基于模型预测控制方法的聚合物电解质燃料电池多输入单输出电压控制

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A multi-input and single-output voltage control for a polymer electrolyte fuel cell system using model predictive control method

基于模型预测控制方法的聚合物电解质燃料电池多输入单输出电压控制

Introduction

Because of its high efficiency, fast start-up, low operating temperature, and environmental friendliness, polymer electrolyte fuel cells (PEFCs) have been more and more employed in portable, stationary, and transport units to replace traditional power sources.

由于其效率高、启动快、工作温度低和环境友好，聚合物电解质燃料电池（PEFCs）已被越来越多地应用于便携式、固定式和运输装置，以取代传统电源。

According to the E4tech Fuel Cell Industry Review,5 the fuel cell (FC) sector showed a remarkable increase in supply chain of more than a gigawatt of shipments with PEFCs dominating the s h i p m e n t s b o t h i n n u m b e r a n d c a p a c i t y .

根据燃料电池行业评论5，燃料电池（FC）行业显示了供应链的显著增长，出货量超过了千兆瓦，其中PEFC占主导地位，包括Nu m b e r和C a p a c i t y。

A l t h o u g h t h e PEFCs have achieved a big progress regarding its development, the reliability issue still remains a major challenge that hinders its commercialization.

尽管PEFC在开发方面取得了重大进展，但可靠性问题仍然是阻碍其商业化的主要挑战。

To ensure reliable operation as a power source, a PEFC system should be able to supply steady voltage in its applications despite any disturbance to the working load, which is an important demand in electrical equipment use.

为了确保作为电源的可靠运行，PEFC系统应能够在其应用中提供稳定的电压，尽管对工作负载有任何干扰，这是电气设备使用中的重要要求。

Efficient and robust control strategies are considered as one of the key solutions to ensure the fuel cell system's high reliability.

A significant number of research has been conducted to develop solid and powerful control algorithms for PEFC systems to provide steady output voltage.

为了给PEFC系统提供稳定的输出电压，已经进行了大量的研究，开发了坚实而强大的控制算法。

高效和强大的控制策略被认为是确保燃料电池系统高可靠性的关键解决方案之一。

controller to stabilize the fuel cell system's voltage by changing the air flow rate under system variations.

控制器通过改变系统变化下的空气流速来稳定燃料电池系统的电压。

It was found that the adaptive controller can effectively achieve the desired command subject to external disturbances.

研究发现，自适应控制器可以有效地实现受外部干扰的理想指令。

Wang et al8 employed H∞ solid control strategy to improve PEFC system's stability, finding that the controller was deemed effective to control the output voltage at the desired value under different loading conditions by adjusting the air flow rate.

Wang等人8采用了H∞固体控制策略来提高PEFC系统的稳定性，发现该控制器被认为在不同的负载条件下通过调整空气流速有效地将输出电压控制在理想值。

In another paper, also by Wang et al,12 multivariable H∞ controllers are proposed to provide steady output response by controlling both the air and hydrogen flow rates.

在另一篇论文中，也是由Wang等人12提出的，多变量H∞控制器通过控制空气和氢气的流量来提供稳定的输出响应。

Chen13 increased the relative stability of a PEFC system by controlling the hydrogen and oxygen input flow rates using the designed state feedback controller.

Chen13通过使用设计的状态反馈控制器控制氢气和氧气的输入流量，提高了PEFC系统的相对稳定性。

Woo and Benziger14 incorporated a standard PID feedback controller in a PEFC system to achieve desired power output by limiting the hydrogen feed.

Woo和Benziger14在PEFC系统中加入了一个标准的PID反馈控制器，以通过限制氢气的供给来实现理想的功率输出。

Fragiacomo and Piraino15 keep the PEFC system working in steady state using a hybrid control algorithm, which combined a fuzzy logic controller and an error minimization control method.

Fragiacomo和Piraino15使用混合控制算法保持PEFC系统工作在稳定状态，该算法结合了模糊逻辑控制器和误差最小化控制方法。

Narjiss et al16 proposed a digital signal process controller to regulate the PEFC system's voltage and current, making it more suitable for transport application.

Narjiss等人16提出了一个数字信号过程控制器来调节PEFC系统的电压和电流，使其更适合于运输应用。

The above mentioned studies have developed various well-qualified control methods to stabilize the PEFC system's performance.

上述研究已经开发了各种合格的控制方法，以稳定PEFC系统的性能。

However, there are still some limitations that can be improved in future studies. For one thing, most previous control systems are only limited to one single input, the effects of other factors in the fuel cell system are neglected; for another, there are still some time delays and oscillation in regard to the control performance. Reliable and highly efficient control algorithms are still much in need.

然而，在未来的研究中，仍有一些限制可以改进。首先，以前的大多数控制系统只限于一个单一的输入，燃料电池系统中其他因素的影响被忽略了；另一方面，在控制性能方面仍然存在一些时间延迟和振荡。可靠和高效的控制算法仍然是非常需要的。

Due to its high complexity, strong nonlinearity, and especially series of constraints on the operation parameters,17 the PEFC system requires much more advanced and robust control algorithms to ensure its reliable and safe operation.

