SOVIET ATOMIC ENERGY VOL. 42, NO. 4

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• Release Decision: RIFPUB
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• Document Page Count: 116
• Document Creation Date: December 23, 2016
• Document Release Date: April 1, 2013
• Sequence Number: 4
• Publication Date: April 1, 1977
• Content Type: REPORT

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Russian Original Vol. 42, No. 4, Ap
SATEAZ 42(4)289-398 (1977)
SOVIET
ATOMIC
ENERGY
ATOMHAR 3HEP11,111
(ATOMNAYA iNERGIYA)
- TRANSLATED FROM RUSSIAN
CONSULTANTS BUREAU, NEW YORK
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? SOVIET
ATOMIC
ENERGY
Soviet Atomic Energy is abstracted or in-
dexed in Applied Mechanics Reviews, Chem-
ical Abstracts, Engineering Index, INSPEC?
Physics Abstracts and Electrical and Elec-
tronics Abstracts, Current Contents, and
Nuclear Science Abstracts.
i ?
Soviet Atomic Energy is a cover-to-cover translation of Atomnaya
nergiya, a publication of.the Academy Of Sciences of the USSR.
An agreement with the Copyright Agency of the USSR (VAAP)
makes available both advance copies of the Russian journal and
original glossy photographs and artwork. This serves to decrease
the necessary time lag between publication of the,original and
publication of the translation and helps to improve the quality
of the latter. The translation began with the first issue of the
Russian journal.
Editorial Board of Atomnaya Energiya:
Editor: 0. D. Kazachkovskii
Associate Editor: N. A. Vlasov
A. A. Bochvar
N. A. Dollezhal'
V. S. Fursov
I. N. Golovin
V. F. Kalinin
A. K. Krasin
V. V. Matveev
M. G. Meshcheryakov
V. B. Shevchenko
V. I. Smirnov
A. R Zefirov
Copyright 0 1977 Plenum Publishing Corporation, 227 West 17th Street, New York,
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Consultants Bureau journals appear about six months after the publication of the
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Published monthly. Second-class postage paid at Jamaica, New York 11431.
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SOVIET ATOMIC ENERGY
A translation of Atomnaya Energiya
October, 1977
Volume 42, Number 4 April, 1977
ARTICLES
The Optimum Ratio between the Operational Component Times of an Emergency
CONTENTS
Engl./Russ.
Protection Drive (EPD) - V. P. Petrov
289
259
Principles Involved in the Design of Automatic Control Systems for the Distribution of
Energy in Channel Reactors - I. Ya. Emel'yanov, E. V. Filipchuk,
P. T. Potapenko, V. G. Dunaev, and N. A. Kuznetsov
294
263
The Approximation Error and the Energy Distribution in a Reactor - L. P. Plekhanov .
300
, 268
Reuse of Power Plutonium and Uranium and Neutron Regeneration in a Thermal Reactor
- P. I. Khristenko
304
272
Pressure Excursion in Fuel Elements with Compact UO2 during the First Power
Emergence of a Reactor - B. V. Samsonov and V. Sh. Sulaberidze
309
277
Field-Emission Microscopy and Emissive Properties of Uranium - A. L. Suvorov
312
280
Creation of a Working Standard of Thermal Neutron Flux Density Based on the F-1
Reactor - E. F. Garapov, A. G. Inikhov, E. P. Kucheryavenko, S. S. Lomakin,
G. G. Panfilov, V. I. Petrov, V. V. Khmyzov, and I. A. Yaritsyna
317
286
Ionizing Radiation Detectors Based on Radiation-Stable Crystalline Semiconductors
of the In2Te3 Type - V. M. Koshkin, L. P. GaUchinetskii, V. N. Kulik,
G. K. Gusev, and U. A. Ulmanis
321
290
SURVEYS
State of the Art and Prospects for the Development of Technology of Atomic Power
Plants Powered by the Water-Moderated-Water-Cooled Reactor VVER-1000
- N. N. Zorev
326
295
DEPOSITED PAPERS
Excited X-ray Fluorescence Analysis of a Substance by an SXRL Program
- S. P. Golenetskii, V. A. Kalugin, V. I. Sedelynikov, and N. I. Sukhlova
334
304
Photoemulsion Method of Personal Neutron Monitoring - M. M. Komochkov
and M. I. Salatskaya
334
304
Some Laws Governing Neutron Spectra behind the Shielding of Proton Accelerators
- V. E. Aleinikov, V. P. Gerdt, and M. M. Komochkov
335
305
Data on the Radioactive Contamination of the Biosphere in Hungary - Szabo Andras
336
306
LETTERS TO THE EDITOR
a-Radiation Measurement of Beryllium Coating Thickness - M. A. Belyakov
and E. P. Terent'ev
338
307
Miniature Fission Chambers for Investigation of Neutron Fields - V. V. Boltshakov. . .
