reactor noise analysis based on the singular value decomposition (svd

Pergamon Ann. Nucl. Energy, Vol. 25, No. 12, pp. 907421, 1998

© 1998 Published by Elsevier Science Ltd. All rights reserved P I I : S0306-4549(97)00090-X Printed in Great Britain

0306-4549/98 $19.00 + 0.00

REACTOR NOISE ANALYSIS BASED ON THE SINGULAR VALUE DECOMPOSITION (SVD)

J. NAVARRO-ESBRi, l G. VERDO, l* D. G I N E S T A R 2 and J .L. M U I ~ O Z - C O B O 1

1Departamento de Ingenieria Quimica y Nuclear, Universidad Politrcnica de Valencia, Camino de Vera S/N, 46071 Valencia, Spain

2Departamento de Matemfitica Aplicada, Universidad Politrcnica de valencia, Camino de Vera S/N, 46071 Valencia, Spain

(Received 4 August 1997)

Abstract--This paper reviews different techniques to analyze BWR's stability regime from neutronic power signals, and alternative methodologies based on the singular value decomposition (SVD) of a given matrix are proposed. The results obtained from experimental signals using two different constructions of the embedding space of the system have been compared. SVD-based AR models have been studied and the results obtained with these methods have been compared with those of a standard AR model for short neutronic power signals. © 1998 Published by Elsevier Science Ltd. All rights reserved

1. INTRODUCTION

Recently, a lot of work has been devoted to the study of BWR's stability regime from the neutronic power signals monitored in the plant. These studies generally follow two kind of approaches. One approach where the phase space of the system is reconstructed and properties of its dynamics are obtained from invariant quantities as the Lyapunov exponents spectrum (Pereira, 1992; Verdfi, 1997) or dimension definitions associated with the attractor of the system (Suzudo, 1993). The other approach considers only a scalar variable of the system dynamics and computes the decay ratio (DR), a parameter which characterizes quantitatively the stability of the system. This parameter is either obtained directly from the autocorrelation function of the signal, from the impulse response of the system (Van der Hagen, 1994) or from the effective transfer function of the system (Kishida, 1990; Sanchis, 1995) calculated fitting an autoregressive model to the signal. Takens and Mafir's theorem (Mafir, 1981; Takens, 1981) asserts that it is possible to reconstruct the phase space of a dynamical system from the observations of an associated

*Author for correspondence. Fax: 34-6-3877639; e-mail: [email protected]

907

908 J. Navarro-Esbri et al.

single scalar variable. In this paper, we review two different methods to obtain the embedding space of a system from short and noisy sets of data: the former one, known as the time delay method (Abarbanel, 1993), and the latter one, based on a projection of the information matrix of the signal using a singular value decomposition technique (Broomhead, 1986). To compare the performance of these methods, we have used the same dynamics reconstruction technique, based on a global fit of the dynamical system by means of orthonormal polynomials (Verdfi, 1997).

We also review the methodology to study the stability of a BWR reactor based on an autoregressive model of the system, and propose a method to calculate the AR parameters based on SVD techniques. The results obtained from experimental neutronic signals using this methodology are compared with the ones obtained with a standard AR model.

The decay ratio and the fundamental oscillation frequency are used as the system parameters to compare the performance of each method. Nevertheless, we have to note that the D R is a parameter defined for a second order linear system, being only in this case a stable measurement of the damping of the system. For nonlinear systems such as nuclear reactors, this concept is not clearly defined (Suzudo, 1993); there exist many definitions and ways to calculate it (Van der Hagen, 1994; Sanchis, 1995). Nevertheless, this parameter is widely used for stability analysis in the linear regime of the reactor.

The paper is organized as follows. In Section 2, we review two dynamics reconstruction techniques, the time delay method and the SVD projection, using a global fit over the whole trajectory of the dynamical system by means of orthonormal polynomials. In Sec- tion 3, we present the SVD based AR model. The numerical results for experimental neutronic signals corresponding to APRM measurements, used for a Stability Benchmark (Lefvert, 1996), are shown in Section 4. The main conclusions of the paper are summar- ized in Section 5.

2. T E C H N I Q U E S F O R T H E R E C O N S T R U C T I O N O F T H E SYSTEM DYNAMICS

Consider a dynamical system formally as:

d)( ~ _ F(X) (1)

dt

where each ) (= (xl, X2 . . . . ) represents a state of the system and may be thought of as a point in a suitably defined space, the phase space of the system.

