# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2022116
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## $\mu$ and $H_\infty$ optimization control based on optimal oxygen excess ratio for the Proton Exchange Membrane Fuel Cell (PEMFC)

 1 Department of Electrical Engineering, Faculty of Electrical and Computer Engineering, Technical and Vocational University (TVU), Tehran, Iran 2 Smart/Micro Grids Research Center, Department of Electrical Engineering, University of Kurdistan, Sanandaj, Iran

Received  November 2021 Revised  May 2022 Early access July 2022

The fuel cell (FC) is a new technology for large-scale power generation. Providing of fast and sufficient air concentration in the FC cathode is a key to prevent the oxygen starvation and extend the life cycle of the FC. The proton exchange membrane fuel cell (PEMFC) parameter variations and output current disturbances can significantly influence on the airflow subsystem performance. For this purpose, this paper addresses the robust airflow control synthesis of the PEMFC based on the $\mu$ and $H_\infty$ control techniques. The presented advanced controllers are designed to regulate the oxygen concentration in the FC cathode on the desired value. Such robust controllers are tuned to guarantee PEMFC performance against parameter variations and output current disturbances. Then, the obtained results are compared with the optimal PID controller tuned by the well-known internal model control (IMC) method. Simulation results show that in comparison of the IMC-based PID controller, the applied robust control methodologies are more effective. The designed $\mu$ and $H_\infty$ controllers efficiently provide the required airflow of the PEMFC on the desired value, and avoid the oxygen starvation. Furthermore, due to the structured type of uncertainties, the $\mu$ controller keeps the airflow on the desired value quite better than the $H_\infty$ controller.

Citation: Mohsen Aryan Nezhad, Hassan Bevrani. $\mu$ and $H_\infty$ optimization control based on optimal oxygen excess ratio for the Proton Exchange Membrane Fuel Cell (PEMFC). Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022116
##### References:
 [1] T. Allag and T. Das, Robust control of solid oxide fuel cell ultracapacitor hybrid system, IEEE Transactions on Control Systems Technology, 20 (2011), 1-10. [2] C. Bao, M. Ouyang and B. Yi, Modeling and control of air stream and hydrogen flow with recirculation in a PEM fuel cell system-Ⅱ. Linear and adaptive nonlinear control, International Journal of Hydrogen Energy, 31 (2006), 1897-1913. [3] H. Bevrani, Robust power system frequency control, Springer, (2014). [4] H. Bevrani, F. Habibi, P. Babahajyani, M. Watanabe, et al., Intelligent frequency control in an AC microgrid: Online PSO-based fuzzy tuning approach, IEEE Transactions on Smart Grid, 3 (2012), 1935-1944. [5] F. D. Bianchi, C. Kunusch, et al., A gain-scheduled LPV control for oxygen stoichiometry regulation in PEM fuel cell systems, IEEE Transactions on Control Systems Technology, 22 (2013), 1837-1844. [6] Q. Chen, L. Gao, R. Dougal and S. Quan, Multiple model predictive control for a hybrid proton exchange membrane fuel cell system, Journal of Power Sources, 191 (2009), 473-482. [7] M. Y. El-Sharkh, A. Rahman, M. S. Alam, P. C. Byrne, et al., A dynamic model for a standalone PEM fuel cell power plant for residential applications, Journal of Power Sources, 138 (2004), 199-204. [8] M. Gerard, J. P. Poirot-Crouvezier, D. Hissel and M. C. Pera, Oxygen starvation analysis during air feeding faults in PEMFC, International Journal of Hydrogen Energy, 35 (2010), 12295-12307. [9] D. W. Gu, P. Petkov and M. M. Konstantinov, Robust control design with MATLAB ®, Springer Science & Business Media, (2005). doi: 10.1007/978-1-4471-4682-7. [10] M. Hinaje, S. Raël, J-P. Caron and B. Davat, An innovating