Introduction

With the increasing demand for green technology, hydrogen and fuel cells are being considered as one of the most promising clean fuel and energy conversion devices, respectively. One of the best examples is Coradia i-Lint, the world’s first hydrogen-powered passenger train which was introduced in 2016 in Germany. Fuel cells are electrochemical devices which convert chemical energy to electrical energy as long as fuel and air/oxygen is supplied. The conversion is based on the concept opposite to electrolysis of water, where hydrogen and oxygen ions are separated from a molecule of water by passing electric current through it. Fuel cells are generally classified according to the type of electrolyte used, apart from factors like the type of fuel used or the operating temperature. Solid oxide fuel cells (SOFC), phosphoric acid fuel cells (PAFC), direct methanol fuel cells (DMFC) and proton exchange membrane fuel cells (PEMFC) are some of the widely used fuel cell technologies. The ultimate reaction between hydrogen and oxygen to form water is the same in all the cases irrespective of the type of fuel cell used.

PEMFC is a typical fuel cell operating at a lower temperature, generally under 100 °C, which gives it an advantage of faster connection time as compared to other fuel cell types. The transport and the portable device sectors are inclining more towards PEMFCs due to their compact structure, noiseless operation and lightweight characteristics. Furthermore, polymer electrolyte is used in a PEMFC, which makes it easy to manufacture. The physical structure of a PEMFC consists of several components like feeding channels, diffusion layer, membrane, catalytic layer, diffusion layer and feeding channels in the cathode. The electrodes, membrane and electrolyte are generally combined in a compact structure which is called membrane electrode assembly (MEA). It is the heart of the fuel cell and is fed with hydrogen and oxygen, generating electrical power with a power density of around 1W cm−2. Hydrogen acts as fuel to the system and is supplied on the cathode side, while air is supplied in most cases instead of pure oxygen on the anode side. The only product of the reaction is water and heat. Auxiliary devices are installed along with the fuel cell for special purposes like thermal management, water management, humidification system and air–fuel flow system.

A single PEMFC produces voltage between 0 and 1 V depending on the size of the load and fuel cell operating conditions. Usually, the load requires large operating current, so more than one fuel cell is connected in series to form a fuel cell stack to produce the desired voltage. A single cell produces voltage in the range of 0.6 to 0.8 V, approximately. Since the load requirement is generally large, more than one fuel cell is used in series to form a fuel cell stack, to get higher voltage. The output voltage of the FC stack is generally calculated by multiplying the number of fuel cells used, with the average voltage of a single fuel cell. As with other electrical devices, there are electrical resistances in the fuel cell. The loss associated with the resistance is dissipated in the form of heat. In other words, heat is released from the fuel cell reaction and is considered as one of the by-products along with water (Pukrushpan 2003).

The literature on PEMFC is quite large and diverse in nature. A lot of research has been conducted from modelling to control of PEM fuel cells. Dutta et al. (2000) and Mann et al. (2000) laid the foundation of steady-state PEMFC stack modelling which were used to derive the aspects of the model related to conductivity and hydration of the membrane. A detailed three-dimensional multi-component PEMFC model was presented by Dutta et al. (2000) with the complicated electrochemical aspects of the model. Coming to the dynamic modelling, Pukrushpan (2003) presented a dynamic PEMFC model to further study and design the control system for air subsystem of the PEMFC model.