由于其高复杂性、强非线性，特别是对运行参数的一系列限制，17 PEFC系统需要更先进和强大的控制算法，以确保其可靠和安全运行。

As an advanced process control method, model predictive control (MPC) controller usually excels in solving control schemes with multi input variables and a bunch of constraints,18-20 which makes it especially suitable for PEFC system's control application.

作为一种先进的过程控制方法，模型预测控制（MPC）控制器通常擅长解决具有多输入变量和一堆约束条件的控制方案，18-20这使得它特别适用于PEFC系统的控制应用。

MPC controllers have been widely used in PEFC system industry thanks to its high robustness and optimal performance. Mengi21 designed three different MPC based hybrid controllers to investigate the elimination of reactive power in a medium-scale PEFC system and found out PIλ-MPC controller presented the best performance.

由于MPC控制器的高鲁棒性和最佳性能，它已被广泛应用于PEFC系统行业。Mengi21设计了三种不同的基于MPC的混合控制器来研究在一个中等规模的PEFC系统中消除无功功率的问题，发现PIλ-MPC控制器呈现出最佳性能。

He et al22 proposed a novel MPC controller to enhance a PEFC system's operation by regulating its hydrogen circulation. Zhang et al23 implemented a novel MPC controller in an open cathode PEFC system to manipulate its stack temperature at the desired value despite the changes on the working load.

He等人22提出了一种新型的MPC控制器，通过调节氢气的循环来提高PEFC系统的运行。Zhang等人23在开式阴极PEFC系统中实施了一种新型的MPC控制器，尽管工作负荷发生了变化，但仍能将其烟囱温度控制在理想值。

Goshtasbi and Ersal20 developed a linear time-varying MPC framework for an automotive PEFC system to resist its degradation. Karimi et al24 developed an adaptive neuro-fuzzy interface-based MPC method for the voltage control of a PEFC system and the results revealed that the new algorithm can effectively stabilize the output voltage with satisfying all constraints.

Goshtasbi和Ersal20为汽车PEFC系统开发了一个线性时变MPC框架，以抵御其退化。Karimi等人24为PEFC系统的电压控制开发了一种基于神经模糊接口的自适应MPC方法，结果显示，新算法可以在满足所有约束条件的情况下有效地稳定输出电压。

Chatrattanawet et al25 proposed both a traditional MPC and a novel robust linear time-varying MPC for PEFC control and found out the novel MPC can ensure the PEFC system's stable operation. Ouyang et al26 proposed a nonlinear MPC controller to supply adequate oxygen for a PEFC system and it showed better performance than the traditional MPC algorithm.

Chatrattanawet等人25提出了一种传统的MPC和一种新型的鲁棒性线性时变MPC用于PEFC控制，发现新型MPC可以保证PEFC系统的稳定运行。Ouyang等人26提出了一种非线性MPC控制器来为PEFC系统提供足够的氧气，它显示了比传统MPC算法更好的性能。

It can be time consuming and rather difficult to conduct experimental measurements within a PEFC system, since the phenomena and reaction inside fuel cells are quite complicated.

在PEFC系统内进行实验测量可能很耗时，而且相当困难，因为燃料电池内的现象和反应相当复杂。

Thus, a reliable and qualified model can be used as an efficient tool to study specific aspects of a PEFC system for different applications.Hence, in this paper, a PEFC system model is firstly developed, then, a novel MPC controller and a novel PID controller are designed to regulate its output voltage by controlling its fuel and air flow rates at the same time, which synthesizes a multi-input and single output (MISO) control scheme.

因此，一个可靠和合格的模型可以作为一个有效的工具来研究PEFC系统在不同应用中的具体方面。因此，在本文中，首先建立了一个PEFC系统模型，然后，设计了一个新型MPC控制器和一个新型PID控制器，通过同时控制其燃料和空气流量来调节其输出电压，这合成了一个多输入和单输出（MISO）控制方案。

The rest of the paper is organized as follows: Section 2 presents the PEFC system model and its performance concerning different operation conditions; the MPC and PID controllers are developed in Section 3; Section 4 shows the effectiveness of the controllers; Section 5 summarizes the conclusions.

本文的其余部分组织如下。第2节介绍了PEFC系统模型及其在不同运行条件下的性能；第3节开发了MPC和PID控制器；第4节展示了控制器的有效性；第5节总结了结论。

PEFC系统模型开发

PEFC模型预分析

The PEFC model developed in this paper is based on previously proposed semiempirical models, which have the capabilities to illuminate the electrochemical behavior of a FC without offering deep apprehension of the underlying phenomena.

