Statistical testing of temperature fluctuations for estimating thermal power in central subassembly of fast reactor

Annals of Nuclear Energy 60 (2013) 406–411

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Statistical testing of temperature fluctuations for estimating thermal power in central subassembly of fast reactor Paawan Sharma ⇑, N. Murali, T. Jayakumar Indira Gandhi Centre for Atomic Research, Kalpakkam 603102, India

a r t i c l e

i n f o

Article history: Received 26 December 2012 Received in revised form 5 April 2013 Accepted 9 April 2013 Available online 19 June 2013 Keywords: Temperature fluctuations Fast breeder reactor KPSS Reverse Arrangement Test (RAT) Runs test Approximate Entropy (ApEn)

a b s t r a c t This paper reports results on the use of various statistical tests for studying the characteristic of temperature fluctuations in central subassembly of the core in a fast breeder test reactor.These tests are useful in establishing a correlation between core thermal power and fluctuations. Tests such as KPSS, Reverse Arrangement Test and runs test are used here to quantify the stochasticity of temperature fluctuations. The use of Approximate Entropy (ApEn) is also highlighted as a measure of complexity. Finally, a model is proposed on the basis of findings of above tests to establish the correlation. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction

2. Statistical tests

Systems in Nuclear reactors must follow stringent norms for their operation. One such requirement is that all safety critical systems in Nuclear Power Plants (NPPs) must be validated using diverse methods. Power measurement in fast reactors is carried out by neutronic methods and calibrated by the core temperature readings and thermal balance. This calibration is required to correct the slight difference in reactor power as estimated by neutronics with that of actual core thermal power. Hence, a diverse system is required to validate the power estimation. The core of Fast Breeder Test Reactor (FBTR) consists of 65 fuel subassemblies with 61 pins per subassembly (Srinivasan et al., 2006). The innermost subassembly known as the central subassembly (CSA), exhibits maximum temperature due to the highest value of neutron spectrum at the center. The present work aims to study the temperature fluctuations in CSA, by performing statistical tests to quantify stochasticity, and to correlate them with actual reactor thermal power. Simple frame statistics is less useful in case of CSA data.The proposed correlation can be utilized as a diverse method to validate the estimated reactor power.

Historically, statistical tests were primarily focused on economic analysis. Recently, they are being used as a tool to analyze real time signals. These tests mainly include KPSS test, Reverse Arrangement Test (RAT) and runs test. Several works such as Brcich and Iskander (2006), Weber et al. (2007), Kay (2008), Bilodeau et al. (1997), and Thexton (1996) have used these tests in signal processing domain. Simple frame statistics as shown in Fig. 1is less useful to derive a direct parameter such as standard deviation, since it varies randomly for different frame lengths at various power levels. These tests are used for determining the stationarity of temperature fluctuations. This is necessary since the variables in a model should be stationary for valid behavioral analysis (asymptotic) and gives confidence in any calculated parameter derived from fluctuations. Additionally, use of such tests indicate their suitability for fluctuation analysis.

⇑ Corresponding author. Tel.: +91 44 27480500x22258; fax: +91 44 27480228. E-mail addresses: [email protected] (P. Sharma), [email protected] (N. Murali), [email protected] (T. Jayakumar). 0306-4549/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.anucene.2013.04.040

2.1. Kwiatkowski–Phillips–Schmidt–Shin (KPSS) Test The NULL hypothesis (Ho) in this test is: an observable time series is stationary around a deterministic trend (Kwiatkowski et al., 1992). Hence, the data is checked for stationarity. GRETL (Baiocchi and Distaso, 2003) tool is used for calculating test statistics. The test statistics are compared against the value at 99% confidence level (0.743). If the value is greater than 0.743, Ho is rejected and data is likely to be non-stationary.

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226 408

225

468

366

Temperature ºC

224

406

223

364

466 404

222 362 221

402 464

220

360

400

219 0

50

100

150

200

time (seconds) Fig. 1. Mean temperature vs standard deviation for different frame lengths.

Fig. 2. CSA temperature profile for increasing power levels.

