The Pennsylvania State University. The Graduate School. Department of Mechanical and Nuclear Engineering

Transcription

1 The Pennsylvania State University The Graduate School Department of Mechanical and Nuclear Engineering STATE FEEDBACK REACTOR CONTROL USING A VANADIUM AND RHODIUM SELF-POWERED NEUTRON DETECTOR A Thesis in Nuclear Engineering by Gokhan Corak 2018 Gokhan Corak Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science May 2018

2 The thesis of Gokhan Corak was reviewed and approved* by the following: Kenan Ünlü Professor of Nuclear Engineering Director of the Radiation Science and Engineering Center Thesis Co-Adviser James A. Turso Associate Research Professor Assistant Director of the Radiation Science and Engineering Center Thesis Co-Adviser Arthur T. Motta Professor of Nuclear Engineering and Materials Science Engineering Chair of the Nuclear Engineering Program *Signatures are on file in the Graduate School. ii

3 ABSTRACT The safe and effective control of nuclear reactors is of significant interest to the research reactor community and nuclear power utility industry. Numerous advanced control algorithms have demonstrated superior reactor control over the past several decades primarily on simulated reactors. Among these, state feedback control has been applied to virtually every type of dynamic system. This thesis focuses on developing an accurate model of the Penn State TRIGA Reactor simulation and creating a state feedback controller/state observer design using self-powered Vanadium and Rhodium neutron detectors (SPND) as feedback sensors. This work is the first attempt to use these type of sensors in a closed-loop feedback system for reactor control. The foundation of the equations in Simulink has been derived from normalized point kinetics equations and core averaged thermal-hydraulic equations. The self-powered detector dynamics may be developed from basic activation/decay balance differential equations. The MATLAB and Simulink suite of software tools are used to develop the TRIGA nonlinear model, selfpowered neutron detector models, state observer and state feedback controller. Results demonstrate that the TRIGA Simulink model developed compares well with the actual TRIGA Reactor data. Self-powered neutron detectors are suited to monitor continuously reactor power. Due to their dependence on radioactive decay after irradiation to produce a current signal, self-powered detectors have significant delay times associated with them, making them not useful for real-time feedback control. A major contribution of this thesis is the development and application of detector inverse models, which null-out delays introduced by the physics of the detector. Results demonstrate that the inverse detector iii

4 models have no delay which is desirable for the reactor closed-loop control. The long delay associated with the normal detector models can only realistically be used for applications where this delay can be tolerated, such as post-accident power monitoring. SPNDs need no external power to produce current levels consistent with ion chambers and may provide to be a vital component for closed-loop nuclear reactor control in the future. Accurate and fast measurement of the reactor power with SPNDs will support this goal, and will reactors to safely operate closer to their operational limits. The successful application of an advanced control algorithm i.e., state-feedback control with selfpowered neutron detectors, demonstrates that this technology may be applied in closedloop nuclear reactor control and safety systems not only for power plant applications, but for space nuclear reactor applications as well. iv

5 TABLE OF CONTENTS List of Figures... viii List of Tables... xi Acknowledgments... xii Chapter 1 - Introduction... 1 Section 1.1- The Point Kinetics Equations (PKEs)... 2 Section 1.2- Derivation of the Point Kinetics Equations and Core Averaged Thermal- Hydraulics Equations... 3 Section 1.2.1: Derivation of Linearized Point Kinetics Equations... 6 Section 1.2.2: Derivation of Linearized Core Averaged Thermal-Hydraulic Equations... 8 Section 1.3- The Penn State Breazeale Reactor (PSBR)... 9 Chapter 2 - Modeling the TRIGA Reactor Using Simulink Section 2.1- Simulink Block Diagrams Section 2.2- Point Kinetics Equations Design in Simulink Section 2.3- Control Rod Modeling in Simulink Section 2.4- Shutdown Reactivity in Simulink Section 2.5- Fuel Temperature Feedback (Core Averaged Thermal-Hydraulic) Design in Simulink Section 2.6- Validation of the TRIGA Reactor model Chapter 3 - Experimental Control Rod (ECR) Characterization and Implementation Section 3.1- Experimental Control Rod (ECR) Design in LabVIEW Section 3.2- Experiment Preparation v

6 Section 3.3- Control Rod Worth (Reactivity Effects) Characterization Chapter 4 - State Feedback Controller Design for TRIGA Reactor Simulation Section 4.1- State-Space Equations Section 4.2- State Space Representation of TRIGA Reactor Plant Section 4.3- State Feedback Controller Implemented as a Linear-Quadratic Regulator (LQR) 35 Section 4.4- State Observer Design Section 4.4.1: Comparison between State Feedback Observer Design and Proportionalplus-Integral (PI) Controller Chapter 5 - Vanadium and Rhodium Self-Powered Neutron Detector (SPND) Model Section 5.1- Vanadium Self-Powered Neutron Detector Model (Forward Model) Section 5.1.1: Vanadium Self-Powered Neutron Detector Rate Equations Section 5.1.2: Vanadium Self-Powered Neutron Detector Model in Simulink Section 5.1.3: Inverse Vanadium Detector Model Section 5.2- Rhodium Self-Power Neutron Detector Model Section 5.2.1: Rhodium Self-Power Neutron Detector Rate Equations Section Section 5.2.2: Rhodium Self-Power Neutron Detector Simulink Model Section 5.2.3: Inverse Rhodium Self-Power Neutron Detector Model Section 5.3- Using Self Powered Vanadium and Rhodium Detectors as Closed-Loop Feedback Signals Chapter 6 - Summary, Conclusions and Future Work Section 6.1- Validation of the TRIGA Reactor Simulink Model Section 6.2- Experimental Control Rod Design and Experimental Results vi

7 Section 6.3- State Feedback Controller Design Section 6.4- Self-Powered Neutron Detector Designs Section 6.5- Self Powered Detector Model in Closed Loop Section 6.6- Conclusion Section 6.7- Future Work References Appendix A: Appendix B: Penn State Breazeale Reactor Standard Operating Procedure Experiment Evaluation and Authorization vii

8 LIST OF FIGURES Figure 1-1. A picture of Penn State TRIGA Reactor Core Figure 2-1. Six Group Delayed Point Kinetics Equations Figure 2-2. Control Rod Reactivity Model for TRIGA Reactor Simulink model Figure 2-3. Fuel Temperature Dynamics with Constant fuel element surface area (UA) 18 Figure 2-4. Fuel Temperature Dynamics with fuel element surface area correlated with fuel temperature Figure 2-5. Final TRIGA Reactor model with Reactivity Feedbacks: Control rod reactivity, point kinetics equations and core averaged thermal hydraulics Figure 2-6. Penn State TRIGA Reactor measured data comparison with Simulink model data Figure 3-1. Experimental Control Rod Drive and Major Components Figure 3-2. LabVIEW Software Development for Experimental Control Rod Figure 3-3. LabVIEW User Interface for Experimental Control Rod Control Figure 3-4. Experimental Control Rod Mounted to Penn State TRIGA Reactor Bridge. 27 Figure 3-5. Experimental Control Rod Motor Drive Connection to LabVIEW crio module ad NI LabVIEW Host Computer Figure 3-6. Experimental Control Rod Reactivity worth Curve Calculated by Digital Reactivity Computer at the PSBR Figure 3-7. ECR model design implementation into TRIGA Simulink Model Figure 4-1. Block diagram representation of state equation and output equation Figure 4-2. TRIGA Reactor State Space Model with State Feedback Controller viii

9 Figure 4-3. Initial State Feedback Observer Design for TRIGA Reactor model Figure 4-4. State Feedback Controller/Observer design implemented on the TRIGA reactor simulation Figure 4-5. State Feedback Controller/Observer Subsystem Including Integral Action.. 41 Figure 4-6. TRIGA Reactor Control using the PI controller Figure 4-7. Reactor Power Change from 1MW to 900 kw and 900 kw to 800 kw with Different Controllers Figure 4-8. Reactor Power Change from 800 kw to 850 kw and 850 kw to 950 kw with Different Controllers Figure 5-1. Self-powered neutron detector components Figure 5-2. Vanadium Decay Mechanism Figure 5-3. Vanadium Self-Power Neutron Detector Model in Simulink Figure 5-4. Vanadium Self-Power Neutron Detector response at power set point change from 1 MW to 900 kw Figure 5-5. Inverse Detector model for Vanadium Self-Power Neutron Detector Figure 5-6. Rhodium Decay Mechanism Figure 5-7. Rhodium Self-Power Neutron Detector Model in Simulink Figure 5-8. Rhodium Self-Power Neutron Detector response step power change from 1MW to 900 kw Figure 5-9. Finalized Model with normal and inverse detector models and the state feedback controller/observer Figure Relative Power and Vanadium detector current with inverse and without inverse detector model ix

10 Figure Relative Power and Rhodium detector current with inverse and without an inverse model x

11 LIST OF TABLES Table 2-1. Delayed Neutron Data for Thermal Fission in Uranium Table 5-1. Vanadium Detector Constants [7] Table 5-2. Rhodium Detector Constants [10] xi

12 ACKNOWLEDGMENTS Foremost, I would like to thank my thesis Co-advisors Dr. James Turso and Dr. Kenan Ünlü for the continuous support to this research project, for their patience, motivation, and in-depth knowledge of the topic. Their guidance helped me to complete this research project and write this thesis. I am very lucky to have great advisors and mentors for my study. I am grateful for the financial support provided by the Radiation Science and Engineering Center and the Department of Energy, Nuclear Energy Enabling Technology joint grant with Westinghouse Corporation. I would like to thank my fellow colleagues: Andrew Bascom, Nuri Beydoǧan, Bryan Eyers, Buǧra Karabulut, Alibek Kenges, Maksat Kuatbek, Onur Murat, Adam Rau, Yucel Saygın, and Can Turgut for the discussions, being a second reader of this thesis, and their support through the process of my research and writing this thesis. Nobody has been more important to me in the process of this thesis than my family members. I would like to thank my parents and my brother, whose love with me in whatever I pursue. Finally, I wish to thank my girlfriend, Özlem, who provide unending love and support in my life. xii

13 Chapter 1 - Introduction This thesis will discuss the design of state feedback reactor control using vanadium and rhodium self-powered neutron detectors. For this purpose, a reactor model exhibiting dynamics consistent with the actual Penn State TRIGA reactor is required to develop advanced control techniques. Historically, ion chamber-type neutron detectors have been used as feedback sensors for reactor control. As a significant contribution of the thesis, self-powered neutron detectors will be introduced, modeled, and applied as feedback signals in an advanced reactor control algorithm, state feedback control, which utilizes internal states of the system to calculate the control system output to the control rod mechanisms. Chapter 1 will discuss background and theory of the Penn State TRIGA reactor dynamics, which is based on point kinetics equations and core-averaged thermal hydraulics. Chapter 2 will focus on modeling the Penn State TRIGA Reactor using the Simulink software. The assumptions, equations that drive the simulation and block diagrams will be described in this chapter. The Experimental Control Rod (ECR), its application to control of the TRIGA reactor, and its control design using the LabVIEW software package is presented in Chapter 3. After successful design in LabVIEW, the ECR is mounted atop the Penn State TRIGA Reactor bridge and characterized using the Penn State digital reactivity computer [1]. The ECR will be used to apply the state feedback control technique to the actual Penn State TRIGA reactor after design and testing of the TRIGA Simulink model. Chapter 4 will provide background on state feedback controller theory as well as the design of the state feedback controller for Penn State TRIGA reactor simulation.

