model parameters estimation and sensitivity by genetic algorithms

Model parameters estimation and sensitivityby genetic algorithms

Marzio Marseguerra*, Enrico Zio, Luca Podofillini

Department of Nuclear Engineering, Polytechnic of Milan, Via Ponzio 34/3, 20133 Milan, Italy

Received 13 February 2003; accepted 14 March 2003

Abstract

In this paper we illustrate the possibility of extracting qualitative information on theimportance of the parameters of a model in the course of a Genetic Algorithms (GAs) opti-mization procedure for the estimation of such parameters. The Genetic Algorithms’ search ofthe optimal solution is performed according to procedures that resemble those of natural

selection and genetics: an initial population of alternative solutions evolves within the searchspace through the four fundamental operations of parent selection, crossover, replacement,and mutation.

During the search, the algorithm examines a large amount of solution points which possiblycarries relevant information on the underlying model characteristics. A possible utilization of thisinformation amounts to create and update an archive with the set of best solutions found at each

generation and then to analyze the evolution of the statistics of the archive along the successivegenerations. From this analysis one can retrieve information regarding the speed of convergenceand stabilization of the different control (decision) variables of the optimization problem.In this work we analyze the evolution strategy followed by a GA in its search for the opti-

mal solution with the aim of extracting information on the importance of the control (deci-sion) variables of the optimization with respect to the sensitivity of the objective function. Thestudy refers to a GA search for optimal estimates of the effective parameters in a lumped

nuclear reactor model of literature. The supporting observation is that, as most optimizatonprocedures do, the GA search evolves towards convergence in such a way to stabilize first themost important parameters of the model and later those which influence little the model out-

puts. In this sense, besides estimating efficiently the parameters values, the optimizationapproach also allows us to provide a qualitative ranking of their importance in contributingto the model output. The results obtained are in good agreement with those derived from a

variance decomposition-based sensitivity analysis.# 2003 Elsevier Science Ltd. All rights reserved.

Annals of Nuclear Energy 30 (2003) 1437–1456

www.elsevier.com/locate/anucene

0306-4549/03/$ - see front matter # 2003 Elsevier Science Ltd. All rights reserved.

doi:10.1016/S0306-4549(03)00083-5

* Corresponding author. Fax: +39-02-2399-6309.

E-mail address: [email protected] (M. Marseguerra).

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1. Introduction

Genetic algorithms are numerical search tools used for the optimization of amultivariate function (called fitness or objective function) (Holland, 1975; Goldberg,1989; Chambers, 1995; Uhring and Tsoulakis, 1999). The search of the optimal solu-tion is basically performed proceeding from one group (population) of possible pointsin the search space to another, according to procedures that resemble those of naturalselection and genetics and designed such as to steer the search towards better solutions.While evolving through its generational steps, the algorithm examines and evaluatesseveral solution points in the search space, before converging to the best solution. Thesignificant amount of data thereby handled contains relevant information on the searchspace and on the model characteristics so that it seems worthwhile to make an effort toappropriately process these data so as to extract additional, spin-off results.This work is motivated in this sense and aims at analysing the evolutionary pro-

cess of the genetic algorithms in order to infer qualitative information on theimportance of the control variables with respect to the objective function. Moreprecisely, we start from the observation that in the process of convergence somevariables tend to set earlier than others. Of course, it is reasonable to expect that anefficient optimization strategy proceeds by adjusting first the most important vari-ables, i.e. those which mostly influence the objective function, and worries only at alater stage about tuning the other less important variables which determine onlyminor variations to the fitness function. Based on this consideration, we can expectto be able to extract information on the importance of the control variables from theanalysis of the evolution process towards convergence.In our work, the convergence analysis is performed on a suitably devised archive

containing a given number of different best solutions, each solution being a vector ofvalues of the control variables, and consists in investigating, for each control vari-able, how the first and second order statistics of the archived best solutions setbehave through the successive generations.Furthermore, in case of a large number of generations, when the genetic algorithm

search satisfies some convergence criterion, the archive contains the overall best indivi-duals encountered throughout the whole evolutionary process. The corresponding valuesof the control variables identify points in an hypervolume around the best among allindividuals: these data can be analysed to infer further information on local sensitivityand on dependencies between variables possibly existing in the underlying model.In this paper the above analyses are performed with reference to a genetic algo-

rithm search of the optimal values of five effective parameters of a nuclear reactormodel of literature (Chernick, 1960; Chernick et al., 1961). The search procedurewhile being effective in identifying a near optimal solution also allows us to rankvariables in importance and to find some correlations between them. The resultsthereby obtained are compared to those of a standard variance decomposition sen-sitivity analysis (McKay, 1995).In the next section, we introduce a few basic concepts on genetic algorithms’ com-

puting. In Section 3 we describe how information on sensitivity and correlationsbetween variables can be inferred from an analysis of the genetic algorithm’s evolution

1438 M. Marseguerra et al. / Annals of Nuclear Energy 30 (2003) 1437–1456

process and of the best solutions archive at convergence. The approach is firstlyvalidated on two simple analytic cases for which the sensitivity measures anddependencies are known a priori. In Section 4 the nuclear reactor model applicationis illustrated. A short discussion and suggestions for future work end the paper.

