£) " (2.6.6a) K = R/£ ^ (2.6.6b) Y = exp(-X'T) (2.6.6c) h = XX , 3T 3£+3X , 2T 2(XX'T£+3)+63(X'T-l)T +6B(X'T-1)(3+Y)-246(X'T-1) (2.6.6d) k = XX' 3T 3£(4-Y)-3X , 2T 2(XX'T£+3)Y-6B(X*T-1)TX ,(2+y) -183(X'T-1)(1+Y)+36B(X'T-1) ' (2.6.6e) m = XX , 3T 3£(1-4Y)-3X' 2T 2(XX ,T£+3)+6B(X ,T-1)TX'(1+2 Y) +6g(X'T-l)(1+3Y)-24B(X'T-1) (2.6.6f) p = [-XX , 3T 3£+3X' 2T 2(XX ,T£+B)-6B(X ,T-l)TX , -6B(X ,T-1)]Y+6B(X'T-1) (2.6.6g) q = XX' 4T 4£+4X' 3T 3(XX 1T£+B)+12BT 3X' 2(X'-1/T) -24B(X !-1/T)T 2X'-24B(X*-1/T)T(4+Y)+120T(X ,-1/T) (2.6.6h) r = XX , 4T 4£(11-Y)+4X , 3T 3(XX'T£+B)(3-Y)-12B(X'-1/T)T 3X , 2(1+Y) +24B(X ,-1/T).T 2X ,(3+Y)+48B(X'-1/T)T(3+2Y)-240B(X'T-1) (2.6.6i) u = 11XX , 4T 4£(1-Y)-12X , 3T 3(XX'T£+B) (l+Y)+12B.(X'-l/T) T 3X , 2(Y-1)-72B(X'-1/T )T 2X'(1+Y)-483(X'-1/T)T(2+3Y) +240B(X'T-1) (2.6.6J) v = XX , 4T 4£(1-11Y)+4X , 3T 3(XX ,T£+B )(3Y~1) +12B(X ,T-1)T 2X , 2(H-Y)+24B(X'T-1)TX'(1+3Y) -!-24B(X'T-1)(1-1-4Y)-120B(X'T-1) (2.6.6k) 15 w = -[AA' 4T 4JlT :4A , 3T 3aX'T£+g)-12gCX ,T-l)T 2A' 2 -24g(A ,T-l)TA ,-24g(A'T-l)]Y+24B(A'T-l) (2.6.61) 2 The o v e r a l l system transfer function for the l i n e a r i z e d hold i s : N z ( z ) _ 3A'TKz(2z-l) (azVbz+c) N o N d z ( z ) 6A^Til 2(z-l) 2(z-Y)+K(2z-l) (dz3+e.z2+f z+g) (2.6.7) where a l l the constants are the same as defined f o r the system with the zero-order hold i n equation (2.6.5). S t a b i l i t y analysis of sampled data systems i s performed by determining the zeros of the c h a r a c t e r i s t i c equation, that i s , the de-nominator of equation (2.1.4), i n the z plane. The c r i t e r i o n ^ i s that the c h a r a c t e r i s t i c equation of the sampled data system have no zeros outside the unit c i r c l e , or, i f A. denotes the i t h root of the charac-t e r i s t i c equation then: J x J * 1 (2.6.8) 2 7 Marciniak , using a program developed by Hafner , found the roots of the c h a r a c t e r i s t i c equations f o r a l l three types of hold for both a thermal and a f a s t reactor. Figures (2.6.1), (2.6.2) and (2.6.3) are the root locus p l o t s of the c h a r a c t e r i s t i c equations of the system transf e r functions of a theraml reactor for a zero-order, f i r s t - o r d e r 2 and l i n e a r i z e d hold r e s p e c t i v e l y . (Based on r e s u l t s of Marciniak ). The sampling period i s 0.1 second and the system parameters are as follows: A = 0.076 se c " 1 , 6 = 0.0064, £ = 10~ 3 sec (2.6.9) Examination of equations (2.6.5), (2.6.6) and (2.6.7) shows that T and K are the only variables f o r a f i x e d reactor. T i s the 16 - i . o : T=0.1 sec \ /3=O.O064- . \ A=0.0?6 s e c - ' X,=0.001 sec 1.03 -1.0 i F i g . 2.6.1 Root Locus Plot of a Thermal Reactor Sampled Data Control System with Zero-Order Hold - 1 . 0 0 F i g . 2.6.2 Root Locus P l o t of a Thermal Reactor Sampled Data Control System with Fi r s t - O r d e r Hold 17 F i g . 2.6.3 Root Locus Plot of a Thermal Reactor Sampled Data Control System with. L i n e a r i z e d Hold sample period and K i s the r e a c t i v i t y rate per unit e r r o r per neutron l i f e t i m e , i . e . K = R/A (2.6.10) But £ i s f i x e d for a c e r t a i n reactor, therefore a sampled data reactor system of the form of figure (2.1,4) can be s a i d to be stable for a spe-c i f i c sample period T, provided the r e a c t i v i t y rate per unit e r r o r R i s l e s s than a c r i t i c a l value R . The value R ensures that a l l poles max max of the c h a r a c t e r i s t i c equation l i e within the unit c i r c l e i n the z plane. The unit error E i s defined as: u E = E/N (2.6.11) u o Marciniak has drawn up tables of R versus sample period T f o r various r max r r reactor types. Table (2.6.1) gives the maximum allowable r e a c t i v i t y rate per unit e r r o r for various sample periods f o r the reactor with parameters as i n equation (2.6.9). From t h i s table i t can be seen that the zero-order hold i s the most stable except for the 0.1 second sampling period. 18 Sample Period R e a c t i v i t y Rate (% <S k/k) / sec T (sec.) Zero-Order Hold F i r s t - O r d e r Hold L i n e a r i z e d Hold 0.1 7.4 8.3 8.4 0.5 1.83 1.45 1.2 1.0 0.99 0.73 0.62 5.0 0.173 0.14 0.123 Table 2.6.1 Maximum Allowable R e a c t i v i t y Rate R per J max Unit Error to Ensure S t a b i l i t y versus Sampled Period. (Thermal Reactor) 6 5> 0 - : 1 1 1 a?=27r/uj>s Zero-Order Hold First-Order Hold -- • • Linearized Hold _ ' . ' *\ • * . , * *. **• ^ \ \ • • • • * s - • t . • » 0 1.0 2.0 3-0 F i g . 2.6.4 Amplitude versus Frequency for Zero-Order, First - O r d e r , and L i n e a r i z e d Holds. Figure (2.6.4) shows the amplitude versus frequency curves for the three holds. Compared to the zero-order and f i r s t - o r d e r holds, the l i n e a r i z e d hold does not act as a very good f i l t e r i n that i t am-p l i f i e s frequencies greater than the sampling frequency. There i s also considerable a m p l i f i c a t i o n of frequencies le s s than the sampling f r e -quency with a f a i r l y steep cut o f f . Examination of fi g u r e (2.6.5), 19 ui r a d / s e c F i g . 2.6.5 Bode Plot of Thermal Reactor plus Control Rod Servo System. the Bode p l o t of the reactor plus c o n t r o l rod, shows that with a 0.1 second sample period the high frequency components are not r e a d i l y passed by any type of hold, as these components have an amplitude of the order of -35 dB. However, with longer sample periods, frequencies above the sample frequency are amplified by the l i n e a r i z e d hold, making the system l e s s s t a b l e . Therefore the l i n e a r i z e d hold should be used only f o r sample periods i n the order of 0.1 second and the zero-order should be used f o r a l l longer sample periods. In Chapter 3 i t w i l l be shown that the r e s u l t s of the s t a b i l i t y a nalysis using l i n e a r power l e v e l s are a p p l i c a b l e to the logarithmic power l e v e l c o n t r o l algorithm developed i n that Chapter. In Chapter 5 the c o n t r o l l e r i s tested using analog and d i g i t a l simulations and i t w i l l be seen that the r e s u l t s of Table 2.6.1 have a 25% safety margin. 20 3. DIRECT DIGITAL CONTROL ALGORITHM The d i g i t a l c o n t r o l algorithm besides maintaining the reactor at a steady state must also be able to change the neutron power from one demand l e v e l to another as quickly and as s a f e l y as p o s s i b l e with a mini-mum of over- or undershoot of the f i n a l demand l e v e l . For safety reasons the rate at which the neutron power l e v e l can change i s constrained and any such power l e v e l change must be c a r e f u l l y c o n t r o l l e d . In t h i s chap-ter a summary of previous algorithms w i l l be given, followed by the de-velopment of an algorithm based on logarithmic power l e v e l that r e s u l t s i n much quicker and simpler computer c a l c u l a t i o n s . A l l algorithms are based on the error sampled closed loop c o n t r o l system described i n Chapter 2. Time optimal d i g i t a l c o n t r o l w i l l be covered i n Chapter. 4, although allowances f o r i t s i n c l u s i o n w i l l be made i n t h i s chapter. 3.1 Constraints on Demand Power Level Changes As mentioned i n section 2.1, the input to the reactor c o n t r o l system of f i g u r e (2.1.4) i s a demand power l e v e l plus a c o n s t r a i n t on the minimum allowable reactor period f o r safety reasons. This minimum period c o n s t r a i n t i s only a p p l i c a b l e to increases i n power l e v e l . The demand power l e v e l N^ must therefore be constrained such that: N,[(n+1)T] < N(riT) exp (T/x) (3.1.1) d - d where T = sample period and x = minimum' allowable period However, during power l e v e l decreases, i t i s often d e s i r a b l e to constrain the negative reactor period to prevent the c o n t r o l rods from being i n -serted too f a r , which would r e s u l t i n tremendous undershoot of the f i n a l 21 demand l e v e l such that N d[(n+1)T] > N d(nT) e x p ( T / - T G ) (3.1.2) where t g i s the demanded period f o r power decreases. Reactors of any appreciable power often have a constraint im-posed upon them by the thermal system i n the form of a l i n e a r rate con-8 s t r a i n t so that : N (nT) - AN < N, [(n+l)T] < N, (nT) + AN (3.1.3) d - d - d where AN = l | f - | m a x (3.1.4) The d i g i t a l c o n t r o l algorithm, besides maintaining the reactor l e v e l must thus also be able to increase or decrease the reactor power l e v e l within the above constr a i n t s . In Chapter 4, time optimal c o n t r o l i s handled and t h i s too w i l l impose some constraints on the reactor period. 3.2 Summary of E x i s t i n g Algorithms 3 Cohn proposed a d i g i t a l c o n t r o l algorithm i n 1966 which was 2 l a t e r modified by Marciniak . The hold used was the l i n e a r i z e d hold ana-lysed i n section 2.2.1 where E'(t) = N' - N = N' -N2/N., nT < t < (n+l)T (3.2.1) d ; f d p i ~ -and E'(t) = e r r o r N' d = N d[(n+1)T] = demand f l u x l e v e l of next sampling N 1 =N[(n-i)T] = measured f l u x l e v e l of l a s t sampling N = N(nT) P = measured f l u x l e v e l of present sampling 22 N f = N'[(n+l)T] = expected f l u x l e v e l of next sampling i f no c o n t r o l a c t i o n taken from present to next sampling. The algorithms were only developed f o r power increases and the f l u x de-mand was given by: N' d = min [N p + K, N d exp(T/x), N j (3.2.2) where K = constant T = sample period x = demanded reactor period N £ = f i n a l f l u x endpoint The f i r s t argument (N^+K) ensures that i n the i n i t i a l stages the demand does not diverge too f a r from the a c t u a l f l u x preventing excessively r a -pid r i s e s at a l a t e r stage. Only bang-bang c o n t r o l a c t i o n was used and i f the erro r exceeded a c e r t a i n deadband, the c o n t r o l rods were driven f u l l speed i n or out depending on the erro r sign. Nuclear reactors have a range covering many decades and t h i s has r e s u l t e d i n the use of f l o a t i n g point arithmetic f o r the c o n t r o l a l -9 gorithm c a l c u l a t i o n s . Cohn tested the speeds of a s e l e c t i o n of .small computers and discovered that f o r computers with hardware f i x e d point arithmetic units the time f o r an a d d i t i o n i s i n the order of 0.5 to 1.0 msec. M u l t i p l i c a t i o n and d i v i s i o n times are also from 0.5 to 1.0 msec, with the logarithmic and sine function times i n the order of 5.0 to 10 msec. The computer i s not only responsible f o r the c o n t r o l of the reactor power l e v e l but also f o r other functions such as safety i n t e r l o c k s and safety scanning, data logging and the co n t r o l of other system components. This has meant that the maximum sample rate has often been set by the time taken i n the c a l c u l a t i o n of the c o n t r o l algorithm and other duties instead 23 of a desired f a s t e r sample frequency. In s e c t i o n 2.6 i t was shown that the sample period set a maximum rate of r e a c t i v i t y change per unit e r r o r to ensure s t a b i l i t y and thus the higher the sample rate the greater the s t a b i l i t y margin. There i s s t i l l some controversy over high sample rates due to the greater frequency of movement of the c o n t r o l rods. However, a sample rate of 10Hz has now been accepted as the maximum acceptable sam-ple rate. Besides the use of f l o a t i n g point arithmetic r e s u l t i n g i n a r e -duction i n the sampling frequency, there i s also the delay between the measurement of the reactor power and the a c t u a l output of c o n t r o l a c t i o n which makes the system l e s s stable. This delay has not been taken i n t o account i n the section on s t a b i l i t y a n alysis (section 2.6). In the extreme, the sampling of the neutron power l e v e l can immediately follow the output of c o n t r o l a c t i o n from the l a s t sampling. In the next sub-section, i t w i l l be shown how the use of logarithmic power l e v e l s can allow the use of f i x e d point arithmetic, greatly increasing the algorithm c a l c u l a t i o n speed. Examination of equation (2.4.1) shows that the gain of a reac-tor i s proportional to the neutron power l e v e l such that A = K'N (3.2.3) o where A = t o t a l gain K' = gain constant - N = a c t u a l power l e v e l o r In order to hold the o v e r a l l system gain constant, a gain term of the or-3 der 1/N q must be added to the e r r o r sampler. Cohn i n h i s system did t h i s by varying the e r r o r deadband i n proportion to the neutron power l e v e l . Using logarithmic power l e v e l s w i l l be seen to compensate the gain auto-m a t i c a l l y . 