Power Level Control of Nuclear Power Plant Based on Asymptotical State Observer under Neutron Sensor Fault

Power level control is one of the critical functions in the instrument and control system of nuclear power plants (NPPs). In most power level control systems of NPPs, the power level or average neutron flux in reactor cores provided by out-of-core neutron sensors are usually measured as feedback of power control systems, while, as critical measuring devices, there is a risk of damage to out-of-core neutron sensors. For improving the operation reliability of NPPs under the neutron sensors’ failure, a power control system based on power observer is developed in this work. The simulation based on NPP simulator shows the power control system based on the observer is effective when neutron sensors fail.


Introduction
Ensuring safe and effective operations of nuclear power plants (NPPs) is the precondition during the development [1]. In terms of reactor power control, the implementation of desirable control for a core in an NPP is an important measure to improve security and effectiveness of the core.
In most power control systems of NPPs, the average neutron flux in reactor cores provided by out-of-core neutron sensors is usually measured as feedback of power control systems, which can be shown as Figure 1 [2]. During the past decades, some core power control methods are developed gradually, including optimal control [3,4], neural network [5][6][7], fuzzy logical control [8], slide mode control [9,10], model predictive control [11][12][13], and fractional order control [14,15]. ese control methods have to depend on the neutron flux measurement, while, as critical measuring devices, there is a risk of damage to out-of-core neutron sensors [16]. e neutron sensors are difficult to be replaced and maintained online. If neutron sensors fail, the power control system cannot work normally, and NPPs will be shut down further. On the contrary, the thermocouple in the secondary loop is convenient for replacement and maintenance. Meanwhile, the coolant temperature in secondary can also reflect the core power level. erefore, it is possible to rebuild the neutron flux and core power level according to the coolant temperature in the secondary loop, when the out-of-core neutron sensors fail.
For improving the operation reliability of NPPs under the neutron sensors' failure, a power control system based on power observer is developed in this work. Firstly, a common NPP dynamic model containing neutron kinetic and thermal-hydraulic is built, and then the formulation of control problem about power level control under neutron sensors' failure is presented; after that, based on the separation principle, the power level observer and power level controller are developed independently; at last, the effectiveness of the power level controller based on the observer is verified.

Modeling
As shown in Figure 2, the whole plant module of NPPs is mainly composed of a reactor core with thermal neutron, pressurized water, U-tube steam generator (UTSG), main pump, necessary connection pipes, and vessels [17]. Since this study focuses on modeling the dynamic response and sensors fault characteristics between the nuclear reactor power and primary coolant as well as proposing a proper control method under the neutron sensors fault to enhance the reliability of the control system, it is reasonable to adopt a simplified model of the lumped parameters core, coolant, and UTSG module, where the corresponding nodalization scheme is shown in Figure 2.

Neutron Kinetic.
According to the lumped parameter method, the nonlinear core model is established via using the point kinetics equations with six groups of delayed neutrons and two reactivity feedback mechanisms [18]. e normalized point kinetics equations for the critical core without external source of neutrons can be expressed as e reactivity feedbacks in this model are expressed as a function of the average temperatures of fuel and coolant. As a result, the total reactivity is the control rod reactivity combined with the feedbacks described as where the subscript 0 refers to a reference condition, taken at the rated power, n means the neutron density, ρ means the reactivity, β means the neutron fraction, Λ means the neutron generation time, λ i means the decay constant of i th group, c i means the delayed neutron precursor concentration of i th group, δ rod means the reactivity of rod, α f means the reactivity feedback of fuel, α coolant means the reactivity feedback of coolant in reactor core, T f means the temperature of fuel, T f means the temperature of fuel at the rated power, and T p means the average temperature of pressurized water of reactor core. It is assumed that the neutron density in steady state is n 0 , delayed neutron precursor concentration of i th group in steady state is c i0 , and then the neutron kinetic model can be linearized as follows: where Δ means the increment.

