Cooperative Control of a Steam Reformer Solid Oxide Fuel Cell System for Stable Reformer Operation

Abstract

Solid oxide fuel cells (SOFCs) have complex characteristics, including a long time delay, strong thermoelectrical coupling, and multiple constraints. This leads to multiple control objectives, such as efficiently controlling the power output of the stack and considering the temperature constraints of multiple high-temperature components. Dealing with multiple objectives at the same time brings challenges to the design of SOFC system control. Based on the verified high-precision system model and aiming to achieve fast response, high efficiency, and thermal management, this paper first designs a generalized predictive controller (GPC) to realize the global optimization of the system. Then, through the actual test of the individual reformer, the reformer characteristics are analyzed, the standby controller to control the reformer temperature is designed, and the thermoelectric cooperative controller is constricted with the GPC. The results show that while fast power tracking, high efficiency, and multiple temperature constraints are realized by the controller, the temperature and methane conversion rate (MCR) of the reformer are stably controlled, providing a basis for further practical experiments of the SOFC system.

1. Introduction

As a popular new-energy technology, solid oxide fuel cells (SOFCs) have been recognized by many research institutions because of their various advantages, such as environmental protection, high efficiency, and strong fuel diversity [1]. In recent years, great progress has been made in materials [2], system design, and integration [3] for SOFCs, but in commercial applications, the lack of operating life is still the main bottleneck. Due to the complex characteristics of SOFCs, the operation of the system has a variety of requirements and limitations. Therefore, a focus for controlling the long operational life of the SOFC system is the design and effectiveness of the system controller.

SOFC systems have three main characteristics: “long time delay”, “strong thermoelectrical coupling”, and “multiple constraints” [4]. A “long time delay” means that when charge is input in the system, the electrochemical reaction in the stack can respond very quickly, but then takes several seconds for the fuel to enter the pipeline, and the temperature change in the system will gradually change after a few minutes, and the strong thermoelectric coupling of the SOFC will make it difficult for the system to enter the steady state quickly. For “strong thermoelectrical coupling”, the intensity of the exothermic electrochemical reaction in the SOFC changes accordingly during power switching, which leads to a temperature change. Meanwhile, the change in the stack temperature will, in turn, affect the discharge performance and lead to a change in output power; that is, the thermoelectric characteristics will highly influence each other. As for “multiple constraints”, in the operation of the system, various temperature constraints should be met to avoid metal fatigue, corrosion and oxidation of various components, thus affecting the working state and life of the system. At the same time, during fast or high power switching, it is necessary to avoid fuel shortage leading to accelerated degradation or even damage of the stack. Therefore, it is a challenge to design a system controller that can effectively solve the problems of load tracking, temperature security, fuel shortage, and efficiency optimization.

In order to drive the system to achieve different objectives, such as power tracking, high efficiency control optimization, and temperature constraint control of the SOFC system, many scholars have designed various control strategies for SOFC systems. Murshed et al. [5] developed two dynamic models of SOFC that could be used to simulate the steady state and dynamic behaviors of the system, and can also be used to design model-based system-wide controllers, such as nonlinear MPCs. Mueller et al. [6] have investigated the power tracking problem via current based fuel flow rate regulation, and Scarlett et al. [7] regulated the power output of the SOFC system by manipulating the inlet hydrogen molar flow rate. Some scholars have optimized system efficiency by partially considering fuel utilization, thermal management, and the flow of various bypass valves [8,9,10]. These show that the SOFC system can possess excellent power tracking capability with an appropriate system control strategy. Stiller et al. [11] controlled the current, fuel flow, and air flow by using a proportional–integral–derivative (PID)-type controller to realize improvements in power tracking, fuel utilization rate, and system efficiency. The temperature of the stack, the core component of the SOFC system, is also an important part of this control. Xi [12] optimized the system performance with the consideration of two temperature constraints: maximum cell temperature gradient and maximum PEN operating temperature. Pohjoranta et al. [13] applied GPC to control the maximum temperature and temperature gradient in the SOFC stack, which reduced the temperature difference and thermal stress over the stack, indicating that the multi-input control enables improved response time compared to single-input control. By controlling the dynamic SOFC model, Fardadi et al. [14] showed in detail the effects of the co-flow, counter-flow, cross-flow, and nonuniform air flow between the channels on the temperature distribution and thermal gradient under transient and steady state responses. These studies fully show that stack temperature and stack temperature gradient have a very important impact on stack performance.

