A Small-Signal Stability Analysis Method for Fuel Cell Hybrid Systems

Abstract

This invention provides a small-signal stability analysis method for a fuel cell hybrid system. First, a control model of the fuel cell is constructed based on the power generation principle of the fuel cell, and a circuit model of a Boost converter, a bidirectional DC-DC converter, and a supercapacitor is constructed. Second, based on Kirchhoff's laws and the theoretical knowledge of linear systems, small-signal models of the fuel cell connected to the Boost converter and the supercapacitor connected to the bidirectional DC-DC converter are constructed. Finally, based on the above small-signal models, the state matrix of the system is obtained, and its eigenvalues ​​are calculated using simulation software. According to Lyapunov's first law, it is determined whether the real parts of the system's eigenvalues ​​are all greater than 0, further determining whether the fuel cell hybrid system is stable.

Description

Technical Field

[0001] This invention relates to the field of fuel cell technology, and in particular to a method for small-signal stability analysis of fuel cell hybrid systems. Background Technology

[0002] Using fuel cells as a distributed power source has become one of the most promising power generation technologies. A fuel cell is an electrochemical energy conversion device that releases the chemical energy from fuel and oxides through an oxidation reaction and directly converts it into electrical energy. However, the output power of a fuel cell is affected by many factors such as reactant flow rate, gas pressure, and temperature, severely limiting the flexibility of fuel cell power control. Most existing methods focus on small-signal analysis of bidirectional DC-DC converters and Boost converters, without performing small-signal analysis on the fuel cell itself. Existing fuel cell modeling methods generally only mathematically model the fuel cell itself, without mathematically modeling the fuel cell hybrid system consisting of a supercapacitor and a bidirectional DC-DC converter. Summary of the Invention

[0003] In view of this, the purpose of this invention is to provide a small-signal stability analysis method for fuel cell hybrid systems, which helps to detect faults and then use appropriate methods to eliminate the impact of faults, thereby improving the stability of the power system.

[0004] To achieve the above objectives, the present invention adopts the following technical solution: a small-signal stability analysis method for a fuel cell hybrid system. First, a control model of the fuel cell is constructed based on the power generation principle of the fuel cell, and a circuit model of a Boost converter, a bidirectional DC-DC converter connected to a supercapacitor is constructed. Second, based on Kirchhoff's laws and the theoretical knowledge of linear systems, small-signal models of the fuel cell connected to the Boost converter and the energy storage device supercapacitor connected to the bidirectional DC-DC converter are constructed. Finally, based on the above small-signal models, the state matrix of the system is obtained, and its eigenvalues ​​are calculated using simulation software. According to Lyapunov's first law, it is determined whether the real parts of the system's eigenvalues ​​are all greater than 0, thereby further determining whether the fuel cell hybrid system is stable.

[0005] In a preferred embodiment, the main circuit topology of the fuel cell hybrid system includes a fuel cell module and an energy storage module. The fuel cell module consists of the fuel cell itself and a Boost converter. The Boost converter in the DC-DC converter acts as a bridge between the load and the fuel cell. The DC power generated by the fuel cell is amplified by the Boost converter to match the subsequent load or network. The energy storage module consists of a supercapacitor and a bidirectional DC-DC converter. The supercapacitor acts as an auxiliary energy storage unit to supplement or absorb the insufficient or surplus energy, enabling the power supply system to quickly achieve power balance. The bidirectional DC-DC converter plays the role of step-up / step-down and impedance transformation, which can improve the energy utilization rate of the supercapacitor and suppress the output current fluctuation of the fuel cell.