2.2. Runs Test Ho in runs test is: Data is from a random process. Test statistic z is calculated as follows (Bradley, 1968):

RR z¼ SR

ð1Þ

R¼ SR ¼

2n1 n2 þ1 n1 þ n2 2n1 n2 ð2n1 n2  n1  n2 Þ 2

ðn1 þ n2 Þ ðn1 þ n2  1Þ

ð2Þ þ1

ð3Þ

513.5

411.5

513.0

411.0

CSA Temperature ºC

where R is the observed no. of runs, R the expected no. of runs, and SR is the standard deviation of no. of runs. The absolute value of test statistic jzj is compared with z-score for normal distribution at 5% significance level (z1a/2 = 1.96). Ho is rejected if jzj > z1a/2 and hence, the data is unlikely to be from a random source. R and SR are calculated as:

412.0

512.5

410.5 512.0 410.0 511.5 409.5 511.0

409.0

510.5

408.5

510.0

408.0 0

50

100

150

200

time (seconds)

where n1 is the no. of positive values in series (or above mean/median) and n2 is the no. of negative values in series (or below mean/ median).

Fig. 3. CSA temperature profile at stable power levels.

2.3. Reverse Arrangement Test (RAT) Ho in RAT is similar to runs test, but is free from any distribution related pre-assumptions. It is helpful in detecting any trends which lead to presence of non-stationarity in time-series. The code for calculating jzj is written in SCILAB (Scilab Enterprises, 2012) with the following relationship (Julius and Bendat, 2010):

      c  NðN  1Þ=4   z ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   3 2  ð2N þ 3N  5NÞ=72

ð4Þ

where N is the time series length and c is the no. of arrangements in time-series for which,

xðiÞ > xðj þ 1Þ where i ¼ 1 to N; j ¼ i to N  1

Fig. 4. Primary coolant flow.

3. Experimental data 2.4. Approximate Entropy (ApEn) ApEn is a method used to estimate regularity and unpredictability of fluctuations in a time-series data (Pincus et al., 1991). The other methods for measuring regularity such as entropy are not suitable for experimental data. The ApEn code is written in Scilab Enterprises (2012) and is tested using fluctuation data from FBTR. The time-series for which ApEn has lower value (less complex) is more predictable, and vice versa.

Performing a number of tests requires vast amount and wide range of data. The thermal power in FBTR ranges from few kilowatts to several megawatts (1 kW Th to 18 MW Th). The full scale is divided in source, intermediate and power range in addition to shutdown state. Central subassembly (CSA) outlet temperature was obtained for full range of operation. The time-series consists of 2400 samples for each power level, with each sample separated by 0.1 s. Also, data was explicitly taken for increasing power cam-

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Table 1 Tests results. Test

Test statistic Increasing

KPSS RAT Runsa a

0.007–0.71 23.26–31.04 0.9–2.75

Critical value z

Conclusion

0.743 1.96 1.96

Ho accepted: hence, data is stationary Ho rejected: hence, data is not from a random source Ho rejected: hence, data is not from a random source

Stable 11 MW T h

18 MW T h

0.0138 31.29 2.6488

0.014 27.8 2.17139

Ho passes after a particular mean level, but corresponding p-value also increases, suggesting against the passing of Ho.

Fig. 5. KPSS test statistic.

Fig. 7. Runs test statistic.

Fig. 6. KPSS test statistic after 1-D.

paign as well as few stable power levels to get good results. The temperature of CSA varies from 210 °C to 520 °C. It is clear from Figs. 2 and 3 that fluctuations increases with mean level value. There are two loops for primary coolant in FBTR viz. east and west loop. Since each loop is monitored by three flowmeters, average is taken for the individual loops.The total coolant flow is calculated as the sum of average flow from east and west loop. It is observed that average total flow is around 657.9 m3/h with variation of 4.66 m3/h for a wider power range. This variation comes around 0.7% and can be assumed nearly constant throughout the operation (Fig. 4).

4. Results and discussion The results are summarised in Table 1.The performed statistical tests have given confidence which is useful in proposing a signal processing scheme to estimate power from temperature fluctuations. KPSS test statistics in Fig. 5 shows that the data is non-stationary for all power levels. Since the aim is to correlate temperature fluctuations with reactor power, the mean level value is removed by performing first order differencing (1-D). A KPSS test run for 1-D data is shown in Fig. 6. This data passes the Ho as explained in Section 2, and hence becomes stationary. Similarly, the test statistics for stable power levels of 11 and 18 MW T h are [12.99, 17.88] (raw data) and [0.013, 0.019] (1st order differenced data) respectively. However, it is necessary to quantify the stochasticity present in the time-series (if any) to make it more useful for

Fig. 8. RAT statistic.

modeling purpose. Runs test and RAT are performed on 1-D data and gives an idea about the stochasticity. The runs test result is shown in Fig. 7. It is observed that the Ho (data is from a random source) passes only after a certain value of mean temperature. However the p-value, which represents the evidence against Ho, also increases 35 folds. To get more insight into the stochasticity measures, RAT proves to be of immense use. The result for RAT is shown in Fig. 8. Complexity statistics by ApEn are shown in Fig. 9.