With the deepening of the research, the ultimate goal of SOFC system control has become to achieve thermal safe, efficient, and fast power tracking. Wu et al. [15] used the non-dominated ranked particle swarm algorithm (NSPSO) for multi-objective optimization of efficiency and cost of SOFC systems under air leakage conditions, providing a basis for the controller to achieve the maximum electrical efficiency and the minimum cost of SOFC systems. Kouramas et al. [16] focused on the design of a model predictive controller (MPC) for controlling stack voltage and stack temperature, which can keep the SOFC voltage and temperature at the required value. Zhang et al. [17] developed a nonlinear MPC controller for a planar SOFC while using the moving horizon estimation (MHE) method, and this controller can control the power output of SOFC without changing fuel utilization or temperature. Murshed et al. [18] used linear and nonlinear MPCs for an independent SOFC system, and reported that nonlinear MPC can control the system to deal with greater load interference, although the calculation time is much higher. Sanandaji et al. [19] designed an MPC based on local linear parameter variation, which could satisfy the load demand, consider the constraints of input and output variables, and provide excellent performance on a very demanding current trajectory. As a type of MPC, GPCs can control complex system well under multiple constraints, and has good adaptability and robustness. Jiang et al. [20] proposed a new control strategy to maintain thermal constraints and optimize system efficiency while performing fast load tracking. The control strategy includes an optimal operating points (OOPs)-based feed-forward controller and a constrained generalized predictive control (CGPC) controller based on a Takagi–Sugeno (TS) fuzzy model.

Papurello et al. [21] demonstrated that adding an external reformer to form a steam reforming SOFC (SR-SOFC) system and using it under stable working conditions could enhance fuel diversity. Vrečko et al. [22] established an external SR-SOFC system model to analyze the temperature distribution in the stack, and designed a current-based temperature control strategy.

Since the SR-SOFC system has multiple components, there may be only individual components that become undesirable when the system is operating. Taking the reformer as an example, despite its operation in the normal zone, the efficiency of the internal reforming reaction may decrease due to internal carbon deposition, catalyst degradation and catalyst poisoning, and the reduction of hydrogen composition seriously affects the operation of the system. In the work of Zhang et al. [23], the impact of MCR has been thoroughly analyzed. Therefore, a separate controller can be designed for individual components or local areas, or the standby controller can be designed for possible situations. Jiang et al. [24] developed a constrained GPC for an independent reformer of SOFC, which can maintain the reforming temperature during fuel flow rate fluctuations caused by load variation. In order to solve the problem of burner temperature fluctuation caused by water flow fluctuation, Xue et al. [25] added a fuzzy controller to the local area of the running SOFC system, which effectively alleviated the influence of water flow fluctuation. Based on the results of fault diagnosis, Wang et al. [26] designed six different standby controllers for fault-tolerant control, so that the system power could be kept at the expected value when the state of the burner or stack became unsatisfactory.

As shown from these studies, the research into SOFC system control is in-depth and numerous; however, with the further development of the actual SOFC system, new system inputs are added for a more comprehensive control system, and more indicators need to be comprehensively considered, such as load tracking speed, temperature constraints of various components, the impact of different inputs on the system. Meanwhile, due to the addition of the reformer in the SR-SOFC system, the working performance of the reformer needs to be taken into account to ensure a high efficiency of the reforming reaction.

Considering these problems, based on the established and verified high-precision kW-class SR-SOFC system model [27], the system characteristics and reformer characteristics are analyzed, and a thermoelectric cooperative controller suitable for SR-SOFC system is designed. Firstly, generalized predictive control is adopted to control the complex SOFC system more effectively under multiple constraints, achieve fast power tracking, multiple temperature constraints, prevent fuel shortage, and realize the global optimization of the system. Second, the additional standby controller is added to keep the reformer temperature within the expected range, maintain a high-efficiency reforming reaction, and ensure the stability of hydrogen concentration in the stack fuel inlet.

The rest of the paper is arranged as follows. In Section 2, the working principle and modeling of the SR-SOFC system are introduced. Then, in Section 3, the characteristics of SR-SOFC system and reformer are analyzed to understand the problems existing in system control. The control strategy and controller are designed in Section 4. The control results are shown in the Section 5 and the conclusion is drawn in Section 6.

