Independent Component Analysis for Redundant Sensor Validation

Independent Component Analysis for Redundant Sensor Validation Jun Ding, J. Wesley Hines, Brandon Rasmussen The University of Tennessee Nuclear Engineering Department Knoxville, TN 37996-2300 E-mail: hines2@utk.edu Abstract Redundant sensors have been widely used in safety critical facilities such as nuclear power and chemical plants. As these industries strive to move towards condition-based sensor calibration practices, on-line calibration verification algorithms must be developed. Independent component analysis (ICA) can be applied for redundant sensor validation. Independent component analysis is a statistical model in which the observed data is expressed as a linear transformation of latent variables ( independent components ) that are nongaussian and mutually independent. The ICA method is able to reduce the redundancy of the original dataset in order to predict the process parameter more accurately. The ICA prediction method is proven to be a robust method that can be used as a non-parametric approach to build a model that can detect faulty and drifted sensors so that they can be scheduled for maintenance. A slow sensor drift case study from a nuclear power plant is presented to show the usefulness of this technique. The ICA based system results are much better than other current methods. Independent component analysis is shown to be a new and effective approach for redundant sensor validation. 1.0 Introduction Redundant measurements are widely used in mission critical applications such as nuclear power plants, chemical facilities and the aerospace industry. Redundant information enhances the reliability of measurement. On the other hand, the redundant information can be utilized to check measurement channel integrity. On-line monitoring is the process of automatically checking component operation while the process is operating. EPRI formed the EPRI/Utility On-Line Monitoring Working Group in 1994 with the goal of obtaining NRC approval of on-line monitoring as a calibration reduction tool for safety-related instruments. Their On-Line Monitoring Cost-Benefit Guide estimates an industry wide cost savings of $40M to $290M over the next 20 years [EPRI 2002]. The report also claims the following benefits of on-line monitoring. Helps eliminate unnecessary field calibrations. Reduces associated labor costs. Limits personnel radiation exposure. Limits the potential for damaging equipment. Various on-line calibration monitoring algorithms have been developed. For example, the Instrument Calibration and Monitoring Program (ICMP) [Wooten 1993] was used for redundant sensor monitoring. It has been implemented at the V.C. Sumner Nuclear plant beginning in 1991 as a performance-monitoring tool. ICMP is a weighted average algorithm,

which assigns a consistency value to each channel. If the measurement is consistent all the time, all measurements will be equally weighted and the algorithm is reduced to simple average. If one of the measurements differs from the others a lot, the weight of that measurement will be reduced due to inconsistency. Thus the parameter estimate will contains less drift due to reduced weight for the faulty channel. Other systems have been developed by suppliers such as Smartsignal Inc. (www.smartsignal.com), PCS (www.pcs-home.com), and EXPERT Microsystems (www.expmicrosys.com). These methods are geared towards the general monitoring of process sensors but not specifically towards redundant sensors. More sophisticated models can be built to fully utilize the redundant information contained in the measurement. A research program using Independent Component Analysis (ICA) shows that ICA model captured essential information in the redundant measurement method and is able to reduce the redundancy of the original dataset in order to predict the process parameter more accurately. ICA prediction is very robust in that faulty sensors do not adversely affect the status of good sensors [Ding, 2003]. In this paper, a slightly different approach using a non-regression ICA modeling for redundant sensor validation is presented using actual plant data set. The results are compared with ICMP. For a description of the ICMP algorithm, one can refer to Rasmussen [2002]. 2.0 Methodology 2.1 System Description A typical redundant sensor validation system is as follows: Parameter Estimate Residuals Sensor Status Redundant Sensor Measurements Estimator Residual Formation Fault Detection Algorithm Figure 2.1 Redundant Sensor Validation System The functional description for each block is as follows: Estimator: the system receives redundant sensor values (redundancy of n = 2, 3, 4...) and processes them to provide a best estimate of the measured parameter. Residual Formation: the parameter estimates are compared to the actual sensor signals and residuals are formed. Fault Detection Algorithm: the residuals are processed to determine if they have significantly changed from zero.