24 3.3 Logarithmic D i g i t a l Control Algorithm Taking the case of the zero order hold define e(t) = l n E (t) = l n N d(nT) - l n N(nT) nT < t < (n + 1)T (3.3.1) instead of the normal l i n e a r case of E(t) = N d(nT) - N ( n T ) , nT < t < (n+l)T (3.3.2) as given i n equation (2.2.3). Therefore E,(t) = N,(nT)/N(nT), nT < t < (n+l)T (3.3.3) l d Let E (t) = 1 + A (3.3.4) Using the approximation that In (1 + A) « A f o r |A| « | l | (3.3.5) then e(t) * A (3.3.6) as N(nT) must deviate only s l i g h t l y from N^(nT). Subst i t u t i o n i n equa-t i o n (3.3.4) gives E 1 ( t ) = 1 + e ( t ) ) nT < t < (n+l)T (3.3.7) Dividing equation (3.2.3) by N(nT) gives N(nT) N(nT) From equations (3.3.3) and (3.3.7) . N (nT) MIA_ _ __d ± (3.3.8) i l l " E l ^ " 1 < 3- 3' 9 ) = £ ( t ) , nT < t < (n+l)T (3.3.10) For the f i r s t order hold the logarithmic er r o r i s defined as follows: e(t) - e(nT) + (e(nT) - e ( ( n - l ) T ) ) (t-nT)/T, nT < t < (n+l)T (3.3.11) 25 where e(riT) = lnN d(nT) - lnN(nT) The e r r o r of the l i n e a r i z e d hold using logarithmic power l e v e l s i s defined as follows: e(t) = l n N d[(n+1)T] - 2 l n N(nT) + In N[(n-1)T], nT < t < (n+l)T ' (3.3.12) Again i t i s easy to prove that for the f i r s t - o r d e r and l i n e a r i z e d holds that e(t) = E(t)/N(nT), nT < t < (n+l)T (3.3.13) where e(t) i s the error using logarithmic power l e v e l s and E(t) i s the e r r o r using l i n e a r power l e v e l s . The unit e r r o r that was defined i n section 2.6 i s the same as equation (3.3.13), therefore the s t a b i l i t y analysis of that section applies 3 to the logarithmic control algorithm as w e l l . Cohn i n h i s system com-pensated for the non l i n e a r gain of the reactor by varying the deadband i n proportion to the neutron power. From equation (3.3.13) i t can be seen that the logarithmic e r r o r sampler automatically compensates f o r this gain v a r i a t i o n . The range of power of a reactor can vary from a minimum of 6 decades for heavy water moderated reactors to as much as 14 decades f o r graphite reactors. I t i s t h i s extreme range that has made f l o a t i n g point arithmetic necessary. When using logarithmic power l e v e l s t h i s range i s reduced to 14 f o r the case of the graphite reactor making i t possible to use f i x e d point arithmetic with tremendous i n c r e a s e s . i n c a l -c u l a t i o n speeds. In'Chapter 5, a PDP-9 computer was used to t e s t ex-perimentally the control algorithm. The t o t a l time elapse from the rea-ding of the neutron power l e v e l to the output of the c o n t r o l action was twice the time taken f o r one addition using the f l o a t i n g point package 26 of the computer. 3.4 L o g a r i t h m i c D i g i t a l C o n t r o l A l g o r i t h m Demand Power L e v e l Changes In s e c t i o n 3 .1 the c o n s t r a i n t s on the change i n r e a c t o r power l e v e l were seen to be a minimum a l l o w a b l e r e a c t o r p e r i o d c o n s t r a i n t and a l i n e a r r a t e c o n s t r a i n t . Time optimal c o n t r o l (which i s covered i n the next chapter) imposes a c o n s t r a i n t on the minimal al l o w a b l e r e a c t o r p e r i o d as the f i n a l endpoint i s approached, so that there i s minimal over- or undershoot. The demand power l e v e l at the next sampling i s given as: N d[(n+ 1 ) T ] = N d(nT) exp ( T / T J ) ( 3 . 4 . 1 ) Therefore l n N [(n+l)T] = l n N , (nT) + T / T , ( 3 . 4 . 2 ) d d d where T , = demanded r e a c t o r p e r i o d , d Let T = minimum all o w a b l e r e a c t o r p e r i o d m = minimum allowable r e a c t o r p e r i o d as imposed by the l i n e a r r a t e c o n s t r a i n t = N(nT)/AN ( 3 . 4 . 3 ) where AN = | | max ( 3 . 4 . 4 ) and T q = minimum a l l o w a b l e r e a c t o r p e r i o d as imposed by the time op t i m a l c o n s t r a i n t . Then T , = max [T , T . , T ] ( 3 . 4 . 5 ) d m 1 o For the case when In N, (nT) < l n N , that i s a power l e v e l i n c r e a s e , d e where N i s the f i n a l endpoint, then e l n N [(n+l)T] = min [ l n N(nT) + C, In N d(nT) + T/x d, In N J ( 3 . 4 . 6 ) 27 where C = constant. (3.4.7) Constant C i s chosen somewhere i n the.order of twice the e r r o r which gives f u l l c o n t r o l rod v e l o c i t y . The power demand ( l n N,) i s used f o r the c a l c u l a t i o n of the d next power demand point instead of ( l n N), to ensure that the demand w i l l r i s e smoothly, unaffected by the s t a t i s t i c a l f l u c t u a t i o n s i n ( l n N). However, the f i r s t term ensures that the demand w i l l not diverge too fa r from the act u a l power i n the i n i t i a l stages of the power l e v e l increase, when the power i s r i s i n g much more slowly than the demand, thus preventing excessively rapid r i s e s at a l a t e r stage. I f there i s a decrease i n power l e v e l , that i s l n N^(nT) > l n N e, then l n N [(n+l)T] = max [ln N(nT) - C, l n N,('riT) - T/x,, l n N ] (3.4.8) d d e Equations (3.4.6) and (3.4.8) w i l l be used i n the development of a p r a c t i c a l d i g i t a l c o n t r o l l e r i n Chapter 4. 3.5 Logarithmic Power Level Measuring C i r c u i t s Two possible methods of measuring the logarithmic power l e v e l are as follows: a) The same c i r c u i t s as i n s e c t i o n 2.5 can be used and the logarithm of the power ca l c u l a t e d d i g i t a l l y ; b) The ion chamber am p l i f i e r s of figures (2.5.1) and (2.5.2) can be replaced by logarithmic ion chamber a m p l i f i e r s . Exceptionally good logarithmic a m p l i f i e r s covering up to seven decades are now a v a i l a b l e . This method i s p r e f e r -able to (a) as i t provides a much more even spread of d i g i t i z e d logarithmic power l e v e l s besides r e q u i r i n g fewer measuring ranges to cover the entire, power operating range. 28 4. TIME OPTIMAL REACTOR CONTROL Due~t,p the approximate nature of the models used i n time op-timal c o n t r o l studies, p r a c t i c a l a p p l i c a t i o n s to r e a l or simulated sys-tems normally r e s u l t i n sub-optimal c o n t r o l . This i s e s p e c i a l l y true of reactor systems which are highly non-linear and complex. Safety standards impose many constraints upon reactor operation, making optimal c o n t r o l more complicated. ; Studies i n time optimal d i g i t a l c o n t r o l of nuclear reactors have thus r e s u l t e d i n time consuming computer c a l c u l a -tions of high complexity. In th i s s e c t i o n , optimal control sequences using switch points w i l l be developed. Simulation techniques w i l l be used i n obtaining the switch points, thereby e l i m i n a t i n g the complex and very approximate c a l c u l a t i o n s . 4.1 Review of Present L i t e r a t u r e Much has been published i n the past twenty years concerning the optimization of continuous and sampled-data control systems. Most notable of these endeavours are the more general theories advanced by Pontryagin et a l ^ and Bellman^. Only i n recent years has much at t e n t i o n been focused on the optimization of nuclear systems, e s p e c i a l l y i n the optimal shutdown of reactors to avoid the poisoning of the reactor by Xenon b u i l d up. L i t e r a t u r e on the design of optimal d i g i t a l or sampled-data control systems for nuclear reactors i s sparse. 12 13 1A-A serie s of papers published by Monta ' 5 was one of the f i r s t complete studies on the optimization of continuous as w e l l as dis c r e t e reactor systems. The analysis was based on the c a l c u l a t i o n of 12 the r e a c t i v i t y using a prompt-jump approximation . This approximation was proven to be inadequate as the reactor, when set for a 25 sec. mini-mum period, increased with an unsafe 16 sec. period. A side e f f e c t was 29 that the minimum sample frequency possible was 0.5 Hz, due to the c a l -4 c u l a t i o n time. L i p i n s k i , who has made a complete l i t e r a t u r e study of papers p e r t a i n i n g to nuclear reactor c o n t r o l systems, proposed a l i n e a r d e t e r m i n i s t i c system using a Kalman f i l t e r . The r e s u l t s from t h i s sys-tem were extremely good; however they were i d e a l i s t i c , because the reac-t i v i t y and delayed neutron precursor d e n s i t i e s were required at each sampling i n s t a n t , r e s u l t i n g i n long c a l c u l a t i o n times. I t was suggested that a hybrid computer system be used, with the analog portion s o l v i n g the d i f f e r e n t i a l equations i n order to speed up the c a l c u l a t i o n time. These studies did not include a l l the constraints imposed on a nuclear reactor such as minimum allowable period, the maximum rate of r e a c t i v i t y i n s e r t i o n and l i n e a r rate constraints. With t h e i r i n c l u s i o n , the com-p l e x i t y of the optimum c o n t r o l algorithms can only be expected to i n -crease. Since t o t a l optimization of the control of a nuclear plant i n -cludes the o v e r a l l performance and cost of the c o n t r o l l e r as w e l l , the question i s r a i s e d whether sub-optimum performance of the reactor i s not desirable. With so l i t t l e p r a c t i c a l experience at present with a c t u a l sampled data reactor control systems, t h i s question i s d i f f i c u l t to answer and might form the basis of an i n t e r e s t i n g future i n v e s t i g a t i o n . 2 Marciniak studied the problem from the side of the constraints imposed upon the system by safety regulations. This study seems to be most applicable to p r a c t i c a l applications and w i l l form the basis of a time optimal study using the logarithmic d i g i t a l c o n t r o l algorithm developed i n the previous chapter. 