LPV Model-Based Fault Diagnosis Using Relative Fault Sensitivity Signature Approach in a PEM Fuel Cell.

Transcription

1 LPV Model-Based Fault Diagnosis Using Relative Fault Sensitivity Signature Approach in a PEM Fuel Cell. S. De Lira V. Puig J. Quevedo Automatic Control Department, Universitat Politecnica de Catalunya (UPC), Rambla de Sant Nebridi, 1, 8222 Terrassa (Spain); Tel:+(34) ; Fax: +(34) salvador.de.lira@upc.edu Abstract: In this paper, the problem of fault diagnosis in a PEM Fuel Cell (PEM FC) system using a model-based approach is addressed. A Linear Parameter Varying (LPV) model is used to catch the non linear dynamics of the PEM FC. A new fault diagnosis method is proposed based on structured residuals generation and sensitivity analysis evaluation. Robustness in fault diagnosis is increased by use of a relative fault sensitivity approach. The proposed methodology is applied to a Polymer Electrolyt Membrane Fuel Cell system. Finally, simulation results using the proposed approach presents a better isolability performance than classical Boolean fault signature approach. Keywords: Fault Detection, Fault Diagnostic, PEM FC. 1. INTRODUCTION The energy generation systems based on fuel cells are complex since they involve thermal, fluidic and electrochemical phenomena. Moreover, they need a set of auxiliary elements (valves, compressor, sensors, regulators, etc.) to make the fuel cell working at the pre-established optimal operating point. For these reasons, they are vulnerable to faults that can cause a emergency shut down or a permanent damage of the fuel cell. To guarantee a safe operation of the fuel cell systems, it is necessary to use systematic techniques, like the recent methods of Fault Tolerant Control (FTC) in Blanke et al. (23), which allow increasing the fault tolerance of this technology. The first task to achieve active tolerant control is based on the inclusion of a fault diagnosis system operating in realtime. The diagnosis system should not only allow the fault detection and isolation but also the fault magnitude estimation. In this paper, a LPV model based fault diagnosis is proposed as a way to diagnose faults in fuel cell systems. The model-based fault diagnosis is based on comparing online the real behaviour of the monitored system obtained by means of sensors with a predicted behaviour obtained using a LPV dynamic model with an observer scheme. In case of a significant discrepancy (residual) is detected between the model and the measurements obtained by the sensors, the existence of a fault is assumed. If a set of measurements is available, it is possible to generate a set of residuals (indicators) that present a different sensitivity This work was supported in part by the Research Commission of the Generalitat of Catalunya (Grup SAC ref. 25SGR537) and by Spanish Ministry of Education (CICYT projects ref. DPI , ref. DPI and ref. DPI ) and Consejo Nacional de Ciencia y Tecnologa (CoNACYT) and cooperation with Department of Universities, Research and Society of Information. to the set of possible faults. The contributions of this paper are: first, the use of a LPV observer for fault detection and second, the use of the relative residual fault sensitivity analysis that allows to isolate faults that otherwise would not be separable. The structure of this paper is the following: in Section 2, the foundations of the proposed fault diagnosis methodology are recalled. In Section 3, LPV model based fault detection methodology is described. In Section 4, the proposed fault diagnosis is presented. In Section 5, the PEM fuel cell system used to illustrate the proposed fault diagnosis methodology is presented with the fault scenarios that can appear. Finally, in Section 6, the application results of the proposed methodology of diagnosis are presented. 2. FOUNDATION OF THE FAULT DIAGNOSIS METHODOLOGY 2.1 Introduction The task of fault diagnosis consists of determining the type of fault with as much details as possible (fault location, time and size). The proposed methodology of fault diagnosis which is used in this paper is mainly based on classic FDI theory of model-based diagnosis described in Gertler (1998), Staroswiecki and Comtet-Varga (21) and Isermann (25). Model-based diagnosis can be divided in two subtasks: fault detection and fault isolation. The principle of model-based fault detection is checking the consistency of measured and predicted behaviors. The consistency check is based on computing residuals r k. The residuals are obtained from the discrepancy of measured input (u k ) and outputs (y k ) at operating point (ϑ k ) using the set of sensors installed in the process with an analytical relationship which is obtained by system modelling:

