Benefits of Evolutionary Strategy in Modeling of Impedance Spectra

8/3/2019 Benefits of Evolutionary Strategy in Modeling of Impedance Spectra

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Electrochimica Acta 51 (2006) 14531461

Benefits of evolutionary strategy in modeling of impedance spectra

O. Kanoun, U. Troltzsch, H.-R. Trankler

University of the Bundeswehr Munich, Institute for Measurement and Automation, 85579 Neubiberg, Germany

Received 23 June 2004; received in revised form 25 October 2004; accepted 25 February 2005

Available online 2 September 2005

Abstract

The more effects and mechanisms are represented in an impedance spectrum, the more unknown parameters are needed for accurate

modeling. The inverse identification problem corresponding to this parameter extraction process becomes thereby more difficult to solve andneeds generally a lot of a priori knowledge and trials. The use of evolutionary strategies (EV) can significantly contribute to increase efficiency

during this process. With selected examples, many benefits of evolutionary strategy are shown by means of simulations with more than 100

trials.

For simple models, the main advantage of evolution strategy is to be nearly insensitive to the chosen search region. For models with a big

number of unknown parameters, the combination with the LevenbergMarquardt (LevMq) method reaches very good results compared with

each method alone. The combined method profited of positive properties of both methods. The evolution strategy is, also for an ill-posed

inverse problem, able to calculate a parameters vector near the optimum. The LevenbergMarquardt method profits from it and is then able to

calculate the optimum more exactly, because it is already in its next neighborhood. The resulting process is robust and reaches good modeling

results even, if only a few a priori knowledge is available concerning expected parameter values.

2005 Elsevier Ltd. All rights reserved.

Keywords: Impedance spectroscopy; Evolutionary strategy; Genetic algorithm; Inverse problem; Sensitivity analysis

1. Introduction

Modeling of impedance spectra needs suitable methods

for parameter extraction, in order to facilitate investigation

of models and calculation of corresponding parameter val-

ues. Especially in this field, convergence difficulties and

strong dependencies on the start values are observed even

with relative simple impedance models. This is due to their

typical strong parametric non-linearity [1]. The stronger the

non-linear behavior of a model, the weaker is the conver-

gence of the corresponding regression and the more is theprobability to trap into local minima of the quality func-

tion.

Particularly at investigation stage, knowledge about the

suitability of models and their typical parameter values

is restricted. Local non-linear regression algorithms, such

as GaussNewton (GN) or LevenbergMarquardt (LevMq)

Corresponding author. Tel.: +49 89 6004 3740; fax: +49 89 6004 2557.

E-mail address: [email protected] (O. Kanoun).

method, need therefore a lot of trials in order to reach

meaningful results. This is due to the fact, that these

gradient-based methods are suitable only for models with

low parametric non-linearity. In the field of impedance spec-

troscopy, many quantities should be simultaneously mod-

eled, such as electrode dimensions, diffusion coefficients,

reaction rates, charge transfer coefficients and concentra-

tions. Impedance models have therefore generally many

unknown parameters and a higher probability of paramet-

ric non-linearity. Local algorithms find generally the closest

optimum to the starting point. The parameter extraction pro-cedure should be therefore tailored for every case. Thereby,

strategies are developed, e.g. holding some parameters con-

stant at the beginning, choosing good initial estimates of

the parameters or using some suitable weighting or penalty

functions.

Under these conditions, the investigation of a model suit-

ability becomes time consuming and unreliable. It depends

strongly on the number of unknown parameters, the available

time for modeling and the experience of the model developer.

0013-4686/$ see front matter 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.electacta.2005.02.123


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1454 O. Kanoun et al. / Electrochimica Acta 51 (2006) 14531461

The parameter extraction process represents mathemati-

cally an inverse identification problem and is therefore asso-

ciated with certain difficulties. Depending on the application

considered, it can be evaluated with many features, such as

thequality of fit, thecalculation time androbustness.This last

is inalienable for the application considered in our case, in

which model parameters should be automatically calculatedonline without any interventions [2].