We will briefly describe the methodologies for reconstructing the system phase space having only information on scalar measurements of one of the variables of the system, x ( n ) = x ( t o + nr,), where rs is the sampling period. In order to simplify the notation, we assume that time is measured in sampling period units, thus, the scalar set we know from system dynamics can be represented by:

x ( n ) , n = 1 . . . . . N .

Thus, from this set of data, we will discuss two different techniques to obtain the space known as the embedding space, which is diffeomorfic to the phase space of the system.

Reactor noise analysis based on the singular value decomposition (SVD) 909

2.1. Time delay method

In accordance with Takens' theorem (Takens, 1981), we can use time delay methods to construct the embedding space of the system using the time series x(n). In order to do this, we will obtain a matrix, X, called the embedding matrix, whose rows consists of d- dimensional vectors of the form:

.~(n) ---- Ix(n), x(n + 7) . . . . . x(n + (d - 1 ) 7)]

where T is an integer called the delay time lag and d is the dimension of the embedding space.

From the theoretical point of view, the reconstruction is general for every T, and for every d such that d > 2dA + 1, where dA is the dimension of the attractor of the system. But from the practical point of view (Abarbanel, 1993), to obtain an appropriate reconstruction, we have to face two fundamental questions. The election of the dimension, d, of the embedding space, and the problem of choosing the time lag, T.

2.1.1. Choosing the embedding dimension

The criterion used for choosing the embedding dimension is the method of the nearest false neighbors (Abarbanel, 1993). We select the embedding dimension as d, if in a d- dimensional embedding space the number of false neighbors for all the points of the signal falls under 1% of the number of samples considered for the signal.

2.1.2. Choosing the time lag

We have considered a criterion based on the average mutual information function (Fraser, 1989), defined as:

u , F e ( ~ ( j ) , ~(j + 7)) ], Id(7) = Z P (x (J ) 'Y (J +

j=l (2)

where ~ ( j ) is the d-dimensional delay vector reconstructed in the point j, P(£(/) ,£(j)) is the joint probability density of £(i) and £ ( j ) , and P(Yc(j)) is the probability density of vector £( j ) , (Fraser, 1989).

Fraser (Fraser, 1989a), gives a recursive and higher dimensional generalized algorithm to evaluate la (T), but he notes that the number of samples required for the algorithm to converge in a typical case in dimension 3 or 4 can be of millions. For this reason, Abar- banel (Abarbanel, 1993) chooses 11 (T) as a tool to obtain a useful time lag, T, suggesting as a prescription to choose this lag as the T where the first minimum of Ii (T) occurs. This last criterion being the one used in this paper. For more details of the selection methods see Abarbanel (1993) and Verdfi (1997).

2.2. SVD projection

D.S. Broomhead and P. King (Broomhead, 1986) introduced a SVD approach alternative to the time delay method to obtain a good reconstruction of the system phase space.


This method is based on the construction of an information matrix, A, whose rows are built by the application of a nw-window to the time series, x(n), of N samples, resulting in a matrix of the form:

A ----

x(1) x(2) X(nw) ] x(2) x(3) x ( n , + l )

J x ( N - (n,,.- 1)) x ( N - nw) x(N)

(3)

where the window length, nw, is a parameter to be chosen. The embedding space is achieved by a projection of the information matrix onto a subspace, whose basis and dimension also must be selected.

In order to choose the dimension of the projected subspace and the basis for the projection, we will use the singular value decomposition as main tool. The information matrix, A, admits a decomposition of the form:

A = U. E . V 7 (4)

where:

U = [1/1, I/2 . . . . . blq] 6T~qxq

V = [121, 1) 2 . . . . . 1)n.]6T~ n"xn"

E = diag(aj, a2 . . . . . a , . ) ~ q x " "

being q = N-n , , + 1, az the singular values of A, ui and vi the ith left and the ith right singular vectors associated with the singular value a~.

Here, we will select as the dimension of the projection subspace the number of significant singular values of the matrix A and the basis of this subspace will be the corresponding singular vectors.

In absence of noise, the dimension of the subspace containing the embedded manifold would be the rank of the covariance matrix of the signal, defined as E = ATA (Broomhead, 1986). However, due to the noise present in the measurements, E is a full rank matrix. We must note that the singular values of the information matrix, A, are the positive square roots of the covariance matrix singular values. Thus, the noise causes all the singular values of the information matrix to be non-zero, and we need a method to distinguish the fundamental information subspace from the noise subspace.

