# Decoupled inputs sliding mode controllers for a fuel cell-supercapacitor module in hybrid generation applications

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## Abstract

The development of Multiple Input/Multiple Output (MIMO) sliding mode control setups for a fuel cell/supercapacitor module is presented in this paper. The main objective of the proposed controllers consists in simultaneously satisfying the demand and regulating the DC bus voltage, even in the presence of model uncertainties and strongly varying operating conditions. Two design approaches are utilized to synthetise different control setups, each one capable to robustly deal with such control challenges: on one hand, variable-gains first-order sliding mode and, on the other, supert-wisting second-order sliding mode control. The stability of the nonlinear controlled system is formally analysed. Extensive simulations are conducted, to comparatively assess the performance of the proposed MIMO sliding mode controllers. Both control setups exhibited highly satisfactory results, demonstrating robustness to external disturbances and parameter variations, proving to be more suitable than classic linear PID controllers.

## Keywords

Fuel cell Supercapacitors Hybrid system Sliding mode control## Introduction

During the last years, hybrid electric power generation systems based on renewable sources have been strongly studied and developed all over the world. In this sense, Fuel Cell (FC)-based systems result a promising choice due to its high efficiency, zero pollution, and sustainability [1, 2, 3]. In particular, the Proton-Exchange Membrane (PEM) FC-based systems are being extensively studied for transport [4] and stationary applications [5].

One of the main problems associated with the PEM FC is that abrupt changes in its current must be avoided, to prevent irreversible damages to the membranes, improve the \(H_2\) consumption, and reduce voltage drop transients [6]. For this reasons, FC are often complemented with a module capable to rapidly respond to fast load variations. To this end, supercapacitors (SC) show to be an efficient solution [7]. These FC/SC modules are frequently combined with renewable energy sources and a hydrogen production device to conform an auto-sustainable zero-emission Hybrid System (HS). Hydrogen production may be achieved then by taking advantage of the excess power generated by the renewable modules [8].

In an external control level, to optimize the HS performance as a whole, several supervision and energy management strategies that ensure effective operation have been implemented (e.g., [9, 10, 11]). Then, in an inner level, to fully control the HS power converters, diverse Sliding Mode (SM) techniques have proven to be adequate. Through a proper design, these techniques are able to provide excellent dynamic behaviour, robustness, and finite time convergence [12, 13, 14]. For instance, in [15], SM techniques are used for the control of the converter of a single fuel cell stack. In [16], a control strategy based on a linear PI and SM combination is presented for a FC/SC hybrid module. In [17], the control of a stationary autonomous hybrid electric power system is assessed, considering a relative degree approach and inductive loads, while [18] deals with a similar case, but applied to hybrid electric vehicles.

In such context, the authors have developed in [19] a sliding mode control setup for a FC/SC-based hybrid module topology like the one tackled in the present paper. In that preliminary work, a double SISO design approach has been followed to synthetize two complementary fixed-gains First-Order Sliding Mode (FOSM) controllers. The successful results obtained showed the suitability of sliding mode to robustly deal with this topology, in the presence of exacting operating conditions, such as highly variable load demand and varying wind.

Those encouraging results have thrusted the endeavours, leading to the enhanced control proposal presented in this article. Several new contributions have been developed. Among them, a diffeomorphism is utilized to deal with the inherent coupling of the system, allowing a MIMO more-efficient design approach. In addition, chattering reduction is attained. This undesirable phenomenon, particularly present in classic first-order sliding mode controllers, is successfully mitigated by developing a Second-Order Sliding Mode (SOSM) control setup. Another important issue, the stability of the nonlinear controlled system, is established by providing a formal analysis of its zero dynamics, which was omitted in [19].

Summarizing, this paper addresses the development of two different MIMO control setups for the FC/SC module, which is assumed to be part of an already existent HS that also comprises renewable power sources and an electrolyzer. Both proposed MIMO controllers, one based on variable-gain FOSM and the other on SOSM, prove to be robust. Simulation results considering parameter variations and variable load power demand are thoroughly assessed and discussed, including comparisons between both MIMO setups and with a classic PID control structure.