2. Genetic algorithms: some basic principles

The genetic algorithms (Holland, 1975; Goldberg, 1989; Chambers, 1995; Uhringand Tsoulakis, 1999) are a modern soft computing technique which owe the nameto their operational similarities with the biological and behavioral phenomena ofliving beings. The primary target of a genetic algorithm search procedure is theoptimization of an assigned objective function (called fitness) depending on severalparameters, typically non linear and subject to constraints. The practical details ofthe procedural steps governing the optimization search procedure as well as thetheoretical foundations can be found in the flourishing literature on the subject(Holland, 1975; Goldberg, 1989; Chambers, 1995).Recall that the target of the optimization problem is that of finding a point in the

control variables space which gives rise to the optimum value of fitness. The searchis performed by handling a population of ‘‘chromosomes’’ (bit-strings). Each indi-vidual of the population is made up of as many ‘‘genes’’ (sub-strings) as the numberof control parameters to be determined. Each gene codes a possible value of aparameter so that each chromosome represents a possible solution to the problem.The initial population is generated by random sampling the bits of all the strings.This procedure corresponds to uniformly sampling each control factor within itsrange. The chromosome creation, while quite simple in principle, presents somesubtleties worth to mention: indeed it may happen that the admissible hypervolumeof the control factors is only a small portion of that resulting from the cartesianproduct of the ranges of the single variables, so that one must try to reduce thesearch space by resorting to some additional condition. Indeed, it may happen thatthe value sampled for a gene drastically reduces the admissible range for the suc-cessive gene of the chromosome. By adopting this condition, which give rise to asampling procedure which we call conditioned sampling, the search hypervolume canbe drastically reduced. The upshot is that the in the course of the population crea-tion, a chromosome should be accepted only if suitable criteria are satisfied. Thisremark also applies to the step of chromosome replacement, described below. Thepopulation of chromosomes, then, evolves collectively in successive generationsaccording to the following breeding algorithm which defines the way in which the(n+1)-st population is generated from the n-th previous one.

The first step of the breeding procedure is the generation of a temporary newpopulation. Assume that the user has chosen a population of size N (generally aneven number).The population reproduction is performed by resorting to the Stan-dard Roulette Selection rule: to find the new population, the cumulative sum of thefitnesses of the individuals in the old population is computed and normalized to sumto unity. The new population is generated by random sampling individuals, one at a

M. Marseguerra et al. / Annals of Nuclear Energy 30 (2003) 1437–1456 1439

time with replacement, from this cumulative sum which then plays the role of acumulative distribution function (cdf) of a discrete random variable (the position ofan individual in the population). By so doing, on the average, the individuals in thenew population are present in proportion to their relative fitness in the old popula-tion. Since individuals with relatively larger fitness have more chance to be sampled,most probably the mean fitness of the new population is larger.

The second step of the breeding procedure, i.e. the crossover, is performed asindicated in Fig. 1: after having generated the new (temporary) population as abovesaid, N/2 pairs of individuals, the parents, are sampled at random without replace-ment and irrespectively of their fitness, which has already been taken into account inthe first step. In each pair, the corresponding genes are divided into two portions byinserting at random a separator in the same position in both genes (one-site cross-over): finally, the first portions of the genes are exchanged. The two chromosomes soproduced, the children, are thus a combination of the genetic features of their par-ents. A variation of this procedure consists in performing the crossover with anassigned probability pc (generally rather high, say pc50.6): a random number R isuniformly sampled in (0,1] and the crossover is performed only if R<pc. Viceversa,if R5pc, the two children are copies of the parents.

The third step of the breeding procedure, performed after each generation of a pairof children, concerns the replacement in the new population of two among the fourinvolved individuals. The simplest recipe, again inspired by natural selection, justconsists in the children replacing the parents: children live, parents die. In this caseeach individual breeds only once.

Fig. 1. Example of crossover in a population with chromosomes constituted by three genes.


The fourth and last step of the breeding procedure eventually gives rise to the final(n+1)-st population by applying the mutation procedure to the (up to this timetemporary) population obtained in the course of the preceding steps. The procedureconcerns the mutation of some bits in the population, i.e. the change of some bitsfrom their actual values to the opposite one (0!1) and viceversa. The mutation isperformed on the basis of an assigned mutation probability for a single bit (gen-erally quite small, say 10�3). The product of this probability by the total number ofbits in the population gives the mean number m of mutations. If m<1 a single bit ismutated with probability m. Those bits to be actually mutated are then located byrandomly sampling their positions within the entire bit population.