In the simple case of white noise, the existence of a non-zero constant background or noise floor is a notable characteristic which can be used to determine the fundamental component of the signal. But in experimental observations we will find more difficulties due mainly to the nature of the noise taking part of the neutronic signals. So, in practice we have to face the selection of the fundamental subspace dimension in order to avoid the subspace associated with the noise and to obtain an appropriate embedding space. Moreover, there is the mentioned problem of choosing the window length, nw, of the information matrix, A.


2.2.1. Choosing the window length (nw)

The criterion used to choose the window length of the information matrix is based on the analysis of the Singular Value Ratio (SVR) spectrum (Kanjilal and Palit, 1995). Thus, we define a new matrix from the time series data, x(n), in the following form:

y =

x(1) x(2) X(nw) x(nw+l) X(nw+2) x(2nw)

J x ( ( k - 1)nw+l) x ( ( k - 1)nw+2) x(knw)

being k the integer part of the ratio N/nw. We also define the parameter r, called SVR, as the ratio between the two largest sin-

gular values of the data matrix, Y, trl/tr2. This parameter has been used for detecting the periodic components of periodic or almost periodic signals, occurring high values of r at the actual period length as well as its higher multiples (Kanjilal and Palit, 1995).

We will use a similar methodology, taking advantage of the SVR spectrum. In this way, we will construct the data matrix Y with window lengths of nw = 2,3 .... , plotting the SVR spectrum (SVR versus nw), and choosing the window length of the information matrix, nw, as the nw where r takes the first significant maximum, discarding the initial range where the SVR decreases monotically (see Fig. 1).

nw times rs can be defined as the effective period length of the signal. For the case shown in Fig. 1, the window length taken has been 26. We have to note that in some of the signals analyzed we have found difficulties in selecting this maximum because of the irregular shape of the spectrum produced by this kind of signal, but in general, the election of the appropriate window length is quite clear.

2.2.2. Choosing the projection subspace dimension

We deal with experimental neutronic power signals where the neutronic oscillation is masked among other noisy contributions. In this way, the noise masks the signal component useful for the stability analysis. So, by choosing the appropriate dimension for the

3

2.5

~- 2

1.5

10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

r l w

Fig. 1. APRM3 signal (Cycle 24 of Ringhals NPP) SVR spectrum.

912 J. Navarro-Esbrl et al.

projection we will be able to obtain an appropriate subspace associated with the fundamental information of the signal.

There is not a clear criterion to select this dimension. There are some criteria based on the assumption that the smallest singular values can be identified as noise dominated, even when the noise is not white. This kind of method has the inconvenience that it is not always possible to measure the noise separately from the signal to select the magnitude of a singular value to be associated with the noise dynamics. For this reason, there is a general interest in the signal processing community in developing techniques where the data itself may be used to estimate the noise level.

In the following, we propose a criterion to select the dominant singular values of the information matrix constructed with the appropriate window length. As we can see in Fig. 2, for a typical example the first singular values are one or two orders of magnitude bigger than the rest of them. The smallest singular values are non-significant - versus the first ones, so in order to select the dominant singular values associated to the fundamental information, we propose a criterion based on the ratio between every two consecutive singular values, ai/ai+]. Taking as dimension, d, of the reconstructed phase space of the system the value of i where this ratio produces a value higher than 2, starting at the end of the spectrum (see Fig. 3). However, it is also possible to use the criterion based on the nearest false neighbors, exposed before, in order to calculate the embedding dimension, yielding similar results to the ones obtained with the first criterion.

Once we have chosen the values of the two parameters n,. and d, we perform the following projection:

d A d = Z uk" c~k. v[ (5)

k=l

selecting as embedding space of the system the first d columns of the matrix A a.

2.3. Global f i t by means of orthonormal polynomials'. Lyapunov exponents

As the information we have of the system is given by a sampled signak we assume that its dynamics can be approximated by a discrete map:

160 140 120 ~ % ~

"~ 100 80

{ 60 U3 40

20

1 3 5 7 9 11 13 15 17 19 21 23 25 i

Fig. 2. Singular values of the information matrix, A from APRM3 signal.


2.5

2

1.5

1

0.5

0 1 3 5 7 9 11 13 15 17 19 21 23 25

i

Fig. 3. Spectrum of ratios (tTi/Cri+ 1) from APRM3 signal.

