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
ii
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.
iii
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, self-
powered 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
iv
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 self-
powered neutron detectors, demonstrates that this technology may be applied in closed-
loop nuclear reactor control and safety systems not only for power plant applications, but
for space nuclear reactor applications as well.
v
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 ................................................. 12
Section 2.1- Simulink Block Diagrams ....................................................................................... 13
Section 2.2- Point Kinetics Equations Design in Simulink ......................................................... 13
Section 2.3- Control Rod Modeling in Simulink ........................................................................ 15
Section 2.4- Shutdown Reactivity in Simulink ........................................................................... 17
Section 2.5- Fuel Temperature Feedback (Core Averaged Thermal-Hydraulic) Design in
Simulink ..................................................................................................................................... 17
Section 2.6- Validation of the TRIGA Reactor model ................................................................ 20
Chapter 3 - Experimental Control Rod (ECR) Characterization and Implementation ...... 22
Section 3.1- Experimental Control Rod (ECR) Design in LabVIEW ............................................ 22
Section 3.2- Experiment Preparation ........................................................................................ 26
vi
Section 3.3- Control Rod Worth (Reactivity Effects) Characterization ..................................... 28
Chapter 4 - State Feedback Controller Design for TRIGA Reactor Simulation ................. 31
Section 4.1- State-Space Equations .......................................................................................... 31
Section 4.2- State Space Representation of TRIGA Reactor Plant ............................................ 33
Section 4.3- State Feedback Controller Implemented as a Linear-Quadratic Regulator (LQR) 35
Section 4.4- State Observer Design ........................................................................................... 37
Section 4.4.1: Comparison between State Feedback Observer Design and Proportional-
plus-Integral (PI) Controller .................................................................................................. 41
Chapter 5 - Vanadium and Rhodium Self-Powered Neutron Detector (SPND) Model .... 45
Section 5.1- Vanadium Self-Powered Neutron Detector Model (Forward Model) .................. 47
Section 5.1.1: Vanadium Self-Powered Neutron Detector Rate Equations ......................... 47
Section 5.1.2: Vanadium Self-Powered Neutron Detector Model in Simulink ..................... 49
Section 5.1.3: Inverse Vanadium Detector Model ................................................................ 50
Section 5.2- Rhodium Self-Power Neutron Detector Model .................................................... 52
Section 5.2.1: Rhodium Self-Power Neutron Detector Rate Equations Section .................. 53
Section 5.2.2: Rhodium Self-Power Neutron Detector Simulink Model............................... 55
Section 5.2.3: Inverse Rhodium Self-Power Neutron Detector Model................................. 56
Section 5.3- Using Self Powered Vanadium and Rhodium Detectors as Closed-Loop Feedback
Signals ....................................................................................................................................... 57
Chapter 6 - Summary, Conclusions and Future Work ...................................................... 63
Section 6.1- Validation of the TRIGA Reactor Simulink Model ................................................. 63
Section 6.2- Experimental Control Rod Design and Experimental Results ............................... 63
vii
Section 6.3- State Feedback Controller Design ......................................................................... 64
Section 6.4- Self-Powered Neutron Detector Designs .............................................................. 65
Section 6.5- Self Powered Detector Model in Closed Loop ...................................................... 66
Section 6.6- Conclusion ............................................................................................................. 67
Section 6.7- Future Work .......................................................................................................... 68
References ........................................................................................................................ 69
Appendix A: ....................................................................................................................... 71
Appendix B: Penn State Breazeale Reactor Standard Operating Procedure – Experiment
Evaluation and Authorization ........................................................................................... 74
viii
LIST OF FIGURES
Figure 1-1. A picture of Penn State TRIGA Reactor Core ............................................... 11
Figure 2-1. Six Group Delayed Point Kinetics Equations ................................................ 15
Figure 2-2. Control Rod Reactivity Model for TRIGA Reactor Simulink model ............ 16
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 ........................................................................................................ 19