2 r k ψ (y k, u k, ϑ k ) (1) where ψ is the residual generator function that depends on the type of detection strategy used (parity equation Gertler (1998) or observer (Chen and Patton (1999)). At each time instance, k, the residual is compared with a threshold value (zero in ideal case or almost zero in real case). Robust residual evaluation allows obtaining a set of fault signatures, φ(ϑ) [ φ 1 (ϑ), φ 2 (ϑ),..., φ nφ (ϑ) ] where each indicator of fault is obtained as follows: { if ri τ φ i (ϑ) i (2) 1 if r i > τ i where τ i is the threshold associated to the residual r i (k). Fault isolation consists of identifying the fault that affects the system. It is carried out by using the fault signatures, φ (generated by the detection module) and its relation with all the considered faults, f(k) {f 1 (k), f 1 (k),..., f n (k)} The method most often applied is a relation defined on the Cartesian product of the set of faults FSM φ f, where FSM is the theoretical fault signature matrix (Gertler, 1998). An element F SM ij of this matrix will be one, if a fault f j (k) is affected by the residual r i (k). In this case, the value of the fault indicator φ i (k) must be equal to one when the fault appears in the monitored system. Otherwise, the element F SM ij will be zero. However in practice, the existence of parametric uncertainty forces the use of robust methods, that can be active if they act at the residual generation stage or passive if they act in the decision-making process. 3. LPV FAULT DETECTION 3.1 Lineal Parameter Varying Model There exists different ways to obtain LPV models (Murray- Smith and Johansen (1997) and Barahyi and Patton (23)). Some methods use non-linear equations of the system to derive a LPV model such as state transformation, substitution of functions and methods using the well known Jacobian linearization. Another kind of methods uses multi-model identification that consists basically in two different steps. Fist a set of LTI model is identified at different equilibrium points by classical methods (online or off-line). As second part of this methodology a multi-model is obtained by using an interpolation law that commutes the local LTI model according to the operating point. A LPV system in discrete time state space in a fault free case is represented by; x k+1 A(ϑ k )x k + B(ϑ k )u k + w k (3) y C(ϑ k )x k + D(ϑ k )u k + v k where x k R nx, u k R nu and y k R ny are, respectively, the state, input, and output vectors. The process and measurement noises are w k R nx and v k R nv. The type of LPV systems considered in this paper assumes an affine dependence within the parameter vector Θ space Θ { ϑ k R Θ } ϑ k ϑ k ϑ k Lineal Parameter Varying Observer A Linear Parameter Varying (LPV) observer with Luenberger structure for the state estimation of the system described in (3) is given by ˆx k+1 A(ϑ k )ˆx k + B(ϑ k )u k + (4) L(ϑ k )(y k ŷ k ) + w k ŷ k C(ϑ k )ˆx k + D(ϑ k )u k + v k where L is the observer gain to be designed to guarantee stability for ϑ k Θ. This gain is computed at operating point ϑ k using LMI formulation for Pole-Placement within a wide class of pole clustering regions that is founded in an extended Lyapunov Theorem (see Chilali (1996)). The motivation for seeking pole clustering in specific regions inside the unitary circle is to obtain a fast observer dynamics for all considered operating points. 4. FAULT ISOLATION 4.1 Fault Sensitivity Analysis The isolation approach presented in Section 2 uses a set of binary detection (Boolean) tests to compose the observed fault signature. However, the use of binary codification of the residual produces a lack of information that can lead to wrong diagnosis when applied to dynamic systems. This derives in some faults that are not isolabable because they present the same theoretical binary fault signature (Puig et al., 26). To avoid this problem is possible to use other additional information associated with the relationship between the residuals and faults, such as sign, sensitivity, and order or activation time, to improve the isolation results (Puig et al., 26). In this work, a new method for fault diagnosis that