Abstract
The aim of this work is the study of the effectiveness of different parameter estimation techniques for proton exchange membrane fuel cells (PEMFC) under normal operating conditions. The procedure for the estimation of the real time parameters described herein in good approximation meets the requirements posed regarding such techniques, i.e., to obtain data without interrupting the functioning of the fuel cell and to be as simple as possible. In this approach a fuel cell combined with a step-down voltage regulator system was analyzed and the potential relaxation was used to determine PEMFC electrochemical properties. A screening measurement technique was applied using the analysis of variance (ANOVA) to elucidate the effect of the different operating conditions on the obtained parameter, hence to validate the developed algorithm. The results show that the operating conditions of the step-down regulator device affect some parameters, but the weak correlations predict that the method can be used in real applications.
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Acknowledgement
Financial support by the National Office of Research and Technology (OMFB-00356/2007) and the Hungarian Scientific Research Fund (OTKA T031762)(G.I.) are acknowledged. Dr. Branko Popov and the University of South Carolina (USA, SC) are greatfully acknowledged for providing an opportunity for experimental work. Dr. Jong-Won Lee and Seh Kyu Park are also acknowledged for their practical advice.
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Appendix A: Factorial ANOVA
Appendix A: Factorial ANOVA
The analysis of variance, or briefly ANOVA, refers broadly to a collection of experimental situation and statistical procedures for the analysis of quantitative responses from experimental units. It involves the analysis either of data sampled from more than two numerical distributions (populations) or of data from experiments in which more than two treatments have been used. The characteristics that differentiate the treatments one from another are called factors, and the different treatments are referred to as levels of the factors. The evaluation of ANOVA is based on the following statistical definitions:
Confidence interval and confidence level is used to calculate and report an entire interval of plausible values of an event A. A confidence level of 95% implies that 95% of all samples are in an interval that includes any parameters being estimated and only 5% of all samples can be found outside (Fig. A.1). The higher the confidence level, the goodness of the estimation is obviously better.
Hypothesis testing is a method for deciding which of two (or more) contradictory claims about the parameter is correct. The main question to be answered is that the different populations are originated from the same random variable or not.
The hypothesis test procedure is specified by the test statistics and the confidence level. The test statistics is dependent on a parameter estimated (true average, deviation), the sample size and the preliminary known parameters, respectively. If the distribution is normal and the standard deviation (σ) is known, the test statistic is:
where \(\bar{\hbox{X}}\) is the average values of the samples, μ is the mean value and n is the number of the selected samples. Z is the probability of the deviation of the calculated average (\( {\text{\ifmmode\expandafter\bar\else\expandafter\=\fi{X}}} \)) and its expected value (μ).
Test statistics for comparing two population variances is based on the F-distribution, that is
where \( \ifmmode\expandafter\bar\else\expandafter\=\fi{\sigma }^{2}_{{{\text{A,B}}}} = {\sum\limits_{i = 1}^{{\text{P,Q}}} {\frac{{{\left( {x_{i} - \ifmmode\expandafter\bar\else\expandafter\=\fi{x}_{{{\text{A,B}}}} } \right)}}} {{\sigma _{{{\text{A,B}}}} }}} } \), and P, Q are the number of populations A and B, respectively, and σ is the standard deviation of the populations.
Mean squares: The samples standard deviations will generally differ somewhat even when the corresponding σ values are identical. The analysis of the structural effects of the different factors can be done to compare the between-samples variation with the within-samples variation (Fig. A.2). This can be done efficiently by using mean squares instead.
where K is the number of the population and L is the number of the measurement (observation) in each population. SST is the total sum of squares, SSTr is the treatment sum of squares (between-samples) and SSE is the error sum of squares (within-samples). SST = SSE + SSTr is always valid. If the factors have no effects, the values of the individual sample mean values should be close to one another and therefore close to the grand mean, resulting in a relatively small value of SSTr. However, if the μI values are quite different, because of the effects of the factors, the average of the populations should substantially differ from the average of all samples. The probability of the deviation follows the F-distribution and can be estimated by
Using the confidence level, the probability of the difference can be qualitatively estimated. If the ratio \( \frac{{{\text{SSTr}} \cdot {\text{K(L}} - 1)}} {{{\text{MSE}} \cdot {\text{(L}} - 1)}} \) is higher than F value at 95% (or else) confidence level, the parameter is affected by the factor used. Otherwise the factor has no significant effect on the parameter investigated.
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Kriston, Á., Inzelt, G. Estimation of the characteristic parameters of proton exchange membrane fuel cells under normal operating conditions. J Appl Electrochem 38, 415–424 (2008). https://doi.org/10.1007/s10800-007-9454-6
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DOI: https://doi.org/10.1007/s10800-007-9454-6