Figure 2-5. Final TRIGA Reactor model with Reactivity Feedbacks: Control rod
reactivity, point kinetics equations and core averaged thermal hydraulics .............. 20
Figure 2-6. Penn State TRIGA Reactor measured data comparison with Simulink model
data ............................................................................................................................ 21
Figure 3-1. Experimental Control Rod Drive and Major Components ............................ 23
Figure 3-2. LabVIEW Software Development for Experimental Control Rod ................ 24
Figure 3-3. LabVIEW User Interface for Experimental Control Rod Control ................. 25
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 ................................................................. 27
Figure 3-6. Experimental Control Rod Reactivity worth Curve Calculated by Digital
Reactivity Computer at the PSBR............................................................................. 29
Figure 3-7. ECR model design implementation into TRIGA Simulink Model ................ 30
Figure 4-1. Block diagram representation of state equation and output equation ............ 33
Figure 4-2. TRIGA Reactor State Space Model with State Feedback Controller ............ 36
ix
Figure 4-3. Initial State Feedback Observer Design for TRIGA Reactor model .............. 38
Figure 4-4. State Feedback Controller/Observer design implemented on the TRIGA
reactor simulation...................................................................................................... 40
Figure 4-5. State Feedback Controller/Observer Subsystem Including Integral Action .. 41
Figure 4-6. TRIGA Reactor Control using the PI controller ............................................ 42
Figure 4-7. Reactor Power Change from 1MW to 900 kW and 900 kW to 800 kW with
Different Controllers ................................................................................................. 43
Figure 4-8. Reactor Power Change from 800 kW to 850 kW and 850 kW to 950 kW with
Different Controllers ................................................................................................. 44
Figure 5-1. Self-powered neutron detector components ................................................... 46
Figure 5-2. Vanadium Decay Mechanism ........................................................................ 47
Figure 5-3. Vanadium Self-Power Neutron Detector Model in Simulink ........................ 49
Figure 5-4. Vanadium Self-Power Neutron Detector response at power set point change
from 1 MW to 900 kW.............................................................................................. 50
Figure 5-5. Inverse Detector model for Vanadium Self-Power Neutron Detector ........... 52
Figure 5-6. Rhodium Decay Mechanism .......................................................................... 53
Figure 5-7. Rhodium Self-Power Neutron Detector Model in Simulink .......................... 55
Figure 5-8. Rhodium Self-Power Neutron Detector response step power change from
1MW to 900 kW ....................................................................................................... 56
Figure 5-9. Finalized Model with normal and inverse detector models and the state
feedback controller/observer ..................................................................................... 60
Figure 5-10. Relative Power and Vanadium detector current with inverse and without
inverse detector model .............................................................................................. 61
x
Figure 5-11. Relative Power and Rhodium detector current with inverse and without an
inverse model ............................................................................................................ 62
xi
LIST OF TABLES
Table 2-1. Delayed Neutron Data for Thermal Fission in Uranium-235 .......................... 13
Table 5-1. Vanadium Detector Constants [7] ................................................................... 48
Table 5-2. Rhodium Detector Constants [10] ................................................................... 54
xii
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.
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.
2
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 time-
dependent 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
3
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
𝜐
𝑑𝜙
𝑑𝑡− ∇. D∇𝜙 + Σ𝑎𝜙(𝑟, 𝑡) = 𝜈Σ𝑓𝜙(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) = ∑𝐴𝑛exp (−𝜆𝑛𝑡)𝜓𝑛(𝑟)
𝑛
1-2
4
The spatial eigenfunctions can be determined for a specified geometry by:
∇2𝜓𝑛 + 𝐵𝑛2𝜓𝑛(𝑟) = 0 1-3
And the time eigenvalues of 𝜆𝑛:
𝜆𝑛 = 𝜐𝐷𝐵𝑛2 + 𝜐Σ𝑎 − 𝜐𝜈Σ𝑓 1-4
If the higher-order modes are assumed to die out rapidly, neutron flux can be written:
𝜙(𝑟, 𝑡) = 𝐴1𝑒𝑥𝑝 [(𝑘−1
𝑙) 𝑡] 𝜓1(𝑟)
where 𝑙 = [𝜐Σ𝑎(1 + 𝐿2𝐵𝑔2]
−1 mean lifetime of neutron
and 𝑘 =𝜐Σ𝑓/Σ𝑎
1+𝐿2𝐵𝑔2 =
𝑘∞
1+𝐿2𝐵𝑔2 Multiplication factor
1-5
The flux may be assumed to be separable in space and time
𝜙(𝑟, 𝑡) = 𝜐𝑛(𝑡)𝜓1(𝑟) 1-6
Finally, substituting equation 1-6 into one-speed neutron diffusion equation (1-1) will give:
𝑑𝑛
𝑑𝑡= (
𝑘 − 1
𝑙) 𝑛(𝑡)
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, 𝐶𝑖(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
𝜆𝑖𝐶𝑖(𝑟, 𝑡) 1-8
5
With the number of precursors produced being
𝛽𝑖𝜈Σ𝑓 𝜙(𝑟, 𝑡) 1-9
where 𝛽𝑖 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:
𝑑𝐶𝑖
𝑑𝑡= −𝜆𝑖𝐶𝑖(𝑟, 𝑡) + 𝛽𝑖𝜈Σ𝑓 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)𝜈Σ𝑓 𝜙(𝑟, 𝑡) + 𝜆𝑖𝐶𝑖(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
𝑙𝑛(𝑡) + ∑𝜆𝑖𝐶𝑖(𝑡)
6
𝑖=1
𝑑𝐶𝑖
𝑑𝑡= −𝜆𝑖𝐶𝑖(𝑡) + 𝛽𝑖
𝑘
𝑙𝑛(𝑡)
1-12
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
6
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
Λ𝑛(𝑡) + ∑𝜆𝑖𝐶𝑖(𝑡)
6
𝑖=1
𝑑𝐶𝑖
𝑑𝑡=
𝛽𝑖
Λ𝑛(𝑡) − 𝜆𝑖𝐶𝑖(𝑡) 𝑖 = 1,… 6.