End of the search. The sequence of successive population generations is usuallystopped according to one of the following criteria:

1. when the mean fitness of the individuals in the population increases above an
assigned convergence value;
2. when the median fitness of the individuals in the population increases above
an assigned convergence value;
3. when the fitness of the best individual in the population increases above an
assigned convergence value. This criterion guarantees that at least one indi-vidual is good enough;
4. when the fitness of the weakest individual in the population increases above
an assigned convergence value. This criterion guarantees that the wholepopulation is good enough;
5. when the assigned number of population generations is reached.

The main feature of this approach is that the searching technique proceeds fromone batch of tentative solutions (the chromosome population) to another (the suc-cessive chromosome population) instead of from one single tentative solution toanother. Another feature, computationally important, is that there is no need ofcomputing the derivatives of the successive solutions, as in the case of most of thealternative optimization methods, generally based on a gradient descent algorithm.Of particular relevance for our analysis is the construction and management of an

archive devised so as to hold a given number of different best solutions, i.e. withhighest values of fitness, encountered in the successive generations. The management ofthe archive proceeds as follows. In each generation, the fitness value of each generatedchromosome is compared to those of the individuals already present in the archive:

� if the archive’s capacity is not filled up, then the chromosome is inserted;� otherwise, if the chromosome’s fitness value is larger than, and sufficientlydifferent from, that of at least one of the already archived individuals, thechromosome is added to the archive and the archived chromosome withsmallest fitness is discarded.

The archive, thus, turns out to dynamically collect the best chromosomesencountered in the evolution process. Through this archive, the evolution processtowards convergence of the best chromosome-solutions can be analyzed. Note that


this archiving procedure is rather different from that employed in other geneticalgorithms in which the procedure is performed after each generation and applied tothe current best chromosome only.In any case, the archive set up amounts to considering all the best chromosomes-

solutions ever encountered throughout the generations, ranking them in increasingorder of fitness, discarding the replicas (i.e. those chromosomes which appear in thelist more than once) and keeping only a given number of the different individualswith the highest fitness values in the list. In the limit of a large number of genera-tions, the search algorithm efficiently spans the search space and the solutions in thearchive cluster around the best value.

3. Inferring sensitivity information from GAs evolution process

During its evolution, the genetic algorithm evaluates the objective function sev-eral times, in correspondence of the various proposed chromosomes which codethe control parameters values. In this section we examine closely the evolutionaryprocess undertaken by the algorithm and try to infer some qualitative informationon the sensitivity of the objective function with respect to the coded controlvariables.

3.1. Analysis of the evolutionary process

As briefly mentioned in the previous section, in the genetic approach, thesuccessive generations evolve in such a way as to favour the survival of the best, orfittest, individuals. As the evolutionary process approaches convergence, the fittestindividuals tend to dominate and the algorithm tends to narrow its search, even-tually focusing on small regions around the fittest individuals.The analysis of this evolutionary process can provide us with qualitative infor-

mation on the importance of the control variables with respect to the objectivefunction. Indeed, as it will be shown shortly, convergence is not reached simulta-neously or randomly by all coded variables. By performing several repetitions of theprocedure with different random chains, we noticed that the various variablesreached convergence always in the same order, some of them early in the processand some others at later generations. Indeed, as most other optimisation proceduresdo, the genetic algorithm proceeds by adjusting first the most important variables,i.e. those which lead to higher gains of the objective function, worrying only at asecond stage about the tuning of the other less important variables which candetermine only minor improvements of the fitness function value.The above mentioned convergence process, and the associated feasibility of

extracting qualitative information on the importance of the control variables withrespect to the objective function, is illustrated first with reference to a simple caseregarding the maximization of an analytic function in (x, y, z) chosen in such a waythat the sensitivity of the objective function to these control variables is a prioriknown. In the genetic search, each chromosome was made up of as many genes as


the number of variables to be determined, plus an additional gene introduced tocode a fictitious variable from which the function to be maximized was actuallyindependent. The reason for introducing this latter variable was to check the pro-cedure’s capability to recognise its fictitious character. The function to be maximizedis a gaussian distribution in (x, y, z):

fðx
; y; zÞ ¼1
ð2pÞ3=2sxsyszexp �

1

2

x � mx

sx

� �2þ

y � my

sy

� �2þ

z � mz

sz

� �2" #( )ð1Þ

where mx=my=mz=5 are the mean values and sx=1, sy=5, sz=10 are the stan-dard deviations of the corresponding variables. Note that the chosen values ofstandard deviations allow us to establish an a priori sensitivity ranking of the vari-ables, the function value being more sensitive to an equal change in x than in y thanin z. Four genes are set up to code the three significant variables x, y, z and theadditional fictitious one w from which the function is actually independent. Therange spanned by the search is [1, 10] for all four variables. The evolving populationis made up of 50 chromosomes and it is allowed to evolve for 20 generations whichwere found to lead the search sufficiently close to convergence. The results of thegenetic maximization procedure are reported in Table 1 and show the satisfactoryconvergence of the algorithm for the meaningful control variables x, y, z, whereasthe value found for w bears no significance.During the computation an archive of the currently best 100 individuals found by

