Online Fault Detection and Diagnosis of In-Core ... - IEEE Xplore

IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 62, NO. 6, DECEMBER 2015

3311

Online Fault Detection and Diagnosis of In-Core Neutron Detectors Using Generalized Likelihood Ratio Method Yellapu Vidya Sagar, A. K. Mishra, A. P. Tiwari, and S. B. Degweker

Abstract—Vanadium self-powered neutron detectors (VSPNDs) have been proposed to be used in the Advanced Heavy Water Reactor for flux mapping purposes. However, response of VSPNDs to neutron flux variations is slow, and they might also develop faults. This paper proposes a hybrid scheme for state estimation based on the Kalman filtering approach and fault detection, diagnosis, and correction based on the generalized likelihood ratio (GLR) method for VSPNDs. The Kalman filter estimates prompt neutron flux variations from the delayed signal of VSPND, while the GLR method analyzes the innovations to detect fault in VSPND. Subsequently, the scheme also corrects for the step changes in the measurement emanating from fault in VSPND. Performance of the proposed hybrid scheme has been evaluated from simulation of neutron flux variations occurring due to simultaneous movement of regulating rods and demand power variations. It is shown that the magnitude and time of occurrence of step change in signal due to the fault are effectively estimated, and thus the step change can be corrected online. Index Terms—Additive biases, dynamic compensation, fault detection and diagnosis, generalized likelihood ratio (GLR) method, Kalman filter, vanadium self-powered neutron detectors (VSPNDs), whiteness test (WT).

I. INTRODUCTION

F

AULTS are defined as any unpermitted deviations in one or more of the characteristic properties of the system from the acceptable, usual, and standard conditions [1]. Faults can be either incipient or abrupt changes in the components of the system, and the same are reflected in the signals associated with the system. The fault detection and diagnosis (FDD) system, when installed on a component, generates features corresponding to its normal operation and compares them

Manuscript received April 17, 2015; revised August 10, 2015; accepted October 20, 2015. Date of current version December 11, 2015. Y. V. Sagar is with the Homi Bhabha National Institute, Anushaktinagar, Mumbai-400094, India (e-mail: [email protected]; [email protected]). A. K. Mishra is with the Reactor Control Systems Design Section, Bhabha Atomic Research Centre, Trombay, Mumbai-400085, India (e-mail: [email protected]). A. P. Tiwari is with the Reactor Control Systems Design Section, Bhabha Atomic Research Centre, Trombay, Mumbai-400085, India and also with the Homi Bhabha National Institute, Anushaktinagar, Mumbai-400094, India (e-mail: [email protected]). S. B. Degweker is with Mathematical Physics and Reactor Theory Section, Bhabha Atomic Research Centre, Trombay, Mumbai-400085, India (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TNS.2015.2496599

with the actual behavior, in order to recognize the existence of anomalies responsible for deviations. The FDD system can either be model based, which uses the first principles for the generation of the residuals; data based, which relies on the process history; or signal based, which performs time domain and frequency domain analysis on the signals. Faults, if not detected and diagnosed, establish the need for maintenance activities, endanger the system reliability, activate safety systems, cause operational upsets, and reduce safety margins. However, all the above problems can be overcome with the design of a proper FDD system. FDD systems have several potential applications in nuclear reactors, one of which is the health monitoring of neutron detectors. In large heavy water reactors, knowledge of axial, azimuthal, and radial flux distribution is required for monitoring spatial flux transients, flow changes in coolant channels, and reactivity devices’ movements. These functions are usually performed by a flux mapping system. Signals from self-powered neutron detectors (SPNDs) which are placed at strategic locations within the core are useful in implementing both the flux mapping systems and flux tilt control. However, these SPND signals are corrupted by random noise because of the probabilistic nature of the neutron flux hitting the emitter material of the SPNDs and other factors related to operation of SPNDs [2]. Apart from random errors, these SPNDs might also develop failures. Failures in SPNDs can be broadly classified as hard failures (complete loss of signals, e.g., sheath failures) and soft failures or faults (signal changes gradually or suddenly by a relatively small amount). Hard failures in SPNDs can be easily identified, and those SPNDs can be replaced. However, soft faults are difficult to detect since they produce degraded signals over a period of time, resulting in changes in the sensitivity, improper calibration, and systematic biases. Random errors and faults make the SPND signals inaccurate, which results in degraded performance of flux mapping and flux tilt control systems. Fault detection of in-core detectors and other nuclear reactor components are widely reported [3]–[13]. However, most of the methods applied to the FDD of neutron detectors are data based. A faulty SPND shows a different reading than the others, with which it is highly positively correlated during its healthy condition. All these SPNDs are then termed as spatially redundant, and their signals are compared for the detection of any anomaly. When the reactor is in steady state or near to it, a static data reconciliation [14] procedure can be adopted, which uses spatial redundancy of the SPNDs in terms of constraint models. Minimization of random errors and FDD of SPNDs may be done with a data reconciliation scheme [15]. However, the reactors are seldom at steady state and always undergo