本文开发的PEFC模型是基于以前提出的半经验模型，这些模型有能力说明FC的电化学行为，而不提供对基本现象的深入理解。

Besides, the PEFC model developed in this work is on the system scale. Therefore, the focus is on the macro-performances of the PEFC system, the details of fluid flow in the porous area, the temperature distribution across the cell area, and the reaction distribution on the catalyst layer are not considered.

此外，本工作中开发的PEFC模型是在系统规模上的。因此，重点是PEFC系统的宏观性能，没有考虑流体在多孔区的流动细节，整个电池区的温度分布，以及催化剂层上的反应分布。

However, the electrochemical reaction determines the transfer of electrons and protons, which will affect the stack temperature and the FC performance. In the temperature range of a PEFC system (332-352 K), we may assume the reactant gases to follow the ideal gas law, for example, comparing the operating temperature to the critical temperature of each component.

然而，电化学反应决定了电子和质子的转移，这将影响烟囱温度和FC的性能。在PEFC系统的温度范围内（332-352K），我们可以假设反应物气体遵循理想气体定律，例如，将操作温度与每个组件的临界温度进行比较。

The water generated in the cathode may involve two-phase flow pattern, which is beyond the scope of this paper, and the water can be drained from the system, thus not considered.

阴极产生的水可能涉及两相流模式，这超出了本文的范围，水可以从系统中排出，因此没有考虑。

PEFC系统模型

Through a pair of redox reactions, the hydrogen fed PEFC converts the chemical energy from hydrogen and oxygen into electricity with only heat and water as byproducts. Its typical output voltage is usually less than the ideal value because of some losses occurred inside the fuel cells.Thus, to get higher voltage, a number of cells are usually combined in series and the net output voltage of a PEFC is given as follows

通过一对氧化还原反应，供给氢气的 PEFC 将氢气和氧气的化学能转化为电能，副产品只有热和水。由于燃料电池内部发生一些损失，其典型输出电压通常小于理想值。因此，为了获得更高的电压，通常将多个电池串联组合，PEFC 的净输出电压如下：
$$
V_{fc} = n_{cell}(E_{nernst}-V_{act}-V_{ohmic}-V_{con})
$$
here, VFC, ncell, Enernst, Vact, Vohmic, and Vcon denote output voltage of the fuel cell system, cell numbers, reversible voltage, activation voltage drop, ohmic voltage drop, and concentration voltage drop. Enernst is calculated based on the Nernst equation

其中，VFC、ncell、Enernst、Vact、Vohmic、Vcon分别表示燃料电池系统的输出电压、电池个数、可逆电压、活化压降、欧姆压降、浓度压降。Enernst是根据能斯特方程计算的：
$$
E_{nernst} =1.229-0.85*10^{{-3}(T_{stack}-298.15)+4.3085*10}T_{stack}[ln(P_{H_2})+0.5ln(P_{O_2})]
$$
here, Tstack, PH2, and PO2 are stack temperature, hydrogen partial pressure, and oxygen partial pressure. The activation voltage drop Vact occurs due to the activation of the electrodes, and it is defined as

此处，Tstack、PH2 和 PO2 分别是烟道温度、氢气分压和氧气分压。由于电极的激活而产生激活压降Vact，定义为:
$$
V_{\text {act }}=-\left[\xi_{1}+\xi_{2} T_{\text {stack }}+\xi_{3} T_{\text {stack }} \ln \left(C_{\mathrm{O}{2}}\right)+\xi T_{\text {stack }} \ln (I)\right] \
C_{\mathrm{O}{2}}=\frac{P{2}}}{5.08 * 10^{6} * e\left(\frac{-498}{T{\text {stack }}}\right)}
$$
here, ξ is the semi-empirical coefficient, CO2 is the oxygen concentration, and I is the current. The ohmic voltage drop Vohmic comes from the resistance to the electrons transfer and protons transfer. It is given as:

其中，ξ 为半经验系数，Co2 为氧气浓度，I 为电流。欧姆电压降 Vohmic 来自电子转移和质子转移的电阻。它被给出为：
$$
\begin{array}{c}
V_{\text {ohmic }}=I\left(R_{m}+R_{C}\right) \
R_{m}=\frac{\rho_{m} l}{A} \
\rho_{m}=\frac{181.6\left[1+0.03(i)+0.062\left(T_{\text {stack }} / 303\right)^{2}(i)\right]}{(\lambda-0.643-3 * i) \exp \left(4.18\left(\frac{T_{\text {stack }}-303}{T_{\text {stack }}}\right)\right)}
\end{array}
$$
here, Rm, RC, ρm, l, A, i, λ represent membrane resistance, equivalent contact resistance to electron conduction, membrane resistivity, membrane thickness, membrane active area, actual current density, and adjustable parameter dependent on membrane water content of the membrane. The concentration voltage drop Vcon is because of the mass transfer, which reduces the reactants pressure, and it is determined as

