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Investigation of a Frequency Dependent Transfer Function and Its Application to Control Rod Localizaton By SE9700201 CTH-RF-124 February 1997 Investigation of a frequency dependent transfer function and its application to control rod localizaton by N. S. Garis and I. P&zsit CHALMERS UNIVERSITY OF TECHNOLOGY DEPARTMENT OF REACTOR PHYSICS ISSN 0281-9775 CTH-RF-124 February 1997 Investigation of a frequency dependent transfer function and its application to control rod localizaton by N. S. Garis and I. PAzsit Department of Reactor Physics Chalmers University of Technology S - 412 96 GOteborg, Sweden ISSN 0281-9775 CTH-RF-124 Control rod localization Investigation of a frequency dependent transfer function and its application to control rod localization by N. S. Garis and I. Pizsit ABSTRACT Control rod vibrations can be detected via the fluctuations they generate in the neutron flux, i.e. the neutron noise. In a previous paper, a neural network-based algorithm for locat­ ing a vibrating control rod from the measured neutron noise was developed. The transfer function used for the core model was based on the so-called power-reactor approximation resulting in a simple, real-valued solution which means that the phase delay of the signal propagation is neglected. In the present work a more realistic transfer function is used, with­ out the approximations present in the previous model. The transfer function is calculated from the Fourier transformed diffusion equation with a complex, frequency dependent buckling, leading to a complex solution. In physical terms, this means that the phase delay of the signal propagation is accounted for. Using such a complex core model, the present paper investigates the effectiveness of applying neural networks for control rod localisation. -1- CTH-RF-124 Control rod localization 1. INTRODUCTION The detection of vibrations caused by for example control rods via noise measurements is relatively simple, while the localization of which control rod is vibrating is much more complicated. Both the direct task, i.e. calculation of the neutron noise caused by vibrations and the indirect one, i.e. localization of the vibrating control rod by unfolding the neutron noise, are associated with certain difficulties. Solving the direct task requires mainly the calculation of the Green’s function or trans­ fer function of the system. Traditionally, analytical solutions have been preferred. One rea­ son for this was that both the use of the transfer function in the direct task as well as application of the traditional localisation algorithm required analytical manipulations and easy and fast numerical evaluations. In order to get such a simple solution, the core model from which the transfer function was calculated has been kept extremely simple. In all pre­ vious work the so-called power reactor approximation was used, which means that the fre­ quency dependent complex buckling was neglected in the defining equation of the transfer function. The result was a real-valued, frequency-independent transfer function in a simple analytical form. Physically, such a function describes the spatial attenuation of the noise in a certain frequency range in reactors of a certain size (actually the range of power reactor sizes and frequencies, hence the name), but does not account for the phase delay of the sig­ nal propagation. The problems of the unfolding stage are only technical, but somewhat lengthier to describe and here we only refer to the discussion in Ref. [1]. One way to overcome the problems concerning the traditional unfolding procedure was to use unfolding methods based on artificial neural networks (ANNs). An ANN-based procedure for the localization of a vibrating control rod was implemented and investigated thoroughly, see Ref. [2], In this work, just as in the previous ones, the frequency independent transfer function, based on the power reactor approximation, was used. The results have shown that the use of a neural net­ work is an effective and fast method which can be applied on-line. One advantage of the neural network-based localization technique is however that one is not constrained to use simplified core models. It is therefore a logical step to extend the method by applying a more realistic transfer function. The transfer function of the same 2-D cylindrical reactor model as the one used earlier, but now without the previous approxima­ tions, has been recently calculated and investigated. Physically, this means that the phase delay effects in the signal propagation are accounted for. The main purpose of this paper is to report on the performance of the ANN-based localisation technique with this more realis­ tic, complex transfer function. However, the transfer function itself will also be described in some detail because only certain of its