Surrogate modelling of compressor characteristics for fuel-cell applications

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APPLIED ENERGY Applied Energy 85 (2008) 394–403 www.elsevier.com/locate/apenergy

Surrogate modelling of compressor characteristics for fuel-cell applications R. Tirnovan, S. Giurgea *, A. Miraoui *, M. Cirrincione L2ES – UTBM Laboratory, University of Technology of Belfort-Montbe´liard (UTBM), Thierry Mieg, 90010 Belfort Cedex, France Accepted 14 July 2007 Available online 26 November 2007

Abstract The compressor is an important auxiliary for fuel-cell (FC) operation. Growing fuel-cell system efficiency involves an optimal fuel cell energy management and the air management is a key issue. Thus, a good modelling for static and dynamic operation of all components of the FC system, and in particular of the compressor, is required. The difficulties, due to a lack of information about the performance of compressors, demand predictive and modern approximation methods to be used for compressor modelling. To overcome these issues, the paper proposes and presents a moving least squares (MLS) algorithm for obtaining a surrogate model of the centrifugal compressor. The experimental data provided by manufacturers are used for this task. The results can be used for the development of an off-design model or the overall dynamic simulation of the behaviour of a FC system.  2007 Elsevier Ltd. All rights reserved. Keywords: Compressor; Characteristic map; Moving least-squares; Surrogate model; Fuel-cell

1. Introduction Fuel cells (FCs) could be a great alternative for replacing traditional distributed-power sources for transportation applications (cars, buses, trains, ships, etc.), stationary applications (residential supply, hospitals supply, reserve electric-supply sources, electrical energy supply), or portable applications (technology for mobile phone operators, etc.) [1,2]. They are electrochemical power generators, which utilize hydrogen or hydrogen-rich fuels, as a combustible reactant, and oxygen (air) as a combustive reactant, to produce electricity without combustion, providing high efficiency and unmatched environmental benefits. Their electrical efficiency ranges between 40% and 49%, in comparison with competing technologies, which display electrical efficiencies about 30–35%. In the cogeneration mode, fuel-cell systems can achieve efficiencies close to of 85% [1,2].

*

Corresponding authors. Tel.: +33 384 58 36 40; fax: +33 384 58 34 13. E-mail addresses: [email protected] (S. Giurgea), [email protected] (A. Miraoui).

0306-2619/$ - see front matter  2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.apenergy.2007.07.003

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Nomenclature T absolute temperature (K) N mechanical spool speed (RPM) p total pressure (atm) w mass flow (kg s1) T1 inlet compressor’s, absolute temperature (K) p1 inlet compressor’s total pressure (atm) T2 exit compressor’s absolute temperature (K) p2 exit compressor’s total pressure (atm) P pressure ratio, P = p2/p1 g efficiency pffiffiffiffiffiffi Nc corrected spool speed, N c ¼ Np = ffiffiffiffiffiTffi 1 corrected mass flow, wc ¼ w  T 1 =p1 wc Pref reference pressure ratio gref reference efficiency Nc,ref reference corrected spool speed wc,ref reference corrected mass flow Prel relative pressure-ratio Prel = P/Pref grel relative efficiency grel = grel/gref Prel relative pressure ratio Prel ¼ ðP  1Þ=ðPref  1Þ Nc,rel relative corrected spool speed, Nc,rel = Nc/Nc,ref wc,rel relative corrected mass flow, wc,rel = wc/wc,ref e rel approximated relative pressure ratio P ~grel approximated relative efficiency wc,rel,trans transformed relative corrected mass flow

A fuel cell power generating system is composed of a fuel cell stack associated with the auxiliaries, which must ensure essential functions: the transport of the reactants, evacuation of the secondary products, control of the temperature of the stack, etc. These auxiliaries are generally electrical power consumers, and must be considered as parasite charges, which directly influence the electrical energy delivered by the stack. Thus, a reduction of the net electrical power, delivered to the consumer, is the consequence of these parasite charges. The compression subsystem is essential for FC operation and it consists mainly of either a fan, a blower or a compressor. Fans and blowers can supply large air mass flows but at a weak pressure ratio (below 2), while compressors are the more efficient solutions for FC air supplies at increasing electric power density. The application, either stationary or automotive, can require the use of volumetric or centrifugal compressors. Thus, the modelling and simulation of these devices become necessary [3–9]. The centrifugal (turbo) compressor is a device, which operates at a very high speed, around 100 kRPM. It ensures an air pressure between 1 and 5 atm. A compressor is more expensive than a blower, but it allows pressure levels that permit a high-density power operation of the fuel cell (i.e. increasing the power/volume ratio). However it can ensure a continuous FC air supply, without oil pollution. Its non-linear characteristic, and its speed mass flow control, make the turbo compressor ideal for proton exchange membrane fuel cell (PEMFC) applications. The turbo compressor, however, has a poor turn-down characteristics at low flows. Moreover they have the qualities that will make them useful in fuel cell automotive applications [3,7,9]. Thus, the choice of an optimal fuel cell system architecture requires the analysis of the application and of all the fuel cell subsystems. To obtain a good performance, the modelling and the simulation of all components and subsystems, for the complete fuel cell system, are recommended. This paper presents a new analysis and prediction method for the compressor’s characteristic map based on the method of moving least squares. The results and the method can be used both in dynamic and in stationary simulations of fuel cell systems.