~(n+ l) =~(£(n)). (6)

We want to find the global complex Lyapunov exponents of the system (Verdfi, 1997) as a consistent average of all local complex Lyapunov exponents, evaluated on each point of the attractor along the trajectory, in order to obtain the fundamental oscillation frequency and the DR parameter, defined as:

Re(~.) D R = exp (2zr ~ )

where k is the dominant complex Lyapunov exponent. We will suppose that the map (6) is unknown but, nevertheless, the function ~ can be

approximately reconstructed as an expansion in terms of orthogonal polynomials on the attractor of the system (Giona, 1991; Brown, 1993). To construct these polynomials, we use the Gram-Schmidt procedure (Verdfi, 1997). Denoting the orthonormal polynomials by Jr (/), we approximate each component of the map function, .~, as:

npt,...,npa

)vj = Z f~/) zre) (7) il ,-..,/ 'd=0

where npj is the maximum degree in xj considered for the polynomials and (/) = (/j, ..., it). Thus, knowing the reconstruction of the phase space of the dynamical system and the

orthonormal polynomials on the attractor, it is possible to obtain an approximation for the discrete dynamical system (6) (Giona, 1991; Brown, 1993). And from the approximation of each component of the discrete map, jrj, we obtain the Jacobian matrix of the function on each point of the attractor, £(n).

In order to obtain the global complex Lyapunov exponents, we recover the local complex Lyapunov exponents as the neperian logarithms of the eigenvalues of the Jacobian matrix at each point of the trajectory, and we finally obtain the global complex Lyapunov exponents as a consistent average of all local complex Lyapunov exponents.

The main disadvantage of this method is that it is limited to low dimensional embedding spaces (spaces of dimension lower than 5 or 6). We have tried to develop this


methodology for higher dimensions but the CPU time spent for the calculations makes it prohibitive.

3. SVD-AR MODEL

The AR model method offers many advantages, mainly the simpler linear relationship in the time domain between the signal at time n and its past values up to a specified number p (AR order). The time series x(n) is said to be an AR process of order p, if it is generated according to the following relationship:

P

x ( n ) + y ~ a k x ( n -- k ) = b o e ( n ) k=l

(8)

where e(n) is a normalized white-noise process. The autocorrelation function of the process is:

rx(m) = lim 1 M - ~ 2 M + 1 x(k - m)x(k)*. k= M

(9)

In practice, only a finite number of points for the signal are available. So, the autocorrelation sequence is rarely known, and must be estimated with the finite data. Assum- ing N data samples, a discrete-time autocorrelation estimate for the signal x(n) (Oppenheim, 1989) will have the following form:

i N - m

- - Z x ( k - m)x(k)* 0 < m _< N - 1 (10) rx(m) - N - m k=l

where rx ( - m ) = rx (m). Using the Yule-Walker equations it is easy to achieve the following relationship (Cad-

zow, 1982) between the AR(p) autocorrelation elements:

r { r x ( n ) + Z a k r x ( n - - k ) = Ib°12 , n = O

k=l 0 , n > 1 (ll)

using the matrix notation:

rx(O) l F rx(-1) + [ x(O)

r~)J Lr~(p'- I)

rx(-2) r~(-p) rx(-1) rx(l -p)

rx(p - 2) rx(O)

a l

p

,2


Eliminating the first row of this system, we find that:

rx(O) rx(1) rx(1--p)] [ a l rx(1) rx(O) rx(2--p) , a2

rx(p--1) rx(p--2) rx(O) _] Lap

['rx(l) 7

[.rx(p)J

or more compactly:

R . a = - r .

(12)

Once here, we have to face the problem of choosing the appropriate order p for the AR model. With the proposed methodology, based on SVD, we begin selecting an order p large enough to assure that the real AR model order is smaller than the chosen p.

The solution of system (12) could be obtained as:

a= -(R-1).r.

However, with this procedure we are assuming that the matrix R is invertible. Although it could be the case, the matrix can be ill-conditioned and this way of solving the system would not provide accurate results. So, we will use a k order pseudoinverse of the matrix R to solve the system (12), related with the problem of finding the matrix which best approximates R.