tries to better exploit the information provided by the fault residual is proposed for diagnosis system designing. According to Gertler (1998), the sensitivity of the residual to a fault is given by S f r f which is a transfer function that describes the effect on the residual (r) of a given fault (f ). Sensitivity provides quantitative information about the effect of the fault on the residual and qualitative information in their sense of variation (sign). The use of this information at the stage of diagnosis allows separating faults that although show the same theoretical binary fault signature, they present different sensitivity values. In order to perform diagnosis, the algorithm will use a theoretical fault sensitivity matrix F SM sensit. Each value of this matrix, denoted as S rif j, contains the sensitivity of the residual r i to the fault f j. Although sensitivity depends on time in case of a dynamic system, here the steady-state value after a fault occurrence is considered as it is also suggested in Gertler (1998). However several simulation results show a dynamic dependency on operating points for S rif j matrix elements. In order to perform real time diagnosis, the observed sensitivity Sr o if j should be computed using the current (5)

3 value of the residual ri (k) at operating point ϑk when a fault f (k) is detected. ri (k) Sroi (ϑk ) f (k) (6) 4.2 Relative fault sensitivity approach From (6), it is possible to observe that using F SMsensit, in real time requires the knowledge of the fault magnitude or an estimation of it at a specific operating point. To solve this problem, this paper attempts to design the diagnosis using a new concept called relativity sensitivity rather than absolute sensitivity given in (5). The observed relative fault sensitivity is defined as: Sroi fj (ϑk ) ri (ϑk ) Srrel,o (7) i rl Srol fj (ϑk ) rj (ϑk ) which corresponds to the ratio of one residual ri (k) with respect to another specific one (rl ). Then, the relative sensitivity will be insensitive to the unknow magnitude of the fault. A new F SM, called F SMsensitrel (see Escobet et al. (29)), which corresponds to theoretical fault signature matrix based on relative sensitivity is then introduced. The elements of this matrix, is given by i rl fj i rl,fj Srti fj (ϑk ) Srtl fj (ϑk ) ri (k)/ fj (k)(ϑk ) rl (k)/ fj (k)(ϑk ) 5.1 PEM Fuel Cell System Fuel cells are electrochemical devices that not only convert the chemical energy of a gaseous fuel directly into electricity and are widely regarded as a potential alternative stationary and mobile power source but also are environmentally friendly (EESI, 2),(Yacobucci and Curtright, 24). Polymer Electrolyte Membrane Fuel Cells (PEM FC) are at developement stage for automotive applications because of its capability in high power density, solid electrolyte, long cell and stack life. The overall reaction of the fuel cell is described by 2H2 + O2 2H2 O (11) In this paper, a PEM FC model developed in Pukrushpan et al. (22) is used as the case study for fault diagnosis. The overall FC system could be partitioned in two subsystems: fuel cell stack and auxiliary components. Stack, anode, cathode and membrane hydration belong to fuel cell stack subsystem, while compressor, supply and return manifold, cooling and humidifier are the auxiliary components. Figure 1 presents a conceptual diagram of the FC system. (8) In this case, for a set of n faults, a relative fault sensitivity matrix F SMsensitrel should be used as shown in Table 1. r2 /rl r3 /rl rm /rl Sr r,f 2 l 1 3 rl, Sr r,f 2 l 2 3 rl, r,f r,f m l 1 m l 2 fn Sr r,f 2 l n 3 rl,fn r,f m l n Table 1. Theoretical fault signature matrix using relative sensitivity respect to rl at ϑk Fig. 1. PEM FC process diagram The diagnostic algorithm used to assess real-time observed relative sensitivities using (7), is based on a ratio of residuals, will provide a vector in relative sensitivities space. The vector generated will be compared with vectors of theoretical fault places stored into the relative sensitivity matrix F SMsensitrel. The theoretical fault signature vector with a minimum distance with respect to the fault observed vector is postulated as the possible fault: min ds (k),..., dsfn (9) where the distance is calculated using the Euclidean distance between vectors 2 rel,o Sr r,f (ϑk ) Sr2 r1, (ϑk ) (1) dsfn (k) sqrt Srrel,o (ϑ ) S (ϑ ) k k r,f r r,f m 1 n m 1 n 5. CASE STUDY In order to test the fault diagnosis methodology proposed in this paper, a PEM Fuel Cell is used as real process application case study. A model-based dynamic analysis gives an insight of the subsystem interaction and control design limitation because of different control objectives (see Pukrushpan and Stefanopoulou (24)). 