1-15
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.
7
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
Λ𝑛(𝑡) + 𝜆𝑐(𝑡)
𝑑𝑐
𝑑𝑡= 𝜆𝑛(𝑡) − 𝜆𝑐(𝑡)
1-16
The small deviations about equilibrium for neutron population, delayed neutron precursors, and
reactivity are defined as:
𝑛(𝑡) = 𝑛0 + 𝛿𝑛(𝑡)
𝑐(𝑡) = 𝑐0 + 𝛿𝑐(𝑡)
𝜌(𝑡) = 𝜌0 + 𝛿𝜌(𝑡)
1-17
where 𝑛0 , 𝑐0 𝑎𝑛𝑑 𝜌0 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
𝑑𝑛0
𝑑𝑡+
𝛿𝑛(𝑡)
𝑑𝑡=
𝜌0 + 𝛿𝜌(𝑡) − 𝛽
Λ(1 + 𝛿𝑛(𝑡)) +
𝛽
Λ(1 + 𝛿𝑐(𝑡))
𝑑𝑐0
𝑑𝑡+
𝛿𝑐(𝑡)
𝑑𝑡= 𝜆(1 + 𝛿𝑛(𝑡)) − 𝜆(1 + 𝛿𝑐(𝑡) )
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−
𝛽
Λ𝛿𝑛(𝑡) +
𝛽
Λ𝛿𝑐(𝑡)
𝛿𝑐(𝑡)
𝑑𝑡= 𝜆(𝛿𝑛(𝑡) − 𝛿𝑐(𝑡))
1-19
8
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
𝑑𝑇𝑓
𝑑𝑡=
𝑃
𝑀𝑓𝐶𝑓𝑛(𝑡) −
𝑈𝑓𝐴𝑓
𝑀𝑓𝐶𝑓(𝑇𝑓 − 𝑇𝐶)
𝑑𝑇𝐶
𝑑𝑡=
𝑈𝑓𝐴𝑓
𝑀𝐶𝐶𝐶(𝑇𝑓 − 𝑇𝐶) − 2. �̇�𝐶𝐶(𝑇𝑐 − 𝑇0)
1-20
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:
𝑇𝑓(𝑡) = 𝑇𝑓0 + 𝛿 𝑇𝑓(𝑡)
𝑇𝑐(𝑡) = 𝑇𝑐0 + 𝛿 𝑇𝑐(𝑡)
1-21
Using equation 1-21 and equation 1-20 will give
𝑀𝑓𝐶𝑓 (
𝑑𝑇𝑓0
𝑑𝑡+
𝛿𝑇𝑓(𝑡)
𝑑𝑡) = 𝑃(𝑛0 + 𝛿𝑛(𝑡)) − 𝑈𝐴(𝑇𝑓0 + 𝛿 𝑇𝑓(𝑡) − 𝑇𝑐0 − 𝛿 𝑇𝑐(𝑡))
𝑀𝐶𝐶𝐶 (𝑑𝑇𝐶0
𝑑𝑡+
𝛿𝑇𝐶(𝑡)
𝑑𝑡) = 𝑈𝐴 (𝑇𝑓0 + 𝛿 𝑇𝑓(𝑡) − 𝑇𝑐0 − 𝛿 𝑇𝑐(𝑡)) − 2. �̇�𝐶𝐶(𝑇𝑐0 + 𝛿 𝑇𝑐(𝑡) − 𝑇0)
1-22
The linearized thermal-hydraulic equations (1-22), valid about the equilibrium condition
specified become
9
𝑀𝑓𝐶𝑓 (
𝛿𝑇𝑓(𝑡)
𝑑𝑡) = 𝑃( 𝛿𝑛(𝑡)) − 𝑈𝐴(𝛿 𝑇𝑓(𝑡) − 𝛿 𝑇𝑐(𝑡))
𝑀𝐶𝐶𝐶 (𝛿𝑇𝐶(𝑡)
𝑑𝑡) = 𝑈𝐴 (𝛿 𝑇𝑓(𝑡) − 𝛿 𝑇𝑐(𝑡)) − 2. �̇�𝐶𝐶(𝛿 𝑇𝑐(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, 1955. 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 1960. 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 ft-
deep 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 𝐷2𝑂 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]
10
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 1991. The control system manufactured by Atomic Energy of Canada Limited
(AECL). [2]
11
Figure 1-1. A picture of Penn State TRIGA Reactor Core
12
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.