339
308
Detection Efficiency of Si (Li) Detectors for 0.05-1.25-MeV y Rays - V. A. Kozhemyakin
and G. I. Shul'govich
342
309
Use of 11-MeV Protons for Activation Analysis - B. V. Zatolokin, I. 0. Konstantinov,
and N. N. Krasnov
344
311
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Fast-Neutron Radiative Capture Cross Section for 242Pu - A. A. Druzhinin,
V. K. Grigoriev, A. A. Lbov, S. P. Vesnovskii, N. G. Krylov,
CONTENTS
(continued)
Engl./Russ.
and V. N. Polynov
348
314
Analysis of Heavy Elements through the Absorption Jump in an Interfering Element
- M. B. Enker, L. I. Shmonin, G. E. Kolesov, and V. I. Cherevko
350
315
Calculation of Electron Conversion into Positrons at 0.2-2 GeV - V. A. Tayurskii
352
317
Determination of Absolute y Yields in a Decay of 243AM - D. I. Starozhukov,
Yu. S. Popov, and P. A. Privalova
355
319
Variation of Thermal Conductivity of a Gas Mixture under the Fuel-Element Jacket
during Burn-Up - V. S. Yamnikov and L. L. Malanchenko
358
322
Optimal Conditions for Shutting Down a Reactor for a Given Shutdown Time
- S. A. Vasil'ev, V. I. Pavlov, and V. D. Simonov
361
324
Fast-Electron Transfer in Lamellar Materials - B. A. Kononov, Yu. M. Stepanov,
and A. P. Yalovets
363
326
Testing the Biological. Shielding (of the Reactor of the Bilibinsk Nuclear Power Plant
- V. Yu. Ifraimov, V. N. Mironov, A. P. Suvorov, Yu. V. Kharizemenov,
S. G. Tsypin, and A. I. Shul'gin
366
328
Analysis of Solid Lubricating Coating by Proton-Stimulated X Rays - A. G. Strashinskii,
G. K. Khomyakav, N. V. Serykh, I. T. Ostapenko, and R. V. Tarasov
368
329
Nonstationary Distribution Functions of Particles Retarded in Matter - Yu. A. Medvedev
and E. V. Metelkin
370
331
Energy Losses of Slow Ions in Organic Medium for Elastic Nuclear Collisions
- S. P. Kapchigashev and V. V. Duba
373
333
Determination of the Effective Energy Cut-Off of Gadolonium and Cadmium
- I. R. Merkushev
375
335
Automatic Eight-Channel Unit for Recording Tracer Activity - S. S. Volkov,
V. P. Koroleva, N. I. Kurakov, E. B. Martynov, and L. A. Chernov
376
336
The Fine Structure of the Fission Yields of Heavy Nuclei - K. A. Pietrzak (Petrzhak),
E. V. Platygina, and V. F. Teplykh
379
337
Hydraulic Resistance to Forced Two-Phase Flow of Helium in Narrow Channels
- V. I. Deev, Yu. W. Gordeev,- A. I. Pridantseir, V.1. Petrovichev,
and V. V. Arkhipov
381
339
INFORMATION
Fourth Session of the Joint Soviet-American Commission on Cooperation
- V. A. Vasil'ev
384
341
CONFERENCES AND MEETINGS
Fifth All-Union Conference on Charged-Particle Accelerators - V. A. Berezhnoi
386
342
Third Session of the Soviet -American Coordination Commission on Fast Reactors
- A. F. Arifmetchikov
388
344
Soviet-Canadian Seminar on the Vibration of Steam-Generator Tubes and Fuel-Element
Assemblies in Nuclear Power Stations - I. N. Testov and V. G. Federov
391
346
Conference of IAEA Experts on the Protection of the Population in Radiation Accidents
-Yu. V. Sivintsev
392
347
Third International Summer School on Radiation Protection - N. G. Gusev
393
347
BOOK REVIEWS
D. Eadie et al. - Statistical Methods in Physics. Reviewed by G. A. Ososkov
396
349
Fachworter der Kraftwerkstechnik. Teil II. Kernkraftwerke. Deutsch -Englisch
397
350
W. Oldekor - Einfiihrung in Die Kernreaktor- und Kernkraftwerkstechnik
398
350
The Russian press date (podpisano k pechati) of this issue was 3/23/1977.
Publication therefore did not occur prior to this date, but must be assumed
to have taken place reasonably soon thereafter.
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ARTICLES
THE OPTIMUM RATIO BETWEEN THE OPERATIONAL
COMPONENT TIMES OF AN EMERGENCY PROTECTION
DRIVE (EPD)
V. P. Petrov UDC 621.039.5:62-7
Emergency protection drives (EPD's)arewidelyusedin nuclear reactors. Electromagnets (EM's) are
used in retaining the safety rods outside the reactor core, while springs attempt to insert them into the core.
The operation time of such an EPD is made up of the EM pick-up time (i.e., the time from the instant the
the voltage is applied to the EM to the instant the EM armature begins to move) and the travel time of the EPD
mechanical system, which amounts to the travel time of the safety rods. As the spring force increases, the
travel time of the EPD mechanical system tms is reduced, while the EM pick-up time increases, as a conse-
quence of which the latter has to be designed for larger forces. Therefore, there is a basis to assume that
an optimum ratio between the components exists, which ensures a minimum operating time top of the EPD.
The aim of this article is to determine this ratio.
It would be helpful to obtain the results for cases in which the displacement of the safety rod is first a
function of the instant at which the limiting permissible reactor power is achieved under emergency conditions
and, second, a function of a given quantity (a variant that is frequently encountered when developing EPD's).
Let us consider the optimum ratio for the first case. Let us assume that as a result of some cause
(e.g., operator error), a surge of positive reactivity arises in the reactor, and the reactor power rises
sharply. Figure 1 shows the reactor-power curve for this case. The operating time of the automatic equip-
ment in the emergency protection control system is small in relation to the EPD operating time, and can be
assumed to be zero. The EM is energized at instant ti. At t2 the EM armature moves and the safety rods
begin to enter the reactor core. Because of this the rise rate of the reactor power declines. At instant tk
a negative reactivity equal to Pk is introduced by the safety rod, which is sufficient to ensure an end to the
power rise. A sharp decrease in power occurs at tk as the negative reactivity continues to be introduced,
e.g., by a brake on the safety rod.