the algorithm was filled and updated. At the end of each generation we then eval-uated the sample means and variances of the four control variables from therespective 100 values contained in the current archive. Fig. 2 shows the behaviour ofthe square of the coefficient of variation (defined as the ratio between the standarddeviation and the mean, Ang and Tang, 1975) as a function of the generations. Thevariable x, to which the objective function is most sensitive, is shown to convergeafter only few generations, when the individuals in the archive are characterized bysimilar values of this variable so that the corresponding variance drops to very smallvalues close to zero. Similarly, the second most sensitive-important variable y is thesecond to be adjusted and then z is the third (the figure does not give proper credit tosuch ordering since after the quicker descent of the coefficient of variation of y itseems that z overcomes y and reaches convergence earlier; however the actual

Table 1

Genetic algorithms data and true and estimated optimal values of the arguments in the gaussian dis-

tribution maximization

Gaussian distribution
Genetic algorithm
Variable
Optimal value Range No. bits GA value Relative error (%)
x
5 1–10 10 4.995 0.1
y
5 1–10 10 4.995 0.1
z
5 1–10 10 5.005 0.1
w
– 1–10 10 9.067 –

numbers show that it is y who settles first). Finally, w keeps a significant variabilitywithin the solutions of the archive throughout the 20 generations: this reflects thefact that the objective function does not depend on this variable so that there is noreason for convergence to any particular value. Similar insights are provided by themeans of the variable values characterizing the individuals in the archive. Fig. 3confirms that the variables x, y, z, stabilize to their final best values at differenttimes, in accordance to the sensitivity of the function to these variables, and that thefictitiously introduced variable w does not converge to any particular value. Notethat the means of all the variables (even w) apparently converge to their respectivefinal values: this is due to the fact that the mean values of the archive solutions ofone generation are correlated to the mean values of the previous generations since,as previously mentioned, when the evolutionary process is approaching con-vergence, the fittest individuals in the population have spread their genetic contentover the whole population so that the appearance of new greatly different indivi-duals becomes rarer and rarer.This simple analytical case allows us to draw some interesting considerations.

First of all, the adopted evolutionary rules impose to the individuals of the popula-tion a survival strategy which leads to settling first those variables which contributethe most to their fitness and therefore to their survival. This mirrors the approach toevolution of living species which first worry about developing those skills whichensure their survival and only later devote their efforts to improving their lifestyle.

Fig. 2. Behaviour, as a function of the generations, of the square of the coefficient of variation (variance/

squared mean) of the four control variables contained in the current archive.


From a dual point of view, instead, it seems possible to infer from the evolu-tionary process information useful for ranking the sensitivity of the objective func-tion to the control variables.

3.2. Analysis of the archive at convergence

As mentioned in Section 2, at convergence of the genetic algorithm search, thearchive contains the best different individuals encountered throughout the wholeevolutionary process. These best individuals give rise, in general, to similar values ofthe objective function, depending on the resolution of the binary coding of the con-trol variables. The points corresponding to the values of the control variables tendto cluster in a small hypervolume and, in correspondence, the objective functions areclose to the best solution: these data can be analysed to infer local sensitivity infor-mation. Indeed, if the values in the archive corresponding to a given variable arehighly dispersed, then quite different values of such variable possibly give rise tosimilar values of the objective function: this would indicate that this variable doesnot need to be identified precisely as different values affect little the objective func-tion. On the contrary, if the values in the archive pertaining to a given variable aretightly concentrated around the best value, this means that the objective function ishighly sensitive to such variable and only values in a much narrower range aroundthe best allow to obtain high objective function values: in other words, to achievehigh values of fitness the variable must take values not very different from the opti-

Fig. 3. Behaviour, as a function of the generations, of the means of the four control variables contained in

the current archive normalized with respect to their value at the last generation.


mal one. Therefore, we can expect that the values in the archive be less dispersed forthose variables to which the objective function is more sensitive.To validate the above arguments we consider the maximization of a bivariate

gaussian distribution:

fðx;
yÞ ¼
exp�1

2ð1� r2Þ

x� mx

sx

� �2

þy� mysy

� �2

�2�x� mx

sx

� �y� my

sy

� �" #( )

ð2pÞ3=2sxsy

ffiffiffiffiffiffiffiffiffiffiffiffiffi1� r2

p ð2Þ

where r is the correlation coefficient, mx=my=0 and sx and sy will take on differentvalues.In the genetic algorithm search, we employ two-genes chromosomes for coding

the values of the two variables x, y. The range spanned by the search is [�5, +5]for both variables. The population has been set to 30 chromosomes and the evo-lution process lasts for 20 generations deemed sufficient for reaching a reasonableconvergence. In Fig. 4 we plot the (x, y) values of the 200 overall best individualsidentified by the genetic algorithm during its evolution and stored in the final archiveat convergence. The four cases presented refer to r=0 and various sx and sy values.The top lefthand plot refers to the case in which sx=sy=5: the variables values inthe archive appear to be similarly distributed in the x- and y-directions. Indeed, byconstruction, (x, y) values at the same distance from the optimal value (0,0) give rise

