Point Kinetics Model Research Articles

An improved control rod selection algorithm for core power control at TRIGA PUSPATI Reactor

The 1 MWth TRIGA PUSPATI Reactor known as RTP undergoes more than 37 years of operation in Malaysia. The current core power control utilized Feedback Control Algorithm (FCA) and a conventional Control Rod Selection Algorithm (CRSA). However, the current power tracking performance suffers and increase the workload on Control Rod Drive Mechanism (CRDM) if the range between minimum and maximum rod worth value for each control rod has a significant difference. Thus, it is requiring much time to keep the core power stable at the power demand value within the acceptable error bands for the safety requirement of the RTP. In conventional CRSA, regardless of the rod worth value, the lowest position of the control rod is selected for up-movement to regulate the reactor power with 2% chattering error. To improve this method, a new CRSA is introduced named Single Control Absorbing Rod (SCAR). In SCAR, only one rod with highest reactivity worth value will be selected for coast tuning during transient and the lowest reactivity worth value will be selected for fine-tuning rod movement during steady-state. The simulation model of the reactor core is represented based on point kinetics model, thermal-hydraulic models and reactivity model. The conventional CRSA model included with control rod position dynamic model and actual reactivity worth curve data from RTP. The FCA controller is designed based on Proportional-Integral (PI) controller using MATLAB Simulink simulation. The core power control system is represented by the integration of a reactor core model, CRSA model and FCA controller. To manifest the effectiveness of the proposed SCAR algorithm, the results are compared to the conventional CRSA in both simulation and experimentation. Overall, the results shows that the SCAR algorithm offers generally better results than the conventional CRSA with the reduction in rising time up to 44%, workload up to 35%, settling time up to 26% and chattering error up to 18% of the nominal value.

Dhruva simulator: core neutronics and thermal hydraulics modules

It is widely recognized that simulators play an essential and extremely important role in training programs for nuclear power plants (NPPs). The full-scope training simulators provide such in-depth training tools for reactor operators. A full-scope simulator has been developed for research reactor, Dhruva at BARC. The details about the development activities are covered in this paper. Safety analysis and operational experience consistently indicate that human error is one of the key contributors to the risk of a severe accident in a nuclear reactor. The use of full-scope simulator in the training of operators is an essential element to reduce human error. This paper presents the core neutronics and thermal hydraulics of model of Dhruva reactor incorporated in the Dhruva simulator. The core neutronics module is based on point kinetics model which estimates the reactor power depending on the core reactivity. The simulator can reproduce the different operational scenario, i.e., reactor operation under equilibrium and non-equilibrium condition, xenon transients, loss of regulation incidents, etc. The simulator will also be used to study the role of human reliability in safety of nuclear plants. The simulator was so designed that over and above training of the plant personnel, it will be used for safety studies in the area of reactor physics and thermal hydraulics. The interface was designed to support the interaction of the operators with the simulator through virtual panels supported by touch screens with high-fidelity graphic displays. Important Postulated Initiating Events (PIEs) like Loss of Offsite Power, Loss of Coolant Accident, Loss of Regulation Incident, etc. have been modeled. This information will provide an insight to verify and validate the plant Emergency Operating Procedures (EOPs). The simulator will also be used for development of human factor models as well as the design of future Visual Display Unit based control rooms. This will also enable the operators to align themselves from the traditional panel-based Human–Machine Interface to personal computers.

Neutron point kinetics model with precursors’ shape function update for molten salt reactor

The circulation of liquid fuel is a feature of innovative nuclear reactors, such as Molten-Salt Reactors, and requires a specific neutronics model to take it into account. In this paper we propose a point kinetics model, for a hypothetical one-dimensional reactor, that is based upon and improves one that already exists in the literature. The model is therefore solved numerically and compared to spatial kinetics, which is considered to be the exact solution. The approximations that were done, which best portray the phenomena associated to the problem, are discussed to explain the improvements obtained with the proposed model.