One of the earliest works done in the field of control was by Lorenz et al. (1997), where they presented a method to control power of an electric drive unit of a vehicle. Pukrushpan’s (2003) PhD thesis is considered as one of the major contributions, where the principal focus is on air-in flow control. In a decentralized MIMO control system, basic feedback PID controllers were designed and implemented by Serra et al. (2005) to control stack voltage and hydrogen–air pressure difference by manipulating air stoichiometric ratio and hydrogen stoichiometry, respectively. The implemented control strategy gave the best controllability indices in comparison to other control-manipulated variable pairs. A LQG regulator was developed for a dynamic air supply system presented (Rodatz et al. 2003); apart from faster response time, pressure trace was successfully decoupled from the mass flow trace. Under small load variations, a H-infinity controller was designed (Sedghisigarchi and Feliachi 2004) to maintain the output voltage. By control of hydrogen flow rate, variations in the output voltage were kept under 5% during simulations. For a PEMFC system, under varying current conditions, control of air supply system was achieved (Caux et al. 2005). Simulations proposed a species balance model to maintain constant pressure on the cathode (oxygen) compartment and to follow a desired air flow rate. For a small PEMFC system, using a microcontroller, a novel cascade strategy with a static feed-forward control was proposed (Kim et al. 2010). Hou et al. (2020) presented an extensive review on the control strategies and varied controllers for air supply in the PEM fuel cell system, which emphasized on the importance of air flow system with respect to load output voltage.

For a validated non-linear PEMFC model (Danzer et al. 2009), a constrained feed-forward model predictive control (MPC) was developed of air flow rate. The desired action was achieved by comparing the predicted and actual output current, reducing the error and limiting the oxygen excess ratio to avoid oxygen starvation. Since the MPC for a non-linear system requires a lot of computational time, a linearized model was used (Arce et al. 2007) to reduce the computational time and to generate a faster response for maximum efficiency and starvation control by manipulating the air flow rate. Nejad et al. (2019) proposed genetic algorithm optimization-based approach to obtain optimum control parameters for a traditional lead-lag controller for fuel cell voltage control. The controller was able to achieve less voltage deviation and higher overall efficiency. For a two-input two-output system, Sankar et al. (2019) developed an experimental hybrid non-linear control structure to maintain the fuel cell operating temperature coupled with an airflow cooling fan, using sliding mode controller and reduced order sliding mode observer. In terms of disturbance rejection and set point tracking, the proposed controller was compared with the traditional PI controller and observed to give better results.

Based on distributed deep reinforcement learning, an optimal controller developed by Li and Yu (2021) showed high robustness and adaptability to regulate the output voltage; the controller was designed with multi-delay deterministic policy gradient. Ahmed Souissi (2021) synthesized an adaptive sliding mode control on super twisting algorithm to get the maximum power tracking and keeping the hydrogen and oxygen partial in check, for minimum damage to membrane. Mahali et al. (2022) developed fuzzy logic-based strategies to control load voltage output from a PEMFC system. With a feed-forward and conditional switch, the system showed better performance in terms of current variation and regulate oxygen excess ratio. For a reduced order state-space model obtained by dynamic simulations, a multi-parametric model predictive controller designed by Zhang et al. (2018) was performed effectively. A maximum efficiency point tracking algorithm was proposed by Artal-Sevil et al. (2020), for an enhanced energy management of a PEM fuel cell system. The results for the proposed controller showed better results when compared with only maximum power point tracking (MPPT) in loop.

While in the arena of fractional order systems, there has been considerable amount of work done in regard to PEMFC. Taleb et al. (2017) proposed a Warburg impedance based a fractional order fuel cell transfer function model. This model was transformed to implement FOM identification. The method used to identify the model’s parameters was based on the least square method extended to fractional order models. Two-loop fractional order proportional integral (FOPI) designed to regulate output voltage based on hybrid optimization was validated under operating condition and load perturbations, by Bankupalli et al. (2018).

Since PEMFCs are mostly employed in the transportation and portable device sectors, the load requirement is quite dynamic in nature according to external conditions. So, a fuel cell is supposed to follow the same trend in the voltage produced, which calls in the control of stack voltage. Stack voltage depends on many factors like stack temperature, moisture content of the membrane and partial pressure of hydrogen and air, out of which inlet rate of hydrogen and air are quite important factors as it affects the rate of reaction and hence the voltage produced. This solves one of the major control problems of a fuel cell that is control of voltage by manipulating the inlet rates of hydrogen and air simultaneously. On the same idea Wang et al. (2007) described the dynamics of PEMFC and modelled it as MISO system, with the fixed output resistance to control the output voltage by tuning the hydrogen and air flow rates through a multivariable robust controller. From the experimental results, the proposed robust controller was deemed to achieve robust performance and to reduce hydrogen consumption of the system.