13
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
(𝑙𝑖 𝑠𝑒𝑐−1)
Energy
(keV)
Neutrons per
Fission
Fraction
(𝛽𝑖)
1 55.72 0.0124 250 0.00052 0.000215
2 22.72 0.0305 560 0.00346 0.001424
3 6.22 0.111 405 0.00310 0.001274
4 2.30 0.301 450 0.00624 0.002568
5 0.610 1.14 - 0.00182 0.000748
6 0.230 3.01 - 0.00066 0.000273
Total Yield:0.0158
Total delayed fraction (𝛽):0.0065
14
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.
15
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(𝐵4𝐶) ,
silver, indium or cadmium. Typical TRIGA Reactors have four different control rods as
described in Section 1.3.
16
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
17
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 thermal-
hydraulic 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
18
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)
19
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.
20
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
21
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.
22
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
23
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/ on-
off) 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
24
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
25
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.
26
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 3-5.
27
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
28
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 near-
perfect 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 3-6.
29
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.
30
Figure 3-7. ECR model design implementation into TRIGA Simulink Model
31
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
32
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 ≥ 𝑡0 once the
state at 𝑡 = 𝑡0 is given and input u (t) for 𝑡 ≥ 𝑡0 is specified.
State space is the n-dimensional space whose coordinate axes consist of the states i.e., 𝑥1
axis, 𝑥2 axis . . . , 𝑥𝑛 axis. Any system state can be represented in the state space.
Consider the n-th order system:
𝑦𝑛 + 𝑎1𝑦𝑛−1+ . . . 𝑎𝑛−1�̇� + 𝑎𝑛𝑦 = 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
𝑥1̇ = 𝑥2
𝑥2̇ = 𝑥3
.
.
.
𝑥𝑛−1̇ = 𝑥𝑛
𝑥�̇� = −𝑎𝑛𝑥1− . . . 𝑎1𝑥𝑛 + 𝑢
4-2
Which can also be written in matrix form as �̇� = 𝐴𝑥 + 𝐵𝑢 where
33
x = [
𝑥1
𝑥2
⋮𝑥𝑛
] , 𝐴 =
[ 0 1 0 ⋯ 00 0 1 ⋯ 0
⋮⋮
0 0 0 ⋯ 1−𝑎𝑛 −𝑎𝑛−1 −𝑎𝑛−2 ⋯ −𝑎1]
, B =
[ 00⋮01]
,
4-3
with the output of the system being:
y = [1 0 ⋯ 0] * [
𝑥1
𝑥2
⋮𝑥𝑛
] , y = Cx where C = [1 0 ⋯ 0]
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.
34
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
Λ𝛿𝑛(𝑡) +
𝛽
Λ𝛿𝑐(𝑡)
𝛿𝑐(𝑡)
𝑑𝑡= 𝜆(𝛿𝑛(𝑡) − 𝛿𝑐(𝑡))
4-5
𝑀𝑓𝐶𝑓 (
𝛿𝑇𝑓(𝑡)
𝑑𝑡) = 𝑃( 𝛿𝑛(𝑡)) − 𝑈𝐴(𝛿 𝑇𝑓(𝑡) − 𝛿 𝑇𝑐(𝑡))
𝑀𝐶𝐶𝐶 (𝛿𝑇𝐶(𝑡)
𝑑𝑡) = 𝑈𝐴 (𝛿 𝑇𝑓(𝑡) − 𝛿 𝑇𝑐(𝑡)) − 2. �̇�𝐶𝐶(𝛿 𝑇𝑐(𝑡))
4-6
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.
�̇� = 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.
35
[ 𝛿�̇�𝛿𝑐1̇
𝛿𝑐2̇
𝛿𝑐3̇
𝛿𝑐4̇
𝛿𝑐5̇
𝛿𝑐6̇
𝛿𝑇�̇�
𝛿𝑇�̇�
𝛿�̇� ]
=
[
−𝛽
Λ
𝛽1
Λ
𝛽2
Λ
𝛽3
Λ
𝛽4
Λ
𝛽5
Λ
𝛽6
Λ
−∝𝑇
Λ0
1
Λ
𝜆1 −𝜆1 0 0 0 0 0 0 0 0𝜆2 0 −𝜆2 0 0 0 0 0 0 0𝜆3 0 0 −𝜆3 0 0 0 0 0 0𝜆4 0 0 0 −𝜆4 0 0 0 0 0𝜆5 0 0 0 0 −𝜆5 0 0 0 0𝜆6 0 0 0 0 0 −𝜆6 0 0 0𝑃
𝑀𝑓𝐶𝑓0 0 0 0 0 0 −
𝑈𝑓𝐴𝑓
𝑀𝑓𝐶𝑓
𝑈𝑓𝐴𝑓
𝑀𝑓𝐶𝑓0
0 0 0 0 0 0 0𝑈𝑓𝐴𝑓
𝑀𝑐𝐶𝑐−
𝑈𝑓𝐴𝑓
𝑀𝑓𝐶𝑓− 2�̇�𝐶𝐶 0
0 0 0 0 0 0 0 0 0 0]
.