## System description

### Model of the FC/SC module

It is assumed that the power references for the controllers of each module are computed, at a higher level, by an external supervisory control that coordinates the power flows interaction over the entire system.

*i*th converter switch, as indicated in Fig. 2. Matrix functions

*f*(

*x*,

*t*) and

*g*(

*x*) are

*A*is the Tafel equation constant; and

*m*,

*n*are the constants of the mass transfer overvoltage.

### Supervisory control strategy

From this figure, \(P_{{\rm tot},r}\) is the total power demand in the DC bus; \(P_{\rm L}\) is the load power demand; \(P_{\rm E}\) and \(P_{{\rm E}_{{\rm {ref}}}}\) are the electrolyzer actual power and its reference; \(P_{\rm W}\), \(P_{{\rm W}_{{\rm {ref}}}}\) and \(P_{{\rm {W,max}}}\) are the WECS actual power delivered to the DC bus, the WECS power reference, and the maximum available wind power, respectively; \(P_{\rm {FC}}\) and \(P_{{\rm {FC}}_{{\rm {ref}}}}\) are the FC actual power and its reference; and \(P_{{\rm SC}_{\rm ref}}\) is the SC power reference, obtained from the power balance in the DC bus.

## FC/SC module SM controllers design

The FC/SC module control main objectives are to guarantee a constant DC bus voltage and to satisfy the module’s power demand, even under heavy load variations.

Besides, some practical restrictions on the FC current must be taken into account. In addition to a maximum admissible value for that current, it is also convenient to impose a bound to the FC current slew rate. Three main reasons can be cited to support the latter. Firstly, the inherent stack dynamics limits the velocity at which power can be delivered to the load, i.e., FC’s power cannot be set arbitrarily fast. Secondly, fast current changes may produce thermal stress at the catalyst surface, reducing the membrane lifetime. Finally, an adequate slew rate limitation may contribute to a fuel cell hydrogen consumption reduction [20].

To overcome such constraints, the FC is complemented by the SC to accurately maintain the power balance. This requires the SC converter to act rapidly to reject abrupt load variations and provide the so called peak shaving capability. Once the fast transient is extinguished, the control must enable the SC bank to recharge, slowly, to avoid FC overload.

Accordingly with the SM control theory, this set of control objectives must be formalized into the construction of suitable sliding variables \(S=[s_1\,s_2]^T\), comprising current references based on the power references computed online by the supervisory control. The objectives are successfully attained when the SM controllers establish sliding mode regimes on the surfaces defined by \(S=0\).

### Objective 1 into fuel cell sliding variable

### Objective 2 into supercapacitors sliding variable

### Control action design

#### Nominal control term design

It should be emphasized that the objective of the nominal control actions \(u_{iN}\) (\(i=1,2\)) is not to force the trajectories to reach the sliding surfaces, but to bring them to the region near to \(S=0\). The sliding mode control actions designed in the sequel will be in charge of ensuring reaching and permanence on \(S=0\).

#### Sliding mode control action decoupling

To simplify the design the sliding mode control inputs and switching conditions, an input to surface decoupling methodology is proposed. This objective is achieved by a suitable diffeomorphism defined as follows.

*T*(

*x*) a diffeomorphism that defines a new control action \(w_{{\rm SM}}\):

*S*yields

*T*(

*x*), as defined, results:

#### Variable-gains first-order sliding mode design

The design of gains \(W_{ci}\) and \(W_{ai}\) can be done by bounding the terms in (18) and conducting an error propagation analysis. In this case, the independence between both control actions (\(w_{1 {\rm SM}}\) and \(w_{2{\rm SM}}\)) allows to use this method without any major complications. It must be remarked that the initial set of gains obtained through this method are usually too conservative. For suitable application, further offline in-silico iterative tuning is required to simultaneously ensure the existence of the sliding mode regime while minimizing the output chattering. In this particular case, for the nominal system in the "Appendix", the obtained values of \(W_{ci}\) and \(W_{ai}\) were 5000 and 1000, respectively.