As has been remarked in the literature, (Goulub, 1965; Cadzow, 1982), given a mxn matrix, .4, the unique mxn matrix of rank k<_rank(A) which best approximates .4 in the Frobenious norm sense is given by:

A (k) = U . ~ k . vT

where:

U = [ul, u2 . . . . . u~]ETZ m~m,

V = [Vl, v2 . . . . . Vn]ET~ nxn, Ek = diag(tyl, 02 . . . . . Ok, 0 . . . . . O)ET'~ mxn

being t~i the singular values of A, ui and vi the ith left and the ith right singular vectors associated with the singular value t~i.

Ek is obtained from E (diagonal matrix formed by all the singular values of A) by set- ting to zero all its elements but its k largest singular values.

The degree to which A (k) approximates A is dependent on the sum of the (n - k) smallest singular values squared. As k approaches n, this sum will become progressively smaller and it will eventually go to zero at k--n.

So, in order to solve (12) in a consistent way, we obtain a singular value decomposition of the autocorrelation matrix, R:

R = UR. ER. VR r (13)


where: UR ~ VR ~- [ P l , 192 . . . . . t)p] EJ'~pxp ER = diag(~rl, ~r2 . . . . . crp)~7~ pxp.

We will assume that there are k dominan t singular values, and knowing that R is a real symmetric matrix with UR = VR, we construct a pseudoinverse o f R (Goulub, 1965; Cad- zow, 1982) in the following form:

k ( R ( k ) ) # = ~ - - ~ 7 ~ . . - t', v,~ (14)

i= l

comput ing the parameters vector, a, as:

a = - ( R (k)) # - r. ( 1 5 )

Once we have selected the order p o f the A R model, we must choose the order k of the autocorrela t ion matrix pseudoinverse. In order to do this, the autocorrelat ion matrix singular value decomposi t ion is performed as in (13).

Now, we compute the ratios between every two consecutive singular values in the following form:

ratioi =--°i-I , i = 2, 3, . . . , p. O" i

And we propose a t runcation criterion similar to the one proposed in section 2 to obtain the embedding space dimension, but now dealing with the autocorrela t ion matrix instead o f with the informat ion matrix. In Figs 4 and 5 we have plotted the singular values and the ratios versus i, respectively, for an experimental neutronic power signal. We can observe that the last singular values are very small compared with the first one. So, if we want to obtain a consistent solution o f the system (12), we would choose the largest singular values as the dominan t ones. But, where must we stop? In Fig. 5, we observe that starting at i = p and going to decreasing values o f i the ratios remain almost constant, producing values o f the ratio near 1, but when we arrive to the smallest values o f i these

r~

Z

10

9

8

6 5 4 a

2 1

o 11 21 31

N

41

Fig. 4. Spectrum of singular values of the autocorrelation matrix, R (APRM3).


4

~, 2

1

I i I I I I I I I

2 12 22 32 42

I

Fig. 5. Ratios between every two consecutive singular values of the R (APRM3).

ratios grow above 2. We take as truncation criterion to obtain the k dominant singular values, to select k as the first value of i where ratioi is higher than 2. In this way, this truncated SVD-AR method is quite independent of the order p of the autoregressive model selected to begin with.

The decay ratio (DR) will be obtained using the theoretical model (Sanchis, 1995). This model is based on the following. Once the AR(p) model associated with the signal has been obtained, we have a transfer function of the AR model in the z plane:

H(z) = z-~(ap + a p _ : + . . . + : ) "

The next step consists of computing the roots of equation:

z e + a l z P-l + . . . +ap = 0

choosing the two complex conjugate dominant roots,/~ and/3*, and transforming them into the s plane by means of the following expressions.

a = ~ln(/3), or* = In(/3*).

The theoretical D R will be calculated as:

{ 2~r Re(ct)'~ ORtheor. = exp[k ~ ).

4. NUMERICAL RESULTS

In this section, we study experimental signals corresponding to APRM-mean measurements of the neutronic power at the beginning of Cycle 14 from the Swedish Ringhals 1 BWR reactor. These measurements were part of a Stability Benchmark (Lefvert, 1996).


The duration of each measurement was I I min and the sampling frequency was 12.5 Hz.: so the number of samples recorded were 8250. We must declare our interest in the extraction of information from short and noisy data sets, for this reason, we will only use a subset of the data available.

First, we will compare the results obtained from the techniques for the reconstruction of the system dynamics studied above, the time delay technique and the SVD reconstruction of the embedding space. After this, we will present the obtained results with the SVD-AR approach, comparing them with the ones obtained with a standard AR model.