[0006] In a preferred embodiment, the control block diagram and small-signal model of the fuel cell are as follows: the partial pressures of each gas in the two electrodes of the fuel cell should satisfy the ideal gas law:

[0007] pV = nRT (1)

[0008] Differentiating formula (1) with respect to time yields:

[0009]

[0010] Where p is the partial pressure of the gas, V is the volume of the container, R is the universal gas constant, T is the gas temperature, and Δq represents the equilibrium value of the molar flow rate of the gas; for hydrogen, oxygen, and water, equation (2) can be rewritten as equation (3):

[0011]

[0012] in, These are represented as the partial pressures of hydrogen, oxygen, and water, respectively. V represents the molar flow rates of the input and output electrodes and the gases participating in the reaction, respectively; an V is the anode volume. ca The cathode volume;

[0013] The molar flow rate of the gas participating in the reaction and the output current I of the fuel cell SC There must be a connection between them:

[0014]

[0015] Where K r =NI fc / (4F) is the reaction constant; N is the number of cells connected in series in the fuel cell stack; F is the Faraday constant;

[0016] Regarding the relationship between the molar flow rate of the gas discharged from the electrode and the partial pressure of the gas, at a constant temperature, the mixed gas should satisfy:

[0017]

[0018] Where W is the mass flow rate, p is the gas pressure in the channel, K is the valve constant, the value of which is mainly related to the pore area, and M is the molar mass of the mixed gas.

[0019] Since the fuel input to the fuel cell cannot be fully utilized, the concept of fuel utilization rate is introduced here. It represents the degree to which the input hydrogen is utilized, and can be expressed by the molar flow rate of hydrogen participating in the reaction and at the input electrode.

[0020]

[0021] Using fuel utilization rate to express the proportional relationship between reactants and reaction products, equation (5) can be written as equation (7):

[0022]

[0023] Considering that the molar flow rate of any gas passing through the valve is proportional to its partial pressure within the channel, the following expression holds:

[0024]

[0025]

[0026] Based on equations (8) and (9), equation (7) can be rewritten as:

[0027]

[0028] Substituting equations (4) and (8) into equation (10) and performing a Laplace transform, we obtain the partial pressure equation for hydrogen:

[0029]

[0030] in Let be the hydrogen gas flow response time constant; similarly, the water pressure and oxygen partial pressure can be expressed as:

[0031]

[0032] The terminal voltage of a stack of N fuel cell cells connected in series is:

[0033] V fc =N(EV) loss (13)

[0034] Given the partial pressures of the reactants and products, the electromotive force of a single fuel cell can be expressed using the Nernst equation:

[0035]

[0036] Based on the above equations, the fuel cell stack model can be obtained:

[0037] make And its Laplace transform is:

[0038]

[0039] Depend on Figure 2 The stack model yields:

[0040]

[0041] After simplification, we get:

[0042]

[0043] Equation (17) can be obtained by performing an inverse Laplace transform.

[0044]

[0045] After sorting, we can get

[0046]

[0047] Similarly and The expression is as follows:

[0048]

[0049]

[0050]

[0051] Equation (20) can be written in matrix form as follows:

[0052]

[0053] Small-signal analysis of equation (21) yields:

[0054]

[0055] Wherein, the state matrix

[0056] In a preferred embodiment, the small-signal model of the Boost converter is as follows: The small-signal analysis method adopts the state-space averaging method. According to Kirchhoff's laws, the state equation of the Boost converter can be obtained as follows:

[0057]

[0058] Where d is the duty cycle, the small-signal model of the Boost converter can be written according to equation (23):

[0059]

[0060] Wherein, the state matrix

[0061] Where D and I fc These represent the duty cycle and the steady-state value of the inductor current when the fuel cell outputs rated power, respectively. Load It is the equivalent load resistance.

