Research on Temperature Control of Fuel-Cell Cooling System Based on Variable Domain Fuzzy PID

Abstract

To ensure the energy conversion efficiency of a proton-exchange membrane fuel cell (PEMFC), it is necessary to establish a water-cooled cooling system to keep the inlet temperature of fuel-cell coolant and the temperature difference between the inlet and outlet temperature within the set range. First, a semi-empirical and semi-mechanism model was built in Simulink. Then, a variable-universe fuzzy PID controller was designed to adjust the quantization factor and scaling factor by scaling factor α₁, α₂ and β to improve the accuracy of the control results. Finally, the cooling system model with controller is simulated. The results show that compared with other control methods, the simulation results of the variable-universe fuzzy PID controller have smaller maximum overshoot, faster response speed and more gentle temperature fluctuation.

1. Introduction

In the context of the profound adjustment of the global energy pattern, accelerated reconstruction of the energy governance system and the vigorous rise of a new round of energy revolution, China announced for the first time in 2020 that it would strive to realize carbon neutrality by 2060 and achieve zero CO₂ emissions in a relative sense. Proton-exchange membrane fuel-cell (PEMFC) vehicles have become one of the most promising new energy vehicles in the 21st century due to their high conversion rate, renewable energy and zero pollution emission. According to the document released in 2020, the number of PEMFC vehicles will reach about 100,000 by 2023 and 1 million by 2025 [1]. The proportion of fuel-cell vehicles in the future automobile market will gradually increase.

Temperature is an important parameter of a fuel cell. Temperature control is an important link to ensure the efficiency of a fuel cell. A cooling system is a key component that directly affects the performance, reliability and durability of a fuel cell [2]. If the temperature of the PEMFC is too high or too low, the normal use of fuel cells cannot be guaranteed, and the service life of the battery even be affected [3,4]. At the same time, to ensure the uniform distribution of temperature in the battery and avoid excessive local temperature, it is necessary to maintain the temperature difference between the inlet and outlet of the stack within a certain range [5].

Some scholars have made many attempts at a temperature control strategy, such as the flow-following current-control strategy [6], adaptive control strategy [7,8], neural network active disturbance rejection control strategy [9], control strategy of compensation through active disturbance rejection control [10], real-time power optimization strategy of active temperature control [11], and fuzzy control strategy [12]. Among them, fuzzy control is established based on expert knowledge and practical experience, which has low dependence on the controlled model, simple structure, and good robustness [13], and is frequently used in the temperature control system of a fuel cell [14,15]. The control accuracy of fuzzy control is guaranteed by fuzzy rules, but the number of fuzzy rules is limited [16]. To improve the steady-state accuracy, this paper adopts the variable-universe fuzzy PID controller. The fuzzy universe is adjusted by the dilation factor, and the number of fuzzy rules is locally increased to improve the use rate of fuzzy rules, to achieve a better control effect.

First, a cooling system model is built on Simulink software, and the temperature control strategy, taking the inlet temperature and the temperature difference between inlet and outlet of the fuel cell as the control object, is proposed in this paper. Then, the model is simulated and compared with other control results. Finally, conclusions are drawn by analyzing the simulation results.

2. Structure of Cooling System

2.1. Structure Description

The cooling system structure diagram of PEMFC is shown in Figure 1.

The main components of the cooling system are the stack, circulating water pump, radiator, controller and diverter valve. The control object is the inlet temperature in the fuel cell, and the main executive components are circulating water pump, radiator and diverter valve. The cooling water pump provides power to the cooling liquid in the system, so that the cooling liquid flows in the pipeline, absorbs heat when passing through the stack, and releases heat when passing through the radiator, so that the stack is in a stable temperature environment. The cooling system is divided into two cycles by the diverter valve. When the inlet temperature of the fuel cell is lower than 55 °C, the diverter valve is closed, to shorten the temperature rise time in the cell, and the coolant will enter a small cycle, i.e., it will directly flow back to the cell without heat dissipation. When the temperature reaches 57 °C, the diverter valve will be fully opened, leaving only a large circulation in the cooling system loop, and the cooling water with heat will flow into the radiator. In the radiator, the cooling liquid in the pipeline transfers heat with blown air, transferring the heat carried by the cooling liquid to the air, reducing the temperature of the cooling water, and then sending the low-temperature cooling water into the battery for the next cycle. The control module adjusts the rotating speed of the water pump and the opening of the radiator according to the temperature and current values fed back from the control loop. The rotating speed of the water pump determines the power that can be provided, and then determines the flow rate in the pipeline. The greater the flow rate, the more heat is brought out from the stack, avoiding the phenomenon of excessive local heat caused by heat accumulation in the stack and ensuring uniform temperature distribution in the fuel cell. The opening degree of the radiator directly affects the air volume entering the radiator. The larger the opening degree, the more heat is dissipated, so that the coolant is always in a low-temperature state when flowing into the battery, thus achieving the purpose of fuel-cell temperature control.