0018-9499 © 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

3312


Fig. 1. (a) Cross-section of the AHWR core, showing the location of ICDHs (schematic) [15]. (b) Placement of seven VSPNDs along an ICDH in AHWR (all the dimensions are in mm) [27].

variations. Data-driven techniques can give erroneous information regarding the faults in SPNDs under such conditions. On the other hand, model-based FDD techniques possess inherent capability of fault detection, even during transients, though their performance would be strongly dependent on the accuracy of the model. A review of the applications of different FDD schemes to nuclear plants is given in [9]. For detecting the faults, threshold tests like -sigma and sequential probability ratio tests [4], [16] are more popular due to their simplicity and ease of implementation. The outcomes of these threshold tests are fault occurrence time and fault identification and localization, but they do not give any information regarding the magnitude of the fault. Therefore, these tests are not utilized in the applications where online fault correction is essential. An FDD method capable of obtaining the unknown fault magnitude from the ensemble of measurements is highly desirable. In maximum likelihood estimation (MLE), unknown parameter values of the random population are selected which maximize the joint probability density of the ensemble of observations. The MLE procedures have found their use in radiation detection in the applications of localization of the radiation sources [17]–[21]. The generalized likelihood ratio (GLR) method uses the MLE in the sense that a likelihood ratio is derived from the probability density functions of the observations under the possible hypothesis and the unknown parameters are obtained as the MLEs which best explain the likelihood ratio. A GLR-based FDD scheme relying on the temporal redundancy in the data is proposed in [16], [22] to overcome the limi-

tations of threshold-based FDD schemes. This scheme gives all the information regarding the fault, i.e., fault occurrence time, fault location, and magnitude of the fault, but at the cost of increased computational effort. Therefore, it is further modified [23] to reduce its computational burden so that it can be efficiently used for online FDD and fault correction. However, the online implementation of the GLR-based FDD scheme requires prior knowledge of probable fault modes (state jump, state step, sensor jump, sensor step, hard-over actuator or sensor, increased actuator or sensor noise, dynamic shift, etc.) and associated fault signature matrices of the system for which it is designed [16], [22], [23]. Vanadium self-powered neutron detectors (VSPNDs) are known for their delayed response characteristics [24], [25] and need to be compensated for representing the prompt behavior of the flux in the reactor. A Kalman filter-based dynamic compensation for the VSPNDs with unknown input has been developed in [26]. In this work, the Kalman filter framework has been extended such that it also performs FDD using the GLR method. The novelty of this work lies in the coupling of the dynamic compensation proposed in [26] with the GLR-based Kalman filtering [23]. The hybrid scheme proposed in this paper dynamically compensates the VSPND signal for promptness; minimizes the random errors through Kalman filter; performs detection and diagnosis of the faults with GLR method; and corrects the faulty measurements online. The performance of the proposed strategy is established through simulations using the mathematical model

VIDYA SAGAR et al.: ONLINE FAULT DETECTION AND DIAGNOSIS OF IN-CORE NEUTRON DETECTORS

[27] of the Advanced Heavy Water Reactor (AHWR), which uses VSPNDs for in-core measurement purposes. The rest of the paper is organized as follows. Section II explains the locations of VSPNDs in the core of AHWR and mentions the general possible faults that can grow with time in VSPNDs. Mathematical formulation of the proposed method is presented in Section III, and its application to VSPNDs of AHWR is given in Section IV. In Section V, results of the analysis are presented for two different operating conditions of the reactor, and in Section VI, conclusions are drawn. Computation of fault signature matrices that are needed by GLR method, tuning aspects, and test for whiteness of innovations are provided in Appendix A and Appendix B.