14 Self-powered neutron detector models will be implemented into Simulink and will be discussed in Chapter 5. First, a vanadium detector model will be presented and two Simulink implementations will be introduced: the forward and inverse sensor models. Additionally, a Rhodium self-powered neutron detector model will be presented. Chapter 5 will also demonstrate closed-loop control, using a common proportional-plus-integral feedback controller, using the vanadium and rhodium-type self-powered neutron detectors as feedback sensors. Finally, Chapter 6 will summarize the created model, results from the closed-loop control application, and suggestions for future work and conclusion. Section 1.1- The Point Kinetics Equations (PKEs) The one-speed diffusion equation will be used to introduce the point kinetics equations. This model sufficient to describe qualitatively and, to some degree, quantitatively, the timedependent behavior of a small, closely-coupled nuclear reactor such as the Penn State TRIGA reactor. However, the multi-group three-dimensional version of the model is too computationally demanding for real-time reactor calculations with model-based controllers when effects such as 3D temperature, fission product poisons, and fuel burnup-related feedback mechanisms are included. In most transient applications, the one-speed diffusion equation may be reduced under an assumption that the spatial dependence of the neutron flux in the reactor can be represented by a single (fundamental) spatial mode, with the higher order modes rapidly dying out over time. This assumption allows the response to be separated in time and space, with the solution of the spatial dependence of the diffusion equation being straightforward for simple geometries such as a right circular cylinder. The time dependence forms the basis for the point kinetics equations (PKEs). The PKEs describe the dynamics of a nuclear reactor in terms of prompt and delayed 2

15 neutron behavior, using reactivity as the input parameter. Reactivity is dependent on reactor material property changes, which ultimately determine the dynamic response of the reactor. Delayed neutrons have significance in reactor time behavior. Since prompt neutron lifetime is very short, reactor period predicted by prompt neutrons alone is on the order of 10-5 seconds essentially impossible to control with conventional control rod mechanisms. Delayed neutrons provide a much-needed delay to the rector response and permit control by available mechanisms. Section 1.2- Derivation of the Point Kinetics Equations and Core Averaged Thermal-Hydraulics Equations The simplest method the deriving point kinetics equations come from neutron diffusion equation. In this derivation, some assumptions are made. One of the most important assumptions is there is no angular dependence of the neutron flux. The second assumption, called one-speed approximation, is that there is no energy transfer between scattering events. This results in only one energy group of equations and removes the energy dependence. Applying these assumptions to the neutron transport equation gives the one-speed diffusion equation with delayed neutrons. An additional assumption is that the spatial and energy dependence of cross-sections can be approximated by selecting average cross-sections. This gives the following one-speed (single energy) description of the neutron diffusion equation (1-1) 1 dφ υ dt. D φ + Σ aφ(r, t) = νσ f φ(r, t) 1-1 The one-speed diffusion model is capable of describing the time-dependent behavior of the system. Neutron flux can be written as eigenfunctions of time and position dependence: φ(r, t) = A n exp ( λ n t)ψ n (r) n 1-2 3

16 The spatial eigenfunctions can be determined for a specified geometry by: 2 ψ n + B 2 n ψ n (r) = And the time eigenvalues of λ n : λ n = υdb 2 n + υσ a υνσ f 1-4 If the higher-order modes are assumed to die out rapidly, neutron flux can be written: φ(r, t) = A 1 exp [( k 1 ) t] ψ l 1 (r) 1-5 where l = [υσ a (1 + L 2 B g 2 ] 1 and k = υσ f/σ a 1+L 2 B g 2 = k 1+L 2 B g 2 mean lifetime of neutron Multiplication factor The flux may be assumed to be separable in space and time φ(r, t) = υn(t)ψ 1 (r) 1-6 Finally, substituting equation 1-6 into one-speed neutron diffusion equation (1-1) will give: dn dt = (k 1 ) n(t) l 1-7 n(t) can be defined as the number of neutrons per cubic centimeter in the reactor at time t. The preceding equation does not include the effect of delayed neutrons. Reactor power may be determined by scaling n(t) to watts for the reactor under consideration. The effect of delayed neutrons may be incorporated by adding an additional source term to the neutron kinetics equation and developing a balance equation for each delayed neutron precursor group. If the number of delayed neutron precursors (i.e., specific fission products), defined as, C i (r,t) of the i-th kind, in a volume at the specific position, decays by emitting a beta particle and a subsequent delayed neutron, the delayed neutron source may be defined as λ i C i (r, t) 1-8 4

17 With the number of precursors produced being β i νσ f φ(r, t) 1-9 where β i is delayed neutron fraction for specific fission product. The dynamic behavior of a delayed neutron precursor group is described by the following a balance equation: dc i dt = λ ic i (r, t) + β i νσ f φ(r, t) 1-10 Equation 1-10 can be used inside the one-speed diffusion equation 1-1 by defining the total fission source as separate prompt and delayed neutron sources: S(r, t) = (1 β)νσ f φ(r, t) + λ i C i (r, t) 1-11 Delayed neutrons can be produced by 200 different precursor groups. Due to the computational burden imposed by using all 200 delayed neutron precursor groups, a 6 delayed neutron group model is most common in reactor calculations and will be used in this thesis. These equations are incorporated into the one-speed diffusion equation to give the Point Kinetics Equations with delayed neutrons (1-12): dn dt = k(1 β) 1 l 6 n(t) + λ i C i (t) i= dc i dt = λ ic i (t) + β i k l n(t) The PKE can also be written by defining the mean generation time between the birth of neutron and absorption in the fission as Λ = 1 k 1-13 and defining the reactivity 5

18 ρ(t) = k(t) 1 k(t) 1-14 Substituting equations 1-13 and 1-14 into the point kinetics equations (1-12) gives a more common form of the PKE. dn dt = ρ(t) β Λ 6 n(t) + λ i C i (t) i= dc i dt = β i Λ n(t) λ ic i (t) i = 1, 6. Equation 1-15 is nonlinear due to the product of ρ(t) and n(t) with feedback effects the ultimately are driven by n(t), such as fuel temperature and Xenon poison. Although the solution of these equations does not lend itself to conventional methods developed for linear systems, many advanced control algorithms are based on linear control theory. These equations need to be linearized in order to apply advanced linear control techniques. Section 1.2.1: Derivation of Linearized Point Kinetics Equations In order to obtain a linear approximation of the nonlinear system, linearization has to be performed around local equilibrium points. In the point kinetics equations, due to the product of ρ(t) and n(t), feedback that is quantified by ρ(t) are ultimately are driven by n(t), such as fuel temperature and xenon poison. These products need to be approximated by linear relationships in order for the system to be linear. Once again, the application of linear advanced control theory requires application on a linear system. For this purpose, linearized point kinetics equations are derived from the nonlinear model. For demonstration, one group of delayed neutrons will be incorporated in the model, however, the full six delayed neutron group model will be used for controller design and testing. 6

19 Development of the state space equations (required for state feedback control design) will be described in Chapter 4. Given the one delayed neutron group point kinetics equations dn dt ρ(t) β = n(t) + λc(t) Λ dc = λn(t) λc(t) dt 1-16 The small deviations about equilibrium for neutron population, delayed neutron precursors, and reactivity are defined as: n(t) = n 0 + δn(t) c(t) = c 0 + δc(t) 1-17 ρ(t) = ρ 0 + δρ(t) where n 0, c 0 and ρ 0 are initial conditions for each of the states of the point kinetics equations. Inserting these equations into nonlinear point kinetics equation (1-16) will give dn 0 dt + δn(t) dt = ρ 0 + δρ(t) β (1 + δn(t)) + β (1 + δc(t)) Λ Λ dc 0 dt + δc(t) = λ(1 + δn(t)) λ(1 + δc(t) ) dt 1-18 Equation 1-18 is simplified by assuming that deviations δn and δρ about an equilibrium point are small, so products of deviations may be neglected. Also, derivatives of equilibrium conditions are also zero. Simplifying these two equations (1-18) with assumption gives the linear version of the point kinetics equations (1-19) δn(t) dt = δρ(t) Λ δc(t) dt β Λ δn(t) + β Λ δc(t) = λ(δn(t) δc(t))

20 Section 1.2.2: Derivation of Linearized Core Averaged Thermal-Hydraulic Equations The Penn State TRIGA Reactor uses uranium zirconium hydride (U-Zr-H) fuel which has very large and prompt negative fuel temperature coefficient of reactivity. This implies that as the temperature of the core increases, the core reaction rate will decrease due to the large negative temperature feedback effect, maintaining the reactor stability. This unique feature of U-Zr-H allows the Penn State TRIGA Reactor to safely withstand events that would significantly damage reactor cores. It also provides safely pulsing the reactor up to 2000 MW [2]. The nonlinear core averaged thermal-hydraulic equations can be written as dt f dt = P n(t) U fa f (T M f C f M f C f T C ) f 1-20 dt C dt = U fa f M C C C (T f T C ) 2. m C C (T c T 0 ) Once again, these nonlinear equations can be linearized by approximating the state variable (i.e., temperature) by the temperature about an equilibrium point added to a time-dependent deviation: T f (t) = T f0 + δ T f (t) 1-21 Using equation 1-21 and equation 1-20 will give T c (t) = T c0 + δ T c (t) M f C f ( dt f0 dt + δt f(t) ) = P(n dt 0 + δn(t)) UA(T f0 + δ T f (t) T c0 δ T c (t)) 1-22 M C C C ( dt C0 dt + δt C (t) ) = UA (T dt f0 + δ T f (t) T c0 δ T c (t)) 2. m C C (T c0 + δ T c (t) T 0 ) The linearized thermal-hydraulic equations (1-22), valid about the equilibrium condition specified become 8

21 M f C f ( δt f(t) ) = P( δn(t)) UA(δ T dt f (t) δ T c (t)) M C C C ( δt C(t) ) = UA (δ T dt f (t) δ T c (t)) 2. m C C (δ T c (t)) 1-23 Section 1.3- The Penn State Breazeale Reactor (PSBR) The Penn State Breazeale Reactor (PSBR) is the first licensed university research reactor in the USA. The PSBR reached criticality on August 15, The PSBR was initially designed as a Materials Testing Reactor (MTR) which used plate-type fuel and was licensed for a power level of 100 kw(th). The PSBR was later upgraded to 200 kw(th) in In 1965, Penn State received a license that allowed for the conversion of the reactor from highly enriched MTR fuel to a to TRIGA (Training, Research, Isotopes, General Atomics) design. This design requires low-enriched uranium fuel and provides steady state power of 1 MW, with the capability to pulse the reactor up to 2000 MW. [2] The movable core has no fixed reflector and is located in a 24 ftdeep pool with ~71,000 gallons of demineralized water. Figure 1-1 shows a picture of Penn State TRIGA Reactor core. A variety of dry tubes and fixtures are available in or near the core for irradiating samples. A pneumatic transfer system is also available for irradiation of samples. When the reactor core is placed next to the D 2 O tank and graphite reflector assembly near the beam port locations, thermal neutron beams become available for neutron transmission and neutron imaging measurement from two of the seven existing beam ports. The other beam ports are not currently utilized due to their geometrical alignment with respect to the existing reactor core structure but a project is undergoing to incorporate new core moderator assembly and beam ports design into Penn State TRIGA reactor. [3] 9

22 The PSBR has four standard control rods, three of which (the Shim, Safety, and Regulating control rods) can be placed in automatic control. The fourth control rod (the Transient control rod) is permanently in manual mode and is used to pulse the reactor. An Experimental Control Rod and Drive (ECRD) may be positioned over the core and used for control experiments. The standard control rods have a removal distance of 15-inches which is the same length as the TRIGA fuel. The Safety, Shim, and Regulating rods have two different regions. They have a 15-inch Boron Carbide neutron absorber section which is located below 15 inches of TRIGA fuel. Withdrawal of the control rod from the core will insert the fuels lower region of the control rod into the core, increasing the positive reactivity effect of removing a control rod and eliminating excessive thermal power peaking in the control rod channels. The Penn State Radiation Science and Engineering Center (RSEC) was one of the first university reactor facilities to install a digital control and monitoring system while all safety systems remain as an analog system. The new reactor instrumentation and control system was licensed in The control system manufactured by Atomic Energy of Canada Limited (AECL). [2] 10