4.1. Recons truc t ion ~?[the sys t em dynamics

We present the results for the time delay method calculating the embedding dimension and the time lag as explained above. For the SVD projection, we have calculated the optimal window length for the information matrix for each case using the proposed SVR criterion. And finally the embedding space is obtained projecting the information matrix onto a subspace with the appropriate dimension. Three thousand samples were used in all the cases for this first study.

Once the embedding space has been reconstructed, using orthonormal polynomials of degree one to fit the dynamical system, the results for the fundamental oscillation frequency and the DR parameter obtained for each technique are shown in Table 1.

As we can see, the method of time delay produces poor results as it was previously remarked in (Verdfi, 1997). To obtain better results we need to filter the signals or to refine them using a modal decomposition method.

As we have mentioned above, we use the decay ratio to compare the performance of each method. In this way, the SVD projection produces better results than the time delay method, but in some cases (APRM5, APRM6) the relative error suggests that the embedding space needs to be enlarged.

Also, we must point out that the reconstruction of the system dynamics is not the best method to determine the decay ratio of the system. There are better techniques as the one we have explained in Section 3, but nevertheless the study of the dynamic behavior of the original system using the invariants associated to the embedding space is an important subject, mainly when the behavior of the system becomes more nonlinear.

Table I. Numerical results for dynamic reconstruction techniques

Method of time delay SVD projection Reference results

Case Lag Frequency DR dnw Frequency DR Frequency DR (Hz) (Hz) (Hz)

APRM3 8 0.36 0.29(0.58) 526 0 .47 0.68(-0.01) 0.43 0.69 APRM4 8 0.37 0.31(-0.61) 523 0.56 0.77(0.02) 0.55 0.79 APRM5 7 0.47 0.27(-0.60) 5 26 0.54 0.80(0.19) 0.51 0.67 APRM6 6 0.35 0.36(0.44) 524 0 .53 0.79(0.23) 0.52 0.64 APRM8 7 0.50 0.62(0.20) 524 0 .51 0.82(0.05) 0.52 0.78 APRM9 6 0 .53 0.55(-0.37) 6 24 0 .53 0.80(0.00) 0.56 0.80 APRMI0 5 0.48 0.65(0.08) 524 0 .51 0.82(0.15) 0.50 0.71

0: Relative error (DR - DRref)/DRref. DRref is DR from the reference results (Lefvert, 1996).


4.2. S V D - A R modeling

We will assume an order p equal to 50 (Van der Hagen, 1994) to start with for the AR model, taking only 3000 samples in all the cases. In Table 2, we compare the numerical results, solving the system (12) using the pseudoinverse of matrix R, taking as significant all the singular values of the autocorrelation matrix (Full SVD-AR Model) with the results using the truncation criterion (Truncated SVD-AR Model) in order to obtain the low order pseudoinverse, and with the ones obtained using a standard AR model (Wis- consin Multiple Time Series Program Package, WMTS-1). In all the cases the DR parameter has been calculated using the theoretical model.

The DRs of the signals APRM3, APRM5 and APRM6 produced with the Full SVD- AR Model have a large relative error when they are compared with the reference results. But note that the reference results were obtained using 8250 samples, and our calculations only use 3000 samples, and it is known that increasing the number of samples it is possible to improve the results. Otherwise, comparing these results with the ones obtained by the Standard AR Model, we can see that they are practically the same.

Now, if we observe the Truncated SVD-AR results, we can conclude that they are very good. In fact, the pseudoinverse matrix constructed from the k dominant singular values has better properties due to the elimination of the spurious singular values associated with the noise.

To check the influence of the number of samples, we have repeated the calculation taking 8000 samples. In Table 3, we show a comparison of the results obtained by the Truncated SVD-AR Model with the Full SVD-AR Model and with the results obtained in the Stability Benchmark (Lefvert, 1996) for the APRM-mean. We also include the results for the signals associated with the fundamental mode (Verdfi, 1997) calculated with a standard AR model.

The signals analyzed with the SVD-AR models have been the APRM-means pro- vided by Vattenfall (Lefvert, 1996). We notice again that the results obtained using the Full SVD-AR Model are almost the same as the ones obtained with a standard AR model.