Estimation of kinetics parameters by Monte Carlo fixed-source calculations for point kinetic analysis of accelerator-driven system

ABSTRACT The point kinetics model for an initially subcritical system such as the accelerator-driven system (ADS) requires the adjoint-weighted kinetics parameters to be estimated in its reference subcritical state with external sources. Monte Carlo (MC) algorithms are newly developed to calculate the kinetics parameters weighted by the adjoint function, which is the solution to the inhomogeneous adjoint equation, in the MC fixed-source mode calculations. The developed algorithm is verified through a simple two-group problem by comparing with analytic solutions. The one-point kinetics analyses using the estimated kinetics parameters are applied for beam trip transient problems and a control rod drop experiment in ADS at the Kyoto University Critical Assembly.

Improvement of MARS-KS sub-channel module using CTF code

In this work, the sub-channel analysis module of MARS-KS nuclear safety analysis code was upgraded using CTF sub-channel analysis code for realistic prediction of the multi-component and multi-dimensional nuclear core thermal hydraulics. The semi-implicit pressure coupling method was used to couple these two codes. Both codes use the finite difference method and semi-implicit numerical scheme and hence they were coupled together by exchanging volume/cell pressures through the boundary interfaces between 1D (i.e., system) and 3D (i.e., sub-channel) computational regions via implicit phasic velocities at the boundary. The pressure linkage was simply accomplished by designating the 1D and 3D regions as implicit sinks in the CTF sub-channel module and time dependent volumes in the MARS-KS code, respectively. The heat structure component and point kinetics models of the MARS-KS code were also coupled with the CTF core sub-channel module for the analysis of the whole reactor system. Automatic initialization of the 3D vessel was performed with CTF null-transient calculation to initialize all computational cells using a single set of initial and boundary conditions. The improved MARS-KS code has been verified and validated using various problems and tests. This paper selectively presents the results of three analyses, including a conceptual manometer oscillation, a blowdown test, and a 3 × 3 hot rod sub-channel experiments, with the improved code. All the analyses showed good predictions and this confirms the semi-implicit pressure coupling as well as the capability of the improved MARS-KS code. In the future, parallel processing should be implemented for the improved MARS-KS core sub-channel module to perform full core sub-channel analysis.

Derivation of the Two-Exponential Probability Density Function for Rossi-Alpha Measurements of Reflected Assemblies and Validation for the Special Case of Shielded Measurements

Rossi-alpha measurements of fissionable assemblies are used to estimate the prompt neutron decay constant α. Reactivity can be inferred from α if the values of the neutron generation time and effective delayed neutron fraction are assumed. If multiple measurements are performed on an assembly near delayed critical, one can determine α at delayed critical and directly infer reactivity (without needing to assume values for the neutron generation time or effective delayed neutron fraction). Previous works have demonstrated that two-exponential fits for Rossi-alpha measurements of reflected assemblies have better fit metrics than those of one-exponential fits; however, the two-exponential probability density function that is needed to obtain α from the fit parameters has not been derived. This paper derives the two-exponential fit based on a two-region point kinetics model for Rossi-alpha measurements of reflected assemblies, a generalization of the current, one-region model (one-exponential fit). The new model is validated for shielded assemblies, a special case of reflected assemblies where the reflector-to-core leakage is negligibly small. The validation is performed using shielded, fissionable assemblies (highly enriched uranium with keff ≈ 0.95 and weapons-grade plutonium with keff > 0.77). The results show that the two-exponential model can (1) predict the constant α within two standard deviations, and (2) deconvolve α and the time a neutron spends in the reflector region, neither of which is possible with the one-exponential model.

Computational effort comparison of genetic algorithm and particle swarm optimization algorithms for the proportional–integral–derivative controller tuning of a pressurized water nuclear reactor

Power level control is important in nuclear power plants, and some dynamic parameters depend on the output power. One of the popular controllers is proportional–integral–derivative (PID), that the researchers are interested in tuning its gains. In this paper, computational effort comparison of genetic algorithm (GA) and particle swarm optimization (PSO) algorithm is aimed to tune the PID gains for the typical pressurized water reactor (PWR) load-following, based on the point kinetics model with and without the parametric uncertainties. The integral of time-weighted square error (ITSE) performance index is used in the GA and PSO metaheuristic algorithms. The number of function evaluations (NFE) is used for the computational effort comparison as the novelty of this work. The results of both optimization methods show the high performance of the closed-loop metaheuristic–PID controller for the desired load-following with robustness to the parametric uncertainties. However, PSO achieves the optimal solution in less NFE.