2. System Operation Principles and Modeling

The kW-class SR-SOFC system is mainly composed of four parts: stack, high-temperature components, normal-temperature components, and control components [28]. The stack is the core component of the system, which determines the power output of the system. The high-temperature components include the burner, reformer, and air heat exchanger, all three of which work at higher temperatures and ensure the heat supply to the stack. The normal-temperature components include the blower, gas flow meters, desulphurizers and other equipment, and the control components include controllers and external loads. The structure and operating principle of the SR-SOFC system are shown in Figure 1.

The main components of the SOFC system work at high temperatures [29]. In order to avoid deformation or even fracture of the material structure of the components caused by high-temperature thermal stress, in practical application, all components have a corresponding operating range [30]. According to previous research, the operating life of commercial SOFC systems should be more than 40,000 h [31]. In order to achieve efficient and long-term operation, SR-SOFC systems should meet the range of parameters shown in

Table 1. kW-class Steam Reformer SOFC System Input and Output Ranges.

Input	Definition	Range
U	Voltage for a single cell in a SOFC stack (V)	0.7–0.8
F_fuel	Flow rate of the fuel (102 mol/s)	0.1–5.0
F_air	Flow rate of the air (mol/s)	0.1–0.5
BP1	Ratio of fuel bypass to main flow	0–0.2
BP2	Ratio of air bypass to main flow	0–0.2
BP3	Air bypass of the reformer gas inlet (102 mol/s)	0.1–0.5
S/C	Ratio of the water to carbon	1.8–3.0
Pr	Ratio of the burner exhaust to the reformer	0–0.6
Output	Definition	Range
T_air	Temperature of the stack air inlet (K)	<1173
T_fuel	Temperature of the stack fuel inlet (K)	<1173
Tb	Temperature of the burner (K)	<1273
Tr	Temperature of the reformer (K)	<1173
Max.TPEN	Max PEN temperature (K)	[873, 1173]
Max.ΔTPEN	Max PEN temperature gradient (K/cm)	<8
ΔTin	Difference between the inlet gases to the stack (K)	<200
SE	System electrical efficiency (%)	>30
Pnet	System power output (W)	[0, 5000]

.

From Table 1, many temperature constraints show the importance of thermal management to the system. Since electrical management determines the power generation capacity of the system, it is very important for the development of SOFC system control to coordinate thermal management and power tracking while optimizing system efficiency.

3. System Characteristics Analysis

3.1. System Characteristics Analysis

Due to the addition of the reformer, the SR-SOFC system has more complex nonlinear and thermoelectric coupling characteristics. The increase in predictive time domain will make the control behavior more conservative and increase the stability, but will greatly increase the amount of calculations and improve the convergence time.

The fuel and preheated steam are reacted in the reformer to form a reforming gas containing hydrogen. The fuel flow rate changes during the power tracking, and as the reforming reaction is an endothermic reaction, the temperature of the reformer Tr will fluctuate. This will interfere with the operating state of the stack because the stack performance is very sensitive to temperature.

In addition, for SR-SOFC, the reforming reaction in the reformer is the beginning of the system operation process and the tail gas heat transfer in the reformer is the end of the system operation process. At the same time, due to strong thermoelectrical coupling, some system states, such as temperature and power, will oscillate during power switching. Therefore, the addition of the reformer makes the long time delay of the SR-SOFC system more serious and complex. That is, after the fluctuation of the input, the system enters the steady state for a longer time, which has a greater demand for fast and stable properties of the controller.

To sum up, the control of the system needs to be adjusted according to the characteristics of the large time delay. The GPC should be able to better control the response speed of the system and reduce the influence of the system oscillation. Meanwhile, it is necessary to cooperate with the thermal management of the reformer.

3.2. Reformer Characteristic Analysis

Due to the addition of the reformer, methane is the fuel of the SR-SOFC system. The MCR of the reformer determines the hydrogen content of the reformer gas entering the stack, which will greatly affect the electrochemical reaction in the stack and lead to changes in the electrothermal characteristics of the stack. When the MCR is too low, it may not only cause fuel shortage of the stack, but also increase the temperature gradient on the fuel side of the stack and threaten the temperature safety of the stack [32]. It may even lead to carbon deposition, deformation, and rupture of the single cell, seriously affecting the life of the stack [33]. Therefore, it is necessary to keep the MCR as high as possible. The of MCR is defined in Equation (1).

$$MCR=1-\frac{N_{C{H}_4,out}}{N_{C{H}_4,in}}$$

(1)

where$N_{C{H}_4,out}$ is the methane flow rate at the fuel outlet of the reformer and $N_{C{H}_4,in}$ is the methane flow rate at the fuel inlet of the reformer.