In order to find out a robust parameter extraction method,

we propose to make use of global methods like evolution-

ary strategy. Evolutionary and genetic algorithms have been

already applied in many fields for optimization problems

[3,4], mainly due to their ability to avoid trapping into local

optima of the quality function. They make use of a pop-

ulation and use therefore a search region instead of start

parameters. Genetic algorithms imitate directly biological

genetic variation mechanisms with more cross over mech-

anisms than mutation processes [5]. The necessary binary

coding of the variable vector makes them laborious for con-

tinuous problems [4,6]. Evolutionary strategy (EV), whichis considered in this paper, was in contrast specially devel-

oped for parameter extraction. It uses the parameter values

in their original format and allows a targeted deterministic

selection of best individuals. In some literature, evolutionary

strategies are termed as genetic algorithms with real num-

bers.

Although evolution strategy and genetic algorithms are

already well established in a lot of application fields [3,4,7],

their use in modeling of impedance spectra was not widely

used until now. Even in literature, only a few contribu-

tions [6,810] were dedicated to the application of these

methods. In [6], a comparison of genetic algorithms withQuasi-Newton method (QN) was carried out with emphasis

on calculation time, convergence behavior and sensitivity to

measurement noise. The results will be considered all through

this paper. In [8], a hybrid method was developed using

alternately one generation of the genetic algorithm followed

by one attempt with the GaussNewton method in order to

improve convergence. This method benefits only partly from

the advantages of evolutionary strategy as will be explained

in Section 4.

The main aim of this paper is to investigate benefits of evo-

lution strategy for impedance spectroscopy, to give a general

overview of them and to explain them based on consistent

simulations for different classes of models.

2. Investigated models

In order to investigate different methods of parameter

extraction, we defined two impedance models with different

levels of complexity (Fig. 1). The investigations are carried

out using typical models in impedance spectroscopy, espe-

cially the fundamental Randles model [11] with a constant

phase element for representation of the Warburg impedance.

Model1 has four unknown parameters. Model2 consists of

Fig. 1. Investigated models.

two Randles models in series, which can represent the two

electrodes of an electrochemical battery [12] and has alto-

gether seven unknown parameters.

A sensitivity analysis was carried out in order to char-

acterize the investigated models for non-linear parameter

extraction. This method can provide considerable insight intoan estimation problem [13,14]. Although it is a common

method in parameter estimation, it was rarely used in elec-

trochemical impedance spectroscopy as criticized in [15].

We can generally distinguish global and local methods

of sensitivity analysis [15]. Global methods, such as Fourier

amplitude sensitivity test and stochastic sensitivity analysis

[15], produce measures of parametric importance over entire

regions in parameter space. In this case, the sensitivity infor-

mation is averaged on the parameter region considered. Local

methods, such as partial derivates or sensitivity densities,

determine the effect of parameter variation of the solution

at a single point of the parametric domain. These are moresuitable for our investigation, because we know the original

parameter values, which were used for generation of the sim-

ulated data subjected to fit processes. In this paper, we use

partial derivates of the model function as sensitivity coeffi-

cient [13].

SCP =

ZModk

P

(1)

where P is the parameter, ZMod the impedance model and kis the angular frequency.

A scaling is performed by multiplying the sensitivity coef-

ficients by the parameters values. The so scaled sensitivity

coefficients have the same unit and allow their comparison to

each other.

Scaled SCP = P

ZModk

P

(2)

In general, the scaled sensitivity coefficients are desired to be

large for each parameter and uncorrelated (linearly indepen-

dent) for different parameters. A correlation (linear depen-

dency) between the sensitivity coefficients is available, if

there exists a set of parameters a1, a2, . . ., an so that the

relationship (3) is fulfilled for the whole observed frequency

range. In this case, there is no unique minimum and all the


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O. Kanoun et al. / Electrochimica Acta 51 (2006) 14531461 1455

Fig. 2. Scaled sensitivity coefficients for Model1.

parameters cannot be simultaneously and uniquely estimated

[13].

a1 SCP1 + a2 SCP2 + + an SCPn = 0 (3)

where SCPi is the sensitivity coefficient corresponding to the

parameter Pi and n is the number of unknown parameters.