4.1. Mathematical model It is observed with the help of time–frequency analysis of CSA temperature data (Sharma et al., 2012) that the no. of fluctuations as well as their magnitude increases with thermal power. However, these fluctuations are non-linearly localized in time. Hence,

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Let the time-series under analysis be X, such that

X ¼ x1 ; x 2 ; x 3 ; . . . ; x N The AR(1) term is equivalent to the first order difference of the time series,

D ¼ X n  X n1 ;

where n ! ð2 to NÞ

Pure fluctuation levels can be extracted as, A = jDj or rms(D) The MA(w) term is calculated as,

MAðk; wÞ ¼

X 1 kþw1 AðiÞ; w i¼k

  N1 where k ! 1 to w

Fig. 9. ApEn statistic.

The window length w is large owing to the response time of thermocouple. But with appropriate digital signal processing, this effect can be minimized. For e.g., if a window length of 1000 samples is required, then initially all the data elements (1000) in the memory can be initialized with the first sampled signal. Subsequently, parameter calculation can be achieved by giving a left shift to the data set adding the newly sampled data to the set. The results in Figs. 10 and 11 based on proposed model (Fig. 12) shows the variation of derived parameter (running moving average) at different temperature levels. 4.2. Effect of thermophysical properties of sodium Fig. 13 shows variations in various thermophysical properties of sodium (coolant) with temperature (Bobkov et al., 2008). Heat transfer correlations in liquid metal fast breeder reactors are based on Peclet number Pe (Waltar and Reynolds, 1981), given as;

Pe ¼ Pr  Re Fig. 10. Derived parameter: increasing power.

ð5Þ

where Pr and Re being Prandtl and Reynolds no. respectively. The coolant flow through subassemblies is turbulent forced convection type. This is due to the high value of Reynolds no. for liquid sodium. The value of Pr is very small (Pr < 0.01). The Pe value is high and more than 100 for LMFBRs. At low Pe, molecular diffusion (conduction) is the prominant heat transfer mechanism whereas at higher Pe, mechanical mixing (convection) dominates. However, Nusselts no. which defines the overall heat transfer characteristics, is given as,

Nu ¼ a þ bðPe Þc

ð6Þ

where a is the contribution due to conduction and the remaining portion is due to forced convection. Presence of a is due to high thermal conductivity of sodium. As shown in Fig. 13, value of thermal conductivity (k) decreases with increasing temperature. Hence, the heat transfer contribution due to a further decreases. Fig. 14 shows that Pr value, which is already low for liquid sodium, further decreases with temperature. There is a relative 70% decrease in Pr value from 0.0075 (200 °C) to 0.0044 (470 °C). This results into slight enlargement of thermal boundary layer (Fig. 15). The same correlation can be understood by the relationship between Pe and thermal diffusivity (a) as,

Fig. 11. Derived parameter: stable power.

a stationary model consisting of only autoregressive term AR(1) is not sufficient for signal processing. A moving average component MA(w) is necessary to establish a correlation, where w denotes the window length. A mathematical model consisting of AR(1) and MA (w) can be represented as follows.

Pe ¼

UD

a

ð7Þ

where U is the velocity, and D is the characteristic dimension. From Fig. 13, it can be observed that a remains fairly constant in the range 200–470 °C (0.73% decrease). However, as temperature further in-

Fig. 12. Proposed signal processing model.

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Fig. 13. Thermophysical properties of sodium.

Fig. 14. Prandtl number vs temperature.

creases, value of a falls by 5% (480–700 °C). The net effect is a relative increment towards Pe value after a certain temperature. Hence with increasing temperatures, more convective heat transfer takes place. It is concluded from the above discussion that the convective heat transfer dominates the one due to conduction, and this domination increases with temperature increase. The radial core temperature distribution depends on core neutron flux profile, flow through each subassembly, etc. Measures are adopted to maintain a flat radial temperature distribution in the core. Since for the present study, CSA temperature profile is utilized; the fluctuations so observed are mainly due to the coupled effect of turbulent flow and slight temperature differences amongst the fuel pins. The thermal power is directly proportional to the temperature difference and coolant flow rate. Since the flow rate is almost maintained constant (Fig. 4), the increase in thermal

Fig. 15. Boundary layers.