In order to explore the reforming characteristics of the reformer, the reformer of the kW-class SR-SOFC system is tested separately.

The experimental conditions were set as follows: A premixed and preheated steam and methane mixture gas was used, in which the methane fuel flow rate varied from 7 to 24 slpm. The ratio of water to carbon was 2.8 to ensure sufficient steam to avoid carbon deposition. The temperature was from 650 °C to 870 °C; in total, nine temperatures were tested. The reformer gas generated at different temperatures is obtained, and the gas composition is analyzed to obtain the reforming characteristics of the reformer.

From Figure 2, the reformer has a different MCR at different reforming temperatures. When the reforming temperature is higher than 720 °C, the hydrogen composition in the reformer gas is more than 70%. With the increase in temperature, the proportion of methane is close to zero, indicating that methane is almost completely consumed after the reformer. This is because the reforming reaction is endothermic; therefore, sufficient heat supply can effectively ensure a higher MCR.

It is worth noting that when the temperature is higher than 780 °C, the composition of hydrogen gradually decreases, while the proportion of carbon monoxide increases gradually. Although this does not affect MCR, there are some hidden dangers of long-term operation. This is because when the temperature is too high, the CH₄ will undergo a cracking reaction, resulting in carbon deposition, which affects the contact area of the catalyst, making the reforming efficiency lower. Moreover, too much carbon deposition will lead to reformer failure [34], affecting the operation of the system. Despite the fact that the problem can be better avoided by maintaining a sufficient amount of steam, but more steam can exacerbate its own fluctuations, leading to fluctuations in the burner temperature, which in turn affect the stable operation of the stack [25].

In previous studies, there is a wide range of temperature constraint for reformer temperature. The lower limit is usually determined by the working temperature of the reforming reaction, and the upper limit is often determined by the material properties of metal components. However, too high a temperature will not lead to higher MCR, but because too much thermal energy is supplied to the reformer, which affects the overall system efficiency and will accelerate the performance aging of the reformer, such as the degradation of materials and catalysts and catalyst poisoning.

To sum up, reformer temperature has a great impact on the MCR, and too low and too high Tr will reduce the MCR. So, it is necessary to effectively control the reformer temperature in the appropriate range while controlling the SOFC system.

4. Design of Control Strategy

4.1. GPC Control of SR-SOFC System

The control of SOFC systems using hydrogen as fuel was described in detail in previous work [20]. However, with the addition of catalytic reformer, the steam reformer systems that use methane as fuel have more complex nonlinear and thermocouple properties; thus, thermal management needs to be improved.

By adding two inputs of fuel bypass (BP₁) and air bypass (BP₂) directly into the burner, the fuel and air flow requirement of system thermal management can be reduced. Specifically, when the heat of the stack is insufficient and more supply is needed, the increase in the flow rate of the main fuel flow rate will cause more heat to be absorbed by the reforming reaction. Although more heat will be generated after hydrogen enters the stack, it will lower the temperature of the stack fuel inlet, leading to the increase in the stack temperature gradient, and the complex characteristics of SOFC thermoelectric coupling will produce thermal fluctuations, affecting the overall efficiency of the system. The addition of BP₁ can make the fuel directly into the burner to release heat, avoiding the overall heat balance being affected by the additional fuel. When the heat released by the electrochemical reaction of the stack can be self-maintained, the burner temperature can also be quickly controlled by the regulation of BP₂ to avoid excess heat supply and bring a greater burden to the temperature constraint of the stack.

