Experimental Evaluation of Parameterized Nonlinear MPC Applied to PEM Fuel Cell

This paper proposes a parameterized nonlinear model-based predictive control (NMPC) strategy to tackle the oxygen excess ratio regulation challenge of a proton exchange membrane fuel cell. In practice, the most challenging part regarding NMPC strategies remains the on-line implementation. In fact, NMPC strategies, at least in their basic form, involve heavy computation to solve the optimization problem. In this work, a specific parameterization of control actions has been designed to address this limitation and achieve on-line implementation. To assess the effectiveness and relevance of the proposed strategy, the controller has been implemented on-line, experimentally validated on a real fuel cell and compared to the built-in controller. Performance of the parameterized NMPC controller in terms of setpoint tracking accuracy, disturbances rejection and computational cost, have tested under several control scenarios. Experimental results have shown the excellent tracking capability, disturbances rejection ability and low computational cost of the NMPC controller, regardless of the operating conditions. Moreover, compared to the built-in controller the proposed strategy has demonstrated better disturbances rejection capability. Overall, the proposed parameterized NMPC controller appears as an excellent candidate to address the oxygen excess ratio regulation issue.


Introduction
For the last decade, to reduce greenhouse gas emissions and fossil fuel dependence, numerous renewable energy technologies have been studied. In regard to How to cite this paper: Damour renewable energies, one of the primary drawbacks is the variability of the supply flows, which raises the key issue of energy storage to counteract the intermittent nature of their conversion. In this context, the hydrogen vector represents a promising alternative, as long as the hydrogen is produced from a renewable energy technology (e.g. electrolytic hydrogen obtained using photovoltaic energy), and stored to optimize potential gaps and surplus of intermittent production. Downstream the hydrogen chain, electricity is generated from fuel cells, genuine zero-emission power generators. Due to their high power density and low operating temperature, proton exchange membrane fuel cells (PEMFC) have proved to be the most suitable fuel cell technology for both transportation and stationary applications [1] [2] [3].
However, several bolts still remain to be removed to improve their reliability and energy conversion, reduce their cost or extend their lifetime. Among them, one of the most important is related to their control. Indeed, global efficiency improvement, optimal hydrogen and air consumption, and reliable and accurate power response remain challenging control goals.
Numerous control strategies have been reported in the literature for PEMFC systems, ranging from PID controllers [4] [5] [6], state feedback linearizing or differential flatness approaches [7] [8], dynamic neural network controllers [9] [10], linear quadratic Gaussian (LG) controllers [11], to model predictive control strategies [12]. Due to its ability to take into account dynamic nonlinearities of the process, to handle just as well state and input constraints as economical constraints, nonlinear model-based predictive control (NMPC) strategy appears as a promising candidate regarding PEMFC control.
Several works dealing with benefits of predictive control strategies regarding PEMFC control have been reported. Wu et al. [13] proposed a multi-loop nonlinear predictive control strategy using a reduced order model to regulate oxygen excess ratio and stack temperature of a fuel cell. However, the controller has only been tested in simulation environment, and as emphasized by authors some devices and design, such as hydrogen storage or power management, have not been considered. Shokuhi-Rad et al. [14] designed an approximate predictive control strategy to regulate the output voltage of a PEMFC. This approach, based on an instantaneous linearization of a neural network model, has been tested in simulation environment, and appeared to be an interesting alternative to achieve real-time control. Gruber et al. [15] developed a model based predictive strategy to regulate the oxygen stoichiometric ratio using compressor motor voltage as manipulated variable. Experimental results showed that the designed controller stabilized the oxygen stoichiometric ratio around the target value five times faster than the built-in controller. Ziogou et al. [16] proposed a nonlinear model-based predictive control (NMPC) approach to track variable load demands while minimizing hydrogen consumption and avoiding oxygen starvation. To achieve on-line application, these authors developed a tailor-made optimization strategy that discretizes control variables and state variables. The proposed approach has To ensure optimal performance of PEMFC, one parameter, namely oxygen excess ratio, requires special attention. The oxygen excess ratio or stoichiometric ratio represents the ratio of inlet oxygen flow to reacted oxygen flow and is widely used to guarantee safety and to reach a high performance. Arce et al. [32] showed that the oxygen excess ratio has a fundamental influence on the efficiency and the safety of the fuel cell system. A Poor control of this variable can increase the starvation phenomenon probability. The oxygen starvation phenomenon occurs when the oxygen partial pressure falls below a critical level at any location at the cathode [33]. It has been experimentally demonstrated that the oxygen starvation phenomenon can cause damages to the electro-catalyst of the fuel cell, as well as reducing its performance [34]. In this context, preventing oxygen starvation to ensure optimal conversion efficiency and avoid performance deterioration remains a challenging control goal.
In this paper, a real-time implementable nonlinear model-based predictive control (NMPC) strategy is developed to tackle the oxygen excess ratio regulation issue. In practice, the most challenging part regarding NMPC strategies remains the on-line implementation. In fact, NMPC strategies, at least in their basic form,