23 Figure 1-1. A picture of Penn State TRIGA Reactor Core 11

24 Chapter 2 - Modeling the TRIGA Reactor Using Simulink The TRIGA Reactor simulation is programmed in the Simulink software package. Simulink is used by control system design engineers, and, being a graphical programming language, has a library of blocks that enable the user to simulate a wide variety of dynamic and control systems. It has a selection of numerical integration types, easy fairly to use and is well suited for transient analysis. As mentioned previously, input reactivity used in Point Kinetics Equations is the net reactivity from several sources, among these, are control rod worth, reactivity from changes in U-Zr-H based fuel temperature, moderator density changes, and fission product poisons. Effects of fission product poisons are excluded from the model since the goal of this thesis mainly focus on short time transients, it requires a long time. Some fission products have a high neutron absorption cross-section, such as Xenon-135 and Samarium-149. These poisons affect the neutron population in the reactor due to their high neutron absorption capacity. Xenon-135 is a product of Iodine-135 which has a 7 hours half-life. During the steady state operation of the reactor, the Xenon-135 concentration will be a build-up to equilibrium value in about 50 hours. Due to longtime requirement to decay process ox Xenon-135, it is mostly not considered in short transient calculations. Only the effect of control rod movement and fuel temperature feedback will be incorporated into Point Kinetics equations based Simulink model. Moderator temperature feedback is not included the model because moderator temperature does not change significantly. Actual TRIGA Reactor operation data will be used to provide realistic control rod worth curves. Derived non-linear point kinetics equations and non-linear thermal-hydraulic equations will be used to build Simulink reactor model which is used to design and test the controllers. 12

25 Section 2.1- Simulink Block Diagrams Simulink has basic mathematical operation blocks as well as continuous time blocks such as derivative and integration. A variety of different blocks have been used to design a new TRIGA Reactor model. MATLAB m-files (scripts) have been written to facilitate calculation of simulation parameters for use in the Simulink TRIGA model. During runtime, Simulink calls input parameters from MATLAB. Section 2.2- Point Kinetics Equations Design in Simulink The point kinetics equations are coded inside the simulation environment. For this purpose, necessary block types have been created inside the Simulink model and connected with wires (signal lines). Figure 2-1 is a representation of point kinetics equations inside the Simulink reactor plant model. The upper part represents the prompt neutron dynamic equation. The lower part of the point kinetics model has the six delayed neutron precursor group equations. Each group has a different decay constant and delayed neutron fraction (Table 2-1 [4]). Table 2-1. Delayed Neutron Data for Thermal Fission in Uranium-235 Group Half-Life (sec) Decay Constant (l i sec 1 ) Energy (kev) Neutrons per Fission Fraction (β i ) Total Yield: Total delayed fraction (β):

26 The Simulink model uses a variety of blocks, which represent functions, such as add, subtract, integration and gain. To solve time-dependent derivative, Simulink model contains several integration blocks, for which initial values need to be defined. The initial conditions of normalized point kinetics equations for prompt and delayed neutrons defined as 1.0. The input of point kinetics equations is total reactivity and output of the model is relative reactor power. Total reactivity is a summation of three different reactivity parameters. These are reactivity coming from the four standard control rods, reactivity feedback due to fuel temperature change and shutdown reactivity. For long-term high-power operation, fission product poisons (such as xenon and samarium) would also be part of the total reactivity. For the scenarios and transients considered as part of this thesis, which occur over a span of minutes, the long-term reactivity effects of the fission product points have been neglected. 14

27 Figure 2-1. Six Group Delayed Point Kinetics Equations Section 2.3- Control Rod Modeling in Simulink Control rods are the primary external control mechanism for the nuclear reactors. Withdrawal or insertion of the control rods will change neutron population in the reactor core since control rods are made of highly neutron absorbing materials such as Boron Carbide(B 4 C), silver, indium or cadmium. Typical TRIGA Reactors have four different control rods as described in Section

28 Creating an accurate TRIGA Reactor model requires realistic input reactivity coming from control rods due to their positions. For this purpose, the actual control rod reactivity characteristics relative to their position in the core (otherwise known as control rod worth curves) have been included in the TRIGA Simulink simulation. This data has converted into MATLAB data for use in the Simulink model. Figure 2-2 shows the Simulink block used to calculate control rod reactivity for input rod positions. The control rod worth data was curve fit in MATLAB, and the resulting equations coded into the blocks in Figure 2-2. Given input control rod height, the model calculates relative control rod reactivity by a MATLAB function block and provides total control rod reactivity to the point kinetics equations model. Figure 2-2. Control Rod Reactivity Model for TRIGA Reactor Simulink model 16

29 Section 2.4- Shutdown Reactivity in Simulink Another reactivity type is called shutdown reactivity has been added the system as a constant. Shutdown reactivity is the amount of reactivity necessary to get the reactor critical at cold, clean (i.e., fission product poison-free) conditions. This number varies depending on the core design, fuel burnup, and is considered a constant for the control system studies presented in this thesis. To get the reactor critical, the control rods need to be withdrawn to the point where they insert this amount of positive reactivity so that the simulation starts at steady-state conditions (i.e., total reactivity is equal to zero). Additionally, in the power-range of operation, the control rods need to override the negative reactivity inserted due to temperature feedback. Section 2.5- Fuel Temperature Feedback (Core Averaged Thermal-Hydraulic) Design in Simulink The final reactivity mechanism comes from fuel temperature feedback. Core thermalhydraulic equations have been coded into the Simulink. Figure 2-3 shows the initial implementation of the fuel temperature dynamics which contains constant, overall heat transfer coefficient multiplied by fuel element surface area (UA) at 1MW operation. Since this parameter changes with temperature, a correlation was developed to obtain UA as a function of fuel temperatures. To accomplish this, a steady-state heat balance using a core-averaged fuel centerline temperature (approximately) and a core-averaged coolant channel temperature (all measurements available). Various power levels are divided by the difference in fuel/coolant temperatures to give values of the overall heat transfer coefficient multiplied by fuel element 17

30 area (i.e., UA). Using this data, a curve fit has been performed using fuel temperature as the input. Figure 2-3. Fuel Temperature Dynamics with Constant fuel element surface area (UA) 18

31 Figure 2-4. Fuel Temperature Dynamics with fuel element surface area correlated with fuel temperature Figure 2-4 is the updated version of Figure 2-3 which contains the UA value correlated with fuel temperature. The input of the core-averaged thermal-hydraulics part of the TRIGA model is relative reactor power and the output is core-averaged fuel temperature. This value will be subtracted from initial fuel temperature (at zero-power conditions) and multiplied by the fuel reactivity coefficient ( f ) to yield reactivity due to fuel temperature feedback. Figure 2-5 shows a high-level view of the completed plant model in its entirety. The MATLAB code shown in Appendix A which contains constants for Simulink TRIGA Reactor model. 19

32 Figure 2-5. Final TRIGA Reactor model with Reactivity Feedbacks: Control rod reactivity, point kinetics equations and core averaged thermal hydraulics Section 2.6- Validation of the TRIGA Reactor model After implementation of the TRIGA Reactor simulation, the power output will be compared to measured TRIGA reactor operating data in order to validate the simulation. Using identical inputs from the control rods, the output of the model should show a similar response when compared to actual reactor. The model validation was done using measured data from the reactor to update reactivity coefficients for temperature and the overall heat transfer coefficient. Two separate reactor runs were used to collect data. The first was comprised of several power increases to observe how temperature and power changed together, and the second was a step change in power done by SCRAMMING the reactor from full power. This data was put into the model and the necessary coefficients and constants were updated to 20

33 continue improving model accuracy. With these included, the model matched the behavior of the reactor within an acceptable range. Figure 2-6. Penn State TRIGA Reactor measured data comparison with Simulink model data Figure 2-6 shows a comparison between PSBR TRIGA Reactor measured data and simulation results. This figure shows that the designed TRIGA Reactor Simulink model gives an acceptable range of power level with comparing to TRIGA reactor operation data and it would give a chance to use Simulink model in future system development. 21

34 Chapter 3 - Experimental Control Rod (ECR) Characterization and Implementation To prove the design of the controller works in a real-world environment, an experimental setup is needed for basic transient testing. An experimental control rod drive mechanism was mounted to the TRIGA reactor to accomplish this. Before mounting the control rods to the system, a rod control algorithm was developed. ECR worth approximately $0.9 of reactivity, which can significantly affect power transients in TRIGA reactors. Hence, proper design of the ECR controller was necessary to characterize the rod worth curve and then use as part of a closed-loop controller. LabVIEW has been used to design the ECR drive controller. LabVIEW software was used due to its user-friendly interface, available block sets for controller implementation, and hardware interface capability. Section 3.1- Experimental Control Rod (ECR) Design in LabVIEW Figure 3-1 shows components of the ECR drive mechanism system. The limit switch is designed to limit ECR movements between 0-15 inches. The lead screw/ ball nut converts rotational motion, provided by the drive motor, to vertical motion for inserting/removing the ECR. A plate couples the ball nut to the experimental control rod. ECR will change position depending on lead screw movements. The position is acquired by a position sensor (linear potentiometer) which is connected to LabVIEW CompactRIO (crio) data acquisition/control system. This position sensor has a linear potentiometer and output signal conditioner that provides a 0-20mA signal to the crio. The ma output is scaled to maximum and minimum control rod positions, with a corresponding linear calibration function implemented in 22

35 LabVIEW. The motor drive provides power to the motor. LabVIEW is used to provide an ECR speed demand signal (+/-10V) to the motor drive. Figure 3-1. Experimental Control Rod Drive and Major Components Figure 3-2 is the control system design for the ECR in LabVIEW. The three main inputs are power setpoint entered into the user interface by the operator, the power measurement from the reactor, and the ECR velocity demand, which may be manually or automatically controlled by the operator. In manual operation, the operator manually operates the ECR using the interface shown in Figure 3-3. The operating mode can be changed via Boolean inputs (switching function/ onoff) in the LabVIEW interface. When automatic control of the rod drive enabled, the ECR LabVIEW controller will measure the power, compare it with the setpoint power, and adjust the system appropriately. The calculations in the LabVIEW program represents basic mathematical 23

36 calculations. Controller constants for the Proportional-Plus-Integral controller are determined by a trial error. Figure 3-2. LabVIEW Software Development for Experimental Control Rod 24

37 Figure 3-3. LabVIEW User Interface for Experimental Control Rod Control The LabVIEW program was tested prior to mounting the system to Penn State TRIGA Reactor. In manual mode, ECR repositioned using LabVIEW interface. In automatic power control mode, power setpoint is changed and compared to the actual reactor power signal. This was tested on the PSBR at lower powers (approximately 50 kw). The results suggest that the ECR LabVIEW controller design is capable of controlling ECR position movements and reactor power successfully. 25

38 Section 3.2- Experiment Preparation After the successful design of Experimental Control Rod positioning system in LabVIEW, an experiment was performed using the Penn State TRIGA Reactor. The required forms for performing experiments at the PSBR were submitted by the experimenters, including a Standard Operating Procedure (SOP) document that discusses experimental procedures, material activation data, and removal of the ECR (Appendix B). After submitting the necessary documentation to PSBR management, the ECR was attached to the control rod drive mechanism. It has its own control rod drive mechanism, motor, power, and signal connections so there was no direct electrical/signal connection between the licensed Penn State TRIGA Reactor control system and ECR drive mechanism. Figure 3-4 shows the ECR mounted to the TRIGA Reactor. A National Instruments crio FPGA-based system was connected to both PC and the ECR drive mechanism in Figure

39 Figure 3-4. Experimental Control Rod Mounted to Penn State TRIGA Reactor Bridge Figure 3-5. Experimental Control Rod Motor Drive Connection to LabVIEW crio module ad NI LabVIEW Host Computer 27

