Optimal Control of Distributed Nuclear ReactorsThis book is devoted to the mathematical optimization theory and modeling techniques that recently have been applied to the problem of controlling the shape and intensity of the power density distribution in the core of large nuclear reactors. The book has been prepared with the following purposes in mind: 1. To provide, in a condensed manner, the background preparation on reactor kinetics required for a comprehensive description of the main problems encountered in designing spatial control systems for nuclear reactor cores. 2. To present the work that has already been done on this subject and provide the basic mathematical tools required for a full understand ing of the different methods proposed in the literature. 3. To stimulate further work in this challenging area by weighting the advantages and disadvantages of the existing techniques and evaluating their effectiveness and applicability. In addition to coverage of the standard topics on the subject of optimal control for distributed parametersystems, the book includes, at amathemati cal level suitable for graduate students in engineering, discussions of con ceptsoffunctional analysis, the representation theory ofgroups, and integral equations. Although these topics constitute a requisite for a full understanding of the new developments in the area of reactor modeling and control, they are seidom treated together in a single book and, when they are, their presenta tion isoften directed to the mathematician.They are thus relatively unknown to the engineering community. |
Contents
Introduction | 11 |
Some Mathematical Optimization Techniques | 11 |
222 Matrices | 11 |
23 Calculus of Variations Ref 26 | 11 |
231 The Isoperimetric Problem Ref 25 | 12 |
24 Dynamic Programming and the Principle of Optimally Ref 21 | 13 |
241 Dynamic Programming for Continuous Processes Ref 21 | 15 |
25 Pontryagins Maximum Principle Refs 22 26 and 29 | 17 |
495 Construction of Asymptotic Expansions | 107 |
496 Practical Numerical Example | 112 |
497 Concluding Remarks | 114 |
410 System Tau Method Ref 415 | 116 |
4101 Calculational Method | 117 |
4102 Comparison of Various Computational Techniques | 124 |
4103 Practical Application of STM for Optimal Control of Nuclear Reactors | 125 |
4104 Concluding Remarks | 130 |
26 Minimum Norm Problems of Functional Analysis Refs 25 27 and 28 | 18 |
262 Minimum Norm Problems | 20 |
References | 24 |
Distributed Reactor Modeling | 25 |
32 The Multigroup Diffusion Equations | 27 |
321 Solution of the Diffusion Equation | 29 |
322 The Method of Degenerate Kernels | 30 |
323 Practical Example | 33 |
33 The Model Expansion | 37 |
331 The Functional Relation | 42 |
332 Practical Example | 43 |
34 Model Decomposition Techniques | 47 |
341 Symmetry Reduction | 48 |
Decoupling of the Optimality Conditions Refs 325328 | 60 |
Reduction of a Cylindrical Reactor Model Ref 329 | 63 |
37 TimeScale Separation Ref 329 | 67 |
References | 68 |
Optimal Control of Distributed Nuclear Reactors | 71 |
42 The Reactor Core Model | 74 |
43 The Optimal Control Problem | 76 |
A Fredholm Integral Equation | 78 |
OneNeutronGroup Diffusion Equation | 79 |
46 Discussion | 83 |
47 Method for Computing the Optimal Control | 85 |
48 Maximum Principle Approach Ref 41 | 88 |
481 Problem Formulation | 90 |
482 Regulator Problem | 91 |
483 Servomechanism Problem | 92 |
484 Spatial Discretization Scheme | 93 |
485 SpaceTime Discretization Scheme | 94 |
486 Some Examples for Optimal Control Computations | 95 |
487 Concluding Remarks | 99 |
49 Singular Perturbation Theory Ref 413 | 100 |
491 State Equation | 101 |
492 Problem Formulation | 102 |
493 Modal Expansion | 103 |
494 Criticality Conditions and Applicability of Singular Perturbation Theory | 106 |
References | 132 |
Control of Distributed Reactors in LoadFollowing | 135 |
52 Multistage Mathematical Programming Ref 52 | 137 |
522 Objective Function | 138 |
523 Reactor Core Model | 139 |
524 Numerical Solution | 147 |
525 Operational Use | 150 |
526 Results | 153 |
527 Concluding Remarks | 159 |
531 The Reactor Feedback Model | 160 |
532 The SteadyState Solution | 164 |
534 Performance Index and the Optimal Control Problem Refs 522525 | 166 |
535 Calculation Scheme and Results | 170 |
536 Concluding Remarks | 175 |
54 Multilevel Methods Ref 54 | 176 |
542 Problem Formulation | 184 |
543 Solution Algorithm | 188 |
544 Results | 190 |
545 Concluding Remarks | 193 |
Appendix A | 194 |
Appendix B | 199 |
References | 201 |
Application of the Minimum Norm Formulation to Problems in Control of Distributed Reactors | 203 |
62 The Nuclear Reactor Model | 204 |
63 Optimal Control of the State Distribution with PowerLevel Adjustment | 205 |
631 Necessary and Sufficient Conditions of Optimality | 206 |
64 Optimal Control of the State Distribution During LoadFollowing | 213 |
65 Optimal Control of the State Distribution with Fixed End State | 215 |
651 Suboptimal Control | 217 |
652 The Minimum Norm Formulation | 219 |
653 Practical Example | 221 |
228 | |
Conclusions | 231 |
72 Future Work | 232 |
233 | |
Other editions - View all
Optimal Control of Distributed Nuclear Reactors G.S. Christensen,S.A. Soliman,R. Nieva Limited preview - 2013 |
Optimal Control of Distributed Nuclear Reactors G.S. Christensen,S.A. Soliman,R. Nieva No preview available - 2013 |
Optimal Control of Distributed Nuclear Reactors G. S. Christensen,S. A. Soliman,R. Nieva No preview available - 2014 |
Common terms and phrases
A₁ A₂ adjoint algorithm applied approximation axial boundary conditions calculated canonical polynomials coefficients computational control function control model control rod control variable control vector coolant defined denotes diffusion equation distributed nuclear reactor distributed parameter eigenvalues elements feedback Figure flux deviations follows functional analysis given H₁ H₂ Hilbert space initial condition inner product integral equation iteration L₂[V linear load-following matrix maximum principle method minimize minimum norm problem modal expansion mode neutron flux nodal nodes nonlinear Nucl nuclear reactor core objective function obtained operator optimal control problem ordinary differential equations performance index perturbation theory Pontryagin's maximum principle positive-definite matrix power density power distribution problem of controlling pseudoinverse reactor core reactor model representation satisfies simulator singular perturbation singular perturbation theory solution solved subsystem t₁ total power trajectory transformation value problem xenon and iodine Zo(r Σ Σ