Shut-down level
100
Fig. 1. Diagram showing the
shut-down of the reactor after
operation of the emergency
protection.
Translated from Atomnaya gnergiya, Vol. 42, No. 4, pp. 259-262, April, 1977. Original article sub-
mitted December 15, 1975.
This material is protected by copyright registered in the name of Plenum Publishing Corporation, 227 West 17th Street, New York, N.Y. 10011. No part
of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying,
microfilming, recording or otherwise, without written permission of the publisher. A copy of this article is available from the publisher for $7.50.
289
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Fig. 2. Diagrammatic repre-
sentation of a displacement of a
safety rod.
The generation of energy which is proportional to the area below the curve (see Fig. 1) represents a
danger to the reactor itself. If this energy exceeds the permissible level, then damage could result. Accord-
ing to [1], this energy is proportional to the maximum power Nmax. Therefore, the tk represents the instant
of danger to the reactor, which corresponds to the attainment of Nmax.
When deriving the optimum ratio, we assume that the onset of tk to a first approximation is conditioned
only by the negative reactivity Pk and does not depend upon the introduction rate.
Let us consider the displacement of the safety rod, of the pull rod connected to the EM, under the in-
fluence of the spring force and the mass of the moving system comprising the EPD (Fig. 2). The spring acts
over gap /, which equals the displacement of the safety rod in the reactor core up to the initiation of braking.
We signify the force of the spring stretched over the length / by F. We signify the mass of the moving system
of the EPD together with the safety rod by m, while the corresponding force is P. Let us now consider the
displacement of the safety rod in gaseous and liquid media.
If we ignore the effect of the resistance in the gaseous medium on the rod velocity (which is quite per-
missible, due to the fact that this resistance is small in relation to the spring force), then the expression for
determining the travel time of the EPD mechanical system up to the instant the safety rod is inserted by a
length xi, corresponding to the introduction of a negative reactivity equal to pk into the reactor core, has the
form
where
tms c, V-13- arccos (1? 0+ ) 1 (1)
Cj=j17; (2)
13=P/F; (3)
4=40; (4)
where g is the acceleration due to gravity. Equation (1) is obtained by solving the equation of motion of mass
m under the influence of the gravitational and spring forces.
To avoid exceeding the allowable EM surge voltage rating when this is switched off, its winding is
shunted by a resistor (Fig. 3a) or stabilitron (Fig. 3b), or other, less usual, connection schemes may be
employed. It follows from [2] that the EM pick-up time is directly proportional to the EM constant T:
C2T,
where C2 is a function of the EM pick-up current in per unit terms and the magnitude of the surve voltage K,
i.e., the ratio of the EM surge amplitude at switch-off to the EM voltage source.
In order to find the relationship between T and the spring force, let us equate the EM pull force Fpu,
expressed in terms of the pick-up current Ipu, to the sum of spring force F and P. For an unsaturated Mag-
netic system in the EM, this condition can be described in the form [3]:
290
F= ?0 51puEM
' L /6 = ?(F P),
pu?
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(6)
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a
+
where LEm is the EM inductance and 60 is the operating air gap in the EM core.
Let us characterize the pull magnitude due to the EM at rated current Irtd in relation to the pull at pick-
up current 'Pu by margin km:
Dst
Do
Fig. 3. The most commonly used
scheme for disconnecting the EM
using: a) a resistance; b) a sta-
bilitron.
km= Fad ./Fpu (4td //d2. (7)
Then from (6), and taking (7) into consideration, we obtain
= (2.5,1cm/PEm ) (F P), (8)
where PEM is the EM power; we can define C2 from the data given in [2] and Eq. (7) (for the scheme of Fig. 3a
and b, respectively):
where
C2R = , 17T1r-n- , (9)
C2 i
st = n
K?1 (10)
From Eqs. (5), (3), and (8) we find that
tpu= Cs (6 ? 1)/13, (11)
c3?c9- 260km p. (12)
PEM
We find top by summing the right-hand sides of Eqs. (1) and (11). By studying these expressions at the
extremum for variation of fl, we find the conditions for minimum top (at spring force Fe, i.e., f3 = ge)
SIpze [ I arecos ( 1 4 \ alc-Vf:
CI 2 Ire; f3e+1 i (4-0, i) 11 2x* e
(13 ? 1) ? *2 ] (13)
k xk
By defining C3 from Eq. (13) in terms of C1, we are able to find the optimum ratio between the operational
component times of an EPD for motion of a safety rod in a gas:
tpu.e e+1
2
tmo.e
Pe rk
.
arccos-11)
(14)
The relationship of the extremum point ge to the ratio C3/C1 and xrt, and also the optimum ratio of /3e
and xk, are illustrated by the curves of Fig. 4, which were plotted in accordance with Eqs. (13) and (14). When
f3e = 0, which corresponds to neglecting the small mass of the EPD moving system in relation to the spring
force, the optimum ratio is 0.5. The effect of the gravitational force P upon the optimum ratio, up to a tenfold
increase of P over the spring force F, is sufficiently small; overall, the optimum ratio is reduced by 18% and
equals 0.5 over the range of g from 0 to 0.1.