TABLE I PLACEMENT OF 200 NUMBER OF VSPNDS ( DENOTES TH VSPND

3313

IN

32 ICDHS )

II. IN-CORE NEUTRON DETECTORS IN THE AHWR AHWR [28] is a vertical, pressure tube type, heavy water moderated, and boiling light water cooled natural circulation reactor designed to generate 920 MW (thermal) power. AHWR uses 200 VSPNDs distributed in different layers in 32 in-core detector housings (ICDHs) at interlattice locations. Locations of ICDHs are shown in Fig. 1(a). Each ICDH can accommodate up to sevenVSPNDs at and positions as indicated in Fig. 1(b). The VSPNDs, numbered as , are placed in ICDHs at positions given in Table I. Only eight ICDHs, i.e., ICDHs numbered as 5, 8, 13, 14, 19, 20, 25, and 28 contain seven VSPNDs each while the remaining 24 ICDHs contain six VSPNDs each. It can also be observed that each zone [one of the quadrants of the core, shown in Fig. 1(a)] contains 50 VSPNDs. A typical VSPND, as shown in Fig. 2, has a coaxial configuration with four parts, viz., emitter, insulation, sheath (or collector), and mineral insulated coaxial cable. The material of the central electrode called emitter is vanadium in a VSPND. Generally, magnesium oxide (MgO) or aluminium oxide (Al O ) are used as insulator and inconel as the sheath [24]. As discussed in Section I, these VSPNDs are susceptible to failures. The following are the different fault modes of the in-core SPNDs [3], [29]. a) Sensitivity changes: Sensitivity ( ) is the current signal produced per unit neutron flux received in a unit length of the SPND. With increase in fluence, the sensitivity is observed to reduce because of burn-up of emitter material. It can be compensated by increasing the amplifier gain. b) Reduction in insulation resistance: Insulation resistance reduces from the nominal value (of the order of ) mainly due to moisture contamination. This increases the leakage current thereby reducing the signal from the SPND. This has a gradual effect on the SPND signal. c) Failure of seal: Detectors are filled with inert gases such as helium and sealed in order to avoid corrosion. When the seal gets deteriorated, conducting media like air and moisture replace the inert gas thereby corrupting the signal. d) Corrosion: Because of oxidizing and nitriding atmosphere surrounding the SPND assembly, embrittlement of the inconel sheath may happen when encapsulation deteriorates. This gradually deteriorates the signal. e) Ageing and activation products: Due to activation products, sheath and electrode alloys are degraded leading to signal degradation.

Fig. 2. Basic configuration of VSPNDs [24].

f) Clipping and modulation: The SPND signal may be restricted at a particular level or the signal may be modulated because of saturation of either the SPND or the associated electronic circuitry. These effects are similar to faults in the SPNDs. g) Bias error: Sudden jumps in the signal due to degradation of either the SPND or the associated electronic circuitry can also manifest as faults in the SPNDs. In the subsequent sections, a hybrid scheme for state estimation based on the Kalman filtering approach and FDD and correction scheme based on the GLR method is being introduced to handle these faults in VSPNDs.

3314


the input matrix; is the output matrix, represents the uncertainty involved in process model on addition of which the modelled state reaches the true one, and is the measurement noise, which accounts for the uncertainty in measurement. The last term in (2) constitutes the measurement bias model or fault model, where is the magnitude of bias which occurs at a hypothesized time instant and persists thereafter, is a vector with zeros in all positions except for a 1 corresponding to the corrupted measurement . represents a step change, to account for the fact that the bias occurs at the time instant , with the properties

TABLE II STATISTICAL PROPERTIES OF INNOVATIONS AT TIME INSTANT

if if The uncertainty vectors and are assumed to be independent and normally distributed with the following properties: (3) where and are respectively the process and measurement uncertainties, and is Kronecker delta. Based on the above model, a Kalman filter can be designed for the estimation of the states assuming that the system is observable. The following recursive equations represent the Kalman filter [30]: (4) (5) (6) (7) (8) (9) (10)

Fig. 3. Flowchart of the GLR method for dynamic systems.

III. GENERAL FRAMEWORK OF THE GLR-BASED FDD The GLR method [22] is designed to detect abrupt changes either in the state variables or in measurements. It quantifies the type and magnitude of faults along with the time of occurrence. This section describes the formulation of the GLR-based FDD scheme in linear time-invariant (LTI) systems in a stochastic framework, with a Kalman filter used for state estimation. The equations characterizing the dynamics of an LTI system are given as (1) (2) where vector; vector,

is the discrete time instant, is the state is the input vector, is the output is is the state transition matrix;

where is the estimated state, is the covariance of the error in the estimated state. is the innovation, which reflects the discrepancy between actual measurement ( ) and predicted measurement ( ), and represents additional information available to the filter as a consequence of the new observation . When the model of the system under consideration is accurate and the VSPND used to update the estimates is working properly, the innovation sequence is expected to be a zero mean Gaussian white noise process with covariance given by (7) [26]. In (8), is the Kalman gain, which blends the innovation with the predicted state as indicated in (9). In the set of equations (4)–(10), subscripts and indicate the conditional estimates at instant when measurements are available up to instants and respectively. Innovations from the Kalman filter can be used for detection of anomalies in the system. When a fault is present, the whiteness property of innovations is lost. This feature of innovations can be exploited for FDD. A GLR method utilizing this phenomenon [16], [22], [23] is developed, with which sudden jumps in the measurements and states can be detected and estimated. A. Effect of a Step Change in the Measurements on Innovations A dual-hypothesis method is adopted for fault detection, as per which the null hypothesis stands for no fault condition and stands for existence of fault.


3315

Fig. 4. Overall scheme of the Kalman filter-based FDD using GLR for four VSPNDs in different zones.