其中，Rm、RC、ρm、l、A、i、λ分别表示膜电阻、电子传导等效接触电阻、膜电阻率、膜厚度、膜活性面积、实际电流密度，以及取决于膜含水量的可调参数膜。浓度电压降 Vcon 是由于质量传递，降低了反应物压力，确定为
$$
V_{\text {con }}=-\beta \ln \left(1-i / J_{\max }\right)
$$
Here, β is a parametric coefficient related to the fuel cell operating condition, Jmax denotes the maximum current density. The dynamic behavior of a PEFC is largely affected by a “charge double layer” phenomenon. The charge layer on the interface electrode/electrolyte acts as an electrical capacitor. There is always a delay for the charge layer to follow the current changes. This delay only affects the activation and concentration voltage drop, which can be described as the following equations

这里，β是一个与燃料电池工作条件有关的参数系数，Jmax表示最大电流密度。PEFC的动态行为主要受 "电荷双层 "现象的影响。界面电极/电解质上的电荷层充当了一个电容器。电荷层跟随电流的变化总是有一个延迟。这种延迟只影响到激活和浓缩电压降，可以用以下公式来描述
$$
\frac{d V_{a}}{d t}=\frac{I}{C}-\frac{V_{a}}{R_{a} C} \
R_{a}=\frac{V_{\text {act }}+V_{\mathrm{con}}}{I}
$$
here, C and Ra denote the equivalent capacitance of the system and the equivalent resistance. Thus, the output voltage of the PEFC can be rewritten as:

这里，C和Ra表示系统的等效电容和等效电阻。因此，PEFC的输出电压可以改写为。
$$
V_{fc} = n_{cell}(E_{nernst}-V_{a}-V_{ohmic})
$$

parameter	means
Vfc	燃料电池系统输出电压
Ncell	电池个数
Enernst	可逆电压
Vact	活化压降
Vohmic	欧姆压降
Vcon	浓度压降
T_stack	烟道温度
P_H_2	氢气分压
P_O_2	氧气分压

​

阳极和阴极体积模型

All the reactant gases in this study are considered as ideal gases. In the anode volume, hydrogen is delivered as fuel to the channel. Its mass flow rate can be calculated as follows based on the mass conservation principle

本研究中所有的反应气体都被认为是理想气体。在阳极体积中，氢气作为燃料被输送到通道中。根据质量守恒原理，其质量流速可以计算如下
$$
\frac{d m_{\mathrm{H}{2}}}{d t}=\dot{m}{2}, \text { in }}-\dot{m}{2}, \text { rea }}-\dot{m}{2}, \text { out }} \
\dot{m}{2}, \text { rea }}=\frac{N I}{2 F} M{2}} \
\dot{m}{2}, \text { out }}=k\left(P_{\mathrm{H}{2}}-P{\mathrm{amb}}\right) M_{\mathrm{H}_{2}}
$$
here, mH2 , _mH2,in , _mH2,rea , _mH2,out MH2 , ka, and Pamb are hydrogen mass in the anode volume, inlet hydrogen mass flow rate, reacted hydrogen mass flow rate, outlet hydrogen mass flow rate, hydrogen molar mass, anode flow coefficient, and ambient pressure, respectively.

这里，mH2、_mH2,in、_mH2,rea、_mH2,out MH2、ka和Pamb分别是阳极体积中的氢气质量、进口氢气质量流量、反应氢气质量流量、出口氢气质量流量、氢气摩尔质量、阳极流量系数和环境压力。

In the cathode volume, air is delivered to the channel.Similar to the anode volume, the oxygen and nitrogen mass flow rate can be calculated as follows

在阴极体积中，空气被输送到通道中。与阳极体积类似，氧气和氮气的质量流速可以计算如下

$$
\frac{d m_{\mathrm{O}{2}}}{d t}=\dot{m}{2}, \text { in }}-\dot{m}{2}, \text { rea }}-\dot{m}{2}, \text { out }} \
\dot{m}{2}, \text { rea }}=\frac{N I}{4 F} M{2}} \
\dot{m}{2}, \text { out }}=k \frac{m_{\mathrm{O}{2}}}{m{2}}+m{2}}}\left(P{\mathrm{ca}}-P_{\mathrm{amb}}\right) M_{\mathrm{O}{2}} \
\frac{d m{2}}}{d t}=\dot{m}{2}, \text { in }}-\dot{m}{2}, \text { out }} \
\dot{m}{2}, \text { out }}=k \frac{m_{\mathrm{N}{2}}}{m{2}}+m{2}}}\left(P{\mathrm{ca}}-P_{\mathrm{amb}}\right) M_{\mathrm{N}_{2}}
$$
here, mO2 , _mO2,in , _mO2,rea , _mO2,out , MO2 , _mN2 , _mN2,in , _mN2,out , kc, and Pca are oxygen mass in the cathode volume, inlet oxygen mass flow rate, reacted oxygen mass flow rate, outlet oxygen mass flow rate, oxygen molar mass, nitrogen mass in the cathode channel, inlet nitrogen mass flow rate, outlet nitrogen mass flow rate, cathode flow coefficient, and cathode pressure, respectively.