The GPC algorithm can be divided into three parts: output prediction, rolling optimization, and feedback revision. The future system output within the defined prediction range $\left(k+N_1,k+N_2\right)$ is calculated by the following equation, which is derived from the CARIMA model.

$$\widehat{y}\left(k+j\right)=G_j\left(z^{-1}\right)\Delta U\left(k+j-1\right)+F_j\left(z^{-1}\right)y\left(k\right)+H_j\left(z^{-1}\right)\Delta U\left(k-1\right)$$

(2)

where $\widehat{y}\left(k+j\right)$ is (k+j)th system prediction output, $G_j\left(z^{-1}\right)$, $F_j\left(z^{-1}\right)$, and $H_j\left(z^{-1}\right)$ are the polynomial of Diophantine equation, $y\left(k\right)$ is (k)th system output, $\Delta U\left(k+j-1\right)$ is the system input control increment sequence in (k + j − 1)th instance, and $N_1$ and $N_2$ are the minimum and maximum prediction time domain, respectively. The three polynomials can be calculated from the prospective control increment sequence, the known output sequence, and the known input increment sequence separately.

The reference trajectory can be designed as follows:

$$\left\{\begin{array}{c}{y}_r\left(k\right)=y\left(k\right)\\ y_r\left(k+j\right)={\alpha }^j\cdot y\left(k\right)+\left(1-{\alpha }^j\right)\cdot s\end{array}\right\}$$

(3)

where $y_r\left(k\right)$ is the reference value for the kth instance, $s$ is the setting value, and $\alpha$ is the smoothing factor. In order to make the output of the system approach the reference trajectory perfectly, the optimal control increment sequence can be obtained by minimizing the multi-level objective function:

$$J=E\left\{{\displaystyle \sum }_{j=1}^{N_2}{[\widehat{y}\left(k+j\right)-y_r\left(k+j\right)]}^2+{\displaystyle \sum }_{j=1}^{Nu}{\eta }_j{[\Delta U\left(k+j-1\right)]}^2\right\}$$

(4)

where $E\left\{\ast \right\}$ is the expected operator, ${\eta }_j$ is the control-weighted sequence that limits the amplitude of the control sequence, and $Nu$ is the control time domain.

In the TS-CGPC control method based on the optimal operating point (OOP) of the SOFC system, the TS fuzzy model is corrected by the least square method at each operating cycle, and the CARIMA model polynomial used by the GPC algorithm can be obtained from the consequent parameters of the TS fuzzy model. Therefore, the GPC prediction model can be updated and corrected at each operating cycle, ensuring the accuracy of the prediction output and control precision. This control method can achieve fast and accurate power tracking, while the optimization of thermal management and system efficiency can be realized simultaneously, and the occurrence of fuel deficit can be avoided. The specific design of the controller has been introduced in detail in previous work [20].

The definition of system efficiency, $SE$, is as follows:

$$SE=\frac{Pnet}{F\_fuel\cdot LH{V}_{fuel}}$$

(5)

where $Pnet$ is the system output power, $F\_fuel$ is the total fuel flow into the system, and $LH{V}_{fuel}$ is the lower heating value of the fuel.

Due to the complex characteristics of the SR-SOFC system, it is necessary to accurately adjust the control time domain and predictive time domain of the controller. With the increase in the control time domain Nu, the prediction error can be reduced, but excess Nu can easily increase the jitter of the control and reduce the stability. The increase in prediction time domain Pd will make the control behavior more conservative and increase the stability, but it will greatly increase the amount of calculation and improve the convergence time. In this paper, after testing, Nu = 2 and Pd = 4 are the most suitable parameters, and the remaining parameters of the controller are shown in

Table 2. System controller parameters.

Parameter	Description	Value
N1	Minimum prediction time domain	1
N2	Maximum prediction time domain	4
Nu	Control time domain	2
α	Smooth factor	0.95
λ	Control-weighting constant	6
c	Fuzzy rules number	4
μ	Study ratio	0.98
m	Fuzzy factor	2

. The computer configuration is 3.0 GHz CPU, 16 GB RAM, Windows 64 bit system; the software is MATLAB. The simulation is carried out on the computer and the sampling time is 0.1 s.

4.2. Fuzzy Control of Reformer Temperature

In the process of power tracking, because of the “large time delay” of the SOFC system, the power response of the stack is very fast, while the rate of temperature change is slower. Therefore, it is necessary to avoid directly affecting the working state of the stack in the control. That is, under the premise of ensuring the power tracking of the stack, the Tr should be controlled in the appropriate range.

The controllable variables of the SOFC system are limited, and the fuel and air in the main pipe will directly affect the temperature of the stack and the burner, resulting in changes in the electrothermal characteristics of the stack and affecting the power tracking of the system. Other control variables should be taken into account, such as the air bypass valve BP₃, which passes directly into the reformer gas inlet, and the ratio of the burner exhaust to the reformer Pr.