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5. Conclusion Collective analysis of test results suggests that temperature fluctuations becomes stationary after removing the mean level. Fluctuations are not random, though their likeliness of becoming random increases with power. These evolve with more complexity at higher power, but very less than that for a pure random process. With the help of these results, the proposed method for fluctuation processing gives good results and can be utilized for obtaining a pointer to thermal power in subassembly of fast reactor. Moreover, reactor power estimation would be possible by the use of temperature fluctuations, if data is available for all fuel subassemblies. References Fig. 16. Thermal power vs CP.

power is attributed to increase in core temperature. Any increment in CSA temperature means a slight increase in temperature difference amongst individual fuel pins within the CSA. This gives rise to increase in magnitude and frequency of temperature fluctuations at the outlet of subassembly, and gets reflected in the signal as sensed by the thermocouple. 4.3. Model validation The proposed mathematical model in sub section 4.1 is validated for fluctuation data obtained from CSA. Fig. 16 shows the variation of calculated parameter with thermal power. The lower curve represents the CP with no weights multiplied to the CP. In order to spread the magnitude of CP, it is multiplied with a linearly increasing weight in correlation with thermal power. The results so obtained agree with the views and suggestions on statistical properties of fluctuations as in Ghiaasiaan (2011), Muramatsu and Toshiharu (1991), Ogino et al. (1984), etc., where it has been emphasized that the random looking like fluctuations contain information about the system, and it is possible to mathematically model the fluctuation information. Generation of fluctuations depend on core configuration and coolant flow rate. However, if similar metrics is calculated for all the fuel subassemblies and correlated, the relative magnitude of CP would point to the individual thermal power contributions as per core configuration. Similarly, an increased flow rate results into decrease in fluctuation level. Hence, proper coefficients are to be multiplied in accordance with the flow rate. Since a data driven model based design is being utilized here, further analysis with respect to above mentioned points would need more data and its study.

Baiocchi, G., Distaso, W., 2003. Gretl: econometric software for the gnu generation. J. Appl. Econ. 18, 105–110. Bilodeau, M., Cincera, M., Arsenault, A., Gravel, D., 1997. Normality and stationarity of EMG signals of elbow flexor muscles during ramp and step isometric contractions. J. Electromyogr. Kinesiol. 7, 87–96. Bobkov, V. et al., 2008. Thermophysical Properties of Materials for Nuclear Engineering: A Tutorial and Collection of Data. IAEA, Vienna. Bradley, J.V., 1968. Distribution-Free Statistical Tests. Prentice-Hall (Chapter 12). Brcich, R., Iskander, D., 2006. Testing for stationarity in the frequency domain using a sphericity statistic. In: IEEE International Conference on Acoustics, Speech and Signal Processing, 2006. ICASSP 2006 Proceedings. IEEE. pp. III–III. Ghiaasiaan, S.M., 2011. Convective Heat and Mass Transfer. Cambridge University Press. Julius, S., Bendat, A.G.P., 2010. Random Data: Analysis and Measurement Procedures. Wiley. Kay, S., 2008. A new nonstationarity detector. Signal Processing, IEEE Transactions on 56, 1440–1451. Kwiatkowski, D., Phillips, P., Schmidt, P., Shin, Y., 1992. Testing the null hypothesis of stationarity against the alternative of a unit root. J. Econometrics 54, 159– 178. Muramatsu, T., Ninokata, H., 1994. Development of analytical evalution method for thermal striping pheomena. J. Atom. Energ. Soc. Jpn. 36, 1152–1163. Ogino, T., Inujima, H., Haga, K., 1984. Feasibility study of LMFBR local core anomaly detection by use of temperature and flow fluctuations. J. Nucl. Sci. Technol. 21, 172–186. Pincus, S., Gladstone, I., Ehrenkranz, R., 1991. A regularity statistic for medical data analysis. J. Clin. Monit. Comput. 7, 335–345. Scilab Enterprises, 2012. Scilab: Free and Open Source software for numerical computation. Scilab Enterprises. Orsay, France. Sharma, P., Murali, N., Jayakumar, T., 2012. A time–frequency analysis of temperature fluctuations in a fast reactor. In: 2012 5th International Congress on Image and Signal Processing (CISP). IEEE, pp. 1546–1550. Srinivasan, G., Suresh Kumar, K., Rajendran, B., Ramalingam, P., 2006. The fast breeder test reactor – design and operating experiences. Nucl. Eng. Des. 236, 796–811. Thexton, A., 1996. A randomisation method for discriminating between signal and noise in recordings of rhythmic electromyographic activity. J. Neurosci. Methods 66, 93–98. Waltar, A.E., Reynolds, A.B., 1981. Fast Breeder Reactors. Pergamon. Weber, E., Molenaar, P., Molen, M., 2007. A nonstationarity test for the spectral analysis of physiological time series with an application to respiratory sinus arrhythmia. Psychophysiology 29, 55–62.