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2. Compressor mapping [3,9–12] 2.1. Characteristic curves For a turbo compressor, these are defined by: • The compression ratio P versus the mass flow w and speed N, or versus the corrected mass flow wc and corrected speed Nc; • The relative compression ratio Prel versus relative corrected mass flow wc,rel and relative corrected speed Nc,rel; • The relative compression ratio Prel versus relative corrected mass flow wc,rel and relative corrected speed Nc,rel. This approach makes the comparison between different compressors maps easier [11]; • The efficiency of the compressor g as a function of the mass flow w and the speed N, or versus the corrected mass flow wc and the corrected speed Nc, or versus the relative corrected mass flow wc,rel and the relative corrected speed Nc,rel. The main sources of information are the manufacturer’s sales brochures and some data from published journals. These curves are given for a precise gas, for an inlet pressure and for an temperature of the aspirated gas. The examination of these characteristic curves suggests two observations: • The surge line marks a limitation in compressor operation. If an operating point is chosen above the surge line, then it goes into surge, which is an unstable state with pressure oscillations, with possible damage of the machine and the fuel cell [3,10]; • If the rotation speed in a given compressor increases, the speed of the fluid increases; the curve of the flow inflects and passes a maximum. This means that there is sonic blocking or that the limiting smothering flow is reached [10]. 2.2. Short overview of compressor map fitting The above observations, whereby only some data about operating compressor points are available, suggest that most of the information should be predicted. The use of these data entails to take into account the following issues [3,9–12]: • The interpolation and extrapolation of data near certain stable states where the corrected speed is given; • The lack of the characteristic parameters of the compressor under the processes of start-up and shutdown; • The surge limit of the compressor must be considered. Some remarks about the modelling process of the compressor should be made. Few methods are described in the literature, in particular: • The general method employed is a two-dimensional linear interpolation [3,13]. This method is simple and it allows a rapid computation, but its precision is low and the results can be poor. Because of the non-linear compressor performance and the small operating domain, the interpolation method has only a limited use; • The use of polynomial curve fitting, based on third order polynomial approximation [14]; • The introducing of auxiliary coordinates (line b), having no physical meaning, superimposed on the characteristic curves [11]; • Other approaches combine theoretical implicit relations with measured data by using least square methods [9,15]; • A generalized compressor map, developed by Saravanamuttoo and MacIssac [16], has been used by Zhu and Saravanamuttoo [17]; • A neural-network based method for the analysis and prediction of the characteristic performance map of a compressor is developed in [12] etc.

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Fig. 1. Manufacturer’s experimental data with relative corrected rotation speed parameter [3]: (a) relative pressure ratio function of relative corrected mass flow and (b) efficiency versus relative corrected mass flow.

2.3. The proposed compressor modelling approach This paper proposes and develops a numerical method for modelling compressors based on moving least squares (MLS) and starts from the experimental data set presented in [3,9]. Fig. 1 shows an experimental characteristic map of a turbo compressor developed for FC applications. The proposed model belongs to the family of surrogate models. The use of MLS is justified by the fact that it is a powerful method for approximating a function when few data are available, as in the case under study. 3. The moving least squares (MLS) approximation based model There are a large number of methods for function approximation [18] and regression models actually represent the most widely spread class of approximating and exploit basis functions to fit data. Among these techniques, the one based on the polynomial regression, the moving least squares (MLS) method, provides both an accurate local function approximation and a continuous gradient approximation [19–22]. However other approximation techniques could be employed like those based on feed-forward neural-networks [23,24]. A short presentation of the MLS method is presented below. Considering the sampled dataset {(yi,xi)}i=1. . .N, where: • y = {yi}i=1. . .N = {y(xi)}i=1. . .N is the vector of function values of the experimental points; • xi 2 D is the vector of the coordinates of the i experimental points, with D the experimental domain containing N samples. According to MLS, the local character is ensured by the following weight function d(x, xi) defined for a support region B(xi) around xi:  P 0 8x 2 Bðxi Þ dðx; xi Þ ¼ dðx  xi Þ ð1Þ ¼ 0 if not with x the vector of the coordinates of a generic point. This weight function defines a finite domain of influence e around any experimental point xi, where D e is the approximation domain. Bðxi Þ  D,