[0062] In a preferred embodiment, a bidirectional DC-DC converter is used to connect the supercapacitor and the load. The supercapacitor serves as an auxiliary energy storage device. The instantaneous power of the equivalent circuit of the supercapacitor stack is:

[0063] p sc =v sc i sc =(v sc0 -r sc i sc )i sc (25)

[0064] The bidirectional DC-DC converter can determine its operating mode based on the conduction status of the upper and lower transistors. When the upper transistor is completely off and the lower transistor is periodically switched on and off, the circuit operates in the Boost model, and the supercapacitor discharges. When the lower transistor is completely off and periodically switched on and off, the circuit operates in the Buck model, and the supercapacitor charges. Assuming the duty cycle of the lower transistor is d, using the state-space averaging method, its state equation is:

[0065]

[0066] Its small-signal model is:

[0067]

[0068] Its state matrix

[0069] In a preferred embodiment, the state-space expression of the fuel cell hybrid system is as follows:

[0070]

[0071] State matrix of fuel cell hybrid system:

[0072]

[0073] Substituting the required data into matrix A and calculating the eigenvalues, the small-signal stability of the fuel cell hybrid system can be obtained; it can be seen from equation (28) that when i sc When changes occur, the bus voltage u of the fuel cell hybrid system BUS This will change, and consequently the state space matrix will change, affecting the small-signal stability of the system. Therefore, it can be explained that supercapacitors have a significant impact on the stability of fuel cell hybrid systems.

[0074] Compared with existing technologies, this invention has the following advantages: Based on the modeling of the fuel cell system, this invention employs the state-space method to construct small-signal mathematical models of the fuel cell, Boost converter, supercapacitor, and bidirectional DC-DC converter. By substituting known data into the state matrix, the small-signal stability of the fuel cell system can be further analyzed through the small-signal models of each module. Analyzing the small-signal stability of the fuel cell system helps improve the energy utilization rate of new power systems, aids in fault detection, and allows for the adoption of appropriate methods to eliminate the impact of faults, thereby improving the stability of the power system. Attached Figure Description

[0075] Figure 1 A flowchart of a preferred embodiment of the present invention;

[0076] Figure 2 This is a preferred embodiment of the fuel cell hybrid system topology of the present invention;

[0077] Figure 3 This is a fuel cell stack model according to a preferred embodiment of the present invention;

[0078] Figure 4 This is a preferred embodiment of the Boost converter topology of the present invention;

[0079] Figure 5 This is the equivalent circuit of the supercapacitor stack in a preferred embodiment of the present invention;

[0080] Figure 6 This is a topology diagram of a supercapacitor stacked bidirectional DC-DC converter according to a preferred embodiment of the present invention;

[0081] Figure 7 This is a detailed topology diagram of a fuel cell hybrid system according to a preferred embodiment of the present invention. Detailed Implementation

[0082] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0083] It should be noted that the following detailed descriptions are illustrative and intended to provide further explanation of this application. Unless otherwise specified, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this application pertains.

[0084] It should be noted that the terminology used herein is for the purpose of describing particular implementations only and is not intended to limit the exemplary implementations according to this application; as used herein, the singular form is intended to include the plural form as well, unless the context clearly indicates otherwise; furthermore, it should be understood that when the terms “comprising” and / or “including” are used in this specification, they indicate the presence of features, steps, operations, devices, components and / or combinations thereof.

[0085] This invention discloses a small-signal stability analysis method for fuel cell hybrid systems, with reference to... Figure 1-7 First, a control model for the fuel cell is constructed based on its power generation principle, along with circuit models of a Boost converter, a bidirectional DC-DC converter, and a supercapacitor. Second, based on Kirchhoff's laws and the theory of linear systems, small-signal models of the fuel cell connected to the Boost converter and the supercapacitor connected to the bidirectional DC-DC converter are constructed. Finally, based on the aforementioned small-signal models, the system's state matrix is ​​obtained, and its eigenvalues ​​are calculated using simulation software. According to Lyapunov's first law, it is determined whether the real parts of all eigenvalues ​​are greater than 0, further assessing the stability of the fuel cell hybrid system.

