Guidelines to determine the optimal variables for the MTC measurement by noise analysis

Annals of Nuclear Energy 38 (2011) 1924–1929

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Guidelines to determine the optimal variables for the MTC measurement by noise analysis Griet Monteyne a,⇑, Peter Baeten b, Johan Schoukens a a b

Vrije Universiteit Brussel, Department of ELEC, Pleinlaan 2, 1050 Brussel, Belgium SCK-CEN, Belgian Nuclear Research Centre, Boeretang 200, 2400 Mol, Belgium

a r t i c l e

i n f o

Article history: Received 17 February 2011 Received in revised form 4 May 2011 Accepted 5 May 2011 Available online 31 May 2011 Keywords: Moderator temperature coefficient Guidelines Noise analysis

a b s t r a c t In this paper guidelines will be given in order to determine the required measurement time for a specified precision of the Moderator Temperature Coefficient (MTC) estimate by noise analysis. Until now the discussion of the precision of the MTC estimate was neglected. We will study the relation between the precision, the coherence, the amount of temperature sensors and the measurement time. Based on this relation guidelines to determine the optimal measurement time will be given. Simulations in MATLAB will be used to verify the theoretical analysis. Realistic values for the different influencing variables for a specific measurement setup will be given by use of the analysis of a measurement at a Nuclear Power Plant in Belgium. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction The MTC is an important safety parameter in Pressurized Water Reactors (PWRs) that reflects the reactivity feedback of a change in temperature of the water (moderator). It is defined as the change in reactivity (dq) due to a change in the average moderator temper  ature dT amv e as described in the ANSI standard (ANSI, 1997)

MTC ¼

dq dT amv e

ð1Þ

For safety reasons it should always remain negative and not go below a reference value. The application of a noise analysis method to the variations of the temperature and neutron flux would allow estimating the MTC on-line and at any moment during the cycle without perturbation of the reactor core. The most frequently used estimator is based on the division of the Cross and Auto Power Spectral Density (Bendat and Piersol, 1980):

1 CPSD/d/0 ;dT amv e MTC ¼ G0 APSDdT amv e

ð2Þ

In Eq. (2) CPSD denotes the Cross Power Spectral Density, APSD denotes the Auto Power Spectral Density, G0 denotes the open loop transfer function, d/ denotes the neutron flux variation and /0 denotes the static flux. ⇑ Corresponding author. Tel.: +32 2 6292949; fax: +32 2 6292850. E-mail address: [email protected] (G. Monteyne). 0306-4549/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.anucene.2011.05.004

Until now all attention went to the discussion of the bias of the MTC estimate (Demazière and Pázsit, 2002). It was shown that the underestimation of the MTC is mainly due to the radially loosely coupled character of the moderator temperature noise throughout the core. A new estimator was proposed and demonstrated to correct for this bias. This estimator requires the knowledge of the core average moderator temperature noise. The discussion of the precision of the estimate was neglected. However, so far the MTC was never experimentally estimated in nominal operating conditions with the required uncertainty, for example, with an uncertainty of 5%. That is why this article will discuss the relation between the precision of the MTC estimate by noise analysis, the measurement time and the coherence. In Section 2 the theoretical analysis will be given. Section 3 verifies the results on simulations and Section 4 on experiments. Guidelines for the practical user are given in Section 5.

2. Theoretical analysis In this section we introduce the dominating variables that influence the precision: the amount of measurement blocks, the coherence between the measured signals and the requested frequency resolution. These variables determine the required measurement time in order to obtain the specified precision. A simplified scheme of the MTC measurement setup is represented in Fig. 1. A part dTm of the average temperature variations dT amv e is measured, e.g. only dT 1m is measured, while the other dT im ; i – 1 are not known. These will act as an additional disturbance in the MTC measurement. The neutron flux variations d/ as well as the static neutron flux /0 are measured. These neutron

G. Monteyne et al. / Annals of Nuclear Energy 38 (2011) 1924–1929

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the radial correlation length (Demazière and Pázsit, 2002). It is the limited radial correlation length that mainly causes the presence of different temperature variation regions (the dT im in Fig. 1, where i = 1, . . . , N). We assume that there exist Nr uncorrelated temperature variation regions. Every region has the same volume and APSDdT m . Thus

APSDdT amv e ¼ Nr APSDdT m Fig. 1. Simplified scheme of MTC measurement setup.