Fig. 4. Distribution of the (x,y) values of the best 200 individuals identified by the genetic algorithm

during its evolution, for four pairs of �x and �y values.


to the same objective function value, Eq. (2), and thus there would be no reason tohave points lying on a given preferential direction: the isofitness curves arecircumferences centered at (0,0) and individuals representing points on the samecircumference have the same fitness and thus the same probability of survival in theevolutionary process. In the top righthand plot we have a case in which sx=5,sy=10 and correspondingly the variables values in the archive appear to be moredispersed in the y- than in the x-direction: this gives an indication that the objectivefunction is less sensitive to the variable y than to x. The isofitness curves in this caseare elliptical with major axis along the y-direction. The bottom plots refer to twomore cases with sx=1, sy=5 and sx=8, sy=3, respectively: the distributions ofthe points in the archive are consistent with the objective function sensitivity to the xand y variables as given by the respective variances.In Fig. 5 we examine the effects of correlations between the two variables: the best

values in the archive appear to be distributed according to the sign of r6¼0 since theisofitness curves are oriented along the direction of the correlation.From the above analysis we can draw the conclusion that the distributions of

the best individuals’ variables values stored in the final archive after 20 genera-tions, provide us with qualitative local sensitivity and correlation information onthe variables at stake. Table 2 contains the relevant statistics of the final archivesamples with reference to the bivariate gaussian maximization for different valuesof r, sx and sy. It can be seen that the standard deviations �

ax, �

ay estimated from the

Fig. 5. Distribution of the (x,y) values of the best 200 individuals identified by the genetic algorithm

during its evolution. The cases relate to different correlation coefficient values.


values in the archives reflect, in the trend but not in the actual values, the differentsensitivities of the model with respect to the various variables, as given from the truestandard deviations: the most important variable is always characterized by a smal-ler estimated standard deviation �a than that pertaining to the other variable. Thisresult seems worthy of a further comment. Recall that the values of �a

x, �ay are

estimates of the dispersion of the values of the control variables which define thebest solutions and that these are concentrated in a tight region around the max-imum. It is thus not surprising that they do not provide accurate estimates of theparameters sx and sy of the bivariate Gaussian distribution which give a measureof the dispersion of the variablesand y within their entire ranges. Instead, it can benoticed that the estimated correlation coefficients �a describe with reasonable accu-racy the true correlations between the two variables, even if the estimates are carriedout on a small portion of the variables space around the maximum.In summary, we have seen from this simple analytical case that an analysis of the

distributions of the variables’ values contained in the final archive of the best solu-tions at convergence allows us to draw significant information on the local sensitiv-ity of the objective function with respect to the control variables and on the possiblepresence of correlations among them. Such information is qualitative in nature andlocal, as the area of the search space spanned within the final archive is localized in asmall region around the best individual.

4. Application to a nuclear reactor model

In this section the above analyses are applied to the determination of the opti-mal values of five effective parameters of a nuclear reactor model for use in

Table 2

Estimated standard deviations and correlation coefficients obtained from the values in the final archives

for bivariate gaussian distributions maximizations in cases differing due to different values of the standard

deviations and correlation coefficient

�x
�y � �ax �a
y
�a
5
5 0 8.60.10�2 9.26.10�2 �4.02.10�2
5
10 0 7.44.10�2 1.32.10�1 9.60.10�2
1
5 0 1.32.10�2 6.85.10�2 �0.10
8
3 0 9.79.10�2 4.91.10�2 �1.41.10�2
5
5 0.5 7.06.10�2 7.52.10�2 0.52
5
5 �0.5 8.92.10�2 6.61.10�2 �0.45
5
5 0.9 1.75.10�1 1.72.10�1 0.91
3
8 0.5 7.60.10�2 1.66.10�1 0.49
3
8 0.9 6.88.10�2 1.79.10�1 0.89
3
8 �0.75 3.61.10�2 2.61.10�1 �0.66

reactivity control (Chernick, 1960; Chernick et al., 1961). This Section isstructured as follows. In the first part we describe the reactor model. In the sec-ond part we present briefly the parameters determination procedure. The appli-cation to this case study of the evolution and convergence analyses previouslyillustrated is presented in the last part of the Section. A more detailed and com-plete description of the principles of the estimation approach is reported in Mar-seguerra and Zio (2001).