2.2. Cooling System Model

Based on the working principle and test data, the simulation model is built on Simulink software, which is divided into a stack module, pump module, pump control module, radiator module, radiator control module and diverter valve module, as shown in Figure 2.

2.2.1. Simulink Model of Stack

Heat calculation of fuel cell and cooling water heat dissipation are included in the stack model. Generally speaking, the thermal efficiency of the fuel cell is between 50% and 60%. During the research, it is assumed that the working thermal efficiency of the fuel cell is 50%, i.e., the output power of the fuel cell is the same as the value of the generated thermal power. Therefore, the heat generation model of the stack can be simplified as:

$$Q_{hot}=E_{st}·I_{st}$$

(1)

where $Q$_hot is thermal power generated by stack, and E_st and I_st are output voltage and current of fuel cell.

To ensure that the fuel cell is in thermal equilibrium, the heat generated by the fuel should be equal to the heat dissipation. The heat of the fuel cell is dissipated mainly in the following three ways: heat is taken away by the circulating cooling water in the system; heat is transferred to a lower temperature environment through radiation; heat is carried out of the stack through the remaining gas and the generated water vapor. Due to the small proportion of radiant heat and residual gas heat, to simplify the simulation process, the heat taken away by them is ignored. Therefore, only the heat dissipated through the cooling water is taken into account in this paper.

According to the heat balance equation, the heat carried out by the cooling water can be obtained from the following formula:

$$Q_{cool}=W_c·C_{Pc}(T_{w,out}-T_{w,in})$$

(2)

where W_c is the flow rate of cooling water, C_Pc is the heat capacity of water, and T_w,out and T_w,in are the outflow and inflow temperature of the cooling water.

The difference between the heat generated by the stack and the heat brought by the cooling water is the heat ∆$Q$_st that causes the temperature change of the stack. Since graphite is the material of the main structure in the fuel cell, then the heat capacity of graphite is used to replace the heat capacity parameters of the fuel cell. According to the heat balance equation, the change of temperature ∆T is:

$$\Delta T=\Delta Q_{st}/(m_{st}\times C_{Pst})$$

(3)

where C_Pst is the heat capacity of graphite, m_st is the quality of the fuel cell.

The system starts to change with the ambient temperature as the starting value, and then obtains the outlet temperature T_out of the fuel cell through ∆T.

2.2.2. Simulink Model of Radiator

The radiator is an important heat dissipation component in the thermal management system of PEMFC. It dissipates a large amount of heat in cooling water into the surrounding environment through fan. The calculation formula of thermal power transmitted from the radiator to the environment is:

$$Q_{rad}={Ah}_{rad}(T_{rad,in}-T_0)$$

(4)

where A is the effective area of the radiator, h_rad is the heat conduction coefficient of the radiator, T_rad,in is the temperature of cooling water flowing into radiator, and T₀ is the ambient temperature.

Moreover, h_rad and W_air can be fitted from the experimental data in

Table 1. Experimental data of heat dissipation coefficient.

Opening of Radiator %	Air Flow kg/s	Thermal Conductivity W/(m2·°C)
10	0.51	35.63
20	0.65	42.1
30	0.78	53.46
40	0.92	59.47
50	1.17	66.35
60	1.22	66.74
70	1.48	73.46
80	1.66	79.31
90	1.76	84.1

, which are a function of the air flow W_air through the radiator and the opening of radiator θ:

$$h_{rad}=-15.13W_{air}^2+71.74W_{air}+4.033$$

(5)

W_air = 1.616θ + 0.3192

(6)

After calculation, the temperature value T_in(R) after heat dissipation can be obtain.

2.2.3. Simulink Model of Water Pump

In the cooling system, when the coolant flows through the radiator and pipeline, it will produce flow resistance. When the power provided by the water pump P_p is the same as the resistance consumption P_r, it can ensure a stable flow rate.

The relationship between flow resistance and flow rate P_r ~ W, can be fitted from the experimental data in

Table 2. Experimental data of resistance coefficient.

