A robust data treatment approach for fuel cells system analysis


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This paper describes the implementation of a practical approach for fuel cells system data analysis. A number of data treatment techniques such as data management and treatment, data synchronization, and data reconciliation are introduced and discussed in order to solve the issues raised in the practical case. These techniques are integrated in a software environment which provides user a fast, efficient, and rational electrochemical investigation. The performance of the approach is illustrated using an industrial fuel cell stack test system.

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A robust data treatment approach for fuel cells system analysis

  1. 1. ISA Transactions 51 (2012) 841–847 Contents lists available at SciVerse ScienceDirect ISA Transactions journal homepage: www.elsevier.com/locate/isatrans A robust data treatment approach for fuel cells system analysis D. Wang n, Y. Zhen Institute of Chemical & Engineering Sciences, Agency for Science, Technology and Research, 1 Pesek Road, Jurong Island, Singapore 627833, Singapore a r t i c l e i n f o abstract Article history: Received 15 June 2011 Received in revised form 14 March 2012 Accepted 23 May 2012 Available online 20 June 2012 This paper describes the implementation of a practical approach for fuel cells system data analysis. A number of data treatment techniques such as data management and treatment, data synchronization, and data reconciliation are introduced and discussed in order to solve the issues raised in the practical case. These techniques are integrated in a software environment which provides user a fast, efficient, and rational electrochemical investigation. The performance of the approach is illustrated using an industrial fuel cell stack test system. & 2012 ISA. Published by Elsevier Ltd. All rights reserved. Keywords: Fuel cells testing Electrochemical analysis Data handling Data synchronization Data reconciliation 1. Introduction A fuel cells stack is a collection of individual cells electrically connected in series. In an ideal fuel cells stack, every cell would be subjected to identical operation conditions and the overall stack performance would be obtained as the sum of the identical output of individual cells. However, due to manufacturing variability of components, stack configuration, and degradation with use, the individual cells in a real stack will typically show some variation in performance and so will the stack. Hence, in addition to overall system performance, individual cell performance needs to be detected in a fuel cells system. Several cell voltage monitoring systems have been developed for the fuel cells systems and battery systems [1,8], where the focus is given either on the design of cell voltage monitoring system, or the investigation of cell performance by means of the voltage monitoring. A cell voltage monitoring system has been designed in our research group for a fuel cells system measurement. A tapping method is adopted to establish electrical connection to selected individual cells in a fuel cell stack to explore the coupling between adjacent cells. Knowledge of individual cells and power loss related to the interconnect between adjacent cells (i.e., bipolar plate) is thus obtained in combination with electrochemical performance of the overall fuel cell stack. The analysis of data that are measured by the voltage monitoring system takes an important role in obtaining the useful information of a fuel cells system. Polarization curve is generally n Corresponding author. Tel.: þ65 67963959; fax: þ 65 63166185. E-mail address: david_wang@ices.a-star.edu.sg (D. Wang). characterized for a fuel cell stack as it reflects the various sources of polarization (reduction of voltage from the thermodynamically reversible level) in a fuel cell. The polarization behavior for each individual cell provides important clues about the system’s performance and efficiency. Even though the voltage monitoring system has been investigated by fuel cell designers, the data analysis in the fuel cells stack testing still remains a challenge. There are some obstacles that hinder one to obtain the required information easily and appropriately. One of the problems is with the ease of data handling. There is a large amount of data collected from several measuring devices in the voltage monitoring system, the data are usually stored in several different files (which can be accessed with spreadsheet software, for e.g. MS Excel). The data have to be integrated into a statistical software environment (e.g. Origin) in order to undertake electrochemical calculation, plotting