4.1. The reactor model

While in the past the nuclear reactors were generally operated at a constant max-imum power, nowadays the need of economic competitiveness dictates that thereactor power should comply with the time-varying load demand from the electricalgrid. However, along with the generated power, also the generated neutron poisons,such as iodine and xenon vary in time. Since the possibility of performing ademanded increase of the reactor power depends on the reactivity margin and this,in turn, depends also on the poison content, the estimate of these poisons in realtime is of paramount importance.In order to predict the reactivity variation necessary to adapt the power produc-

tion to the demands of the electrical grid, we consider the Chernick’s nuclear reactormodel (Chernick, 1960; Chernick et al., 1961). In this model, proper account is givento the main reactivity feedback effects through Xenon and Iodine balance equationsand a one-group, point kinetics equation with nonlinear power reactivity feedback(Chernick, 1960; Chernick et al., 1961). This latter feedback occurs over a time per-iod which is much smaller than the times of interest, as dictated by the Xenon andIodine decay constants, and it can therefore be treated as prompt. For the samereasons, the effects of the delayed neutrons are ignored.In order to predict the system behavior starting from a given time t0, we use

pseudo experimental (i.e. obtained by simulation) power and reactivity data takenfrom the past reactor history back to a time t preceding t0 by, say, 5 days (severalcharacteristic times of the Xe and I dynamics). The interval t; t0ð Þ is furtherdivided into two subintervals: t; tð Þ during which we only need the reactorpower, and t; t0ð Þ during which both the reactor power and the reactivity changes,measured by the variation of the control rods positions with respect to their posi-tion at t, are used as experimental data. The model equations, written in terms ofthe available data, i.e. the reactor power P ¼ �f� and the reactivity variationD� tð Þ ¼ � tð Þ�� tð Þ, are:

�dP

dt¼ � tð Þ þ D��

�X

c�fXe �

�

�fP

P ð3Þ

dXe

dt¼ �xP þ lII � lxXe �

�x

�fXeP ð4Þ


R

dI

dt¼ �IP � lII ð5Þ

where: � is the reactivity; (cm�2 s�1) is the flux; Xe (cm�3) and I (cm�3) are theXenon and Iodine concentrations, respectively; �f (cm

�1) is the effective fissionmacroscopic cross section; �x (cm

2) is the effective Xenon microscopic cross section;�x and �I are the Xenon and Iodine fission yields, respectively; lx (s

�1) and lI(s�1)

are the Xenon and Iodine decay rates; �(s) is the effective neutron mean generationtime.The temperature and Xenon feedbacks are modeled via the introduction of two

lumped parameters, namely: the temperature feedback coefficient, � (cm2 s) and theadimensional conversion factor of Xenon concentration to reactivity, c.The values of the nuclear constants are: �x=0.003; �I=0.061; lx=2.09*10�5 s�1;

lI=2.87*10�5 s�1 and the effective values to be estimated are the quintuplet ofparameters sx

�f, �

�f, c, �; � tð Þ plus the Xe and I concentration at t. The model is

admittedly very simple but it is potentially very reliable provided the effective para-meters can be updated in real time. In addition to the effective parameters, the initialconcentrations of Xe and I should also be estimated.

4.2. A practical procedure for reactivity forecasting

The details of the procedure for parameter estimation can be found in Marse-guerra and Zio (2001). Here we outline only the main steps with reference to anumerical example.We consider chromosomes made up of five genes, each one coding one of the five

effective parameters. Table 3 contains the relevant data. Each population is com-posed of 1000 chromosomes.As objective function, we consider the inverse of the average squared residuals R

between the experimental power history P exp and the computed one Pc normalizedto the nominal power P, summed over a succession of N=2001 discrete time pointsti 2 ðt; t0Þ, viz.

¼NP2

�N

i¼1P exp tið Þ � Pc tið Þ½

2

; ti ¼ t þ i � 1ð Þt0 � t

N; i ¼ 0; 1; . . . ; N ð6Þ

The objective function R evaluates the capability of a proposed solution to repro-duce the pseudo-experimental power profile P exp in t; t0ð Þ. For a given quintuplet ofvalues proposed by a chromosome, we proceed as follows. As previously stated, weuse the pseudo-experimental power and reactivity data of the reactor history withinthe time interval t; t0ð Þ. Precisely, from the model Eqs. (4) and (5) and from theexperimental power profile P exp during t; tð Þ we get Xe tð Þ and I tð Þ; then, utilizingthe experimental reactivity variation profile in t; t0ð Þ and the pseudo-experimentalinitial value P(t*), we numerically solve the three model equations over the timeinterval t; t0ð Þ, thus getting the power profile Pc in that interval. This is inserted in