[ 𝛿𝑛𝛿𝑐1
𝛿𝑐2
𝛿𝑐3
𝛿𝑐4
𝛿𝑐5
𝛿𝑐6
𝛿𝑇𝑓
𝛿𝑇𝑐
𝛿𝜌 ]
+
[ 000000000Δ𝜌
Δz]
. 𝜌
y=
[ 1000000000] 𝑇
.
[ 𝛿𝑛𝛿𝑐1
𝛿𝑐2
𝛿𝑐3
𝛿𝑐4
𝛿𝑐5
𝛿𝑐6
𝛿𝑇𝑓
𝛿𝑇𝑐
𝛿𝜌 ]
+0.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
36
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
37
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.
�̂̇� = 𝐴�̂� + 𝐵𝑢 + 𝑜𝑏𝑠(𝑦 − �̂�)
�̂� = 𝐶�̂�
Where obs is observer gain vector. The error of the observer
�̇� = �̇� − �̂̇� = (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
38
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
39
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
40
Figure 4-4. State Feedback Controller/Observer design implemented on the TRIGA reactor simulation
41
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.
42
Figure 4-6. TRIGA Reactor Control using the PI controller
43
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
44
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.
45
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
46
[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.
47
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
48
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]
𝑑𝑁51(𝑡)
𝑑𝑡= −𝜎51𝑁51(𝑡)𝜙(𝑡)
𝑑𝑁52(𝑡)
𝑑𝑡= 𝜎51𝑁51(𝑡)𝜙(𝑡) − 𝜆52𝑁52(𝑡)
𝑖(𝑡) = 𝑘𝑝𝑣𝜎51𝑁51(𝑡)𝜙(𝑡) + 𝑘𝑔𝑣𝜆52𝑁52(𝑡)
5-2
Where
𝑁51𝑎𝑛𝑑𝑁52 : Atomic densities of Vanadium-51 and Vanadium-52
𝜎51 : Microscopic neutron absorption cross-section of Vanadium-51
𝜆52 : decay constant of Vanadium-52
𝑖(𝑡): Current from SPND
𝑘𝑝𝑣 𝑎𝑛𝑑 𝑘𝑔𝑣: 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
𝑁51 6.86 × 1022 𝑐𝑚−3
𝜎51 4.90 × 10−24 𝑐𝑚2
𝜆52 0.0036 𝑠−1
𝑘𝑝𝑣 3.49 × 10−21 𝐴 𝑠
𝑘𝑔𝑣 3.85 × 10−20 𝐴 𝑠
49
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.
50
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)
𝑑𝑁51(𝑡)
𝑑𝑡= −𝜎51𝑁51(𝑡)𝜙(𝑡)
𝑑𝑁52(𝑡)
𝑑𝑡= 𝜎51𝑁51(𝑡)𝜙(𝑡) − 𝜆52𝑁52(𝑡)
𝑖(𝑡) = 𝑘𝑝𝑣𝜎51𝑁51(𝑡)𝜙(𝑡) + 𝑘𝑔𝑣𝜆52𝑁52(𝑡)
Use the third equation to solve for the neutron flux,
51
𝜙(𝑡) =
𝑖(𝑡)
𝑘𝑝𝑣𝜎51𝑁51(𝑡)−
𝑘𝑔𝑣𝜆52𝑁52(𝑡)
𝑘𝑝𝑣𝜎51𝑁51(𝑡)
5-3
Inserting Equation 5-3 into Equation 5-2 will gives
𝑑𝑁52(𝑡)
𝑑𝑡= 𝜎51𝑁51(𝑡) × [
𝑖(𝑡)
𝑘𝑝𝑣𝜎51𝑁51(𝑡)−
𝑘𝑔𝑣𝜆52𝑁52(𝑡)
𝑘𝑝𝑣𝜎51𝑁51(𝑡)] − 𝜆52𝑁52(𝑡)
5-4
Simplifying Equation 5-4 gives
𝑑𝑁52(𝑡)
𝑑𝑡= [
𝑖(𝑡)
𝑘𝑝𝑣−
𝜆52𝑁52(𝑡)(𝑘𝑔𝑣 + 𝑘𝑝𝑣)
𝑘𝑝𝑣]
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.
52
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
53
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.
