Annals of Nuclear Engergy, Vol. 2, pp. 241 to 249. Pergamon Press 1975. Printed in Northern Ireland

THE IMPACT OF TWENTY YEARS OF ON NUCLEAR POWER PLANT DESIGN,

AND CONTROL

NOISE RESEARCH INSTRUMENTATION

LESLIE G. KEMENY

School of Nuclear Engineering, University of New South Wales, Box 1, Post Office, Kensington, N.S.W., 2033, Australia

Abstract--Some twenty years have elapsed since the first technical papers began to appear in a general field which can be loosely described as the statistical nature of the nuclear fission reaction and its influence on the criticality and dynamics of nuclear power systems. A few years subsequently, the first zero energy "neutron noise" measurements were reported in the scientific literature. These investiga- tions clearly demonstrated that the time constants and the dynamic characteristics of low energy nuclear systems could be elegantly determined by the correlation or spectral analysis of fluctuating signals from ion chambers and proportional counters. The analyses of the time series information and the multi-filtering operations in the frequency domain were time consuming and tedious projects due to the non-availability of suitable data processing equipment.

During the last decade, the significant advances in the field were the recognition of the advantages of the two-channel cross-correlation technique and the realisation that the dynamic behaviour of nuclear power plant at power could be monitored and studied in depth by the cross-correlation of mechanical, thermal and hydrodynamic signals with neutronic information. The former concept gave the spur to the development of theoretical models for spatial and energy dependent noise fields within a nuclear system. The latter technology, in principle at any rate, opened a floodgate of poten- tial advances in nuclear power plant design optimization, control and safety instrumentation, and control and safety diagnostic systems.

The present decade has seen the interaction of workers in the reactor noise field with workers investigating general vibrational phenomena and the structural mechanics of nuclear power plant. De- spite this, it is sobering to reflect in retrospect that neither design, nor instrumentation and control concepts arising from noise research, have found any great measure of practical acceptance in current nuclear technology. This paper explores the reasons for this unsatisfactory situation, surveys the few available practical examples of accepted noise technology, and makes some definitive proposals with regard to the future implementation of nuclear power plant design, instrumentation, and control procedures, based on the concept of stochastic models and noise analysis.

1. I N T R O D U C T I O N

Some eighty years ago, the theory of probabil i ty developed as a branch of pure mathematics. The relationships between the well established methods of statistical inference and the newly emerging stoch- astic models became apparent. The applications of probabil i ty theory to physics and technology have now become so widespread that there is scarcely a branch to which it does not make a significant contribution. Towards the end of the nineteenth century the establishment of the statistical nature of the second law of thermodynamics and the de- velopment of statistical mechanics were all the results of applying probabili ty theory and statistics to an atomic population. This was followed by the solution of the wave particle paradox when atomic physics became rooted in the theory of probabili ty [1].

In the twentieth century, at a more applied level, electrical noise phenomena, diffusion, Brownian movement, sea wave propagation, atmospheric

and hydrodynamic turbulence, radio wave prop- agation and neutron motion, are examples of fields in which stochastic problems are attracting considerable attention. The apparatus of the mathematician has been appropriated with grati- tude by the physicist and the engineer [2]. The manipulation of probabili ty distributions by means of generating functions and characteristic functions, time series, correlation and spectral analysis and the use of the Wiener-Khintchine theorem are proving to be extremely powerful techniques for analysing the results of statistical measurements and relating statistical parameters to stochastic models [3 ].

The tremendous growth of scientific and engineer- ing interest in this interdisciplinary field is largely due to advances in data acquisition and computer technology. Without the development of special purpose data recorders and analysers, some of which will be described later in this paper, the prac- tical applications of time series analysis could not be realistically implemented. With the use of such

la 241

242 L. G. KEMENY

devices the technique is extending its role from measurement and analysis to diagnosis, control, prediction and optimization [4, 5].

Systematic developments in this important, interdisciplinary field were somewhat restricted prior to the advent of digital computer technology. The computation of correlation and power spectral density functions from large quantities of experi- mental data was a formidable task, even with the aid of the somewhat clumsy electro-mechanical aids which were then available. An ingenious, though rather inaccurate wave or periodogram analyser was built and operated by the National Institute of Oceanography, Witley, Surrey, in the early 1940's [6] and represented a significant break- through in the automation of the spectral analysis of continuous, optically recorded data.