Eq. (6), together with P exp, to give the value of fitness corresponding to the chro-mosomes.In the present application, an acceptable convergence was reached after 15 gen-

erations: the values found by the genetic algorithm are reported in the last column ofTable 3. The computing time was equal to 20 min on an Alpha Station 250 4/266.The resulting power profile Pc from t to t0 is shown in Fig. 6 by the solid line. Thetwo vertical dotted lines in the figure identify the time intervals considered in thedescribed procedure: the time interval t; tð Þ in which the experimental powerprofile P exp is used to get the concentrations of Xe and I at time t; and the timeinterval t; t0ð Þ in which both the power profile P exp and the reactivity variationprofile D� tð Þ are used to calculate the objective function R. The good agreement withthe corresponding (pseudo)experimental profile (dotted line) underlines the properchoice made by the genetic algorithm for the five effective parameters. For com-pleteness, in Fig. 6 we also report the computed Xe and I evolutions (solid line) andthe corresponding true values (dotted line), even if these are not observable quan-tities. Again, the agreement is quite evident.Now, at the present time t0, we have available a ‘good’ model, all its effective

parameters and the Xe and I values, so that we can predict the reactivity changesneeded to satisfy future power load variations. Fig. 6 shows in dotted line a reques-ted power profile after t0 (i.e. in the time region at the right of the vertical dotted linein correspondence of t0), the corresponding reactivity predicted with the modelparameters found by the genetic algorithm (solid line) and the reactivity profileactually followed by the operator (dotted line), here simulated making use of theknown true parameters’ values.Note that the described genetic algorithms procedure leads to very good results

even though one of the parameters, namely the generation time �, has a relativeerror close to 20% (Table 3), thus suggesting a scarse sensitivity of the results tosuch parameter. The larger error in the estimation of the mean generation time � isdue mainly to the fact that in Eq. (3) this parameter multiplies the derivative of thepower which is zero most of the time, except during the power variations: since theselast only half an hour, not much information is provided for the estimation of �.Other computations, not reported here, have shown that also the parameter � canbe estimated properly in case of suitable power transients, such as sinusoidal ones.

Table 3

Genetic algorithms data and true and estimated parameters values for the reference daily power load

profile

Simulation phase
Genetic algorithms
Parameter
True value Range (�50%) No. bits GA value Relative error (%)
C
1.3606 0.6–2.0 10 1.2938 5.0
g/�f*1016 (cm3 s)
8.3802 4.0–12.0 10 8.7148 4.0
�x/�f*1018 (cm3)
3.5358 1.75–5.25 10 3.5290 0.2
�*102 (s)
8.3000 4.0–13.0 10 9.8652 20.8
r0*102
1.5063 0.75–2.5 10 1.3695 10.6

4.3. Analysis of the genetic algorithm’s evolution process

Let us apply the analyses described in Section 3 to the genetic algorithmsapproach for estimating Chernick’s model parameters. The purpose of these ana-lyses is to extract some information on the relative sensitivity of the model to thedifferent parameters.Fig. 7 shows the behaviour of the square of the coefficient of variation as a

function of the generations. The variable �x

�fis shown to converge first, followed by

�(t*), ��f, and c. The parameter � is shown to converge much more slowly, thus

confirming a relatively smaller sensitivity of the model to such parameter. Similarconclusions can be drawn by an analysis of the behaviour through the successivegenerations of the sample means of the parameter values characterizing the indivi-duals in the archive (not reported for brevity).An analysis of the distribution of the parameters’ values corresponding to the 200

overall best individuals encountered during the evolution process and recorded inthe final archive confirms these indications. The second column of Table 4 reportsthe estimated squared coefficients of variation of the five parameters from thearchived values: the qualitative sensitivity ranking previously found is clearlyreflected also here.

Fig. 6. Power load, reactivity prediction, xenon and iodine profiles for the reference case of daily power

load variations: dotted line=real (simulated) evolution; solid line=optimal GA-predicted evolution.


The qualitative sensitivity ranking thus far obtained by the analysis of the geneticalgorithm procedures is in agreement with the results obtained by a standard var-iance decomposition sensitivity analysis (McKay, 1995) of the fitness function Eq.(6), whose sensitivity coefficients values �̂2, reported in the third column of Table 4,are defined as:

�2xs¼ V E yjxsð Þ½ =V½y

These coefficients measure how much of the unconditional variance V[y] of amodel output y (in our case, the fitness function) is accounted for by the varianceof the expected value of y conditioned on the input parameter xs (in our case, anyof the five effective parameters, taken one at the time). Note that for the highlynonlinear Chernick’s model, the first order sensitivity coefficients do not sum up tounity as they can only partially account for the overall output variability. In thissense, higher-order interactions of the input parameters should be accounted forbut a thorough sensitivity analysis of these effects is beyond the scope of thepaper.As a partial analysis of correlations among variables, Table 5 gives the estimates

of the correlation coefficients between the parameters. Only some of these corre-lations are easily traceable from the model equations. For example, multiplying by cthe Eq. (3) we obtain:

Fig. 7. Behaviour of the square of the coefficients of variation as a function of the generations.


c�dP

dt¼ c�ðtÞ þ cD��

sX

�fXe � c

�

�fP

P

in which the products c ��, c�(t*) and c�=�f appear explicitly, thus giving rise to anegative correlation between c and �, c and �(t*) and c and �=�f. Furthermore, Eq.(3) contains explicitly the ratio between c and sx=�f, thus supporting the positivecorrelation identified in the analysis. The other correlations suggested in Table 5 arenot readily derivable analytically.