It should be noted that because the electrical efficiency of the stack is directly related to its temperature, and the burner temperature will quickly affect the stack temperature. In order to ensure the performance of stack power tracking, keeping the stack temperature stable with the burner temperature is a priority. Under this premise, the control of the reformer temperature Tr is considered from two schemes, respectively.

When the Tr is too low, the Pr can be adjusted to increase the heat supply of the reformer. By ensuring the stability of the reforming reaction efficiency, avoid affecting the stack temperature due to the increase of the reforming endothermic reaction, and restrain the influence of fuel fluctuation on the stack, which will not have a great impact on the thermal balance of the system.

When the Tr is too high, the flow rate of the BP₃ can be adjusted to directly reduce the temperature of the gas inlet, so as to quickly affect the Tr and avoid the influence of high temperature on the material and catalyst of the reformer. However, since the increase in BP₃ will increase the parasitic power of the blower, when Tr is too low, the flow rate of BP₃ needs to be considered first in the control strategy to avoid adding excess parasitic power if it is already 0.

On this basis, the adjustment of BP₃ and Pr will not have a direct effect on the stack temperature and burner temperature.

To implement this control strategy, it is more appropriate to add an additional standby controller to cooperate with the GPC. Due to the nonlinear and strong coupling characteristics of SOFC system, and in the system experiment, it often needs to be adjusted according to experience. As a method of using expert knowledge to control the controlled object, fuzzy control is essentially a kind of nonlinear control that belongs to intelligent control and is suitable for the characteristics of SOFC.

According to the above analysis, in this paper, the fuzzy controller C1 with input Tr and output BP₃ and the fuzzy controller C2 with input Tr and output Pr are designed, respectively. The judgment will be made first, and controller C2 will be started if and only if the value of BP₃ is 0, otherwise controller C1 will be started first. The design steps are as follows:

• Set the input and output domain of the controller

Based on the analysis in Section 3.2, the domain of Tr is set in the range of [720, 830] °C. The domain of BP₃ is set in the range of [0.1, 0.5] 10² mol/s, and the domain of Pr is set in the range of [0, 0.6] based on the characteristics and parameters of the SOFC system. Both the input and output domains are divided into seven levels.

• The division of fuzzy subset of domain and the establishment of membership function.

The fuzzy subsets of Tr, BP₃, and Pr are expressed as {L3, L2, L1, OK, H1, H2, H3}. L1, L2, and L3 denote small, medium, and large deviations that are lower than the ideal value; OK represents the expected value; and H1, H2, and H3 represent small, medium, and large deviations higher than the expected value, respectively.

In addition, the membership function of the domain needs to be determined. The membership function is used to describe the membership relationship of an element to a fuzzy subset on the set. As can be seen from the experimental results in Figure 2, when the Tr is lower than the constraint range, it will have a greater impact on the MCR, so when designing the Tr membership function, the values of each fuzzy subset are uneven. Therefore, according to the practical experience, the triangular membership function is selected by the membership functions of Tr, BP₃, and Pr in this paper, as shown in Figure 3.

• Design fuzzy control rules according to control strategy.

After analysis, it is known that Tr has a subsequent relationship with BP₃ or Pr, so in order to keep Tr in the constraint range, it is necessary to quickly reduce BP₃ or increase Pr when Tr is too low, and slightly reduce BP₃ or slightly increase Pr when Tr is slightly low, and vice versa. Similarly, when the Tr is within the expected range, the controller does not need to adjust the output to avoid continuous temperature fluctuations. In the meantime, in the subsequent control, the control rules can be constantly adjusted according to practical experience.

Table 3. Fuzzy control rules of controller.

Number	Fuzzy Control Rules of C1	Fuzzy Control Rules of C2
1	If (Tr is L3) then (BP3 is H3)	If (Tr is L3) then (Pr is H3)
2	If (Tr is L2) then (BP3 is H2)	If (Tr is L2) then (Pr is H2)
3	If (Tr is L1) then (BP3 is H1)	If (Tr is L1) then (Pr is H1)
4	If (Tr is OK) then (BP3 is OK)	If (Tr is OK) then (Pr is OK)
5	If (Tr is H1) then (BP3 is L1)	If (Tr is H1) then (Pr is L1)
6	If (Tr is H2) then (BP3 is L2)	If (Tr is H2) then (Pr is L2)
7	If (Tr is H3) then (BP3 is L3)	If (Tr is H3) then (Pr is L3)

shows the control rules of the designed fuzzy controller.