To take this further, the present study proposes a novel control structure, where both fuel and air inlet rates are manipulated simultaneously to control the stack voltage. For this control objective, the integer order multiple-input single-output (MISO) model developed by Wang et al. (2007) is reduced to a fractional order MISO PEMFC model using Indranil Pan’s method based on genetic algorithm as the optimization technique. Using both the integer and the fractional order models, model-based controllers like the proportional integral (PI), proportional integral derivative (PID), model predictive control (MPC) and predictive PID (PPID) are designed for individual feedback loops and for the proposed two-input single-output (TISO) control unit. The main novelty, contribution and highlights of the study have been summarized below:

  • Researchers have considered the SISO system with an integer order model for the PEM fuel cell system in the literature. In the present study, fractional order model development uses a genetic algorithm as the optimization technique, followed by model-based control strategies for a MISO system.

  • They proposed a new control structure based on feedback control loop and gain percent contribution for a MISO system focusing on the control of stack voltage as in open literature considered SISO configuration.

  • Control performance was evaluated based on set point tracking, disturbance rejection, inverse response and time delay compensation in terms of integral absolute error (IAE), integral square error (ISE) and total variation (TV).

  • Most of the work done in the literature focused primarily on one inlet at a time, with flow optimization or air pump performance; this study focuses on output voltage tracking while keeping in check both the inlet flow rates, which show better efficiency in terms of performance and load disturbance rejection.

PEMFC MISO model

The model used here is a two-input two-output model adopted from the literature. The inputs of the system are hydrogen and air, while the outputs are cell voltage and current. The fuel cell used consists of 15 cells with an active area of 50 cm2 on each. The cells are connected with a pre-treated membrane — Nafion®112 — by hot press for optimum conditions. Platinum loading is about 0.2 mg/cm2 at anode and 0.4 mg/cm2 at cathode. The maximum efficiency of the fuel cell stack is 37% (LHV) under dry H2/air and humidification-free conditions (Wang et al. 2007).

The system dynamics are non-linear and time varying and are influenced by many factors, including the diffusion dynamics, the Nernst equation, proton concentration dynamics and cathode kinetics as follows (Wang  et al. 2007).

Diffusion equation:

$${R}_{ohm}={R}_{ref}+{\propto }_{t}\left(T-{T}_{ref}\right)$$
(1)

where \({R}_{ohm}\) is the ohmic resistance, \({R}_{ref}\) is the reference resistance at room temperature, \({\propto }_{t}\) is the thermal coefficient, \(T\) is the cell temperature, and \({T}_{ref}\) is the reference temperature.

Nernst equation:

$$E={E}_{ref}+\frac{dE}{dT}\left(T-{T}_{ref}\right)+k\frac{RT}{2F}\mathrm{ln}({P}_{{H}_{2}}{P}_{{O}_{2}}^{0.5})$$
(2)

where \({E}_{ref}\) is the open-circuit voltage at room temperature, \({P}_{{H}_{2}}\) is the partial pressure of hydrogen, \({P}_{{O}_{2}}\) is the partial pressure of oxygen, \(E\) is the reversible or open-circuit voltage, \(F\) is Faraday’s constant, and \(R\) is the gas constant.

Proton concentration dynamics:

$$u\left(\frac{-\partial {C}_{{H}^{+}}}{\partial t}\right).\frac{\partial {C}_{{H}^{+}}}{\partial t}+\frac{{C}_{{H}^{+}}}{{\tau }_{{H}^{+}}}=\frac{1+{\propto }_{{H}^{+}}{(j}^{3})}{{\tau }_{{H}^{+}}}$$
(3)

where \({C}_{{H}^{+}}\) is the proton concentration, \({\propto }_{{H}^{+}}\) is the relational parameter, \({\tau }_{{H}^{+}}\) is the time constant, and \(j\) is the current density.

