## 1. Introduction

A solid oxide fuel cell (SOFC) is a promising fuel cell technology that can be used in co-generation systems for widespread commercial applications [

1]. No moving parts, quiet operation, low pollution, and high efficiency are the advantages of fuel cells. Many researchers have discussed the considerable environmental benefits of fuel cell technology [

2]. Using SOFC technology also involves the depletion of greenhouse gas emissions when compared with traditional energy generation methods. Moreover, there is interest in the development of the fuel cell technology as a substitute for internal combustion [

3]. In general, a SOFC is operated over a wide temperature range, from 873–1273 K, which leads to high energy conversion efficiency, fuel flexibility, and the possibility for combined heat and power systems [

4].

The SOFC can use various fuel types, such as methane, methanol, ethanol, and other hydrocarbons, due to its high operating temperature range. Even though high-chain hydrocarbons, such as n-dodecane, can be used as a fuel for the SOFC system, methane is generally considered for SOFC operation, due to its availability, highest hydrogen to carbon ratio in hydrocarbon substances and low cost [

5,

6]. As the long-chain hydrocarbon fuel contains high carbon and it has a low hydrogen-to-carbon ratio, fuel processing is required for breaking this fuel down into small substances and increasing the hydrogen-to-carbon ratio for the avoidance of a carbon formation in SOFC [

7]. In general, methane can be synthesized, as a major product or a by-product, from many chemical processes, or even formation process [

8]. In addition, methane in biogas can be directly fed to SOFC under dry conditions; however, there is a risk that is associated with contaminants in biogas when it is introduced to commercial SOFCs while using Ni-YSZ anode [

9,

10]. Therefore, either commercialized material development or fuel processing with cleaning technologies is required. There are many hydrogen production processes to convert methane into hydrogen-rich gas, such as steam reforming, partial oxidation, and autothermal reforming. However, the methane steam reforming process is perhaps the most well-established technology and it is widely used to produce hydrogen in the conventional SOFC system [

11]. Internal reforming process can occur within the fuel cell to directly convert hydrocarbon fuel into hydrogen-rich gas since SOFC is operated at high temperatures [

12]. The direct internal reforming (DIR) includes the reforming and water gas shift reaction rates and enthalpies, with these reactions occurring on the surface of the anode. The DIR of methane in an anode of the SOFC can possibly be due to the high temperatures that are present in the SOFC anode and it enables high energy conversion efficiency for the system [

13]. However, the complete DIR-SOFC showed poorer performance when compared to the DIR-SOFC with partial external reforming, and thus using the pre-reformer with DIR-SOFC might be a suitable operational option [

14]. Nevertheless, the internal reforming reaction that occurred at the anode leads to complicated dynamic behavior. Additionally, the steam reforming process is a highly endothermic reaction [

15]. The endothermic cooling effect creates a temperature gradient inside the fuel cell stack. The thermal gradient in the cell stack is significantly managed to minimize, because this gradient results in thermal stresses that leads to cell degradation and failure [

16,

17]. Consequently, efficient control is needed for preventing thermal cracking and ensuring system stability for this process.

The model of SOFC in cell, stack, and system levels has been proposed, and each type of the SOFC model is employed for different purposes, i.e., design, improvement, control, and optimization [

18]. Dynamic modeling is especially beneficial for dynamic system analysis, as well as in control design. SOFC operations are often subjected to transient conditions, and, as a result, the fuel cell dynamics have been increasingly considered in modeling activities [

19]. Several published works have concentrated on the dynamic modeling and the control of solid oxide fuel cells [

20,

21]. Li and Choi [

22] studied the control of the power output of an SOFC by applying proportional-integral (PI) controllers to maintaining fuel utilization and voltage as the current of the stack changed. To keep the voltage output under load changes, Chaisantikulwat et al. [

19] developed an SOFC dynamic model and a feedback control scheme with a PI controller to control cell voltage by manipulating the concentration of H

_{2}. The low-order dynamic model that was derived from the step responses was used for designing feedback control. The results showed that the feedback PI controller was able to maintain a constant SOFC voltage for small step changes in the current load. Furthermore, a dynamic model was used to investigate the dynamics of the SOFC stack and design control strategies [

23]. A proportional-integral-derivative (PID) controller was implemented to maintain the outlet fuel temperature and the fuel utilization of a planar anode-supported, direct internal reforming solid oxide fuel cell under intermediate-temperature operation. The feed air was used to maintain the outlet fuel temperature when a disturbance in the current density occurred. A control strategy must be more effective in avoiding oscillatory control action as well as in preventing potentially damaging temperature gradients under a higher magnitude of load changes. Stiller et al. [

24] developed a dynamic model for the control of an SOFC and gas turbine hybrid system. The SOFC power, fuel utilization, air flow, and cell temperature were controlled while using a proportional–integral–derivative (PID) type controller. However, the conventional PID controller cannot guarantee stability and performance when large disturbances occur.