40 Section 3.3- Control Rod Worth (Reactivity Effects) Characterization Prior to use for control algorithm testing, the ECR control rod worth had to be determined. It is essential to quantify how much reactivity would be inserted into the reactor core for a given control rod position. From previous experiments, it was known that the ECR had less than $1 reactivity, but a rod worth determination had not been performed for the current core. The TRIGA Reactor digital reactivity computer was used to determine the ECR reactivity worth. The TRIGA Reactor Digital reactivity computer was developed by Dr. James Turso using NI LabVIEW [1]. The reactivity computer determines control rod worth curves for the TRIGA Reactor standard control rods on a yearly basis. The procedure in reference [1] was followed for the ECR worth calculation. First, the reactor operator moved the control rod to make the reactor critical at 100W. Second, a nearperfect step change in reactivity was performed using the ECR and the LabVIEW software developed for ECR position control. Due to the ECR withdrawal from the reactor core, the reactor experienced a power increase. Several minutes were needed to wait until the reactor power rose to a stable power level. After a stable power level was obtained, the reactivity computer calculated the difference in the reactivity inserted by the control rod. This test was performed between 0-inches to 15-inches with 1-inch intervals, and reactivity worth curves were plotted using the digital reactivity computer as shown in Figure

41 Figure 3-6. Experimental Control Rod Reactivity worth Curve Calculated by Digital Reactivity Computer at the PSBR The curve fit of the reactivity worth of ECR has been implemented in the TRIGA Reactor Simulink model, and will eventually be used to test the controller designs of this thesis. The ECR reactivity model has been added to Simulink model Figure 3-7. As a linear model for use in state feedback controller design (discussed later), the model uses ECR velocity as an input and gives the related reactor power as an output. This way, state feedback control algorithms can control all five control rods depending on the experiment. In Chapter 4, ECR control rod will be used as the actuator in a state feedback closed-loop controller. 29

42 Figure 3-7. ECR model design implementation into TRIGA Simulink Model 30

43 Chapter 4 - State Feedback Controller Design for TRIGA Reactor Simulation In this chapter, the theoretical background of state feedback control will be presented, as well as the methods for design of a TRIGA Reactor state feedback controller. The main control mechanism of the TRIGA Reactor is control rod movement to insert positive or negative reactivity, which results in more or less fission produced in the core. The current PSBR control system was developed by Atomic Energy Canada Limited (AECL) using the PROTROL block diagram language. In this thesis, state feedback controller design will be employed, eventually using self-powered neutron detectors as feedback signals. A similar study was performed by Dr. James Turso in the early 1990 s using ion chambers as the power feedback signal [1]. State Feedback control is a method to place closed-loop poles of a plant in arbitrary locations in the s-plane [5]. This method is very useful because the locations of the poles are the eigenvalues of the system which characterize the stability and the response of the system. This method can only be applied to controllable, linear systems. The state feedback controller of TRIGA reactor will use output relative power (initially from the model developed as part of this thesis) and will change the input of this model, ECR control rod reactivity. For designing the State Feedback controller, it is necessary to create a state space representation of TRIGA Reactor. Section 4.1- State-Space Equations Most dynamic systems with a finite number of lumped elements can be described by ordinary differential equations where time is an independent variable. By use of vector-matrix 31

44 notation, an n-order differential equation can be expressed by a group of first-order differential equations. These equations can also be arranged in matrix form. If n elements of a vector are a set of state variables, then the vector matrix differential equations are called state equations [6]. A State vector describes the n state variables that are needed to describe the dynamic behavior of the (linear) system. The state vector determines the system state x (t) for any time t t 0 once the state at t = t 0 is given and input u (t) for t t 0 is specified. State space is the n-dimensional space whose coordinate axes consist of the states i.e., x 1 axis, x 2 axis..., x n axis. Any system state can be represented in the state space. Consider the n-th order system: y n + a 1 y n a n 1 y + a n y = u 4-1 With initial y parameters and u (t) for time, t 0 will determine the future behavior of the system. Define the system state differential equations x 1 = x x 2 = x 3 x n 1... = x n x n = a n x 1... a 1 x n + u Which can also be written in matrix form as x = Ax + Bu where 32

45 x 1 x 2 x = [ ], A = x n [ a n a n 1 a n 2 a 1 ] 0 0, B =, 0 [ 1] 4-3 with the output of the system being: x 1 x 2 y = [1 0 0] * [ ], y = Cx where C = [1 0 0] x n 4-4 The state and output equations (4-3) and (4-4) are represented in the block diagram of Figure 4-1 [6] Figure 4-1. Block diagram representation of state equation and output equation The B matrix is also called input matrix and would be non-zero if there are time-dependent inputs to the system states. Section 4.2- State Space Representation of TRIGA Reactor Plant To obtain a state feedback design of the system, TRIGA reactor point kinetics equations (1-19) and thermal-hydraulic equations (1-23) should be implemented in state space form. For this purpose, these equations will be linearized by hand and will be implemented in the Simulink model using state space block. 33

46 In Chapter 2, a TRIGA Reactor model is created using a variety of Simulink blocks, one of which will be used to implement the reactor state space representation into the model. The necessary equations have been derived in Chapter 1 Section 1.2 and will be used to create the state space representation in matrix form. Consider the linearized point kinetics equations (1-19) and thermal-hydraulic equations (1-23) δn(t) dt = δρ(t) Λ β Λ δn(t) + β Λ δc(t) 4-5 δc(t) dt = λ(δn(t) δc(t)) M f C f ( δt f(t) ) = P( δn(t)) UA(δ T dt f (t) δ T c (t)) 4-6 M C C C ( δt C(t) ) = UA (δ T dt f (t) δ T c (t)) 2. m C C (δ T c (t)) For the purposes of model-based controller design (to enhance the accuracy of the model), the 6 delayed neutron group point kinetics equations and core-averaged thermal-hydraulic equations for TRIGA reactor may be put into state spate space form. x = Ax + Bu and y = Cx + Du 4-7 Where A is state matrix B input matrix, C is output matrix and D=0. u(t) is the input to the system which is reactivity (or control rod velocity converted to reactivity in the model) and x are the individual states of the system. Each state derivative has to be integrated to determine the state at a point in time. 34

47 δn δc 1 δc 2 δc 3 δc 4 δc 5 δc 6 δt f δt C [ δρ ] = [ β Λ β 1 Λ β 2 Λ β 3 Λ β 4 Λ β 5 Λ λ 1 λ λ 2 0 λ λ λ λ λ λ λ λ λ P β 6 Λ T Λ M f C f U fa f M f C f U f A f 0 1 Λ U f A f M f C f U fa f 2m C M c C c M f C C 0 f ] y= [ 0] T. δn δc 1 δc 2 δc 3 δc 4 δc 5 δc 6 δt f δt c +0.ρ [ δρ ]. δn δc 1 δc 2 δc 3 δc 4 δc 5 δc 6 δt f δt c [ δρ ] + [ Δρ Δz]. ρ Section 4.3- State Feedback Controller Implemented as a Linear-Quadratic Regulator (LQR) A state-space model of the TRIGA Reactor is used to design a Linear-Quadratic Regulator (LQR) controller. In control theory, the main concern is the operation of the dynamic system with minimum control effort, given physical constraints on the system behavior. The system dynamics can be described by linear differential equations and the degree which control effort and constraints impact the closed-loop controller performance can be represented by a quadratic performance index (or cost function). The LQR is an established method to design state feedback control systems. The LQR minimizes quadratic cost function with weighting factors which are chosen by the design engineer. This performance index allows specified system states (e.g., temperature and power) to be more or less heavily weighted. The design minimizes 35

48 weighted state deviations in the cost function. States (or control input) that have no weight are interpreted as the controller imposing no constraints on their behavior. Heavily weighted states (or control input) may be interpreted as having the controller tightly controlling their behavior. The LQR design is implemented in a MATLAB script to reduce the effort for optimizing the controller. However, cost function parameters still need to be selected by iteration, typically. The results of each design iteration should be compared with design goals. If they are not within the margin of the desired value, the cost functions should be changed and tested again. An LQR state feedback controller is designed using the matrix A and B matrices from the state space representation of TRIGA model and user-defined Q and R matrixes. Figure 4-2 is a representation of state feedback controller design (using LQR) in Simulink. Figure 4-2. TRIGA Reactor State Space Model with State Feedback Controller 36

49 The value of the state feedback controller gain, K in Figure 4-2 is determined by the LQR design MATLAB script. The controller gains are calculated in MATLAB using the linear quadratic regulator design script i.e. K=lqr (A, B, Q, R) The Q matrix represents the weights (or penalties) imposed on the states in the point kinetics equations, thermal-hydraulic equations, and the control rod position states. The R weight matrix values (penalizing control input) are selected for the best response by iteration. Section 4.4- State Observer Design In most practical examples, not all state variables are measured. These values need an estimated in order to implement a full state feedback controller design. Estimation of an unmeasurable state is referred to as observation. If the observer estimates all state variables of the system, it is called full-order state observer. The observer may be designed by using the place command in MATLAB. This command places the closed-loop system eigenvalues and calculates an observer gain matrix. The observer compares the actual measured output of the system to the estimated output and subsequently minimizes the error in the estimated state. x = Ax + Bu + obs(y y ) y = Cx Where obs is observer gain vector. The error of the observer e = x x = (A obs C)e The observer gain should be selected depending on the system response. Given that the eigenvalues may be arbitrarily placed, the observer may produce state estimates that are faster or 37

50 slower than the actual system. While this may be desirable for certain state feedback control applications, an observer that provides exact estimates of the dynamic state variables is most desirable for nuclear reactor control. Figure 4-3 is a representation of the observer-based controller. For preliminary testing, the system being controlled is a linear version of the TRIGA reactor model. The output of the observer (estimated state vector) is multiplied by the state feedback controller gain vector and is subtracted from the control input to the system. Preliminary results demonstrate that state feedback controller with a state observer can be applied to the TRIGA reactor. Figure 4-3. Initial State Feedback Observer Design for TRIGA Reactor model 38

51 After successful test results from state feedback controller/observer design, the state observer is implemented on the non-linear TRIGA Reactor simulation to control the reactor. The main purpose is to control the reactor using a state feedback controller/observer. The overall input is a change in power setpoint. Figure 4-4 is a representation of the TRIGA reactor controlled by a state feedback controller/observer design. The subsystem for the observer (which marked in a red circle) is shown in Figure 4-5. The design takes as an input reactor power from the TRIGA model. The controller setpoint change is input by the operator, compared to the measured (in this case simulated) power output of the reactor. The state observer uses the difference between the actual and estimated reactor power to tune each of the state estimates. These state estimates are multiplied by a controller gain matrix and used to calculate an overall control signal to the control rod drives. Integrator action is included to robustify the state feedback controller. Given that the state feedback controller is model-based, and that it theoretically will be optimal for only one version of the plant (i.e., the plant it was designed to operate on), minor differences between an actual plant and the design model may result in poor performance. The integral action allows the control system to accommodate plant uncertainties and still provide acceptable performance. The output of the integral action is combined with the output of the state feedback controller to result in a velocity demand sent to control a rod drive mechanism (in this case the Experimental Control Rod (ECR)). In the simulated (and actual) experiment, velocity demand from state feedback controller/observer is directly sent to ECR. The other 4 control rods are left in manual operation and stay at their initial positions 39

52 40 Figure 4-4. State Feedback Controller/Observer design implemented on the TRIGA reactor simulation

53 Figure 4-5. State Feedback Controller/Observer Subsystem Including Integral Action Section 4.4.1: Comparison between State Feedback Observer Design and Proportional-plus- Integral (PI) Controller To demonstrate the advantage of the state feedback controller/observer design, a proportional-plus-integral (PI) controller design is also implemented in the TRIGA Reactor simulation. The proportional and integral gains have been assigned to obtain a similar response to the actual TRIGA reactor for specified power changes. Designing the PI controller was challenging and resulted in an oscillation of the response. Figure 4-6 shows TRIGA Reactor controlled by the PI controller. The power change will be implemented by the ECR using the velocity demand control signal coming from PI controller. The PI controller coded inside the subsystem marked in red circle. 41

54 42 Figure 4-6. TRIGA Reactor Control using the PI controller

55 State feedback observer design has been tested at different power levels. To show the accuracy of the model, simple PI Controller designed to compare with the state feedback controller/observer. Figure 4-7 demonstrates the power change from 1 MW to 900 kw at 100 seconds then, 900 kw to 800 kw at 300 seconds. Both PI controller and state feedback controller/observer follow the power setpoint change by the operator. PI controller follows desired power level with overshoot. However, state feedback controller reaches desired power level without overshoot. Figure 4-7. Reactor Power Change from 1MW to 900 kw and 900 kw to 800 kw with Different Controllers 43