5.2 PEM FC Dynamic Model For simulation purposes the model proposed in Pukrushpan et al. (22) will be used. This model has the following characteristics: The law of mass conservation is used for mass balance estimation. Physic laws and some empirical equations are used. The properties are based on electrochemical, thermodynamic and zero-dimensional fluid mechanics principles. The resulting model dynamic equation set is described by

4 m O2 WO2,i WO2,o WO2,r m H2 WH2,i WH2,o WH2,r m N2 WN2,i WN2,o ω cp Jcp1ωcp (Pcm Pcp ) a P sm γr Vsm (Wcp Tcp Wsm,o Tsm ) m sm Wcp Wsm,o m w,an Wvan,i Wvan,o Wvmbr Trm (Wca,o Wrm,o ) P om Rair Vrm ID Fault Description There is an increase of friction in the compressor motor. Suddenly change in the system pressure because the system is not capable to avoid an atmospheric pressure change. A suddenly obstruction at air inlet pipe is presented. The result is a decrease in the inlet diameter. Air leak in the air supply manifold. A degradation in the humidifier performances is presented at stack upstream. Degradation in the cells at stack is presented because of contact-sensitivity reactions against a reaction killer. The 3 % of the cells suddenly are broken down. The fluid resistance increases due to water blocking, the channels or flooding in the diffusion layer. Hydrogen leak in the hydrogen supply manifold. (12) f3 f4 f5 where the subindex i,o, represent inlet and outlet flow respectibly and subindex an, rm, mbr, sm, cp means anode, return manifold, membrane, supply manifold and compressor, respectively. f6 f7 f8 f9 5.3 PEM FC Linear Parameter Varing Model For fault detection previous model will be approximated by a LPV model that it is scheduled with operating point using the stack current as scheduling variable along the operating range. Table 2. Description of the fault scenarios implemented in FGB. The LPV model has been obtained through an off-line identification process performed along different operating points in the operating range (OPR) of interest, where g (ph pl )/ OP R, since high (ph ) and low (pl ) limits were selected with an increment ( OP R), where OP R Rg. In state space (SS) form, the structure of PEM FC linear model is described by a11 a 31 A(ϑk ) a51 a61 a81 c C(ϑk ) 21 c41 a13 a15 a18 a22 a25 a27 a33 a35 a38 a44 a45 ; B(ϑk ) b41 a53 a54 a55 a56 a63 a64 a65 a72 a75 a77 a83 a88 c14 c23 c24 d22 ; D(ϑ ) k c35 c42 c43 d42 b12 b22 b42 b72 Fig. 2. Fault Generator Block (FGB) and Fault Diagnosis System (FDS) implementation diagram (13) done using MATLAB/SIMULINK environment. The PEM FC Model already developed and described in Pukrushpan et al. (22) was used as case study. A FGB and FDS subsystems blocks were added to the PEM FC simulator developed by Pukrushpan et al. (22). The aim of those where x [mo2 mh2 mn2 ωcp Psm msm mw,an Prm ]T, blocks is to diagnose the faults described in Table 2. u [vcm Ist ]T and y [ωcp ro2 Psm vst ], states and outputs are given in compatible unit magnitudes. The 6.1 Fault Detection Process (FDP) resulting linear matrix system structure was denoted as Using the measured inputs and (13), where each ij-th element is computed as a polynomial Structured residuals. correlation of the scheduling variable from a non-linear outputs presented in Section 5.3 and using the structural analysis methodology (Blanke et al., 23), the following regression. set of residuals can be obtained: r1 ωcp ω cp ; 5.4 Faulty PEM FC scenarios implementation r2 ro2 r O2 ; (14) r3 Psm P sm ; In order to test the proposed methodology in the PEM r4 vst v st ; FC system, a set of common possible fault was selected and implemented in simulation as a Fault Generator Block (FGB )(Figure 2). Through this block it is possible to select Those residuals are implemented using an observer in and simulate them. Each selected fault acts over a specific order to reduce model uncertainty and measurement noise. part of the overall system. But on other hand, interaction occurs because of process dynamics and multiplicative LPV Observer Design. The optimal observer gain L(ϑk ) effects of the faults. The Table 2 describes the set of faults is computed using LMI formulation for Pole-Placement with a region defined by an affix ( q, ) and a radius r which were considered. such that ( q + r) <, where in this case r.1 and q (see (Chilali and Gahinet, 1996)). 6. RESULTS An observability analysis along the operating point range The implementation of the fault diagnosis system (FDS) of interest shows observability problems because of the following the approach proposed in Section 3 has been high condition number for slow eigenvalues, degrading the