Cathodic kinetics:

$$\eta =b.\mathrm{ln}\{\frac{{p}_{I0}{\left[{H}^{+}\right]}_{o}}{{p}_{I}\left[{H}^{+}\right]}.(1+\frac{{j}_{r}}{{j}_{o}{A}_{r}})\}$$
(4)

where, \({j}_{r}\) is the current density at specific area, \({j}_{o}\) is the current density at ambient conditions, \({A}_{r}\) is the active area of cell, \(b\) is the constant parameter, \({p}_{I0}{\left[{H}^{+}\right]}_{o}\) is the partial pressure of hydrogen proton at ambient conditions, and \({p}_{I}\left[{H}^{+}\right]\) is the partial pressure of hydrogen proton.

Integer order model

To describe a system, the author measured the given input signals and the corresponding output signals, and then, using MATLAB, they identified the system by the subspace system identification skills. A chirp signal and a PRBS signal were generated as input signals to drive the air pump and the hydrogen valve of the PEMFC system, respectively, and the corresponding output voltage was recorded. The bandwidth of the chirp signal and the PBRS signal was between 0.01 and 10 Hz. Due to the non-linear properties, the author selected three operating points, namely, 2A, 3A and 4A, and repeated the experiments three times at each operating point to take the system variations into account. The PEMFC system was regarded as a MISO system.

By using the input output data in System Identification toolbox of MATLAB, nine transfer functions for each loop were obtained at different load conditions. Out of these nine pairs of transfer functions, using the gap metric technique, a nominal plant is selected for the PEMFC system, which minimizes the maximum gap between the nominal plant and the perturbed systems. The nominal plant selected for voltage control by manipulating air and hydrogen flow inlet, respectively, is as follow.

$${\mathrm G}_{\mathrm{IO}}=\begin{array}{cc}\overset{\mathrm{Air}\;\mathrm{Pump}}{\lbrack\frac{-0.102s-8.148}{s^2+13.09s+18.58}}&\overset{\mathrm{Hydrogen}\;\mathrm{Valve}}{\frac{0.245s-3.016}{s^2+13.09s+18.58}\rbrack}\end{array}$$
(5)

Fractional order model

Using Pan and Das (2013) method of model reduction, to reduce a higher integer order model to a lower order fractional model, the above-mentioned transfer functions are reduced to fractional order transfer functions, resulting in a fractional order MISO PEMFC model. This method uses optimization techniques to approximate the integer order models to lower order fractional models. The author used Nelder–Mead simplex method as the optimization algorithm, but as suggested by the author that genetic algorithm can also be used, here in this paper, genetic algorithm has been implemented for the reduction process in MATLAB’s Optimization Toolbox (Sharma and Babu 2020). The reduced order non-integer transfer functions using genetic algorithm are given follows.

$${\mathrm G}_{\mathrm{FO}}=\begin{array}{cc}\overset{\mathrm{Air}\;\mathrm{Pump}}{\lbrack\frac{-0.408e^{-0.017s}}{0.529s^{1.143}+1}}&\overset{\mathrm{Hydrogen}\;\mathrm{Valve}}{\frac{-0.706e^{-0.07s}}{s^{1.486}+2.23s^{0.789}+4.133}\rbrack}\end{array}$$
(6)

Novel MISO control structure

A sudden increase in the current demand by the external load connected to PEMFC would lead to a sudden decrease in the stack voltage, thus making voltage control a critical task. Stack voltage depends on many parameters like air–fuel flow rates, temperature of the stack and relative humidity of the stack. Two most important parameters which affect the stack voltage air–fuel inlet rates have been considered here, assuming that the stack operates at a constant temperature and relative humidity. Air–fuel inlet rates directly affect the stack voltage; which is evident from Nernst equation (Eq. 2) where partial pressure of the hydrogen and oxygen term appears as the last term on the right-hand side of the equation. So, in this section, separate controllers are synthesized for regulating hydrogen valve and air pump using the transfer functions obtained in the Eqs. (5) and (6). The proposed control structure is shown in the Fig. 1.