#### Second-order sliding mode design

## Stability analysis: zero dynamics

With the sliding mode regime secured by control (9), the stability of the closed loop must be established by analysing the zero dynamics of the system. That reduced internal dynamics can be obtained from (1) by taking \(S = 0\), which imposes two algebraic restrictions over the states, and by replacing *u* with the equivalent control actions \(u_{\rm {eq}}\) (12)–(13).

It can be concluded then that, given the local stability of the equilibrium point and the existence of an invariant region, where the divergence of the vector field never changes sign, the zero dynamics results stable and converges to the references, within the operating region.

## Simulation results

In this section, the performance of both proposed MIMO control setups, FOSM and SOSM, are assessed. For the analysis, extensive simulation were done, considering the FC/SC subsystem operating as part of the complete hybrid system, i.e., including the WECS, the electrolyzer, and a variable external load. As mentioned, the whole system is managed by a supervisory control that computes the power flow references for each module of the HS [19].

*Comparison with classic PID controller*

To compare the performance of the designed MIMO-sliding mode control setups, simulation results with the most ubiquitous control scheme, namely, Proportional–Integral–Derivative (PID) controller, are presented (most practical feedback loops in industry are PID-type structures). In this case, specifically, one PID controller was designed for the FC power converter to regulate its current. Besides, for the SC boost power converter, two PID controllers were used in nested loops, the inner one to regulate its current and the outer one to regulate the DC bus voltage. The PIDs were, initially, tuned using Ziegler–Nichols methods and, then, were further adjusted through extensive simulation.

As can be appreciated from these results, the PID controller is unable to accurately reject some perturbations, although its response under step-like load variations may be equivalent to the ones of FOSM and SOSM. On the other hand, both FOSM and SOSM show a considerably better disturbance rejection and with comparable computational cost.

## Conclusions

Two different MIMO sliding mode approaches, namely, FOSM and SOSM, were utilized to tackle the robust control of a fuel cell/sSupercapacitor module intended for a hybrid system. In both cases, the proposed control actions for the FC and the SC power converters comprised two terms. The first one is a nominal control action, resulting from a simplified equivalent control computation. The second one, in the FOSM case, is a variable-gain SM term, while in the SOSM case is a super-twisting-based control term.

The use of a decoupling diffeomorphism proved to be a convenient method to simplify the sliding mode control design of this FC/SC topology, which is inherently coupled due to the connection of their components to a common DC bus. The diffeomorphism generates a new set of auxiliary control inputs which are decoupled with respect to the sliding variables.

Both proposed MIMO controllers obtained in this way demonstrated to be suitable solutions to robustly fulfil the flow management requirements of the FC/SC module embedded in a HS. Their performances were intensively assessed through realistic in-silico tests, dealing with varying wind and abrupt load demand changes, and extensive model uncertainties.

From comparative analysis, it was established that faster voltage drop recovery time could be achieved with the variable-gains FOSM controller, but with appreciable chattering. On the other hand, the SOSM exhibited slower recovery time, but with neglectable chattering. This can be observed in the DC bus voltage, and the FC and the SC currents, but more importantly, in the switching surfaces and control actions applied to the power converters.

Further comparisons, in this case against classic PID controllers, allowed to conclude that both developed MIMO controllers setups, FOSM and SOSM, displayed better disturbance rejection and performance under abrupt and sinusoidal load variations than the PID structure. Besides, it is worthy to note that the online computational burden of the formers is not exceedingly larger than the one of the PID algorithms.

In addition, it was formally demonstrated in the paper that the FC/SC module, controlled with the proposed MIMO SM setups, is stable. This could be established by studying the zero dynamics, and analysing the equilibrium point and the resultant vector field in the operation range.

Finally, it can be remarked that the MIMO SM controllers setups developed for the FC/SC turns this subsystem into a versatile module, with enough flexibility to be used in different hybrid topologies. For instance, those that incorporates photovoltaic panels, lithium batteries or flow batteries, to name a few.

## Notes

### Acknowledgements

This research was supported by the Universidad Nacional de La Plata (UNLP), the CONICET and the ANPCyT, from Argentina. Authors also want to thank the support of the Fuel Cell Group at IRI (CSIC-UPC) and of the institute IOC at the Universitat Politècnica de Catalunya, Barcelona, España.

### Compliance with ethical standards

### Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

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