On a point kinetic model for nuclear reactors considering the variation in fuel composition

The present discussion is an extension of a new model recently developed for neutron point kinetics, where the reactivity is decomposed in terms of short and long timescales. The short timescale corresponds to the operational control of the reactor, while the long timescale corresponds to the effects of the chemical composition variation in the nuclear fuel as a consequence of fission and neutron capture processes. This work considers not only the effects of the main neutron poisons but contains also a first analysis of the effects of some transuranium. The proposed model consists of a system of non-linear equations, which is solved by the use of the Adomian decomposition method. This method expands the solution into an infinite series, whose contributions are obtained by the solution of a recursive equation system where the non-linear terms are treated as source terms. These are calculated by so-called Adomian polynomials and are constructed from the solutions obtained in previous recursion steps. An approach is proposed, that diagonalizes the linear matrix in order to circumvent the existing stiffness problem, which is also responsible for instabilities in the computational algorithm. Two case studies are presented, analysing the non-linear behaviour of nuclear reactor kinetics with constant reactivity using input parameters for which the algorithm was stable. Further, a standard point kinetics case with periodic reactivity is performed in order to compare the present model to a standard point kinetics simulation from the literature.

Modelo reducido de un reactor nuclear y su utilización en las asignaturas del ámbito de la Ingeniería Nuclear

<p>En el grado de Ingeniería de la Energía, los alumnos pueden cursar las asignaturas de Tecnología Nuclear y Seguridad Nuclear. En ambas asignaturas se estudia el diseño de un reactor nuclear para que la central funcione en condiciones seguras. Para conseguir que los alumnos entiendan el funcionamiento de un reactor nuclear es interesante el uso de modelos matemáticos que reproduzcan tanto la potencia generada como la evolución de otras variables que afectan a la seguridad de la planta. En este trabajo, se presenta un modelo sencillo de reactor de agua a presión que permite observar la evolución temporal de la potencia del reactor y las temperaturas de los componentes más importantes de la planta. Para obtener la potencia generada se hace uso del modelo de cinética puntual, mientras que para el cálculo de las temperaturas del combustible y del refrigerante se utiliza un modelo sencillo de transmisión de calor. <br />De este modo, el problema se formula mediante un sistema de ecuaciones diferenciales ordinarias que se resuelve utilizando métodos numéricos. El desarrollo del modelo permite a los alumnos de Tecnología Nuclear y Seguridad Nuclear comprender la evolución temporal de alguna de las variables de un reactor y darse cuenta de los efectos estabilizantes que los parámetros de la termohidráulica tienen sobre la potencia generada.</p>

Core Power Control of a Nuclear Research Reactor During Power Maneuvering Transients Using Optimized PID-Controller Based on the Fractional Neutron Point Kinetics Model With Reactivity Feedback Effects

Core power control of a nuclear reactor during power maneuvering transients is an important issue in a nuclear power plant. This paper, for the first time, presents an optimized PID controller for a nuclear research reactor with a new model of the reactor core, which is based on the fractional neutron point kinetics (FNPK) equations. This model preserves the main dynamic properties of the neutron movement in which the relaxation time related to a quick change in the neutron flux includes a fractional order, acting as an exponent of the relaxation time, to show the dynamic characteristics of the nuclear reactor in the best way. The physical justification of the fractional exponent is associated with nonFickian impressions from the neutron diffusion equation point of view. In addition, in this paper, for the first time, feedback reactivity effects are considered for FNPK model. Genetic algorithm (GA) is used to optimize gains of the PID controller. The objective function is according to the minimization of the integral of time-weighted absolute error (ITAE). The simulation result demonstrates that this optimized control method has satisfactory performance and stability during output-tracking process for higher values of fractional order.