[0086] The main circuit topology of a fuel cell hybrid system is as follows: Figure 1 As shown, the system consists of two parts: a fuel cell module and an energy storage module. The fuel cell module comprises the fuel cell itself and a Boost converter. Due to the relatively low stack voltage of the fuel cell, the Boost converter in the DC-DC converter is introduced here as a bridge between the load and the fuel cell. The DC power generated by the fuel cell is amplified by the Boost converter to match the subsequent load or network. The energy storage module consists of a supercapacitor and a bidirectional DC-DC converter. The supercapacitor acts as an auxiliary energy storage unit to supplement or absorb the insufficient or surplus energy, enabling the power supply system to quickly achieve power balance. The bidirectional DC-DC converter plays the role of step-up / step-down and impedance transformation, which can improve the energy utilization rate of the supercapacitor and suppress the output current fluctuation of the fuel cell.

[0087] The partial pressures of the gas at the two electrodes of a fuel cell should satisfy the ideal gas law:

[0088] pV = nRT (1)

[0089] Differentiating formula (1) with respect to time yields:

[0090]

[0091] Where p is the partial pressure of the gas, V is the volume of the container, R is the universal gas constant, T is the gas temperature, and Δq represents the equilibrium value of the molar flow rate of the gas. For hydrogen, oxygen, and water, equation (2) can be rewritten as equation (3):

[0092]

[0093] in, These are represented as the partial pressures of hydrogen, oxygen, and water, respectively. V represents the molar flow rates of the input and output electrodes and the gases participating in the reaction, respectively; an V is the anode volume. ca This represents the cathode volume.

[0094] The molar flow rate of the gas participating in the reaction and the output current I of the fuel cell SC There must be a connection between them:

[0095]

[0096] Where K r =NI fc / (4F) is the reaction constant; N is the number of cells connected in series in the fuel cell stack; F is the Faraday constant.

[0097] Regarding the relationship between the molar flow rate of the gas discharged from the electrode and the partial pressure of the gas, at a constant temperature, the mixed gas should satisfy:

[0098]

[0099] Where W is the mass flow rate, p is the gas pressure in the channel, K is the valve constant, whose value is mainly related to the pore area, and M is the molar mass of the mixed gas.

[0100] Since the fuel input to the fuel cell cannot be fully utilized, the concept of fuel utilization rate is introduced here. It represents the degree to which the input hydrogen is utilized and can be expressed by the molar flow rate of hydrogen participating in the reaction and at the input electrode.

[0101]

[0102] Using fuel utilization rate to express the proportional relationship between reactants and reaction products, equation (5) can be written as equation (7):

[0103]

[0104] Considering that the molar flow rate of any gas passing through the valve is proportional to its partial pressure within the channel, the following expression holds:

[0105]

[0106]

[0107] Based on equations (8) and (9), equation (7) can be rewritten as:

[0108]

[0109] Substituting equations (4) and (8) into equation (10) and performing a Laplace transform, we obtain the partial pressure equation for hydrogen:

[0110]

[0111] in Let be the hydrogen gas flow response time constant. Similarly, the water pressure and oxygen partial pressure can be expressed as:

[0112]

[0113] The terminal voltage of a stack of N fuel cell cells connected in series is:

[0114] V fc =N(EV) loss (13)

[0115] Given the partial pressures of the reactants and products, the electromotive force of a single fuel cell can be expressed using the Nernst equation:

[0116]

[0117] Based on the above equations, the fuel cell stack model can be obtained:

[0118] make And its Laplace transform is:

[0119]

[0120] Depend on Figure 2 The stack model yields:

[0121]

[0122] After simplification, we get:

[0123]

[0124] Equation (17) can be obtained by performing an inverse Laplace transform.