56 Figure 4-8 shows the response of the controllers for a power increase. Similarly, PI controller reaches the desired level with overshoot, state feedback controller achieves with a smooth curve. Figure 4-8. Reactor Power Change from 800 kw to 850 kw and 850 kw to 950 kw with Different Controllers The results show that the State feedback controller design accurately controls the reactor and follows operator setpoint changes. 44

57 Chapter 5 - Vanadium and Rhodium Self-Powered Neutron Detector (SPND) Model Power generation in nuclear reactors is determined by the number of fission reactions occurring inside the reactor core, which is directly related to the number of neutrons available to create fission. Thus, reactor power can be estimated by measuring neutron flux [7]. Self-Powered Neutron Detectors (SPNDs) are used inside the reactor core to obtain neutron flux distributions. These detectors are capable of being embedded in the reactor fuel for in-situ, distributed online monitoring of the system. Unfortunately, these detectors do not provide real-time estimates of reactor power, which would inhibit their use as signals for closed-loop controllers. The signals these detectors create have two components: one is proportional to prompt neutrons (and occurs instantaneously), and the other is related to emission of beta particles following a neutron absorbed in an emitter material which provides the reason for their relatively slow response their output is dependent on the beta decay of the emitter material. New signal processing designs could improve the time response of these detectors, facilitating their use for reactor control and protection-safety purposes. The main advantage of SPND over standard ion chamber-type detectors is that they do not need external power supplies. Bombarding the emitter material in an SPND with neutron flux activates the emitter, which subsequently decays by emitting beta particles [8]. The SPNDs can be built with a relatively small mechanical size, which is advantageous for in-core (i.e., in-fuel element) measurements. Also, they also exhibit high resistance to temperature and pressure. There are some disadvantages when applying SPND. Compensation of the background noise is necessary and due to the decay process, SPNDs exhibit a significantly delayed signal response 45

58 [9]. In this thesis, inverse detector models are designed to compensate for the inherent SPND sensor delay. Delayed response mainly comes from the (n,β) interaction within the emitter i.e., the detector signal will be proportional to the neutron activation of the emitter. A SPND design consist of three main components: an emitter, a collector and an insulator. Figure 5-1 is a typical representation of a SPND: [8] Figure 5-1. Self-powered neutron detector components The detector models (differential equations) for the Vanadium and Rhodium detectors will be developed using equations that represent the balance of production and decay (loss) of the isotopes considered. The rate-of-change in the number of nuclei of isotope X can be described as dx(t) dt = Production of the Isotope X Loss of the Isotope X 5-1 The loss from the decay of the Vanadium and the Rhodium produced will, in turn, be a production of the next stable nuclei in the chain. 46

59 Section 5.1- Vanadium Self-Powered Neutron Detector Model (Forward Model) In this section, sensor modeling and neutron flux estimation will be discussed for the vanadium self-powered detector. The equations and parameters will be developed from the balance equation for the Vanadium isotope considered. The vanadium detector has slightly more advantages compared to other SPNDs. Vanadium has a 1/v characteristic without any resonances in the thermal energy range. It also has a low reactivity load and low burn-up rate, which makes it a strong candidate for in-core applications. The main disadvantage of the vanadium detector is its very slow response time due to the long half-life of Vanadium-52. This isotope decays with a 3.76 min half-life in 99 % of all transitions [8]. If the time to steady-state is typically 5-7 half-lives, then the delay between actual reactor power change and sensor output is on the order of 25 minutes given the short time constants encountered in the dynamics of nuclear reactors (seconds), this delay would prove to be intolerable for closed loop reactor control. Section 5.1.1: Vanadium Self-Powered Neutron Detector Rate Equations Vanadium has the decay scheme shown in Figure 5-2. Vanadium-51 absorbs a neutron and becomes Vanadium-52*, which emits a gamma ray and eventually beta decays to Chromium-52 with a half-life of 3.76 min. Figure 5-2. Vanadium Decay Mechanism 47

60 The rate equations of the Vanadium can be written from the production of and radioactive transition between different isotopes. The Vanadium rate equations are provided in Equation 5-2. The corresponding equation coefficients are provided in Table 5-1. [7] dn 51 (t) = σ dt 51 N 51 (t)φ(t) 5-2 dn 52 (t) = σ dt 51 N 51 (t)φ(t) λ 52 N 52 (t) i(t) = k pv σ 51 N 51 (t)φ(t) + k gv λ 52 N 52 (t) Where N 51 andn 52 : Atomic densities of Vanadium-51 and Vanadium-52 σ 51 : Microscopic neutron absorption cross-section of Vanadium-51 λ 52 : decay constant of Vanadium-52 i(t): Current from SPND k pv and k gv : Probabilities of Vanadium-51 neutron capture and Vanadium-52 decay leading to a current carrying electron φ(t): Input flux from the reactor Table 5-1. Vanadium Detector Constants [7] The constants Values and Units N cm 3 σ cm 2 λ 52 k pv k gv s A s A s 48

61 Section 5.1.2: Vanadium Self-Powered Neutron Detector Model in Simulink The vanadium rate equations (5-2) have been used to create a Vanadium detector model in Simulink with the constants given in Table 5-1. Figure 5-3 shows the Vanadium detector model created in Simulink. The model uses reactor flux from the TRIGA Reactor model as an input. The assumption made here is that the reactor power is directly proportional to the neutron flux. The neutron flux is used in the rate equations to determine the atom densities of Vanadium-51 and Vanadium-52. The model also calculates the corresponding detector output as a current being a function of the neutron flux. The current output would have some amount of delay due to the dependency of the decay of Vanadium-52. Figure 5-3. Vanadium Self-Power Neutron Detector Model in Simulink The Vanadium detector simulation showed that the burn-up rate of Vanadium-51 is negligible, but it was kept in the model for completeness. 49

62 Figure 5-4 shows Vanadium detector model response with step power change from 1 MW to 900 kw. Due to its characteristic decay mechanism, detector responses very slow the reactor power change. The detector current output slowly decreases with a relatively long time which proves that using Vanadium detector without any compensation cannot be used in state feedback controller. Figure 5-4. Vanadium Self-Power Neutron Detector response at power set point change from 1 MW to 900 kw Section 5.1.3: Inverse Vanadium Detector Model In order to deploy SPND for the nuclear reactor control applications, a compensation technique must be developed to eliminate (or at least minimize) the delay introduced by the inherent detector response. An inverse reactor model was developed for overcoming the signal delay. Starting with the Vanadium rate equations (Equations5-2) dn 51 (t) = σ dt 51 N 51 (t)φ(t) dn 52 (t) = σ dt 51 N 51 (t)φ(t) λ 52 N 52 (t) i(t) = k pv σ 51 N 51 (t)φ(t) + k gv λ 52 N 52 (t) Use the third equation to solve for the neutron flux, 50

63 φ(t) = i(t) k pv σ 51 N 51 (t) k gvλ 52 N 52 (t) k pv σ 51 N 51 (t) 5-3 Inserting Equation 5-3 into Equation 5-2 will gives dn 52 (t) i(t) = σ dt 51 N 51 (t) [ k pv σ 51 N 51 (t) k gvλ 52 N 52 (t) k pv σ 51 N 51 (t) ] λ 52N 52 (t) 5-4 Simplifying Equation 5-4 gives dn 52 (t) dt = [ i(t) k pv λ 52N 52 (t)(k gv + k pv ) k pv ] 5-5 Given the measured current output from the detector, Equation 5.5 is used to calculate the time derivative of Vanadium-52, which after integration provides the concentration of Vanadium-52. Equation 5.3 is subsequently used to determine the neutron flux that activated the emitter material. Equations 5-3 and 5-5 are implemented in Simulink to predict neutron flux given the current output from the detector. Figure 5-5 is a representation of the developed inverse detector model in Simulink. 51

64 Figure 5-5. Inverse Detector model for Vanadium Self-Power Neutron Detector Power and detector current response for Vanadium model and inverse detector model will be provided in Section 5.3 using the state feedback controller. Section 5.2- Rhodium Self-Power Neutron Detector Model Due to the high absorption cross-section of Rhodium-103, a Rhodium SPND provides greater signal strength compared to a comparably-sized Vanadium detector. Additionally, the isotope of Rhodium that is the emitter has a shorter half-life and provides a faster response (but still not fast enough for closed-loop control). These properties are well suited for identifying flux maps in the PWR systems. It is expected that rhodium detector model will give faster response comparing to Vanadium detector model. However, the relatively high absorption cross-section of 52

65 Rhodium implies that the Rhodium SPND will burn-out faster than a comparably-sized Vanadium SPND. Section 5.2.1: Rhodium Self-Power Neutron Detector Rate Equations Section Compared to Vanadium, Rhodium has a relatively complicated decay scheme as shown in Figure 5-6. An important detail in rhodium decay is the meta-stable states in its decay mechanisms. [9] Figure 5-6. Rhodium Decay Mechanism Equation 5-6 provides the rate equations developed for the isotopes that result in a Rhodium SPND current signal. The important contributors are Rhodium-104 and Rhodium 104m. dn 104m (t) = σ dt 104m N 103 φ(t) λ 104m N 104m (t) 5-6 dn 104 (t) = σ dt 104 N 103 φ(t) + λ 104m N 104m (t) λ 104 N 104 (t) i(t) = k pv (σ 104 +σ 104m )N 103 φ(t) + k gv λ 104 N 104 (t) 53

66 Where N 103, N 104 andn 104m : Atomic densities of Rhodium-103, Rhodium-104 and Rhodium-104m σ 104 and σ 104m : Microscopic neutron absorption cross-section of Rhodium-104 and Rhodium- 104m λ 104 and λ 104m : decay constant of Rhodium-104 and Rhodium-104m i(t): Current from Self powered neutron detector k pv and k gv : Probabilities of Rhodium-103 neutron capture and Rhodium-104 decay leading to a current carrying electron. φ(t): Input flux from the reactor Table 5-2. Rhodium Detector Constants [10] The constants N 103 N 104m N 104 Values and Units cm cm cm 3 σ cm 2 σ cm 2 λ 104 λ 104m k pv k gv s s A s A s 54

67 Section 5.2.2: Rhodium Self-Power Neutron Detector Simulink Model Equations 5-6 were coded inside Simulink. The input to this model will be reactor flux coming from the TRIGA Reactor model and the output is detector current. Figure 5-7. Rhodium Self-Power Neutron Detector Model in Simulink Figure 5-8 is the representation of the Rhodium detector step power change response. The Rhodium detector has better performance than the Vanadium detector model however, there still is a delay in detector current which is associated with the decay mechanism. 55

68 Figure 5-8. Rhodium Self-Power Neutron Detector response step power change from 1MW to 900 kw Section 5.2.3: Inverse Rhodium Self-Power Neutron Detector Model Similar derivation was followed to derive Inverse Rhodium detector model. dn 104m (t) = σ dt 104m N 103 φ(t) λ 104m N 104m (t) 5-7 dn 104 (t) = σ dt 104 N 103 φ(t) + λ 104m N 104m (t) λ 104 N 104 (t) i(t) = k pv (σ 104 +σ 104m )N 103 φ(t) + k gv λ 104 N 104 (t) Neutron flux can be derived from the current equation: φ(t) = i(t) k pv (σ 104 +σ 104m )N 103 (t) k gv λ 104 N 104 (t) k pv (σ 104 +σ 104m )N 103 (t) 5-8 Inserting the neutron flux Equation 5-8 into Equation 5-7 will gives the rhodium rate equations with respect to detector current: 56