Stability analysis of linear FNPK model considering reactivity feedback effects for a research nuclear reactor

Fractional neutron point kinetics (FNPK) model has been known for some researchers in the field of nuclear science and engineering for less than a decade. This modeling has a better approximation and physical interpretation for some phenomena, specially, when large variations of neutron cross sections occur and the dynamic behavior of neutron is different. This paper, for the first time, presents a fractional neutron point kinetics model which consists of three delayed neutron groups and the reactivity feedback effects (temperature feedbacks and poison feedbacks). Also, Stability Analysis of linear FNPK model considering reactivity feedback effects for a Research Nuclear Reactor is done. The model has been linearized around the equilibrium operating point. Then, the closed-loop linear fractional neutron point kinetics model considering reactivity feedback effects is transformed from S-domain to W-domain. For stability evaluation, two strategies have been implemented. The first strategy is the assessment of the region of closed-loop poles in the principal Riemann sheet (PRS) for the transformed system. The second one is the step response evaluation of the linear fractional-order closed-loop model for different values of the model parameters (including relaxation times and anomalous diffusion coefficients). Simulation results demonstrate that the stability of the system depends on the values of these model parameters and for one scenario, the closed-loop poles are located in the unstable region of PRS and the step response is unstable. However, for the others, it is stable and settles at an equilibrium value. In addition, a smaller anomalous diffusion coefficient (α), leads to a more unstable closed-loop model (have poles closer or inside the unstable region). Comparative analysis is shown that reactivity feedback effects are considerable on the stability of the research nuclear reactor. Specially, feedback by xenon has a negative effect on the stability and reduces the stability of system. Also, the stability highly depends on the values of α and τ.

Verification of a HC-PK-CFD coupled program based a benchmark on beam trip transients for XADS reactor

It is important to employ multi-physics coupling strategy in thermal-hydraulics/safety analysis of nuclear reactors. In general, it is based on couplings of system codes with neutron dynamics codes, which is incompetent to cope with multi-dimensional thermal-hydraulics/safety problems involved in liquid metal cooled pool-type reactors. To cope with it, Computational Fluid Dynamics tools are good choices which have attracted considerable attentions in reactor simulation these years. In order to apply a commercial CFD program in thermal-hydraulics/safety analyses, the research on couplings of Point Kinetics (PK) model and a fuel pin Heat Conduction (HC) model to the commercial CFD platform FLUENT abbreviated as HC-PK-CFD is performed in this study. Verifications are performed by the benchmark simulation on beam interruption transients for an accelerator-driven sub-critical reactor system. It is demonstrated that the results simulated by HC-PK-CFD agrees well with the ones by other ten multi-physics coupling codes participating in this benchmark.

Design and stability analysis of an implicit non-standard finite difference scheme for fractional neutron point kinetic equation

This paper is concerned with the construction of a new numerical technique to obtain approximate solution for an extended fractional neutron point kinetic (EFNPK) model with six groups of delayed neutron precursors. The fractional derivative is approximated by means of Caputo's definition and the shifted Grünwald-Letnikov formula is used to obtain a finite difference approximation for the solution of this model. The discretization leads to an implicit finite difference scheme. The stability analysis of the method is discussed. It is proved that the method is unconditionally stable. Various numerical experiments are carried out for validity and efficiency of the suggested method. The neutron density is calculated for three types of reactivity: step, ramp and sinusoidal reactivity. The influences of anomalous diffusion exponent, reactivity functions, relaxation time and time steps on the neutron density are examined. Comparison is made between the results corresponding to the EFNPK model [4] and the results corresponding to the fractional neutron point kinetic model proposed in [17]. Comparison confirms that the difference between the values of neutron density increases as the anomalous diffusion exponent decreases or computational time increases. Comparison is also made between the approximate solution obtained by the suggested method and exact solution of the classical model for step and ramp reactivity. Moreover, our results are compared with those obtained by other methods.

Validating an adaptive time step scheme for the neutron diffusion equation with the UMass-Lowell Research Reactor kinetics data

Currently, most reactor simulation codes use predefined constant or variable time steps for simulation, which is inefficient. Available adaptive time step methods in the area of neutron transport and reactor kinetics are rather based on multistep and multistage integration methods, which could be time expensive. In this work, an adaptive time step scheme that takes advantage of a single integration per time step is evaluated for linear and non-linear reactor kinetics problem, using the point kinetics model of the UMass-Lowell Research Reactor. The time step selection scheme requires no solution evaluations or operator inversions beyond those already performed in the adaption-free solution, and requires no modifications to the solution algorithm. The solution accuracy from an inverse matrix method implementing the adaptive time step scheme is compared to an analytical solution for the linear problem. And for the nonlinear problem, the comparison is made with a numerical solution from an automated ordinary differential equation solver and the reactor experimental data.