and analysis. Before a further electrochemical analysis, the raw data have to be manually examined, synchronised and then transferred from Excel to Origin. This will be a tedious work and timeconsuming, especially when all the procedures have to be repeated for the new data in the cycle of experiment. Another challenge is that electrochemical analysis of the data that are available after the above processing may not produce reasonable results when comparing the performance between overall fuel cells stack and individual cells. There are two critical phenomena that support this argument. It is found that an event (e.g. voltage change) happened in the experiment is usually recorded at different time instant and with different logging points by different testing devices (names of which are mentioned in the following section). It is difficult to synchronize the data so that the discrepancy in time presents a random fashion 0019-0578/$ - see front matter & 2012 ISA. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.isatra.2012.05.005
  2. 2. 842 D. Wang, Y. Zhen / ISA Transactions 51 (2012) 841–847 Nomenclature Q,C time series m, n number of points in time series qi,ci ith elements/points of time series d(qi,cj) distance d(qi,cj) between the two points qi and cj P a warping path pk ¼ (i,j)k kth element of warping path DTW(Q,C) warping cost for two time series Q,C g(i,j) the cumulative distance in dynamic programming x estimation (not a regular fashion—for that case a constant time shift of data could be enough) as shown in Fig. 1, where two data profiles are horizontally compared. Such a time discrepancy complicates the analysis and (if not resolved) makes the reasonable analysis difficult. Another critical phenomenon that has been found in the investigation of raw data is that the data recorded do not appear sound in term of the first principles. The relationship between the voltage of the whole stack and the voltages of individual cell should satisfy some kind of conservative law, e.g., overall voltage equals to sum of individuals as one would expect in an ideal fuel cell stack. However, this constraint is usually not satisfied as shown in Fig. 1, where two data profiles are compared vertically. Also, there is no consistency of the discrepancy but it appears random. Apparently, directly applying electrochemical investigation methods to the raw data, even in a user-friendly handling environment, will hardly give reasonable results. In this paper, a robust and fast data treatment approach is developed in order to practically process experimental data obtained from the voltage monitoring system and to analyze the complicated fuel cell performance. The approach includes data automation, data synchronization and data reconciliation. Firstly, the focus is given on the alleviation of tedium of data handling cross two software environments. To do this, an 32 Measurement of whole stack Sum of individual cells A shift of Sum of individual cells Design point Start/End by data logger 1 30 Voltage (V) 28 26 24 22 20 18 16 0 200 400 600 800 1000 1200 1400 ^ x estimation y measurement W ¼ diagða2 , a2 ,. . .a2 ,Þ weight matrix with diagonal elements n 1 2 a2 ,. . .a2 n 1 ai standard deviation of ith measurement noise DAE differential and algebraic equation A coefficient matrix of liner equation Voutput voltage measurement of whole stack V cell voltage measurement of ith individual cell i V int voltage measurement of jth individual interconnect j V sic voltage measurement of kth secondary interconnect k interface for data analysis is developed in an Excel file. This interface includes the functionality with which one cannot only configure and process the data, but can also take electrochemical investigation. A computing engine (e.g. MATLAB) is connected with Excel spreadsheet at background where various algorithms of data processing and analysis run. It will not affect usage even though one has little knowledge of MATLAB. This data automation scheme saves much effort and improves efficiency for data handling in fuel cell testing, as it is much convenient for one to undertake analysis in a single software environment where all the data from different testing devices can be directly accessed and analyzed. In order to get the best estimate of data for electrochemical investigation, the raw data need to be further processed. It is desirable to identify the similarity among all the data sequences recorded in the individual testing equipments, and then align the data in term of a same time instant. (Fig. 1 shows a comparison of voltage changes measured by different testing devices. It can be found that the two data sequences have the approximately same overall component