4.2 Time Optimal Power Increases For power increases i t i s desirable that the minimum allox^able reactor period constraint be adhered to and that there be a minimum of 30 overshoot. There i s also the constraint on the maximum allowable r e a c t i v i t y rate imposed e i t h e r by s t a b i l i t y or mechanical design r e q u i r e -2 ments. Taking these into account Marciniak applied the Maximum P r i n c i p l e of Pontryagin"^ to obtain an optimal c o n t r o l sequence for reactor power increases. For the case where the delayed neutron precursors are ignored, the sequence i s as i n figure (4.2.1). The c o n t r o l rods are withdrawn at f u l l speed u n t i l the demanded minimum period i s obtained and the r e a c t i v i t y i s then held constant. At a switch point S the control rods are in s e r t e d m at f u l l speed such that as the f i n a l demand l e v e l i s reached, the t o t a l Time Fi g . 4.2.1 Time Optimal Control Sequence for Prompt Reactor. Time F i g . 4.2.2 Time Optimal Control Sequence with Delayed Neutrons Included. r e a c t i v i t y i s zero. Taking one group of delaj'ed neutrons into account resulted i n the sequence as given i n f i g u r e (4.2.2). Again the c o n t r o l rods are withdrawn from the reactor at f u l l v e l o c i t y u n t i l the demand period i s obtained a f t e r which the r e a c t i v i t y i s held constant. At a switch point S^, the control rods are inserted at maximum rate u n t i l the f i n a l demand i s reached. However, on reaching the f i n a l demand, the t o t a l r e a c t i v i t y i s not zero, and the r e a c t i v i t y i s now decreased exponentially, 31 maintaining the endpoint l e v e l . When the endpoint i s reached, the de-layed neutrons are not i n equilibrium f o r the endpoint l e v e l and the power l e v e l i s held constant by the v a r i a t i o n of the r e a c t i v i t y while precursor-density equilibrium i s attained. The v a r i a t i o n of the r e a c t i v i t y to maintain the f i n a l demand l e v e l can be obtained as follows: The one delayed group k i n e t i c s equations as given i n Appendix A are •dn u - 8 dt £ and n + AC (4.2.1) f = f n - AC (4.2.2) where u = the r e a c t i v i t y or co n t r o l . Solving f o r u ( t ) , when dn/dt i s zero gives u(t) = 8 - (A£C(t)/n e) (4.2.3) where n i s the f i n a l demand l e v e l , e Harrer^^ showed that the r a t i o of C to n when the reactor i s on an asymptotic period T can be given by C _ B (4.2.4) n £(A + 1/T) Therefore i f i t i s assumed that the reactor i s on an asymptotic period T when the f i n a l demand l e v e l n i s reached, then: e e n e 3 C = e £(A + 1/T ) (4.2.5) e Solving the d i f f e r e n t i a l equation (4.2.2) f o r when the demand l e v e l n^ i s reached,gives 8n f exp[-A(t-t )] C ^ = - A ! j 1 - (AT E + i ? > ( 4 ' 2 ' 6 ) where t i s the time when the demand l e v e l n was reached. S u b s t i t u t i n g e e 32 for C(t) into equation (4.2.3) gives u(t) = 7 I 7 + 1 T S X P [ - X ( t _ t e ) ] (4.2.7) D i f f e r e n t i a t i o n gives the r e a c t i v i t y rate du(t) _ -BX dt (xx +i7 e x P I - ^ t - V ^ <4-2-8> The maximum r e a c t i v i t y rate i s needed p r e c i s e l y at the time the f i n a l demand l e v e l i s reached, thus i f the maximum r e a c t i v i t y rate i s known, the minimum allowable asymptotic period x^ at the i n s t a n t the f i n a l demand l e v e l n i s reached, can be c a l c u l a t e d to ensure no overshoot, e X R Therefore, from (4.2.8) R = T (4.2.9) max Xx + 1 e and x = ~r— - v (4.2.10) e R X max where R i s the maximum.rate of r e a c t i v i t y , max As the maximum r e a c t i v i t y r a t e i s normally known f o r a p a r t i -cular reactor, as w e l l as the minimum allowable period, a check using equation (4.2.9) or (4.2.10) can v e r i f y whether the c a l c u l a t i o n of a switch point i s necessary. Examination of equation (4.2.3) shows that the f i n a l demand power l e v e l can only be held constant when R * max XI dC n dt e (4.2.11) Due to the non-linear nature of reactors, the e a s i e s t method f o r deter-mining the switch point i s by simulation methods. In Chapter 5, a-d i g i t a l simulation of a zero-order, s i x delayed group, point k i n e t i c s model i s used f o r t e s t i n g of the d i g i t a l c o n t r o l l e r . The simulated 33 reactor i s set on a power increase and brought to the desired demand period with the a i d of the c o n t r o l l e r . When an asymptotic period has been attained, the c o n t r o l rods are i n -serted at f u l l speed. The condition of equation (4.2.11) i s met when the peak power value N g i s reached, as i n f i g u r e (4.2.3). The switch point can be determined i n • the form of a r a t i o of N to N where N i s the s e s power l e v e l at the switch point. Table (4.2.1) gives the r a t i o of N F i g . 4.2.3 Time Optimal Con-t r o l Switch Point C a l c u l a t i o n to N f or various minimum allowable e periods and maximum r e a c t i v i t y rates f o r the thermal reactor simulated i n Chapter 5. Reactor SWITCH POINT N s/N e Period 0.2mk/sec. O.lmk/sec. 0.05mk/sec. 0.02mk/sec. O.Olmk/sec. (sec). max rate max rate max rate max rate max rate 20 0.983 0.922 0.726 0.278 0.048 30 - 0.976 0.885 0.549 0.213 40 - 0.943 0.722 0.400 50 - - 0.971 0.814 0.548 100 - — — 0.966 0.880 Table 4.2.1 Time Optimal Switch Points For Power Increases 2 Marciniak .developed the switch equation N C r " r + c d + b ( 1" d ) / + e o + R max £- tr - 1=^  + i- 2 Ha K a J 21 A -1 (4.2.12) 34 where a = A + 6 / £ b = X/a c = B/£a d = exp (~at A) U q = t o t a l r e a c t i v i t y at the switch point S^ t^ •= time i n t e r v a l from the switch point to the f i n a l demand l e v e l and where t^ i s ca l c u l a t e d by assuming that when the f i n a l demand i s reached, the period i s asymptotic. Making use of the r e l a t i o n s h i p between asymptotic period and r e a c t i v i t y developed by G l a s s t o n e ^ where the r e a c t i v i t y u i n terms of the period x i s u = (Xx + 1) (4.2.13) as w e l l as u = u - R (t.) e s max A (4.2.14) the time i n t e r v a l i s fcA = F max (Xx + 1) (Xx + 1) (4.2.15) In the der i v a t i o n of equation (4.2.12), use was made of the l i n e a r i z e d one delayed group k i n e t i c s equation (See Appendix A), where dn Bn dt un I  + X C + — (4.2.16) This equation i s only v a l i d i n the v i c i n i t y of n Q . As a r e s u l t , equation (4.2.12) i s only r e l i a b l e f o r switch points i n the v i c i n i t y of the f i n a l demand l e v e l which i s the case f o r minimum periods greater than 80 to 100 seconds or f o r reactors with l a r g e maximum r e a c t i v i t y rates. In Chapter 5, a p r a c t i c a l d i g i t a l c o n t r o l l e r i s developed and 35 use w i l l be made of the switch point f o r time optimal c o n t r o l . 4 . 3 Time Optima] Power Decreases The optimal shut-down of reactors has been well covered i n 17 18 19 20 optimal reactor c o n t r o l studies ' ' ' and w i l l not be covered here. Therefore, for reactor shut-downs, where the cont r o l of the xenon poison-ing i s required, the demand power l e v e l of the reactor w i l l be program-med according to a time optimal sequence as given i n tne above references. The occasion could a r i s e , however, when i t i s required to reduce the reactor power to a predetermined l e v e l f o r a short period of time, such 16 that the xenon poisoning problem can be disregarded. Glasstone has shown that i t i s not possible to reduce the neutron f l u x i n a reactor more r a p i d l y than i s permitted by the most delayed neutron group with the r e l a t i o n s h i p between the r e a c t i v i t y u and period T given as follows: ( 4 . 3 . 1 ) 1 + A T where A^ i s the decay constant of the group having the precursor of longest l i f e . As u increases numerically, (1 + A^T) ->• 0 , thus f o r large negative r e a c t i v i t i e s the stable period T approaches 1 / A ^ . I t must be noted that B i s l a r g e r than usual since the delayed neutrons now c o n s t i -tute a greater proportion of the f i s s i o n neutrons. For most reactors , A^ i s i n the order of 0 .0125 sec \ therefore the stable period f o r large negative r e a c t i v i t i e s tends towards 80 seconds. Due to the con-s t r a i n t on the r e a c t i v i t y rate, i t has thus been customary to constrain the maximum amount of negative r e a c t i v i t y i n order to prevent tremen-dous undershoots of the f i n a l demand l e v e l . A second method i s to l i m i t the allowable negative reactor period f o r power decreases. Figure ( 4 . 3 . 1 ) shows a simple sub-optimal power decrease co n t r o l sequence with a con-36 s t r a i n t on the minimum allowable negative period. The e f f e c t of the precursor with longest delay time can be c l e a r l y seen, as more than 10 minutes i s required for a stable asymptotic period of 100 seconds to be attained. At a switch point S the r e a c t i v i t y i s inse r t e d at maximum ' i 1 1 — — T .1 i i u_i . L 1 1 I I I I 0 5 10 15 20 25 30 35 Time (minutes) F i g . 4.3.1 Power Decrease with 100 Second Period Constraint rate u n t i l the f i n a l demand' l e v e l i s reached. As was seen i n the case for power increases, when the endpoint i s reached, the r e a c t i v i t y i s not c zero. The power l e v e l i s held constant by the v a r i a t i o n of the r e a c t i -v i t y while precursor-density e q u i l i b r i u m i s attained. C a l c u l a t i o n of the switch point S g i s complex, with many approximations and assumptions. Again the e a s i e s t method i s by simulation techniques. A minimum negative reactor period of the order of 100 seconds i s s u i t a b l e , as i t requires only 1.6 to 2.0 mk to maintain i t on a stable period (see fig u r e 4.3.1) 37 and does not d i f f e r too much from the 80 second l i m i t . The main pro-blem i s that the power l e v e l has decreased by as much as s i x decades before a stable asymptotic period i s attained. However, i f the switch point i s determined when this period has been attained, i t w i l l be con-servative f or -power decreases of fewer decades as far as undershoot i s concerned. Using the d i g i t a l simulation and c o n t r o l l e r of Chapter 5, with a 100 second minimum allowable reactor period for power decreases, and a maximum, r e a c t i v i t y rate of 0.02mk/s, the r a t i o of switch point l e v e l N to the f i n a l endpoint N i s 1.188. s r e In the next chapter, the switch points w i l l be used i n the development of a p r a c t i c a l d i g i t a l c o n t r o l l e r which permits f a i r l y good approximations to the time optimal control sequences o u t l i n e d i n the previous sub-sections. 38 5. PRACTICAL DEMAND POWER LEVEL CONTROLLER A p r a c t i c a l d i g i t a l c o n t r o l l e r i s developed using machine language programming and incorporating the time optimal sequence switch points o u t l i n e d i n Chapter 4. D i g i t a l and analog simulations of a thermal reactor are used to test the c o n t r o l l e r f o r o v e r a l l s t a b i l i t y as w e l l as for c o n t r o l l e d power l e v e l changes with various minimum allowable periods and r e a c t i v i t y rates. 