Fig. 1
figure 1

PEMFC control structure

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The aim to control stack voltage is achieved by designing controllers for each loop separately and then by using different combinations of controllers for both the loops to work as a single TISO unit. Output from each loop is multiplied by the percentage gain of how much it would affect the output voltage when the complete systems work as a single TISO unit and then added to give an overall stack voltage output. Percent gain for each loop’s output is calculated by taking the ratio of the steady-state gain of each transfer function. The hydrogen valve loop showed approximately 30% effect on the output voltage, whereas air pump loop showed an effect of 70% on the stack voltage. Figure 2 shows the Simulink diagram for the proposed MISO control structure that is used in this study.

Fig. 2
figure 2

MISO PEMFC control structure

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Performance measures

The performance is measured on the basis of two popular integral error criteria, namely, IAE and ISE. These performance measures are generic and comprehensive criteria which allow comparison between different controller designs or even different controller structures. They are usually calculated using the following mathematical expressions:

$$ISE=\int {(\varepsilon }^{2})dt$$
(7)
$$IAE=\int |\varepsilon |dt$$
(8)

Here, \(\varepsilon\) is the error between the desired and measured output. Along with the errors, control effort (TV) for each controller is also calculated. The total variation is a good measure of the ‘smoothness’ of a signal and should be as small as possible and is mathematically represented as

$$TV=\sum \nolimits_{k=0}^{\infty }|{u}_{k+1}-{u}_{k}|$$
(9)

where u is the manipulated variable.

Results and discussions

Simulation results for individual loops and the TISO unit, integer as well as fractional order, are presented and analysed in this section. For the case of integer order MISO model is inverse response rejection, while for fractional order MISO model, time delay compensation is also compared for different combinations as well as the individual loops.

Hydrogen valve

Firstly, we consider controlling stack voltage by manipulating the hydrogen inlet through hydrogen valve. A basic feedback control structure is implemented here. The integer as well fractional order transfer functions used to design different controllers is

$${G}_{{H}_{2}-IO}= \frac{0.245s-3.016}{{s}^{2}+13.09s+18.58}$$
(10)
$${G}_{{H}_{2}-FO}= \frac{-0.706 {e}^{-0.07s}}{{s}^{1.486}+2.23{s}^{0.789}+4.33}$$
(11)

From the above transfer functions, it is apparent that integer order transfer function has a positive zero which makes it a problem of inverse response, and to study the output performance, the response was compared for feedback loop with PI/PID and PI/PID with inverse response compensator, PPID and MPC. Fractional order system does not have any zero but has a delay term associated with it, so similarly, PI/PID, PI/PID with smith predictor and predicted PID and MPC were synthesized for the fractional system. Figures 3 and 4 show the comparison of controlled response for integer order and fractional order transfer functions associated with hydrogen valve loop, respectively.

Fig. 3
figure 3

Integer order system (hydrogen valve): controlled response by manipulating hydrogen inlet using PI, PID and PI with inverse compensator, PID with inverse compensator, MPC and PPID

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Fig. 4
figure 4

Fractional order system (hydrogen valve): controlled response by manipulating hydrogen inlet using PI, PID and PI with inverse compensator, PID with inverse compensator, MPC and PPID

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The error and TV values are given in the Tables 1 and 2 for integer order and fractional order, respectively. From Fig. 3, it can be inferred that PI works better as compared to PID controller with respect to inverse response rejection. Also, inverse response compensator is able to reduce the inverse response up to a good extent.

Table 1 Integer order system (hydrogen valve): ISE, IAE and TV values for different controllers
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Table 2 Fractional order system (hydrogen valve): ISE, IAE and TV values for different controllers
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The designed MPC in comparison to other controllers shows the best performance in terms of ISE, IAE and TV values. From Fig. 4, it is evident that Smith predictor is able to completely compensate the time delay associated with the fractional order transfer function. Even in the case of fractional order system, MPC gives the best results as compared to other controllers. Also, it can be inferred from Tables 1 and 2 that the values of ISE, IAE and TV are less for fractional system in comparison to integer order from Tables 1 and 2 and that the values of ISE, IAE and TV are less for fractional system in comparison to integer order system.