properties have been reported so far. One would expect that since there is more information in the noise signals when phase delay effects are also accounted for, application of the complex transfer function will improve the performance of the localisation technique. This advantage is somewhat offset by the increased complexity of the algorithm which may lead to performance deterioration. Actually, in all reported work so far neural networks use real valued input signals, and there is no prior information available on their performance with complex inputs. On the reactor physics side, the analysis of the complex transfer function showed that somewhat surprisingly, spatial phase delay effects are very small at plateau frequencies, smaller than at lower frequencies. Thus, there is very little phase delay information availa­ CTH-RF-124 Control rod localization ble at these frequencies. It is not certain that the use of a complex transfer function in com­ bination with a neural network, with its increased network complexity and larger number of nodes will perform better than a simplified (real) transfer function. This question was also investigated in the present report. 2. GENERAL THEORY One-group neutron diffusion theory with one delayed neutron group is used in a bare, homogeneous cylindrical reactor with extrapolated boundary at core radius R . The vibra ­ tions of the control rod are described by a 2-D stochastic trajectory around the equilibrium position rp such that its momentary position will be given by r + e (f) , where |e (t) | « R . According to Feinberg-Galanin theory, the perturbation in the macroscopic absorption cross-section caused by the vibration of the absorber can be written as 5Efl (r,0 = y[5(r-r p-e(0) -5(r-r p)] (1) where y is Galanin’s constant (strength) of the rod. Using the weak absorber approximation, the neutron noise induced by the vibrating rod can be written in the frequency domain as 6 <t>(r,co) = ^e(co) • V^{G(r, rp, to) <|)0(rp)} = = 5 1 {e^(co)Gx (r, ry, co)+ey(co)Gy(r, ry,co)} (2) Here, r is the position of the neutron flux detector, ex (ro) and e (co) are the vibration components in the frequency domain, and Gx and Gy the x and y components of the gradient of the static flux times the Green’s function. The neutron noise given by Eq. (2) is, for small amplitude vibrations, a linear function of the displacement components ex and ey, whereas it is an implicit function of the rod position . The determination of the latter, which is much more complicated, can be per­ formed as follows. One selects 3 neutron detectors at positions r., i - 1, 2, 3 with meas­ ured noise signals (to) = 6 <j) (r., co) . For each detector signal an equation of the form of Eq. (2) is applied. Using two of the equations to eliminate ex and ey, this results in a sym­ metric expression where the only unknown parameter is the rod position r which is given as the root of the equation. In reality, however, it is not the Fourier transforms of the stationary random processes 5<t>. (f) which are used, since the defining integral diverges. Instead, auto- and cross power spectra of 6 c|)- (co) must be used, since they are defined as the Fourier transform of the auto- and cross-correlation functions of the corresponding processes. Likewise, instead of ex (co) and ey (to) , the auto- and cross spectra S xx (co) , S (co) and Sxy (co) of the dis ­ placement components need to be used as input source. With application of the Wiener- Khinchin theorem, the auto- and cross spectra of the detector signals can be expressed from Eq. (2) as -3- CTH-RF-124 Control rod localization APSD6^(co) = s_(m) + |G,y(^,(o)| Syy(m) + D‘ (3) 2Re[Gix(Zp> to) Gify Sxy(®)] > CPSD, As Eqs. (3) and (4) show, calculation of the neutron noise, and thus application of any localisation procedure, requires a model of the displacement component spectra S xx (to) , Syy (to) and S Xy (to) and the transfer function G (r, r , to) . The Green’s function will be described separately below. Regarding the displacement spectra, simple expressions for these were derived in Ref. [3] from a model of random pressure fluctuations as driving forces of the rod motion. It was found that the possible variety of displacement component spectra can be parametrized by two variables, an ellipticity (anisotropy) parameter k G [0, 1 ] and the preferred direction of the vibration a G [0,7t] as S xx « 1 +kcos2a (5) Syy « 1 -k cos 2a (6 ) S Xy K ksin2a (7) For an isotropic vibration k — 0 while vibration along a straight line has k = 1. Between these two extreme values the amplitude distribution of the vibration is an ellipse with the main axis lying in the direction a. As Eq. (7) shows, the cross-spectrum of the displacements, and thus all displacement spectra, are real in this model. The same model will be kept also in the present paper.
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