Since the spring force usually exceeds the gravitational force on the moving system, we are particularly
interested in the case where the EM pick-up time is half of the safety rod travel time. To find the spring force
Fe corresponding to this condition, together with tpu and tine, we introduce the gravitational force P for con-
stant Ct and C3 into Eqs. (2) and (12) and also take (3) into account; we can then rewrite Eqs. (1), (11), and
(13). As a result, we obtain
291
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tpu.e/tms.e
0 44 .C3/(1
o 48 4
0,1
42
42
- 0,4
0,6
Fig. 4. Relationships of ex-
tremum point fie to C3/C1 and
xic% and of optimum ratios of
k
tpu. e/tms. e to xi and
tins = Cjil/F; (15)
C4 = 111M arccos (1? 4); (16)
tpu= C,F; (17)
C, C2 (26pkrn !PEM ); (18)
(19)
Figure 5 shows the variations of tins, tpu, and top with the spring force for arbitrary units of F*:
F* =FIFe (20)
It follows from Fig. 5 that the EPD operating time varies very little within the region F* = 1 (which cor-
responds to Fe). This enables us to somewhat reduce the spring force without greatly affecting the operating
time. For example, reducing F* to 0.65 leads to an increase in the operating time to 5% of its maximum value.
292
Fig. 5. Variation of pick-up
time, safety rod travel time,
and EPD travel time in rela-
tion to the spring force of the
drive.
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To obtain the optimum ratio for the case of a safety rod moving through liquid, we have to take into con-
sideration the resistance of the liquid to the rod motion. The gravitational force acting on the moving system
of the drive and the force ejecting the safety rod from the liquid can be neglected, which is justified for two
reasons:
a) in drives where the rod moves through a liquid, the spring force is usually significantly greater than
the gravitational force acting on the moving system of the drive;
b) the force tending to eject the safety rod from the liquid is partially compensated by the gravitational
force acting on the moving system of the drive, which greatly reduces its share of the resultant force
acting on the rod.
Consequently, the pick-up time of the EM should represent half the travel time of the safety rod for
minimum operating time of the EPD as a whole. In this case, Eqs. (15) and (17)-(19) are justified. The effect
that the viscosity of the medium has on the travel of the safety rod is reflected in the expression for C4, which
takes on the form:
C4.114= 1.411a VT v-21,1(1 _ x*) + m ? (211/ exp (-2?1.z*Im)
0
Here
(21)
tt = 0.5C, pS,
where Cx is the drag coefficient, which depends upon the rod form; p, the medium density; S. end-surface
area of the safety rod [4].
Equation (21) is obtained for turbulent motion of the rod through the liquid when the rod drag coefficient
is proportional to the square of the velocity [4]. Turbulent motion of rod follows from the Reynolds number
which, as can be seen from the dimensions of the rod normally used in water-cooled?water-moderated reac-
tors and its travel time, will be of the order 200,000-400,000, which exceeds the limiting value of the Reynolds
number (20,000) by a number of times at which laminar flow of a liquid remains possible.
Since the working stroke of the safety rod is a given value, the formulas given earlier are valid. We
understand length/ as value xk, i.e., the given length of the operating stroke of the rod. In this case, accord-
ing to [4], xrc = 1; the optimum ratio remains as before; to obtain a minimum drive operating time, the pick-
up time of the EM should equal about half the travel time of the safety rod.
LITERATURE CITED
1. M. A. Schulz, Regulating Nuclear Power Reactors [Russian translation], Izd-vo Inostr. Lit., Moscow
(1957), p. 388.
2. V. P. Petrov, Elektrichestvo, No. 8, 50 (1975).
3. A. K. Ter-Akopov, The Dynamics of High-Speed Electromagnets [in Russian], gnergiya, Moscow (1965),
p. 47.
4. K. A. Putilov, A Course in Physics [in Russian], Fizmatgiz, Moscow (1962), p. 208.
293
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PRINCIPLES INVOLVED IN THE DESIGN OF AUTOMATIC
CONTROL SYSTEMS FOR THE DISTRIBUTION OF
ENERGY IN CHANNEL REACTORS
I. Ya. Einel'yanov, E. V. Filipchuk,
P. T. Potapenko, V. G. Dunaev,
and N. A. Kuznetsov
UDC 621.039.562:621.039.517
During the widespread development of nuclear power over recent years, both in the USSR and abroad,
work has been carried out on the development of computer-based automatic control systems employing reac-
tors. Regardless of the computer's drawbacks as a means of control, including first and foremost its inade-
quate reliability, it is being used more and more in the design of reactors. Almost all reactors built through-
out the world over the last 5-8 years, and all reactors at present under design, have incorporated computers
in their automatic monitor and control systems. Check systems based on "objective" computers have been
used on the second units at the Beloyarsk and Novovoronezh nuclear power stations and a similar system has
been installed on the Leningrad nuclear power station for monitoring the operation of the power unit incor-
porating reactor type RBMK.
One of the important and urgent questions concerning the operation of the computer complexes of auto-
matic control systems for reactors is the creation of a mathematical base for their operation. The difficulty
here lies not only in the size of the program but also in the creation and experimental processing of effective
monitor and control algorithms.
The problems of mathematical processing for the signals from transducers within the reactor have been
well-enough tested out under operational conditions in [1]. Some questions concerned with the design of direct
digital control of reactor power distribution have been considered in [2]. At the same time, there have been
practically no publications in non-Soviet literature-concerned with the development of algorithms for the opti-
mum control of energy distribution.
Statement of the Problem. The balancing of energy distribution, designed to bring the average power
and thermal loading near to the maximum, is the basis of the modern approach to controlling large reactors.