Experimental Setup
In the present study, a 50 cm 2 single home fuel cell assembling, with a commer-

Controller Design
Regarding NMPC control strategies, the first step is to design a model of the system. This model is expected to predict the system behavior several steps ahead.
Among the various PEMFC models that can be consulted in the literature, a very few are dedicated to control purposes. In real-time control context, most of complex and heavy computations mechanistic models cannot be considered. Recently, several works emphasize the interest of ANN to model PEMFC systems.
Here, due to its short computational time, its low sensitivity to noise and its reliability an ANN model is designed and used as predictive model.

ANN Control Oriented Model
The proposed control strategy involves an ANN model of the PEMFC used to predict the system output several steps ahead.
The training phase aims to determine a mapping from the set of training data to the set of possible weights, so that the ANN model produces predictions ( ) The prediction error approach, which is the strategy applied here, is based on the introduction of a measure of closeness in terms of a mean square error criterion: The weights are then found as: The training phase is performed using Levenberg-Marquart algorithm with an error goal of 0.001. The validation results are presented in Figure 2.
Two criteria, namely root mean square error ( With an RMSE of 0.003 and an AME under 0.1 the predictive performance accuracy of the ANN model is more than sufficient, especially since it is dedicated to control purposes (Table 1).

Parameterized Nonlinear MPC for PEMFC
In this study, a parameterized NMPC controller is used to control the PEMFC oxygen excess ratio ( NMPC is an optimization-based multivariable constrained control method that uses a nonlinear model to predict the future behavior of the process. Classical  [36]. However, in their basic form, NMPC strategies involve heavy computations, which could become a bolt to reach on-line implementation.
In this work, to address this problem and reach real-time control goal, a parameterized NMPC strategy is designed. When dealing with classical NMPC strategies, the predictive horizon sets the optimization problem dimension, which could lead to huge problem dimension. The main feature of parameterized NMPC strategy consists in decreasing the optimization problem dimension using a particular parameterization of the control sequence u: where ( ) ( ) where the optimal solution p is the set of parameters that minimizes the cost function J while meeting the problem constraints.
In this context, as the problem dimension depends on the dimension of p, itis straightforwardly linked to the parameterization choice and decoupled from the predictive horizon N y . In this manner, a well-structured parameterization scheme may significantly decrease the optimization problem dimension, which is a quite important feature to fulfilreal-time requirements [37]. However, it is important to highlight that there is no universal parameterization that can be applied to any problem. Each problem must be represented using a specific parameterization.
Here, λ depends on several variables, Gruber et al. [15] showed that the stationary behavior of the system mainly relies onair inlet flow rate ( air m  ) and load current (I). Therefore, during these experiments the stationary behavior is tested with different values for the air inlet flow rate ( air m  ) and the load current (I).
In a second step, these data are gathered to generate a stationary map correlating the load current, the inlet air flow rate and the oxygen excess ratio.
In a last step, based on this stationary map a simplified model is identified to approximate the steady state control action as a function of the oxygen excess ration and the load current: a is a constant parameter that has to be determined off-line.
In this work, a specific parameterization of the control actions based on the approximation of the steady state control action * u and a single scalar parameter p is proposed: Here the cost function is written to minimize the setpoint tracking error and to limit the variations of the control variable: In this context, the optimal value p is obtained by minimizing the cost function J, which is subject to constraints on manipulated variable: Here, the optimization problem that lead to the sequence of future control actions relies entirely on a single scalar parameter, which dramatically reduces the computational effort compared to a classical NMPC strategy [40].