An implicit finite difference method for solving the corrected fractional neutron point kinetics equations

It is well known that Espinosa-parades et al. (Espinosa-Paredes et al., 2011) proposed a fractional neutron point kinetic (FNPK) model with multi-group of delayed neutrons to describe dynamic behavior in a nuclear reactor. In (Aboanber and Nahla, 2016b), Aboanber and Nahla presented an extension of Espinosa-parades et al. FNPK model. This new model is called as the corrected fractional neutron point kinetic (CFNPK) model. The present study is concerned with the numerical solution of the CFNPK model (Aboanber and Nahla, 2016b) with six groups of delayed neutron precursors. The fractional derivative is described in the sense of Grünwald-Letnikov. An implicit finite difference method (FDM) is constructed for the solution of CFNPK model. The stability analysis of the method is carried out. We analyze the results of neutron density for different values of anomalous diffusion order, reactivity function, relaxation time and time step size. In addition, we compare the results corresponding to CFNPK model (Aboanber and Nahla, 2016b) with the results corresponding to the FNPK model proposed by Espinosa-parades et al. (Espinosa-Paredes et al., 2011). It is shown that as anomalous diffusion order decreases or simulation time increases the difference between the values of neutron density increases. We have investigated the effects of each term of the CFNPK.

Modified Point-Kinetics Model for Neutron Precursors and Fission Product Behavior for Fluid-Fueled Molten Salt Reactors

Fluid-fueled nuclear reactors, such as molten salt reactors (MSRs), have recently gained significant interest. These advanced reactors represent a potential revolutionary shift in the implementation of nuclear power, and as a broad class of reactors, they have the potential to directly address many U.S. energy policy objectives. Fuel that is dissolved in the coolant requires methods to account for the birth, decay, and transport of fission products not only in the core but also throughout the loop and any auxiliary systems, such as off-gas, to which liquid fuel flows, gaseous products are carried, or solid particulates plate out. System models are particularly well suited to explore the wide range of phenomena that are associated with fluid-fueled systems, especially for safeguards analysis. However, before system dynamics can be explored, the compositions of fission products of the salt throughout the loop must be determined as they drive the dynamic behavior of a reactor.This paper describes the derivation of a modified point-kinetics model for obtaining a first-order approximation of the behavior of a salt-fueled system in which neutron precursors and fission products are born in the fuel-salt and transported outside the core. This paper also provides verification of the model using a steady-state analytic solution and provides additional cases exploring the response under transient cases. This model establishes a baseline model that can be used to explore the dynamic response of fluid-fueled reactors and to investigate important safeguards issues such as mass accountability of source terms. The model is implemented in the Oak Ridge National Laboratory–developed, Modelica-based TRANSFORM library that was developed to investigate various aspects of advanced energy systems.

An alternative to standard nuclear core control using a multi-objective approach

This paper proposes an alternative to standard nuclear core control. The control objectives are standard and concern the Average Coolant Temperature, the Axial Offset and the rod bank position (Pbank in this study). They were carefully formalized before proceeding to LTI structured controllers synthesis at various operating points, using a multi-objective approach. This study relies on a point kinetic core model, embedding an AO estimator. A Non-smooth optimization technique is used to solve the control problems at each operating point. Then gain scheduling is performed, taking advantage of the parsimonious parameterization of the considered multivariable controller. Finally, simulations are carried out on a highly realistic reactor model including a Xenon model and show the benefits of the new control

Transient analysis of MOX-3600 and MET-1000 sodium-cooled fast reactor using SARAX code system