application of PEM fuel cell: Current source controlled by hydrogen supply, International Journal of Hydrogen Energy, 37 (2012), 12481-12488. [11] C. A. Ibanez, J. M. Valenzuela, O. G. Alarcon and M. M. Lopez, PI-type controllers and $\Sigma$ and $\Delta$ modulation for saturated DC-DC buck power converters, IEEE Access., 9 (2021), 20346-20357. [12] F. Inthamoussou, J. Pegueroles-Queralt and F. D. Bianchi, Control of a supercapacitor energy storage system for microgrid applications, IEEE Transactions on Energy Conversion, 28 (2013), 690-697. [13] J. Jia, G. Wang, Y. T. Cham, Y. Wang and M. Han, Electrical characteristic study of a hybrid PEMFC and ultracapacitor system, IEEE Transactions on Industrial Electronics, 57 (2009), 1945-1953. [14] Q. Li, W. Chen, Z. Liu, A. Guo and S. Liu, Control of proton exchange membrane fuel cell system breathing based on maximum net power control strategy, Journal of Power Sources, 241 (2013), 212-218. [15] Q. Li, H. Yang, Y. Han, M. Li, et al., A state machine strategy based on droop control for an energy management system of PEMFC-battery-supercapacitor hybrid tramway, International Journal of Hydrogen Energy, 41 (2016), 16148-16159. [16] O. Madani, A. Bhattacharjee and T. Das, Decentralized power management in a hybrid fuel cell ultracapacitor system, IEEE Transactions on Control Systems Technology, 24 (2015), 765-778. [17] I. S. Martin, A. Ursua and P. Sanchis, Integration of fuel cells and supercapacitors in electrical microgrids: Analysis, modelling and experimental validation, International Journal of Hydrogen Energy, 38 (2013), 11655-11671. [18] D. I. Martinez, J. J. Rubio, V. Garcia, T. M. Vargas, et al., Transformed structural properties method to determine the controllability and observability of robots, Applied Sciences, 11 (2021), 3082. [19] H. Marzooghi and M. Raoofat, Improving the performance of proton exchange membrane and solid oxide fuel cells under voltage flicker using Fuzzy-PI controller, International Journal of Hydrogen Energy, 37 (2012), 7796-7806. [20] M. Morari and E. Zafiriou, Robust process control, Morari, (1989). [21] S. Nasri, S. B. Sihem, I. Yahyaoui, B. Zafar, et al., Autonomous hybrid system and coordinated intelligent management approach in power system operation and control using hydrogen storage, International Journal of Hydrogen Energy, 42 (2017), 9511-9523. [22] M. A. Nezhad and H. Bevrani, Real-time AC voltage control and power-following of a combined proton exchange membrane fuel cell, and ultracapacitor bank with nonlinear loads, International Journal of Hydrogen Energy, 42 (2017), 21279-21293. [23] M. A. Nezhad and H. Bevrani, Frequency control in an islanded hybrid microgrid using frequency response analysis tools, IET Renewable Power Generation, 12 (2018), 227-243. [24] R. S. Ortigoza, E. Hernandez-Marquez, et al., Sensorless tracking control for a "full-bridge buck inverter-DC motor" system: Passivity and flatness-based design, IEEE Access, 9 (2021), 132191-132204. [25] A. Packard and J. Doyle, The complex structured singular value, Automatica, 29 (1993), 71-109.  doi: 10.1016/0005-1098(93)90175-S. [26] J. T. Pukrushpan, A. G. Stefanopoulou and H. Peng, Control of fuel cell power systems: Principles, modeling, analysis and feedback design, Springer Science $&$ Business Media, (2004). doi: 10.1109/MCS. 2004.1275430. [27] Y. Qin, Q. Du, et al., Study on the operating pressure effect on the performance of a proton exchange membrane fuel cell power system, Energy Conversion and Management, 142 (2017), 357-365. [28] C. Restrepo, T. Konjedic, et al., Simplified mathematical model for calculating the oxygen excess ratio of a PEM fuel cell system in real-time applications, IEEE Transactions on Industrial Electronics, 61 (2013), 2816-2825. [29] D. Shin, K. Lee and N. Chang, Fuel economy analysis of fuel cell and supercapacitor hybrid