5 observer performance (Pukrushpan and Stefanopoulou, 24). A large observer gain is required, producing a large overshoot in residual evaluation being not suitable for fault diagnosis system implementation, because a large observer gain can produce degradation of residual generation. In order to avoid a large observer gain, a reduced observer is designed: a closed-loop observer for fast eigenvalue output estimation should be used while an openloop observer is used for slow eigenvalues output estimation. To implement the reduced output observer, the system matrices are transformed to the canonical modal form: A(ϑk,2 ) PA(ϑk )P 1, B(ϑk,2 ) PB(ϑk ) and C(ϑk,2 ) C(ϑk )P 1, where P is the transformation matrix. Then the structure of A(ϑk,2 ) has the eigenvalues of A(ϑk ) on its diagonal, and then a partition can be performed as two submatrices. The reduced output observer gain uses the first sixth columns and rows of ϑk,2. This assumption involves the sixth first eigenvalues of A(ϑk )(λ1 to λ6 ) to estimate L(ϑk,2 ). The global observer gain is computed by 1 L(ϑk,2 ) L(ϑk ) P (15) Using this gain with reduced order output approach improves the state estimation. Figure 3 describes the estimation quality of outputs, when a set of step current changes is applied. r1 r2 r3 r4 f3 f4 f5 f6 f7 f8 (+)1 (+)1 (+)1 Table 3. PEM FC Fault Binary Signature for the proposed set of faults. r2 /r1 r3 /r1 r4 /r1 Stadistic f3 f4 f5 f6 f7 f8 std mean min max std mean min max std mean min max Table 4. Statistics Analysis of Theoretical fault signature matrix using relative sensitivity respect to r1 for operating point vector; [185, 25]A. In order to improve isolability, the relative fault sensitivity signature method proposed in Section 4 is used. This method takes into accout residual fault sensitivities without requiring to know the fault magnitude. Morevoer, this approach also reduces the sensitivity dependency with the operating point (see Table 4) because of the use of relative sensitivies as a ratio of residuals. Due to lack of space, this section only shows the results corresponding to, which is one of the most usual fault presented in fuel cell system during normal operation conditions (see Escobet et al. (29)). In order to test the LPV model-based fault diagnosis approach, a set of different operating point conditions were selected in the range of [185, 25] Amp. This selection tries to obtain the optimal performance specifications for the fuel cell, that is, λo2 2 and Pnet 4kW ). Fig. 3. Faultless Process vs. Estimated signals and residual performance during a set of operating point(ist ) changes: (Ist, Ts ) (191, ) (189, 2) (194, 5) (192, 8) (191, 11) 6.2 Fault Isolation Process (FIP) Using the set of residuals (14) and the faults described in Table 2, a theoretical binary fault signature matrix (FSM) presented in Table 3 is obtained. Looking at this table considering only the binary signature faults, and f5 are not isolable because they present the same binary signature. However, the rest of faults can be isolated one from another using just binary information. One of the main contributions of the proposed approach in this paper is the capability for isolation of faults that with only using binary information would not be possible. Based on F DP information, the F IP tries to determine the fault cause. In this case, in order to perform a successful fault diagnosis, the F IP block must contain knowledge about the fault that has appeared in the system. The theoretical fault ratios in the selected operating points and its statistics analysis are presented in Table 4. In Figure 5, the theoretical relative fault sensitivity is represented in the space of residual ratios. The elliptical region around each theoretical fault ratio corresponds to an isolation threshold volume that has been calculated by statistical analysis (Velleman and Hoaglin, 1981). This threshold introduces robustness in the isolation process taking into account