Kalman Filter-Based Dynamic Compensator for Vanadium Self Powered Neutron Detectors

Large nuclear reactors employ a wide variety of in-core detectors to determine the neutron flux distribution within the core. Among them Vanadium SPNDs are extensively used for flux mapping applications due to their accuracy. However, the slow response characteristics of Vanadium detectors precludes their direct use for reactor protection and regulation applications. Neverthless, by overcoming their inherent time delay, it is possible to use them in such applications. Moreover, benefits offered by Vanadium SPNDs like better life span, simple response characteristics, easiness in handling the replaced SPNDs etc., make them desirable candidates for such applications. Therefore, a method to improve the response time of Vanadium SPNDs would enable them to be utilized for reactor control applications as well as to fulfill core monitoring and surveillance requirements. In this paper, a Kalman filter-based compensator is proposed for online dynamic compensation of Vanadium SPND. Moreover, compensated flux obtained by Kalman filter is compared with the compensated flux obtained from existing dynamic compensators i.e. Direct Inversion and Tustin, and robustness of proposed algorithm is studied. The compensator is validated using the plant data collected from the 540 MWe PHWR units in India. It is established that the compensated Vanadium SPND signals are in very good agreement with the prompt Cobalt SPND signals. This puts the possibility of using Vanadium SPNDs in lieu of Cobalt SPNDs for reactor protection and regulation applications.

Nomenclature

System matrix

Compensated VSPND signal

Identity matrix

Expectation operator

State transition matrix

Prompt fraction of output signal of VSPND

Delayed fraction of output signal of VSPND

Transfer function of VSPND

Measurement matrix

Null hypothesis

Total output current of VSPND

Steady–state output current of VSPND at time zero

Gain matrix for Kalman filter at time

Number of sample data points

Number of ion chambers, 3

Number of Co SPNDs, 42

Number of steady state sample points

Number of VSPNDs, 102

Dimension of measurement, 1

Power from th Co SPND, % Full Power (FP)

Bulk power from Co SPNDs, % FP

Bulk power from ion chambers, % FP

a–priori error covariance matrix at discrete time

a–posteriori error covariance matrix at discrete time

Uncompensated power from th VSPND, % FP

Uncompensated bulk power from VSPNDs, % FP

Probability that a reaction leads to an electron contributing to the detector current

Probability that a reaction leads to an electron contributing to the detector current

Process noise covariance matrix

Measurement noise covariance

Reference signal (Cobalt SPND/Ion Chamber)

Complex frequency in Laplace transformation

Steady state calibration factor

Sampling time

Scalar measurement noise at time

Process noise vector at time ,

State vector in Kalman filter,

a–priori state estimate vector,

a–posteriori state estimate vector,

Intermediate state variable in DI compensator

Scalar measurement output

Complex number in Z–transform

Chi-square distribution with n degrees of freedom

Kronecker delta function

Decay constant of

Zero mean white noise

Thermal neutron flux sensed by SPND

Thermal neutron flux reconstructed by compensator

Steady–state neutron flux at time zero

Covariance of innovation sequence

Absorption cross - section of , 4.5 barns

Time constant of , 325.47 seconds

Innovation at th time step

State variable in Kalman filter