Experimental Evaluation of the Parameterized Nonlinear MPC
To assess the performance of the proposed control strategy in terms of tracking capability, disturbances rejection, and robustness against plant-model mismatch, series of experiments are performed on the PEMFC system presented in Section 2.
In this aim, several VI's of the SCDA system are modified to replace the built-in controller by the parameterized NMPC controller. With this setup, the optimal control action, computed on-line in Matlab® environment, is sent through a specific VI to be applied to the experimental unit.
Here, to exemplify the controller performance three control scenarios are considered: The first scenario illustrates the tracking capability of the controller for a constant value of load current. The second scenario is designed to study the controller ability to cope with disturbances. Eventually, the third scenario evaluates the controller performance in terms of setpoint tracking accuracy in presence of disturbances. A comparison between the built-in controller and the proposed control strategy is made for the second scenario. Note that this comparison cannot be performed for the first and third scenario due to technical constraints. In fact, the built-in controller of the SCDA system, integrated by Fuel Cell Technologies, is designed to operate at constant oxygen excess ratio. In other word, when using the original setup, the oxygen excess ratio value has to be set before the experiments and cannot be modified during the experiments. The built-in controller is a Proportional Integral (PI) controller. However, the SCDA system provided by Fuel Cell Technologies, Inc does not allow to access the controller parameters or to modify them.
In the sequel, the inlet air flow rate is taken as manipulated variable, whereas the load current is a measured disturbance. It is assumed that all other variables, required to ensure safety operation of the fuel cell, are properly controlled. The cell temperature and the sampling period are set to 75˚C and 3 s respectively.
The optimization problem is solved on-line using Levenberg-Marquardt algorithm. All experiments are performed using control parameters listed in Table 2.

First Case Scenario: Constant Load Current-Variable Oxygen Excess Ratio Setpoint
The tracking capability of the controller is evaluated using an oxygen excess ratio setpoint that covers the whole operating condition. In this control scenario, the load current is set to 15A and kept constant during the entire experiments. Figure  3 shows that the controller is able to track accurately a variable oxygen excess ratio setpoint, while offering an entirely suitable dynamic for the manipulated variable.

Second Case Scenario: Step Changes on Load Current-Constant Oxygen Excess Ratio Setpoint
To assess the controller performance in terms of disturbances rejection, a set of step changes on the load current is performed. The dynamics of the PEMFC is strongly correlated to the power level. Therefore, to verify that the controller performs accurately whatever the power level, the current steps is chosen to cover the whole operating conditions. Several experiments, with different sets of step changes in load current and different oxygen excess ration values, are carried out to compare the performance of the original built-in controller and the proposed NMPC strategy.
In each and every case, the NMPC controller performed better than the built-in controller. Figure 4 illustrates one of these experiments with the oxygen excess ratio set to 6.
The NMPC controller demonstrates that it can maintain the oxygen excess ratio to the desired level, even in presence of disturbance. Moreover, compared to the built-in controller, the proposed controller exhibits significantly better disturbances rejection capability. Indeed, for five steps in current (t = 48 s, t = 138 s, t = 351 s, t = 438 s and t = 528 s) an important overshot is observed with the built-in controller, whereas the proposed controller efficiently rejects the disturbances.