Eqs. (5)–(7) result in the following relation:

flux variations are considered to be a sum of the neutron flux variations caused by the average temperature variations dT amv e and the neutron flux variations coming from other sources d/n//0. This scheme does not consider measurement noise nor feedback. In order to simplify the analysis we assume that the reactor behaves in a point kinetic manner. Thus, the reactivity can be expressed as a function of the relative neutron flux variation as follows:

dq ¼

1 d/  G0 /0

ð3Þ

Up to now the non-parametric estimation of the Frequency Response Function (FRF) is the most commonly used method to estimate the MTC by noise analysis (see Eq. (2)). The estimator divides the CPSD of the neutron flux and temperature variations by the APSD of the temperature variations multiplied by the open loop transfer function. 2.1. The major players 2.1.1. The coherence The standard deviation depends on the coherence between the measured signals. The coherence reflects how much of the output power is linearly related to the input power. When the complete output signal (the neutron flux variations) is linearly related to the measured input signal (the temperature variations) the coherence will be unity. When there is no linear relation at all between input and output the coherence will be zero. Clearly, the coherence should be as close as possible to unity in order to have an accurate measurement since the MTC reflects a linear relationship between the input and output. The coherence between the temperature and neutron flux variations can be calculated as follows:

c2 ¼

 2   CPSD d/ av e   /0 ;dT m 

ð4Þ

APSDdT amv e APSD d/

ð7Þ

c2 ¼

1 Nr

ð8Þ

The amount of uncorrelated temperature variation regions is determined by the spatial temperature correlation length. The shorter the spatial temperature correlation length is, the larger the amount of uncorrelated temperature variation regions will be and the lower the coherence will be. If we neglect the axial dependence of the temperature variations and consider that the radial temperature correlation length is equal to the diameter divided by three (this is probably an overestimation), then there exist nine independent temperature regions (see Fig. 2). This corresponds with N = 9 in Fig. 1. This leads to a coherence <0.2 when only one temperature sensor is used. When four temperature sensors are used we expect a coherence <0.5 between the neutron flux and the sum of the temperature signals, since we measure less than half of the present temperature variations. 2.1.3. The amount of measurement blocks The amount of measurement datapblocks M influences the stanffiffiffiffiffi dard deviation of the result ðr  1= MÞ due to the fact that the CPSD and APSD are estimated by making averages over M blocks (Bendat and Piersol, 1980). The complete measurement record is split into M equally long measurement blocks. For each measurement block the contributions to the CPSD and APSD are calculated. Then these values are averaged over the M measurement blocks (see Eqs. (9) and (10)). M 1 X d/½k ðxÞdT amv e½k ðxÞ M k¼1 /0

ð9Þ

M 1 X dT av e½k ðxÞdT amv e½k ðxÞ M k¼1 m

ð10Þ

CPSD d/ ;dT av e ¼ /0

m

APSDdT amv e ¼

av e½k where dT m ðxÞ denotes the discrete Fourier transform of the kth measurement block of the temperature.

/0

2.1.2. The amount of temperature sensors and the spatial temperature correlation length The coherence between the temperature and neutron flux variations is a function of the number of temperature sensors that is used and of the temperature correlation length (the number N in Fig. 1). The coherence can be calculated as follows (Bendat and Piersol, 1980):

c2 ¼ jMTC  G0 j2

APSDdT m APSD d/

ð5Þ

/0

where dTm denotes the measured temperature in one region. When there is no external noise present, the following relation exists:

APSD d/ ¼ jMTC  G0 j2 APSDdT amv e /0

ð6Þ

We assume that the axial temperature correlation length is infinite, since the axial correlation length is known to be much larger than

Fig. 2. Amount of uncorrelated regions in reactor with a correlation length equal to the diameter divided by 3.