5. System Control and Results

The target power is set between 1000~5000 W as the external load demand of the SR-SOFC system. Three stages t1, t2, and t3 are set, and the target power is 2500 W/4500 W/1500 W, respectively. The target power will be switched in turn with time, so as to test the control effect of the controller. Due to the characteristics of the SR-SOFC system, there is large thermal inertia in the process of power switching. Therefore, the power simulation time of each stage is set to 5000 s, in order to refer to the actual situation in reality and ensure the safety of the system temperature. A single GPC controller and cooperative controller are used to compare the effect of adding fuzzy control to the reformer temperature.

The control effect is shown in Figure 4. Figure 4a,b show the output power and system efficiency of the system, respectively. As can be seen in the Figure 4, the designed TS-CGPC controller can control the output power of the system, tracking the target power quickly and stably, and the use of the cooperative controller has little impact on the performance of the GPC, ensuring the efficient and stable operation of the system.

Figure 4c,d show the temperature and MCR of the reformer. For the cooperative controller, the Tr is controlled between 720 °C to 830 °C most of the time, which shows that the cooperative controller is effective, and the stable and less fluctuating reformer temperature can also ensure the thermal balance of the system. Although there is a short period of beyond the excepted range in the moment of power switching, this is because power tracking is the highest priority, and overly aggressive control can interfere with the performance of power tracking. Specifically, in the t1 stage, because the reformer works in the middle of the expected range, the controllers C1 and C2 do not work, so there is no significant difference in the curve. After entering the t2 stage, due to the target power is close to the upper limit of the system output power, the temperature of the reformer rises rapidly, and the standby controller starts up correspondingly. After briefly exceeding the expected range, it quickly returns to the expected range and gradually converges steadily. In the t3 stage, with the rapid decrease in the target power, the temperature of the reformer also changes. However, because it is just at a higher temperature, the decreasing speed is slower compared to the t2 stage, and the cooperative controller has been effectively controlled before the lower limit of the desired range is exceeded, which eventually leads to a gradual and stable convergence in the expected range. The red curve in Figure 4d also shows that the reformer can stably maintain a high MCR, and the MCR is always above 70% in the appropriate temperature range. The stable MCR can ensure the stability of hydrogen concentration in the stack, reduce the burden of thermal management of the system, and ensure the stable work of the stack.

For single GPC, the temperature and efficiency of the reformer are represented by blue curves in Figure 4c,d. Although the whole system is still working in the working range shown in Table 1, the temperature sometimes cannot converge quickly, which affects the MCR of the reformer and cannot always be stable during the operation of the system.

In spite of the unseen effects in a short period of time, the fluctuation of reforming gas composition will have an impact on the interior of the stack. With the continuous operation of the system, it will lead to accelerated degradation of the stack, and even make the internal electrochemical reaction unbalanced, resulting in structural damage and gas leakage.

6. Conclusions

For a complex SR-SOFC system, effective control of power tracking and compliance with a variety of temperature constraints, as well as system efficiency optimization, determine the high efficiency and long operational life of the SOFC system. To solve the problem of multi-objective optimization of SR-SOFC and ensure the efficient reforming reaction of the reformer, a thermoelectric cooperative controller is designed in this paper.

By analyzing the system characteristics, this paper shows that as the reformer increases, the system characteristics become more complex, so additional attention to reformer temperature and performance stability is needed in addition to more optimization objectives. Considering the new control variables, the GPC for global optimization of the system control was designed. In additional, based on the actual test of the individual reformer, the reformer characteristics were analyzed. It was pointed out that the reformer temperature should not be too high or too low, but should be controlled in a more precise expected range. The standby controller was designed based on fuzzy control to form a thermoelectric cooperative controller with the GPC. The results showed that the thermoelectric cooperative controller can achieve fast system power tracking, system efficiency optimization, and multiple temperature constraints while effectively controlling the reformer temperature and maintaining a stable and high MCR. The influence of hydrogen concentration fluctuations on the stack operating state is avoided, which provides a basis for further practical experiments, thus realizing the stable operation of the stack and the high efficiency and long operational life of the SOFC system.