这里，mO2 , _mO2,in , _mO2,rea , _mO2,out , MO2 , _mN2 , _mN2,in , _mN2,out , kc, 和Pca分别是阴极体积中的氧气质量、进口氧气质量流量、反应氧气质量流量、出口氧气质量流量、氧气摩尔质量、阴极通道中的氮气质量、进口氮气质量流量、出口氮气质量流量、阴极流动系数和阴极压力。

不同操作条件下的PEFC系统性能

The PEFC system model developed in this study is based on the commercial stacked PEFC-NedSstackPS6.All operating parameters needed for the PEFC system are shown in Table 1. The performance of the PEFC system under different operating conditions are investigated.

Figure 1A gives the changes of the PEFC system behavior with the hydrogen flow rate varied from 100 and 400 lpm. While Figure 1B presents the changes of the PEFC system performance with the air flow rates increased from 300 and 700 lpm. The changes of the reactant gas flow rates are made the same as in Monem's model for later comparison. From Figure 1, it can be clearly seen that the increase of hydrogen and air flow rate both slightly improve the PEFC system's voltage.

Moreover, our model shows good agreement with Monem's model when the current is in the range between 0 and 180 A. Above 180 A, the output voltage of our model drops more than Monem's model, which follows better with the fuel cell polarization behavior. Because at high current range, the output voltage of the fuel cell system is mainly affected by the concentration voltage drop, which makes the voltage decline faster with the current increasing to its limiting value.To clarify, in Monem's model, the voltage increase under the air flow rate from 300 to 700 lpm is very small, which almost overlapped, making it difficult to capture. Thus, in Figure 1B, only one line is shown for his model.

本研究中开发的PEFC系统模型是基于商业化的叠加式PEFC-NedSstackPS6.PEFC系统所需的所有操作参数见表1。研究了PEFC系统在不同操作条件下的性能。

图1A给出了氢气流速在100和400升/分钟之间变化时PEFC系统行为的变化。而图1B给出了随着空气流速从300和700升/分钟增加，PEFC系统性能的变化。反应气体流速的变化与Monem的模型相同，以便以后比较。从图1中可以清楚地看到，氢气和空气流速的增加都稍微提高了PEFC系统的电压。

此外，当电流在0-180A之间时，我们的模型与Monem的模型显示出良好的一致性。超过180A时，我们模型的输出电压比Monem的模型下降得更多，这与燃料电池的极化行为有较好的关系。因为在高电流范围内，燃料电池系统的输出电压主要受浓度压降的影响，这使得电压随着电流增加到极限值而快速下降。要说明的是，在Monem的模型中，空气流速从300到700lpm下的电压增加非常小，几乎是重叠的，这使得它难以捕捉。因此，在图1B中，他的模型只显示了一条线。

parameter	value
ncell	65
ξ1	−1.023071
ξ2	3.4760e-3
ξ3	7.7883354e-5
ξ4	−9.54e-5
RC (Ω)	1.62e-4
λ	15.03229
β	(V)
l (μm)	178
A (cm2)	240
Jmax (Acm−2)	0.918
MH2 (gmol−1)	2
MO2 (gmol−1)	3 2
kc (mols−1atm−1)	0.065
ka (mols−1atm−1)	0.065
T (K)	332-342
Fuel flow rate (lpm)	100-400
Air flow rate (lpm)	300-700

PID和MPC控制器的开发

控制目标

A steady output voltage is an important criterion for evaluating a fuel cell system's reliability as an alternative power source in practical applications.In this study, we assume the developed PEFC system can be utilized as a steady power source for an electric vehicle of 48 V. The reactant gas flow rate largely determines the hydrogen and oxygen pressure in the gas channel, which will significantly affect PEFC system's output voltage. Hence, the control task of this study is to stabilize the PEFC system's voltage at 48 V by setting its hydrogen and air flow rates as control variables. The current is defined as the input load in this work. Moreover, in order to capture more details of the real situation in the simulation environment, the measurement errors, which may occur in practical operation are considered as a noise signal applied to the actual output voltage during the whole voltage control process. In this section, two different novel controllers, PID controller and MPC controller, are designed for the PEFC system' voltage regulation. Their control schemes are displayed in Figures 2 and 3, respectively.

稳定的输出电压是评估燃料电池系统在实际应用中作为替代电源的可靠性的一个重要标准。在本研究中，我们假设开发的PEFC系统可以作为48V的电动汽车的稳定电源使用。因此，本研究的控制任务是通过设置其氢气和空气流量作为控制变量，将PEFC系统的电压稳定在48V。在这项工作中，电流被定义为输入负荷。此外，为了在模拟环境中捕获更多的真实情况的细节，在整个电压控制过程中，将实际操作中可能出现的测量误差视为应用于实际输出电压的噪声信号。在本节中，我们为PEFC系统的电压调节设计了两种不同的新型控制器，即PID控制器和MPC控制器。它们的控制方案分别显示在图2和图3中。



PID控制算法

PID controllers are widely used in industrial applications due to its easy implementation and simple operation. The PID controller applies a correction—calculated by three control terms: proportional (P), integral (I), and derivative (D)—to the control variables to minimize the error between the desired setpoint and the measured value.As shown in Figure 2, the difference between the reference voltage and the actual output voltage is set as the input of the PID controller. It can be noticed that two PID controllers are employed, which are responsible for hydrogen and air flow rate regulation, respectively. The practical and simple tuning method “trial and error” is adopted to find suitable coefficients kp, ki, and kd for the P, I, and D terms, respectively.