69 dn 104m (t) i(t) = σ dt 104m N 103 ( k pv (σ 104 +σ 104m )N 103 (t) k gv λ 104 N 104 (t) k pv (σ 104 +σ 104m )N 103 (t) ) 5-9 dn 104 (t) dt λ 104m N 104m (t) i(t) = σ 104 N 103 ( k pv (σ 104 +σ 104m )N 103 (t) k gv λ 104 N 104 (t) k pv (σ 104 +σ 104m )N 103 (t) ) + λ 104m N 104m (t) λ 104 N 104 (t) Simplifying the Equation 5-9 will gives the rhodium detector inverse model rate equations: dn 104m (t) i(t) = ( dt k pv (σ 104 )N 103 (t) k gvλ 104 N 104 (t) k pv (σ 104 )N 103 (t) ) λ 104mN 104m (t) 5-10 dn 104 (t) i(t) = ( dt k pv (σ 104m )N 103 (t) k gvλ 104 N 104 (t) k pv (σ 104m )N 103 (t) ) + λ 104mN 104m (t) λ 104 N 104 (t) The Equation 5-10, the inverse Rhodium detector model, were coded in Simulink model. The comparison between the inverse model and the normal detector model will be performed using the closed-loop state feedback controller in next section. Section 5.3- Using Self Powered Vanadium and Rhodium Detectors as Closed-Loop Feedback Signals The main purpose of this thesis is to design a state feedback controller using SPNDs as a closed loop control system feedback signal. The advantage of a closed-loop system is to use the feedback signal to reduce errors and improve stability. (A system in which the output has no effect on the input signal is called an open loop system; open loop systems don`t have feedback.) The closed-loop system has the system (i.e., the reactor) in its forward path, and incorporates a 57

70 feedback signal path, closing the loop. Closed-loop controllers are designed to automatically maintain the desired system output by comparing the actual measured system output to a setpoint, or desired system output. The state feedback design in Chapter 4 is an example of a closed-loop system that successfully controls the TRIGA Reactor model. The Vanadium and Rhodium detector models have been implemented in the TRIGA reactor state-feedback controller/observer design. The output signal will be converted into neutron flux by assuming that neutron flux is linearly dependent on reactor power. The current output of detector will be converted into a neutron flux estimate by using the exact inversion model for each detector type. Afterward, this signal will be the new input the state feedback observer. Figure 5-9 is a representation of the finalized system showing the SDND current feeding into an inverted sensor model, producing a power estimate used by the state feedback controller. The green arrow represents the normal Vanadium detector model subsystem. The input of reactor flux converted to current in the Vanadium detector model. Vanadium inverse detector model uses Vanadium detector model and inverse Vanadium detector model shown in red arrow. Similarly, for the Rhodium detector model (shown in blue) uses reactor flux as an input and output scaled and used as a state feedback controller/ observer power measurement. The developed Rhodium inverse detector model implemented into Simulink which is shown in purple arrow. Finally, the output of the inverse detector model is scaled and connected to power measurement input of state feedback controller/observer shown with a black arrow. Additional Simulink simulations have the normal detector models connected to the state feedback controller/observer for comparison with the use of the inverse detector models. All the detector 58

71 models use neutron flux data from TRIGA Reactor simulation and, process this data to obtain the error signal which is related the difference between power setpoint and measured power. The state feedback controller sends a velocity demand to the ECR depending on the difference between the power setpoint and the power measurement and controls the TRIGA Reactor simulation. 59

72 60 Figure 5-9. Finalized Model with normal and inverse detector models and the state feedback controller/observer

73 Figure 5-10 shows the power setpoint change response for the Vanadium detector and inverse Vanadium detector model. The power changed from 1MW to 900 kw at 100 seconds and responses observed. Clearly, use of the inverse model allows the control system to follow exactly same power setpoint change without any delay. On the other hand, use of the normal Vanadium detector model exhibits an extremely long delay resulting in a poor transient response, resulting in excessive oscillation and response time and it is not suitable for use in closed-loop reactor control. The bottoming out of the power seen in Figure 5-10, when using the normal detector configuration and not modifying the output by feeding into the inverse model, is due to the ECR reaching its lower limit. This is due to the control system attempting to control a system it perceives, due to the excessive delay of the sensor, to be significantly slower than the actual system. Figure Relative Power and Vanadium detector current with inverse and without inverse detector model 61

74 Similarly, rhodium detector model and inverse rhodium detector model were compared with power setpoint change. Again, inverse detector model successfully follows power setpoint without any delay. Although there is a delay in normal rhodium detector model, the reactor power does not have any delays, only has some overshoot. Figure Relative Power and Rhodium detector current with inverse and without an inverse model 62

75 Chapter 6 - Summary, Conclusions and Future Work Section 6.1- Validation of the TRIGA Reactor Simulink Model Development and validation of a Simulink TRIGA reactor simulation model were accomplished to facilitate design and implementation of advanced feedback controllers (i.e., state feedback controllers) and to incorporate self-powered neutron detectors (SPND) as part of the closed-loop system. This was an essential part of the design process since it was discovered that SPNDs have a significant delay associated with them, and are not inherently suitable for use in closed-loop control. After modeling of the neutronics and thermal hydraulics of the PSU TRIGA reactor, test data was obtained and compared to the output of the simulation. This was primarily power data and control rod worth data. Actual control rod worth curve data was converted to a series of polynomial curves that were implemented in the Simulink TRIGA model. This model was tested by feeding rod positions from the operating console, feeding those positions into the TRIGA model, and comparing the output power of the model to that of the actual reactor. The difference observed between measured data and Simulink model at the higher power levels is primarily due to sampling time that the control rod position data was sampled at (1 second). Nevertheless, the two datasets matched with each other well given the final application of the model, to design closed-loop control systems that are robust to plant uncertainties (i.e., differences between the model used to design the controllers and the actual plant). Section 6.2- Experimental Control Rod Design and Experimental Results Experimental Control Rod (ECR) drive mechanism control was successfully implemented in the LabVIEW environment. The desired position for ECR can be achieved by using the developed LabVIEW program. ECR control rod worth data was obtained and plotted 63

76 using reactivity computer, with the data used to create a polynomial curve used to calculate ECR control rod worth in the Simulink environment for future use in state feedback controller design and testing. The experiment took place in Penn State TRIGA Reactor. Standard Operation Procedure (SOP) was developed for the test (Appendix B). The ECR was mounted atop of the Penn State TRIGA Reactor and the drive mechanism motor controller connected to the LabVIEW control program. The experiment demonstrated that the ECR can be successfully controlled by the LabVIEW program. The resulting ECR integral control rod worth was $0.91. After obtaining the reactivity worth of the ECR, rod worth curve was incorporated into the Simulink environment as part of the ECR control rod dynamics used to control the Penn State TRIGA Reactor simulation. The ECR is used to replicate the actual conditions that would exist when controlling the real reactor. The four standard control rods cannot be used for control experimentation (they are only approved for licensed use and standard operation). Final deployment of state feedback controllers will necessary use the ECR to control the TRIGA reactor. Section 6.3- State Feedback Controller Design A state space representation of the Penn State TRIGA Reactor was necessary to design state observers and state feedback controllers of the TRIGA. The six delayed group point kinetics equations and core-averaged thermal-hydraulic equations were linearized and used to create a state space representation of the Penn State TRIGA Reactor. The response of the state space representation of the TRIGA Reactor was compared to the TRIGA Reactor Simulink model, using identical inputs, and the results show that the state feedback representation was accurately derived. 64

77 Using the state space representation of the TRIGA Reactor, a state observer and feedback controller was designed using Linear Quadratic Regulator (LQR) controller design algorithm. LQR design provides controller gains that are multiplied by estimates of the internal states of a system. These states are not measurable and must be estimated using a state observer. The state feedback controller compares the setpoint power level and measured power, and uses the corresponding error calculated to determine a velocity demand signal sent to the ECR. The state feedback controller/observer was successfully developed by changing design different parameters, such as weighting factors on the states and measurements in a quadratic performance index in the LQR design, and implemented in Simulink. The controller/observer successfully follows power setpoint changes with no overshoot or undershoot in reactor power and fuel temperature and successfully drives the ECR by demanding a speed from the ECR motor drive, which ultimately changes the reactivity associated with the ECR. This was compared to a conventional Proportional-Plus-Integral controller, and showed significantly improved performance. Section 6.4- Self-Powered Neutron Detector Designs Vanadium and Rhodium rate equations were translated into Simulink to obtain detector models. The important issue in the self-powered neutron detectors is that the detector response is generally very slow compared to that of the response of the actual reactor, which makes these detectors unsuitable for closed-loop reactor control. Compensating, or eliminating, this delay is necessary to use the self-powered neutron detectors in real-time reactor control. For this purpose, inverse detector models were developed to minimize the delays due to the inherent isotope decay mechanisms, the timing of which is due to the nuclides half-life. 65

78 Results demonstrate that the inverse detector models have no delay which is desirable for the reactor closed-loop control. The long delay associated with the normal detector models can only realistically be used for applications where this delay can be tolerated, such as post-accident power monitoring. Section 6.5- Self Powered Detector Model in Closed Loop The self-powered detector models were combined with the state feedback controller/observer and deployed in the TRIGA Reactor simulation. The state feedback controller/observer and the detector models are used in a closed-loop application and will be used to estimate the error between measured power and setpoint power, and compensate for the difference. Four different detector models were connected to the state feedback controller/observer to estimate neutron flux and detector current, and to use them as a feedback power signals. Results showed that the inverse detector models successfully allow for the control system to follow the power setpoint changes without any delay and overshoot. On the other hand, normal detector model output current has a long time delay, which makes them unsuitable for reactor control. 66

79 Section 6.6- Conclusion The safe and effective control of the nuclear reactors is the main requirement for nuclear reactor operation. Control algorithms that can support this goal, provide rapid power and temperature response while maintaining all reactor operating limits, will prove to be essential in future, and advanced reactor designs. Simulations of these systems are essential in designing these advanced controllers, prior to deployment on the actual reactor. State feedback control is common controller design in fields outside of reactor control and has been applied to virtually every type of dynamic system (with the exception of nuclear reactors). This thesis developed an accurate model of the Penn State TRIGA Reactor simulation and created a state feedback controller/state observer design using self-powered Vanadium and Rhodium neutron detectors as feedback sensors to the author s best knowledge this work is the first attempt to use these type of sensors in a closed-loop feedback system for reactor control. The linearized state-space equations used to design the control system has been derived from normalized point kinetics equations and core averaged thermal-hydraulic equations. The self-powered neutron detector dynamics may be developed from production/decay balance differential equations. Results demonstrate that the TRIGA model developed compares well with the actual TRIGA Reactor. Due to their dependence on radioactive decay after irradiation to produce a current signal, selfpowered detectors have significant delay times associated with them, making them not useful for real-time feedback control. The development and application of detector inverse models prove that the delays introduced by the physics of the detector could be eliminated by inverse models. The results from the self-powered neutron detectors in closed-loop proves that this design may successfully be applied for use in closed-loop advanced reactor control. 67

80 Section 6.7- Future Work The control system developed in this thesis has been tested solely in the simulation. The next phase of development will implement the controller in the LabVIEW environment. LabVIEW is a commonly used program to develop and deploy experimental control systems, and allows for seamless integration software and hardware. The TRIGA Reactor simulation, ECR, self-powered neutron detector models and the state feedback controller/observer may be easily coded inside a LabVIEW real-time program. Ultimately, the Westinghouse Rhodium self-powered neutron detector signal will be used as a feedback signal, fed to an inverse model of the detector, and used as part of a state feedback controller to obtain desired reactor power while minimizing reactor fuel temperature over/undershoot. The major issues with using actual self-powered neutron detectors in TRIGA reactor control is the significant associated delay and the effect of signal noise - both of which will be addressed by use of an inverse detector model and digital filtering implemented in a suitably-designed LabVIEW program. 68

