A Kalman filtering of the Self-Powered Neutron Detector current to determine excess reactivity

Highlights

• Core external reactivity is to be unfolded by way of the detector reading.

• A coupled reactor flux-detector dynamic model is attempted.

• Response features of Rhodium and Cobalt SPNDs are compared in this regard.

• The rapid dynamics of Cobalt detector excludes the need for a compensator.

• The delayed dynamics of Rhodium detector is compensated through an Extended Kalman filter.

Abstract

Self-powered neutron detectors are common sensory devices for flux mapping purposes in nuclear reactors. A Kalman filtering strategy has been explored in this work to overcome the inherent delay in certain designs thereof enabling their application for the online monitoring and control activities. Determination of the core excess reactivity by way of the current reading in a Rhodium detector has thus been attempted as a generalization to the well-known reactor inverse kinetics problem. Efficient reactivity estimation is reported. Further analysis carried out for a Cobalt SPND excludes the need for additional current compensation as this detector predominantly enjoys a fairly rapid $\gamma$induced electric current.

Introduction

The determination of the core excess reactivity on the basis of the registered power profile is a well-known inverse (output → input) problem in the nuclear reactor instrumentation discipline and has long been addressed by several authors. Among all methods, the classical inverse kinetics (IK) method (Ciftcioglu and Geckinli, 1980) and the Kalman filtering (KF) (Peng et al., 2016) framework are the two most well-known strategies namely applied in the practice of control rod calibration by the way of flux detector reading (Chao et al., 2000). A comparative analysis on the performance of these two methods of reactivity estimation has been presented in (Bhatt et al., 2012) and the KF method is reported for superior performance especially in subcritical, low flux regime subject to more pronounced stochastic fluctuations. In practice however, the reactor flux is measured by means of in-core mounted detectors which display a flux proportionate current signal at the output.

Self-powered neutron detectors (SPNDs) are nuclear sensory devices frequently employed in the flux mapping activities (Goldstein and Todt, 1979) such as noise localization (Seidl et al., 2015) and control rod position monitoring (Hu et al., 2020) at high neutron flux thanks to their rugged design features. While enjoying an accurate steady state response, certain types thereof (e.g. Vanadium or Rhodium detectors) are subject to a relatively large rise time which induces a sensible delay in the measurement current (Mishra and Tiwari, 2014). Whereas Cobalt or Hafnium SPNDs have been employed in the industry as instant flux detectors (Ulybkin et al., 2020), the quest for efficient methods to overcome this delay is an active research issue. Several strategies have thus been proposed in the literature to suit Vanadium or Rhodium detectors for online protection and control objectives. Major lines of interest likewise include the direct inversion approach (Mayo and Yan, 1996), nonlinear Kalman filtering scheme (Mishra et al., 2014) or robust filtering techniques (Peng et al., 2014). In the first strategy, the measured flux and the detector current reading are reasonably interrelated b a transfer function template $(G\left(s\right)=\delta \phi \left(s\right)/\delta i\left(s\right))$which helps reproduce the unknown input flux using integration schemes. In the Kalman filtering strategy resorted in this work, an optimal predictor-corrector state observer is attempted which basically tends to minimize an incumbent Gaussian noise covariance on the estimated state vector. Non Gaussian uncertainty is however accounted for through more generalized robust estimation schemes which tends to minimize the pertaining H₂ or H_∞ norm incurred on the observed states by way the of the linear matrix inequalities framework (LMI). The performance in flux estimation of the inverse kinetics and Kalman filtering methods has been surveyed in certain recent attempts respectively for Vanadium and Cobalt SPNDs (Srinivasarengan et al., 2012) and Vanadium and Rhodium detectors in (Khoshahval et al., 2020). The advantage is expectedly granted to the Kalman filter which maintains better noise robustness features. Similar survey has been carried out in (Tamboli et al., 2016) among the Kalman and robust filtering techniques. The latter scheme however provides more flexibility in practice achieved at the price of a high gain filter. A classical Lead/Lag compensation strategy has meanwhile been explored in (Xu, 2017).

The cited works are all concerned with either flux → reactivity or current → flux unfolding problems. The generalized determination of excess reactivity by way of the detector reading (current → reactivity) combining these two schemes has not been formally reported in the literature yet. Indeed, the development of a closed form formalism such as the inverse point kinetics to explicitly express the core excess reactivity as a function of the registered detector current involves a nested integration chain over nonlinear inseparable terms (e.g.$N\phi$). Hopefully nonlinear estimation techniques such as the extended Kalman filtering serve this end and the work presented herein is a contributing attempt in this regard. The remainder of the manuscript is organized as follows: in the next section, preliminaries are presented for the models employed in the Rhodium and Cobalt detectors to measure the reactor core flux. A point kinetic model with adiabatic power feedback is adopted to describe the reactivity/flux dependence in the reactor core. Kalman filtering scheme is addressed in section three and a suitable template is attempted for the current → flux inverse problem. Results are finally presented in section four and the unfolding performance of both Rhodium and Cobalt detectors are explored in certain rapid reactivity injection scenarios. The paper ends up with concluding remarks.