A similar device for the computation of auto- and cross-correlograms was built by the same Institution, at about the same time. In this case the chart records, with the area under the recorded signal suitably blackened by hand or photographic means, were placed around the circumference of a fixed cylindrical drum. Two photo scanners which can be delayed one with respect to the other, rotate on a central, internal axis and record the reflected light from the black and white time series profiles.

Fluctuating signals from nuclear reactor systems were analysed by the author [7] on the apparatus described above. Such analyses enabled basic neutron physics parameters and reactor system transfer functions and time constants to be assessed. Towards the end of the 1960's it became possible to digitise such records and to carry out the corre- lation and spectral analysis using numerical algo- rithms on the first generation of electronic digital computers.

A significant technological advance of the early 1960's was the development of correlators and spectrum analysers based on closed loop magnetic tape recording techniques (Fig. 1). The charac- teristic features of this machine, which is currently being used by the School of Nuclear Engineering, University of New South Wales, can be summarized as follows:

1. One to three electrical input signals in the frequency range 0-300 Hz may be recorded simul- taneously on a magnetic tape loop.

2. The equipment then computes auto- and cross- correlation functions, power density and amplitude density spectra and first order amplitude distri- bution functions.

3. All computations are performed automatically

within minutes and the results presented graphically on an X- Y recorder.

This equipment is presently being used by the School in the general fields of nuclear reactor system diagnostics, fluid flow and heat transfer investi- gations and system modelling and optimization in conjunction with a hybrid computer.

The present decade has seen the introduction of fully digital, high speed time series analysers, some of which are capable of "on-line" operation in real time. In the SNEDAC [8] system (School of Nuclear Engineering Data Acquisition Computer (Fig. 2) the best features of hard wired and program- mable analysers are combined to make possible the on-line computation and display of such statis- tical estimates as:

1. Mean, variance and standard deviation. 2. Drift phenomena, histogram and stationarity

assessment. 3. Probability density and cumulative probability

density distribution. 4. Auto- and cross-correlation functions. 5. Auto- and cross-power spectral density func-

tions. 6. Intensity, phase and coherence functions. 7. Linear systems analysis programs for single

or multiple time series each of which may have up to 5000 samples for real time analysis or a very much greater sample set for off-line computation.

The presently available program options are being extended to include function fitting by non- linear regression analysis and system identification and optimization programs. The real-time capa- bility of this system has been greatly enhanced by the provision of a hard-wired correlator and Fourier transform unit interfaced with the central processing unit of the computer. For system model- ling and optimization purposes, and for the simu- lation of power reactor conditions, the data acquisi- tion computers ISAC and SNEDAC can be interfaced to a small hybrid computer (Fig. 3).

2. NOISE ANALYSIS AND NUCLEAR POWER PLANT

The development of sophisticated signal analysis and computing equipment such as those described above, over the past twenty years has enhanced the potential application of noise analysis tech- niques to the design, instrumentation and control of nuclear power plant. Unfortunately, however, the impact of these developments on plant design, operation and economics has been, up to the present time, negligible. In the context of this Conference, it is a worthwhile exercise to examine

Fig.

1.

"IS

AC

" in

stru

men

t fo

r st

atis

tical

ana

logu

e co

mpu

tati

on.

Fig.

2.

"SN

ED

AC

" da

ta a

cqui

sitio

n co

mpu

ter.

Fig. 3. The TR48/DES30 hybrid computing system.

Twenty years of noise research on nuclear power plant design, instrumentation and control 243

the reasons behind this sad lack of progress. To do this we shall first consider what nuclear plant noise analysis can do adequately at the present time, and then define some practical and meaningful objectives for future effort.

In a series of excellent review papers, Saito [9-11 ] surveys the reactor noise research field and gives a full bibliography of relevant books and, papers. It is salutary to remind ourselves that as early as the mid-1940's [12, 13] much of the theo- retical and the experimental equipment to handle the problems of neutronic noise and zero energy systems was available. The Monte Carlo method served to emphasize the power of stochastic model- ling and simulation in criticality studies and neutron transport problems. Stochastic renewal theory and the Markoff matrix methods were available for neutron spectrum investigations and, in principle, the spatial aspects of neutronic noise could be handled using integral transport theory, slowing down kernels and collision probabilities.

