Based on the point reactor neutron dynamics with six groups of effective delayed neutron and thermal-hydraulic coupling methods, taking into account the reactivity feedback caused by the temperature change of core coolant, fuel, and cladding, the ordinary differential equations are shown in Eq. 1 to describe the nonlinear model of LFR core (Lorenzi et al., 2013). { d C i ( t ) d t = β i Λ ⋅ N r ( t ) − λ i ⋅ C i ( t ) , i = 1,2 … 6 d N r ( t ) d t = ρ ( t ) − β Λ ⋅ N r ( t ) + ∑ i = 1 6 λ i ⋅ C i ( t ) M f ⋅ C f d T f d t = P ( t ) − U f , c ( T f − T c ) M c ⋅ C c ⋅ d T c d t = U f , c ( T f − T c ) − U c , a v e ( T c − T a v e ) M a v e ⋅ C a v e ⋅ d T a v e d t = U c , a v e ( T c − T a v e ) − 2 G ( t ) ⋅ C a v e ⋅ ( T a v e − T i n ) ρ ( t ) = ρ r o d + α f ( T f − T a v e ) + α c ( T c − T c 0 ) + α a v e ( T a v e − T a v e 0 ) + α i n ( T i n − T i n 0 ) ρ r o d = α r o d △ h (1) where the relationship between relative core power and relative neutron density should conform to Eq. 2. { N r ( t ) = N ( t ) N 0 , P r ( t ) = P ( t ) P 0 N r ( t ) = P r ( t ) (2)

The linear model is established based on the nonlinear reactor core model by introducing small perturbations and ignoring the high-order term (Li and Zhao, 2013; Yinuo, 2021). The ordinary differential equations are shown in Eq. 3 { d δ P r ( t ) d t = ρ 0 − β Λ δ P r ( t ) + ∑ i = 1 6 λ δ C i + ρ r o d + α f δ T f + α c δ T c + α a v e δ T a v e + α i n δ T i n Λ P r 0 d δ C i ( t ) d t = β i Λ δ P r − λ δ C i ( t ) , i = 1,2 … 6 M f C p , f d δ T f d t = δ P r ( t ) P 0 − U f , c [ δ T f ( t ) − δ T c ( t ) ] M c C p , c d δ T c d t = U f , c ( δ T f − δ T c ) − U c , a v e ( δ T c − δ T a v e ) M a v e C p , a v e d δ T a v e d t = U c , a v e ( δ T c − δ T a v e ) − 2 G C p , a v e ( δ T a v e − δ T i n ) (3)

Following Eq. 3, the state space model of the reactor core is obtained as Eq. 4. { x ˙ = A x + B u y = C x + D u (4) where u = [ δ ρ r o d , T i n ] T is the input, y = [ δ P r , δ T f , δ T c , δ T a v e ] T is the output, x = [ x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 , x 8 , x 9 , x 10 ] T = [ δ P r , δ C 1 , δ C 2 , δ C 3 , δ C 4 , δ C 5 , δ C 6 , δ T f , δ T c , δ T a v e ] T is the R^10×10 state array, A is the R^10×10 system matrix, B is the R^10×2 input matrix; C is the R^4×10 output matrix, D is the R^10×2 zero matrix. A , B , C , and D are represented by A = [ ρ 0 − β Λ λ 1 λ 2 λ 3 λ 4 λ 5 λ 6 P r 0 Λ ⋅ α f P r 0 Λ ⋅ α c P r 0 Λ ⋅ α a v e β 1 Λ − λ 1 0 0 0 0 0 0 0 0 β 2 Λ 0 − λ 2 0 0 0 0 0 0 0 β 3 Λ 0 0 − λ 3 0 0 0 0 0 0 β 4 Λ 0 0 0 − λ 4 0 0 0 0 0 β 5 Λ 0 0 0 0 − λ 5 0 0 0 0 β 6 Λ 0 0 0 0 0 − λ 6 0 0 0 P 0 M f C p , f 0 0 0 0 0 0 − U f , c M f C p , f U f , c M f C p , f 0 0 0 0 0 0 0 0 U f , c M c C p , c − U f , c + U c , a v e M c C p , c U c , a v e M c C p , c 0 0 0 0 0 0 0 0 U c , a v e M a v e C p , a v e − U c , a v e + 2 G C M a v e C p , a v e ] B = [ P r 0 Λ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 G C ] C = [ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 ] D = [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ]

Model predictive control (MPC), also known as receding horizon control, has gained attention in academia and industry due to its ability to optimally control nonlinear systems subject to physical constraints (Brian Froisy, 1994; Mayne, 2000; Hu and Ding, 2019). The structure of the MPC controller is shown in Figure 1.

As shown in Figure 2, the I/V converter converts reactor core power into voltage signal V(n) to be the feedback signal of the core power control system with the MPC controller, and the MPC controller regulates the electrical signal and outputs to the control rod drive mechanism. The control rod drive mechanism drives the control rod to move in the reactor according to the relative power deviation. Inserting or withdrawing reactivity in the reactor adjusts the reactor power back to the setting value and finally makes the core power stable (Hu, 2021; Yinuo, 2021). As shown in Figure 3, based on the operation strategy of stable core power with MPC controller, the core power control simulation system with MPC controller is designed by using the MPC design module of MATLAB/Simulink, taking 0.1s as the sample time. The V(n) of the core power system is the Y and the ΔV1 is Δu of the MPC controller.

At 100%FP, the response of the LFR core under the power control system with 20 pcm step reactivity is shown in Figure 4. Figure 5 shows the response of the LFR core power control system under the 5% step down of coolant inlet temperature. With the MPC controller or the PID controller, relative core power, core fuel temperature, core coolant average temperature, and core-cladding temperature finally return to the initial level, introducing step reactivity. In Figures 4, 5, although both have great control characteristics, the dynamic characteristics of the core power control system with MPC controller are better than the PID controller in terms of MO and transient time.

In the frequency domain, the core power control system is linearized using the APP linearize model in MATLAB/Simulink, and the Nyquist diagram, Pole-Zero Map , and Bode diagram are drawn. As shown in Figures 6, 7, the Nyquist diagram and Pole-Zero Map are drawn to judge the absolute stability of the core power control system. According to the Nyquist diagram, the number of times that the Nyquist curve bypasses point (−1,0) counterclockwise is zero, and the number of characteristic roots outside the unit circle is zero. According to the Nyquist criterion, both closed-loop systems are stable. And as shown in Figure 8, the Bode diagram is drawn to compare the relative stability of different core power control systems. It can be seen from the Bode diagram that both phase margin and gain margin are stable in the low-frequency region.