81 References [1] J. A. Turso, "Penn State University TRIGA Reactor Digital Reactivity Computer: Development and Testing," Annals of Nuclear Energy, vol. 114, pp , [2] Penn State University Radiation Science and Engineering Center, RSEC History, [Online]. Available: [Accessed 2018]. [3] K. Ünlü, "The Radiation Science and Engineering Center Utilizations and Future Developments at Penn State University," Transactions of the American Nuclear Society, vol. 116, [4] J. Lamarsh and A. Baratta, Introduction to Nuclear Engineering, Upper Saddle River, New Jersey 07458: Prentice Hall, Inc, [5] E. D. Sontag, Mathematical Control Theory -Deterministic Finite Dimensional Systems, NJ: Springer, [6] K. Ogata, Modern Control Engineering, NJ: Prentice Hall, [7] K. Srinicasarengan, L. Mutyam, M. N. Belur, M. Bhushan, A. Tiwari, M. Kelkar and M. Pramanik, "Flux Estimation from Vanadium and Cobalt Self Powered Neutron Detectors(SPNDs): Nonlinear Exact Inversion and Kalman filter approaches," American Control Conference, [8] F. P. S. H. C. Z.,. L. D.,. Khoshahval, "Vanadium, Rhodium, Silver and Cobalt Self- Powered Neutron Detector Calculations by RAST-K v2.0," Annals of Nuclear Energy, vol. 111, pp ,

82 [9] S. W. H. Todt, Characteristics of Self-Powered Neutron Detectors used in Power Reactors, Switzerland: European Nuclear Society, [10] G.-S. Auh, "Digital Dynamic Compensation Methods of Rhodium Self-Powered Neutron Detector," Journal of the Korean Nuclear Society, vol. 26, [11] Nuclear Regulatory Commision, "Shutdown Margin," [Online]. Available: 70

83 Appendix A: Matlab code for parameters and some calculations: lam52=0.0036; sigma51=4.9*10^-24; N51=6.86*10^22; N52=9.3372e+14; kpv=3.487*10^-21; kgv=3.846*10^-20; % A=[-lam52 sigma51*n51; 0 0]; % B=[1;1]; % C=[kgv*lam52 kpv*sigma51*n51]; % D=0; % % VD=ss(A,B,C,D); % % Vd_dis=c2d(VD,0.1,'Tustin') % % [KEST,L,P] = kalman(vd_dis,1e26,1e-16); Tz = 26; Tp=313; Sv = 1.415e-20; Avd = -(1/Tz); Bvd = (Tz - Tp)/(Sv*Tz^2); Cvd = 1; Dvd= Tp/(Sv*Tz); Beta=0.007; LAMBDA=0.0001; lambda=0.1; Cr0=1.0; Nr0=1.0; rho=0.001; T=LAMBDA/rho + (Beta-rho)/(lambda*rho); %T1=LAMBDA/(rho-rho*0.5) + (Beta-(rho-rho*0.5))/(lambda*(rho-rho*0.5)) %1.377*Beta/(Beta-(-rho*0.5))*exp(30/T1) t1=55.6; t2=22.7; t3=6.22; t4=2.30; t5=0.61; t6=0.23; lambda1=log(2)/t1; lambda2=log(2)/t2; lambda3=log(2)/t3; lambda4=log(2)/t4; lambda5=log(2)/t5; lambda6=log(2)/t6; Beta1= ; Beta2= ; Beta3= ; Beta4= ; Beta5= ; Beta6= ; sumbeta=beta1+beta2+beta3+beta4+beta5+beta6; P = 950/100; MF = 16.44; Cpf = ; Cpc = 1; m_dot_c = 1.055; Mc = 17; 71

84 Tin = 75; Tf0 = ; Tc0 = 80.24; alpha_t=2.5054e-05; %% State feedback A=[(-lam52*(kpv+kgv))/kpv]; B=[1/kpv]; C=[(-kgv*lam52)/(kpv*sigma51*N51)]; D=[1/(kpv*sigma51*N51)]; p1=10^-23; Q=p1*C'*C; R=10^23; [K]=lqr(A,B,Q,R) sys_cl=ss(a-b*k,b,c,d) step(141*10^-9*sys_cl) %% pole poles=eig(a) p2 = [ ]; K1 = place(a,b,p2) l = place(a',c',p2).' sys_cl1=ss(a-b*k1,b,c,d) step(141*10^-9*sys_cl1) %% phi0=10^13; A1=[-Beta/LAMBDA Beta/LAMBDA 0;lambda -lambda 0;sigma51*N51*phi0 0 -lam52]; B1=[1/LAMBDA;0;0]; C1=[kpv*sigma51*N51*phi0 0 kgv*lam52]; D1=[0]; new=ss(a1,b1,c1,d1) poles=eig(a1) k2=place(a1,b1,(poles)) sys_cl2=ss((a1-b1*k2),b1,c1,d1) t = 0:0.01:2000; u = zeros(size(t)); x0 = [1 1 0]; lsim(10^9*sys_cl2,u,t,x0); %% co = ctrb(sys_cl2); controllability = rank(co) ob = obsv(new); observability = rank(ob) Q2 = C1'*C1; R2 = ; K4 = lqr(a1,b1,q2,r2) Ac = [(A1-B1*K4)]; Bc = [B1]; Cc = [C1]; Dc = [D1]; sys_cl5 = ss(ac,bc,cc,dc); t = 0:0.1:2000; r =ones(size(t)); [y,t,x]=lsim(sys_cl5,r,t); plot(t,10^9*y) %% with temp feedback phi0=10^13; 72

85 %zz=mean(ua1(1)) zz= ; A2=[-Beta/LAMBDA Beta/LAMBDA 0 alpha_t 0;lambda -lambda 0 0 0;sigma51*N51*phi0 0 -lam52 0 0;P/(MF*Cpf) zz/(mf*cpf) zz/(mf/cpf);0 0 0 zz/(mc*cpc) -zz/(mc*cpc)-2*m_dot_c/mc]; B2=[1/LAMBDA;0;0;0;0]; C2=[kpv*sigma51*N51*phi0 0 kgv*lam52 0 0]; D2=[0]; Q3 = C2'*C2; R3 = ; K5 = lqr(a2,b2,q3,r3) tempfeed=ss(a2-b2*k5,b2,c2,d2) % [y1,t,x]=lsim(tempfeed,r,t); % plot(t,10^9*y1) t = 0:0.01:2000; u = zeros(size(t)); x1 = [ ]; lsim(10^9*tempfeed,u,t,x1); %% A3=[-Beta/LAMBDA Beta1/LAMBDA Beta2/LAMBDA Beta3/LAMBDA Beta4/LAMBDA Beta5/LAMBDA Beta6/LAMBDA -alpha_t/lambda 0;lambda1 -lambda ;lambda2 0 -lambda ;lambda lambda ;lambda lambda ;lambda lambda ;lambda lambda6 0 0;P/(MF*Cpf) zz/(mf*cpf) zz/(mf*cpf); zz/(mc*cpc) -zz/(mc*cpc)-2*m_dot_c/mc] B3=[1/LAMBDA;0;0;0;0;0;0;0;0] C3=[ ] D3=[0] % A3=[-Beta/LAMBDA Beta/LAMBDA -alpha_t/lambda 0;lambda -lambda 0 0;P/(MF*Cpf) 0 -zz/(mf*cpf) zz/(mf*cpf);0 0 zz/(mc*cpc) -zz/(mc*cpc)-(2*m_dot_c/mc)] % B3=[1/LAMBDA;0;0;0] % C3=[ ] % D3=[0] 73

86 Appendix B: Penn State Breazeale Reactor Standard Operating Procedure Experiment Evaluation and Authorization 1. Valid Period 2/15/2018-5/30/ Supervisor J. Geuther Phone Academic Rank (if applicable) _Assoc. Dir for Operations_ Department (Company)_RSEC_ Address 101 Breazeale Nuclear Reactor 3. Experimenter (s) J. Turso/G. Corak Phone Academic Rank (if applicable) _Assoc. Rescearch Prof. Department (Company) RSEC Address 101 Breazeale Nuclear Reactor 4. Experiment Description ECRD Initial Testing and Worth Determination No. 5. Encapsulation None Needed, can be immersed in reactor pool water 6. Max Time 3 hours Max Power Level 10 kw (900 kw limit/rabbit or CT Osc) 7. Location(s) _Central Thimbal Mechanism Mounted On Bridge at Dedicated Location (200 lb. total limit on experiments supported by grid plate) 8. Attachments (see 17. also) ECRD 2 Dose Rate Calculations for 1 hour, 24 hours, 30 days. 9. Experimental Procedures Appendix A: PENN STATE BREAZEALE REACTOR ECRD 2 INSTALLATION AND REMOVAL PROCEDURE, REVISION 0 Appendix B: ECRD 2 Rod Worth Measurement Appendic C: SOP-5 PSBR Reactivity Measurement Worksheet Appendix D: Safety Evaluation for Experimental Changeable Reactivity Device #2 (ECRD #2) January 18, 2000 NOTE: SAMPLE WILL BE DANGEROUSLY RADIOACTIVE IMMEDIATELY AFTER IRRADIATION. WAIT 7 DAYS PRIOR TO REMOVAL TO EAST WALL. POST IRRADIATION: Place sample in Central Thimble at least 6ft above core to ensure no further irradiation of sample DO NOT COMPLETELY REMOVE FROM CT UNTIL AT LEAST 7 DAYS POST IRRADIATION. Relocate to east wall upon removal from CT. NOTE: REMOVAL FROM Central Thimble SHALL BE PERFORMED SLOWLY, WITH REACTOR SHUTDOWN. USE SURVEY INSTRUMENTS WHILE REMOVING. IF DOSE RATES EXCEED EXPECTED VALUES, CEASE REMOVAL AND PLACE IN ORIGINAL POST- IRRADIATION LOCATION. 74

87 10. Neutron Exposure Data Date Sample ID No. Time Power Fluence Daily MWH Total MWH 11. UIC Authorization (or NRC or Agreement State License ) No. _R-2 Expiration Date

88 12. Material Data (table below or attachment) summary by J. Turso Date 2/8/18 Calc/Est SEE ATTACHED DOSE RATE CALCULATION RESULTS A B C D* E F** G Material & Weight or Isotopes Expected Gamma Gamma Exposure Rate Activity Identification Volume Produced Activity Factor At a Distance Limits * days hours after irradiation is the earliest time that the sample should be released **Rule of thumb for Beta exposure each mci of activity produces 300 rad/hr at 1 cm and 100 mrad/hr at 1 ft. 13. Review Considerations: Radiation Exposure & Effluent Release considered in ALARA (date) x Not Required Ar-41 production (< 1xE-8 Ci/ml est/meas) N/A Ar-41 production meas/cal if estimated dose > 0.01 mrem/24 hrs/exclusion boundary (release N/A Ar-41 production meas/cal if estimated dose > 0.1 mrem/24 hrs/exclusion boundary (no release N/A Total excess reactivity change to the Known Loading (< $1.00 est/meas) TBD Determine via test. (includes fuel, experiments, and experimental facilities) Maximum Cold Core Excess Reactivity with exp/exp fac in place ($7.00 est/meas) N/A Reactivity Worth/Fuel Changes (est/meas) N/A Reactivity Worth/Movable Experiment (< $2.00 est/meas) <$1.00 (negative) Reactivity Worth of Movable Experiments or Movable Portions of a Secured Experiment Plus Maximum Allowed Pulse Reactivity (< $3.50 est/meas) No pulsing authorized during this experiment Unless ECRD is is decoupled from drive and upper Connecting rod is pulled several feet above core. Reactivity Worth/Secured Experiment (< $3.50 est/meas) N/A Reactivity Worth/All Experiments (< $3.50 est/meas) <$1.00 (negative) Failure Mechanisms Considered (corrosion, overheating, impact from projectiles, chemical and mechanical explosions) N/A Off Gas-Sublimation-Volatilization-Aerosol Production N/A Fueled experiment - Iodine inventory (< 1.5 Ci) N/A Results of review if > 5mCi of Iodine N/A Safety System Review X Yes Approval Date 1/18/2000 or, 15. PSRSC Review Date or, X No 76