In [1], a method for assessing non-linearity in impedance

models was developed and tested. The results show, that

impedance spectra have generally, and also the Randles mod-

els, a strong parametric non-linearity [1]. This is one reason

why, we can expect some difficulties, also with this relative

simple model.

The scaled sensitivity coefficients for Model1 (Fig. 2)

are not totally correlated but the sensitivity coefficient for

the parameter is very low. Both facts let us characterize

the process of parameter extraction to be solved as ill-

posed inverse identification problem. The stronger is theill-posedness of an inverse identification problem, the more

dependency on start parameters, the more convergence diffi-

culties, the more probability to trap into local minima of the

quality function and the higher sensitivity to measurement

noise.

In the case of Model2, not only the bigger number of

unknown parametersand the expected higher parametric non-

linearity [1] pose problems. The sensitivity analysis illus-

trates also a low information content in the measurement data

concerning the parameters 1 and 2 (Fig. 3).

The differences between Model1 and Model2 represent a

suitable basis for evaluation and comparison issues as will be

seen in the next sections.

3. Evolutionary strategy

Evolutionary strategy (EV) is classified in contrast to tai-

lored methods to stochastic methods which are more suitable

for general-purpose problems in which the a priori knowl-

edge about parameters is limited [7]. Their advantages are

the easy implementation even of different fitting criteria, easy

consideration of constraints through rejection of undesired

individuals and generally robustness. Their main disadvan-

tage is the calculation expense due to the big number of

function evaluations to be carried out [6].

EV makes use of a population with a certain number of

individuals. The genes of each individual correspond to thevalues of the unknown parameters. Every individual repre-

sents a trial solution to the fitting problem and is evaluated

by thecorresponding fitting error. Beginning with a randomly

generated population, the best individuals are selected, prop-

agated to the original population size and than mutated by

changing the gene of the individuals. This process is very

important for exploration, which consists in finding trial

solutions in new regions of the search space. Selection and

mutation processes are repeated until a certain termination

condition is reached.

Fig. 3. Scaled sensitivity coefficients for Model2.


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Many methods were proposed to design evolutionary strat-

egy for a certain optimization problem. The aims thereby

were, e.g. to minimize calculation time [6], to introducesome

constraints or to adjust selection criterion to improve certain

properties [9]. Forthe comparison carried outin this paper, we

use conscious a standard evolutionary strategy without any

adjustments. The search region, which is the only adjustmentin order to consider the available knowledge about model

parameters, was used in the generation of the initial popula-

tion and in the mutation procedure. The population size was

chosen by 400 individuals. A smaller population can lead to

a more rapid convergence, but thereby the risk of converging

to a local optimum increases, because no sufficient thorough

search of the search space will be done. From every gener-

ation the best 40 individuals were selected by choosing the

minimal values of the quality function:

Quality function (P) =kZMod(P) ZMeas

2(4)

where P is the parameter vector,ZMod the impedance model,

ZMeas the measurement data andk is the angular frequency.

The termination condition was a given maximal number

of 50 generations.

During mutation processes, the parameter values (genes

of individuals) are changed. The mutation was implemented

under consideration of the search region, variable within the

same generation and exponential declining with progress in

order to increment convergence probability (Fig. 4).

gi+1 = gi +g et/

randomnormal (5)

where gi is the genes of the ith generation, g the search

region, the mutation variation factor within a genera-

tion, the mutation declination with processing time and

randomnormal is a normal distributed random number.

The quality function corresponding to selected best indi-

viduals in every generation are shown in Fig. 5. We can see

that increasing the generation number leads to a continuous

decrease of the quality function for the best individuals. The

norms corresponding to selected individuals have a higher

density towards the lowest quality function value. We see

also that through the mutation process even individuals with

lower performance are taken into consideration.