PID控制器因其易于实施和操作简单而在工业应用中得到广泛使用。PID控制器对控制变量施加修正--由三个控制项计算：比例（P）、积分（I）和导数（D）--以最小化期望设定值和测量值之间的误差。3如图2所示，参考电压和实际输出电压之间的差异被设定为PID控制器的输入。可以注意到，采用了两个PID控制器，分别负责氢气和空气流速的调节。采用实用而简单的调谐方法 "试错"，分别为P、I和D项找到合适的系数kp、ki和kd。

MPC控制器算法

As an advanced process controller, the uniqueness of MPC is that it uses the controlled models to predict the future system behavior while keep optimizing the current timeslot, which PID does not have.Another advantage that makes MPC superior to PID is that it can handle a multi input control problem without implementing extra MPC controllers. As presented in Figure 3, the input of the MPC controller contains the reference voltage, the actual voltage, and state vector, which is obtained by linearizing the controlled system. Based on the input signal, the MPC controller predicts future behavior of the PEFC system and calculates the correct hydrogen and air flow rates at the same time by solving an optimization problem to finally achieve the desired voltage. The tuning method of the MPC controller involves a linearization of the original PEFC system model, which is explicitly addressed as follows. Differentiating both sides of Equation (11):

作为一个先进的过程控制器，MPC的独特性在于它使用被控模型来预测未来的系统行为，同时不断优化当前的时隙，这是PID所不具备的。另一个使MPC优于PID的优点是，它可以处理多输入控制问题，而无需实施额外的MPC控制器。如图3所示，MPC控制器的输入包含参考电压、实际电压和状态矢量，这是由被控系统线性化得到的。基于输入信号，MPC控制器预测PEFC系统的未来行为，并通过解决优化问题同时计算出正确的氢气和空气流量，最终实现理想的电压。MPC控制器的调谐方法涉及原始PEFC系统模型的线性化，其明确的处理方法如下。对方程（11）的两边进行微分。

$$
\dot{V}{\mathrm{FC}}=n{\text {cell }} \times\left(\dot{E}{\text {nernst }}-\dot{V}-\dot{V}_{\text {ohmic }}\right)
$$
here _Enernst is a differentiation of Equation (2) assuming Tstack to a constant value of 343 K during one prediction horizon:

这里_Enernst是方程（2）的微分，假设Tstack在一个预测范围内达到343K的恒定值。
$$
\begin{array}{l}
\dot{E}{\text {nernst }}=k \frac{1}{P_{\mathrm{H}{2}}} \dot{P}{2}}+\frac{k{1}}{2} \frac{1}{P_{\mathrm{O}{2}}} \dot{P}{2}}\
k=4.308 \times 10^{-5} \times 343
\end{array}
$$
Combining Equations (12)–(19), _PH2 , _PO2 , and _PN2 are expressed as:

结合公式（12）-（19），PH2、PO2和_PN2表示为。
$$
\begin{array}{l}
\dot{P}{2}}=\frac{k_{2}}{0.005}\left[k_{3} m_{\mathrm{H}{2, \mathrm{in}}}-0.065 P{2}}+0.065-\frac{65 I}{2 \times 96485}\right] \
\dot{P}{2}}=\frac{k{2}}{0.01}\left[k_{4} m_{\mathrm{air}, \mathrm{in}}-\frac{0.065 m_{\mathrm{O}{2}}}{m{2}}+m{2}}} P{2}}\right. \
\left.+\frac{0.065 m{2}}}{m{2}}+m{2}}}-\frac{65 I}{4 \times 96485}\right] \
\dot{P}{2}}=\frac{k{2}}{0.01}\left[k_{5} m_{\text {air,in }}-\frac{0.065 m_{\mathrm{N}{2}}}{m{2}}+m{2}}} P{2}}\right. \
\left.-\frac{0.065 m{2}}}{m{2}}+m{2}}} P{2}}+\frac{0.065 m{2}}}{m{2}}+m{2}}}\right] \
k=0.0821 \times 10^{-3} \times 343 \
k_{3}=\frac{0.0706}{2 \times 60} \
k_{4}=0.21 \times \frac{1.121}{2 \times 60} \
\end{array}
$$

The derivative _V ohmic is simplified as:

导数_V ohmic被简化为。


assuming _Rm is constant during the prediction horizon.