5.1 Control Computer S p e c i f i c a t i o n s and Programming As mentioned i n Chapter 1, the b a s i c power l e v e l c o n t r o l l e r w i l l be assumed to be part of a much l a r g e r system c o n s i s t i n g of a number of mass storage units and mini-computers assigned t h e i r own p a r t i c u l a r tasks. The control computer must therefore be able to communicate with the other system computers as well'as read from and write to the mass storage u n i t s . A hardware f i x e d point arithmetic unit option must be i n s t a l l e d i n the computer. I f output of c o n t r o l action i s d i r e c t l y from the b a s i c c o n t r o l l e r then the necessary equipment must be i n t e r f a c e d to the computer. The range of a nuclear reactor can extend over more than four-teen decades, although under normal operating conditions t h i s would be i n the order of s i x to ten decades, depending on the reactor type. How-ever, i t i s convenient to have the computer extending over the widest range possible, e s p e c i a l l y for i n i t i a l startups and long term shutdowns. Cal c u l a t i o n speed i s important, and as f l o a t i n g point arithmetic units for mini-computers are not r e a d i l y a v a i l a b l e , the logarithmic c o n t r o l alogarithm was developed i n Chapter 3, making the use of f i x e d point 39 arithmetic possible. Of the v a r i a b l e s f o r the algorithm, the demand power l e v e l requires the greatest p r e c i s i o n . From equation ( 3 . 4 . 2 ) the logarithmic power l e v e l i s l n N d[(n+1)T]= l n N d(nT) + T/x d ( 5 . 1 . 1 ) or f o r the log^Q case log N d[(n+1)T]= log N d(nT) + ( T / T ^ log (e) ( 5 . 1 . 2 ) The minimum l i k e l y sample period T i s 0 .1 second (see section 3 . 2 ) . If the smallest maximum rate of r e a c t i v i t y change'R i s 0 . 01 J a max mk/sec, then from equation ( 4 . 2 . 1 0 ) the longest probable period f o r power changes i s 630 sees. This period gives a minimum l i n e a r rate constraint of 0.16% full.power per second which i s more than adequate. Therefore [ ( T / T J ) l o g ( e ) ] min = ( 0 . 1 / 1 6 0 ) 0 . 4 3 5 ( 5 . 1 . 3 ) = 6 . 8 x 1 0 ~ 5 ( 5 . 1 . 4 ) Assuming a 1% accuracy f o r these extremely long periods and taking the 16 decade power range into account gives a p r e c i s i o n requirement f o r log N d of nine decimal d i g i t s or t h i r t y b i t s . This i s an extreme maxi-mum l i m i t . On the other hand i t might only be p o s s i b l e to bbtain a spread of 1000 sample points per decade. Assuming a c a l c u l a t i o n accuracy of 1%, t h i s gives a p r e c i s i o n requirement of seven decimal d i g i t s or twenty-three b i t s . The word length of most mini-computers i s 1 2 , 1 6 , 18 and 24 b i t s . Therefore, for most machines, double p r e c i s i o n f i x e d point arithmetic i s necessary. Depending on the sample period, the power range, accuracy of c a l c u l a t i o n s and maximum required reactor period, i t might be p o s s i b l e to use s i n g l e p r e c i s i o n arithmetic with the 24 b i t machines, which has many advantages. In the development of a p r a c t i c a l d i g i t a l 40 21 c o n t r o l l e r , a D i g i t a l Equipment Corporation PDP-9 computer with a word 22 length of 18 b i t s was used. This computer i s l i n k e d to an EAI 231R analog computer to form a hybrid f a c i l i t y . The hybrid i n t e r f a c e was 23 o r i g i n a l l y developed by Marston with a software package being developed by Crawley 2 4. Most mini-computers are supplied with comprehensive soft-ware packages., i n c l u d i n g a.basic operating system, Fortran, an assembler, e d i -tor and 1 loaders. The development of an o v e r a l l operating system w i l l not be dealt with i n t h i s t h e s i s . In previous reactor control system pro-gramming, much use has been made of Fortran, due p a r t l y to the r e q u i r e -ment of f l o a t i n g point arithmetic. Use of machine language programming usually r e s u l t s i n much fas t e r and smaller programs i n core space than would be attained with Fortran programming. Throughout the development of a p r a c t i c a l d i g i t a l c o n t r o l l e r the PDP-9 Assembly language was used. A f t e r examination of the p r e c i s i o n and mathematical functions required, a double p r e c i s i o n fixed point two's complement arithmetic package was developed. The sub-routines i n the package and t h e i r c a l c u l a t i o n speeds are given i n table 5.1.1. Comparison of the c a l c u l a t i o n speeds with 9 those found by Cohn show how much f a s t e r the fixed point routines are than t h e i r f l o a t i n g point equivalents. Part of the software support 25 package f o r the PDP-9 i s a macro-assembler which can s i m p l i f y tedious machine language programming. A macro d e f i n i t i o n f i l e was developed for the c a l l i n g of the above sub-routines and includes c o n d i t i o n a l as w e l l as s i n g l e p r e c i s i o n arithmetic macros. The form of the macros i s as follows: LABEL FUNCTION VARIABLE 1,-VARIABLE 2, (ANSWER OR CONDITIONAL JUMP . . ADDRESS) 41 Using these macros makes programming much simpler and more r e l i a b l e as w e l l as e l i m i n a t i n g many p i t f a l l s f o r the inexperienced machine language programmer. Function C a l c u l a t i o n Time (ysec) Fixed Point F l o a t i n g Point* Two's Complement 37 -Addition 48 500 Sub t r a c t i o n 55 550 P o s i t i v e M u l t i p l y 130 500 Signed M u l t i p l y 245 500 F r a c t i o n a l P o s i t i v e M u l t i p l y 70 -F r a c t i o n a l Signed M u l t i p l y 190 -Logarithm** 183 4770 A n t i l o g * * 230 -Ten Power X** 140 -* See Reference 9 ** See Reference 26 for Algorithms Table 5.1.1 Arithmetic Sub-routine Functions and C a l c u l a t i o n Times 5.2 Demand Power Level C o n t r o l l e r A flow diagram of the b a s i c power l e v e l c o n t r o l l e r i s shown i n f i g u r e (5.2.1). On the sample period i n t e r r u p t , the neutron power i s sampled and i f more than one c i r c u i t i s used, the readings are then averaged. I f the readings are i n l i n e a r form, the logarithmic value i s found and then scaled and c a l i b r a t e d . The e r r o r between the deman-ded f l u x at that sampling and the actual f l u x i s determined and the 42 / S a m p l e PeriooN v. Interrupt J I ~ Fetch and Calibrate log Neutron .tower Sample Error Calculation Output Control Action I Rext Demand Level Calculation ^Return. ^ F i g . 5.2.1 Basic C o n t r o l l e r Flow Diagram necessary action i s output to the con-t r o l rods. The demanded f l u x f o r the next sampling i s then c a l c u l a t e d before e x i t i n g from the routine. By f a r the longest c a l -c u l a t i o n i s f o r the demand power at the next sampling and i t i s ca l c u l a t e d l a s t , so that the output of c o n t r o l action occurs as soon as possible a f t e r sampling the neutron power. In the following sub-sections, each phase of the algorithm w i l l be dealt with i n d e t a i l . , 5.2.1 Fetching of Neutron Power Sample The precise manner i s which the neutron power i s sampled w i l l de-pend on the o v e r a l l system configuration. I f a separate computer i s used for data a c q u i s i t i o n and logging, as i n Chapter 1, i t can t r a n s f e r the •averaged power l e v e l to the demand power l e v e l c o n t r o l l e r and then i n t e r -rupt i t . I f logarithmic conversion, s c a l i n g and c a l i b r a t i o n are required t h i s can take place i n e i t h e r computer. I f po s s i b l e , logarithmic ion chamber am p l i f i e r s should be used so that the logarithmic neutron l e v e l can be sampled d i r e c t l y , as well as providing an even spread of d i g i t i z e d power l e v e l s (see figures (2.5.1) and (2.5.2)). E x c e l l e n t logarithmic 27 ampl i f i e r s covering up to seven decades, are now a v a i l a b l e . No matter which type of a m p l i f i e r i s used, i t w i l l be necessary to divide the e n t i r e power l e v e l span into overlapping measuring ranges as i n fig u r e (5.2.2). The t r a n s i t i o n from one range to the next i s given by N = (1-a) N.. + aN 1 u (5.2.1) where N i s the power l e v e l , N^ i s the reading from the lower range, N^ i s the reading from the upper range and a i s as i n fig u r e (5.2.2) 43 1 1 H f— Upper Range i i l l l / ^ \ l i i I i i I I I I I -7 - 6 - 5 -4 -3 - 2 - 1 0 1 10 10 10 10 10 10 10 10 10 Power Level ( F u l l Power Units) F i g . 5.2.2 Merging of Upper and Lower Measuring Ranges In the test of the c o n t r o l l e r using the d i g i t a l simulation, t h i s stage i s omitted as the simulation transfers the logarithmic neutron power l e v e l d i r e c t l y to the c o n t r o l l e r . Using the analog simulation, the l i n e a r power l e v e l was sampled by analog to d i g i t a l converters covering two ranges: one from 0 to 10% f u l l power and a second from 0 to 150% f u l l power. A f t e r s c a l i n g , the two readings were merged using equation (5.2.1) before f i n d i n g the logarithmic power l e v e l d i g i t a l l y . A flow diagram i s shown i n fig u r e (5.2.3). 5.2.2 Error C a l c u l a t i o n The equations f o r the error, using the zero-order and l i n e a r -i z e d holds, are given by equations (3.3.1) and (3.3.12) r e s p e c t i v e l y . The f i r s t - o r d e r hold w i l l be neglected because, from the s t a b i l i t y analysis of Chapter 2, i t was seen to be the worst of the holds analysed. From equations (3.4.6) and (3.4.8) the demand power l e v e l can be given by: l n N ' d = l n N d [ ( n + 1 ) T ] (5.2.2) 44 min lnN(nT) + C, lnN.(nT) + T / i j , max 1 — d — d l n N (5.2.3) where min and + are for pox^er increases and max and - are for power de-creases r e s p e c t i v e l y . The l a s t two terms are independent of the sampled Enter ^ Read A / 1 ) Channels 1 and 2 K = Channel 1 = Channel 2 N = K +BIAS * SC ALE1 u u h - Mj+BIAS * SCALE2 Amplifiers are biased so that zero i s -Vref and f u l l scale i s +Vref. =0 LIHP • " V ALPHA » (N 1-L0tf)/(HI-L0W) L1NP = (1-ALPHA )* 11-^  +ALPHA*H„ FLXP e L o g 1 0 (LIHP) ^ Return. ^ F i g . 5.2.3 Flow Diagram of Neutron Power Fetch neutron f l u x and can be calculated and tested before the sampling i n t e r -rupt (see section-5.2.4). Equation (5.2.3) can then be reduced to lnN d[(n+l) T]= ^ JlnN(nT) + C, l n N " d [(n+l)TJ \ (5.2.4) where lnN" d[(n+l)T] = m n I l n N,,(nT) + T/x,, l n N max ] d — d e (5.2.5) A flow diagram of the err o r c a l c u l a t i o n i s given i n fig u r e (5.2.4). 45 |TEKI=FLXP-C| |l'EM1=?IXi'+0 | I F)J(I)~=TEH11 ERBO =FLXD-2FLXD+FLXL (Linearized Hold) or ERR0-FLX11-FLXP (Zero-Order Hold) Return F i g . 