Fig. 4. Mutation variation factor in dependence of the number of the

individual Pi within one generation.

Fig. 5. Quality function for selected individuals (e.g. Model2).

4. Comparison of the evolutionary strategy with the

LevenbergMarquardt method

For comparison with evolutionary strategy, we chose to

use the LevenbergMarquadt method, because it is nowa-

days more often implemented in parameter estimation tools

due to of some specific advantages relative to GaussNewton

method, for example.

For evaluation of both methods more than 100 fitting tri-

als were carried out for every investigation. The frequency

range was selected between 10 mHz and 5 kHz. The gen-

erated impedance spectra using the parameter values from

Fig. 1 consist of 200 data points constantly distributed on the

logarithmic axis.

For comparison purpose, the options of each method, such

as number of individuals for EV and termination conditions

for LevMq, were setto typical values andmaintained constant

independent of the investigated aspect and of the used model.

Therefore, the results should be not considered as the best

reachable in every case.

For LevMq, we used randomly varied start values with a

given relative variance around the original parameter, with

which the simulated spectrum were generated (Fig. 6). For

EV, the search region was incrementally increased with a

certain percentagefrom the original parametervalues (Fig.6).For graphical representation of results, we will make use

of the network representation, which allows the simultaneous

display of results in more than two dimensions in a plane.

In this representation, axes have an intersection on a point,

which is not corresponding to zero. In order to reach a better

differentiation, the relative deviations of parameter from the

original value are represented in percent relative deviation

(Eq. (6)).

100 Pcalculated Porginal

Porginal(6)


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Fig. 6. Comparison of evolution strategy (EV) with LevenbergMarquardt

Method (LevMq).

For example, a value of 0.1 on one axis corresponds in reality

to a 0.0001 relative deviation from the original parameter

value.

The simulationresults for Model1 are represented in Fig.7.

Thesmall deviations between parameter values andtheir esti-

mates observed forthis model arenot alwaysremarked during

modeling processes. If we compare the simulation results for

different parameters relative to each other, we will see that

conform to the results of the sensitivity analysis in Section 1,

the parameter was calculated with more difficulties. This

problem was observed in the results obtained with both meth-

ods, because it is a problem more related to a low information

content in measurement data, than to the used optimization

algorithm.

A main advantage which was observed by EV is the big

search region allowed (Fig. 7). In the case of Model1, the

global optimum was found even by a search region of 1500%

of the original parameter values and beyond. Enlarging thesearch region by maintaining the same population size, as

carried out in this paper, leads to a less precise calculation of

the optimum.

The results obtained by the LevenbergMarquardt method

show a significantly higher dependence on start parameters.

Already by a relative variance of 0.1 (can be compared to

approximately 10% search range by the evolution) some

cases of non-convergence were observed (Fig. 7d). By a rel-

ative variance of 2 non-convergences were observed in 44%

of the 100 trials, which should be critically considered as a

result because of the model simplicity. The main advantage

of LevMq was the more precise calculation of the optimum,

if it found it.Similar results were reported in [6] with the Quasi-Newton

method. It was observed that QN method converged within

a smaller number of function evaluations than genetic algo-

rithms and more precise, but it did not always converge to the

global optimum.

The simulationresults for Model2 are represented in Fig.8.

This model has shown, as it was expected, more difficul-

ties with parameter extraction. Allowed search region and

Fig. 7. Results for Model1: (a) EV for search region 10%; (b) LevMq with start parameters within a relative variance of 0.05; (c) EV for search region

600%; (d) LevMq with start parameters within a relative variance of 0.1.


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Fig. 8. Results for Model2: (a) LevMq with start parameter within a relative variance of 0.1; (b) LevMq with start parameters within a relative variance of 0.3;

(c) EV for search region 20%.

relative variance were significantly smaller than in case of

Model1. Besides, It was observed that meaningful results

were obtained for a much larger search region in the case

of EV (Fig. 8c) as, through the corresponding relative vari-

ance (Fig. 8a) in the case of LevMq. This means, if we havea

certain knowledge concerning the expected range of a param-

eter, we obtain with EV a better estimation. We only noticedthat the quality of parameter estimates was not uniform in

case of EV. Some identification problems were observed for

the parameters 1, 2, Cd1 and Cd2 (Fig. 8c).