5. Conclusions

Genetic algorithms are optimization methods based on procedures which resem-ble those of natural selection and genetics. Such algorithms differ from most opti-

Table 4

Sensitivity ranking of the five effective parameters: estimated squared coefficients of variation obtained

from the values in the archive (column 2) and sensitivity coefficients obtained by a standard variance

decomposition analysis (column 3)

Parameter
�a2=a2 �̂2
�x/�f
6.68.10�3 2.47.10�3
�(t*)
9.02.10�3 2.06.10�3
g/�f
1.06.10�2 1.32.10�3
c
1.48.10�2 1.00.10�3
�
9.65.10�2 3.15.10�4
Table 5

Correlation coefficients between parameters estimated from the values in the final archive at convergence

Parameter xi
Parameter xj �ija
c
g/�f �0.58
c
�x/�f 0.40
c
� �2.72.10�2
c
�(t*) �0.51
g/�f
�x/�f 0.35
g/�f
� 0.20
g/�f
�(t*) 0.31
�x/�f
� 0.17
�x/�f
�(t*) 0.10
�
�(t*) 0.28

mization techniques in that the search proceeds from one group (population) of solutionto another, rather than from an individual solution to another. In other words, thegenetic algorithms perform a global search instead of a hill-climbing search.This paper is originated by the consideration that, while effectively performing

its characteristic procedural steps, the algorithm evaluates several solution pointswithin the search space, thus offering the possibility of disclosing the character-istics of the underlying model object of the optimization. In particular, we haveinvestigated the possibility of retrieving qualitative sensitivity information on theimportance of the optimization control variables with respect to the objectivefunction. Simple analytical examples have illustrated that, similarly to most otheroptimisation methods, the genetic algorithm convergence procedure is effective infirst settling the most important variables to their best values, and only after-wards in focusing on the tuning of the less important variables. Thus, the suc-cession of convergences of the variables throughout the generations gives aqualitative indication of the importance of the variables with respect to the func-tion objective of the optimisation. Furthermore, an analysis of the distribution ofthe overall best different individuals recorded throughout the evolutionary processin an appropriately devised archive has shown that at convergence the moreimportant variables are more tightly clustered around the best values whereas moredispersion is allowed in the less important variables. Thus the computed samplevariances from the final archive also give a qualitative measure of the importanceof the variables. Estimated correlation coefficients can also provide us with infor-mation on dependences between the variables possibly existing in the model.The above analyses were performed on a case study concerning the search for the

optimal values of five effective parameters of a nuclear reactor model. The geneticalgorithm procedure was able to efficiently determine the parameters’ values. Inaddition, the information gained by an analysis of the genetic algorithm evolutiontowards convergence and by the final archive of overall best solutions confirmedthat the algorithm proceeds orderly to convergence according to the importance ofthe parameters with respect to the objective function. In particular, the neutronmean generation time was found of little importance in determining the modelresponse to the particular transient analysed. The importance ranking of the para-meters mirrored that of a standard variance decomposition sensitivity analysis.Moreover, indications on some existing correlations were confirmed by simple ana-lytical treatment.In conclusion, we believe that genetic algorithms, while efficiently searching for

optimal solutions, can offer remarkably insightful information. This information isstill hidden within their powerful optimization procedures: more exploratory effortsare recommended.

References

Ang, A.S., Tang, W.H., 1975. Probability Concepts in Engineering Planning and Design, Vol. I. Wiley,

New York.

Chambers, L. 1995. Pratical Handbook of Genetic Algorithms, Vol. 1 and 2, CRC Press.


Chernick, J., 1960. The dynamics of a xenon-controlled reactor. Nuclear Science and Engineering 8,

233–243.

Chernick, J., Lellouche, G., Wollman, W., 1961. The effect of temperature on Xenon instability. Nuclear

Science and Engineering 10, 120–131.

Goldberg, D.E. 1989. Genetic Algorithms in Search, Optimization, and Machine Learning, Addison-

Wesley Publishing Company.

Holland, J.H., 1975. Adaptation in Natural and Artificial Systems. University of Michigan Press,, Ann

Arbor, MI.

Marseguerra, M., Zio, E., 2001. Genetic algorithms for estimating effective parameters in a lumped reac-

tor model for reactivity predictions. Nuclear Science and Engineering, September 139 (1), 96–104. 2001.

McKay, M.D. 1995. Evaluating Prediction Uncertainty. Tech. Rep. NUREG/CR-6311, U.S. Nuclear

Regulatory Commission and Los Alamos National Laboratory.

Uhrig, R.E., Tsoulakis, L.H. 1999. Soft Computing technologies in nuclear engineering applications.

Progress in Nuclear Energy, Vol. 34 N.1, Pergamon.


model parameters estimation and sensitivity by genetic algorithms

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