2.2 ICA model and algorithm Independent component analysis is a statistical model in which the observed data (X) is expressed as a linear transformation of latent variables ( independent components, S) that are nongaussian and mutually independent. We may express the model as X = A S (2.1) where: X is an (n x p) data matrix of n observations from p sensors S is an (p x n) matrix of p independent components A is an (n x p) matrix of unknown constants, called the mixing matrix The problem is to determine a constant (weight) matrix, W, so that the linear transformation of the observed variables Y = W X (2.2) has some suitable properties. In the ICA method, the basic goal in determining the transformation is to find a representation in which the transformed components, y i are as statistically independent from each other as possible. When random variables with specific non-gaussian distributions are combined, the central limit theorem shows that the sum is more gaussian than the original variables. Therefore, to separate the original variables (S) from a sum (X), we want to choose a transformation (W) that makes them as non-gaussian as possible. This is the assumption used to find the original independent components. Hyvarinen [1999], developed an ICA algorithm called FastICA as described below. It uses negentropy J(y) as the measurement of the non-gaussianity of the components. J ( y) = H( y gauss ) H( y) (2.3) H(y) is the differential entropy of a random vector y. H ( y) = f ( y) log f ( y) dy (2.4) where: f(y) is the density of the random vector y. Based on maximum entropy principle, negentropy J(y) can be estimated: J( y ) ν i 2 c[ E{ G( yi )} E{ G( )}] (2.5) where: G is any nonquadratic function c is an irrelevant constant ν is a Gaussian variable of zero mean and unit variance E{} is the expectation One attempts to maximize negentropy so that a non-linear transformation of y is as far as possible from a nonlinear transformation of a gaussian variable (v). This nonlinear transformation (G) is also called a contrast function. The following is a commonly used

contrast function G and its derivatives: 1 G( u) = log cosh( a1u) a g( u) = tanh( a u) 1 1 (2.6) The FastICA algorithm for estimating several independent components is described below: 1. Center the data to make its mean zero. 2. Whiten the data to give z. 3. Choose m, the number of independent components to estimate. 4. Choose initial values for the w i, i=1,,m, each of unit norm. Orthogonalize the matrix W as in step 6 below. 5. For every i=1,,m, let w i E{zG(w T iz)}-e{g(w T iz)}w, where G and g is defined e.g. as in (2.11) 6. Do a symmetric orthogonalization of the matrix W = ( w 1, w m ) T by W (WW T ) -1/2 W. 7. If not converged, go back to step 5. A concern with using ICA is that it has two ambiguities [Hyvarinen 2001]. One is that the variances (energies) of the independent components cannot be determined. The other is that the order of the independent components cannot be determined. These ambiguities are of concern when performing on-line instrument channel monitoring for two reasons: 1). the components must be scaled back to their original units and 2). the component containing the parameter estimate needs to be selected. In order to scale the components back to their original units we need to calculate the correct scale factor α, and to find the correct scale factor we must select the component corresponding to the parameter of interest. To do this, the mean of the measured parameter is estimated by taking the mean of the medians for each (i) of n channels (see equation 2.12). Next, compute n scale factors by dividing the mean of the parameter by the mean of each component and use those scale factors to give the components the same mean as the measured parameter. The scaled component with the highest correlation coefficient to the raw signals (X) is the component of interest and is the parameter estimate. mean( median( X )) α i = i=1 n (2.7) median( IC ) i where: X is the matrix of n mixed signals IC i is the i th independent component mean and median are MATLAB functions To calculate the correct transformation matrix, rescale the transformation matrix W to W c : W c = sign (α) * α * W (2.8) where: α is the α i that maximizes the correlation between the scaled component and the

parameter value. The parameter estimate is now calculated with: Y= W c X (2.9) Residuals between this parameter estimate and the channel measurements are evaluated to assess the calibration status of the redundant instrument channel sensors. Sensor drifts are suspected when a channel's residual deviates from some nominal value determined using a representative data set. 2.2 Model Justification and Drift Detection The measurement from each channel contains the process parameter, a common source noise and a channel noise. These three components are independent from each other. With the exception of the channel noise, the components seldom have a gaussian distribution. Another assumption is that the transform matrix A is linear and time invariant. This assumption is valid during most conditions, especially for steady state or slowly changing measurements. Moreover, during a fault condition, the fault component is introduced into one or more redundant channels. The fault component is absolutely independent from the process parameter so we use the model for regression and we can build the model when drift is present. A given channel's residuals are defined as the differences between the parameter estimates and the channel measurements. Each channel will have a unique residual with respect to the parameter estimate. The mean values and standard deviations of the residuals will be used to identify out-of-calibration channels via the following rule: If r 2 σ p x r + 2σ, then the th channel is operating within calibration, f k k otherwise the th channel's calibration is suspect. (2.10) where: r is the mean residual between the parameter estimate and the training data for the th channel σ is the standard deviation of the residual between the parameter estimate and the training data for the th channel x k is the th element of the k th observation vector not contained in the training data f p k is the parameter estimate corresponding to x k Detection of an out-of-calibration channel requires a method of determining when a given channel's residual exceeds some nominal value. Methods such as the Sequential Probability

Ratio Test (SPRT) [Wald 1945] can be used to identify when a drift has occurred, but we will employ a much simpler, and less optimal, method in this research. The chosen approach is to suspect an out-of-calibration alarm when a channel's residual falls outside of a ± 2σ band surrounding the residual mean value. 3.0 Results 3.1 Drift Detection The redundant channel data for this case study is from the English Sizewell B nuclear power plant. The data set has nine redundant channel measurements with one experiencing an actual channel drift. We select three signals to simulate a U.S. system made up of three redundant channels. The drifting channel is included as channel #2. Figure 3.1: Redundant channel measurement and ICA estimate An ICA transform was carried out on the select window from 0 to 800. The drift channel was included during model building. Figure 3.1 shows results of ICA estimate and the original data. From visual inspection, we can conclude that the ICA estimate contains no drift. The next three figures (Figure 3.2) show the residuals of three of the sensors with their drift detection error bands. The error bands for each channel are determined from equation (2.10). The variances are estimated from first 100 data points. Figure 3.2 shows drift detection results from residual and error band. Channel 2 is clearly identified as drift channel.