shapes, but these shapes do not line-up in X-axis (time). The steps (ramping) do not happen at same time, nor do they follow a consistent fashion.) Dynamic time warping (DTW) can be used to achieve a better alignment, as it is a method that allows a computer to find an optimal match between two given sequences (e.g. time series) with certain restrictions. It has been successfully used in speech recognition, gesture recognition, alignment of batch profiles and medicine [5,2,3,6,10]. Its application to fuel cell system data analysis will be a novel approach. Another concern is to estimate the ‘‘real’’ values of the measurements from all the devices subject to the first principles. The vertical discrepancy between the two sets of data as shown in Fig. 1 is hardly acceptable and less meaningful in terms of the first principle; at least, the law of conservation should not be breached. Data reconciliation is a procedure of optimally adjusting measured data so that the adjusted values obey the conservation law and other constraints. Although data reconciliation has been applied to chemical processes for process optimization, monitoring and control [4,7], its application to fuel cells system testing has not been reported. 2. Experimental platform Sample time Fig. 1. The fuel cells stack voltages obtained by combining data from several different measuring devices during polarization curve scanning. The dash line represents the whole stack voltages measured by one device; the solid line represents the sum of individual cell voltages that is obtained by combining data from the other two measuring devices; the solid line with cross marker represents the shift of the solid line. The circle sign represents the specified point recorded by data logger file. This figure gives an example of the problems that are to be focused in the work. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.) The fuel cell stack tested in this study contained 30 cells connected in series. The test platform for electrochemical performance measurement of the fuel cell stack consists of a customized 500 W fuel cell test station and a TDI MCL488 electronic load. For individual cell voltage monitoring, conductive wires are connected to the anode or cathode of selected adjacent cells of the stack, which enables the voltage measurement of individual cell and power loss related to the interconnect. The cells’ and
  3. 3. D. Wang, Y. Zhen / ISA Transactions 51 (2012) 841–847 843 Fig. 2. Layout of multi-tapped fuel cell and the sections where voltages are measured. Around 30 individual fuel cells are connected in series. Voltages generated across each cell or some adjacent cells are measured. Voltages over the interconnects are measured. The overall output voltage is measured. These data are measured by three devices and stored in three data files respectively. interconnects’ voltages are recorded by two multi-channel data loggers (Giant, UK), which are controlled by two separate computers, respectively. The schematic diagram of the fuel cell stack testing system in this study is shown in Fig. 2. The overall fuel cell stack performance is generally conducted in galvanostatic mode and the testing current is reached with the use of the electronic load system. The polarization curve is performed from open circuit to 2.0 A with increment of 0.1 A/step and holding time of 40 s/step. The inlet hydrogen gas with 3% relative humidity flow rate is maintained at 4 l/min and air flow is maintained at 100 l/ min regardless of current density. The typical polarization curve of the overall fuel cell stack is presented as red dotted line in Fig. 1. For easiness of data handling, the voltage data at current of 0.9 A will be focused on in this study, where the corresponding cell voltage is among 0.7–0.8 V, which is the typical working potential of fuel cells. Fig. 3. Integration between MS Excel and MATLAB. User can make options in Excel interface to let MATLAB to undertake various data treatment, calculation and plotting. 3. Interface for data automation and manipulation In fuel cell stack testing, it is desirable that the measurements could be easily accessed, pre-processed and manipulated, and the important parameters could be obtained and displayed promptly whenever they are needed for investigation. Given that there are three data files generated by three different devices within each run in the experiment, manually handling these data across MS Excel and Origin will exert much burden. In addition, much effort needs to be put on the treatment of these raw data. Most often, the data have to be pre-processed, aligned and adjusted in order to make them right for use. These procedures are either tedious or hardly completed