89 16. Radiation Protection Office Monitoring of Release Required (Yes/No)? No RWP required by UIC Auth (Yes/No)? No RWP required by this SOP-5 No 17. Approval Conditions or Restrictions: Operations with ECR2 restricted to R1. Excess reactivity determined after step 45, App B 18. Approved by: Date: ECRD 2 Activity 1 Hour Post-Irradiation Target Product Initial_Activity Delayed_Activity Gamma Beta Nuclide Sample(g) Nuclide/Isotope (uci) (uci) (mr/hr) (mr/hr) Al Al Al Mg Al Na Cd Cd-115m Cd Cd-115g Cd Cd-117m Cd Cd-117g Cd Cd-111m Total mr/hr Target Nuclide Sample(g) ECRD 2 Activity 24 Hours Post-Irradiation Product Nuclide/Isotope Initial_Activity (uci) Delayed_Activity (uci) Gamma (mr/hr) Beta (mr/hr) Al Al Al Mg Al Na Cd Cd-115m Cd Cd-115g Cd Cd-117m Cd Cd-117g Cd Cd-111m Total mr/hr ECRD 2 Activity 30-Days Post-Irradiation Target Product Initial_Activity Delayed_Activity Gamma Beta Nuclide Sample(g) Nuclide/Isotope (uci) (uci) (mr/hr) (mr/hr) Al Al Al Mg Al Na Cd Cd-115m Cd Cd-115g Cd Cd-117m Cd Cd-117g Cd Cd-111m Total mr/hr 77

90 Procedure for Installation or Removal of an ECRD I. Purpose: II. III. IV. To define the steps required for installation and removal of the Experimental Changeable Reactivity Device 2, ECRD 2. Precautions: A. This procedure shall be done under the supervision of the duty SRO. B. Ensure all precautions of SOP-1, relating to experimental apparatus are followed. C. When handling an ECRD that has been irradiated use gloves and a survey meter. References: A. SOP-1, Reactor Operating Procedure B. Technical Specifications 3.7 C. Figures A1 and A2, ECRD attachment to Drive Motor, (attached) Special Equipment: A. ECRD 2 B. ECRD restraint padlock C. AP 10, Equipment Tags V. Procedure: 78

91 1. Place a string into the Central Thimble (CT) and retrieve the loose end from the CT cut away below the core bearing. Pass the loose end between the Safety Rod and the instrumented element tubes. 2. Retrieve ECRD from low bay rack (or east pool wall) and remove the Identification Tag. Verify string is securley tied to bolt hole in upper end of ECRD 3. Bring the ECRD toward the CT above the core. 79

92 4. Attach the loose end of the string through the CT to the end of the ECRD string. 5. Pull the string up through the CT until the weight of the ECRD is felt. 6. Raise the ECRD until the lower end swings into the cut away section of the CT. 7. Position the ECRD until the upper end is just above the top of the CT. 8. Remove the attached string and rope and survey. Place in the ECRD parts bag. Ensure someone holds the ECRD until Step From the bag of ECRD parts retrieve an aluminum pin and outer (small) sleeve. 10. Obtain the center and upper sections of the ECRD from the reactor low bay wall. 11. Lower the middle section of the ECRD through the upper hole in the rod drive assembly and the rod drive plate. 12. Place the sleeve onto the middle section of the ECRD and slide it upwards far enough to expose the hole drilled through the end. 13. Align the holes in the lower end of the middle section and the upper end of the ECRD place the aluminum pin through the holes and slid the sleeve down over the pin. 14. Lower the ECRD into the CT ensuring that the lower end remains in the CT. 15. Lower the upper section of the ECRD through the upper hole in the rod drive assembly and the rod drive plate. (see Figure A1) 16. Place the sleeve onto the upper section of the ECRD and slide it upwards far enough to expose the hole drilled through the end. 80

93 17. Align the holes in the lower end of the upper section and the upper end of the middle section, place the aluminum pin through the holes and slid the sleeve down over the pin. 18. Lower the ECRD until the upper end is aligned with the ECRD mounting plate. 19. Refer to Figure A2 to attach the ECRD to the rod drive mounting plate. 20. Secure ECRD electrical power until ready for use. 21. FOR REMOVAL OF THE ECRD USE THE FOLLOWING STEPS. SAMPLE WILL BE DANGEROUSLY RADIOACTIVE IMMEDIATELY AFTER IRRADIATION. WAIT 7 DAYS PRIOR TO REMOVAL TO EAST WALL. POST IRRADIATION: Place sample in Central Thimble at least 6ft above core. 22. Refer to Figure A2 to remove the nuts, washers, and rubber washers from the upper end of the ECRD. ENSURE SOMEONE HOLDS THE ECRD WHILE REMOVING THE HARDWARE. 23. Raise the ECRD through the mounting plate hole and upper rod drive assembly hole until the sleeve connecting the upper and middle sections is exposed. 24. While holding the middle section, slide the sleeve up to expose the aluminum pin. 25. Remove the aluminum pin, separate the sections and remove the aluminum sleeve. Tie the two lower sections of the ECRD to the tower and let deactivate prior to movement to the east pool wall. 26. Remove the upper section by passing it through the upper hole in the rod drive assembly. 27. Raise the ECRD through the mounting plate hole and upper rod drive assembly hole until the sleeve connecting the middle and lower sections is exposed. Survey prior to placing in plastic bag. 28. While holding the lower section, slide the sleeve up to expose the aluminum pin 29. Remove the aluminum pin, separate the sections and remove the aluminum sleeve. Survey prior to placing in plastic bag. 81

94 30. Remove the middle section by passing it through the upper hole in the rod drive assembly. 31. Raise the ECRD until the lower end of the ECRD is at least 6ft above reactor core. Secure and properly tag. Keep in this location for 7 days prior to removal to east reactor pool wall. LEAST 7 DAYS POST IRRADIATION. DO NOT COMPLETELY REMOVE FROM CT UNTIL AT 32. Store ECRD mounting parts in a plastic bag for later use. Provide to Experimenter. 33. When ready to be relocated to reactor pool east wall, raise the ECRD until the lower end of the ECRD can be swung out of the cut away section of the central thimble. 34. Lower the ECRD until the rope can be retrieved from the cut away section of the central thimble below the core bearing. 35. Carefully remove the ECRD from the area above the core without disturbing the instrumented elements, thermocouple connectors and control rods. 36. Hang the ECRD on the pool wall. Ensure that ECRD is properly tagged and secured. 82

95 Figure A1: ECRD Drive and major components 83

96 Figure A2A: ECRD coupling hardware Figure A2B: ECRD coupling and connecting rod mounted on drive Appendix B- ECRD 2 Rod Worth Measurement CAUTION: 84

97 All steps shall be done at 1.5 rps ECRD speed as set in the ECRD Motor Controller. 1. Ensure that the ECRD motor drive is powered-off (i.e., unplugged). 2. Bring the reactor to Standby. 3. Install ECRD #2 in core Central Thimble (CT) position in accordance with Steps 1-20 of the PSBR ECRD 2 Installation and Removal Procedure (Appendix A). 4. (Experimenter) Verify the ECRD motor controller is configured for 1.5 revolutions per second (rps). 5. Verify that the reactor is at Standby. 6. Remove the padlock from the ECRD. 7. (Experimenter) Setup a separate computer and ECRD controller using LabView and the provided National Instruments hardware. 8. With operator s permission, apply power to the ECRD motor controller. 9. (Experimenter) Demonstrate correct connection polarity and LabView controller functionality by carefully cycling the ECRD over its full travel to check for free movement over entire length and a full travel time of approximately 10 seconds, corresponding to a 1.5 rps rod speed. (Initial motion should be in the downward direction.) 10. (Experimenter) Practice moving the ECRD in small increments such as those that will be needed for the reactivity measurements. 11. Position the ECRD to its Lower Electrical Limit (LEL). 12. Remove power to the ECRD motor controller. 13. Setup the Reactivity Computer and CIC IAW CCP-15 Step C.1. CAUTION: 85

98 All steps shall be done at 1.5 rps ECRD speed as set in the ECRD Motor Controller. 14. The Reactor Operator (RO), Reactivity Computer (RCO) operator and the ECRD Computer operator will communicate using headphones during this procedure 15. Verify that the reactor is at Standby. 16. Go to a power level less than 1kW as requested by the RCO. The reactor should stay below 1kW for this entire procedure. 17. Apply power to the ECRD motor controller. 18. (Experimenter) Verify correct functionality of the LabView ECRD software 19. Measure the reactivity worth of ECRD #2 IAW CCP-15 Step C.2 Note: The RO will compensate for ECRD motion by driving in control rods as instructed by the RCO or as needed. Note: Approximately ten (10) reactivity steps should be used in the measurement with an average of 10 cents per move. 20. When the measurement is complete, take the reactor to Standby. 21. Position the ECRD at the Upper Electrical Limit (UEL) and lock it in place with the padlock. 22. Remove power to the ECRD motor controller. 23. Secure the reactor. 24. Plot the Integral and Differential Rod Worth Curves IAW CCP-15 D Determine the ECRD maximum reactivity and move the UEL and LEL to enclose the most reactive 15 of ECRD #2. Note: The ECRD worth must be confirmed following movement of the limit switches. 26. Bring the reactor to Standby. 27. Remove the padlock from the ECRD. 86

99 CAUTION: All steps shall be done at 1.5 rps ECRD speed as set in the ECRD Motor Controller. 28. With operator s permission, apply power to the ECRD motor controller. 29. Position the ECRD to its lower limit (LEL). 30. The Reactor Operator (RO), Reactivity Computer (RCO) operator and the ECRD Computer operator will communicate using headphones during this procedure. 31. Verify that the reactor is at Standby. 32. Go to a power level less than 1kW as requested by the RCO. The reactor should stay below 1kW for this entire procedure. 33. Apply power to the ECRD motor controller. 34. Measure the reactivity worth of ECRD #2 IAW CCP-15 Step C.2 Note: The RO will compensate for ECRD motion by driving in control rods as instructed by the RCO or as needed. Note: Approximately ten (10) reactivity steps should be used in the measurement with an average of 10 cents per move. 35. When the measurement is complete, take the reactor to Standby 36. Position the ECRD at the Upper Electrical Limit (UEL) and lock it in place with the padlock. 37. Remove power to the ECRD motor controller. 38. Secure the reactor. 39. Plot the Integral and Differential Reactivity Worth Curves IAW CCP-15 D.1. 87

100 40. Compare the curves produced in Steps 28 & 43. If the agreement is satisfactory, the value from the second measurement will be the reactivity worth for the ECRD #2 for all future experiments using Core 57A. Mark the UEL and LEL on the bracket for future reference. 41. Calculate the reactivity insertion rate, averaged over full travel, for ECRD #2 in the worst case scenario for both speeds. A. ECRD 1.5 rps = cents/second B. ECRD 4.5 rps = cents/second C. PSTR in 3 Rod AUTO Mode = cents/second D. Sum of b+c (maximum)= cents/second CAUTION: Verify that the value in Step 41.D is less than 90 cents/second. CAUTION: All steps shall be done at 1.5 rps ECRD speed as set in the ECRD Motor Controller. 42. Unlock the ECRD padlock. 43. Apply power to the ECRD motor controller. 44. Perform a 50W Critical Rod Position Measurement with the ECRD fully withdrawn (UEL). Total Rod Worth = $ 45. Perform a 50W Critical Rod Position Measurement with the ECRD fully inserted (LEL). Total Rod Worth = $ 88

101 46. Compare the rod worth measured in step 44 to the integral worth measured by ECRD calibration for Core 57A (ECRD worth at maximum height) 47. Secure Reactor IAW SOP-1. APPENDIX C PSBR Reactivity Measurement Worksheet Date: Core Position: Core Loading Number: Description of Experiment or Experimental Apparatus: ECRD Xenon Reactivity: Log Book Number: Page: Method Used to Determine Reactivity: Difference of Excess Reactivities Regulating Rod Source: IN OUT Rod Calibration Curve Date: Difference of CRPs Experimental Reactivity: $1.00 Difference of CRPs Method Worksheet Initial Control Rod 100W CRP Final Control Rod 100W CRP Date: Worth Worth 89

102 Transient Rod Safety Rod Shim Rod Regulating Rod Total Worth Transient Rod Safety Rod Shim Rod Regulating Rod Total Worth Experimental Reactivity: 90

103 91

104 92