𝑑𝑁104𝑚(𝑡)
𝑑𝑡= 𝜎104𝑚𝑁103𝜙(𝑡) − 𝜆104𝑚𝑁104𝑚(𝑡)
𝑑𝑁104(𝑡)
𝑑𝑡= 𝜎104𝑁103𝜙(𝑡) + 𝜆104𝑚𝑁104𝑚(𝑡) − 𝜆104𝑁104(𝑡)
𝑖(𝑡) = 𝑘𝑝𝑣(𝜎104+𝜎104𝑚)𝑁103𝜙(𝑡) + 𝑘𝑔𝑣𝜆104𝑁104(𝑡)
5-6
54
Where
𝑁103, 𝑁104𝑎𝑛𝑑𝑁104𝑚 : Atomic densities of Rhodium-103, Rhodium-104 and Rhodium-104m
𝜎104 𝑎𝑛𝑑 𝜎104𝑚 : Microscopic neutron absorption cross-section of Rhodium-104 and Rhodium-
104m
𝜆104 𝑎𝑛𝑑 𝜆104𝑚 : decay constant of Rhodium-104 and Rhodium-104m
𝑖(𝑡): Current from Self powered neutron detector
𝑘𝑝𝑣 𝑎𝑛𝑑 𝑘𝑔𝑣: 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 Values and Units
𝑁103 7.26 × 1022 𝑐𝑚−3
𝑁104𝑚 3.04 × 1015 𝑐𝑚−3
𝑁104 6.60 × 1015 𝑐𝑚−3
𝜎104 1.39 × 10−22 𝑐𝑚2
𝜎104 1.10 × 10−23 𝑐𝑚2
𝜆104 1.65 × 10−2 𝑠−1
𝜆104𝑚 2.63 × 10−3 𝑠−1
𝑘𝑝𝑣 2.46 × 10−12 𝐴 𝑠
𝑘𝑔𝑣 3.36 × 10−13 𝐴 𝑠
55
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.
56
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.
𝑑𝑁104𝑚(𝑡)
𝑑𝑡= 𝜎104𝑚𝑁103𝜙(𝑡) − 𝜆104𝑚𝑁104𝑚(𝑡)
𝑑𝑁104(𝑡)
𝑑𝑡= 𝜎104𝑁103𝜙(𝑡) + 𝜆104𝑚𝑁104𝑚(𝑡) − 𝜆104𝑁104(𝑡)
𝑖(𝑡) = 𝑘𝑝𝑣(𝜎104+𝜎104𝑚)𝑁103𝜙(𝑡) + 𝑘𝑔𝑣𝜆104𝑁104(𝑡)
5-7
Neutron flux can be derived from the current equation:
𝜙(𝑡) =
𝑖(𝑡)
𝑘𝑝𝑣(𝜎104+𝜎104𝑚)𝑁103(𝑡)−
𝑘𝑔𝑣𝜆104𝑁104(𝑡)
𝑘𝑝𝑣(𝜎104+𝜎104𝑚)𝑁103(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:
57
𝑑𝑁104𝑚(𝑡)
𝑑𝑡= 𝜎104𝑚𝑁103(
𝑖(𝑡)
𝑘𝑝𝑣(𝜎104+𝜎104𝑚)𝑁103(𝑡)−
𝑘𝑔𝑣𝜆104𝑁104(𝑡)
𝑘𝑝𝑣(𝜎104+𝜎104𝑚)𝑁103(𝑡))
− 𝜆104𝑚𝑁104𝑚(𝑡)
𝑑𝑁104(𝑡)
𝑑𝑡= 𝜎104𝑁103(
𝑖(𝑡)
𝑘𝑝𝑣(𝜎104+𝜎104𝑚)𝑁103(𝑡)−
𝑘𝑔𝑣𝜆104𝑁104(𝑡)
𝑘𝑝𝑣(𝜎104+𝜎104𝑚)𝑁103(𝑡))
+ 𝜆104𝑚𝑁104𝑚(𝑡) − 𝜆104𝑁104(𝑡)
5-9
Simplifying the Equation 5-9 will gives the rhodium detector inverse model rate equations:
𝑑𝑁104𝑚(𝑡)
𝑑𝑡= (
𝑖(𝑡)
𝑘𝑝𝑣(𝜎104)𝑁103(𝑡)−
𝑘𝑔𝑣𝜆104𝑁104(𝑡)
𝑘𝑝𝑣(𝜎104)𝑁103(𝑡)) − 𝜆104𝑚𝑁104𝑚(𝑡)
𝑑𝑁104(𝑡)
𝑑𝑡= (
𝑖(𝑡)
𝑘𝑝𝑣(𝜎104𝑚)𝑁103(𝑡)−
𝑘𝑔𝑣𝜆104𝑁104(𝑡)
𝑘𝑝𝑣(𝜎104𝑚)𝑁103(𝑡)) + 𝜆104𝑚𝑁104𝑚(𝑡)
− 𝜆104𝑁104(𝑡)
5-10
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
58
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
59
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.