Third Case Scenario: Step Changes on Load Current-Variable Oxygen Excess Ratio Setpoint
This control scenario is designed to assess the controller performance when the oxygen excess ratio setpoint is modified on-line according to the load current.
This control scenario, which is a common working scenario in automotive applications, is significantly important. Indeed, several works demonstrated that updating the oxygen excess ratio reference according to the load current could significantly increase the PEMFC efficiency while avoiding oxygen starvation [16] [41] [42].
The proposed controller demonstrates excellent tracking capability even in presence of disturbances ( Figure 5). The ability of the controller to track efficiently an oxygen excess ratio setpoint, calculated according to a variable load current, is a quite important feature. Indeed, this control scenario appears as one of the most promising options to improve the overall PEMFC efficiency, while ensuring safety operation of the fuel cell.

Computational Cost
In order to highlight the benefits of the proposed parameterized NMPC scheme compared to a classical NMPC strategy, both strategies are implemented in Matlab environment and their performance are evaluated in simulation. In this aim, a mechanistic model, which has been presented and validated in a previous work [43], is used as process simulator. To truly highlight the effect of the parametri-  Table 3 summarizes the computational time efficiency of both NMPC controllers for all three tests. In all cases, and for similar accuracy performance, the parameterized approach turns out to be significantly faster than the classical NMPC strategy. Whereas the computational time of the classical approach drastically increases with the dimension of the control horizon, the computational time of the parameterized NMPC strategy remains quite the same. The parameterized NMPC approach allows to decouple the optimization problem dimension from the control horizon dimension, which turns out to be quite important feature when dealing with real-time implementation. Considering a control horizon set to 30, which appears to be reasonable regarding a sampling period of 3s, the parameterized approach is almost 30 times faster than the classical NMPC strategy. Obviously, the computational time depends on what hardware is used to solve the optimization problem. However, the comparison performed here be-Engineering tween both NMPC approaches clearly demonstrates that the parameterized NMPC controller has a significantly better computation efficiency than the classical NMPC controller.
Note that due to its too high computation cost, the classical NMPC controller has not been experimentally tested on the real fuel cell.

Conclusions
The oxygen excess ratio is considered as a performance variable of the system and its regulation is an important issue since this parameter determines the safety of the fuel cell. In this paper, to address oxygen excess ratio control challenge, a parameterized NMPC strategy has been developed. In a first stage, due to its short computational time and its low sensitivity to noise an artificial neural network (ANN) model has been designed. The oxygen excess ratio is a function of inlet air flow rate, load current, relative humidity of air at the cathode inlet, stack temperature and inlet pressure at the cathode. However, regarding real-time control goal a simplified model has been proposed. This ANN model, used as predictor in the control strategy, is expected to predict the oxygen excess ratio of the PEMFC several steps ahead, once the load current and the inlet air flow rate are available. The validation procedure has been performed on experimental data and the model has shown good performance in terms of prediction accuracy. In a second stage, a parameterized NMPC approach has been designed. This approach, based on a particular parameterization of the control sequence, has led to a low-dimensional optimization problem. Indeed, the optimization problem that leads to the sequence of future control actions relies entirely on a single scalar parameter, which dramatically reduces the computational effort compared to a classical NMPC strategy. Eventually, the controller has been implemented on-line and experimentally validated on a real fuel cell. Numerous control scenarios have been experimentally conducted to evaluate the controller performance, especially in terms of setpoint tracking accuracy, disturbances rejection and computational cost. These control scenarios gather all the possible scenarios in which the system would have to work. In each and every case, the controller demonstrated highly satisfactory results since it tracked efficiently the desired oxygen excess ratio value while compensating disturbances, regardless of the operating conditions. Besides, experimental comparison demonstrated that the proposed controller had much better disturbances rejection capability than the built-in controller. Eventually, to emphasize the benefit of the proposed controller in terms of computational time efficiency, a comparison has been performed between the proposed NMPC controller and a classical NMPC controller. In all case scenarios, the computational cost of the proposed parametrized NMPC controller was significantly lower than the one of the classical NMPC controller.
Overall, the proposed parameterized NMPC controller appears as an excellent candidate to address the oxygen excess ratio regulation issue.