systems, International Journal of Hydrogen Energy, 41 (2016), 1381-1390. [30] L. A. Soriano, J. J. Rubio, E. Orozco, D. A. Cordova, et al., Optimization of sliding mode control to save energy in a SCARA robot, Mathematics, 9 (2020), 3160. [31] L. A. Soriano, E. Zamora, J. M. V. Nicolas and G. Hernandez, PD control compensation based on a cascade neural network applied to a robot manipulator, Frontiers in Neurorobotics, (1992), 78. [32] J. Sun and I. V. Kolmanovsky, Load governor for fuel cell oxygen starvation protection: A robust nonlinear reference governor approach, IEEE Transactions on Control Systems Technology, 13 (2005), 911-920. [33] W. Tan, Unified tuning of PID load frequency controller for power systems via IMC, IEEE Transactions on power systems, 25 (2009), 341-350. [34] A. Taniguchi, T. Akita, K. Yasuda and Y. Miyazaki, Analysis of degradation in PEMFC caused by cell reversal during air starvation, International Journal of Hydrogen Energy, 33 (2008), 2323-2329. [35] M. Tekin, D. Hissel, et al., Energy-management strategy for embedded fuel-cell systems using fuzzy logic, IEEE Transactions on Industrial Electronics, 54 (2007), 595-603. [36] S. Tong, J. Fang and Y. Zhang, Output tracking control of a hydrogen-air PEM fuel cell, IEEE/CAA Journal of Automatica Sinica, 4 (2017), 273-279.  doi: 10.1109/JAS.2017.7510526. [37] M. Uzunoglu and M. S. Alam, Dynamic modeling, design, and simulation of a combined PEM fuel cell and ultracapacitor system for stand-alone residential applications, IEEE Transactions on Energy Conversion, 21 (2006), 767-775. [38] B. Wu, M. Parkes, A. Yufit, et al., Design and testing of a 9.5 kWe proton exchange membrane fuel cell-supercapacitor passive hybrid system, International Journal of Hydrogen Energy, 39 (2014), 7885-7896. [39] J. T. Pukrushpan, A. G. Stefanopoulou and H. Peng, Control of fuel cell breathing, IEEE Control Systems Magazine, 24 (2004), 30-46.  doi: 10.1109/MCS.2004.1275430. [40] Z. Yong, H. Shirong, J. Xiaohui, Ye. Yuntao, et al., Performance study on a large-scale proton exchange membrane fuel cell with cooling, International Journal of Hydrogen Energy, 47 (2022), 10381-10394. [41] L. Zhao, J. Brouwer, S. James, J. Siegler, et al., Dynamic performance of an in-rack proton exchange membrane fuel cell battery system to power servers, International Journal of Hydrogen Energy, 42 (2017), 10158-10174.

show all references

##### References:
 [1] T. Allag and T. Das, Robust control of solid oxide fuel cell ultracapacitor hybrid system, IEEE Transactions on Control Systems Technology, 20 (2011), 1-10. [2] C. Bao, M. Ouyang and B. Yi, Modeling and control of air stream and hydrogen flow with recirculation in a PEM fuel cell system-Ⅱ. Linear and adaptive nonlinear control, International Journal of Hydrogen Energy, 31 (2006), 1897-1913. [3] H. Bevrani, Robust power system frequency control, Springer, (2014). [4] H. Bevrani, F. Habibi, P. Babahajyani, M. Watanabe, et al., Intelligent frequency control in an AC microgrid: Online PSO-based fuzzy tuning approach, IEEE Transactions on Smart Grid, 3 (2012), 1935-1944. [5] F. D. Bianchi, C. Kunusch, et al., A gain-scheduled LPV control for oxygen stoichiometry regulation in PEM fuel cell systems, IEEE Transactions on Control Systems Technology, 22 (2013), 1837-1844. [6] Q. Chen, L. Gao, R. Dougal and S. Quan, Multiple model predictive control for a hybrid proton exchange membrane fuel cell system, Journal of Power Sources, 191 (2009), 473-482. [7] M. Y. El-Sharkh, A. Rahman, M. S. Alam, P. C. Byrne, et al., A dynamic model for a standalone PEM fuel cell power plant for residential applications, Journal of Power Sources, 138 (2004), 199-204. [8] M. Gerard, J. P. Poirot-Crouvezier, D. Hissel and M. C. Pera, Oxygen starvation analysis during air feeding faults in PEMFC, International Journal of Hydrogen Energy, 35 (2010), 