using manual manipulation, needless to say that they have to be repeated for the new data set in analysis. This definitely needs a user friendly environment for an easy and fast data handling. Since the data files generated by the devices can be accessed using MS Excel spreadsheet, it would be more convenient to manage, to analysis and to display the data in this environment. Considering the various approaches for data treatment as well as electrochemical computation that are desired, an interface is developed in Excel for various data handling, treatment, and electrochemical computation. This interface connects with MATLAB at background as computational and graphics engine and it brings the results to display. Spreadsheet Link EX connects Excel spreadsheet software with the MATLAB workspace, enabling one to access the MATLAB environment from an Excel spreadsheet [9]. With Spreadsheet Link EX software, one can exchange data between MATLAB and Excel, taking advantage of the familiar Excel interface without leaving the Excel environment while accessing the computational functionality, graphic interface and visualization capabilities of MATLAB (Fig. 3). Fig. 4. Main tasks listed in Excel interface for data treatment and analysis. It displays on spreadsheet with user information for configuration, which allows user to make options and display the results. Given the environment, a standard data processing procedure can be listed in the spreadsheet with brief information on operation for a user. Correspondingly, at each step of procedure the programs are edited that aims to undertake a specific task. User can make options to configure a need in the procedure. Fig. 4 displays an example of spreadsheet interface that includes several
  4. 4. 844 D. Wang, Y. Zhen / ISA Transactions 51 (2012) 841–847 main tasks, where user can work on each cell (denoted by 0) for a specific operation. 32 Measurement of whole stack Consolidation of indiv. cells, and interconnects Design point from whole stack measurement Design point from consolidation 31 30 4. Data pre-processing and synchronization 29 28 27 26 25 24 23 0 50 100 150 200 250 300 Sample time 350 450 500 26 25.8 25.6 25.4 25.2 25 Measurement of whole stack Condolidation of indiv. cells, and interconnects Design point from whole stack measurement Design point from consolidation 24.8 340 350 360 370 380 390 400 410 420 Sample time Fig. 6. A zoom-in of two data sequences in a sample range. To align two sequences using DTW one constructs a m  n matrix where the (ith, jth) element of the matrix contains the distance d(qi,cj) between the two points qi and cj (typically the Euclidean distance is used, i.e., d(qi,cj)¼(qi Àcj)2), each matrix element (i,j) corresponds to the alignment between the points qi and cj. A warping path P is a contiguous set of matrix elements that defines a mapping between Q and C. The kth element of P is defined as pk ¼(i,j)k so one has P ¼ p1 ,p2 ,. . .pk ,. . .,pK maxðm,nÞ r K om þ nÀ1 4.2. Data synchronization 400 Fig. 5. Two data profiles after the pre-treatment. Voltage (V) The quality of data is very important in fuel cell system electrochemical investigation. Suitable pre-treatment of raw data is sometimes crucial for the analysis result. In this study the testing data offer a unique challenge not only in terms of data handling, but also in terms of data quality. Missing data points are not uncommon in measuring device, which make a data file incomplete and its samples wrongly associated with the ones recorded in other data files. There are outliers in the data due to equipment malfunction, and there are no numerical data in the records due to equipment calibration. In addition, it is necessary that the samples from different files need to be aligned to describe the phenomena (e.g. voltage change) that happen at same time instant. These data quality issues need to be carefully addressed prior to further electrochemical analysis. Generally, data are checked visually at first. Anything that appears suspicious from electrochemical point of view is double-checked and carefully considered to determine what treatment is needed. For example, there is a missing record every 10 s in one data file. These missing data need to be reconstructed using interpolation approach, which can be realized by making a choice on the interface and running functions in the background. The interpolation can also be used to reconstruct the missing data due to device calibration. Outliers, which can be simply regarded as the data points that are not consistent with the bulk of data, are common in experiment data set. In the univariate approach, outliers are detected based on visualization together with numerical comparison with