Apart from a few theoretical papers, interest in the field was dormant until the first experiments took place in the late 1950's. By the advent of the first two Noise Conferences at the University of Florida [14, 15] in 1963 and 1966, reactor noise researchers could justifiably claim success in the following areas:

1. The measurement of zero energy nuclear system dynamic parameters and time constants.

2. The impulse response identification of nuclear system components using pseudo-random binary sequence perturbations,

3. Dynamics of coupled core systems assuming a lumped parameter model,

4. A degree of correlation between theoretical models and experimental measurements involving power reactors with temperature and void feedback phenomena relative to their dynamics and stability,

5. A preliminary investigation into the nature of non-nuclear fluctuations in reactor coolant channels with particular emphasis on the statistical analysis of temperature and pressure fields.

Despite the admirable research that has un- doubtedly gone into the above projects, it would not be unfair to state that, in practical terms, the results have been negative. One must quite ruthlessly challenge the reactor noise fraternity with such questions a s - -How many nuclear power plant have noise diagnostic instruments in their control rooms ?

What practical design improvements or system optimization has taken place as a consequence of detailed noise investigation? Where does one find

nuclear power systems monitored with and con- trolled by noise signals in either normal operation or as a possible safety measure in accident con- ditions ? Many more questions of this type could be asked, and in trying to answer them we are often forced to confess that the enthusiasm for the re- search project sometimes filters out a need for a practical objective.

Fortunately the present decade has brought with it a new concern for environmental and ecolog- ical considerations, as well as the need for a careful global assessment of energy resources. This has forced upon us some very practical issues con- cerning noise analysis and its relevance to the integrity and safety of nuclear power plant. The impact of twenty years of noise research on nuc- lear power plant design, instrumentation and control will now be determined by correct, quantitative answers to clearly defined problems such as:

1. Can the dynamic behaviour of a single fuel channel be monitored, in a quantitative sense, by rapid response transducers, in such a way that normal operating conditions can be optimized and accident conditions anticipated and, if possible, prevented? What are the best combination of detectors and where should they be located to en- sure optimal physical and mathematical conditions ? Is there a role for self powered neutron detectors in neutron noise measurement ? Can rapid changes in correlation and spectral density functions be used to anticipate nonlinear transient pheonomena such as loss of coolant ?

2. What are the differences in both amplitude and frequency content of signals from noise de- tectors--neutronic, thermal, mechanical and hydro- dynamic--when placed in different core locations ? Where are the optimal locations from the point of view of subcritical reactivity and absolute power measurements ? Which signals are the most suitable for diagnostic and control applications? What is the best physical description and mathematical model for the neutron, gamma photon, temperature and pressure noise fields? Is it really possible to cross-correlate one with the other and, if so, how do we handle the mathematical models for detailed probability balance? Which signals are best suited for use in control schemes and can these be made adaptive ?

3. How is the structural integrity of nuclear power plant components affected by near and far located noise sources and fields, such as seismic disturbances, mechanical and hydroelastic buffet- ting and thermal fluctuations? Can fatigue failure of fuel elements or cladding, control surfaces and

244 L. G. KE~mNv

other components result from these phenomena? Is there any significant reactivity coupling with these vibrations? What effects do metallurgical imperfections in fuel element manufacture and random dispersion of fuel in a fuel pin have on reaction rates, temperature fluctuations and thermal stressing? How does the micro-structure of a fuel pin influence the process of heat transfer in the coolant channel? Is it possible to detect fuel clad- ding oxidation processes by vibration measurement ?

In the next section of this review paper we shall briefly discuss some work in progress within the School of Nuclear Engineering, University of New South Wales, falling into the above three categories. Whilst experiments are carried out on rigs or in the HIFAR research reactor, the monitoring and control techniques are being developed for potential power reactor application, in particular for fast breeder systems. A brief summary of mathematical models, numerical algorithms and experimental techniques is given in the Appendix.