[0125] After sorting, we can get

[0126]

[0127] Similarly and The expression is as follows:

[0128]

[0129]

[0130]

[0131] Equation (20) can be written in matrix form as follows:

[0132]

[0133] Small-signal analysis of equation (21) yields:

[0134]

[0135] Wherein, the state matrix

[0136] This invention employs a Boost converter to connect the fuel cell and the load network. The Boost converter not only boosts the voltage to match the subsequent load network but also regulates the fuel cell's output current by controlling the inductor current, limiting fuel utilization within a preset range. The topology of the Boost converter and fuel cell stack is as follows: Figure 4 As shown:

[0137] The small-signal analysis method of this invention employs the state-space averaging method. According to Kirchhoff's laws, the following can be obtained: Figure 3 The state equation of the Boost converter is:

[0138]

[0139] Where d is the duty cycle, the small-signal model of the Boost converter can be written according to equation (23):

[0140]

[0141] Wherein, the state matrix

[0142] Where D and I fc These represent the duty cycle and the steady-state value of the inductor current when the fuel cell outputs rated power, respectively. Load It is the equivalent load resistance.

[0143] In fuel cell systems, constrained by the current saturation equation, fuel cells often require a relatively long transient process to meet load demands. Therefore, it is essential to introduce auxiliary energy storage units to supplement or absorb the insufficient or surplus energy, enabling the power supply system to quickly achieve power balance. Since the auxiliary power supply only needs to provide power support during the short transient period, its output and input are both zero in steady state. Therefore, this invention selects supercapacitors as auxiliary energy storage devices. Supercapacitor units have relatively low voltages, similar to fuel cell units. In practical applications, they must be connected in series and parallel to form a supercapacitor stack to meet the requirements for voltage, energy storage capacity, and power. Therefore, a bidirectional DC-DC converter is selected to connect the supercapacitors to the load. The bidirectional DC-DC converter not only performs step-up and step-down functions but also allows for a wider voltage variation range for the supercapacitors, more flexible design, and higher energy utilization.

[0144] The topology of a supercapacitor stack is as follows Figure 5 As shown:

[0145] Its instantaneous power:

[0146]

[0147] The topology of a bidirectional DC-DC converter connected to a supercapacitor stack is as follows: Figure 6 As shown:

[0148] A bidirectional DC-DC converter can determine its operating mode based on the conduction status of the upper and lower transistors. When the upper transistor is completely off and the lower transistor is periodically switched on and off, the circuit operates in a Boost model, and the supercapacitor discharges. When the lower transistor is completely off and periodically switched on and off, the circuit operates in a Buck model, and the supercapacitor charges. Let the duty cycle of the lower transistor be d. Using the state-space averaging method, its state equation is:

[0149]

[0150] Its small-signal model is:

[0151]

[0152] Its state matrix

[0153] according to Figure 2 The topology diagram of the fuel cell hybrid system can be obtained Figure 7 , Figure 7 By organically combining fuel cell stacks with supercapacitor modules, and controlling the bus voltage by controlling the charging and discharging of the supercapacitors, the fuel cell hybrid system can achieve balance.

[0154] according to Figure 7 The state-space expression for the fuel cell hybrid system can be obtained as follows:

[0155]

[0156] State matrix of fuel cell hybrid system:

[0157]

[0158] Substituting the required data into matrix A and calculating the eigenvalues, the small-signal stability of the fuel cell hybrid system can be obtained. Equation (28) shows that when i sc When changes occur, the bus voltage u of the fuel cell hybrid system BUS This will change, and consequently the state space matrix will change, affecting the small-signal stability of the system. Therefore, it can be explained that supercapacitors have a significant impact on the stability of fuel cell hybrid systems.