60
Figure 5-9. Finalized Model with normal and inverse detector models and the state feedback controller/observer
61
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 5-10. Relative Power and Vanadium detector current with inverse and without inverse detector
model
62
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 5-11. Relative Power and Rhodium detector current with inverse and without an inverse model
63
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
64
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.
65
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.
66
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.
67
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, self-
powered 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.
68
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.
69
References
[1] J. A. Turso, "Penn State University TRIGA Reactor Digital Reactivity Computer:
Development and Testing," Annals of Nuclear Energy , vol. 114, pp. 561-568, 2017.
[2] Penn State University Radiation Science and Engineering Center, RSEC History, [Online].
Available: http://www.rsec.psu.edu/History.aspx. [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, 2017.
[4] J. Lamarsh and A. Baratta, Introduction to Nuclear Engineering, Upper Saddle River, New
Jersey 07458: Prentice Hall, Inc, 2001.
[5] E. D. Sontag, Mathematical Control Theory -Deterministic Finite Dimensional Systems,
NJ: Springer, 1998.
[6] K. Ogata, Modern Control Engineering, NJ: Prentice Hall, 1990.
[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, 2012.
[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. 644-659, 2018.
70
[9] S. W. H. Todt, Characteristics of Self-Powered Neutron Detectors used in Power Reactors,
Switzerland: European Nuclear Society, 1998.
[10] G.-S. Auh, "Digital Dynamic Compensation Methods of Rhodium Self-Powered Neutron
Detector," Journal of the Korean Nuclear Society, vol. 26, 1994.
[11] Nuclear Regulatory Commision, "Shutdown Margin," 2017. [Online]. Available:
https://www.nrc.gov/reading-rm/basic-ref/glossary/shutdown-margin.html.
71
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=0.00021;
Beta2=0.00141;
Beta3=0.00127;
Beta4=0.00255;
Beta5=0.00074;
Beta6=0.00027;
sumBeta=Beta1+Beta2+Beta3+Beta4+Beta5+Beta6;
P = 950/100;
MF = 16.44;
Cpf = 0.0028;
Cpc = 1;
m_dot_c = 1.055;
Mc = 17;
72
Tin = 75;
Tf0 = 964.58;
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 = [-0.0433];
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 = 1000000;
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;
73
%zz=mean(UA1(1))
zz=0.01075;
A2=[-Beta/LAMBDA Beta/LAMBDA 0 alpha_t 0;lambda -lambda 0 0 0;sigma51*N51*phi0 0 -lam52 0 0;P/(MF*Cpf) 0 0 -
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 = 1000000;
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 = [1 1 0 0 0];
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 -lambda1 0 0 0 0 0 0 0;lambda2 0 -lambda2 0 0 0 0 0 0;lambda3 0 0 -lambda3 0
0 0 0 0;lambda4 0 0 0 -lambda4 0 0 0 0;lambda5 0 0 0 0 -lambda5 0 0 0;lambda6 0 0 0 0 0 -lambda6 0 0;P/(MF*Cpf) 0 0 0 0 0 0
-zz/(MF*Cpf) zz/(MF*Cpf);0 0 0 0 0 0 0 zz/(Mc*Cpc) -zz/(Mc*Cpc)-2*m_dot_c/Mc]
B3=[1/LAMBDA;0;0;0;0;0;0;0;0]
C3=[1 0 0 0 0 0 0 0 0]
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=[1 0 0 0]
% D3=[0]
74
Appendix B: Penn State Breazeale Reactor Standard Operating Procedure – Experiment Evaluation and Authorization 1. Valid Period ______2/15/2018-5/30/2018__________________________________________
2. Supervisor ____J. Geuther______________________________ Phone 814-863-2745
Academic Rank (if applicable) _Assoc. Dir for Operations_ Department (Company)_RSEC_
Address __101 Breazeale Nuclear Reactor_________________________________________
3. Experimenter (s) ___J. Turso/G. Corak_____________________ Phone _____863-2820____
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.
75
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___2029____
76
12. Material Data (table below or attachment) summary by ___J. Turso________ Date __2/8/18___ Calc/Est
SEE ATTACHED DOSE RATE CALCULATION RESULTS
A
Material &
Identification
B
Weight or
Volume
C
Isotopes
Produced
D*
Expected
Activity
E
Gamma
Factor
F**
Gamma Exposure Rate
At a Distance
G
Activity
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
Review
(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
intended)
N/A
Ar-41 production meas/cal if estimated dose > 0.1 mrem/24 hrs/exclusion boundary (no release
intended)
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
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 131-135 inventory (< 1.5 Ci) N/A
Results of review if > 5mCi of Iodine 131 - 135 N/A
Safety System Review X Yes
(
s
e
e
1
No
14. 50.59 Approval Date 1/18/2000 or,
N
o
t
R
15. PSRSC Review Date or,
N
o
t
X
77
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
Auth (Yes/No)?
