"Applied Science, Faculty of"@en .
"Electrical and Computer Engineering, Department of"@en .
"DSpace"@en .
"UBCV"@en .
"Harvey, Geoffrey Alan"@en .
"2011-03-30T20:07:52Z"@en .
"1973"@en .
"Master of Applied Science - MASc"@en .
"University of British Columbia"@en .
"A direct digital control algorithm for low power reactors is proposed using logarithmic power level as input. The logarithmic power levels allow the use of fixed point arithmic resulting in faster calculation speeds than are obtainable with algorithms using floating point arithmetic. A stability analysis for various sampled data hold types is shown to have a 25% safety margin. A time optimal control sequence for power increases is derived using switch points. The switch points are determined using simulation techniques, eliminating the use of complex and approximate calculations. A practical demand level controller is developed using machine language programming to minimize the delay from the sampling of the neutron power to the output of control action. The controller is tested with digital and analog simulations of a thermal reactor showing that a successful, near time-optimal, control algorithm with general applications to low power reactors has been developed."@en .
"https://circle.library.ubc.ca/rest/handle/2429/33068?expand=metadata"@en .
"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 \u00C2\u00AB 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 \u00C2\u00AB ) 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 ) \u00C2\u00A9 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 \u00E2\u0080\u00A2 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 \u00C2\u00A3 (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 \u00C2\u00AB(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 \u00C2\u00A3 (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 \u00C2\u00AB 1 ( 2 . 2 . 1 4 ) p d p 1 p 1 and N 1 = N(t-T) = N^ ( 1 + Y ^ , \Y\u00C2\u00B1\ \u00C2\u00AB 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) \u00C2\u00AB 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) \u00C2\u00A3sls + X + B/A] U . 4 . \u00C2\u00B1 ; 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 \u00E2\u0080\u00A2 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/\u00C2\u00A3 (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\u00C2\u00A3Y+2BTX'Y+2BY-2B (2.6.5f) d = XX , 3T 3\u00C2\u00A3+3BT 2X'-6BTX i-6B(3+Y)+248 (2.6.5g) e = XX' 3T 3\u00C2\u00A3(4-Y)-3X , 2T 2BY+6BTX'(2+Y)+18B(1+Y)-36B (2.6.5h) f = XX' 3T 3\u00C2\u00A3(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 ) = . \u00E2\u0080\u00A24X,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/\u00C2\u00A3) \" (2.6.6a) K = R/\u00C2\u00A3 ^ (2.6.6b) Y = exp(-X'T) (2.6.6c) h = XX , 3T 3\u00C2\u00A3+3X , 2T 2(XX'T\u00C2\u00A3+3)+63(X'T-l)T +6B(X'T-1)(3+Y)-246(X'T-1) (2.6.6d) k = XX' 3T 3\u00C2\u00A3(4-Y)-3X , 2T 2(XX'T\u00C2\u00A3+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\u00C2\u00A3(1-4Y)-3X' 2T 2(XX ,T\u00C2\u00A3+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\u00C2\u00A3+3X' 2T 2(XX ,T\u00C2\u00A3+B)-6B(X ,T-l)TX , -6B(X ,T-1)]Y+6B(X'T-1) (2.6.6g) q = XX' 4T 4\u00C2\u00A3+4X' 3T 3(XX 1T\u00C2\u00A3+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\u00C2\u00A3(11-Y)+4X , 3T 3(XX'T\u00C2\u00A3+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\u00C2\u00A3(1-Y)-12X , 3T 3(XX'T\u00C2\u00A3+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\u00C2\u00A3(1-11Y)+4X , 3T 3(XX ,T\u00C2\u00A3+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\u00C2\u00A3+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, \u00C2\u00A3 = 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 \u00C2\u00A3 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 (% ~~ 0 - : 1 1 1 a?=27r/uj>s Zero-Order Hold First-Order Hold -- \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 Linearized Hold _ ' . ' *\ \u00E2\u0080\u00A2 * . , * *. **\u00E2\u0080\u00A2 ^ \ \ \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 * s - \u00E2\u0080\u00A2 t . \u00E2\u0080\u00A2 \u00C2\u00BB 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 \u00C2\u00A3 = 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) \u00C2\u00AB A f o r |A| \u00C2\u00AB | 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 \u00C2\u00B1 (3.3.8) i l l \" E l ^ \" 1 < 3- 3' 9 ) = \u00C2\u00A3 ( 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 \u00E2\u0080\u00A2dn u - 8 dt \u00C2\u00A3 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\u00C2\u00A3C(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 \u00C2\u00A3(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 \u00C2\u00A3(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\u00E2\u0080\u0094 - 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 \u00E2\u0080\u00A2 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 - \u00E2\u0080\u0094 \u00E2\u0080\u0094 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 \u00C2\u00A3- tr - 1=^ + i- 2 Ha K a J 21 A -1 (4.2.12) 34 where a = A + 6 / \u00C2\u00A3 b = X/a c = B/\u00C2\u00A3a 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^ \u00E2\u0080\u00A2= 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 + \u00E2\u0080\u0094 (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) ->\u00E2\u0080\u00A2 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 \u00E2\u0080\u0094 \u00E2\u0080\u0094 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 \u00E2\u0080\u00A2averaged 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\u00E2\u0080\u0094 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 \u00E2\u0080\u0094 d \u00E2\u0080\u0094 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 \u00E2\u0080\u00A2 \" V ALPHA \u00C2\u00BB (N 1-L0tf)/(HI-L0W) L1NP = (1-ALPHA )* 11-^ +ALPHA*H\u00E2\u0080\u009E 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 \u00E2\u0080\u0094 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 \u00E2\u0080\u0094 i 1 \u00E2\u0080\u0094 i \u00E2\u0080\u0094 i r -r d ~ ^ - d \u00C2\u00AB 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\u00C2\u00BBKod(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) <\u00C2\u00BB > U G> % o (k 6D o H i y^ /y i\u00E2\u0080\u0094Demand - pi ' Power l \u00E2\u0080\u0094 A c t u a l / 1 t 1 1 Power I l +> T m m T ' \u00C2\u00A3 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 \u00C2\u00B0 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 \u00C2\u00AB 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 \u00E2\u0080\u00A202mk/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 \u00C2\u00AB 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'\u00C2\u00BB/SD-FLXD)/( SWSD-FLXE) TEK1 =DHRT- ( DKRrJ-DERT ) TSH2 Load Kew Endpoint and Switchuoints. Reset EP Flag. LIMD=AKTILOG(FLXD) FLXT=FLXE - FLXD |PERD\u00E2\u0080\u0094TEMll [ PERD=-TEM2J [PERD=-LNRT/LIKD |PERfl=-DRAT| T \u00E2\u0080\u00A2IEM2=( FLXD-SW8T- )/(FLXE-SV;ST ) TEH1 ='Ji4HT-( UFlR'X'-UERT ) TEM2 FERD=URAT PERD=LKRT/LIHD| IF\u00C2\u00A3RD=T\u00C2\u00A3M1| |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 \u00E2\u0080\u00A2 ( 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 \u00C2\u00A3 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). \u00E2\u0080\u00A2'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\u00C2\u00A3MP4=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 \u00E2\u0080\u00A2finding 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 \u00E2\u0080\u00A2 \u00C2\u00B0 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 \u00E2\u0080\u00A2\u00E2\u0080\u00A2 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 \u00E2\u0080\u00A2 1 _. ... ,.\u00E2\u0080\u009E. 1 \u00E2\u0080\u0094 T \u00E2\u0080\u0094 1 \u00E2\u0080\u0094r~ t \u00E2\u0080\u0094 i 1 . 4 i X i ^ i 1 Cn I \u00E2\u0080\u00941\u00E2\u0080\u0094 2 3 4 ! Time (min). ( a ) . Conservative Switch Point. \"I \"T\" 1 -..., 1 i ' ! - - 1 > - --.r V \u00E2\u0080\u00A2 i \u00E2\u0080\u00A2 1 i ..1 t n i ' i 1 1 1 1 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 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 \u00E2\u0080\u0094 I \u00E2\u0080\u0094 \u00E2\u0080\u0094 i i 1 1 \u00E2\u0080\u00A2 \u00E2\u0080\u0094 i \u00E2\u0080\u0094 , i i i i I I ! ^ \u00E2\u0080\u0094 r ! i 1 ... i \u00E2\u0080\u0094 i 1 i \u00E2\u0080\u0094 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 \u00C2\u00BB \u00E2\u0080\u00A2 o \u00E2\u0080\u00A2p 03 n -\u00E2\u0080\u0094\ CO \u00E2\u0080\u00A2H \ [> M \u00E2\u0080\u00A2H B \u00E2\u0080\u00A2P O n! Q) ft +R\u00E2\u0080\u009E V ' 1 ,i \u00E2\u0080\u00A2rH \u00E2\u0080\u00A2P o o> \u00C2\u00AB 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 _ . \u00E2\u0080\u009E r 1 1 'I i \u00C2\u00AB | i ' i 'I \u00E2\u0080\u0094r-! ^ 1 j > I I 1 1 ' - I \u00E2\u0080\u00A2\u00E2\u0080\u00A2\" I--- i 1 l I 1- ^ \u00E2\u0080\u00A2 :' j - \u00E2\u0080\u00A2-\"I--: \u00E2\u0080\u00A2 - 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'': \u00E2\u0080\u0094 = n + I A;C. + S (A. 1.1) dt \u00C2\u00A3 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 \u00C2\u00A3 = 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 \u00C2\u00A3 n + AC (A. 1.3) and where f = f * - AC (A.l.4) / 6 A = 8 / I 6 \u00C2\u00B1/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^: \u00C2\u00A3 --f \u00E2\u0080\u009E + j V l t \u00C2\u00AB ^ . D ( A . 2 . 1 ) 1 = 1 and dC. 3. -~ = \u00E2\u0080\u0094 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\"*: \u00C2\u00A3 . _ ! n + X C + i | \u00E2\u0080\u009E 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 , \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 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 \u00E2\u0080\u00A2 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. \u00E2\u0080\u00A225. 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. "@en .
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