By utilizing the linearity property of the system under consideration, conditional state estimates, measurements, and innovations can be expressed respectively as (10) (12) (13) where

,

, and

denote the respective variables cor-

, , and responding to the healthy condition while denote the effects of step change in the measurement. Lemma III.1: Let represent the measurement bias occurring and in one VSPND at a time instant . Then the components are given by (14) (15) and are fault signature where matrices obtained by the recursive solution of (16) (17) where is the identity matrix of appropriate dimension. Proof: Refer to Appendix A. The innovations given by (6) are of a zero-mean white noise process under healthy condition and have a different mean given by (14) under faulty condition . However, they have the same covariance in both situations. The statistical properties of the innovations are also summarized in Table II for the hypotheses and . Thus the magnitude of bias is obtained from , but the time instant needs to be specified. B. Estimation of Time of Occurrence of Bias During a fault, the innovation experiences a change in its mean value, which may exceed certain threshold value. A fault detection test (FDT) can be conducted based on the test statistic defined as [23] (18)

In the above, given by (7) is invertible as the measurement errors are assumed to be independent of each other. Note that is a quadratic term of innovations normalized by their covariance at the th instant amounting to the sum of the squares of independent standard normal random variables. Hence, follows a central distribution of degrees of freedom [23], [31]. It can be inferred that is the possible instant of bias occurrence if . Here, represents the value of distribution curve spans the the statistic at which the portion of the total area under the curve, where is a measure for false detection. Since the decision process is based on the distribution known under healthy condition, it is natural that the fraction of the curve that represents the abnormalities also erroneously declares the faults if the statistics under healthy condition exceed the threshold because of random errors. Hence is known as false detection rate and is generally chosen as 0.05 [14], [23]. Apart from random errors, some occasional outliers (not step changes) also make the FDT statistic exceed the threshold. One way for reliable detection of bias is to conduct a fault confirmation test (FCT) on the FDT statistics in a time window of certain number of samples. According to this, is declared as the estimate of , the time of occurrence of step change, if

where

, FCT statistic, is defined as (19)

is the number of sampling instants in the time where window. The FCT statistic follows a central distribution of degrees of freedom. Thus the computation of fault signature matrices can be initiated from , where is the estimate of . This procedure is followed for each and every sampling instant. C. GLR Test for Fault Identification Isolation of faulty VSPND and identification (estimation) of magnitude of bias needs to be accomplished subsequent to the denotes the joint probability density function of FDD. If

3316


Using the expression for the multivariate normal probability density function given in [30] and from Table II, we have

(21)

Fig. 5. Position of RRs during open-loop transient.

where (22)

TABLE III GLR OUTCOMES FOR THE OPEN-LOOP TRANSIENT

For simplicity, define (23) where

(24) The MLE is obtained by equating the first derivative of (24) with respect to to zero. Thus

(25)

Substituting the value of in (22) and manipulating (21), (22), and (24), we get

(26)

Fig. 6. Different GLR outcomes during open-loop transient: FCT statistic on FDT rejections, bias estimate, and identified faulty VSPNDs at FCT rejection instants.

the -variate innovations from time to generalized likelihood ratio can be written as

, then the

(20)

Note that is computed for all . If a bias is detected by FCT, the VSPND that generated maximum is declared as the faulty VSPND . The estimate computed from (25) corresponding to the faulty VSPND is declared as the fault magnitude . However, incorrect isolation of the VSPNDs is still possible because of occasional rejections by fault detection and fault confirmation tests, due to the measurement noise content in the healthy VSPND readings. Nevertheless, these erroneous rejections are associated with nonzero bias estimates close to zero. Such isolations and identifications have negligible impact on the overall performance because of very small bias estimates. D. Online Correction for Bias Step faults in the measurement data not only corrupt the quality of the data but also make the innovations nonwhite, which might hinder the detection of subsequent faults, if any. An online scheme for correction of the bias in the measurements


is of utmost importance. The FDD outcomes such as isolation and estimation of bias can be used for online correction for the bias in the measurement data. For an isolated VSPND , the corrected measurement vector is given by

3317

and the remaining two states correspond to the state-space representation of first order power-series approximation for unknown input flux, given by (34)