Flow Resistance Test of Large Circulation			Flow Resistance Test of Small Circulation
Flow LPM	Inlet Pressure of Thermostat kPa	Outlet Pressure of Large Circulation kPa	Flow LPM	Inlet Pressure of Thermostat kPa	Outlet Pressure of Small Circulation kPa
0	115.1	114	0	115.1	112.8
21.6	119.5	117.1	8.22	122.5	119.6
47.7	119.5	115.5	19.6	121.8	117.8
58.9	121.5	116.5	28.5	127.7	121.5
71	123.6	118.1	39.1	135	126.1
92.8	129.6	121.8	47.6	143.2	131.1
104.4	133	123.5	55.2	151.9	137.2
123	140.2	127.6	62.5	162	143.7
134.3	144.2	130.3	67.4	169	148.1
154.5	152.4	135	74.8	181	55.9

:

Pr = ξW²

(7)

where ξ is resistance coefficient of cooling system, which can be calculated from the above formula.

The rated power P_p of the water pump can be obtained from the following formula:

P_p = γWH

(8)

where γ is unit weight of cooling water, H is the pump head.

Therefore, when the flow rate of cooling water is stable, P_r can be used to calculate the head of the water pump. Then the differential pressure of the water pump under different power states can be calculated by the following formula:

P_diff = γ(H − h_w)

(9)

where h_w is head loss of pump, which can be calculated by the following formula:

h_w = v²/(2g)

(10)

where v is velocity in the pipe.

Through the experiment, the interpolation data relationship between differential pressure and the flow of the water pump at the rated speed of 6500 rpm can be obtained in

Table 3. Experimental data of differential pressure and flow.

Differential Pressure kPa	Flow LPM
83.8	223.93
100.6	215.07
116.9	200.74
135.6	180.18
149.4	160.11
159.4	140.04
168.1	119.96
178.1	99.89
179.4	80.29
180.6	59.74
181.3	39.67
182.5	20.55

.

However, in the actual work, the speed N of the water pump will not be stable at the rated speed N₀, so a similar principle can be used to calculate the flow value of the water pump at different speeds, as shown in the following formula:

W_c = NW₀/N₀

(11)

2.2.4. Simulink Model of the Diverter Valve

The function of the diverter valve is to divide the cooling system into two cycles. When it is closed, the coolant flows directly into the stack without heat dissipation. When the inlet temperature reaches the set value, it will be opened, and the cooling water enters the large cycle and dissipates the heat through the radiator, to realize cooling.

We avoid frequently opening and closing the diverter valve in case of temperature fluctuation, and add a transition stage between the opening and closing of the diverter valve to delay the control, i.e., the diverter valve is only opened when the temperature reaches 57 °C to enter the large cycle, and closed when the temperature drops to 55 °C to enter the small cycle, as shown in Figure 3. This process can be realized in Simulink using the delay module.

2.3. Model Validation

To verify the accuracy of the Simulink model of the fuel-cell cooling system, it is necessary to measure the current and temperature online [17], and the parameters of each component in the experiment are the parameters used in the modeling of the simulation research. The actual test was carried out according to the working principle of the cooling system. A temperature sensor was set at the cooling water inlet and outlet of the fuel cell. Through the monitoring of the temperature, the opening of the radiator was determined [18], so that the temperature of the cooling circulating water outlet was stabilized at about 70 °C. When the fuel cell is in steady-state operating conditions of 8 kW, 16 kW, 20 kW, 24 kW, 28 kW, 30 kW, 32 kW, 36 kW and 40 kW, feed-forward control is carried out on the water pump and PID control is carried out on the radiator, to obtain the inlet coolant temperature under this control strategy. The experimental data are processed and compared with the results obtained by software simulation, as shown in Figure 4 below.

Specific results of simulation data and test data are shown in

Table 4. Data comparison.

Power/kW	Test Data °C	Simulation Data °C	Relative Error %
8	70	69.962	0.054
16	70.32	69.957	0.516
20	69.505	69.971	0.671
24	70.147	69.968	0.255
28	70.147	69.969	0.254
30	69.396	69.968	0.824
32	70.334	69.965	0.525
36	70.244	69.972	0.387
40	70.631	69.968	0.939

.