3 . S O M E C O N T E M P O R A R Y O B J E C T I V E S F O R R E A C T O R N O I S E R E S E A R C H

(a) Structural mechanics and reactor component vibrations

This project is concerned with the elastic vibration of pin ended tubes and rods [16] due to stationary random surface pressure fluctuations. The par- ticular problem of interest is the significance and validity of Bernouilli-Euler or simple beam theory for the prediction of displacements and strains, or the interpretation of measured differential axial surface strain in the general field of flow induced vibration of rods and tubes.

The vibration can be due to far field effects such as pumps or near field effects, particularly local turbulent flow boundary layer pressure fluctuation (Fig. 4) or boiling. The question of the interpreta- tion of measurements must be carefully investigated. It is possible to measure the displacement by optical means outside of the core, but in reactors and when dealing with rod clusters this becomes very difficult. The most common approach is to use strain gauges which are very often attached without any thought being given to mathematical theory or physical interpretation. The interpretation is usually based on BernouiUi-Euler theory which immediately raises the following questions:

i. How valid is simple beam theory for deflections where the surface pressures fluctuate rapidly in time and space?

ii. What is the relationship between the power

Pump impeller vibrofiott

Turbulent flow vibrotion

60

5O

m 40 "o

14 16 18 20 22 24 2 6 28 30 32 34 36 38 40 42 l I I l I I I l I I I I I I 1

1

= 5O

10

0 2 5 ~ 613 I I I f f I I f I 1 I I00 160 250400650 I'OK ~'6K2.SK40K65K IOK / K

Thi rd-octove-band center frequency, Hz

Fig. 4. Hydroelastic vibration of fuel element showing far and near field effects.

spectral density of the differential axial strain as measured by diametrically opposed strain gauges ?

In the HIFAR reactor the vibrational behaviour of fuel elements, pumps, control arms and reactor structural components is being monitored, in collaboration with Australian Atomic Energy Commission staff, in such a way (Table 1) that the effects of undesirable oscillations can be understood and, in principle, minimized or eliminated by design modifications.

(b) Core and fuel channel identification and safety Work is continuing on the development of a

system for measuring subcritical reactivity using the two detector cross-correlation technique. High sensitivity ion chambers are being used for this project. In the course of making the measurements it has become possible to determine, by on-line noise analysis, the dynamic characteristics of the reactor. Typical results are shown in Figs. 5 and 6. A family of statistical estimates including auto- and cross-correlation and auto- and cross-spectral density functions, coherence functions and transfer functions, have been measured at a series of power levels ranging from source power to 10MW. Most of these results show the influence of a

Twenty years of noise research on nuclear power plant design, instrumentation and control

Table 1. Acceleration levels of HIFAR structural vibrations

Fuel Elements Centre Side A1 4.5 3.5 A2 8.5 3.5 A3 3.7 2.5 A4 7.0 3.5

B1 7.0 3.0 B2 4.0 2.0 B3 3.0 1.0 B4 6.0 3.0 B5 4.5 2.5 B6 5.0 3.0

CI 5.0 2.5 C2 4.5 2.5 C3 5.0 2.0 CA .6.0 4.0

D1 4.0 1.5 D2 5.0 3.0 D3 4.5 1.0 D4 6.5 1.5 D5 3.5 3.0 D6 3.0 2.5

E1 3.5 2.5 E2 3.5 3.0 E3 2.0 1.5 E4 4.5 2.5

Other Parts o| Structure CCA2 3.0 CCA5 1.8

4V5 2.7 SR1 2.0

Heavy water weir overflow valve - - 12

L I l I I I I I I

0.1HZ I0 HZ Iogjo frequency

I '1 I [ I" I I I

"noisy" control system which can be traced back to a control arm oscillation resonating at approxi- mately 6.3 Hz in the heavy water.

A single fuel element in the H I F A R reactor of the Australian Atomic Energy Commission is being instrumented in order that multiple cross- correlation of neutronic, thermal, hydraulic and vibrational phenomena will become possible. Boil- ing will be simulated with Helium gas and the effects of transients on the experiment will be observed. The spectral density of typical tem- perature fluctuations in such an experiment is shown in Fig. 7.

The instrumentation utilized will include self powered neutron detectors, fission chambers, hydrophones, accelerometers and strain gauges. Signal analysis and display will be carried out on the SNEDAC system. A powerful multiple cross- correlation programme with facilities for the analy- sis of up to eight time series will support the off-line calculations.