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2.1.4. The frequency resolution As known, the measurement time determines the frequency resolution. When you split the complete measurement record into M measurement blocks, the frequency resolution will be determined by the length of one measurement block. The following formula gives the relation between the complete measurement time, the amount of measurement blocks and the requested frequency resolution:

T measure ¼

M fres

ð11Þ

where Tmeasure denotes the measurement time in s and fres denotes the user required frequency resolution in Hz. Since there are only useful signals present at very low frequencies in an operational power plant (up to 0.2 Hz) we expect to require a frequency resolution of 0.05 Hz. One measurement block thus corresponds with 20 s.

2.2. The precision of the MTC estimate The relative standard deviation of the estimated MTC, namely

e(MTC) can be calculated as follows (Pintelon and Schoukens, 2001):

eðMTCÞ ¼

rMTC jMTCj

¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  c2 pffiffiffiffiffi jcj M

ð12Þ

In this equation rMTC represents the one sigma standard deviation. When the coherence is low, the standard deviation of the estimate increases. A graphical presentation of this formula is shown in Fig. 3. Using the calculated coherence and the desired standard deviation of the MTC estimate, one can calculate the required number of measurement blocks by use of Eq. (12) or Fig. 3. For example, to achieve a standard deviation of 10% when the coherence equals 0.2, the measurement should contain 400 measurement blocks. This corresponds to a measurement time of 2 h 10 min if the requested frequency resolution is equal to 0.05 Hz. A lower coherence and a smaller standard deviation will require more measurement blocks. For example, if the coherence equals 0.1 and we want to obtain a standard deviation of 1% then we need to measure 90.000 blocks. This corresponds to a measurement time of more than 20 days. Clearly, low coherence between signals and high precision demands can lead to unrealistic measurement times.

3. Verification by simulations In the previous section we introduced the dominating variables that influence the precision and showed that it is possible to theoretically predict the required measurement time. This is done by calculating the coherence between the signals and taking the requested frequency resolution and precision of the estimate into account. The results of the previous section are tested on simulations. 3.1. Simulation setup The simulations are based on the analytical Green’s function in one dimension and with one energy group that was derived in Pázsit (2007). This Green’s function describes how neutron absorption cross section variations result in neutron flux variations. Since we used a one dimensional model in the radial direction of the reactor core, axial effects are neglected. This choice was made for its simplicity and easiness of interpretation. First the temperature variations were simulated at 41 radial positions. We considered a reactor core radius of 1.5 m and a spatial temperature correlation length of one tenth of the reactor radius. The standard deviation of the temperature variations was chosen to be equal to 0.1 °C. No spatial dependence of the standard deviation was considered. The 41 temperature variations were low-pass filtered in order to simulate the time correlation of the temperature variations. Since the low-pass filter removes the high frequency contributions of the signal, the temperature variation at a certain time instant will be correlated with the temperature variation before this time instant. Starting from these 41 temperature variations, we calculated the neutron absorption cross section variations, by use of the following formula:

dRa ðrÞ ¼ MTC  m  Rf  dTðrÞ

ð13Þ

where m denotes the average number of neutrons emitted by fission, Rf denotes the macroscopic fission cross section and dRa(r) denotes the variation of the absorption cross section at radial position r. Next, we used these cross section variations to calculate the neutron flux variations based on the Green’s function as described in Pázsit (2007). The Green’s function is a function of the open loop transfer function G0 (see Eqs. (14) and (15)).

8 <  sin½BðxÞðaþri Þsin½BðxÞðarj Þ if ri 6 rj BðxÞsinð2BðxÞaÞ

Gðr i ; r j ; xÞ ¼

:  sin½BðxÞðari Þsin½BðxÞðaj þrÞ if r > r i j BðxÞsinð2BðxÞaÞ

ð14Þ

With

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 BðxÞ ¼ B0  1  q 1  G 0 ð xÞ

ð15Þ

In Eqs. (14) and (15), a denotes the radius of the reactor, rj denotes the position of the temperature variation in the reactor, ri denotes the position of the resulting neutron flux variation, B0 denotes the geometrical buckling parameter, q1 denotes the reactivity and G0(x) denotes the open loop transfer function as a function of the frequency x. The Green function was then used to calculate the neutron flux variations as follows:

d/ðri ; xÞ ¼ Fig. 3. Amount of measurement blocks as a function of the coherence.