Reducing the radial coefficient of imbalance in energy distribution enables us to achieve practically the
same increase in reactive power, increased burnup, and ensure the maintenance of a given margin up to
critical thermal loads.
Among the control problems that make it necessary to use manual optimization of the energy distribution
in reactors type RBMK, we would do well to pick out the following:
1. obtaining a basic optimally balanced energy distribution;
2. controlling the regulation of the rods so as to maintain the regulating rods within their most effective
range, particularly when load is being reapplied after a shut down;
3. the compensation of local disturbances caused by overloading fuel channels, when the location of the
disturbance and its magnitude are usually known beforehand;
In general, the problem of controlling energy distribution consists in maintaining some desired distribu-
tion under steady-state and transient conditions, which is assigned to the computer as the conditions for
achieving maximum power and reliability from the heat-engineering standpoint.
Translated from Atomnaya Energiya, Vol. 42, No. 4, pp. 263-267, April, 1977. Original article sub-
mitted May 3, 1976.
This material is protected by copyright registered in the name of Plenum Publishing Corporation, 227 West 17th Street, New York, N.Y. 10011. No part
of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying,
microfilming, recording or otherwise, without written permission of the publisher. A copy of this article is available from the publisher for $7.50.
294
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As the optimum condition for the type RB1VIK reactor is a uniform distribution, the control problem can
best be formulated as the minimization of any deviations of energy generation in each discrete control region
from the mean level. As a measure of this deviation, we use the quadratic form
1=
12=1
where k = 1, 2,...; N is a discrete time; cli(k) and 41(k) are vectors of deviation from the mean value with a
scale defined by the number of check points; Q is the positively defined weight matrix for the forms of energy
distribution, a special case of which is equal to the identity matrix.
We should note that the quadratic relationship of the functional to the value of {41)(k) ? 1-,(k)} has the effect
that the functional varies little with small variations in energy output. This is an advantage insofar as small
(apparent) variations in energy output can be due to inaccuracies in measurement.
In [3] are formulated a number of safety requirements, which must be observed when developing
specific control schemes for reactors. When we are considering control functions suitable for assignments
to the computer, we should take safety in reactor operation as the basic criterion.
The computer must be entrusted to carry out those functions which, while they are sufficiently important,
can still be carried out manually or by means of analog regulators should the computer fail.
From this standpoint, the energy distribution control problem is very suitable for assignment to the
computer. On the one hand, the balancing of energy distribution for large reactors is a necessary condition
of their operation, while optimizing the position of the regulating rods is a difficult multidimensional problem,
whose solution reduces essentially to sorting through a large number of variants. The quality and speed of
solution depends in large measure on the operator test. The use of a computer for generating recommenda-
tions on optimum strategy for control to a great extent takes the burden from the operator and enables a more
uniform distribution of energy to be achieved. On the other hand, failure of the computer or its control pro-
gram should not give rise to a fault situation, as its function in this case would override the operator.
From consideration of the safety of the regulation system, it would be advisable to base this on the
heirarchical principle, using self-consistent regulating equipment with a self-consistent memory in the lowest
level of the heirarchy. The normal state of the computer as an element of the higher level of the control heir-
archy would be one in which there is no output signals [3]. From this point of view, the best operating mode
for the computer would be the "operator advisory" mode.
To design energy distribution control systems we must have enough information on the state of the object
being controlled. We must then postulate that all questions connected with the mathematical processing of the
signals from the transducers within the reactor are solved on the basis of method [1], while the reactor has a
computer-based system of monitoring which operates in real time.
The problem consists in developing a control algorithm for a given number of regulating rods whose
position is known, which enables us to find the optimum control, taking engineering limitations into considera-
tion, for a reasonable expenditure in computer time and data storage capacity.
Formulating the Control Problem. Let us assume that the energy distribution is measured digitally in n
regions, while r regulating rods are used for control purposes.
For the steady-state condition, the process of measurement at the monitor points gives rise to displace-
ments of the regulating rods. This can be described by the equation
(1)
where 4)(k) is the n-dimensional vector of energy distribution; ou(k) is the r-dimensional vector of control;
C(k) is the static transfer matrix of the zone.
Under these circumstances, due to the corresponding choice of time interval [tk, tk+i] between two
sequential regulating influences, the rapid processes linked, e.g., with temperature variations of the fuel
elements, with the displacements of the rods, and the inertia of the transducers, are completed during this
interval, whereas the contamination by xenon and the temperature of the graphite have not changed.
The transfer matrix C(k) can be obtained for a given physical design of reactor or it can be determined
from measurements by means of transducers within the reactor. The variations of state 64i(k) = C(k)(5u(k),
observed at the end of the transient, represent linear conversions of the measured positions of the rods,
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P b
0,8
0,6
0,4
0,2
Fig. 1. Calibration charac-
teristic of the control chan-
nel.
expressed in reactivity units Su(k), with conversion operator
sot 601 ?
Sui ? ' ' Su
C=I r
SO i
. . . -75-17 . . . .
SO ? ow 11
Su 1 ? ? ? Sur.
The elements of the transfer matrix C = 64,-/Su? can be determined from known pairs of vectors {Sil,(k);
ou(k)} [4].
We can see from the physical concepts involved that matrix C(k) exists and is unique for any conditions
of operation of the reactors. However, inaccuracies in measurement and significant spread of the vector pair
{64,(k); ou(k)} affect its numerical dependence. Furthermore, the transfer matrix depends upon the actual
state of the process; we therefore have to restore (adapt) its elements periodically, replacing the initial data
by new data which more closely approximate to the real conditions of operation. When doing this, we stipulate
that vectors 05(1) and Su belong to regions of linearity and effectiveness, respectively, and that among vectors
Su we have at least one r that is linearly independent.