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e let, ~y ðxÞ be the moving least squares approximation given by: For every point x of the domain D, m X ~y ðxÞ ¼ pj ðxÞaj ðxÞ

ð2Þ

j¼1

where pj(x) is the j-th basis (a monome), m (m < N) is the number of the bases, and aj(x) are the coefficients of each base. The m basis terms form the following m-dimensional vector: m

P T ðxÞ ¼ fpj ðxÞgj¼1 ;

ð3Þ

m
The vector a(x) = [a1(x), . . . am(x)] of the coefficients is obtained by solving a regression problem using the weighted least-squares error J(a(x)) for the N sampling points, defined as follows: " #2 N m X X dðx  xi Þ pj ðxi Þaj ðxÞ  y i ð4Þ J ðaðxÞÞ ¼ i¼1

j¼0

The minimization of the error J(x) with respect to the coefficients aj(x) gives: aðxÞ ¼ A1 ðxÞBðxÞy

ð5Þ

where the matrix A and B are defined by: AðxÞ ¼

N X

dðx  xi Þpðxi ÞpT ðxi Þ

ð6Þ

i¼1

BðxÞ ¼ ½dðx  x1 Þpðx1 Þ . . . dðx  xn Þpðxn Þ

ð7Þ

To guarantee the non-singularity of A, for a point x, an adequate support B(xi) for each sample point xi is needed so that d(x  xi) 5 0 for at least m experimental points [21]. The expression of the global approximation is ~y ðxÞ ¼ PT ðxÞA1 ðxÞBðxÞy

ð8Þ

4. Applications 4.1. Adapting the experimental domain for the MLS approximation The main goal of this work is to realize a compressor model that can be used efficiently within the global model of the fuel cell system. Moving least squares has been employed to approximate two continuous functions, which define the main compressor characteristics: e rel ðwc;rel ; N c;rel Þ approximates the value of the relative compression ratio Prel versus the relative corrected • P mass-flow wc,rel and the relative corrected speed, Nc,rel; • ~ grel ðwc;rel ; N c;rel Þ estimates the value of the relative compressor efficiency versus the relative corrected mass flow wc,rel and the relative corrected speed, Nc,rel; e defined by (wc,rel,Nc,rel) points, represents the Moreover, the two-dimensional approximation domain D, same domain for the two considered functions. Some relative corrected mass flow wc,rel points have been considered for 11 different operation speeds. The experimental domain D (Fig. 1) is a discrete one and is defined by the bi-dimensional points x(wc,rel, Nc,rel) 2 D. It is evident that the experimental domain is included in the approximation domain e To adapt experimental data with the MLS approximation, a support domain B(xi) is considered D  D. around each experimental point. Thus, an entity is introduced to associate each experimental point xi with a scalar r value by which the support domain can be defined it. This support can be constructed either by using a sphere S I ¼ fxjjx  xI j < rg

ð9Þ

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or a n-dimensional cube: S I ¼ fxjjxi  xIi j < r; 1 6 i 6 ng:

ð10Þ

To ensure the computational feasibility on an experimental domain D, each point of the approximation doe must be inside m compact supports built around the experimental points, where m represents the nummain D ber of bases. On the other hand, a too large support SI of the function d (x  xi) will corrupt the local character of the approximation, resulting in less precision of the approximation. Another problem is that each point of the domain of approximation must be included in the compact supports which should contain experimental points as uniformly distributed as possible along each coordinate to ensure both the convergence and the accuracy of the method [21]. Considering these criteria, the initial experimental domain has been transformed to obtain a new domain (Fig. 2a and b), where the distances between the experimental points are more uniformly distributed along the coordinates. Because the relative corrected speed, Nc,rel is normalized and uniformly distributed only the relative corrected mass flow wc,rel has been transformed. The transformation is given by the following linear relationship: n o w wc;rel arctg N c;rel  Min arctg N c;rel c;rel ðw e c;rel ;N c;rel Þ2 D n o n o wc;rel;trans ðwc;rel ; N c;rel Þ ¼ ð11Þ wc;rel w Max arctg N c;rel  Min arctg N c;rel c;rel e ðwc;rel ;N c;rel Þ2 D ðwc;rel ;N c;rel Þ2e D The expression of the transformation function has been chosen to have an almost rectangular transformed domain. It is a linear angular transformation, as highlighted in Fig. 2a, with wc;rel aðwc;rel ; N c;rel Þ ¼ arctg N c;rel In addition, this new domain allows spherical or cubic supports to be used, which are more efficient than polyhedral supports (Fig. 3). The approximation method has been applied in the new transformed domain. Then the inverse operation is made to return to the initial domain. 4.2. Results After the preparation of the experimental domain, the MLS method can be applied to approximate the characteristic curves of the compressor, using all the experimental points. Fig. 4 shows the approximated results for the 11 considered speeds.