(27) Note that the correction is delayed by samples after the fault occurrence. Too large a value for makes the detection procedure delayed by the same number of samples, while too small a value makes the bias estimate erroneous. So a trade-off is required in the selection of , and a reasonable value for it is known only by simulations with different choices of and looking for the best accurate bias estimate. The flowchart of the GLR method is shown in Fig. 3. At every instant, the Kalman Filter generates the innovation vector from (6). This vector is used for conducting FDT based on the statistic given by (18). The FDT statistic is computed, and FDT is carried out. If is false, the computations are continued over the new cycle. If is found true by FDT at an instant , the instant is declared as the FDT rejection instant . number of fault signature matrices are computed recursively from (16) and , the FCT statistic is computed using (19), (17), where and FCT is conducted. Note that a rejection of null hypothesis in FDT does not guarantee the same in FCT, which means that FCT rejection instants are a subset of FDT rejection instants which in turn are a subset of entire set of observations. If FCT also declares the fault at , then the instant is denoted as . The outcomes of the GLR test such as and (refer to Section III-C) are used for the online correction of the faults using (27). IV. APPLICATION OF THE GLR-BASED FDD SCHEME TO VSPND The method discussed in Section III is used for detection and diagnosis of step changes in VSPND signals. Its application necessitates a mathematical model of VSPNDs. Modeling of VSPNDs is well explained in [24], [26], according to which the standard LTI state-space formulation for a VSPND , in a continuous time domain is given by (28) (29)

and are arbitrary constants. where The above continuous time model is discretized to obtain the following state and measurement equations for the th VSPND: (35) (36) is the state transition matrix. The models where given by (35) and (36) for all the 200 VSPNDs can be arranged as (37) (38) where

For simulating a realistic behavior, a white Gaussian noise having standard deviation of the order of 2% of the nominal signal around the full power steady-state is superimposed on the VSPND output signal. Since the reactor is assumed to be operating at the power level of 1.0 fractional full power (FFP), the measurement noise variance obtained from the simulated data is about 0.04 for each VSPND. The process noise covariance matrix for the th VSPND is considered as [26]

where (30) represents the output, which represents the state vector, is identical to the current generated by the VSPND. and respectively denote the system and output matrices, given as (31) (32) where is the sensitivity of VSPND; and are respectively the prompt and delayed fractions of the output signal of VSPND, and is the time-constant of . is identical to the inverse of the decay rate constant of . In (30), we have (33)

In the above, the first diagonal element of is considered zero, which corresponds to the fact that the VSPND model is known with reasonably high accuracy as evident from the validation results given in [24]. The remaining two diagonal elements of the process covariance matrix are large because of poor confidence in approximation of neutron flux near the VSPND by a first-order power-series [26]. Now, the covariance matrices for the process noise and measurement noise for the composite Kalman filter are

Optimality of the Kalman filter with these is addressed in Appendix B.

and

matrices

3318


V. RESULTS The methods described in the preceding sections can be generalized for FDD of all the 200 VSPNDs of the AHWR. However, for the purpose of illustration while retaining simplicity, four different VSPNDs from the four quadrants are considered, which are on the same layer. Specifically, VSPNDs , , , and in layer are considered. Individual VSPND signals are passed through the dynamic compensations. In this case, since the number of measurements , the fault signature matrix is a diagonal matrix of size . Innovations from the Kalman filter combined with this fault signature matrix are used for the identification of faulty VSPND and estimation of bias magnitude, using the GLR method as already described in Section III. The overall schematic of the proposed method for this case is shown in Fig. 4, in which the block named “GLR method for FDD” works on the basis of the algorithm given in Fig. 3. is taken as 50, and the level of significance ( ) is taken as 0.05 for all tests. In this section, results are presented for two different simulated operating conditions of the AHWR. In the first transient, biases are introduced in the VSPND measurement data during the open-loop operation, and in the second one, biases are introduced while the reactor is maintained under closed loop control. Simulations are done for 300 and 200 s respectively in the first and second cases, with a sampling duration of 0.02 s. VSPND signals are generated with the help of their dynamic models. Results are explained in the following. A. Open-Loop Regulating Rod (RR) Transient When the reactor is critical, all the RRs are at 66.66%-in position. Starting from this configuration, simultaneous movement of the RRs in nodes 2, 4, 6, and 8 was simulated [27]. At time s, control signals to the drives of these RRs are chosen such that the rods move linearly into the reactor core and take 100%-in position in 120 s, i.e., at s. Then the RRs are held at this position till s. Fig. 5 shows the position of the RRs during the transient. Additive biases are introduced into the VSPND data, in a sequential manner. Initial th length of data corresponds to the case of no fault. Bias equivalent to 5% of the steady state value is added to signal of , from the 2500th sampling instant. Similarly biases of different magnitudes are added in the simulated measurement data of , , and at the observation indices , as given in Table III, in which the GLR statistics [estimates of the bias magnitude ( ) and bias occurrence instant ( )] are also given. From the 6500th sampling instant, biases of magnitude 1.5, 2, 2.5, and 3% are introduced in the simulated measurement data of , , , and for every 1000 sampling instants till , as given along with the GLR statistics in Table III. The FCT statistic is computed on the rejection of FDT, and on every FCT rejection faulty VSPNDs are identified and the bias magnitude is estimated. From Table III, it is clear that for a bias magnitude greater than the standard deviation of measurement errors, i.e., 2% (refer to Section IV), and are close to their actual values. Fig. 6 shows the plot of the FCT statistic computed from (19) for all the instants for which is rejected in FDT, estimate of bias, , and the identified faulty VSPND . The sudden jumps in the FCT statistic, at the observations 2500, 3500, 4500, and 5500, are in response to the injected additive biases in the signals

