1. Modeling of Core Protection Calculator System Software February 28, 2005 Kim, Sung Ho Kim, Sung Ho

2. 1 TABLE OF CONTENTS q Introduction q Classification Modeling q Dependency Modeling q Application of Modeling to CPCS q Further Works q References Kim, Sung Ho

3. 2 Introduction q Current CPCS Software  performs the calculation of DNBR and LPD for reactor trip generation  uses process variables and operator-addressable constants for calculation  does not have any formal method to express system functions and software quality  requires some formal expressions for the system functions and quantitative software reliability which can be used to improve the software quality Core Protection Calculator System Software CEA position Excore neutron flux signal Hot leg temperature Cold leg temperature RCP shaft speed Pressurizer Pressure DBNT Trip LPD Trip Coefficients of Equations CWP

4. 3 Kim, Sung Ho Introduction q Modeling of the CPCS Algorithm  is recommended for formal expression of the system functions  can be used for the evaluation of relationships between variables  can be used for the software test coverage to suggest the software reliability  Bayesian Way of System Modeling  Formalization of the software The relationship between process input variables and output values of the CPCS can be modeled using Bayesian concept of conditional probability  Test case confirmation The variances of input values and corresponding variances of output values can be used to check the coverage of test cases for the system software  Some modeling methods for trial classification modeling dependency modeling etc.

5. 4 Kim, Sung Ho q What is Classification Modeling class  To divide the variables of a system into a class variable and the rest as predictor predictor variables  To find a model such that if the values of predictor variables are given, then the values of the class variable are inferred  To use as a prediction model for the values of class variable using the values of predictor variables  Bayesian theory can give us tools to merge many quantitative models to build a classification model by adopting the concept of probabilities of the parameters  search for certain kind of Bayesian network structures  Naïve Bayes’ model is considered to be optimal for classification modeling due to its special form that connects all predictors to the class variable, can be said that all the predictors are dependent on the class variable, and can be said that all the predictors are independent of each other once we know the value of the class variable Classification Modeling

6. 5 Kim, Sung Ho q Example of Classification Modeling: Detecting Heart Disease  Class variable and predictor variables Variable NameData Type AgeNumerical SexMale/Female Chest pain type1, 2, 3, 4 Resting blood pressureNumerical Serum cholestoralNumerical Fasting blood sugarYes/No Resting electrocardiographic results0, 1, 2, 3 Max. heart rate achieveNumerical Exercise induced anginaYes/No ST depression induced by exercise relative to restNumerical Slope of the peak exercise ST segmentNumerical No. of major vessels colored by fluoroscopy0, 1, 2, 3 ThalNormal/Fixed defect/Reversible defect Heart diseaseAbsence/Presence Class variable Predictor variables Classification Modeling

7. 6 Kim, Sung Ho q Best Classification Model  The classification model showing the best possible predictive accuracy a model that most accurately classifies unclassified data vectors  Training classifiers is building the best classifiers  Searching procedure for the best classification model pick one model at random and declare it the best model so far pick another model at random and compare the predictive accuracy of this model to that of the best model so far select more accurate model as the best model and continue  Estimation of predictive accuracy of the classifier (Leave-one-out cross validation)  Remove one data vector from N data matrix and train the classifier with (N-1) data vectors  Insert the removed data vector for value of class  If the classifier pops the correct class, the classifier gets one point  Return the removed data vector to the data matrix and repeat this procedure by removing some other vector for each data vector in the matrix  Sum up the points and divide the points by the number of data vectors to get the average predictive accuracy (in %) Classification Modeling

8. 7 Kim, Sung Ho Classification Modeling q Result of Classification Modeling(1)  Evaluated 37293 models for the sample case  Last 36400 models have not resulted in finding better models  The classification accuracy is 86.67 %  The variables Age, Sex, Cholestoral, Sugar, Max, Angina have been left out from the model Arcs (from Class) Reduction of Accuracy #Vessels7.41 % Thal6.67 % Old Peak6.30 % Pain5.56 % Pressure1.85 % Rest1.48 % Slope Peak1.11 % Importance of the variables