measurement noise and model uncertainty. The residual evolution when fault appears at sample 95 is shown in Figure 5. Note that the system suffers a step change in operating point (stack current change from 191 to 194 A) at sample 4, producing an impulse in the residual not signficantly enough to cross the residual threshold. However, when the fault appears at 9, the residual r3 crosses above upper threshold value and r4 crosses under lower threshold value, while r1 and r2 does not cross their threshold. Then, according the observer fault signature and the binary fault signature presented in Table 3, the candidate faults are, and f5. On the other hand, the evolution of the observed relative fault sensitivity in the space of theoretical relative fault sensitivity signature is shown in Figure 5. From this figure,

6 knowhow sharing, specially to J. Riera, M. Serra and D. Feroldi. REFERENCES Fig. 4. Residual evolution. it can be noticed that the evolution of the observed relative sensitivity pass around several fault places after the fault is detected and the isolation process is initiated. Finally, when residuals reach steady state, the observed relative sensitivity enters in the region of uncertainty around fault. According to the isolation criterion (9), this is the final candidate fault. Fig. 5. Evolution of observed relative sensitivity ratio in space of residuals ratios in case of fault. 7. CONCLUSION In this paper, a new LPV model-based fault diagnosis methodology based on the relative fault sensitivity has been presented and tested. An advantage of this new methodology is twofold: first, the variation of the dynamics with the operating point is considered by using an LPV observer when generating residuals. Second, a fault isolation algorithm based on the relative fault sensititivy concept is proposed. This method allows isolating faults that are not isolable considering only a binary (or a sign) fault signature matrix. To prove this methodology, a PEM fuel cell case study well-known in the literature has been used. The case study was modified to include a set of possible fault scenarios that try to reflect the most common faults. All the considered faults have been tested with the new diagnosis methodology, which has diagnosed correctly all of them. ACKNOWLEDGEMENTS The authors would like to tank to the Instituto de Robotica i Informatica Industrial (IRI/UPC-CSIC) for its D. Tikk Y. Yam Barahyi, P. and R. Patton. From differential equations to pdc controller design via numerical transformation. Computers in Industry, 51: , 23. M. Blanke, M. Kinnaert, J. Lunze, and M. Staroswiecki. Diagnosis and Fault-Tolerant Control. Springer-Verlag Berlin Heidelberg, 23. J. Chen and R.J. Patton. Robust Model-Based Fault Diagnosis for Dynamic Systems. Marcel Dekker, M. Chilali and P. Gahinet. h design with pole placement constraints: an LMI approach. IEEE Transactions on Automatic Control, 41(3): , EESI. Fuel cell fact sheet. Environmental and Energy Study Institute, pages 2 25, 2. T. Escobet, D. Feroldi, S. de Lira, V. Puig, J. Quevedo, J. Riera, and M. Serra. Model-based fault diagnosis in pem fuel cell systems. Journal of Power Sources, (169): , 29. J. Gertler. Fault Detection and Diagnosis in Engineering Systems. Marcel Dekker, New York, ISBN R. Isermann. Model-based fault-detection and diagnosisstatus and applications. Annual Reviews in Control, 29 (1):71 85, 25. R. Murray-Smith and T. A. Johansen. Multiple model approaches to modelling and control. London, UK: Taylor and Francis., 2nd edition, V. Puig, J. Quevedo, T. Escobet, and J. Meseguer. Towards a better integration of passive robust intervalbased f di algorithms. 6th IFAC SAFEPROCESS, 16 (5), 26. J. Pukrushpan, H. Peng, and A. Stefanopoulou. Pem fc dynamic system. Journal of Dynamics Systems, Measurement, and Control, 126:14 25, 22. T. Pukrushpan and A. G. Stefanopoulou. Control-oriented modeling and analysis for automotive fuel cell systems. Automotive Research Center,Department of Mechanical Engineering., 126:14 25, 24. M. Staroswiecki and G. Comtet-Varga. Analytical redundancy relations for fault detection and isolation in algebraic dynamic systems. Automatica, 375(5): , 21. P.F. Velleman and D.C. Hoaglin. Applications, Basics, and Computing of Exploratory Data Analysis. Duxbury Press, Brent D. Yacobucci and Aimee E. Curtright. A hydrogen economy and fuel cells: An overview. CRS Report for Congress, RL32196:the CRS Web, 24.