It is clear from the steady-state response and ISE, IAE and TV values that MPC performed better than other model-based controllers. Not only it reduced the inverse response in case of integer order system but has the smoothest control effort along with least integral error values in both the cases. But there is always a trade-off between either a shorter rise time or a good inverse response rejection while designing the MPC. Here, the most optimal values were chosen to get considerably good performance in both the respects.

Fractional order systems are considered to be more realistic in comparison to integer order systems because dynamically they behave more closely to the real systems. Looking upon the figures and tables, it can be observed that the controlled voltage response has better results for fractional order system.

Predictive PID is also designed for the above system, by the method proposed by Moradi et al. (2001). In terms of control efforts, PPID is smooth in comparison to normal PID controller because of its predicting capabilities.

Air pump

Here, voltage control is achieved by manipulating oxygen inlet rate. The transfer functions used to design different controllers are

$${G}_{Air-IO}=\frac{-0.102s-8.148}{{s}^{2}+13.09s+18.58}$$
(12)
$${G}_{Air-FO}=\frac{-0.408 {e}^{-0.017s}}{{0.592s}^{1.143}+1}$$
(13)

From the above transfer functions, PID, MPC, PPID and PID with Smith predictor for fractional order system are designed and the system output that is compared. Figure 5 shows the comparison of the designed controllers for the desired voltage of 5 V for the integer order transfer function, while Fig. 6 shows the responses of output voltage for the fractional order system. From the figures and the performance indices and from Tables 3 and 4, it is evident that MPC gives the best controlled response in comparison to PID and PPID in both the cases. Also, PPID is seen to give better results as compared to PID in terms of performance indices. Fractional order system has lower error as well as controller effort values.

Fig. 5
figure 5

Integer order system (air pump): controlled response by manipulating hydrogen inlet using PID, MPC and PPID

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Fig. 6
figure 6

Fractional order system (air pump): controlled response by manipulating hydrogen inlet using PID, MPC and PPID

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Table 3 Integer order system (air pump): ISE, IAE and TV values for different controllers
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Table 4 Fractional order system (air pump): ISE, IAE and TV values for different controllers
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TISO system

Since the output of both the loops is FC stack voltage, so to function as a single TISO unit, output from each loop is multiplied by the percent of their respective gain and then added to get the desired output. Below are the simulation results for different combinations of controllers to work as a single TISO PEMFC unit and are described along with the ISE, IAE and TV values for each case.

Integer order TISO model

Firstly, the integer order model adopted from the literature is used to form the TISO unit. Figure 9 shows comparison in case 1 when PID controller is used for the air pump loop, and different controllers are used for the hydrogen valve loop.

From Fig. 7 and the performance index values from Table 5, it is evident that the best combination is when PID with inverse response compensator is used in the hydrogen valve loop along with PID in air pump loop.

Fig. 7
figure 7

Integer order TISO system case 1: PID for air pump loop and PI, PID and PI with inverse response compensator, PID with inverse response compensator, MPC and PPID for hydrogen valve, for a desired output voltage of 5 V along with disturbance rejection

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Table 5 Integer order TISO system case 1: ISE, IAE and TV (control effort) values, where ‘H’ is used for hydrogen valve loop and ‘A’ is used for air pump loop. Also ‘IRC’ is used for inverse response compensation
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Secondly, the case is when MPC is used for air pump loop with all the designed controllers for hydrogen valve loop. Figure 8 shows the output response for case 2 for a set point of 5 V. Performance is compared on the basis of calculated IAE, ISE and TV values for each loop, which can be observed in Table 6.

Fig. 8
figure 8

Integer order TISO system case 2: MPC for air pump loop and PI, PID and PI with inverse response compensator, PID with inverse response compensator, MPC and PPID for hydrogen valve, for a desired output voltage of 5 V along with disturbance rejection

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Table 6 Integer order TISO system case 2: ISE, IAE and TV (control effort) values, where ‘H’ is used for hydrogen valve loop and ‘A’ is used for air pump loop. Also ‘IRC’ is used for inverse response compensation
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In case 2, it is clear enough that when MPC is used for both the loops, the best result is obtained in terms of ISE, IAE and TV values. Even when compared with first case, this combination works the best. Also, the inverse response due to hydrogen valve loop has completely vanished.