Z

a

a

Gðr i ; rj ; xÞ  Sðr j ; xÞdrj

ð16Þ

G. Monteyne et al. / Annals of Nuclear Energy 38 (2011) 1924–1929

With

/0 ðr j Þ  dRa ðrj ; xÞ D r  p j /0 ðr j Þ ¼ cos 2a

Sðrj ; xÞ ¼

ð17Þ ð18Þ

In Eqs. (16)–(18), S(rj, x) denotes the fluctuation source term, /0 denotes the static flux and D denotes the neutron diffusion coefficient. The frequency dependence of the Green’s function was neglected by simplifying the open loop transfer function as follows:

G0 ¼

1 b

ð19Þ

where b denotes the delayed neutron fraction. Using Eqs. (13)–(19) we calculated the neutron flux variations starting from 41 temperature variations. The spatial dependence of the Green’s function translates in this way the spatial dependence of temperature variations into neutron flux variations. Since we neglected the frequency dependence of the Green function in the simulation, we evaluated MTC at only one frequency. 3.2. Simulation results The previous paragraphs described how we simulated the temperature variations and the resulting neutron flux variations at 41 radial positions. From these simulated data we estimated MTC by use of the non-parametric estimator (see Eq. (2)) that was analyzed in Section 2. We assumed that we have only access to the signal coming from one single temperature sensor and one single neutron flux sensor located at exactly the same radial position. First we calculated the coherence between the simulated temperature and neutron flux signals. As mentioned in Section 2, the precision increases with the coherence of the measured signals. In Fig. 4 the coherence between a single temperature measurement and a single neutron flux measurement as function of the measurement position is plotted. We used exactly the same measurement position for the temperature variations as for the neutron flux variations. We plotted the estimate only in one half of the reactor since the simulated reactor core is symmetric. The coherence varies between 0 and 0.3. It is never equal to 1 since we measure the signal coming from a single local temperature sensor and a single neutron flux sensor. These signals are not 100% linearly related since the neutron flux signal also contains contributions of uncorrelated temperature variations at other positions in the reactor.

Fig. 4. Coherence and 2r standard deviation as function of the measurement position.

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In order to increase the coherence one can try to measure all the temperature variations in the different regions. In this way one would measure all the input signals leading to the measured neutron flux signal. In the next step we used the calculated coherence and the amount of measurement blocks (M = 256) to calculate the predicted standard deviation (see Eq. (12)). In Fig. 5 the predicted and simulated relative standard deviation of the estimated MTC are plotted as function of the measurement position. The simulated standard deviation was calculated by taking the relative standard deviation of 100 simulated MTC estimates. It can be seen that the predicted and the simulated relative standard deviation agree very well. This confirms that we can predict the relative standard deviation by use of the calculated coherence and the amount of measurement blocks. This also means that we can predict the required amount of measurement blocks in order to achieve a certain standard deviation when we know the coherence (see Fig. 3). 4. Experimental verification In order to have an idea of the coherence and thus the required measurement time, we also analyzed measurements coming from the Nuclear Power Plant Doel (Belgium), Unit 4. On 18 December 2008, six temperature sensors and eight ex-core ionization chambers were measured during more than 12 h (see Fig. 6). The temperature sensors (PT200) were situated in each of the three bypasses of the cold respectively hot legs. The traveling time of the water is indicated on Fig. 6. The eight ionization chambers

Fig. 5. Predicted and simulated relative 1r standard deviation as function of the measurement position.

Fig. 6. Simplified scheme of experimental setup.

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G. Monteyne et al. / Annals of Nuclear Energy 38 (2011) 1924–1929

that low coherences can result in unrealistic measurement times during which the reactor has to stay stationary. 5. Guidelines for the user In this section guidelines are given in order to calculate the required measurement time for a certain demand on the precision of the estimate.