All the results are very similar and very close to the reference ones except for the results of the APRM3 signal. In this case we believe that the noise corrupts partially the signal,

Table 2. Numerical results for the AR model

Truncated SVD-AR Full SVD-AR Standard AR Reference Model Model Model results

Case Frequency DR Frequency DR Frequency DR Frequency DR (Hz) (Hz) (Hz) (Hz)

APRM3 0.44 0.61(-0.116) 0.43 0.49(-0.290) 0.43 0.48(-0.30) 0.43 0.69 APRM4 0.55 0.77(-0.025) 0.54 0.74(-0.063) 0.54 0.73(-0.076) 0.55 0.79 APRM5 0.55 0.65(-0.029) 0.54 0.54(-0.194) 0.54 0.53(-0.209) 0.51 0.67 APRM6 0.52 0.72(+0.125) 0.52 0.74(+0.156) 0.52 0.73(+0.14) 0.52 0.64 APRM8 0.50 0.80(+0.025) 0.50 0.79(+0.013) 0.50 0.80(+0.025) 0.52 0.78 APRM9 0.56 0.85(+ 0.062) 0.56 0.85(+ 0.062) 0.56 0.85(+ 0.062) 0.56 0.80 APRMI0 0.49 0.78(+0.098) 0.50 0.76(+0.07) 0.50 0.76(+0.07) 0.50 0.71

O: Relative error (DR-DRref)/DRref.


Table 3. Comparing results taking 8000 samples

Truncated SVD-AR Model (8000 samples)

Full SVD-AR Model (8000 samples)

Standard AR Fundamental Mode

Standard AR (APRM-mean)

Case Frequency D R Frequency D R Frequency D R Frequency D R (Hz) (Hz) (Hz) (Hz)

APRM3 0.43 0.65 0.43 0.58 0.43 0.64 0.43 0.58 APRM4 0.55 0.79 0.55 0.77 0.54 0.77 0.54 0.77 APRM5 0.52 0.64 0.52 0.65 0.52 0.67 0.52 0.67 APRM6 0.51 0.67 0.52 0.68 0.52 0.68 0.52 0.68 APRM8 0.51 0.78 0.51 0.79 0.51 0.79 0.51 0.79 APR9 0.56 0.79 0.56 0.80 0.56 0.80 0.56 0.79 APRM10 0.50 0.72 0.50 0.71 0.50 0.71 0.50 0.71

having a negative influence in the results. Again, the Truncated SVD-AR or the modal decomposition ameliorate the results.

I f we compare now the results produced by theTruncated SVD-AR Model for 3000 and 8000 samples, we notice that they are also very close. This means that with this method, it is possible to extract true information from short and noisy data sets. The reason for this is that the autocorrelation matrix, R, is in essence the covariance matrix ~ (identically, when the number of points is infinite), which is related with the information matrix. Thus, the construction of the pseudoinverse matrix from the dominant singular values is an efficient mechanism to eliminate the noise present in the signal and to separate the t rue

information subspace from the subspace mainly associated with the noise.

5. C O N C L U S I O N S

This paper presents several methods based on the SVD technique to analyze neutronic power signals, as alternative to the standard methods. Two different techniques to reconstruct the embedding space associated with the dynamics of the reactor have been compared, the time delay method and the reconstruction based on the singular value decomposition of the information matrix of the signal. It has been observed that the SVD technique gives better results for the analysis of the experimental signals used in the study. This is because this method has been proven successful extracting the appropriate information from the signal. However, care has to be taken in the determination of the window length used to construct the information matrix in view of its large influence on the results. The best results for the stability analysis have been achieved when as window length the effective period length, calculated with the SVR criteron, has been used and are those presented in the paper. What remains as an open question for study is the embedding space reconstruction in higher dimensions performing local fits of the dynamics to obtain the dominant Lyapunov exponents of the system.

Two SVD-AR models for the neutronic power signals have been proven successful, the Full SVD-AR Model and the Truncated SVD-AR Model. It has been checked that the selected AR model order, p, has a weak Influence on the results, whenever it is chosen higher than the effective period length of the system (Van der Hagen, 1994). The Full SVD-AR Model gives similar results to those obtained with a standard AR model. The


results obtained with the Truncated SVD-AR Model using either 3000 or 8000 samples have been very close to the ones given as a reference. It is then concluded that the Truncated SVD-AR Model affords a useful method when only short and noisy data sets are available.

Acknowledgement--The authors are indebted to the DGCYT for the financial support received from project number PB94-0537.

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reactor noise analysis based on the singular value decomposition (svd

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