Case 3 for integer order TISO unit is when the predictive PID is used for air pump loop. Figure 9 shows the controlled voltage for this case when different controllers are used for hydrogen valve loop along PPID in air pump loop. The output response of each combination is compared on the basis of the values of performance indices given in Table 7. From the values, it is understandable that the best combination in third case is when PID with inverse response compensator is used for the hydrogen loop along with PPID for air pump loop.

Fig. 9
figure 9

Integer order TISO system case 3: PPID for air pump loop and PI, PID and PI with inverse response compensator, PID with inverse response compensator, MPC and PPID for hydrogen valve, for a desired output voltage of 5 V along with disturbance rejection

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Table 7 Integer order TISO system case 3: ISE, IAE and TV (control effort) values, where ‘H’ is used for hydrogen valve loop and ‘A’ is used for air pump loop. Also ‘IRC’ is used for inverse response compensation
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Considering the best combinations from all three cases, Figs. 10 and 11 represent the combinations when PID, MPC and PPID are used for the air pump loop and PID with inverse response compensator, MPC and PID with inverse response compensator for hydrogen valve loop are implemented, respectively. Table 8 compares the ISE, IAE and TV values for the best three combinations for integer order TISO system. Clearly, when MPC is used in both the loops, we get the best results in terms of least ISE, IAE and TV values for both the loops. Large TV values when PID is used with or without any compensator can be justified in Figs. 11 and 12, which show how the manipulated variables, hydrogen inlet and air inlet are varying, respectively. The negative inlet rate is observed because of the fact that while getting the mathematical model of the TISO PEMFC system, the process transfer functions have a negative gain so the controller gains are also negative, hence the negative manipulated variable. Also, it is clearly evident that the third case, that is, the one where predictive PID is used for air pump loop, gives better results when compared to basic PID in the first case; this shows that PPID works better because of its predicting capabilities.

Fig. 10
figure 10

Stack voltage response for integer order PEMFC TISO system best combinations

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Fig. 11
figure 11

Hydrogen inlet trend for integer order PEMFC TISO system best combinations

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Table 8 ISE, IAE and TV values of the best combinations from three cases for integer order PEMFC TISO system
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Fig. 12
figure 12

Air inlet trend for integer order PEMFC TISO system best combinations

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Fractional order TISO model

Consider the cases when fractional order transfer function is used for both the loops to form a single TISO PEMFC unit. Unlike the cases of integer order model, here both the systems have a time delay associated with them, and there are no zeros associated with the transfer functions, which account for no inverse response. In this scenario instead of inverse response compensator, Smith predictor is used with PI/PID for time delay compensation. Case 1 has six different combinations, fixing PID as the base controller for air pump loop. Controlled voltage for different combinations by using varied controllers in hydrogen loop is compared in Fig. 13 and Table 9.

Fig. 13
figure 13

Fractional order TISO system case 1: PID for air pump loop and PI, PID and PI with Smith predictor, PID with Smith predictor, MPC and PPID for hydrogen valve, for a desired output voltage of 5 V along with disturbance rejection

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Table 9 Fractional order TISO system case 1: ISE, IAE and TV (control effort) values, where ‘H’ is used for hydrogen valve loop and ‘A’ is used for air pump loop. Also ‘SP’ is used for Smith predictor
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From Table 9 and Fig. 13, it can be said that in case 1, the best combination in terms of error and TV values is when PID with Smith predictor is used for the hydrogen valve loop and PID for air pump loop. In case 2, PID with Smith predictor is fixed for the air pump loop, and controllers for hydrogen valve are varied. Figure 16 shows the controlled voltage response for six different combinations in this case, and Table 10 consists of the value of the performance indices.