The capability of transient analysis has been implemented in the SARAX code system recently. Alternatively, two calculation schemes using the point-kinetics model and 3-D transport-based spatially-dependent model have been developed. Based on these, two transient benchmarks of sodium fast reactor (SFR) suggested by the Working Party on Reactor and System (WPRS) were calculated, which include the large size core loaded with oxide fuel (MOX-3600) and the medium size core loaded with metallic fuel (MET-1000). Two typical states including the state of begin-of-cycle (BOC) and state of end-of-cycle (EOC) were considered. The unprotected loss of flow (ULOF) transient, unprotected transient over power (UTOP) transient, and unprotected control rod withdrawal (UCRW) were calculated. For the large size MOX-fueled SFR, the ULOF transient is the most severe accident, which leads to the core damage in a very short time. At the EOC state, the thermal margin is smaller because of the change of fuel compositions due to the depletion. Besides, the comparisons show that it is very important to use the spatially-dependent model in predicting the power level in the UCRW transient, where the point-kinetics model significantly underestimates the power level by 7–17%.

Developed mathematical technique for fractional stochastic point kinetics model in nuclear reactor dynamics

Fractional stochastic kinetics equations have proven to be valuable tools for the point reactor kinetics model, where the nuclear reactions are not fully described by deterministic relations. A fractional stochastic model for the point kinetics system with multi-group of precursors, including the effect of temperature feedback, has been developed and analyzed. A major mathematical and inflexible scheme to the point kinetics model is obtained by merging the fractional and stochastic technique. A novel split-step method including mathematical tools of the Laplace transforms, Mittage–Leffler function, eigenvalues of the coefficient matrix, and its corresponding eigenvectors have been used for the fractional stochastic matrix differential equation. The validity of the proposed technique has been demonstrated via calculations of the mean and standard deviation of neutrons and precursor populations for various reactivities: step, ramp, sinusoidal, and temperature reactivity feedback. The results of the proposed method agree well with the conventional one of the deterministic point kinetics equations.

Sodium-cooled fast reactor pin model for predicting pin failure during a power excursion

Within the framework of the Generation IV Sodium-cooled Fast Reactor (SFR) in which the CEA (French Commissariat à l’Energie Atomique et aux Energies Alternatives) is involved, the French innovative reactor design behavior under severe accidents conditions has to be assessed. Such accidents have mainly been simulated with mechanistic calculation tools (such as SAS-SFR and SIMMER-III). As a complement to these codes, which provide reference accidental transient calculations, a new physico-statistical approach is being developed at CEA; its final objective being to derive the variability of the main results of interest to quantify the safety margins. This approach requires fast-running tools to simulate extended accident sequences, by coupling models of the main physical phenomena with advanced statistical analysis techniques. The tool enables to perform a large number of simulations in a reasonable computational time and to describe all the possible scenario progressions of the hypothetical accidents. This general approach, combining mechanistic codes and evaluation tools, has already been conducted for some accidental initiator families (USAF – Unprotected SubAssembly Fault (Marie et al., 2016) and ULOF – Unprotected Loss Of Flow (Droin et al., 2017). In this context, this paper presents a physical tool (numerical models and result’s assessment) dedicated to the simulation of the beginning of the primary phase of the Unprotected Transient OverPower accidents (i.e. before failure of sub-assembly wrapper).At the beginning of this primary phase, the fast increase of nuclear power induces a strong temperature rise in the fuel pellets leading to strong mechanical and thermal loads on the cladding which could lead to clad failure or/and fuel meltdown. These phenomena are described and modelled analytically in single pin geometry in accordance to the level of details required to catch all the decisive phenomena.Slow power increase transients, such as control rod withdrawal, and fast power increase transients have been investigated in the past. Experimental validation on CABRI (experimental reactor dedicated to safety studies) and CESAR (Circuit d’Etude de l’ébullition du Sodium lors d’un Accident de Réactivité) experiments were carried out focusing on the amount of molten fuel formed during the transient, on the propagation of the void front in the channel in case of sodium boiling and on the pin failure mechanisms. Furthermore, a comparison of the physical tool calculation results was performed against reference accident SIMMER-III calculations.The tool is demonstrated to be able to predict the radial propagation of the molten zone in the fuel pin, the pin failure mechanism and void front propagation with a discrepancy of less than 10%. In the future, this physical tool, associated with a point kinetic neutronic model, will be used to simulate the global core behavior under an UTOP transient following a ramp excursion.