of VSPNDs , , , and . The bias correction equivalent to is made in the VSPND signal using (27) from the instant . The innovation sequence, VSPND output, and the estimated output for the four VSPNDs are shown in Fig. 7(a)–(d), for both before bias correction and after bias correction, obtained using (2) and (27) respectively. The true reactor flux variation is also shown. In these plots, it can be observed that the sudden jumps as a result of faults in VSPND outputs before bias correction are eliminated after bias correction. The plot of innovation sequences shown in Fig. 7(a)–7(d) have abrupt jumps that can be related to the sign and magnitude of the bias in the corresponding VSPND. The actual and estimated outputs of the VSPNDs also experience a similar jump following the introduction of fault if bias correction is not incorporated. However, with correction of bias the jump is appreciably reduced, establishing the effectiveness of the proposed method. B. Demand Power Change In this transient, the demand power changes from 1.0 to 0.9 FFP, and all the four RRs meant for automatic regulation act to reach the new steady state corresponding to the new demand power. The variation of total power of the reacto in response to the change in demand is depicted in Fig. 8. In this case, additive biases are introduced in the signals of VSPNDs , , , and according to the magnitudes of biases and time instants at which they are introduced as given in Table IV. It should be noted that when a bias is introduced in a VSPND at a particular time instant, it is persistent thereafter. Table IV gives the corresponding statistics, and Fig. 9 shows the characteristics such as FCT statistic, bias estimate, and the identified faulty VSPND. Fig. 10(a)–10(d) show the innovation sequences, reactor flux, and delayed and estimated values of all four VSPND signals. In this case also, it can be observed that the fault-driven jumps in VSPND delayed outputs are not present after bias correction obtained using (27). On overall basis, Table IV and Fig. 10 make it clear that the proposed hybrid strategy for dynamic compensation and FDD performed well even for faults during the transient condition. Summarizing the results obtained in these cases, it is possible to say that the developed strategy with bias correction shows significant improvement in the performance in terms of successful detection and diagnosis of step changes in the measurement signals, satisfactory tracking, and the dynamic compensation as compared to the case in which no FDD scheme is employed. VI. CONCLUSIONS When vanadium or rhodium SPNDs are used for in-core neutron flux monitoring and control, a system that can compensate for the delayed response of the SPNDs could be deployed. If such a system is based on a very accurate model of the SPNDs, it can also be exploited for detection and diagnosis of faults, if any, in the SPNDs. In this paper, a Kalman filter-based dynamic compensator is coupled with the GLR method for dynamic systems. Apart from dynamic compensation, the method performs simultaneous detection and correction of step changes in the signals of the VSPND or the associated circuit as a result of faults, which might be experienced during their operation. The fault correction is facilitated by the GLR method by virtue of its ability to quantify the fault magnitude. As established through simulation of realistic transients, the proposed hybrid method is


Fig. 7. Open-loop RR transient. Innovation sequence and actual, delayed, and estimated signals before and after bias correction for (a) . (d)

effective in obtaining prompt neutron flux variations from the delayed signal of the VSPND as well as in estimation of magnitude of step change in the signal along with the time of its

3319

, (b)

, (c)

, and

occurrence. Moreover, when correction of step change is incorporated, the estimated output matches closely with the true neutron flux variation. The proposed technique would be useful to

3320


Fig. 8. Variation of total power during demand power change.

other instruments like resistance temperature detectors, which are used extensively in nuclear reactors. If the fault correction is deployed along with closed loop control, overall accuracy and availability will improve. APPENDIX A PROOF TO LEMMA III.1 From (2), (6), and (13), just before and at the instant of occurrence of step change, we have (A1) as the term

is equal to zero, since it is the component

of fault before its occurrence. At , we can also compute which helps obtaining the innovation as a function of bias, at immediately next sampling instant . From (9) and (11), we have (A2)

Fig. 9. Different GLR outcomes during demand power change. FCT statistic on FDT rejections, bias estimate, and identified faulty VSPNDs at FCT rejection instants.