9. 8 Kim, Sung Ho Classification Modeling q Result of Classification Modeling(2) – probabilistic classifier

10. 9 Kim, Sung Ho Dependency Modeling q What is Dependency Modeling  Finding the model of the probabilistic dependencies of the variables  Dependencies can be used to speculate about causalities that might cause them  A good dependency model is one with high probability  Each dependency model is a set of statements about the dependencies between sets of variables A and B are dependent on each other if something about C or D are known A and C are dependent on each other on matter what about B or C are known …

11. 10 Kim, Sung Ho q Dependency Modeling using Bayesian Belief Network  Expressing the causalities between variables  Setting up the decision making rules based on formal method  Updating the probability based on the new information  Intelligent components to deduce the non-deterministic conditions in complex system  A dependency model can be represented in simple graphical form using Bayesian Network having the only one Bayesian network representation, or many slightly different Bayesian network representations  But in some case, there does not exist a Bayesian network representation A B DC Dependency Modeling

12. 11 Kim, Sung Ho q Predicting with Dependency Models(1)  A dependency model can be used to calculate conditional probabilities of unknown future observations Dependency Modeling

13. 12 Kim, Sung Ho q Predicting with Dependency Models(2) Dependency Modeling

14. 13 Kim, Sung Ho q Classification Modeling of the CPCS Algorithm  To divide the input variables into DNBR/LPD/CWP classes q Dependency Modeling of the CPCS Algorithm  To know whether or how the variables in CPCS algorithm depend each other  To know the conditions for the dependencies between variables (e.g., some knowledge on other variables)  To assess the software designer’s certainty about the true existence and nature of dependencies between variables using probability theory  CPCS Model to be used for the software test  Model the system using input/output variables and internal constants, if needed to find out the dependencies between input variables, output variables, and constants  Select one input variable  Change the values of the selected variable, and monitor the change of values of constants and outputs  Confirm if we can predict the output values when we know the input values  Confirm if we can generate all the possible range of input process values and constants to cover all the possible conditions of the software Application of Modeling to CPCS

15. System Outputs Process Input Signals Addressable constants by the operators 14 Kim, Sung Ho q Core Protection Calculator System Algorithm Core Protection Calculator System Software CEA position Excore neutron flux signal Hot leg temperature Cold leg temperature RCP shaft speed Pressurizer Pressure DBNT Trip LPD Trip Coefficients of Equations Application of Modeling to CPCS CWP

16. 15 Kim, Sung Ho q Modeling of CPCS Algorithm  Considered to have combined connections of converging and diverging which need: partition of variables for each class variable after classification modeling manipulation of common variables for each class variable Application of Modeling to CPCS CEA position T Cold P PZR T Hot DNBR Trip LPD Trip CWP

17. 16 Kim, Sung Ho q Modeling of CPCS Algorithm  Considered to be modeled using divorcing to reduce the number of cases for state distributions Application of Modeling to CPCS CEA position T Cold P PZR T Hot DNBR Trip LPD Trip CWP M1M1

18. 17 Kim, Sung Ho q Data Vector for CPCS Modeling Application of Modeling to CPCS

19. 18 Kim, Sung Ho Further Works q Data Vectors for the CPCS variables should be generated  Test cases should be generated  Test results should be generated using the test cases q Dependency Modeling of the CPCS using data vectors should be performed  Dependencies of input variables can be shown for the output values of DNBR, LPD, and CWP  Formal expressions for the CPCS algorithm and software quality should be generated q Test coverage should be checked using the model

20. 19 Kim, Sung Ho References q Finn V. Jensen, “An Introduction to Bayesian Networks”, 1996 q Unit Test Cases for CPCS for SKN 1&2 q http://b-course.cs.helsinki.fihttp://b-course.cs.helsinki.fi

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