12295-12307. [9] D. W. Gu, P. Petkov and M. M. Konstantinov, Robust control design with MATLAB ®, Springer Science & Business Media, (2005). doi: 10.1007/978-1-4471-4682-7. [10] M. Hinaje, S. Raël, J-P. Caron and B. Davat, An innovating application of PEM fuel cell: Current source controlled by hydrogen supply, International Journal of Hydrogen Energy, 37 (2012), 12481-12488. [11] C. A. Ibanez, J. M. Valenzuela, O. G. Alarcon and M. M. Lopez, PI-type controllers and $\Sigma$ and $\Delta$ modulation for saturated DC-DC buck power converters, IEEE Access., 9 (2021), 20346-20357. [12] F. Inthamoussou, J. Pegueroles-Queralt and F. D. Bianchi, Control of a supercapacitor energy storage system for microgrid applications, IEEE Transactions on Energy Conversion, 28 (2013), 690-697. [13] J. Jia, G. Wang, Y. T. Cham, Y. Wang and M. Han, Electrical characteristic study of a hybrid PEMFC and ultracapacitor system, IEEE Transactions on Industrial Electronics, 57 (2009), 1945-1953. [14] Q. Li, W. Chen, Z. Liu, A. Guo and S. Liu, Control of proton exchange membrane fuel cell system breathing based on maximum net power control strategy, Journal of Power Sources, 241 (2013), 212-218. [15] Q. Li, H. Yang, Y. Han, M. Li, et al., A state machine strategy based on droop control for an energy management system of PEMFC-battery-supercapacitor hybrid tramway, International Journal of Hydrogen Energy, 41 (2016), 16148-16159. [16] O. Madani, A. Bhattacharjee and T. Das, Decentralized power management in a hybrid fuel cell ultracapacitor system, IEEE Transactions on Control Systems Technology, 24 (2015), 765-778. [17] I. S. Martin, A. Ursua and P. Sanchis, Integration of fuel cells and supercapacitors in electrical microgrids: Analysis, modelling and experimental validation, International Journal of Hydrogen Energy, 38 (2013), 11655-11671. [18] D. I. Martinez, J. J. Rubio, V. Garcia, T. M. Vargas, et al., Transformed structural properties method to determine the controllability and observability of robots, Applied Sciences, 11 (2021), 3082. [19] H. Marzooghi and M. Raoofat, Improving the performance of proton exchange membrane and solid oxide fuel cells under voltage flicker using Fuzzy-PI controller, International Journal of Hydrogen Energy, 37 (2012), 7796-7806. [20] M. Morari and E. Zafiriou, Robust process control, Morari, (1989). [21] S. Nasri, S. B. Sihem, I. Yahyaoui, B. Zafar, et al., Autonomous hybrid system and coordinated intelligent management approach in power system operation and control using hydrogen storage, International Journal of Hydrogen Energy, 42 (2017), 9511-9523. [22] M. A. Nezhad and H. Bevrani, Real-time AC voltage control and power-following of a combined proton exchange membrane fuel cell, and ultracapacitor bank with nonlinear loads, International Journal of Hydrogen Energy, 42 (2017), 21279-21293. [23] M. A. Nezhad and H. Bevrani, Frequency control in an islanded hybrid microgrid using frequency response analysis tools, IET Renewable Power Generation, 12 (2018), 227-243. [24] R. S. Ortigoza, E. Hernandez-Marquez, et al., Sensorless tracking control for a "full-bridge buck inverter-DC motor" system: Passivity and flatness-based design, IEEE Access, 9 (2021), 132191-132204. [25] A. Packard and J. Doyle, The complex structured singular value, Automatica, 29 (1993), 71-109.  