adjacent data. The suspicious outliers can be replaced with median values of the adjacent data in the approach. Another important pre-treatment is to align the data from three files to describe the phenomena that happen at same time. Without such treatment, the data could not be used for electrochemical investigation with confidence. It is observed in Fig. 1 that the plot of raw data clearly misrepresent the real phenomena, because the step changes do not happen at same time instant. In order to alleviate this, the variable measurement that is most sensitive to such step change and trustworthy is selected as the benchmark for such alignment based on a prior knowledge. The samples of other variable measurements are shifted accordingly to align with this data set. The plot of data after above pre-treatment is shown in Fig. 5as an example. The two profiles are approximately coinciding in general, but still a discrepancy exists. A closer look at current of 0.9 A as shown in Fig. 6 demonstrates the discrepancy in both X-axis (sample) and Y-axis (value) although the above treatment is implemented. Voltage (V) 4.1. Data pre-processing ð2Þ The warping path is typically subject to several constraints: A better alignment in sample time can be achieved by using dynamic time warping (DTW) technique, which is used to find the similarity between time series that have approximately the same overall component shapes but these shapes do not line up in Xaxis. The following is a brief description on DTW algorithm. For two time series Q and C, of length m and n respectively, where: Q ¼ q1 ,q2 ,. . .qi ,. . .,qm C ¼ c1 ,c2 ,. . .ci ,. . .,cn ð1Þ 1. Boundary conditions: w1 ¼(1,1) and pK ¼(m,n). This required the warping path to start and finish in diagonally opposite corner cells of the matrix 2. Continuity: given pk ¼(i,j) then pk À 1 ¼(i’,j’) where iÀ i’ r1 and j Àj’r1. This restricts the allowable steps in the warping path to adjacent cells. 3. Monotonicity: given the pk ¼(i,j) then pk À 1 ¼ (i0 ,j0 ) where i À i’Z0 and j À j0 Z0. This forces the points in P to be monotonically spaced in time.
  5. 5. D. Wang, Y. Zhen / ISA Transactions 51 (2012) 841–847 There are many warping paths that satisfy the above conditions; however, one is interested in the path which minimizes the warping cost: vffiffiffiffiffiffiffiffiffiffiffiffiffiffi u K uX ð3Þ pk =K DTWðQ ,CÞ ¼ mint and the warped data are aligned in X-axis. This indicates that the data from different measuring devices are being synchronized to describe the phenomena that happen at the same sample time. The warping path and matrix of distance are depicted in Fig. 8, where the distances between the data, the warping path, and two data sequences are presented. It is worthwhile to note that DTW may increase the number of the warped data comparing with the original ones, as DTW tries to map the similarity between the two sequences so that one point in a sequence may map with several points in the other. Fig. 9 overlays the warped data on the original ones. It can be seen that there are four more data point in both warped data sets. These added data do not affect the investigation and can be ignored if the data section at specified point is focused. It is also noticed that the warping usually happen where there is a step change. This advises that in the electrochemical calculation that follows, it would be better to select the data that is located in the middle section of the steps (e.g. the first and the last three samples of the data in a step can be ignored). k¼1 This path can be found very efficiently using dynamic programming to evaluate the following recurrence which defines the cumulative distance g(i,j) as the distance d(i,j) found in the current cell and the minimum of the cumulative distance of the adjacent elements: gði,jÞ ¼ dðqi ,cj Þ þ mingðiÀ1,jÀ1Þ, gðiÀ1,jÞ, gði,jÀ1Þ 845 ð4Þ Given the data section in Fig. 6, applying DTW to the data gives the warped data as shown in Fig. 7, where the original is shown in the left part of the figure, the warped data is shown in the right part. It can be seen that the pessimistic dissimilarity is eliminated, Original data Warped data 26 Output V Calculated V 25.8 25.8 25.6 25.6 Voltage (V) Voltage (V) Output V Claculated V 25.4 25.4 25.2 25.2 25 25 24.8 24.8 20 40 60 80 20 40 60 80 Samples Samples Fig. 7. Comparison of original data and the warped data. 0 74 10 63 20 52 42 40 50 31 60 Distance Samples 30 21 70 10 80 10 20 30 40 50 60 70 0 V 28 26 24 24 26 28 V 80 Samples Fig. 8. Distance matrix of DTW, where the two data sequences are shown in the left and underneath. The rightmost bar indicates the distance value. The diagonal curve is the warping path that maps the data in the two sequences.