No
17. Approval Conditions or Restrictions: Operations with ECR2 restricted to R1. Excess reactivity determined after step 45, App B.
to R1. Provide excess
18. Approved by: Date:
ECRD 2 Activity 1 Hour Post-Irradiation
ECRD 2 Activity 24 Hours Post-Irradiation
ECRD 2 Activity 30-Days Post-Irradiation
Target
Nuclide Sample(g)
Product
Nuclide/Isotope
Initial_Activity
(uCi)
Delayed_Activity
(uCi)
Gamma
(mR/hr)
Beta
(mR/hr)
Al 264.00 Al-28 12316857.59 6.59 0.06 4.60
Al 264.00 Mg-27 101885.85 1645.39 9.32 817.89
Al 264.00 Na-24 3307.15 3157.33 64.55 1329.57
Cd 47.50 Cd-115m 264.40 264.23 0.00 106.76
Cd 47.50 Cd-115g 9721.04 9595.84 2.35 3877.18
Cd 47.50 Cd-117m 3204.76 2613.71 0.17 1035.25
Cd 47.50 Cd-117g 5580.32 4224.38 0.70 1673.21
Cd 47.50 Cd-111m 31408.80 13324.26 19.10 0.00
96.26 8844.45
Total 8940.71 mR/hr
Target
Nuclide Sample(g)
Product
Nuclide/Isotope
Initial_Activity
(uCi)
Delayed_Activity
(uCi)
Gamma
(mR/hr)
Beta
(mR/hr)
Al 264.00 Al-28 12316857.59 0.00 0.00 0.00
Al 264.00 Mg-27 101885.85 0.00 0.00 0.00
Al 264.00 Na-24 3307.15 1087.07 22.22 457.77
Cd 47.50 Cd-115m 264.40 260.32 0.00 105.18
Cd 47.50 Cd-115g 9721.04 7121.98 1.74 2877.62
Cd 47.50 Cd-117m 3204.76 24.04 0.00 9.52
Cd 47.50 Cd-117g 5580.32 7.00 0.00 2.77
Cd 47.50 Cd-111m 31408.80 0.00 0.00 0.00
23.97 3452.87
Total 3476.84 mR/hr
Target
Nuclide Sample(g)
Product
Nuclide/Isotope
Initial_Activity
(uCi)
Delayed_Activity
(uCi)
Gamma
(mR/hr)
Beta
(mR/hr)
Al 264.00 Al-28 12316857.59 0.00 0.00 0.00
Al 264.00 Mg-27 101885.85 0.00 0.00 0.00
Al 264.00 Na-24 3307.15 0.00 0.00 0.00
Cd 47.50 Cd-115m 264.40 165.87 0.00 67.02
Cd 47.50 Cd-115g 9721.04 0.86 0.00 0.35
Cd 47.50 Cd-117m 3204.76 0.00 0.00 0.00
Cd 47.50 Cd-117g 5580.32 0.00 0.00 0.00
Cd 47.50 Cd-111m 31408.80 0.00 0.00 0.00
0.00 67.37
Total 67.37 mR/hr
78
Procedure for Installation or Removal of an ECRD
I. Purpose:
To define the steps required for installation and removal of the Experimental
Changeable Reactivity Device 2, ECRD 2.
II. 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.
III. References:
A. SOP-1, Reactor Operating Procedure
B. Technical Specifications 3.7
C. Figures A1 and A2, ECRD attachment to Drive Motor, (attached)
IV. Special Equipment:
A. ECRD 2
B. ECRD restraint padlock
C. AP – 10, Equipment Tags
V. Procedure:
79
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.
80
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 19.
9. 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.
81
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.
82
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. DO NOT COMPLETELY REMOVE FROM CT UNTIL AT
LEAST 7 DAYS POST IRRADIATION.
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.
83
Figure A1: ECRD Drive and major components
84
Figure A2A: ECRD coupling hardware
Figure A2B: ECRD coupling and connecting rod mounted on drive
Appendix B- ECRD 2 Rod Worth Measurement
CAUTION:
85
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:
86
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.1.
25. 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.
87
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.
88
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 #2 @ 1.5 rps = _____________cents/second
B. ECRD #2 @ 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 = $____________
89
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:
Source: IN OUT
Difference of Excess Reactivities
Regulating Rod Rod Calibration Curve Date:
Difference of CRPs Experimental Reactivity: $1.00
Difference of CRPs Method Worksheet
Initial Control Rod Worths @ 100W CRP Final Control Rod Worths @ 100W CRP
Date: Worth Worth
90
Transient Rod Transient Rod
Safety Rod Safety Rod
Shim Rod Shim Rod
Regulating Rod Regulating Rod
Total Worth Total Worth
Experimental Reactivity:
91
92
93
94
95
