1. Axially Variable Strength Control Rods for The Power Maneuvering of PWRs 2002. 3. 25. KIM, UNG-SOO Dept. of Nuclear and Quantum Engineering

2. 2 Table of Contents  Introduction  Simulation Optimization  Optimization of AVSCRs  Further Study

3. 3 Introduction  Optimization for the worth shape of the AVSCRs  Minimizing AO variation and power deviation during power maneuvering  Simulation optimization  Using response surface methodology  Objective function for optimization of the AVSCRs  Relationship between AO variation(or power deviation) and worth shape  Analytic objective function does not exist.  Response for input can be only be evaluated by computer simulation.

4. 4 Simulation Optimization  What is simulation optimization?  The process of finding the best input variable value from among all possibilities without explicitly evaluating each possibility  Optimization problem where the objective function and some constraints are responses that can only be evaluated by computer simulation  An analytic expression of the objective function or the constraints do not exist.  This eliminates the possibility of differentiation or exact calculation of local gradients.  Precise evaluation of the objective function is computationally very costly. This makes the efficiency of the optimization algorithms more crucial.

5. 5 Simulation Optimization  Advantages in using simulation in optimization  Complexity of the system being modeled does not significantly affect the performance of the optimization process.  For stochastic systems, the variance of the response is controllable by various output analysis technique.  A general simulation model  n-input variables and m-output variables  Simulation optimization entails finding optimal settings of the input variables which optimize the output variables

6. 6 Simulation Optimization  A simulation optimization model  The output of a simulation model is used by an optimization strategy to provide feedback on progress of the search for the optimal solution.

7. 7 Simulation Optimization  Simulation optimization method  The six major categories  Gradient based search method  Stochastic optimization  Response surface methodology  Heuristic methods  A-teams  Statistical methods

8. 8 Simulation Optimization  Gradient based search methods  Estimate the response function gradient to assess the shape of the objective function  Finite differences  To estimate the gradient at a specific value of x, at least n+1 configuration of the simulation model must be run.  The crudest method of estimating gradient  Likelihood ratios  Estimate the gradient of the expected value of an output variable with respect to an input variable.  Suitable for transient and a regenerative simulation optimization problems

9. 9 Simulation Optimization  Perturbation analysis  If an input variable is perturbed by an infinitesimal amount, the sensitivity of the output variable to the parameter can be estimated by tracing its pattern of propagation.  All partial gradients of an objective function are estimated from a single simulation run.  Frequency domain method  Selected input parameters are oscillated sinusoidally at different frequencies during one long simulation run.  If the output variable is sensitive to an input parameter, the sinusoidal oscillation of that parameter should induce corresponding oscillations in the response.

10. 10 Simulation Optimization  Stochastic optimization  The problem of finding a local optimum for an objective function whose values are not known analytically but can be estimated or measured  Mimics the gradient search method in a rigorous statistical manner that takes into account the stochastic nature of the system model  Response surface methodology  A procedure for fitting a series of regression models to the output variable of a simulation model and optimizing the resulting regression function  Starts with a first order regression function and the steepest ascent/descent method.  Higher degree regression functions are employed after reaching the vicinity.

11. 11 Simulation Optimization  Heuristic methods  Genetic algorithms  Search strategy that employs random choice to guide a highly exploitative, striking a balance between exploration of the feasible domain and exploitation of “good” solutions  Selection, reproduction, crossover, and mutation  Evolutionary strategies  Algorithms that imitate the principles of natural evolution  Two membered ES   Multi-membered ES   Extended multi-membered ES 

12. 12 Simulation Optimization  Simulated annealing  A stochastic search method analogous to the physical annealing process where an alloy is cooled gradually so that a minimal energy state is achieved.  Tabu search  Optimizing an objective function with a special feature designed to avoid being trapped in local minima  A fixed-length of the Tabu moves that are not allowed at the present iteration is maintained.  Simplex search  Starts with points in a simplex consisting of p+1 vertices in the feasible region.  Proceeds by continuously dropping the worst point in the simplex and adding a new point determined through the centroid of the remaining vertices.

13. 13 Simulation Optimization  A-teams  A process that involves combining various problem solving strategies so that they can interact synergistically  Inherently suitable for multi-criteria simulation optimization problems  Statistical methods  Important sampling method  To achieve significant speed ups in simulations involving rare event  Simulating system under a different probability measure so as to increase the probability of typical sample paths involving the rare event of interest  Ranking and selection  Have the ability to treat the optimization problem as a multi-criteria decision problem.  Multiple comparisons with the best  The problem to select the best of a finite number of system designs.

14. 14 Optimization of AVSCRs  Optimization of AVSCRs through simulation optimization  RSM as an optimization strategy  Automation of the sequential process of RSM  Response surface methodology  A procedure for fitting a series of regression models to the output variable of a simulation model and optimizing the resulting regression function  Sequential nature of RSM  Starts with a first order regression function and the steepest ascent/descent method.  Higher degree regression functions are employed after reaching the vicinity.

15. 15 Optimization of AVSCRs  First stage

16. 16 Optimization of AVSCRs  Approximate the response surface function locally by first-order model  Usually a fractional two-level factorial design of resolution-III is used.  The number of design points is small compared to other types of two-level factorial design

17. 17 Optimization of AVSCRs  Test the first-order model for adequacy  Lack of fit test with the analysis of variance(ANOVA) table  Perform a line search in the steepest descent direction  A line search is performed from a center point of current region of interest to find a point of improved response.  The steepest descent direction is given by  To end this type of search, a stopping rule is applied.  The most straightforward rule ends the line search when an observed value of the simulation response function is higher than the preceding observation.

18. 18 Optimization of AVSCRs  Solve the inadequacy of the first-order model  Approximating the simulation response function in the region of interest by a second-order polynomial (usually)  To reduce the size of the region of interest by decreasing the step size (alternative)  To increase the simulation size used to evaluate a design point or to increase the number of replicated observation done in the design points (alternative)

19. 19 Optimization of AVSCRs  Second stage

20. 20 Optimization of AVSCRs  Approximate the objective function in the current region of interest by a second-order model  The central composite design(CCD) is most popular.  It can be transformed orthogonal by choosing a specific number of replicated observation in the center point.  Solve the inadequacy of the second-order model  To reduce the size of the region of interest  To increase the simulation size used in evaluating a design point  In RSM it is not customary to fit a higher than second-order polynomial.

21. 21 Optimization of AVSCRs  Perform canonical analysis  To determine the location and the nature of the stationary point of the second-order model  If the stationary point is a minimum, the stationary point is accepted as the center of the next region of interest.  Otherwise the stationary point is rejected.  Perform ridge analysis  A search for a new stationary point on a given radius R such that the second-order model has a minimum at this stationary point  Accept the stationary point  Dependent on the result of the canonical analysis  A minimum is found : a new second-order approximation  A maximum or saddle point : a new first-order approximation

22. 22 Optimization of AVSCRs  Results of first stage  Size of interest region=1.0,

23. 23 Further Study  Obtain optimum worth shape of axially variable strength control rods  Enhancement of the operation strategy for the AVSCRs  to extract optimal performance of the AVSCRs to be developed  applying T_avg signal