Fig. 7. Coherence between 1 cold leg temperature measurement and 1 ionization chamber measurement.

a. Calculate coherence By use of the temperature variation signal and the relative neutron flux variation signal, the coherence can be calculated by use of Eq. (4). b. Determine the required frequency resolution and precision It will be seen that the coherence is the largest at frequencies smaller than 1 Hz. The higher the required frequency resolution, the longer the required measurement time will be (see Eq. (11)). However a good precision also requires a high amount of measurement blocks and thus a long measurement time (see Eqs. (11) and (12)). One should make a trade-off between the uncertainty on the MTC estimate and the frequency resolution in order to make the required measurement time as short as possible. c. Calculate the measurement time Once the frequency resolution, the coherence and required precision are known, one can calculate the required measurement time by use of Eq. (12). 6. Conclusion

Fig. 8. Amplitude MTC estimate and calculated 2r standard deviation.

were measured on four different radial positions. On each radial position one ionization chamber is positioned at the level of the upper half of the reactor core and the second is positioned at the level of the lower half of the reactor core. In Fig. 7 the coherence between 1 cold leg temperature measurement and 1 ionization chamber measurement is plotted. The coherence is the highest for frequencies smaller than 0.1 Hz and never exceeds 0.1. By use of Fig. 3 we can estimate the amount of measurement blocks in order to obtain a certain standard deviation. Since there is only a statistical significant coherence at low frequencies, a frequency resolution of 0.05 Hz is required. By use of the amount of measurement blocks (640) and the coherence we calculated the standard deviation of the MTC estimate (see Eq. (12)). In Fig. 8 the MTC estimate and the 2r standard deviation is plotted. For the estimation we used 640 measurement blocks of more than 80 s. For frequencies higher than 0.1 Hz the predicted standard deviation is very uncertain. Thus, only the MTC estimates for frequencies smaller than 0.1 Hz are meaningful. In that region the two standard deviation is larger than 24%. A lower demand on the frequency resolution would lead to a more precise result. Thus, one should make a trade-off between the different demands. The coherence can be used to calculate the required measurement time in order to obtain a certain precision. The coherence is in the range of 0.1 at a frequency lower than 0.1 Hz. This results in a measurement time of more than 9 days if we want to achieve a standard deviation of 1%. A standard deviation of 5% can be achieved with a measurement time larger than 11 h. This indicates

Guidelines were given in order to estimate the required measurement time for a specified precision of the Moderator Temperature Coefficient (MTC) estimate by noise analysis. The relation between the precision, the coherence, the amount of temperature sensors and the measurement time was studied. Simulations in MATLAB and measurements at a Nuclear Power Plant in Belgium were used to verify the theoretical analysis. It was explained that a low coherence results in a large standard deviation. The standard deviation can be decreased by increasing the measurement time. The measurement at a Nuclear Power Plant in Belgium showed that in practice we expect a coherence of 0.1 at a frequency lower than 0.1 Hz. This results in a measurement time of more than 9 days if we want to achieve a standard deviation of 1%. A standard deviation of 5% can be achieved with a measurement time larger than 11 h. This indicates that low coherences can result in unrealistic measurement times during which the reactor has to stay stationary. Acknowledgements The authors thank everyone who contributed to this work. More specifically we thank KC Doel, Laborelec and Tractebel Engineering GDF Suez for their help with the realization and interpretation of the noise measurements at the Doel 4 PWR. This research was part of a project defined by Tractebel Engineering and was supported by GDF Suez in the framework of the GDF-Suez/ SCK-CEN collaboration agreement. This work is sponsored by the Fund for Scientific Research (FWO-Vlaanderen), the Flemish Government (Methusalem), and the Belgian Federal Government (IUAP VI/4). References ANSI, 1997. Calculation and Measurement of the Moderator Temperature Coefficient of Reactivity for Water Moderated Power Reactors, An American National Standard. American Nuclear Society, ANSI/ANS-19.11-1997.

G. Monteyne et al. / Annals of Nuclear Energy 38 (2011) 1924–1929 Bendat, J., Piersol, A., 1980. Engineering Applications of Correlation and Spectral Analysis. A Wiley-Interscience Publication, New York, USA. Demazière, C., Pázsit, I., 2002. Theoretical investigation of the MTC noise estimate in 1-d homogeneous systems. Annals of Nuclear Energy 29, 75–100.

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Pázsit, I., 2007. Transport Theory and Stochastic Processes. Lecture Notes, Chalmers University of Technology, Göteborg, Sweden. Pintelon, R., Schoukens, J., 2001. System Identification. A Frequency Domain Approach. IEEE Press, Piscataway, USA.