We can conclude that LevMq was more sensitive to the

difficulties related with ill-posed optimization problem. A

higher sensitivity on start parameters and higher rate of non-

convergence and trapping in secondary optimawere observed

in every case. Already at a relative variance of 0.1 (Fig. 8a)

the results were in reality unacceptable at all.

We also notice that only EV has difficulties by the calcu-

lation of the parameters Cd1 and Cd2. This was also the case

by the results of Model1, but the problems to calculate the

parameter were more serious. This can be explained by the

relative flat behavior of the quality function in dependence of

these parameters (Fig. 9). In this case, the selection process

is not working well related to these parameter.

The sensitivity to measurement noise was comparatively

investigated in [6]. The results show that evolutionary strat-

egy is less sensitive to noise than a gradient-based method

(QN). This is mainly due to noise sensitivity of the gradients

calculation. With a similar model to Model1 and at a noise

level of 1% both methods converged to the right optimum.

At a noise level of 2%, the gradient-based method converged

frequently to a local optimum corresponding to a 5% fitting

error. The evolution strategy was able to find a significantly

better fit.

The longer calculation time of EV can be some times crit-

ically considered. Our results can weaken this expression.

They have shown, that using a today standard computer with

a processor frequency of 1500 MHz the average calculation

time was in case of evolution strategy approximately constantand amounted to 17 s (Model1). This is due to the determin-

istic calculation procedure. In the case of LevMq method, the

calculation time was variable depending on the start param-

eters and was at average approximately 7 s (Model1).

A comparative study of calculation time was carried out in

[6] between genetic algorithms with real numbers, which are

similar to evolutionary strategy, and Quasi-Newton method

using a slight differentmodelto Model1. Thecomparison was

made on the basis of the number of function evaluations and

show, that fitting error decreasesquadratic with the number of

function evaluations. For genetic algorithms the convergence

Fig. 9. Dependence of the quality function on the space charge capacity

(Model2).


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results were slower at the beginning of the fit algorithm and

at the end. They need 20000 function evaluations. The fit-

ting error of the best individual improves in discrete jumps.

The QN method needs approximately 2000 function evalua-

tions before it reached convergence. These include the func-

tion evaluations required for calculation of gradient at each

point.The counter-argument of high calculation time should

be perhaps critically considered by modeling dynamically

changing processes, but not in every case. At investigation

stages, the differences in calculation time are no really rele-

vant because of the high calculation power of nowadays pro-

cessors. Besides, if we consider the number of trials needed

for guessing start values for local optimization methods, we

will see that evolutionary strategy has even benefits concern-

ing the whole modeling and parameter extraction time.

5. Combination of evolutionary and

LevenbergMarquardt method

The main idea in this section to combine a stochastic

globalmethod with a local method[7]. The stochastic method

should increase the probability of landing in the neighbor-

hood of the global optimum and the local method can meet

precisely the right value of the optimum.

The proposed combination of both evolution strategy

and LevenbergMarquardt method is shown in Fig. 10. The

results of the EV are taken as start parameter vector for the

LevMq method. In reality, this procedure can be repeated as

a loop with few iterations in cases, where we should be really

sure of meeting the global optimum [2]. In thispaper, we willonly evaluate one direct process.

There are some main differences to the method pro-

posed by Yang et al. [8] using genetic algorithms and

GaussNewton method. The hybrid method by Yang et al.

uses alternately one generation of the genetic algorithm fol-

lowed by one attempt with the GN method. On one hand,

this method is better than local optimization methods and can

realize a good solution if the corresponding inverse problem

to be solved is not very ill-posed and the calculation time is

critical. On the other hand, this method does not profit from

real advantages of global search in evolutionary strategy and

is therefore less sure against trapping in local optima.The results for Model1 are shown in Fig. 11. They are

better than for EV alone because of the advantages of LevMq

in meeting the real value of the optimum. But the difference

Fig. 10. Combination of evolution strategy with LevenbergMarquardt

method.