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wc,rel,trans 0.55 0.5 0.45 0.4 0.35 0.3

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Fig. 3. The recovering domain with spherical compact supports-detail.

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wc,rel Fig. 4. Approximate compressor map for all experimental points.

To estimate the precision of the method, the average of the global error for all the experimental points is given. The expression of this error is, in percentage:    N e 100 X  P rel ðxi Þ  Prel ðxi Þ eg ¼    N i¼1  P rel ðxi Þ

ð12Þ

where xi = (wc,rel, Nc,rel) 2 D, and N the number of the experimental points. The computed value of this error is eg ¼ 1:2%. However, the true interest of the method is to predict and to describe the compressor behaviour in all of the points of the approximation domain. Since available data are few, part of the data cannot be kept aside for validation. Thus the procedure of ‘‘cross-validation’’ [25] can be used. The experiment set is split into S distinct [26] segments. Then the approximation is made using data from S-1 of the segments and its performance is tested using the remaining segment. Thus another experiment is made by excluding experimental points. Starting from this reduced experimental domain, the characteristic curve is reconstructed. This reconstruction includes the curve corresponding to the excluded points:

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wc,rel Fig. 5. Approximate map with the fifth curve points excluded.

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wc,rel Fig. 6. Approximated map with the eleventh curve points excluded.

• First the excluded experimental points correspond to Nc,rel = 0.5. Fig. 5 shows the obtained curves. In Fig. 5, the grey points (5th curve) are the excluded experimental points. In this situation, the average error for the 5th curve is about 0.87%, i.e. double the same error computed in the initial case. However this increase of the error is insignificant, and is less than the global average error; • Secondly, the excluded experimental points correspond to Nc,rel = 1.05. This point is outside the boundaries of the new reduced experimental domain. Fig. 6 shows the obtained curves. In Fig. 6, the grey points are the excluded experimental points. In this situation, the average error for the 11th curve is about 2.3% while it was 1.8% in the initial case. However, this increase of the error is insignificant. This experiment proves that the method allows experimental data to be extrapolated; • Fig. 7 represents the approximate characteristic curves for 20 different speeds. The predicted curves (with red dotted lines) preserve the same aspect as the approximated curves (coloured in grey) which denotes the predictive character of the proposed method; • The approximation results for the compressor efficiency map are shown in Fig. 8. The approximation of the compressor efficiency, with this method, follows the same steps as in the case of the pressure ratio. Consequently, one does not present these aspects in this paper. The curves presented in Fig. 8 demonstrate that the method can be applied with good results to approximate and predict the compressor efficiency.

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wc,rel Fig. 7. Predicted compressor map for 20 speeds.

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wc,rel Fig. 8. Approximate efficiency for all experimental points.

5. Conclusions This paper proposes and develops a numerical method for modelling compressors performance based on moving least squares (MLS). The use of MLS is justified by the fact that it is a powerful method for approximating a function when few data are available, as in the case under study. The proposed model belongs to the family of surrogate models. In conclusion, one can state that: • The proposed methodology can be adapted for modelling centrifugal compressors starting from weak experimental dataset; • The method covers all the experimental domain, but in the same time the extension of it, with a good accuracy; • The initial experimental domain must be transformed to obtain a new domain, where the distances between experimental points are more uniformly distributed along the coordinates, to ensure both the convergence and the accuracy of the method; • The validation of the method has been made using the procedure of ‘‘cross-validation’’. It is based on the graphical results and relative errors estimation. The level of the relative errors shows that the method is accurate;

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• The method can be improved to obtain a better precision in the neighbourhood of the domain boundary by adjusting the compact supports associated with the experimental points; • The compressor characteristic map can be predicted and used for the development of a surrogate model, which can be used for fuel cell system modelling.

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