  6. 6. 846 D. Wang, Y. Zhen / ISA Transactions 51 (2012) 841–847 26 26 Cal. V Out. V Cal. V Out 25.6 Out. V Voltage (V) 25.8 25.6 Voltage (V) 25.8 Cal Out Cal 25.4 25.2 Cal. V 25.4 25.2 25 25 24.8 24.8 24.6 24.6 0 10 20 30 40 50 60 70 80 90 10 20 30 40 50 60 70 80 90 Sample Sample Fig. 9. The two original data sequences and the two warped data sequences are superimposed in one figure. The warping happens at the step changes. More data points are included in warped data. 0 Fig. 10. Data reconciliation of the measurement data. The pentagrams represent the reconciled data for both data sequences that are adjusted to be equal in quantity. 1 5. Data reconciliation x s:t: DAE model and inequality constraints ð5Þ where e¼yÀ x, f(e) is the probability density function of the error. If the sensor errors are assumed to follow Normal distributions, the objective function Àlog f ðeÞ is quadratic and the conventional weighted least squares formulation is obtained, min ðyÀxÞT W À1 ðyÀxÞ x s:t: ð6Þ 0.9 0.8 Voltage (V) DTW tries to explain the variability in Y-axis by warping the X-axis so that two data sequences can be aligned in terms of sample time. However, DTW could not make the data aligned in terms of quantity. As can be seen in Figs. 7 and 9, there is still discrepancy in value between the two data sequences before and after DTW applies, which conflicts the energy balance. It is necessary to eliminate the errors before the data is rationally used for electrochemical investigation. Data reconciliation is a procedure of optimally adjusting measured data so that the adjusted values obey the conservation laws and other constraints. The essence of data reconciliation is that: given the measurement vector y, one wants to estimate its state x, which satisfies the models from first principles. This problem can be approached with the laws of probability and maximum likelihood principle by minimizing the negative logarithm of the probability function of the difference between measurement y and estimation x, subject to the constraints [11], _ ¼ argmin½Àlogf ðeފ x 0.7 0.6 0.5 cell voltage using the approach cell voltage without using the approach 0.4 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Cell position Fig. 11. Typical graph for cell voltage vs. cell position of fuel cell stack. It shows that the voltages obtained without using the approach are out of the reasonable range (0.7–0.8 V), while the voltages obtained using the approach are within the reasonable range. The values appear the same among the adjacent cells are the averages since the voltages across the adjacent cells are measured. where Voutput is the stack voltage measurement, V cell is individual i cell voltage measurement, V int is the measurement for individual j interconnect, V sic is the measurement for secondary interconnect. k Re-arranging the constraint equation and choosing a set of suitable standard deviations for the measurement, the estimated data can be obtained as shown in Fig. 10, where the reconciled data for both profiles are coincided. DAE model and inequality constraints where W ¼ diagða2 , a2 ,. . .a2 ,Þ and ai is the measurement noise n 1 2 standard deviation. Furthermore, for linear equality constraints Ax ¼0 and in the absence of inequality constraints, this problem has a closed-form solution via Lagrange-multipliers, ^ x ¼ yÀWAT ðAWAT ÞÀ1 Ay ð7Þ In fuel cell stack testing problem, given the measurement data from three devices, one wants to estimate their ‘‘real’’ values subject to the following energy balance: V output ¼ 30 X i¼1 V cell À i 29 X j¼1 V int À j 2 X k¼1 V sic k ð8Þ 6. Fuel cell system performance analysis Given the data being processed, the electrochemical calculations can be implemented within the interface so that various plots can be provided for visualization and analysis. Fig. 11 gives an example of plot of some parameters. From the individual cell I–V–P curves data, take the average of the voltage values for each individual cell obtained when the current is 0.9 A. The voltage values obtained at this current value are the voltages at specified point current. The voltage values are tabled against cell positions on tube. For the cells that are connected and measured as a whole, an average will be taken for each cell. Plot the voltage values