5.2.4 Flow Diagram of E r r o r C a l c u l a t i o n 5.2.3 Output of Control Action The precise form of the c o n t r o l a c t i o n , i . e . moderator l e v e l , c ontrol rods, depends e n t i r e l y on the design of the reactor system. How-ever, the input drive i n a l l cases i s a v e l o c i t y s i g n a l and the maximum rate of change of r e a c t i v i t y i s l i m i t e d . There are three b a s i c forms of v e l o c i t y s i g n a l : (a) Bang-bang control with deadband. The r e a c t i v i t y rate i s e i t h e r zero, f u l l speed withdrawal or f u l l speed i n -s e r t i o n . (b) A d i s c r e t e number of r e a c t i v i t y i n s e r t i o n and withdrawal rates. (c) A continuously v a r i a b l e r e a c t i v i t y rate, with or without deadband. The three forms of s i g n a l are shown schematically i n figure (5.2.5). The most commonly used i s the f i r s t , due to i t s s i m p l i c i t y and the low 46 i — i 1 — i — i r -r d ~ ^ - d « J ! l i t ' ' Error (#) ( a ) . Bang-Bang Error (#) (b). Discrete m ~ " m Err o r (#) ( c ) . Continuous F i g . 5.2.5 R e a c t i v i t y Rate Signal Types frequency of rod movement. The discre t e system has b e e n ( l i m i t e d to two or three r e a c t i v i t y rates, while the continuous systems have always r e -quired f a i r l y complex feedback control, i . e . tachometer. With nuclear q u a l i t y stepping motors now r e a d i l y a v a i l a b l e , and coupled with d i r e c t d i g i t a l c o n t r o l , the di s c r e t e and continuous systems are now e a s i l y r e a l i z a b l e without the -need 'for complex feedback systems. Detailed 5 15 coverage of these systems i s given by Schultz and Harrer . Flow d i a -grams for a l l three types of system are given i n f i g u r e ( 5.2.6). As the maximum r e a c t i v i t y i n s e r t i o n rate for a p a r t i c u l a r reactor i s usually f i x e d (R ), the s t a b i l i t y of the reactor i s ensured by choosing J max J J O the appropriate c o n t r o l l e r gain (see section 2.6). The v a r i a b l e GAIN sets the required e r r o r between the actu a l neutron power l e v e l and the demanded power l e v e l to give a maximum r e a c t i v i t y rate s i g n a l . 5.2.4 Demand Power Level C a l c u l a t i o n Resides maintaining the reactor on a steady state reactor power, the c o n t r o l l e r must be able to change the power from one l e v e l to another, maintaining the performance within the constraints as given i n s ection 3.1. The l i n e a r rate constraint, although applicable only to the higher powered reactors, w i l l be included to give a complete 47 ^ Enter ^) |0?EM1=GAli»Kod(ERRO) IERRG = Sign(ERnO)*TEMl| / Output 7 / Error / ^ Return ^ (a). Basic Flow Diagram. For Analog Simulation Error Output on D/A#1. For D i g i t a l Simulation Error Value Transferred. Only " i f Deadband A ) Required. TEH1 = hAXE (b). Bang-Bang with Deadband. Quantasize ^Right S h i f t or Int. Division) (d). Continuous with and without Deadband. ( c ) . Discrete F i g . 5.2.6 Flow Diagram of Control Action Output general c o n t r o l l e r . From equation (5.2.3) the demand power l e v e l i s seen to be a function of the f i n a l endpoint and the demanded reactor period. The f i r s t term i s to ensure that the demand does not diverge too f a r from the actual power during the i n i t i a l stages of l e v e l changes. Taking the l i n e a r rate constraint i n t o account r e s u l t s i n the inverse demand period being a function of the power l e v e l as given i n f i g u r e (5.2.7). Below the switch point the demand period x^ i s the minimum allowable reactor period T . Above the switch point the inverse demand m period i s : l / x d = l / x 1 (5.2.6) 48 where (5.2.7) (5.2.8) Log Neutron Power F i g . 5.2.7 Inverse Period for Log and Linear Constraints. The inverse demand period, besides being p h y s i c a l l y measurable, i s used as i t i s the form of the period required by equation (5.2.3). In Chapter 4, dealing with time-optimal c o n t r o l , i t was seen that for a power l e v e l increase, the control rods are i n s e r t e d at maxi-mum rate on reaching a switch point u n t i l the f i n a l demand l e v e l i s attained. The e a s i e s t method i s t o open the c o n t r o l loop on reaching the switch point, and output a maximum co n t r o l rod v e l o c i t y s i g n a l , c l o s i n g the loop again when the f i n a l demand power i s reached. This method i s only s u i t a b l e under i d e a l conditions. I f , on reaching the switch point S^, the reactor i s r i s i n g on a slower period than the minimum allowable period, then with the maximum rate of control rod i n s e r t i o n , the f i n a l endpoint w i l l never be reached. The problem as to what c o n t r o l procedure must be followed also arises i f the i n i t i a l power l e v e l i s above the switch point. With these problems, i t i s doubtful whether the c o n t r o l l e r would pass the s t r i c t safety regulations with an open loop c o n t r o l band about the f i n a l endpoint. The i d e a l s o l u t i o n i s . to have dynamic time 49 optimal c o n t r o l . The study by L i p i n s k i has shown th i s to be p o s s i b l e , but the complexity of the c a l c u l a t i o n s required a f t e r each sampling r e s u l t s i n extremely long c a l c u l a t i o n times, even when a hybrid computer i s used. It was seen i n section 4.2 that for a maximum r e a c t i v i t y rate R , there i s a corresponding minimum allowable period T , such that max e for reactor periods greater than x , no switch point i s required. The r e l a t i o n s h i p between R a n d T i s given by equations (4.2.9) and (4.2.10). max e Therefore the minimum allowable period at the f i n a l endpoint must be greater than x g i f the neutron power i s to be held constant. A p o s s i b l e method of obtaining maximum control v e l o c i t y , while s t i l l maintaining closed loop c o n t r o l , i s to increase the demand period to x & at the switch point, as i n figure (5.2.8). Examination of equation (5.2.3) shows that for power increases the minimum of the three terms i s chosen as the demand power l e v e l l n N^. The neutron power w i l l therefore increase r a p i d l y compared to the demand, with the r e s u l t i n g e r r o r giving f u l l c o n t r o l rod v e l o c i t y . From the e r r o r c a l c u l a t i o n flow diagram ( f i g u r e 5.2.4), i t can be seen that as the neutron power reaches the f i n a l end-point, the demand power In i s automatically set to the f i n a l endpoint value In The disadvantages of t h i s system are that the f i n a l endpoint power must be attained, otherwise the demand l e v e l l n w i l l continue to r i s e on a period T , instead of being set to the f i n a l endpoint. As a r e s u l t , the neutron power l e v e l w i l l turn around and decrease u n t i l the demand l e v e l i s reached, a f t e r which the f i n a l endpoint w i l l be approached on a period (see figure (5.2.9)). I f the i n i t i a l power l e v e l i s above the switch point, then the demand period w i l l be kept constant at x , which i s far from optimal. 50 Log Keutron Power F i g . 5.2.8 Inverse Period f o r Time Optimal Power Increase. (Step Increase i n Period) ( a ) . Ideal Case (b)."Conservative Switch Point F i g . 5.2.9 Neutron Power Level Increase with Step Period Change Log Power Level F i g . 5.2.10 Inverse Period f o r Time Optimal Power Increase. (Continuous Increase of Period) <» > U G> % o (k 6D o H i y^ /y i—Demand - pi ' Power l — A c t u a l / 1 t 1 1 Power I l +> <d as Time F i g . 5.2.11 Delay i n A t t a i n i n g R• with Continuous Period max Increase Case. 51 If the demand period i s gradually increased from T at the switch point to T G at the f i n a l endpoint as i n f i g u r e (5.2.10), i t should be possible to maintain an error s i g n a l giving maximum r e a c t i v i t y r ate, while not allowing the demand l e v e l to l a g too f a r behind the actual neutron power l e v e l , as was the case i n f i g u r e (5.2.9). This method also permits f a s t e r power l e v e l changes i f the i n i t i a l l e v e l i s above the switch point. However, there i s now a s l i g h t delay a f t e r reaching the switch point, before the e r r o r i s s u f f i c i e n t to output a maximum con t r o l s i g n a l as shown i n figure (5.2.11). This delay i s dependent on the c o n t r o l l e r gain; the higher the gain, the shorter the delay. The switch points given i n table 4.2.1 w i l l have to be compensated to make up f o r the delay. The delay can be shortened by varying the demanded reactor period as shown i n f i g u r e (5.2.12), as i t i s often impossible to increase the system gain due to i n s t a b i l i t i e s a r i s i n g . I d e a l l y , the demand neu-tron power should reach, the f i n a l endpoint at the same insta n t as the neutron power. The time taken f o r F i g . 5.2.12 Inverse Period the demand neutron power l e v e l to for Time Optimal Power Increase. (Continuous reach the endpoint from the switch plus Step Increase of Period). point can be c a l c u l a t e d as follows: E ' e Log Power Level Let l n N, - l n N d £ L = l n N - l n N e e s ln = l o g demand power l e v e l l n N g = log switch point power l e v e l 52 ln = l o g endpoint power l e v e l T = minimum allowable reactor period m f" = allowable reactor period at endpoint e A = 1/T - 1/T m e From figure (5.2.10) i t can be seen that f o r the continuous case dt (5.2.9) Therefore dL AL - 7 ^ = 1 / 1 (5.2.10) dt m L e The s o l u t i o n of t h i s equation i s : L,(t) = L / T A f l - exp(-tA/L ) ] (5.2.11) d e m L . r v e J The time taken f or L^ to reach L i s required, therefore the following equation i s solved f o r t: t = a L /A (5.2.12) e where a i s solved from exp(-a) = 1 - AT m (5.2.13) = T / T (5.2.14) .me For the case where the period i s va r i e d as i n fig u r e (5.2.12), i s replaced by T\ The act u a l times taken by the reactor from the switch point to the endpoint under i d e a l conditions were measured when the switch points of table 4.2.1 were determined. Table 5.2.1 gives these times for various minimum allowable periods and maximum r e a c t i v i t y rates. The table also gives the times f or the demand power l e v e l to r i s e from 53 the switch point to the endpoint using equations (5.2.12) and (5.2.14). The endpoint period f o r the various r e a c t i v i t y rates was determined using equation (4.2.10). I t can be seen that, except f o r those cases MINIMUM TIME (SECONDS) PERIOD (sec) . ^max Reactor lmk/ s Demand ^max Reactor .05mk/s Demand Rmax _ • Reactor 02mk/s Demand ^max -Reactor .Olmk/s Demand 20 3.2 2.6 15 13.6 71 75 162 219 30 2.3 .95 11 6.7 50 46.5 121 149 40 1.7 .425 8 3.8 38 31 96 109 50 - - 5 2.2 29 22 80 84 100 - - - - 11 5.8 37 28.4 Table 5.2.1 Times for Reactor and Demand to Reach Endpoint from Switchpoint where the maximum r e a c t i v i t y rate i s 0.01 mk/second, and where the 20 second period i s combined with R = 0.02 mk/second, the time f o r the max demand power l e v e l to reach the endpoint i s shorter than that f o r the reactor. The period can therefore be va r i e d as i n f i g u r e (5.2.12), to compensate f o r the delay i n a t t a i n i n g a maximum r e a c t i v i t y rate s i g n a l , By choosing appropriate values f o r T 1 and x' e such that: and T > T m m T ' £ T e e (5.2.15) (5.2.16) the reactor and demand power l e v e l s can reach the f i n a l endpoint at the same in s t a n t . Table 5.2.2 gives values f o r T ' and T ' using the switch m e ° points of table 4.2.1. The problem s t i l l e x i s t s for those cases where the demand power l e v e l takes longer than the reactor to reach the endpoint. I f a s l i g h t l y shorter value f o r T can be toler a t e d , the time f o r the demand to reach e 54 the endpoint can be made the same as. the reactor. The appropriate values of T ' were included i n table 5.2.2. Whether these values can be t o l e r -e ated w i l l be determined when the c o n t r o l l e r i s tested i n section 5.5. The preceding analysis has been f o r power increases. S i m i l a r l y , for power l e v e l decreases the same procedure can be followed, with the reactor period v a r i e d as i n figure (5.2.13). Table 5.2.3 gives the switch point values, the time for the reactor to reach the endpoint and appro-p r i a t e values of T ' and x 1 . r m e o O U /-> N « V -H o n u o 0) w e e Log Neutron Power F i g . 