(c) Instrumentation and control

With the sampled da ta- -neut ron density, tem- perature, pressure void fraction or v ibra t ion- -

245

! 1 ! O.H-~ IOHz

log frequency

Fig. 5. HIFAR neutron noise signal used for identifica- tion of noisy control system at 5000 W (top) and 100,000

W (bottom).

stored either in the core of the computer, on the incremental tape deck or in the correlator in the form of coefficients, SNEDAC is being programmed to carry out a fourfold task as follows.

Mode 1: Diagnostics. Functions computed and displayed include

Vertical Axis Horizontal Axis

Correlation coefficients Time Spectral density Frequency Coherence Function Frequency Phase Frequency Log modulus Log Frequency

as well as the usual statistical estimates cited above. Mode 2: Monitoring. Abrupt changes in the

normal shapes, amplitudes and frequency content of the above statistical estimates can be recognized as the onset of unwanted reactor conditions. Typically, boiling reactor channel hot spots, fission product release and structural vibration can be detected by comparison with standard, stored statistical parameters. The appropriate monitoring action might then be to actuate alarm circuits or release a control rod.

Mode 3: Identification and optimization. The dynamic analysis section of SNEDAC incorporates

246 L . G . KErCmNY

I I I h L

Channel I 4 H'TC I

signals 480 mm/~min

t I l I I I I I

_ Criticality - increasing

T IO msec ~ IO00 W ,..,Delaye,

~critical

I I 0"I

.... I . i I I f I T

'Time, see

Fig. 6. HIFAR sub-critical reactivity on-line display by cross-correlation of neutron noise signals channels

1 and 3.

a device for generating pseudo-random binary signals. Using a cross-correlation technique, both impulse response identification and optimization become possible. The mathematical algorithms for constraining impulse responses of reactor system components to a desired shape include regression analysis, Gaussian iteration and least squares functional minimization. Mathematical aspects of this work will be published elsewhere.

Mode 4: System control. The preliminary stage of reactor control consists of the identification and elimination of unwanted noise signals in the reactor control circuits and instrumentation which may cause unnecessary trips, false alarms and shut- downs. SNEDAC has been designed with this in mind. The Fokker-Planck equation of a reactor counting channel can be solved in terms of proba-

o&

Prot l 'ed s e t t i n g Y a x i s , I O O r n V / c r n I O H z

Fig. 7. Metal temperature fluctuations at surface of HIFAR fuel element.

bility distribution generating functions which enable the mean and variance of control circuit signals to be predicted under all operating conditions. With a knowledge of time dependent changes in the probability density of fluctuating variables, the control of the system for a constant mean value and variance becomes possible.

It is envisaged that ultimately SNEDAC will also be utilized in digital control investigation embracing the full start-up, shut-down and accident dynamics of a nuclear reactor. Appropriate algorithms are being investigated and programmed.

4. CONCLUSION

It may yet be premature to say, in a 1974 Confer- ence review paper, that the reactor noise field has become an accepted technology vital to the con- struction and operation of nuclear power plant. What is obvious is that, allied with borderline dis- ciplines such as structural mechanics, it will con- tinue to play an important role in aiding the acceptance of nuclear power as a safe and econom- ical energy source.

Acknowledgements--The author gratefully acknowledges profitable discussion with Professors J. J. Thompson and Z. J. Holy, of the School of Nuclear Engineerin.g, and members of the staff of the Australian Atomic Energy Commission. Financial assistance for the project is being provided by the Australian Institute of Nuclear Science and Engineering.

REFERENCES

1. Domb, C., Fluctuation phenomena and stochastic processes. Nature 184, 509 (1959).

2. Shannon, C. E., A mathematical theory of com- munication. Bell Syst. Techn. 3. 27, 379 (1948).

3. Wax, N. (editor), Selected Paper on Noise and Stochastic Processes. Dover Books, New York (1954).

4. Wiener, N., The Extrapolation, Interpolation and

Twenty years of noise research on nuclear power plant design, instrumentation and control 247

Smoothing of Stationary Time Series. Wiley, New York (1949).

