# Digitalization of the Ex-Core Neutron Flux Monitoring System for APR1400 Nuclear Power Plant
## 2 Design of the Digital ENFMS for APR1400
### Application of the Extended Kalman Filter Algorithm  
Extended Kalman filter (EKF) is utilized for optimal estimation of neutron flux level and reactor period information. The neutron flux and the reactor period have the relationship [17; 18]:  
\[P(t)=P_{0}e^{\frac{t}{T_{e}(t)}} \tag{18}\] \[=P_{0}10^{SUR(t)\times t}\] (19) \[T_{e}(t)=\frac{26.06}{SUR(t)} \tag{20}\]where \(P(t)\) is reactor power at time \(t\) [\(\%\)], \(P_{0}\) is reactor power at \(t=0\) [sec. ], \(T_{e}(t)\) is the reactor period [sec.] (time required for reactor power to change by a factor \(e\), controlled by plant operators through movement of control rods and changing the boron concentration), and \(SUR(t)\) is the reactor start-up rate [min\({}^{-1}\)] (number of factors of ten that reactor power changes in a minute). When the logarithmic reactor power \(E(t)\) is assumed to have a logarithmic relationship with \(P(t)\) and ranges from 0 to 10 corresponding to the reactor power from \(2\times 10^{-8}\) to \(2\times 10^{2}\) [\(\%\)], it gives:  
\[E(t)=\log_{10}[P(t)]+8-\log_{10}2=\frac{\log_{10}e}{T_{e}(t)}t+\log_{10}P_{0} +7.7. \tag{22}\]  
The discrete-time version of Equation (22) and the inverse of reactor time can be expressed as:  
\[E(k+1)=E(k)+\log_{10}e\left\{\frac{(k+1)\Delta t}{T_{e}([k+1]\Delta t)}-\frac {k\Delta t}{T_{e}(k\Delta t)}\right\} \tag{23}\]  
\[\frac{1}{T_{e}([k+1]\Delta t)}=\frac{1}{T_{e}(k\Delta t)+dT_{e}(k\Delta t)} \tag{24}\]  
where \(\Delta t\) is the discrete-time sampling interval. If \(\Delta t\) is too small, Equations (23) and (24) can be approximated as:  
\[E(k+1)\simeq E(k)+\frac{\Delta t}{T_{e}(k)}\log_{10}e \tag{25}\]  
\[\frac{1}{T_{e}(k+1)}\simeq\frac{1}{T_{e}(k)}-\frac{dT_{e}(k)}{T_{e}(k)^{2}}. \tag{26}\]  
The perturbed process can be described by the following discrete-time state space realization:  
\[\left[ \begin{array}{c}E(k+1)\\ \frac{1}{T_{e}(k+1)}\end{array} \right]=\left[ \begin{array}{cc}1,&\log_{10}e \times\Delta t\\ 0,&1\end{array} \right]\left[ \begin{array}{c}E(k)\\ \frac{1}{T_{e}(k)}\end{array} \right]+\left[ \begin{array}{c}0\\ -\frac{dT_{e}(k)}{T_{e}(k)^{2}}\end{array} \right]+\left[ \begin{array}{c}w_{ E}(k)\\ w_{T_{e}}(k)\end{array} \right] \tag{27}\]  
where \(w_{E}(k)\) and \(w_{T_{e}}(k)\) are the state error covariances. With a two-dimensional state vector, \(x(k)\) is defined as  
\[\mathbf{x}(k)=\left[ \begin{array}{c}E(k)\\ \frac{1}{T_{e}(k)}\end{array} \right] \tag{28}\]  
and Equation (27) can be expressed as  
\[\mathbf{x}(k+1)=\mathbf{F}\mathbf{x}(k)+\mathbf{B}\mathbf{u}(k)+\mathbf{w}(k) \tag{29}\]  
where \(\mathbf{F},\mathbf{B},\mathbf{u}(k)\) and \(\mathbf{w}(k)\) are the state transition matrix, input coupling matrix, input vector, and state error vector. The reactor power measurement with measurement noise can be described as  
\[\left[ \begin{array}{c}E_{m}(k)\\ \frac{1}{T_{e-m}(k)}\end{array} \right]=\left[ \begin{array}{cc}1,&0\\ 0,&1\end{array} \right]\left[ \begin{array}{c}E(k)\\ \frac{1}{T_{e}(k)}\end{array} \right]+\left[ \begin{array}{c}v_{E}(k)\\ v_{T_{e}}(k)\end{array} \right] \tag{30}\]  
\[z(k)=\mathbf{H}\mathbf{x}(k)+\mathbf{v}(k) \tag{31}\]  
where \(\mathbf{H}\) and \(\mathbf{v}(k)\) are the observation matrix and the measurement error vector.