5.2.13 Inverse Period for Power Level Decrease (Continuous Increase of Period). MINIMUM x' and x' (sec) m e PERIOD R max .lmk/s R max .05mk/s R max •02mk/s R max .Olmk/s (sec) . T 1 m x' e x' m x' e x' m T 1 e x' m x' e 20 30 51 25 110 24 180 24 165 30 94* 94* 67' 115 35 280 35 245 40 182* 182* 137* 137* 56 307 46 315 50 - - 167* 167* 75 307 57 465 100 - - - - 314* 314* 151 627 *Note: A = 0, therefore x ' = x' m e - Vt Table 5.2.2 x' and x' for Simultaneous A r r i v a l of m e Reactor and Demand at Endpoint 55 Switch Time to R Point Endpoint T ' T* max 1 m e mk/sec N /N (sec) (sec) (sec) .05 1.008 1.9 211 211 .02 1.188 37 150 310 .01 1.63 114 140 475 Table 5.2.3 Parameters for Power Decreases with 100 second Minimum Period Constraint Log Power Level F i g . 5.2.14 Inverse Period as a Function of Power Level I f the reactor i s at steady state, the v a r i a t i o n of the demand period about the steady state l e v e l i s given i n f i g u r e (5.2.14). There i s a deadband of value C on e i t h e r side of the steady state l e v e l and i f the neutron power l e v e l remains i n this deadband, the demand power l e v e l i s held at the steady state l e v e l . I f the neutron power l e v e l should deviate outside the deadband the demand power l e v e l l n N^' i s set to the value l n N + C, depending on whether i t i s below or above the steady state. (See the er r o r flow diagram of f i g u r e (5.2.4)). The demand power l e v e l l n N^1 i s then returned to the steady state l e v e l with the demand period varying as i n f i g u r e (5.2.14). Figure (5.2.15) shows the v a r i a t i o n of the demand period when the f i n a l endpoint i s i n the range where the l i n e a r rate constraint i s a c t i v e . 56 -d m « 01 V H O n) a> o) S w w O 0) H F i g . 5.2.15 Inverse Period as a Function of Power Level (Linear Constraint) A flow diagram of the demand power l e v e l c a l c u l a t i o n i s given i n f i g u r e (5.2.16). I t must be remembered that the switch points given i n table 4.2.1 are for i d e a l conditions. I f there i s any delay i n a t t a i -ning a maximum r e a c t i v i t y rate s i g n a l a f t e r the switch point, the switch point must be compensated to avoid over- or undershoot of the f i n a l l e v e l . I t i s probable that the c o n t r o l l e r developed w i l l have a sub-optimal response. Just how sub-optimal i t i s , w i l l be determined i n subsection 5.5, where the c o n t r o l l e r i s tested using a d i g i t a l simulation of a thermal reactor. The r e s u l t s obtained f o r power l e v e l increases and decreases- w i l l be compared to the i d e a l r e s u l t s of Chapter 4 and necessary adjustments i n the switch points w i l l be determined. 5.2.5 New Endpoint and Switch Point Calculations The only probable i n t e r a c t i o n between the safety system and the c o n t r o l l e r would be the> s e t t i n g of the f i n a l endpoint by the safety system for c o n t r o l l e d power reversals and power l e v e l l i m i t setbacks. Program or operator i n i t i a t e d power l e v e l holds can also be expected. A l o g i c diagram of possible endpoint p r i o r i t i e s i s given i n f i g u r e (5.2.17). The switch points associated with a p a r t i c u l a r endpoint are TEM2=( S'»/SD-FLXD)/( SWSD-FLXE) TEK1 =DHRT- ( DKRrJ-DERT ) TSH2 Load Kew Endpoint and Switchuoints. Reset EP Flag. LIMD=AKTILOG(FLXD) FLXT=FLXE - FLXD |PERD—TEMll [ PERD=-TEM2J [PERD=-LNRT/LIKD |PERfl=-DRAT| T •IEM2=( FLXD-SW8T- )/(FLXE-SV;ST ) TEH1 ='Ji4HT-( UFlR'X'-UERT ) TEM2 FERD=URAT PERD=LKRT/LIHD| IF£RD=T£M1| |PERD=TEM2 FLXD=FLXD+FERD TEK1=FLXD-FLXE I ^Return. ^ O i F i g . 5.2.16 Flow Diagram of Demand C a l c u l a t i o n 58 Flaps Power Hold Desired Endpoint Demand at Flag Set e.g. H0% P u l l Power e.g. 1CT 6 F u l l Power Power Limit Power Setback Actual Endpoint F i g . 5.2.17 Endpoint P r i o r i t y Chain' simple to c a l c u l a t e . As the maximum r e a c t i v i t y rate R and the minimum J max allowable reactor period x are f i x e d for a reactor, the switch points r m fo r time optimal control f o r a p a r t i c u l a r endpoint N 1 are given as follows: l n N . = l n N' + In S,. s i e d i (5.2.9) where i = 1 i s for power increases i = 2 i s f o r power decreases and S , = ratio, of N /N as determined by simulation methods described d s e J . i n Chapter 4.-The l i n e a r rate switch point remains f i x e d , independent of the f i n a l endpoint and i s given by N = x dn/dt| 1 m 1 ' max (5.2.10) With the occurrence of a reactor scram or emergency shutdown, the end-point i s set to the minimum power l e v e l , the c o n t r o l l e r output discon-nected and the demand power l e v e l allowed to f l o a t down x^ith the neutron power l e v e l . 59 5.2.6 General Remarks Throughout the development of the c o n t r o l l e r , an attempt was made to minimize the c a l c u l a t i o n time from the sample i n t e r r u p t to the output of the control a c t i o n . A l l c a l c u l a t i o n s not dependent on the mea-sured neutron f l u x , were completed p r i o r to the sample i n t e r r u p t . The algorithm as i t stands i s by no means complete. Areas such as shim and regulator rod con t r o l , maintaining the regulator rod at maximum e f f e c t i v e -ness and many others have not been included as i n most cases they are de-pendent on the i n d i v i d u a l reactor type. From the s t a b i l i t y analysis of Chapter 2, i t was seen that except for extremely fast sampling rates, the zero-order hold was the best of the three hold types. Examination of equation (3.3.1) shows that at steady state there i s only proportional c o n t r o l , with no rate c o n t r o l . I f the error equation (3.3.12) f o r the l i n e a r i z e d hold i s broken down, the following form can be obtained: e n ( t ) = [ln N d(nT) - l n N(nT).] + JT/TJ - [lnN(nT)-lnN(n-l) T] (5.2.11) Therefore, i t can be seen that the. l i n e a r i z e d hold gives p r o p o r t i o n a l plus rate c o n t r o l . 5.3 D i g i t a l Simulation of a Nuclear Reactor The c o n t r o l l e r requires as input a logarithmic neutron power l e v e l and outputs control action i n the form of a l i m i t e d r e a c t i v i t y rate s i g n a l . The point reactor k i n e t i c s equations with six- groups of delayed neutrons are given i n Appendix A by equations (A.1.1) and (A.1.2). These equations r e s u l t i n a l i n e a r neutron power. D i v i d i n g through by n i n both 60 equations gives: and f / n = ^ - f + I ^ C . / n + S/n (5.3.1) 1=1 dc . 8. "dt / n " "I " A i V n • ( 5 ' 3- 2 ) Let ra = / n (5.3.3) dt V = C./n (5.3.4) i i and w = S/n (5.3.5) Su b s t i t u t i o n i n t o equations (5.3.1) and (5.3.2) gives ,m = - | + I A . V + .w (5.3.6) i = l and dV. 3. ~ = - y - V. (m + A.) (5.3.7) d t £ l i The quantity / n i s the inverse reactor period. Also l o g e n = Jm dt (5.3.8) Therefore with the change i n the v a r i a b l e , the logarithmic neutron power can be obtained d i r e c t l y from the simulation, with the simulation input of r e a c t i v i t y being retained. Equations (5.3.6) to (5.3.8) are more s u i t e d to d i g i t a l than to analog simulation techniques. The s i x equations of the form of (5.3.7) require extremely accurate and r e l a t i v e l y f a s t mul-t i p l i e r s which are not always a v a i l a b l e . The main problem i s the "open loop" in t e g r a t o r of equation (5.3.8). Both of these problems are e l i m i -61 nated with d i g i t a l simulation techniques. The change i n v a r i a b l e has the added advantage of normalizing the equations, with the delayed neutron precursors being t r a n s f e r r e d i n t o r a t i o s instead of absolute values extending over the range of power l e v e l of a nuclear reactor. I t i s therefore possible to use the f i x e d point arithmetic routines devel-oped f o r the c o n t r o l l e r , with much f a s t e r c a l c u l a t i o n times p o s s i b l e than would be the case with f l o a t i n g point arithmetic. Much c r i t i c i s m has been l e v e l l e d at d i g i t a l simulation techni-ques due to the inherent quantization and s e r i a l operation. Many sophi-s t i c a t e d methods for the numerical s o l u t i o n of d i f f e r e n t i a l equations have been proposed i n an attempt to reduce the errors incurred by d i g i t a l techniques. One of the s o - c a l l e d "unsophisticated" i n t e g r a t i o n methods, the trapezoidal i n t e g r a t i o n method, was used i n the d i g i t a l simulation of the nuclear reactor because of i t s s i m p l i c i t y , ease of programming and 2 8 excell e n t s t a b i l i t y properties . The form of trapezoidal i n t e g r a t i o n i s as follows: y x i = y " T f i (5.3.9). •'n+l n 2 n 2 n-1 where y n = the output of the int e g r a t o r at time nT f = sum of inputs to the integra t o r at time nT and T = sample period. The accuracy of the d i g i t a l simulation as a function of sample period T was tested against the analog simulation described i n s e c t i o n 5.5 for step inputs i n r e a c t i v i t y . With a sample period of 0.1 second, there was a noticeable e r r o r i n the order of 5% during the i n i t i a l stages a f t e r the step where the influence of the prompt neutrons was the greatest. With a 0.05 second sample period, this e r r o r was reduced to 1%, while at 62 a sample period of 0.01 second, the e r r o r could not be distin g u i s h e d i n the noise of the analog computer simulation. For the longer sample periods, the error was only detectable i n the i n i t i a l stages where the prompt neutrons were e f f e c t i v e . Although the simulation was accurate when the e f f e c t of the delayed neutrons became prominent, the i n i t i a l e rror was c a r r i e d forward and remained. A flow diagram of the basic d i g i t a l simulation i s given i n figur e (5.3.1). Using a sample period of 0.01 second, the simulation T£MP4=0-KEACT=REACT+T*RRATE DO F O R X=1 TO 6 TEKP1 =LAMDA ( X ) * DELAY ( X ) TEMP4 = TEKT4+ TEMPI TEMP3=BETA(X)-TEMPI-HRATE* DELAY(X) DELAY ( X )=DELAY ( X )+J * T/2 * TEHP3-T/2 * FUR C ( X ) FUNC(X)=TE!'iP3 ' TEMP4=TEMP'l + REACT-BETAT P0V,'ER=P0WER+ 3 * T/2 * TEMP4-T/2 * NRATE NRATE=TEMP4 ^ R e t u r n . ^ F i g . 