Fig. 11. Combination of evolution strategy and LevenbergMarquardt

method with a search region 700% for Model1.

observed can be in reality not significant, because it will lie

under the measurement noise.

The results for Model2 (Fig. 12) are better than the resultsof EV or LevMq alone. Relative to EV we can say, that

no problems concerning Cd1 and Cd2 were observed at all

(Fig. 12b and c). The main improvement relative to LevMq

was the better handling of the problem related with the cal-

culation of1 and 2 (Fig. 12a and c).

The explanation of these results is that evolution

brought the parameters near the optimum and the

LevenbergMarquardt method was therefore able to find the

optimum easier and more precisely. The difficulties of Evo-

lution with flat quality functions could be corrected using the

gradient-based method. The difficulties of LevMq method by

ill-posed optimization problem could be overcome with goodstart parameters. This hybrid solution profited from different

sensitivities of both methods and allowed a better handling

of difficult inverse identification problems.

The quality function consists of the sum of all contribu-

tions of fitting errors. Although it does not give a differentiate

consideration of the calculated parameter values, it gives an

insight into an optimization solution. The values of the qual-

ity function are represented in Fig. 13 for all attempts in case

of Model2. The results for evolution are the worst, because of

the combination of difficulties in calculation the parameters

1, 2, Cd1 and Cd2. This can be surely avoided, if a bigger

population number is chosen, but this kind of adjustments

was not subject of investigation in this paper. The results for

LevMq are generally better. The only problem thereby is the

very unstable behavior, which leaded in practice to the need

of several iterations with this method. The results for the

combination show a definitively better result, which is reli-

able over the whole number of trials and makes only some

non-significant fluctuations.

These results demonstrate a significant reduction of time

expenditure during model selection and parameter calcu-

lation processes for a given model/data set. The proposed

parameter extraction method was tested for several models

with 11 and more unknown parameters [2].


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Fig. 12. Results for Model2: (a) LevMq with start parameters within a relative variance of 0.1; (b) EV for a search region 10%; (c) combination of EV and

LevMq with a search region 10%.

Fig. 13. Quality function for Model2 and different methods (corresponds to

Fig. 12).

6. Conclusion and outlook

Evolution strategy has a lot of benefits for parameter esti-

mation in impedance spectroscopy. With its stochastic prop-

erties, it is more able to find out global minima. A compari-

son between evolutionary strategyand LevenbergMarquardt

method has been carried out basing on simulations with a sta-

tistically significant number of trials (>100).

Comparing models with different complexity levels, we

found out, that evolution alone can be a profitable alternative

in case of models with lower complexity. The main advan-

tages thereby are the large search region allowed, the low

sensitivity to noise and the robust convergence to the right

optimum of the quality function. Difficulties were only in

case of flat behavior of the quality function in dependence of

some parameter values. In this case the selection process was

not working well considering these parameters.

For models with higher complexity level and moreunknown parameters, the inverse parameter extraction prob-

lem to be solved becomes more difficult. In this case, we

found out, that using the results of evolution strategy as start

parameter vector for LevenbergMarquardt method reached

better and more reliable results compared to both meth-

ods alone over the whole number of attempts. The evo-

lution brought the parameters near the optimum and the

LevenbergMarquardt method profited from it and was there-

fore able to find the optimum easier and more precisely. This

hybrid solution profited also from different sensitivities of

both methods and allowed a better handling of difficult mod-

eling problems.

An easily realizable implementation of evolutionary strat-egyin standard impedanceevaluation software involves many

advantages for users. As an alternative or in combination with

other parameter extraction methods, it can significantly con-

tribute to a shortening of time expenditurefor model selection

and mechanisms investigations.

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Benefits of Evolutionary Strategy in Modeling of Impedance Spectra

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