The linearized continuous-time state-space model is written as:

假设_Rm在预测范围内是恒定的。

线性化的连续时间状态空间模型被写成：。


where the state vector x:

其中，状态向量x。


and the input u is:

而输入的u是。


the output y:

输出Y。


the state-space matrices are:

的状态空间矩阵是。




这里A13-16分别为


A quadratic programming (QP) problem will be solved at each time step to obtain the optimal control inputs:

在每个时间步骤中，将解决一个二次编程（QP）问题，以获得最佳控制输入。


subject to:

受制于。


here Ad, Bd, and Cd are state-space matrices in discrete-time; Hp and Hu are prediction and control horizon length; r is the control reference; Q and R are weight tuning parameters for reference tracking and control inputs; ulb, uub, xlb, and xub are the lower bounds and upper bounds of inputs u and states x; xinit is the latest measured value, the state feedback.

这里Ad、Bd和Cd是离散时间的状态空间矩阵；Hp和Hu是预测和控制视界长度；r是控制参考；Q和R是参考跟踪和控制输入的权重调谐参数；ulb、uub、xlb和xub是输入u和状态x的下限和上限；xinit是最新测量值，即状态反馈。

结果和讨论

In this section, two different test cases are conducted to investigate the performance of the proposed MPC and PID controllers under different working conditions: (1) typical disturbance applied on the working load; (2) any random perturbation applied on the working load.

在本节中，我们进行了两个不同的测试案例，以研究所提出的MPC和PID控制器在不同工作条件下的性能：（1）在工作负载上施加典型的干扰；（2）在工作负载上施加任何随机扰动。

test case 1

In this case, the current load is interrupted by one sudden increase and one sudden decrease step change as presented in Figure 4. Figure 5 shows hydrogen and air flow rate profiles under the control of PID and MPC controllers. After the system starts, the MPC reacts slower than PID in terms of increasing hydrogen and air flow rate.

The reason is that the MPC has a penalty term to limit the inputs change rate. Meanwhile, MPC calculates and acts every 1 second, and PID acts continuously. After 25 seconds, the current load jumps from 110 to 120 A, according to the polarization theory, the output voltage of the PEFC system will drop with this increased current load. To resist the voltage drop, after receiving the input signal, the controller will try to adjust the hydrogen and air flow rates at a higher level to increase the hydrogen and oxygen pressure at the anode and cathode volume, thus, keeping the voltage at desired 48 V. When referring to the underlying true value, the rising time for MPC to arrive at 47.9 V is 6.01 seconds, while it takes PID 7.66 seconds. The overshoot for MPC is 0.08 V, and for PID it is 0.17 V. At 50 seconds, the current load drops from 120 to 115 A, the PEFC system's output voltage is supposed to rise with this decrease in the current load.

To stop the voltage increase, the controllers then regulate the hydrogen and air flow rates at a lower level to reduce the reactant gas pressure, which will stop the voltage raise and keep the voltage at the reference value 48 V.

在这种情况下，如图4所示，电流负载被一个突然增加和一个突然减少的步骤变化所打断。图5显示了在PID和MPC控制器控制下的氢气和空气流速曲线。系统启动后，MPC在增加氢气和空气流量方面的反应比PID慢。

原因是MPC有一个惩罚项来限制输入变化率。同时，MPC每1秒计算一次并采取行动，而PID则连续行动。25秒后，电流负载从110A跳到120A，根据极化理论，PEFC系统的输出电压将随着电流负载的增加而下降。为了抵抗电压下降，在接收到输入信号后，控制器将尝试将氢气和空气流量调整到更高的水平，以增加阳极和阴极体积的氢气和氧气压力，从而使电压保持在所需的48V。当参考基本的真实值时，MPC到达47.9V的上升时间为6.01秒，而PID需要7.66秒。MPC的过冲是0.08V，PID的过冲是0.17V。在50秒时，电流负载从120A下降到115A，PEFC系统的输出电压应该随着电流负载的减少而上升。

为了阻止电压上升，控制器将氢气和空气流量调节到较低水平，以降低反应气体压力，这将阻止电压上升，并将电压保持在参考值48 V。

The voltage profile is exhibited in Figure 6. It seems obvious that both PID and MPC controllers can keep the PEFC system's output voltage at desired 48 V, but clearly the MPC shows superior performance with faster response and lower oscillation.

图6中显示了电压曲线。很明显，PID和MPC控制器都能使PEFC系统的输出电压保持在所需的48V，但显然MPC表现出更快的响应和更低的振荡。




test case 2

In this case, the current load is disturbed with any random perturbation, which contains almost all stochastic step changes in current load, as shown in Figure 7. The current load changes the following: 110 ! 112 ! 115 ! 110.5 - 120 - 117 - 113 - 111 - 116 ! 114 ! 118 ! 119 ! 111.5. Figure 8 displays the hydrogen and air flow rate profiles. The corresponding voltage profile is demonstrated in Figure 9, from which it is clear that for any random disturbance, both MPC and PID are qualified to control the voltage at 48 V, but still, the MPC controller shows more superior performance with smaller overshoot and faster response for each step change in the current load.

