Sensor position optimization for flux mapping in a nuclear reactor using compressed sensing

Highlights

• Novel approach for sensor position optimization based on compressed sensing (CS).

• Estimation error minimized by designing an optimal sensing matrix.

• Sensing matrix selection index considered for deriving optimized sensing matrix.

• Formulation of CS based method for optimization of sensor position in AHWR.

Abstract

The paper presents a novel approach to sensor position optimisation based on compressed sensing (CS) and its application to neutron flux estimation in a nuclear reactor. An online flux mapping system (FMS) with a suitable algorithm is needed to continuously monitor neutron flux distribution from the measurements of in-core detectors in large nuclear reactors. In the present work, sensor positions are optimized by designing a sensing matrix $\mathbf{\Phi }$ for the CS system in which the dictionary is known in advance. The sensing matrix is optimized by maximising the incoherence of the dictionary matrix and incorporating the estimation error in the optimization process. The proposed method uses the CS recovery technique, Basis Pursuit De-Noising (BPDN), for flux mapping along the vertical direction at the detector housing locations (1-D). In the end, the method suggests the optimized positions and number of sensors for efficient estimation of neutron flux inside the reactor core.

Introduction

The safe, reliable and economic operation of nuclear power plants essentially calls for accurate monitoring of neutron flux in the reactor core. In a reactor, core averaged flux is monitored by using out of core neutron detectors. However, in addition to this, Spatio-temporal variations in flux distribution encountered in large or loosely coupled reactors due to online refuelling, movements of reactivity devices, xenon induced oscillation etc. emphasise the need for flux (Sonavani et al., 2012) and power (Dubey et al., 1998) distribution monitoring inside the core. Flux distribution monitoring is crucial to ensure that safety limits imposed on fuel pellets and fuel clad barriers are not violated during the reactor operation. Hence, many of the commercial power reactors are equipped with a large number of in-core neutron detectors distributed inside the reactor core for accurate monitoring of the flux distribution. These detectors generate electrical signals proportional to the neutron flux in their close vicinity. Thus their reading represents localized measurements, which are processed by a suitable online flux mapping algorithm in order to generate the 3-D core power or flux distribution.

Reliable computation of 3-D flux map demands a large number of such in-core detectors inside the reactor core. Nevertheless, factors like reactivity load contributed by the detectors, physical and dimensional restrictions, safety implications of a large number of penetrations in the reactor vessel, and maintenance issues, put forth restriction on the number of such detectors. Hence, the optimum number and location of such in-core detectors need to be determined carefully, with a judicious trade-off between the economy, safety and measurement accuracy. Efficient flux mapping algorithms have been evolved in recent years with a goal to improve accuracy and to reduce computational effort. One of the methods presented in (Vashaee et al., 2008) estimates the neutron flux distribution and reactivity by using space–time kinetics model of the reactor and Kalman filtering technique. Some attempts towards optimizing the position and number of in-core detectors are evidenced in the literature. Application of generalized empirical interpolation method for systematic, simplified and accelerated detector placement in nuclear reactors is addressed in (Argaud et al., 2018). In another work (Li, 2019), three optimization methodologies namely Knight Moving Law, Random Forest and Correlation Coefficient (CC) are explored for a specific number of detectors to achieve the appropriate detector arrangement. Information theory is used to evaluate their performance and to bring out the superiority of the CC method over others. The recent work presented in (Anupreethi et al., 2020) optimised the locations of the in-core detector in AHWR by using a clustering approach. Here strongly correlated in-core detectors are grouped together to form a cluster and one in-core detector represents the rest in the cluster.

Beyond this, limited literature is found for sensor position optimization in a nuclear reactor. Most of the other literatures available are for sensor placement optimization related to sensor network (Akbarzadeh et al., Aug 2014); (Iqbal et al., July 2015) and (Nguyen et al., Dec 2016) for completely different application areas, such as battlefield surveillance, factory automation and environmental monitoring etc. These methods use gradient descent optimization with a realistic model (Akbarzadeh et al., Aug 2014), generic multi-objective optimization problem relating to wireless sensor network (WSN) (Iqbal et al., 2015) and a multi-objective firefly algorithm for estimating the location of nodes in WSNs (Nguyen et al., 2016). Most of the aforesaid methods use model based approach and experience of experts from different fields.

A solution to the sensor position optimisation problem can be achieved based on the CS approach (Bahuguna et al., 2018) by appropriately designing the sensing matrix. The work presented in (Lihi et al., 2011) proposes a framework for sensing matrix design that improves the ability of block-sparse approximation techniques to reconstruct and classify signals. The methods presented in (Abolghasemi et al., 2010) and (Li et al., 2015a) highlight the fact that small mutual coherence between the sensing matrix and representing (basis) matrix is essential for estimation using CS. The work presented in (Abolghasemi et al., 2010) proposed an algorithm where optimization is mainly applied to a random Gaussian matrix and shows higher reconstruction quality compared to those of the un-optimized case. The approach for sensing matrix optimization, presented in (Li et al., 2015a) takes the sparse representation error into account and hence leads to a more robust CS system. In another work, optimal sensing matrix design for the CS system is presented in (Li et al., 2015b) where the dictionary is known a priori and gradient based algorithm is derived to solve the optimization problem. In general, a dictionary is a redundant system consisting of prototype signals that are used to express other signals. Normally the sparsest representation is preferred for simplicity and easy intelligibility (Chen and Needell, 2016). A novel approach for the optimization of sensing matrix using CS is presented in (Xu and He, 2017) for hyperspectral unmixing. The orthogonal gradient descent method for sensing matrix optimization is used here to achieve optimal incoherence. An iterative procedure is developed in (Li et al., 2017) for searching the optimal dictionary, in which the dictionary update is executed using a gradient descent based algorithm. Another method of sensing matrix optimization is proposed in (Oey, 2018) by introducing the relative sparse representation error (SRE) parameter and incorporates it into the optimization problem. In most of the above mentioned methods, it is aimed to optimize the sensing matrix by minimizing the mutual coherence of the equivalent dictionary or SRE or both.

In CS, sensor position optimization problem can be posed as designing of a suitable sensing matrix to improve the signal reconstruction accuracy keeping the number of measurements same. The work presented in (Duarte-Carvajalino and Sapiro, 2009) and (Elad, 2007) has demonstrated that the performance of compressed sensing can be improved by carefully designing sensing matrices rather than choosing the arbitrary one. To account for maximum coverage with an optimum number of sensors a new approach has been suggested in this paper. The sensing matrix optimization based alternate approach has been presented for optimization of sensor position for flux mapping in a nuclear reactor. The sensing matrix is optimized here by minimizing the mutual coherence of the dictionary matrix and incorporating the estimation error in the optimization process. All possible combinations of sensor placement i.e. sensing matrices are considered for a given layout of the reactor and finally the sensing matrix, which minimizes mutual coherence and estimation error simultaneously, is selected.

The remaining part of the paper is arranged as follows. The basic theory underlying CS along with sensing matrix optimization is explained in Section 2. Approach for proposed method is presented in Section 3. The extension of CS theory for sensor position optimization in AHWR is presented in Section 4. Section 5 presents the test data set on which the performance of the proposed method is evaluated. Results and discussion on the proposed optimized position of the sensors for AHWR core with simulated test data set are presented in Section 6. In Section 7, the conclusion on the reported work along with the future scope of work is presented.