Model predictive control (MPC), which is a multivariable control algorithm, computes a controller action while using a process model to predict the processes output trajectory in the future [

25]. The implementation of MPC requires the identification of an internal process model. Therefore, applying model-based controllers, such as MPC, is more challenging when compared to PI or PID controllers for which explicit controller equations exist [

26]. Zhang et al. [

25] developed the nonlinear MPC controller (NMPC) for a planar SOFC while using the moving horizon estimation (MHE) method. The current density and molar flow rates of fuel and air were manipulated variables to control the output power, fuel utilization, and cell temperature. The proposed NMPC controller can drive the SOFC following the desired output trajectory when the power output was changed under constant fuel utilization and temperature. In addition, many real chemical processes involve a high degree of parameter uncertainty. Some studies have focused on the development of a robust MPC to handle nonlinear systems and guarantee system stability, as a traditional MPC algorithm is unable to address plant model uncertainties [

27,

28]. Kothare et al. [

29] synthesized a robust MPC algorithm that explicitly incorporated plant model uncertainties. The state feedback control law was obtained by minimizing the worst-case performance cost. This worst-case scenario was used by the simultaneous design and control methodologies to evaluate the process cost function and constraints that were considered in the process [

30]. Manthanwar et al. [

31] studied the derivation of the explicit control strategy while using a min–max formulation to safeguard against the worst-case uncertainty problem. This guarantee process feasibility as well as process stability for efficient plant operation. A convex optimization problem with linear matrix inequalities (LMIs) constraints was formulated. Bumroongsri and Kheawhom [

32] proposed a robust MPC for uncertain polytopic discrete-time systems. In addition, an ellipsoidal off-line MPC strategy for linear parameter varying (LPV) systems was studied. The smallest ellipsoid that contained the present measured state was determined in each sequence of ellipsoids and the scheduling parameter for LPV was measured on-line. Pannocchia [

33] also developed a robust MPC algorithm to stabilize the system that was described while using a linear time-varying (LTV) model. Kouramas et al. [

34] focused on the design of an MPC controller to control the cell voltage and cell temperature. The results showed that the controller was able to maintain the SOFC voltage and temperature at the desired values. A comprehensive model of SOFC behavior involves numerous complex phenomena, which include electrochemical reaction and the thermal and mechanical properties of the materials. Thus, the SOFC model involves a great deal of parameter uncertainty; the control design should take the model uncertainty into account. For an on-line synthesis approach, the optimization requirement leads to significant amounts of on-line MPC computational time. When MPC incorporates the model uncertainty, the resulting on-line computation will significantly grow with the number of vertices of the uncertainty set. As a result, an off-line synthesis approach is a focus for generating a robust MPC for an uncertain model. With the off-line approach, the computation of a robust MPC is significantly reduced, with only minor losses in its control performance. Wan and Kothare [

35] implemented an off-line LMIs for robust MPC while using an asymptotically stable invariant ellipsoid. An off-line robust output feedback MPC approach can certify the robust stability of the closed-loop system in the presence of constraints and it can stabilize both polytopic uncertain systems and norm bound uncertain systems.

A model-based control system should be designed, while taking uncertain parameters into account, to avoid physical damage and achieve high energy efficiency, as the dynamics of SOFC, especially with a DIR operation, are complicated and its model consists of many key parameters. This study concentrates on control design for the DIR SOFC fed by methane-rich gas. The investigation of transient responses of an SOFC by changing the current density, the air and fuel inlet temperatures, and the air and fuel inlet molar flow rates in terms of velocity, is reported. A MIMO control approach using an off-line robust MPC algorithm with LTV system is implemented to control the SOFC with uncertainty in cell voltage. The paper is organized, as follows:

Section 2 presents the mathematical model for the SOFC;

Section 3 gives a brief review of the robust MPC algorithm;

Section 4 and

Section 5 outline the application, and results and discussion of the proposed off-line robust MPC to the SOFC; and lastly,

Section 6 presents the conclusions.