The most important cause of variation in matrix C is variation in the effectiveness of the rods with
changes in the height of energy distribution. In this regard, there is significant variation in the characteris-
tics of the control channel (Fig. 1). As the height of the energy distribution is a monitored quantity, we can
vary the characteristic in relation to variations in the neutron flux and further increase its accuracy with
respect to experimental results.
For reactors with zero or positive power coefficients of reactivity, the necessary condition for the
existence of a quaeistatic C matrix is the presence of an automatic power regulator, while for a reactor with
an unstable spatial energy distribution, zone (local) regulators are needed. Therefore, in the future we will
propose that the reactor be equipped with an automatic power regulator and (or) a system of zone regulators,
and matrix C be obtained as the result of measurements with the automatic regulators in circuit.
As we have already stated, the aim of control consists in minimizing in the general case over an infinite
interval of time a functional of the form
= E (k) ? Q (k) ? ?Ti (k)}
h= 1
However, if we take into consideration the properties of the object (1) the unknown control strategy breaks
down into a series of partial strategies. Over each interval of time, the aim consists in minimizing the sum
of the squares of the energy distribution deviations from the mean value, taking the limitations into considera-
tion.
We can assume that the solution to this optimum problem is equivalent to a solution by the least-squares
method of the system of equations
MD= C Au,
where ticl? is the error vector; Au is the unknown displacement of the regulating rods.
? The Control Algorithm. We know that in the absence of limitations, the solution to system (2) can be
found with the aid of the pseudoinverse matrix C+ [5]:
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(2)
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Auo = c+ (3)
where
?(CTC)-1 CT.
This solution possesses the important property that its norm is minimal. This is particularly important
when applied to reactor control, as the optimum equation not only ensures the best equalization of energy gener-
ation, but also guarantees the minimum possible displacement of the regulating rods.
In practice, the unknown optimum equation must satisfy a series of engineering limitations, among which
the following are the most important:
1. the optimum equation so obtained must not vary the total power of the reactor;
2. limitation on the regulating action. A typical calibration characteristic for a control channel is
given in Fig. 1. The nonlinearity we are considering is connected with the fact that the rods in the
general case can only be introduced into a channel that has first been filled with water;
3. the power (output steam content) of the individual fuel channels must not exceed the limiting permis-
sible value corresponding to critical heat exchange.
Let us consider the algorithm for operation of the system taking limitations into account. We will not
specify the reasons leading to the need to calculate the optimum displacements of the regulating rods; we
merely assume that the aim consists in calculating a technologically practicable system of control for a given
vector of errors 4, ?
The First Step. We calculate, in accordance with (3), the "ideal" control vector
u? u+ Au? = u+ (CTC)-1 CT A4),
where u is the initial position of the rods.
The Second Step. Taking (1) into account, we determine the desired energy distribution at each monitor
point (lid = + Cau? and the total power 9 = cei4,d, where cri are the weight coefficients.
i=1
n r
E Ecip
i=i 3-1
If 9 exceeds the given total power 90, all the control actions are reduced by a value 6Au? = (9 ?
The Third Step. The values of control action so obtained are compared to the permissible actions (see
Fig. 1) and the engineering realization of the rod displacement AuP is determined in the following manner:
if al Au?_>? b,thenAur b ? ui;
if aui+6,14< b,then Au'? = Au7;
if a/2 < ui Au? < a,thenAuf = a ? ui;
if ? < a/2,then Aur
The Fourth Step. We determine the new desired value of the error vector 6.4,d = CAuP, which further
minimizes the variation in control action, corresponding to the effective range of rods
AuP=(ere)ter Atpd
where .6 is the matrix obtained from C by plotting those columns which correspond to the boundary values of
the control effect (points 0, a, and b in Fig. 1).
The Fifth Step. We again determine the desired energy distribution (I)d = 4) + CAuP and compare this
with the limiting permissible values. For those channels in which an increase is observed, recommendations
are given regarding the increase in the flow of coolant or reduction in the appropriate settings.
As we have already stated, in addition to the problem of calculating the position of the control actuators
to ensure the base regime, the need also arises to compensate for disturbances brought about by overloaded
fuel channels and maintaining the rods by means of an automatic regulator in the effective range. In similar
cases, we have observed undesirable deviations of energy generation in small local zones, and for control
purposes, it is advisable to employ criteria of the form
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(Start)
(Stop)
Fig. 2. Structural schematic of control algorithm: 1) in-
put initial position of regulating rods; 2) form matrix of
object; 3) determine the error vector; 4) calculate the
"ideal" control; 5) summated deviation of energy genera-
tion within the dead zone of the total power regulator;
6) calculate the synchronous displacements of the rods;
7) equation in given limits; 8) form the "real" equation;
9) power of channels riot exceeding limitation; 10) print
numbers of channels with excessive power; 11) calculate
displacement of rods; 12) print.
111
where i = 1, 2,...; m is the number of monitor points belonging to the local zone..
This problem, in essence, can bosolved by the mathematical modeling method, or by prediction of re-
sults which lead to the carrying out of the control actions. We should stress that from the calculation point of
view, all the problems we are considering are of the same type. Figure 2 shows the structural schematic of
the control algorithm we are describing.