Figure 3.2: Drift detection from residual and error band The ICMP result is displayed as Figure 3.3 for comparison. Figure 3.3: ICMP results for drift detection ICMP algorithm identifies ch#2 as a drifting channel. Unfortunately, it also identifies ch#3 as a drifting channel. The reason for the incorrect identification of the drifted channel is that ICMP estimate is drifting. This is commonly termed spillover. The ICA method has the advantage of being resistant to spillover, thus making it a robust technique.

4.0 Conclusion Two methods for on-line redundant sensor calibration monitoring were compared using actual nuclear power plant data. The variance components of the parameter estimate uncertainty were used as the measures of performance for the actual plant data. In this case the ICA method outperformed the ICMP method. The maor advantages of using the ICA algorithm over the ICMP are its ability to not have any spillover from faults in other channels. The ICA redundant sensor estimation technique (RSET) has significant advantages over other methods commonly employed for redundant sensor calibration monitoring. The researchers are currently investigating the use of these techniques for sensor calibration reductions in the nuclear power industry. 5.0 Acknowledgements The authors would like to acknowledge the Electric Power Research Institute (EPRI) and the Department of Energy's Nuclear Energy Plant Optimization (NEPO) program for funding this research. 6.0 References Rasmussen, Brandon and J. Wesley Hines, Monte Carlo Analysis and Evaluation of the Instrumentation and Calibration Monitoring Program, Proceedings of the Maintenance and Reliability Conference (MARCON) 2002, Knoxville, TN. Ding, Jun, Andrei Gribok, J. Wesley Hines and Brandon Rasmussen, (2003) "Redundant Sensor Calibration Monitoring Using ICA and PCA", Special issue of Real-time Systems on Applications of Intelligent for Nuclear Engineering, Kluwer Academic Publishers (accepted for publication) EPRI, On-line Monitoring News Letter, August, 2002. EPRI, TR-1003572, On-Line Monitoring Cost-Benefit Guide, June 2002. Hyvarinen, A., (1999), Fast and Robust Fixed-Point Algorithms for Independent Component Analysis, IEEE Transactions on Neural Networks, Vol. 10, No. 3, May 1999. Hyvarinen, A., J. Karhunen, and E. Oa (2001), Independent Component Analysis, pp 1-11 & 125-137. Wald, A., "Sequential Tests of Statistical Hypothesis", Ann. Math. Statist., 16, (1945), 117-186. Wooten, B., (1993) "Instrument Calibration and Monitoring Program Volume 1: Basis for the Method," EPRI TR-103436-V1.

Biography Mr. Jun Ding is a Ph.D student in the Nuclear Engineering Department at The University of Tennessee. He received the BS degree in Nuclear Physics and Technology from University of Science and Technology of China in 1992, and was an engineer and proect manager in the Instrumentation and Control Department at Shanghai Nuclear Research and Design Institute for seven years. He then received a M.S in nuclear engineering from the University of Cincinnati and a M.S in Physics Entrepreneurship Program from Case Western Reserve University. His research interest is in artificial intelligence in nuclear engineering application. Dr. J. Wesley Hines is an Associate Professor in the Nuclear Engineering Department at the University of Tennessee. He received the BS degree in Electrical Engineering from Ohio University in 1985, and was a nuclear qualified submarine officer in the Navy for 5 years. He then received both an MBA and an MS in Nuclear Engineering from The Ohio State University in 1992, and a Ph.D. in Nuclear Engineering from The Ohio State University in 1994. Dr. Hines was the UT Maintenance and Reliability Center Education Program Coordinator for five years and is currently the UT College of Engineering Extended Education Coordinator. Dr. Hines teaches and conducts research in advanced statistical and artificial intelligence applications in process monitoring and diagnostics. Brandon Rasmussen (BS, health physics, Francis Marion University, 1995; MS nuclear engineering, The University of Tennessee, 2002) is a graduate student in the department of nuclear engineering at the University of Tennessee in Knoxville. He is currently finishing the requirements for a PhD in nuclear engineering under a Department of Engineering Nuclear Engineering Fellowship. His background includes research and development of large-scale empirical process models for instrument channel calibration monitoring using advanced statistical and artificial intelligence techniques.