5.3.1 D i g i t a l Simulation of Nuclear Reactor-Flow Diagram. (One Sampling Only) time was about h a l f the r e a l time. A handler was developed to c o n t r o l the d i g i t a l simulation and c o n t r o l l e r and pass the necessary v a r i a b l e s between the two programs. The handler provides on l i n e graphic readout of the logarithmic power l e v e l j r e a c t o r period, r e a c t i v i t y rate and t o t a l r e a c t i v i t y , and also p r i n t s out f i n a l r e s u l t s on a s t r i p chart recorder. Program i n t e r r u p t i o n and r e - i n i t i a l i z a t i o n or the s e t t i n g of any v a r i a b l e i s possible without d i s t u r b i n g the continuous simulation sequence. The three sample periods, simulation, c o n t r o l l e r and readout are independent of each other and can be set to the required values. The handler simu-63 l a t e d the neutron power measuring c i r c u i t s (see se c t i o n 2.5) by averaging the neutron power from one c o n t r o l l e r sampling to the next, before pas-si n g the neutron power l e v e l to the c o n t r o l l e r . A thermal reactor with parameters as given i n Appendix A.4 was simulated f or the t e s t i n g of the d i g i t a l c o n t r o l l e r . 5.4 Analog Simulation of a Nuclear Reactor The point k i n e t i c s equations with s i x groups of delayed neutrons are given i n Appendix A by equations (A.1.1) and (A.1.2). The analog computer c i r c u i t diagram i s given i n fig u r e (5.4.1). The analog simu-l a t i o n was set up on an EAI, PACE 231R analog computer which i s i n t e r -faced to the PDP-9 computer used f or the d i g i t a l c o n t r o l l e r . The neutron power l e v e l measuring c i r c u i t described i n s e c t i o n 2.5 i s simulated by i n t e g r a t i n g the neutron power l e v e l from one sampling to the next and by i n i t i a l i z i n g the i n t e g r a t o r a f t e r each sampling. Fortunately the 231R analog computer i s equipped with e l e c t r o n i c switching for the integra t o r modes and the shorter sample periods of 0.1 second can be e a s i l y handled. Two measuring ranges are used; one up to 10% and the second to 150% f u l l power. The two ranges are merged using the tech-nique described i n section (5.2.1). The outputs of the two in t e g r a t o r s are sampled by a multiplexed analog to d i g i t a l converter. No timing pro-blems were encountered as a l l the inputs to the multiplexer are preceded by sample and hold units and the integrators are i n i t i a l i z e d immediately a f t e r sampling and holding the two s i g n a l s . The i n i t i a l conditions of the two integrators are biased, so as to allow f u l l use of the A/D. sampling range of + Vref, thereby gaining double the number of d i g i t i z e d power l e v e l s . 64 F i g . 5.4.1 Analog Simulation of Nuclear Reactor The c o n t r o l action i n the form of a r e a c t i v i t y rate or a t o t a l r e a c t i v i t y s i g n a l i s returned to the analog simulation from the c o n t r o l l e r by means of a d i g i t a l to analog converter. The same reactor parameters were used as for the d i g i t a l simu-l a t i o n . (See Appendix A.4). 5.5 Test of D i g i t a l C o n t r o l l e r Both analog and d i g i t a l simulations described i n the previous 65 subsections w i l l be used i n t e s t i n g the d i g i t a l c o n t r o l l e r . The ad-vantage of the d i g i t a l simulation i s that i t covers the e n t i r e range of possible reactor power l e v e l s . Another advantage i s that no a d d i t i o n a l external equipment i s necessary for the t e s t i n g of the d i g i t a l c o n t r o l l e r . The analog computer simulation provides the best r e a l time conditions with the monitoring and c a l c u l a t i o n delays, as would be expected i n an actual reactor system. The disadvantage i s that the range i s l i m i t e d to about 2 decades of operation. Automatic r e s c a l i n g i s possible with power-f u l and advanced analog systems, but they are not always a v a i l a b l e . 5.5.1 C a l c u l a t i o n Time of Control Algorithm Using the analog simulation, the time taken from the moment the sampling of the neutron power i s begun, to the output of the c o n t r o l a c t i o n i s 0.8-1.1 ms. More than h a l f of t h i s time i s required i n the "sampling, "merging "and •finding the logarithm of the neutron power. These problems encountered using a "hybrid" simulation are the same as would be encountered i n a true on-line system. The longer c a l c u l a t i o n times of 1.1 ms are required when two measuring ranges are merged. The t o t a l time from the sampling to the e x i t a f t e r c a l c u l a t i n g the next demand l e v e l i s 1.5 - 1.8 ms. Even with the shortest sample period of 0.1 second, the e f f e c t of the c a l c u l a t i o n delay before the output of the control a c t i o n can be neglected. The use of the logarithmic c o n t r o l algorithm and f i x e d point arithmetic can be seen to give exceptionally f a s t and simple c a l c u -l a t i o n s . The time taken from sampling to the output of control a c t i o n i s about twice as long as the time for an addition using the computer's f l o a t i n g point package. 5.5.2 S t a b i l i t y Test of C o n t r o l l e r The o v e r a l l system s t a b i l i t y was analysed i n section 2.6. For 66 a p a r t i c u l a r sample period T, the s t a b i l i t y of the reactor could be en-sured by maintaining the r e a c t i v i t y , rate per unit e r r o r l e s s than a maximum value R . Table 2.6.1 gives R per unit error for various max max sample periods for the thermal reactor of the analog and d i g i t a l simu-l a t i o n s . The most convenient method of t e s t i n g the o v e r a l l s t a b i l i t y i s to use the analog simulation. By means of adjusting the c o n t r o l rod gain potentiometer, the gain can be increased slowly u n t i l s t a b i l i t y i s l o s t . For both the l i n e a r i z e d and zero-order holds, the values given i n table 2.6.1 were conservative. The value of R , where s t a b i l i t y max was l o s t , was 25 to 35% greater than the t h e o r e t i c a l values for a l l four sample periods. This i s an i d e a l s i t u a t i o n from the safety point of view, as the t h e o r e t i c a l c a l c u l a t i o n s of section 2.6 then have a safe 25% margin. When using the d i g i t a l simulation, the r e s u l t s d i f f e r e d by no more than 2% from those obtained using the analog simulation. 5.5.3 Power Level Increases Using the d i g i t a l simulation, the c o n t r o l l e r was tested f o r power increases using the switch points of table 2.6.1 and the respective values of T' and x' as i n table 5.2.2. A sample period of 0.1 second m e r r was used throughout the t e s t i n g , as one of the main reasons for the l o g -arithmic c o n t r o l algorithm was to allow the use of these f a s t sample frequencies. Figure (5.5.1) shows power increases with a 20 second minimum allox^able period and a maximum r e a c t i v i t y rate of 0.02 mk/second. With the c o n t r o l l e r gain such that a .1%/decade err o r between the power l e v e l and the demand gave a maximum r e a c t i v i t y rate s i g n a l , the overshoot was never more than 0.5% of the f i n a l endpoint, f o r a l l the combinations 67 F i g . 5.5.1 Time Optimal Power Increase 68 of period and R given i n table 2.6.1. When using a lower c o n t r o l l e r max • ° gain, such that an error of 1%/decade r e s u l t e d i n R out, the overshoot max i n a l l cases was between 5 and 7% of the f i n a l endpoint. This greater overshoot i s not only due to the delay i n a t t a i n i n g a maximum r e a c t i v i t y s i g n a l but also because an overshoot of 2.3% of the f i n a l endpoint i s required to obtain a maximum output s i g n a l . These r e s u l t s are e x c e l l e n t , with the overshoot being a l i t t l e over twice the er r o r required f or a maximum r e a c t i v i t y s i g n a l . Examination of figure (5.5.1) shows that the period becomes shorter than the minimum allowed j u s t p r i o r to a t t a i n i n g an asymptotic period. During the i n i t i a l stages of start-up, the demand does not deviate too f a r from the power l e v e l , due to the f i r s t term of equation (3.4.6). This peak i n the inverse period occurs as the power f i n a l l y catches up with the demand. The amount of peaking can be reduced by reducing C of equation (3.4.6). The most s u i t a b l e value of C was found to be i n the order of one-and-one-half times to twice the e r r o r required f or a maximum r e a c t i v i t y s i g n a l . In tables 5.2.1 and 5.2.2, i t was seen that f or the smaller values of R , T ' had to be smaller than the desired f i n a l endpoint max e period x , so that the demand and power l e v e l s reached the endpoint to-gether. To see whether these values of x 1 could be t o l e r a t e d , the reac-tor was set on a power increase with a minimum allowable period of T 1 . On a t t a i n i n g an asymptotic period, the c o n t r o l rods were inse r t e d at maximum v e l o c i t y and the overshoot measured. For the case of T*~ = 165 •• J e seconds, (instead of the desired 627 seconds) and R = 0.01 mk/second, max the overshoot was 6.5%. With a x' of 245 seconds, t h i s overshoot r e -e duced to 2.5%. These overshoots were much l e s s than those obtained when 69 there was a s i z a b l e delay i n a t t a i n i n g a maximum r e a c t i v i t y rate s i g n a l a f t e r the switch point. Fortunately, the p r o b a b i l i t y of any reactor having a 20 second allowable period i s small, e s p e c i a l l y i f i t only has a r e a c t i v i t y rate of 0.01 mk/second and the values of x can therefore e be t o l e r a t e d where necessary. The greater overshoot w i l l only be found i n those cases where: (a) the switch point i s conservative, (b) the minimum allowable period has not been attained and (c) the i n i t i a l power l e v e l i s above the switch point, a l l of which are shown i n f i g u r e (5.5.2). As was expected, the c o n t r o l l e r i s suboptimal. However, the higher the c o n t r o l l e r gain, the c l o s e r to the i d e a l i s the performance. Taking as examples the cases shown i n f i g u r e (5.5.1), the i d e a l time from switch point to endpoint i s 71 seconds. With a c o n t r o l l e r gain of .1%/ decade for an output of R , the time for the power f i n a l l y to s e t t l e max r J within .23% of the endpoint i s 71.5 seconds. When the gain i s 1%/decade, the corresponding time to s e t t l e within 2.3% of the endpoint i s 86 seconds. This longer time i s due to the greater overshoot, which i s a d i r e c t r e -s u l t of the delay i n a t t a i n i n g a maximum r e a c t i v i t y s i g n a l . For these lower gain cases, the time could be shortened s l i g h t l y by making the logarithmic switch point conservative by about 2% of a decade. I t can be seen i n figure (5.5.1(a)) from the spike i n the r e a c t i v i t y rate s i g n a l , how the demand and reactor power reach the endpoint simultaneously, followed by an instantaneous maximum s i g n a l which tapers o f f while pre-cursor density e q u i l i b r i u m i s attained. Figure (5.5.3) shows a power increase when a l i n e a r rate con-s t r a i n t of 1% f u l l power per second i s imposed. The reactor i s f i r s t constrained by the minimum allowable period, followed by the l i n e a r rate constraint and f i n a l l y the time optimal c o n s t r a i n t . 