ermal-Hydraulic
Core. e reactor core is assumed to be a lumped cylindrical area as a function of the power [19]. e fission energy is released in the fuel elements in the form of heat.
is heat is then transferred to the upcoming pressurized water. is process is described as follows: where ρ f and ρ c mean the fuel density and the primary coolant density in the reactor core, respectively, V f means the fuel volume, P means the reactor power, U t means the heat transfer coefficient between fuel and primary coolant, A c means the fuel assemblies area, C f and C p the mean specific enthalpy of fuel and primary coolant, respectively, T f and T c mean fuel and primary coolant temperatures, respectively, and G p means the mass flow rate of primary loop. Accordingly, the increment form of equations (4) and (5) can be given as

Steam Generator.
Usually, steam generators in NPPs will be of U-tubes and counter current type. Single-phase pressurized water and two-phase water are in shell-side and tube-side of USTGs, respectively. In order to give a simple description for the control-oriented modeling, one-section model is adopted to describe the main thermodynamics of UTSGs: where T sgp means the average coolant temperature in the primary side of UTSGs, T sgpin and T sgpout mean inlet and outlet coolant temperatures in the primary side of UTSGs, T sgΔ means the temperature difference between the coolant flows in the two sides of UTSGs, which can be given by T sgΔ � T sgp − T sgs , where T sgp and T sgs mean average coolant temperatures in primary and secondary sides of UTSGs, respectively, T fw means the feedwater temperature, T sgsout means the outlet temperature in secondary side of UTSGs, U sg and A sg mean the heat transfer coefficient and area between the two sides of UTSGs, respectively, and V sgp and V sgs mean primary and secondary side volume of UTSGs, respectively. Accordingly, the increment form of equations (7) and (8) can be given as 2.3. Coupling Process. In the NPP model, (Δn, ΔP), (ΔT cout , ΔT sgpin ), and (ΔT sgpout , ΔT cin ) are three pairs of couple variables: ΔP � Σ f ΔnU e Β 3.125 × 10 10 Δn, where τ 1 is the time delay from reactor core outlet to UTSG inlet, and τ 2 is the time delay from UTSG outlet to reactor inlet.

Science and Technology of Nuclear Installations
If the control rod reactivity and feedwater temperature are regarded as system input, the coupling process of the NPPs can be shown as Figure 3, and the NPP model can be converted to state space (11): where

Formulation of Control Problem.
Generally, the full state feedback can be used for the whole plant control system. However, in this work, it is assumed that the control system cannot depend on neutron flux feedback from neutron sensors in core or out of core, so that, as a part of state feedback, the nuclear core power should be reconstructed by the output state. As mentioned in Section 1, the accuracy of temperature sensors in the secondary loop can be guaranteed, because maintenance and replacement of temperature sensors in the secondary loop are more convenient than the temperature sensors in primary loop. So, the outlet temperature increment is regarded as the output variable of system (11), as shown as the following equation: 3.125 × 10 10 0 · · · 0 .
e control problem in this work can be given as follows: For system (11) and output (14), design a control system to make the state x converge to 0, where Δn or ΔP cannot be measured.

Observer Design.
Accordingly, the observer of core power and core temperature can be given as follows: x � O(y). (17) In fact, equation (17) is an open loop observer. ere are two disadvantages for this observer: (1) e initial state x 0 is necessary (2) If x 0 or x 0 has deviation caused by system perturbation, the error e ≜ x − x will be unstable because the state matrix A has eigenvalues with a positive real part To improve observation performance, the closed loop observer needs to be built. Accordingly, the closed loop observer model can be given as follows [20]: e observation of equation (17) is asymptotically stable. x can be regarded as the estimate of x at any time. Actually, the key problem is the eigenvalues configuration of the matrix (A − LC). e system A T C T is controllable if the system A C is observable. Further, it can be shown that eigenvalues of (A T + C T K) can be configured arbitrarily. If L � −K T , one has where K means the state feedback matrix. It is obvious that eigenvalues of (A − LC) can also be configured arbitrarily.