5. Jenkins, G. M. and Watts, D. G., Spectral analysis and its application. Holden-Day, San Francisco (1969).

6. Barber, N. F., A frequency analyser used in the study of ocean waves. Nature 158, 329 (1948).

7. Kemeny, L. G. and Samaun, Random fluctuations in a nuclear fission reactor. Nature 189, 130 (1961).

8. Kemeny, L. G., The monitoring, identification and control of nuclear power systems with an on-line computer. Austr. Computer J. 4, 170 (1972).

9. Saito, K. On the theory of power reactor noise-I. Ann. nucl. Sci. Engng, 1, 31-38 (1974).

10. Saito, K., On the theory of power reactor noise-II. Ann. nucl. Sci. Engng, 1, 107-128 (1974).

11. Saito, K., On the theory of power reactor noise--HI. Ann. nucl. Sci. Engng 1,209-221 (1974).

12. de Hoffman, F., Statistical aspects of pile theory. The Science and Engineering o f Nuclear Power, pp. 103-119. Addison-Wesley, New York.

13. Courant, E. D. & Wallace, P. R. Fluctuations of a number of neutrons in a pile. Physical Review, 72, 1947, 1038-1048.

14. U.S. Atomic Energy Commission, Division of Technical Information. Noise analysis in nuclear systems. Proceedings of a Symposium held at the University o f Florida, 4-6 November 1963. Edited by R. E. Uhrig. Oak Ridge, Tennessee, U.S. Atomic Energy Commission, 1964. AEC Symposium Series No. 4.

15. U.S. Atomic Energy Commission, Division of Technical Information. Neutron noise, waves and pulse propagation. Proceedings of a Symposium hem at the University of Florida, 1966. Edited by R. E. Uhrig. Oak Ridge, Tennessee, USAEC, 1967. AEC Symposium Series No. 9.

16. Thompson, J. J. and Holy, Z. J. Random pressure induced vibration of pin ended cylindrical rods. 2nd lnt. Conf. on Structural Mechanics in Reactor Technology. Paper D3/6. Berlin (1973).

17. Astr6m, K. J. Introduction to Stochastic Control Theory. Academic Press, New York (1970).

18. Wiberg, D. M. Identification of the LMFBR dy- namic state for detection of coolant boiling. Kern- forschungszentrum, Karlsruhe. Report No. KFK 1911 (1974).

APPENDIX

Reactor noise research in Australia was initiated in 1967 by the School of Nuclear Engineering, University of New South Wales, with the financial encouragement and technical support of the Australian Institute of Nuc- lear Science and Engineering, and the Australian Atomic Energy Commission.

Experimental techniques are being developed on the HIFAR reactor of the Australian Atomic Energy Commission. This is an 11 MW materials testing reactor with annular plate fuel elements and heavy water moderator. Various power reactor noise conditions can be simulated in HIFAR and appropriate monitoring, identification and control schemes can be implemented using a range of detectors and the ISAC and SNEDAC statistical analysers, as well as the TR48/DES30 hybrid computer. Theoretical interest in stochastic modelling and noise analysis is centred on the criticality, dynamics

and safety problems of fast breeder reactor systems. Where possible, experimental simulation and numerical calculation is aimed at the development of practical fast reactor monitoring and control technology based on noise signal analysis.

From the outset of the research programme it became apparent that both theoretical modelling and experimental measurement should have as an objective the attainment of practical goals and the development of useful hardware in the fields of power reactor design, operation, monito- ring and control. The experimental equipment was designed and assembled in such a way that it can be utilised in a threefold capacity of data acquisition, signal analysis and on-line monitoring and control. The constraints on the experimental measurements are set by the transducer employed; by the bandwidth of the data processing equipment, and by the sophistication of the mathematical algorithms used for analysis or control. Some features of the experimental apparatus are detailed below.

Table A. 1

Transducers

Data acqui- sition bandwidth

Data analysis

Neutron and gamma ion chambers Proportional and fission counters Self powered neutron detectors Accelerometers, hydrophones, strain gauges, thermometers, ultrasonic transmitters and receivers. d.c. to 250,000 Hz for stochastic variates neutron, gamma and fission product number-density; tempera- ture and pressure. All time series operations up to the multiple cross-correlation of eight stochastic variates. Identification and control algorithms.