According to the data comparison, the relative error between the simulation results and the test results is less than 1% under multiple working conditions. This shows that the established simulation model has strong credibility. It can be seen from the data in Table 1 that the relative error is small when the output power of the stack is small. The relative error increases when the power increases. This phenomenon can be explained from the causes of errors. The model established in this paper does not take into account the radiation heat of the stack and the heat brought out by the exhaust gas. When the output power of the stack is small, the temperature of the fuel-cell system is low, and less heat is ignored, so the relative error is also small. However, in the case of high power, the fuel-cell temperature is much higher, and the proportion of neglected heat to the total heat dissipation increases, so the relative error also increases.

3. Control Strategy of Cooling System

In this paper, when the temperature rises into a big cycle, the actuators in the cooling system are mainly radiators and water pumps, and the main control quantities are cooling water flow and radiator opening. To ensure the working efficiency and safety of fuel cells, the temperature of the fuel cell achieves the following requirements:

• the fuel cell can rapidly heat up from the ambient temperature (25 °C) to the suitable temperature range of 60–80 °C;
• after the temperature is stabilized, the inlet temperature of cooling water is as close as possible to 70 °C, and the steady-state error is approximately 1 °C;
• the temperature difference between the entrance and exit will be controlled within 5–7 °C to ensure uniform temperature in the battery.

The control of inlet temperature of PEMFC is mainly realized by adjusting the opening of the radiator, and the inflow of air volume increases with the increase of the opening. Based on PID control method, fuzzy controller is added to adjust the scale factor and modify the control quantity. Moreover, based on the fuzzy PID, this paper presents a variable-universe fuzzy PID (VUFP) algorithm to achieve better control effect, based on fuzzy control to add the appropriate expansion factor. After fuzzy domain and the corresponding expansion factor multiplication, fuzzy domain corresponding to compression, and their membership function, will also be compressed. Then, originally the smaller error will be regarded as big at this moment, and the fuzzy control will become more sensitive, which can improve the accuracy of the control.

The temperature difference between inlet and outlet in PEMFC is controlled by adjusting the speed of the water pump. Under the control of the following function, the flow rate is adjusted with the change of current. To make the adjustment more stable and faster, a PID controller is added to realize the combined control of feed-forward and PID (Figure 5). Because the cooling system has a long-term lag phenomenon, the dynamic changes caused by the current can be ignored if the load current suddenly changes in a short time.

4. Design of Variable-Universe Fuzzy PID Controller

Controlling the water pump speed adjusts the flow of cooling water. When the rotation speed is constant, the flow of cooling water system is also certain. Therefore, we achieve stack-suitable inlet temperature mainly by controlling the opening of the radiator to a certain flow of cooling water for cooling.

In the actual working state, there are many and complex interference factors, which cannot meet the stable control environment required by traditional PID control. The PID parameters can be adjusted online by adding a controller [19]. However, the number of rules in the controller is limited, which limits control accuracy. The ultimate goal of fuzzy control is to stabilize the error near the “zero point”, so it is necessary to increase the number of rules near the “zero point” to ensure control accuracy. However, it is difficult to add rules, and increasing the number of rules in the controller will slow down the operation speed. To solve the contradiction between control accuracy and calculation difficulty, this paper introduces the concept of variable universe based on fuzzy PID controller, which is equivalent to adding the interpolation function [20] in fuzzy control, changing the range of fuzzy universe, improving the use rate of fuzzy rules and improving the control results.

To adjust the fuzzy universe online, expansion factors α₁, α₂ and β are added to the original fuzzy PID model. If the expansion factors are controlled separately, the building process is complicated, and a controller with two inputs and three outputs [21] can be used to adjust the expansion factors. The input values are e and ec, and the output values are expansion factors α₁, α₂ and β. The working principle is shown in Figure 6.

When designing the controller, the temperature rises from the ambient temperature of 25 °C to the ideal temperature of 70 °C, so the physical domain of temperature error is [−45, 45] °C. In this paper, if the initial fuzzy domain of error and error rate is [−6, 6], then we can calculate its quantitative factor Ke to be 0.133. In addition, the maximum error rate is 29.1 °C/s, so the physical domain is [−29.1, 29.1] °C/s and the quantization factor Kec is 0.2.

Considering the properties of expansion factors, the fuzzy universe of output is divided into four fuzzy subsets—{Z, S, M, B}—in which the membership functions of expansion factors α₁ and α₂ are trigonometric functions, and the non-uniform trigonometric functions can achieve better control effect, while ensuring that the independent variables will not exceed the original range after changing, adjust the trigonometric functions, and the abscissa of the peak points of membership functions are {0.25, 0.62, 1.2, 1.35}, respectively. However, the membership function of β is not out of range, and uniformly distributed trigonometric functions are adopted. The abscissa of the peak points of the membership function are {0.5, 0.67, 0.83, 1}, as shown in Figure 7 and Figure 8.