Discussion of the Results. The proposed automatic control system for the energy distribution in a
reactor type RBMK consists of two basic functional units: systems for identifying the parameters and systems
for calculating the optimum position of the regulating rods.
It is assumed that the value of energy generation is estimated by the method described in [1]. The inputs
to the system take the form of the indications of transducers in the energy distribution monitor system and the
position of the rods, and also the maximum possible local energy generation and the range of rod positions. A
characteristic feature of the solution to the problem consists in the significant nonlinearity of the control chan-
nel characteristic, which is connected with the fact that the rod can only be introduced into a channel that has
first been filled with water. The proposed general algorithm includes the case in which there is no column of
water in the safety rod channel, as was the case in the first type RBMK reactor.
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The external data for the system comprises the total power of the reactor and the height of the distribu-
tion of energy generated. The output data is assumed to be the calculated optimum displacement of the rods
and the data for the operator on the desired distribution of energy.
The mathematical description of the process, based upon a static matrix model of the reactor, the ele-
ments of which, representing the variation in energy generation at the monitor points, give rise to displace-
ments of the rods, has been experimentally refined by operation of the reactor itself.
The essence of the proposed algorithm for calculating the optimum control system lies in solving by the
least-squares method the square matrix equation
AO= C Au,
where A4) is the vector of deviation of energy generation from the mean value; Au is the unknown displace-
ment of the regulating rods.
The limitations given in the form of inequalities are taken into consideration by means of mathematical
modeling of the reaction system using the boundary values of the control influences with subsequent solution of
the unconditional minimization problem of least dimensionality. Digital modeling of the operation of the sys-
tem in a typical situation has confirmed the efficiency of the control algorithm.
It should be stressed that the present article has formulated only the basic principles of designing an
automatic control system for the energy distribution in a type RBMK reactor. The proposed algorithm can be
transformed when enough experience in the use of computers has been accumulated and when a large number of
experimental results have been obtained. However, in the initial stage of using computers in the "advisory"
mode, the proposed concentration of equations can be very useful, and the algorithm for calculating the opti-
mum positions of the regulating rods could prove to be an effective means of assisting the operator.
LITERATURE CITED
1. I. Ya. Emel'yanov et al., At. Energ., 34, No. 5, 331 (1973).
2. E. V. Filipchuk et al., At. Energ., 39, No. 1, 12 (1975).
3. I. Ya. Emel'yanov and P. A. Gavrilov, in: Experience in the Operation of Nuclear Power Stations and
Ways of Furthering the Development of Atomic Power Engineering [in Russian], Vol. 2, Obninsk (1974),
p. 288.
4. P. T. Potapenko, At. Energ., 41, No. 1, 25 (1976).
5. F. R. Gantmacher, Applications of the Theory of Matrices, Wiley (1959).
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THE APPROXIMATION ERROR AND THE ENERGY
DISTRIBUTION IN A REACTOR
L. P. Plekhanov UDC 621.039.61
The operating efficiency of nuclear reactors, in many respects, depends on the quality of the in-reactor
monitoring, to which much attention has recently been devoted [1-7]. References [1, 2, 4-7] consider various
aspects of obtaining the detailed energy distribution in the active zone on the basis of physical calculations and
approximation of the readings of sensors [1, 2] and their statistical analysis [4-7]. These methods are general
and are effective for detailed monitoring of the energy distribution and involve a large volume of computational
and experimental work. Reference [3] gives the upper estimate of the error of discrete monitoring, an esti-
mate that is valid for a fairly large number of sensors and positive deviations of the energy distribution from
the nominal value.
The present paper proposes a method for estimating the discrete monitoring error which differs from
the above-mentioned methods. It is based on the limited nature of the deviations of the unmonitored param-
eters which determine the neutron field.
Formulation of the Problem. Let the stationary neutron field (1,(r) be described by the boundary-value
problem
L 1), 00=0; D (r) = 0, r ES, (1)
where L is an operator; a(r) = a0(r) + Aa(r) is a parameter, and S is the boundary of the region.
We take 6.41 = (I) ? (Do to be the deviation of the field from the base distribution (1)0(r), obtained for a = ao,
and we rewrite the problem as
(Y1), cto) f (Acc); (r) = 0; r E S, (2)
where Lo is a linear operator in Arl, and f is an operator. This representation is obtained either directly from
Eq. (1) or after expansion of Eq. (1) in powers of 6.a and 6.4..
The right-hand side of Eq. (2) is subject to a constraint in one of two forms
fz [Zia (r)] dV A2.
(3a)
(3b)
Henceforth all integrals are taken over the region bounded by S. A constraint in the form (3b) is con-
venience to use, e.g., when analyzing "point" interactions in a reactor, expressed in terms of (5 functions.
Let us consider the following points:
1. Approximation of the neutron-field deviation 6.(I)(r) by a system of orthonormal functions
(r) hE1 c hcp ?, (r) R (r),
(4)
where Rn(r) is the remainder (error) and ck = ck[6,4,(r)] is the expansion coefficient, defined as a linear func-
tional of the approximated function.
2. Reproduction of the field from the measurements of 1 sensors:
(r) = hEi coph (0+ Rnt (r, ri), (5)
Translated from Atomnaya E'nergiya, Vol. 42, No. 4, pp. 268-271, April, 1977. Original article sub-
mitted October 23, 1975.
390
This material is protected by copyright registered in the name of Plenum Publishing Corporation, 227 West 17th Street, New York, N.Y. 10011. No part
of this publication may be reproduced, "stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying,
microfilming, recording or otherwise, without written permission of the publisher. A copy of this article is available from the publisher for $7.50.