The experimental and theoretical research programmes have made steady progress subject to funding and staffing limitations. A brief summary of projects undertaken and mathematical techniques used is given in Table A.2 below.

Table A.2

Noise source identification

Time series analysis Probability-density analysis Fourier and Walsh transforms Zero crossing analysis

Reactor core, system component and coolant channel identifica- tion

Cross-covariance analysis Optimal filtering Optimal impulse response technique Integral equation and Green's function techniques

Noise signal analysis for control and predication

Variance minimization Spectral optimization Sampled data and z transform Convolution transform Linear, non-linear, stationary and non-stationary prediction techniques

248 L. G. KErCmNY

As an illustration of theoretical modelling some typical simple models and algorithms are given below.

1. Noise source identification The precise statistical description of reactor noise

sources and fields can be used to optimise the engineering design of reactor structural components, On a micro- scopic level the statistical analysis of dispersion phenomena such as the random distribution of fuel in fuel pins has an important bearing on structural mechanics and integrity. In terms of reactor rate and neutron physics it influences system criticality.

Hydroelastic vibrations affect all components in a reactor core. In the case of the identification of the vibrational behaviour of a cylindrical rod or tube in parallel flow, it is possible to develop a solution, based on exact three dimensional elasticity theory [16] for the cylindrical rod modelled on simple beam theory as

a ~ ay (m, + m/)~-~ + EI~a--Yx4 + C - ~ = q ( x , t ) (A1)

where q(x, t) is the spatially homogeneous and stationary random load field arising from a stationary surface pressure field p(x, O, t).

Consider a cylindrical rod or tube bounded by a cylindrical surface S and ends E, subjected to purely normal pressure fluctuations over S. The boundary conditions on E, in cylindrical coordinates, are u¢ = uo = a,, = 0. Then for a linear system

4'k(el, o)1, f~, ~o2)

= ~8 ~S ~ ('1, 0)1, '2,092)/-/'k(Fa, '1, 0)1.) Hk*(e,, ~a, ~o,)dSx dS~ (A2)

where ~b/~(~l, ~ol, ~¢2, co~) is the cross-spectral density of the surface pressure p(~¢, t) acting on S and ~bk(?x, co~, f~,co2) is the cross-spectral density of the response Rk of type k at any two points fx, ra. Also Hk(f, ~, to) is the frequency response Green's function defined by

Hk(F, S, to) = t~ ~ e ~'°) hk(f , ~, t) dt (A3)

in terms of the real time weighting function h,(f, ~, t) defined by

f f: Rk(~, t) = p(,-q, r)hk(~, ~q, t -- ~-) d~" dS (A4)

In particular the spectral density at ~ is

~(e, o~)

(A5)

The response of interest refer to the surface r = a and the problem reduces to the calculation of the three frequency response functions H~(a, O, z, 0~, zx, o9) where k = r, 0, Z referring respectively to radial surface displacement u~, tangential surface displacement u/~ and the surface axial strain differential AZffi~ = Z , ~ ( 0 ) - Z~(0 + ~r). A degree of correlation has been shown to exist between the theoretical predictions of this model and experimental tests (Fig. 4) carried out on a HIFAR fuel element liner in a water tunnel.

The above example highlights the complex problem of identifying a pressure induced noise source. Other sources and fields under investigation are the stochastic variates neutron and gamma photon density and tem- perature.

2. Reactor core, system component and coolant channel identification

Work has been completed on reactor core identification in the time and frequency domains using two channel cross-correlation and spectral analysis. Transfer and coherence function techniques (Fig. 6) have been used to identify control system and reactor component [8] vibrations. The cross-correlation function technique has been used to assess reactivity (Fig. 7) and power levels and as a basis for the measurement of subcritical reactivity.

Impulse response and optimal filtering methods are being developed to identify reactor core dynamic be- haviour using inherent noise signals.

For identification purposes, if an impulse response g(t) is first calculated from an experimental time series measurement, then a desired, optimal response h(cq t) can be obtained by minimizing the functional

= f~l'{<~p [g(t) -- h(cq t)12 > dt} (A6)

where ~ is a vector of unknown parameters and ~ is an arbitrary weighting vector representing the importance of each component of h and g'. A possible optimization scheme will then be to vary the parameters ~, a process equivalent to a Gauss-Seidel iteration scheme for solving the set of linear equations

a~ a~--~ = 0 (i = 1, 2 . . . . j ) (A7)

In the case of power reactor system coolant channel monitoring, it should, in principle be possible to detect deviations from normal operating states in an optimal manner by using an estimator of the state vector such as the Kalman filter.