Table 10 Fractional order TISO system case 2: ISE, IAE and TV (control effort) values, where ‘H’ is used for hydrogen valve loop and ‘A’ is used for air pump loop. Also ‘SP’ is used for Smith predictor
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From Fig. 14 and Table 8, it is clear that the best combination in case 2 is when PID with Smith predictor is used for both the loops. Similar to the above cases, Figs. 15 and 16 show the control response of different combinations for case 3, where MPC is used for air pump loop, and case 4, where PPID is used for air pump loop, respectively. Tables 11 and 12 consist of performance index values for case 3 and case 4, respectively.

Fig. 14
figure 14

Fractional order TISO system case 2: PID with Smith predictor for air pump loop and PI, PID and PI with Smith predictor, PID with Smith predictor, MPC and PPID for hydrogen valve, for a desired output voltage of 5 V along with disturbance rejection

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Fig. 15
figure 15

Fractional order TISO system case 3: MPC for air pump loop and PI, PID and PI with Smith predictor, PID with Smith predictor, MPC and PPID for hydrogen valve, for a desired output voltage of 5 V along with disturbance rejection

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Fig. 16
figure 16

Fractional order TISO system case 4: PPID for air pump loop and PI, PID and PI with Smith predictor, PID with Smith predictor, MPC and PPID for hydrogen valve, for a desired output voltage of 5 V along with disturbance rejection

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Table 11 Fractional order TISO system case 3: ISE, IAE and TV (control effort) values, where ‘H’ is used for hydrogen valve loop and ‘A’ is used for air pump loop. Also ‘SP’ is used for Smith predictor
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Table 12 Fractional order TISO system case 4: ISE, IAE and TV (control effort) values, where ‘H’ is used for hydrogen valve loop and ‘A’ is used for air pump loop. Also ‘SP’ is used for Smith predictor
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Out of all four cases for fractional order model, consisting of twenty-four combinations, it is found that the best result in terms of ISE, IAE and TV values is achieved when MPC is used in both the loops. This can be inferred from Fig. 17 and Table 13, which show the best combinations from each case for fractional order model. Also, case 4 where PPID is used for air pump loop shows better results in terms of less error and TV values in comparison to PID or PID with Smith predictor. Figures 18 and 19 show how manipulated variables, hydrogen inlet and oxygen inlet are varying as the stack voltage is being controlled by the best four combinations.

Fig. 17
figure 17

Stack voltage response for fractional order PEMFC TISO system best combinations

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Table 13 ISE, IAE and TV values of the best combinations from four cases for Fractional order PEMFC TISO system
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Fig. 18
figure 18

Hydrogen inlet trend for fractional order PEMFC TISO system best combinations

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Fig. 19
figure 19

Air inlet trend for fractional order PEMFC TISO system best combinations

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Conclusions

In this paper, a novel MISO control structure has been proposed for a PEMFC system. Integer order MIMO PEMFC system from literature was used and treated as a TISO system by fixing the output resistance, and therefore, the only controlled output was fuel cell voltage. The adopted model was then transformed into a fractional order TISO PEMFC model using genetic algorithm as the optimization technique. Model-based controllers, namely, PI, PID, MPC and predictive PID, were designed individually for both the loops, hydrogen valve and air pump, and then different combinations of controllers were used to work a single TISO unit for both integer and fractional order system. In the case of integer order MISO system, transfer function associated with hydrogen valve showed inverse response due to the presence of one right hand zero. From the simulation results, it was observed that PI controller and MPC works better in case of inverse response rejection as compared to other model-based controllers. The fractional order system had a time delay term associated with the transfer functions of both the loops; hence, Smith predictor was designed for time delay compensation. In the case of ultimate TISO unit, it was evident that when MPC was used for both the loops, better results were observed in terms of least ISE, IAE and TV value for integer order as well as fractional order systems. Also, whenever PPID was used in the air pump, it delivered better performance as compared to the basic PID controller. Fractional order model performed better in comparison to integer order system. For further scope, a single economic and non-linear MPC can be designed for the TISO system for even better performance. Also, temperature control management of the fuel cell stack can be considered as one of the parameters to be regulated, to increase the scope of work.