The division of the universe takes into account the monotonicity and zero-preserving property of the expansion factor. In the given fuzzy universe, after the change, it will not exceed the initial universe [−6, 6], which is consistent with the coordination. The fuzzy rules [22] used in control can meet the duality and normality of the expansion factor. The method of barycenter is used to disambiguate, which can obtain better control effect, and is convenient for calculation.

The scaling factors α₁ and α₂ are adjusted by the controller, and the new quantization factors Ke and Kec are obtained. The scaling factor β acts on the scaling factor, and the new sum scaling factors Kp, Ki and Kd are calculated. The calculation formula is shown in Equation (12) as follows to realize the online regulation of the fuzzy universe.

$$\{\begin{array}{l}Ke=\frac{K{e}^{\prime }}{{\alpha }_1}\hfill \\ Kec=\frac{Ke{c}^{\prime }}{{\alpha }_2}\hfill \\ K_p=K_{p0}+\Delta K_p\times \beta \hfill \\ K_i=K_{i0}+\Delta K_i\times \beta \hfill \\ K_d=K_{d0}+\Delta K_d\times \beta \hfill \end{array}$$

(12)

5. Simulink Result

To verify the performance of the designed expansion factors controller, the cooling system model of PEMFC is built in Simulink software, in which the 40-kW fuel cell is composed of 216 cells connected in series.

When the water pump is controlled by feed-forward and PID, the maximum overshoot is less than the control result only using the flow-following function. To keep a single variable, PID control, fuzzy PID control and variable-universe fuzzy PID control are used to control the radiator, respectively. The simulation results of different controllers are compared and analyzed, among which the results of PID control used in the experiment are compared.

In the simulation, the change of current causes the chemical reaction in the fuel cell, which leads to the change of temperature in the cell. The test signal of load current is shown in Figure 9.

By analyzing the simulation results, it is found that the maximum overshoot under different control methods, the control accuracy, and the temperature difference under stable conditions are all different. The simulation results are shown in Figure 10 and Figure 11.

Through the analysis of simulation results, it can be seen that the inlet temperature reaches the maximum in about 4 s. After fuzzy PID or variable-universe fuzzy PID control, the maximum overshoot of the inlet temperature is significantly reduced, the temperature fluctuation is reduced, and the adjustment time for entering the steady state is shortened. Compared with the traditional PID control with the maximum overshoot of 80.47 °C, the maximum overshoot of inlet temperature decreased to 78.19 °C by 3.28 °C after adding a fuzzy controller, and the time to enter steady state was shortened from 22.1 s to 17.7 s, which was 4.4 s ahead of schedule. The temperature change trend of fuzzy PID control with variable universe is similar to that of fuzzy PID control, and the maximum temperature is 77.74 °C. Compared with fuzzy PID control, the maximum overshoot of inlet temperature is reduced by 0.45 °C, and the time to enter steady state of 17.4 s is slightly earlier. The temperature finally stabilized at about 70 °C and the steady-state error was within 1 °C. When the current changes, the fluctuation range of variable-universe fuzzy PID is the smallest and tends to become stable the fastest. The temperature difference between entrance and exit is stable within 4 °C. Compared with traditional PID control and the fuzzy PID control method, the temperature-difference fluctuation after adding a variable-universe controller is smaller, and the temperature-difference value after stabilization is also the smallest.

6. Conclusions

Effective thermal management is the key to ensure the high reliability and long-life operation of PEMFC. In this paper, the inlet temperature and the temperature difference between the inlet and outlet of cooling water are the main control objects, and a water-cooled cooling system with deionized water as the cooling liquid is built on Simulink software.

Based on fuzzy PID, by introducing the expansion factor, the variable-universe fuzzy PID control algorithm is designed to realize the online adjustment of the fuzzy universe, to improve the use of fuzzy rules and obtain better control effect. Compared with PID and fuzzy PID, variable-universe control can reach the set inlet temperature of 70 °C at a faster speed, its adjustment time is shortened by 4.7 s, and the maximum overshoot is reduced by 4.73 °C. Moreover, the steady-state error and the temperature difference between inlet and outlet can be controlled within ±1 °C and 4 °C, respectively. Therefore, variable-universe fuzzy PID control has faster response speed, smaller overshoot, and more stable temperature-difference fluctuation.