70 rl o U W I 0.05 0.0 1 . _ j • 1 _. ... ,.„. 1 — T — 1 —r~ t — i 1 . 4 i X i ^ i 1 Cn I —1— 2 3 4 ! Time (min). ( a ) . Conservative Switch Point. "I "T" 1 -..., 1 i ' ! - - 1 > - --.r V • i • 1 i ..1 t n i ' i 1 1 1 1 • • 1 1 Time (min). (b ) . Period Longer Than Demand Period on Reaching Switch Foint. 0 1 2 3 L J ' Time (min). ( c ) . I n t i a l Power Level Above Switch Point. F i g . 5.5.2 Power Level Increases 71 — I — — i i 1 1 • — i — , i i i i I I ! ^ — r ! i 1 ... i — i 1 i — i 1 " . i .... i , 1 Time (min). F i g . 5.5.3 Power Level Increase with Linear Rate Constraint 72 5.5.4 Power Level Increases with. Noisy Reactor The analog simulation was used f o r t e s t i n g the c o n t r o l l e r with a noisy reactor. This made i t easy to add various noise s i g n a l s ; f u r t h e r -more, the simulation included the d i g i t i z i n g e f f e c t of the analog to d i -g i t a l converters. Figure (5.5.4) shows a power increase with a white noise s i g n a l with R.M.S. value of 3% of f u l l power. The overshoot of the f i n a l endpoint was only .5% greater than the reactor without noise. Removal of the f i l t e r i n g c i r c u i t before the A/D converter r e s u l t e d i n much poorer performance, e s p e c i a l l y at low power, due to the low s i g n a l to noise r a t i o . A l i n e a r power s i g n a l i s read by the A/D converters and the logarithmic power i s d i g i t a l l y c a l c u l a t e d . The r e s u l t i n g uneven spread of d i g i t i z e d power l e v e l s can be c l e a r l y seen: i n the r e a c t i v i t y rate s i g -n a l of figu r e (5.5.4). As stated previously, the use of logarithmic i o n chamber amplifiers w i l l a l l e v i a t e t h i s problem. 5.5.5 Power Level Decreases A power l e v e l decrease with a 100 second period constraint i s shown i n figure (5.5.5). The undershoot was found to be twice the e r r o r required for a maximum output s i g n a l , which was i d e n t i c a l to the r e s u l t s for power increases. As mentioned i n section 4.3, time optimal power decreases were not dealt with due to the wealth of e x i s t i n g l i t e r a t u r e . 73 * i o o ft ft 0,-i O H M 3 +> ft 100 » • o •p 03 n -—\ CO •H \ [> M •H B •P O n! Q) ft +R„ V ' 1 ,i<f 7 1 " > •rH •P o o> « Time (sec) 1^/Decade Error = R m a x Output 40 Second Period R ,,=0.02ink/s max F i g . 5.5.4 Power Level Increase with Noisy Reactor 0.1#/Decade Error=R Output 0.0 0.05 1 " T ! 1 _ . „ r 1 1 'I i « | i ' i 'I —r-! ^ 1 j > I I 1 1 ' - I ••" I--- i 1 l I 1- ^ • :' j - •-"I--: • - j ! i ' i i i 1 . i i , 1 1 . ! . I I i i 0 2 4- 6 8 10 12 14 16 Time (min). Power Level Decrease with 100 Second Period 75 6. CONCLUSIONS A basic e r r o r sampled data c o n t r o l system f o r a nuclear reactor was developed. The con t r o l system was analysed f o r s t a b i l i t y with various sampled data holds and sample frequencies. The r e s u l t s obtained, when compared to those measured with d i g i t a l and analog simulations, proved safe, with a 25% margin. A d i g i t a l c o n t r o l algorithm, using the logarithmic neutron power l e v e l as input, was developed, which allowed the use of f i x e d point a r i t h -metic. The c a l c u l a t i o n speeds of the algorithm were seeri to be much f a s t e r than algorithms using f l o a t i n g point a r i t h m e t i c . Time optimal power i n -creases were studied, and a time optimal c o n t r o l sequence using switch points was derived. The determination of the switch points was done by simulation techniques, e l i m i n a t i n g the use of complex and very approximate c a l c u l a t i o n s . A p r a c t i c a l demand power l e v e l c o n t r o l l e r was developed, using machine language programming. A l l c a l c u l a t i o n s not r e q u i r i n g the sampled neutron f l u x were c a l c u l a t e d p r i o r to the sample i n t e r r u p t , i n an attempt to minimize the delay from the sampling to the output of con t r o l a c t i o n . The a c t u a l delay was found to be from 0.8 to 1.1 ms, which i s the time required f o r approximately two f l o a t i n g point additions. Time optimal power increases were tested using a d i g i t a l simulation of a thermal reactor. The.overshoot of the f i n a l endpoint was seen to be twice the e r r o r required for a maximum r e a c t i v i t y rate s i g n a l which i s most s a t i s f a c t o r y . The c o n t r o l l e r , although sub-optimal, approached the i d e a l time optimal t r a -j e c t o r y as the c o n t r o l l e r gain was increased. A c o n t r o l l e r gain of .1%/de-cade f o r a maximum r e a c t i v i t y rate s i g n a l r e s u l t e d i n near time-optimal r e s u l t s . 76 It can be concluded that a successful, near time-optimal con-t r o l algorithm has been developed with general a p p l i c a t i o n s to low power reactors. 77 APPENDIX A. REACTOR KINETICS EOUATIQNS A . l General Reactor K i n e t i c s Equations The space independent reactor k i n e t i c s equations f o r s i x groups of delayed neutrons are'': — = n + I A;C. + S (A. 1.1) dt £ h , i i i = l dC 8. -~ = ~r n - A.C. (A.1.2) dt 1 i i and where 3 n = neutron density (neutrons/cm ) 8 = t o t a l f r a c t i o n of delayed neutrons 6k = r e a c t i v i t y £ = mean e f f e c t i v e l i f e t i m e of a neutron (sec) 3 C^ = concentration of neutrons i n the i t h delayed group (neutrons/cm ) A_^  = decay constant of the i t h delayed group (sec ^) 8^ = f r a c t i o n of neutrons i n the i t h delayed group 3 S = source strength (neutrons/cm /sec) The space independent reactor k i n e t i c s equations i n the absence of an external source f o r one group of delayed neutrons are"': dn :6k- 8 dt £ n + AC (A. 1.3) and where f = f * - AC (A.l.4) / 6 A = 8 / I 6 ±/X i (A.l.5) / i = l 78 A. 2 Lin e a r i z e d Reactor K i n e t i c s Equations L i n e a r i z e d k i n e t i c s equations ahout a power l e v e l n are as o follows^: £ --f „ + j V l t « ^ . D ( A . 2 . 1 ) 1 = 1 and dC. 3. -~ = — n - A.C. (A.2.2) dt J , i i For the s i n g l e delayed group model the l i n e a r i z e d k i n e t i c s equations i n the the absence of an external source are"*: £ . _ ! n + X C + i | „ o (A.2.3) and f = f n - A C (A.2.4) A. 3 Reactor K i n e t i c s Transfer Function Using the l i n e a r i z e d k i n e t i c s equations the reactor t r a n s f e r n , • • jr i i ' 2 , 4 function i s as follows : The t r a n s f e r function for the one delayed group model i s : n k(s) ls(s + X + B/A) (.A.J.^; o A.4 Thermal Reactor Parameters The parameters of the delayed neutron groups of the thermal reactor used throughout t h i s study are given i n table A.4. The t o t a l f r a c t i o n of delayed neutrons i s : B = 0.0064 79 the mean e f f e c t i v e neutron l i f e t i m e i s : H = 10~ 3 sec From equation (A.1.5) the decay constant f o r the s i n g l e delayed neutron group case i s : A = 0.076 sec - 1 Group Fr a c t i o n of Number Decay Constant ' Delayed Neutrons i A. (sec "*") B. I • x_ 1 0.0124 0.00024 2 0.0305 0.00140 3 0.1110 0.00125 4 0.3010 0.00253 5 1.1400 0.00074 6 3.0100 0.00027 Table A.4 Parameters of Delayed Neutron Groups of a Thermal Reactor 80 REFERENCES 1. Pearson, A., "The Future of the D i g i t a l Computer i n Power Reactor Instrumentation", Trans. Am. Nucl. S o c , 9, 266, 1966. 2. Marciniak, T.J., "Time-Optimal D i g i t a l Control of Zero-Power Nuclear Reactors", ANL-7510, October 1968. 3. Cohn, C.E., "Further Use of an On-Line Computer i n Reactor Physics Experiments", Trans. Am. Nucl. S o c , 9, 262, 1966. 4. L i p i n s k i , W.C., "Optimal D i g i t a l Computer Control of Nuclear Reactors", ANL-7530, January 1969. 5. Schultz, M.A., "Control of Nuclear Reactors and Power Plants", 2nd ed., McGraw-Hill Book Co., New York 1961. 6. Tou, J.T., " D i g i t a l and Sampled-Data Control Systems", McGraw-Hill Book Co., New York, 1959. 7. Hafner, W.L., " A l l Roots of Polynomial Equations with Real C o e f f i c i e n t s " , ANL-C252, March 1966. 8. Pearson, A., Lennox, CG.', "Sensing and Control Instrumentation", "The Technology of Nuclear Reactor Safety", Volume 1, Eds. Thompson and -Becker-ley, The M.-I.T. Press, -Cambridge, Massachusetts, 1964. 9. Cohn, C.E., "Speed Tests on Some Control Computers", Trans. Am. Nucl. S o c , 13, 177, 1970. 10. Pontryagin, L.S., et a l . , "The Mathematical Theory of Optimal Pro-cesses", Interscience Publishers, New York, 1962. 11. Bellman, R., "Adaptive Control Processes", Princeton U n i v e r s i t y Press, 1961. 12. Monta, K., "Time Optimal D i g i t a l Computer Control of Nuclear Reactors, (I) :- Continuous Time System", J . Nucl. S c i . Technol. 3(6), 227, June 1966. 13. Monta, K., "Time Optimal D i g i t a l Computer Control of Nuclear Reactors, (II) :- Discrete Time System", J . Nucl. S c i . Technol., 3(10), 419, October 1966. 14. Monta, K., "Time Optimal D i g i t a l Computer Control of Nuclear Reactors, (III) :- Experiment", J . Nucl. S c i . Technol., 4(2), 51, February 1967. 15. Harrer, J.M., "Nuclear Reactor Control Engineering", D. van Nostrand Co., Princeton, 1963. 16. Glasstone, S., " P r i n c i p l e s of Nuclear Reactor Engineering", D. van Nostrand Co.., Princeton, 1955. 81 17. Ash., M., "Nuclear Reactor K i n e t i c s " , McGraw-Hill Book Co., New York, 1965. 18. Rosztoczy, Z.R., "Optimization Studies i n Nuclear Engineering", Ph.D. Thesis, The U n i v e r s i t y of Arizona, Univ. Microfilms, No. 64-10, 458, 1964. 19. Woodcock, G.R., Babb, A.L., "Optimal Reactor Shutdown Programs f o r Control of Xenon Poisoning", Trans. Am. Nucl. Soc.'., 8, 235, 1965. 20. Rosztoczy, Z.R., Weaver, L.E., "Optimum Reactor Shutdox-m. Program for Minimum Xenon Buildup", Nucl. S c i . Eng., 20, 318, 1964. 21. PDP-9 Users Handbook, D i g i t a l Equipment Corporation, Manyard, Massa-chusetts. 22. EAI - 231R Analog Computer Console, E l e c t r o n i c Associates Inc., Long Branch, New Jersey. 23. Marston, G.P., "Design of Medium Scale Hybrid Interface", M.A.Sc. Thesis, U n i v e r s i t y of B r i t i s h Columbia, 1967. 24. Crawley, B., "Software for Medium-Scale Hybrid Computer", M.A.Sc. Thesis, U n i v e r s i t y of B r i t i s h Columbia, 1969. •25. PDP-9 Macro 9 Assembler Manual,, .Di g i t a l Equipment Corporation, Manyard, Massachusetts. 26. Hastings, C , "Approximations f o r D i g i t a l Computers", Princeton Uni-v e r s i t y Press, 1955. 27. I n s t r u c t i o n Manual for Linear Logarithmic Monitor Sperry Gyroscope D i v i s i o n , Sperry Rand Canada Ltd., Montreal. 28. Beckey, G.A., Karplus, W.J., "Hybrid Computation", John Wiley and Sons, Inc., New York, 1968. 

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