在这种情况下，电流负载受到任何随机扰动的干扰，其中几乎包含了电流负载的所有随机阶跃变化，如图7所示。电流负载的变化情况如下。110 ! 112 ! 115 ! 110.5 ! 120 ! 117 ! 113 ! 111 ! 116 ! 114 ! 118 ! 119 ! 111.5. 图8显示了氢气和空气流速曲线。图9显示了相应的电压曲线，从中可以看出，对于任何随机干扰，MPC和PID都有资格控制48V的电压，但MPC控制器仍然显示出更优越的性能，对于电流负载的每一步变化，超调更小，响应更快。




It can be seen from these two studies that despite any random disturbance on the working load, the proposed MPC controller can stabilize the output voltage at the reference value with faster response and smaller oscillation compared to the PID controller. The MPC controller first linearizes the PEFC system model, then captures any disturbance occurred in the system during the control process, finally regulates the voltage at the desired value as fast as possible, thus is more robust and fast, making it more suitable for control of the complicated and coupled systems.

从这两项研究中可以看出，尽管工作负载上有任何随机干扰，但与PID控制器相比，所提出的MPC控制器可以将输出电压稳定在参考值上，而且响应速度更快，震荡更小。MPC控制器首先将PEFC系统模型线性化，然后在控制过程中捕捉系统中发生的任何干扰，最后尽可能快地将电压调节到期望值，因此更加稳健和快速，使其更适合于复杂和耦合系统的控制。

In both cases, MPC tends to use the hydrogen flow more actively during the control transient. The primary reason is the weight tuning parameter R puts a higher cost in adjusting oxygen flow. By adjusting the MPC parameters, one can achieve various desired system behaviors. For the PID controller, even though it is less efficient than MPC controller, its controlling performances are still acceptable. Besides, PID controllers can handle a wide range of systems and from the view of economy and practicality, PID is the primary choice if it can solve one control problem good enough.

在这两种情况下，MPC倾向于在控制瞬态期间更积极地使用氢气流量。主要原因是权重调整参数R使调整氧气流量的成本更高。通过调整MPC参数，人们可以实现各种期望的系统行为。对于PID控制器，即使它的效率低于MPC控制器，其控制性能仍然是可以接受的。此外，PID控制器可以处理广泛的系统，从经济性和实用性的角度来看，如果PID能够很好地解决一个控制问题，它是首要选择。

conclusion

In this work, a dynamic PEFC system model is firstly developed. Its performances under different operating conditions are investigated.

在这项工作中，首先建立了一个动态PEFC系统模型。研究了它在不同操作条件下的性能。

The simulation results show good agreement was achieved between our model and the results from previous studies, which addressed the reliability of the proposed PEFC system model.

仿真结果显示，我们的模型和以前的研究结果之间取得了良好的一致，这解决了所提出的PEFC系统模型的可靠性。

In order to improve the fuel cell system's stability, a novel MPC and a novel PID controllers are designed to control the PEFC's voltage at the desired value by regulating both hydrogen and air flow rates, which simulates a MISO control problem.

为了提高燃料电池系统的稳定性，我们设计了一个新型的MPC和一个新型的PID控制器，通过调节氢气和空气的流量将PEFC的电压控制在理想值，这模拟了一个MISO控制问题。

Two test cases are carried out to evaluate the effectiveness of the designed controllers under different operating conditions: (1) typical interruption on the input current; (2) random fluctuations on the input current.

为了评估所设计的控制器在不同工作条件下的有效性，进行了两个测试案例：（1）输入电流的典型中断；（2）输入电流的随机波动。

Moreover, the measurement errors in real operation are employed as a random noise on the actual output stack voltage, which is fed back to the controller as an input in the whole system, making the simulation environment closer to the actual experimental situation.

此外，实际操作中的测量误差被用作实际输出堆栈电压的随机噪声，在整个系统中作为输入反馈给控制器，使仿真环境更接近实际的实验情况。

The simulation results show that the proposed MPC and PID controllers are both qualified at stabilizing the PEFC system's output voltage at the desired value despite any perturbation and noise applied on the input load.

仿真结果表明，尽管在输入负载上施加了任何扰动和噪声，拟议的MPC和PID控制器都能将PEFC系统的输出电压稳定在期望值。

Meanwhile, the MPC controller is much superior compared to the traditional PID controller in restraining system disturbance and tracking the reference voltage with faster response and smaller overshoot. Furthermore, the designed MPC and PID control algorithms can be employed in various control applications for fuel cell systems to enhance its performance, which will eventually accelerate the fuel cell's commercialization.

同时，与传统的PID控制器相比，MPC控制器在抑制系统干扰和跟踪参考电压方面更有优势，反应更快，过冲更小。此外，所设计的MPC和PID控制算法可用于燃料电池系统的各种控制应用，以提高其性能，这将最终加速燃料电池的商业化。