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where ck are the approximate values of the coefficients calculated from the measurements and ri are the co-
ordinates of the sensors.
The remainders in Eqs. (4) and (5) must be evaluated with constraint (3) and with allowance for the
measuring error and for the indeterminacy of the sensor coordinates.
Estimation of the Approximation Error and the Choice of Coordinate Functions. The results of the
classical theory of approximation [8] are inadequate for solving the problem formulated since only poly-
nomials or trigonometric functions are considered as (pk and the estimates either' give only the order of
magnitude by which the remainder decreases as n -.00 or are too high.
For example, the corollary to the second Jackson theorem for the one-dimensional case states that in
the approximation with polynomials of order up to n we have max I Rn(x) I M ? 6k/nk, where k 1 is the order
of the differentiability of 4(x); M I(I)(k)(x). For the given problem, k = 2 and for n 5_ 6 this estimate is too
high.
The solution of problem (2) can be written as
iD (r) --= TO? (r) AeDi (r); (6)
6o:Di (r) = f [Act (a)] G (r, a) clV (a). (7)
Here, 4,?(r) is the solution of the homogeneous problem (2) if it is nonzero; y is a number found from auxiliary
conditions (the controller equation, feedback, etc.) and the criticality condition for the perturbed state; and
G(r, is the generalized Green's function, orthogonal to (1)0(r). If 40(r) a 0, then G(r, 4) is the ordinary
Green's function of problem (2) [9].
Let us consider separately the estimation for both terms in Eq. (6), respectively, with superscripts 0
and 1:
I .11, (r) I < max I y HO? (r) Cj, [o(r)] (ph (r) I. (9)
For the second term from Eqs. (4) and (7) we find
(r) = f [Aa P,, (r, a) nV (a), (9)
where the expression Pn(r, = G(r, 4) ? ck[G(r, )1(pk(r) is the remainder of the approximation of the
Green's function G(r, in the variable r by a chosen system of coordinate functions.
Using the Cauchy ?Bunyakovskii inequality, we obtain the following estimates from Eq. (9):
lil,(r)1.< F P,, (r, a) I
I (r) I < A [ (r , dv (a)] "2.
The system of coordinate functions (pk(r) can be chosen from various considerations such as convenience
of calculations and physical meaning. Estimates (8) and (10) permit optimization of this choice from the con-
dition of minimum approximation error. The first optimal function is 4.0(r), provided it is nonzero; then
R(r) = 0. The other functions can be found by minimizing any positive functional of the right-hand sides of
inequalities (10). It can be shown, e.g., that in every particular case of self-adjoint operator Lo and quadratic
functional on the right-hand side of inequality (lob), the eigenfunctions of problem (2), taken in the order of
increasing modulus of their eigenvalues, are optimal.
Estimation of the Reproduction Error. Suppose that there are 1 sensors whose maximum measuring
error is Si and whose coordinates are ri, with a maximum indeterminacy of Sri.
We write the equations for determining the coefficients in Eq. (4) from the measured values of the neu-
tron flux with allowance for the errors
(10a)
(10b)
A a), + xi = ECh (ri) Rn (r,), i =1, 2, ... 1,
i
where A.Ixi is the value of the deviation of the field at the points of measurement and xi is the sensor error
( xi 5_ oi). We solve system (11) by the least-squares method. Then, taking the error of sensor coordinates
(11)
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0,015
0,005
95
Fig. 1. Estimates of approxi-
mation error [Mr(p) is the
right-hand side of inequality
(10b); the numbers denote the
value of n].
zi into account (retaining only the first degree of smallness), we get
ch= E uh, Rn(ri)]+ Eiij3z3 MD,,
.1=1 3=1
(12)
where uki are coefficients determined by the least-squares method and uicij are the derivatives of these coef-
ficients with respect to the coordinate rj. It can be shown that uiij = E ukjgom(rpurni.
By di (r) = E uki(pk(r) we denote the "weighting" function of the i-th measurement in reproducing the
k=1
field. Substitution of the coefficients ck from Eq. (12) into Eq. (4) yields
(r) = EA(Didi (r)+R?: (r) ? 1?? (r i) di (r)? zd (r) +
1
-1-EEET;n (ri) umizi 4cD1di (r).
3-1
(13)
In this equation the first term is the calculated value of the field, the next two represent the error of discrete-
ness of measurements, which we denote by di Rs cr(r), and the last two terms are, respectively, the reproduc-
tion? error, due to the sensor error Itsens(r), and the indeterminacy of the coordinates, 11coord (r). For each
?
error we can give the estimate:
iRdiscr V) I max I 171.1 CD? (r) (Do (r i) di NI; (14)
/
Wdiscr (r) I < (a) IP n (r a) 2 P n (r, t) (r) jdV (a); (15a)
I Raiscr A { S[P ? (r , ? P? (r ,)di (r)r dV (t)}112 ; (15b)
I Rsens (r) ai I di (r) I; (16)
i=1
1 1 fl
I Rcoord (r) I < 6r/ E At I d (r) E (P;n(r (17)
t=i m=l
where Ai is the maximum possible deviation of the field at the point ri; usually Ai is known from practice or
can be estimated from the inequalities
302
TABLE 1. Moduli of First Eigenvalues Xkj
2
0
1
2
0
20,8
66,8
4,85
39,7
96,0
13,2
61,6
127
24,8
39,5
86,4
114
161
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Af F G (r i)jdV
(18a)
Ai