Algorithms are being developed, based on the time series measurements of stochastic variates such as neutron density and temperature to compute the con- ditional mean of a state vector ~(t) when we have access to a measured time history #(t) in the range to _< T < t. Here #(t) is an n dimensional measurement vector. For a typical power reactor system ~ and # may be related through reactor dynamic equations such as

d~ = ~ + fly + y~ (A8) dt

and fi = A~ + g (A9)

where)7 is a multi-dimensional input of known structure; is a multi-dimensional noise source with zero mean and is a multi-dimensional noise source with zero mean. The optimal estimate for ~(t) can now be obtained

[17] by digital or analogue computer through the solution of an equation of the form

d__~ = a t + tip + ~/(# -- A~) (AI0) dt

Twenty years of noise research on nuclear power plant design, instrumentation and control 249

where ~, d, 7 and ~ are compatible matrices defined for the reactor system being identified. In essence the vector if(t) minimizes any convex function of the error

~(t) = ~(t) -- ~(t) (Al l )

Whilst this approach to utilizing power reactor noise signals for monitoring and malfunction detection is somewhat complex and tentative, with a suitable com- bination of analogue and digital computing elements it will undoubtedly find application in the safety philos- ophy of future power reactors. It would appear to have particular relevance to fast reactor diagnostics [18]

3. Reactor noise signal utilization f o r control and instru- mentation purposes

Twenty years of noise analysis have not, to any marked extent, influenced the design philosophy of nuclear power plant control and instrumentation. For full range start-up and shut-down procedures, a strong case can now be made out for direct digital control based on a sampled data input device, an on-line computer and, possibly, an impulse motor to apply the control signal.

In such a system the value of noise analysis would be in the recognition of spurious or unwanted noise signa!s in the operation of an integrating data sampler. Especi- ally at low power the r.m.s, neutron noise signal has a high level relative to the mean and without a correctly designed and filtered integrating sampler the system may easily experience unnecessary trips.

The interaction between sampling frequency and a noisy low power signal is being investigated in connection with the H/FAR research programme. The relative levels of noise from source power to criticality have been measured and start-up procedures are being simulated prior to possible on-line experiments utilizing SNEDAC (Fig. 2). This project has an obvious bearing on period meter design and the correct selection of reactor control instrumentation and transducers.

For power reactors used as irradiation facilities and propulsion units and possibly in fusion systems of the future the maintenance of a steady mean neutron density is important.

For control about a steady state, the general approach in a "noisy" system is one of variance minimization or spectral optimization. A control algorithm in a non- stationary and non-linear system can be modelled on the solution of a Fokker Planck equation of the form

aG(x, t) a o ' ( t ) "G(x, t) Ot 8x [ ¢(x, t) G(x, t)] + T Ox -------7- (A12)

where the dynamics of the system are described by the equation

dx d--t = ~(x, t) + Q(t) (A13)

The left-hand side of equation (AI2)is a time varying probability density function and describes the statistics of the state x of the system. The term ~(x, t) represents the non-stationary non-linear dynamics of the control system and Q(t) is a non-stationary, zero mean Gaussian noise source with a cross-spectral density matrix of az(t).

An appropriate computer solution of equation (A12) can be carried out by using finite difference approxima- tions of the Crank-Nicholson type. The derivatives in the equation then become:

G(x, t) -+ Gl(n)

aG(x, t) --~ Gx(n + 1) -- Gl(n -- 1) Ox 2Ax

02G(x, t) Ga(n + 1) -- 2Ga(n) + G~(n - - 1) (A14) Ox 2 ~ A x 2

and G(x, t) --~ G,+~(n) -- Gx(n)

Ot A t

Time series techniques can also be applied to the general problems of system control through the develop- ment of algorithms which numerically effect a convolu- tion or a de-convolution transformation. However such numerical methods, including the two cited above, necessitate the use of large, fast computing equipment.