So (A8a)

Substituting

from (A1) in (A2) yields

(A8b) (A3)

. Hence, by comparing (A1) and (A3) respectively with (14) and (15), we have

From (A6), (A7), and (A8), it can be written that

as

or (A9)

(A4a) (A4b)

and from (A9)

The component of innovation for bias subsequent to the instant of occurrence of step change is given by

Substituting

(A5)

(A10)

(A6)

Hence, the variation of fault signature matrices and at any other time are expressed by the set of recursive equations

from (A3) yields

The conditional state estimate after correction through Kalman filter, when measurements up to instant are available, is as follows:

(A7)

(A11a) (A11b) For a hypothesized bias occurrence time (with reference to Section III-B), the above fault signature matrix is calculated for any instant after , as follows: (A12)


Fig. 10. Demand power change transient. Innovation sequence and actual, delayed, and estimated signals before and after bias correction for (a) . (c) , and (d)

3321

, (b)

,

3322


REFERENCES

TABLE IV GLR OUTCOMES FOR CHANGE IN THE DEMAND POWER

Fig. 11. Whiteness statistic.

The GLR test explained in Section III-C is applied to obtain the maximum likelihood estimate for the magnitude of bias over all choices for faulty VSPND . APPENDIX B TUNING OF THE KALMAN FILTER Whiteness of innovations of one of the VSPNDs is tested [32] for the different values for the tuning matrices. During the learning phase of the Kalman filter, a consistent estimate of covariance of innovations is computed as (B1) where is the th element of the innovation vector and is the number of observations in the time-window. If the innovations are white, the autocovariance (B2) has the following properties:

and (B3) is satisfied with a 5% violation. This test is called a whiteness test (WT), while the threshold of is called WT threshold. If the violation exceeds 5%, then the innovation sequence is declared as nonwhite. Fig. 11 shows the autocorrelation characteristic as a function of the lag index , when maximum value for it, , is taken as 1000 for . Violation of the WT threshold by this characteristic is calculated to be 0.16%. Since this is less than the 5% tolerance, innovation sequence can be declared white. Hence, the Kalman filter is found to be optimal and no more tuning is required [26].

[1] R. Isermann, “Process fault detection based on modeling and estimation methods- a survey,” Automatica, vol. 20, no. 4, pp. 387–404, 1984. [2] N. Tsoulfanidis and S. Landsberger, Measurement and Detection of Radiation, 3rd ed. Washington, DC: Taylor & Francis, 2010. [3] B. R. Upadhyaya and M. Skorska, “Fault analysis of in-core detectors in a pwr using time-series models,” in Proc. 22nd IEEE Conf. Decision and Control, Dec. 1983, vol. 22, pp. 758–763. [4] F. D. Maio, P. Baraldi, E. Zio, and R. Seraoui, “Fault detection in nuclear power plants components by a combination of statistical methods,” IEEE Trans. Reliab., vol. 62, no. 4, pp. 833–845, Dec. 2013. [5] R. A. Razak, M. Bhushan, M. N. Belur, A. P. Tiwari, M. G. Kelkar, and M. Pramanik, “Data reconciliation and gross error analysis of self powered neutron detectors: Comparison of PCA and IPCA based models,” Int. J. Adv. Eng. Sci. Appl. Math., vol. 4, pp. 91–115, 2012. [6] N. Kaistha and B. R. Upadhyaya, “Incipient fault detection and isolation of field devices in nuclear power systems using principal component analysis,” Nucl. Tech., vol. 136, pp. 221–230, 2001. [7] B. R. Upadhyaya, K. Zhao, and B. Lu, “Fault monitoring of nuclear power plant sensors and field devices,” Prog. Nucl. Energy, vol. 43, pp. 337–342, 2003. [8] B. Lu and B. R. Upadhyaya, “Monitoring and fault diagnosis of the steam generator system of a nuclear power plant using data-driven modeling and residual space analysis,” Ann. Nucl. Energy, vol. 32, pp. 897–912, 2005. [9] J. Ma and J. Jiang, “Applications of fault detection and diagnosis methods in nuclear power plants: A review,” Prog. Nucl. Energy, vol. 53, no. 3, pp. 255–266, 2011. [10] K. Roy, R. Banavar, and S. Thangasamy, “Application of fault detection and identification (FDI) techniques in power regulating systems of nuclear reactors,” IEEE Trans. Nucl. Sci., vol. 45, no. 6, pp. 3184–3201, Dec. 1998. [11] K. Roy, R. Banavar, and S. Thangasamy, “Temporal redundancy methods using risk-sensitive filtering and parameter estimation to detect failures in neutronic sensors during reactor start-up and steady-state operation,” IEEE Trans. Nucl. Sci., vol. 47, no. 6, pp. 2706–2714, Dec. 2000. [12] J. Ning and H. Chou, “Construction and evaluation of fault detection network for signal validation,” IEEE Trans. Nucl. Sci., vol. 39, no. 4, pp. 943–947, Aug. 1992. [13] F. Li, B. Upadhyaya, and S. Perillo, “Fault diagnosis of helical coil steam generator systems of an integral pressurized water reactor using optimal sensor selection,” IEEE Trans. Nucl. Sci., vol. 59, no. 2, pp. 403–410, Apr. 2012. [14] S. Narasimhan and C. Jordache, Data Reconciliation and Gross Error Detection: An Intelligent Use of Process Data. Houston, TX, USA: Gulf, 2000. [15] Y. V. Sagar, A. P. Tiwari, and S. B. Degweker, “Application of data reconciliation and fault detection and isolation of SPNDs in AHWR,” in Proc. 3rd Nat. Symp. Advances in Control Instrumentation (SACI), Nov. 2014, pp. 542–551. [16] A. S. Willsky, “A survey of design methods for failure detection in dynamic systems,” Automatica, vol. 12, no. 6, pp. 601–611, Nov. 1976. [17] B. Deb, J. Ross, A. Ivan, and M. Hartman, “Radioactive source estimation using a system of directional and non-directional detectors,” IEEE Trans. Nucl. Sci., vol. 58, no. 6, pp. 3281–3290, Dec. 2011. [18] B. Deb, “Iterative estimation of location and trajectory of radioactive sources with a networked system of detectors,” IEEE Trans. Nucl. Sci., vol. 60, no. 2, pp. 1315–1326, Apr. 2013. [19] D. Cooper, R. Ledoux, K. Kamieniecki, S. Korbly, J. Thompson, M. Ryan, N. Roza, L. Perry, D. Hwang, J. Costales, and M. Kuznetsova, “Intelligent radiation sensor system (IRSS) advanced technology demonstration (atd),” in Proc. 2010 IEEE Int. Conf. Tech. Homeland Security (HST), Nov. 2010, pp. 414–420. [20] D. Cooper, R. Ledoux, K. Kamieniecki, S. Korbly, J. Thompson, J. Batcheler, S. Chowdhury, N. Roza, J. Costales, and V. Aiyawar, “Intelligent radiation sensor system (irss) advanced technology demonstrator (atd),” in Proc. 2012 IEEE Int. Conf. Tech. for Homeland Security (HST), Nov. 2012, pp. 511–516. [21] D. Cooper, R. Ledoux, K. Kamieniecki, S. Korbly, J. Thompson, J. Batcheler, and J. Costales, “Integration of inertial measurement data for improved localization and tracking of radiation sources,” in Proc. 2013 IEEE Int. Conf. Tech. for Homeland Security (HST), Nov. 2013, pp. 613–617.