Controller Design.
e controller designed in this work needs to depend on the estimate x.
e system with asymptotical state observer (17) can be rewritten as follows: Under the linear transformation where T � I n 0 It is obvious that T cannot change feature frequencies, so that the feature spectrums of (20) and (21) can be given as where According to equation (24), system (21) consists of a closed loop subsystem with state feedback x and an estimate subsystem. If the system A C is observable, the feature spectrum σ(A − LC) of the estimate subsystem can be configured arbitrarily by the matrix L; if the system A B is controllable, the feature spectrum σ(A + BK) can also be configured arbitrarily by the matrix K.
Because of the characteristic of separation design, the feedback matrix can be designed according to the algorithm of pole assignment: (1) If A c B c C c is the controllable form of system (11), calculate the characteristic polynomial P(s) where α � α 0 . . . α 12 , α � α 0 . . . α 12 , and

Reference Design.
To verify the efficiency of the control strategy and the controller in Section 3, the CPR1000 is regarded as the reference design. CPR1000 (Chinese pressurized reactor) is a three-loop 1000 MWe pressurized water reactor (PWR). It is an advanced, mature, safe, and economical nuclear reactor which is an updated version of Daya Bay Nuclear Reactor. e CPR1000 utilizes the proven technology, which builds on nearly 40 years of operating PWR experience. e major parameters of CPR1000 are listed in Table 1 [21].
Meanwhile, the simulator with CPR1000 type is used for accurate dynamic simulation. e structure of CPR1000 simulation model is shown schematically in Figure 4. For ease of the verification, three components of the model structure are extracted: the reactor core, primary loop, and USTG. e steam turbine, condenser, third loop, and other auxiliary components are instead of temperature and pressure boundaries.

Poles Distribution.
According to the control strategy in Section 3.1, the closed loop poles without the controller in Section 3.2 can be given as Figure 5. It is obvious that the system is not stable, because there are a pair of complex poles in the right half of the plane (20.8 ± 25.5i), so that we should reconfigure the unstable poles according to the method in Science and Technology of Nuclear Installations Section 3.2. e pair of complex poles is reset at 0.22 ± 0.47i, which can be shown in Figure 6. Meanwhile, three pairs of complex poles, 0.22 ± 0.47i, 0.324 ± 0.00836i, and 0.361 ± 0.101i, become the predominant poles. Accordingly, the feedback matrix can be given according to equation (26). Furthermore, the observer of core power can also be designed according to equation (18).

Dynamic Response.
For verifying the effectiveness clearly, two dynamic responses are calculated: one is the response of the linear model equation (11), and the other is the response of the model in the CPR1000 simulator.
In the initial condition, the core power is at 80% FP, and the mass flow of primary loop is kept constant. At the 1 st sec, the setting power is changed from 80% FP to 100% FP. It is assumed that the out-of-core neutron sensors fail completely.
e neutron flux feedback can be given by the observer equation (18), where the input contains core outlet coolant temperature and UTSG outlet coolant temperature in secondary side. e dynamic response of core power, core outlet temperature increment, and UTSG outlet temperature increment can be shown as Figures 7-9. Before 100 th sec, the core power and temperature increment dynamic response of system equation (11) have an overshoot of less than 10% and 5%, respectively. After 400 th sec, the core power and temperature increment tend towards stability. In comparison, the power and temperature increment dynamic responses     have same steady state although that they have longer response time.
is is due to that some high-frequency components in the simulator are neglected in the linear model equation (11).

Conclusion
In this work, a power control system of NPPs based on power observer is developed. When the out-of-core neutron sensors fail, the traditional power level control system cannot work because accurate neutron flux and total power in the reactor core cannot be provided. However, because of the separation principle, the neutron flux or total power in the reactor core can be observed by the coolant temperature in the secondary loop and independent from power controller design, so that the state observer and controller are designed, respectively, and the effectiveness of the power level controller based on the state observer is verified by the NPP simulator. e analysis shows consistency of results between the linear model and NPP simulator, which indicates that the power level controller based on the state observer is effective, although there is difference in the dynamic response, which is caused by the truncation of the high frequencies.
In the future work, we will focus on the nonlinear core power observer and eliminate the dynamic deviation caused by the truncation for high frequencies.
Data Availability e nuclear data used to support the findings of this study have not been made available because the limit of the law in China.

Conflicts of Interest
e authors declare that they have no conflicts of interest.  Science and Technology of Nuclear Installations 7