doi: 10.1016/0005-1098(93)90175-S. [26] J. T. Pukrushpan, A. G. Stefanopoulou and H. Peng, Control of fuel cell power systems: Principles, modeling, analysis and feedback design, Springer Science $&$ Business Media, (2004). doi: 10.1109/MCS. 2004.1275430. [27] Y. Qin, Q. Du, et al., Study on the operating pressure effect on the performance of a proton exchange membrane fuel cell power system, Energy Conversion and Management, 142 (2017), 357-365. [28] C. Restrepo, T. Konjedic, et al., Simplified mathematical model for calculating the oxygen excess ratio of a PEM fuel cell system in real-time applications, IEEE Transactions on Industrial Electronics, 61 (2013), 2816-2825. [29] D. Shin, K. Lee and N. Chang, Fuel economy analysis of fuel cell and supercapacitor hybrid systems, International Journal of Hydrogen Energy, 41 (2016), 1381-1390. [30] L. A. Soriano, J. J. Rubio, E. Orozco, D. A. Cordova, et al., Optimization of sliding mode control to save energy in a SCARA robot, Mathematics, 9 (2020), 3160. [31] L. A. Soriano, E. Zamora, J. M. V. Nicolas and G. Hernandez, PD control compensation based on a cascade neural network applied to a robot manipulator, Frontiers in Neurorobotics, (1992), 78. [32] J. Sun and I. V. Kolmanovsky, Load governor for fuel cell oxygen starvation protection: A robust nonlinear reference governor approach, IEEE Transactions on Control Systems Technology, 13 (2005), 911-920. [33] W. Tan, Unified tuning of PID load frequency controller for power systems via IMC, IEEE Transactions on power systems, 25 (2009), 341-350. [34] A. Taniguchi, T. Akita, K. Yasuda and Y. Miyazaki, Analysis of degradation in PEMFC caused by cell reversal during air starvation, International Journal of Hydrogen Energy, 33 (2008), 2323-2329. [35] M. Tekin, D. Hissel, et al., Energy-management strategy for embedded fuel-cell systems using fuzzy logic, IEEE Transactions on Industrial Electronics, 54 (2007), 595-603. [36] S. Tong, J. Fang and Y. Zhang, Output tracking control of a hydrogen-air PEM fuel cell, IEEE/CAA Journal of Automatica Sinica, 4 (2017), 273-279.  doi: 10.1109/JAS.2017.7510526. [37] M. Uzunoglu and M. S. Alam, Dynamic modeling, design, and simulation of a combined PEM fuel cell and ultracapacitor system for stand-alone residential applications, IEEE Transactions on Energy Conversion, 21 (2006), 767-775. [38] B. Wu, M. Parkes, A. Yufit, et al., Design and testing of a 9.5 kWe proton exchange membrane fuel cell-supercapacitor passive hybrid system, International Journal of Hydrogen Energy, 39 (2014), 7885-7896. [39] J. T. Pukrushpan, A. G. Stefanopoulou and H. Peng, Control of fuel cell breathing, IEEE Control Systems Magazine, 24 (2004), 30-46.  doi: 10.1109/MCS.2004.1275430. [40] Z. Yong, H. Shirong, J. Xiaohui, Ye. Yuntao, et al., Performance study on a large-scale proton exchange membrane fuel cell with cooling, International Journal of Hydrogen Energy, 47 (2022), 10381-10394. [41] L. Zhao, J. Brouwer, S. James, J. Siegler, et al., Dynamic performance of an in-rack proton exchange membrane fuel cell battery system to power servers, International Journal of Hydrogen Energy, 42 (2017), 10158-10174.
Algorithm for design of $\mu$ and $H_\infty$ controllers
Robust control configuration with FC perturbed model
Closed-loop structure of FC air control with lumped input multiplicative uncertainty
FC uncertainty variations and the weighting function W2(s)
Bode plot of $G_{z2w}$
Airflow closed-loop model for using the D-K iteration algorithm
Robust stability condition based on Nyquist diagram
S(s) and T(s) for the FC with $H_\infty$ controller
S(s) and T(s) for the FC with $\mu$ controller
Comparison between original (dashed) and reduced-order (solid) $H_\infty$ -controller
Comparison between original (dashed) and reduced-order (solid) $\mu$-controller
Comparison among sensitivity functions of the designed controllers
Comparison among complementary functions of the designed controllers
S(s) and T(s) of the FC perturbed model with $H_\infty$ controller
S(s) and T(s) of the FC perturbed model with $\mu$ controller
S(s) and T(s) of the FC perturbed model with PID controller
The OER variations for step increase in the output current for PID controller
The OER variations for step increase in the output current for $H_\infty$ controller
The OER variations for step increase in the output current for $\mu$ controller
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