[22] A. Willsky and H. Jones, “A generalized likelihood ratio approach to the detection and estimation of jumps in linear systems,” IEEE Trans. Autom. Control, vol. AC-21, no. 1, pp. 108–112, Feb. 1976. [23] S. Narasimhan and R. S. H. Mah, “Generalized likelihood ratios for gross error identification in dynamic processes,” AIChE J., vol. 34, no. 8, pp. 1321–1331, 1988. [24] A. K. Mishra, S. R. Shimjith, T. U. Bhatt, and A. P. Tiwari, “Dynamic compensation of vanadium self powered neutron detectors for use in reactor control,” IEEE Trans. Nucl. Sci., vol. 60, no. 1, pp. 310–318, Feb. 2013. [25] R. A. Razak, M. Bhushan, M. N. Belur, A. P. Tiwari, M. G. Kelkar, and M. Pramanik, “Clustering of self powered neutron detectors: Combining prompt and slow dynamics,” IEEE Trans. Nucl. Sci., vol. 61, no. 6, pp. 3635–3643, Dec. 2014. [26] A. K. Mishra, S. R. Shimjith, T. U. Bhatt, and A. P. Tiwari, “Kalman filter-based dynamic compensator for vanadium self powered neutron detectors,” IEEE Trans. Nucl. Sci., vol. 61, no. 3, pp. 1360–1368, Jun. 2014.

3323

[27] Y. V. Sagar, A. P. Tiwari, S. R. Shimjith, M. Naskar, and S. B. Degweker, “Space-time kinetics modeling for the determination of neutron detector signals in Advanced Heavy Water Reactor,” in Proc. 2013 IEEE Int. Conf. Cont. Appl. (CCA), Aug. 2013, pp. 1224–1229. [28] R. K. Sinha and A. Kakodkar, “Design and development of the AHWR - the indian thorium fuelled innovative nuclear reactor,” Nucl. Eng. Design, vol. 236, no. 78, pp. 683–700, 2006. [29] L. A. Banda, “Operational experience in rhodium self-powered detectors,” IEEE Trans. Nucl. Sci., vol. 26, no. 1, pp. 910–915, Feb. 1979. [30] P. S. Maybeck, Stochastic Models, Estimation and Control. New York, NY, USA: Academic, 1982, vol. 1. [31] R. A. Johnson and D. W. Wichern, Applied Multivariate Statistical Analysis. Englewood Cliffs, NJ, USA: Pearson Prentice Hall, 2002, vol. 1. [32] P. Stoica, “A test for whiteness,” IEEE Trans. Autom. Control, vol. 22, no. 6, pp. 992–993, Dec. 1977.