  7. 7. D. Wang, Y. Zhen / ISA Transactions 51 (2012) 841–847 versus the cell positions on tube, and the voltage distribution at the specified point current (0.9 A) against cell position on tube will be obtained. Fig. 11 shows the voltage distribution generated by each cell at current of 0.9 A (denoted by the stars). A comparison is given by superposing the data that were obtained initially when the proposed treatment was not applied (denoted by circles). It can be seen that the previous voltages are higher than that with the treatment, and they are out of the reasonable range (0.7–0.8 V). Many other data analysis and plots can be produced with the developed approach, but they will not be displayed for the brevity. With this user-friendly interface and proposed data treatment, one can undertake a fast, efficient, and rational electrochemical investigation of fuel cell stack testing. 7. Conclusions Data handling and processing are ubiquitously topics in virtually every scientific discipline and industrial practice. While the basic techniques are by now well known, their sensible selection, integration and deployment in a real problem still remains a challenge. Before undertaking electrochemical analysis in industrial setting, there is always a need to process the raw data in order to make the analysis either efficient or rational, or both. The issues to be considered in this practical case also imply that there is no single technique that can solve the practical data analysis problem. It needs a number of techniques, well selected and seamlessly integrated into a software tool. Even though the techniques being employed in the approach are not originally created in this work, they are judiciously selected and they are firstly applied and customized in this practical case to solve the issues. Even though this work presents a solution, there still is a room to improve. The interface needs to be upgraded to facilitate its 847 usage, the precision in synchronization and reconciliation needs to be improved. For example, the non-measured interconnects are not considered in data reconciliation, which may have an impact on the analysis. This could be eliminated either by an estimation of these interconnects, or a re-design of the measurement layout. In addition, suggestion can be given on the current system architecture that a single computer connecting the data loggers would minimize the synchronization problem. The authors intend to pursue these concerns in the future. References [1] Brunner D, Prasad AK, Advani SG, Peticolas BW. A robust cell voltage monitoring system for analysis and diagnosis of fuel cell or battery systems. Journal of Power Sources 2010;195(24):8006–12. [2] Godin C, Lockwood P. DTW schemes for continuous speech recognition: a unified view. Computers Speech and Language 1989;3:169–98. [3] Kassidas A, MacGregor John F, Paul ATaylor. Synchronization of batch trajectories using dynamic time warping. AIChE Journal 1998;44(4):864–75. ´ [4] Prata Diego Martinez, Schwaaba Marcio, Luis Lima Enrique, Carlos Pinto Jose. Nonlinear dynamic data reconciliation and parameter estimation through particle swarm optimization: application for an industrial polypropylene reactor. Chemical Engineering Science 2009;64(18):3953–67. [5] Rabiner LR, Juang B. Fundamentals of speech recognition. Prentice-Hall, Inc.; 1993. [6] Ramaker Henk-Jan, van Sprang Eric NM, Westerhuis Johan A, Smilde Age K. Dynamic time warping of spectroscopic BATCH data. Analytica Chimica Acta 2003;498:133–53. [7] Ramamurthi Y, Sistu PB, Bequette BW. Control-relevant dynamic data reconciliation and parameter estimation. Computers & Chemical Engineering 1993;17(1):41–59. [8] Santis M, Freunberger SA, Papra M, Wokaun A, Buchi FN. Experimental investigation of coupling phenomena in polymer electrolyte fuel cell stacks. Journal of Power Sources 2006;161:1076–83. [9] Spreadsheet Link EX, /http://www.mathworks.com/products/excellink/S. 2010. [10] Tormene P, Giorgino T, Quaglini S, Stefanelli M. Matching incomplete time series with dynamic time warping: an algorithm and an application to poststroke rehabilitation. Artificial Intelligence in Medicine 2009;45(1):11–34. [11] Wang D, Romagnoli JA. A framework for robust data reconciliation based on a generalized objective function. Industrial & Engineering Chemistry Research 2003;42:3075–84.