Claims

1. A method for small-signal stability analysis of a fuel cell hybrid system, characterized in that, First, a control model for the fuel cell is constructed based on its power generation principle, along with circuit models of a Boost converter, a bidirectional DC-DC converter, and a supercapacitor. Second, based on Kirchhoff's laws and the theory of linear systems, small-signal models of the fuel cell connected to the Boost converter and the supercapacitor connected to the bidirectional DC-DC converter are constructed. Finally, based on the aforementioned small-signal models, the system's state matrix is ​​obtained, its eigenvalues ​​are calculated using simulation software, and Lyapunov's first law is used to determine whether the fuel cell hybrid system is stable. The main circuit topology of the fuel cell hybrid system includes a fuel cell module and an energy storage module. The fuel cell module consists of the fuel cell itself and a Boost converter. The Boost converter in the DC-DC converter acts as a bridge between the load and the fuel cell. The DC power generated by the fuel cell is amplified by the Boost converter to match the subsequent load or network. The energy storage module consists of a supercapacitor and a bidirectional DC-DC converter. The supercapacitor acts as an auxiliary energy storage unit to supplement or absorb the insufficient or surplus energy, enabling the power supply system to quickly achieve power balance. The bidirectional DC-DC converter plays the role of step-up / step-down and impedance transformation, improving the energy utilization rate of the supercapacitor and suppressing the output current fluctuation of the fuel cell. The state-space expression of the fuel cell hybrid system is as follows: (28) State matrix of fuel cell hybrid system: Substitute the required data into the matrix Calculate the eigenvalues ​​to obtain the small-signal stability of the fuel cell hybrid system; from equation (28), it is known that when When changes occur, the bus voltage of the fuel cell hybrid system This will change, and consequently the state space matrix will change, affecting the small-signal stability of the system. This indicates that supercapacitors have a significant impact on the stability of fuel cell hybrid systems.

2. The small-signal stability analysis method for a fuel cell hybrid system according to claim 1, characterized in that, The control block diagram and small-signal model of a fuel cell are as follows: the partial pressures of the gas in the two electrodes of the fuel cell satisfy the ideal gas law: (1) Differentiating formula (1) with respect to time, we get: (2) in, Let V be the partial pressure of the gas, V be the volume of the container, R be the universal gas constant, and T be the gas temperature. The equilibrium value characterizing the molar flow rate of the gas; for hydrogen, oxygen, and water, equation (2) is rewritten as equation (3): (3) in, , , These are the partial pressures of hydrogen, oxygen, and water, respectively. , , These represent the molar flow rates of the input and output electrodes, as well as the gases participating in the reaction; For the anode volume, The cathode volume; The molar flow rate of the gas participating in the reaction and the output current I of the fuel cell fc There must be a connection between them: (4) in is the reaction constant; N is the number of cells connected in series in the fuel cell stack; F is the Faraday constant; Regarding the relationship between the molar flow rate of the gas discharged from the electrode and the partial pressure of the gas, at a constant temperature, the mixed gas should satisfy: (5) in For quality flow, The pressure of the gas in the channel. This is the valve constant, and its value is mainly related to the orifice area. The molar mass of the gas mixture; Since the fuel input to the fuel cell cannot be fully utilized, the concept of fuel utilization rate is introduced here. It represents the degree to which the input hydrogen is utilized, expressed as the molar flow rate of hydrogen participating in the reaction and at the input electrode. (6) Using fuel utilization rate to describe the proportional relationship between reactants and reaction products, equation (5) can be written as equation (7): (7) Considering that the molar flow rate of any gas passing through the valve is proportional to its partial pressure within the channel, the following expression holds: (8) (9) Based on equations (8) and (9), equation (7) can be rewritten as: (10) Substituting equations (4) and (8) into equation (10) and performing a Laplace transform, we obtain the partial pressure equation for hydrogen: (11) in , where is the hydrogen gas flow response time constant; water pressure and oxygen partial pressure are expressed as: (12) The terminal voltage of a stack of N fuel cell cells connected in series is: (13) Given the partial pressures of the reactants and products, the electromotive force of a single fuel cell is expressed using the Nernst equation: (14) Based on the above equations, the fuel cell stack model is obtained: make And its Laplace transform is: (15) From the stack model: (16) After simplification, we get: (17) After performing an inverse Laplace transform on equation (17), we get... (18) After sorting (19) Similarly and The expression is as follows: (20) (21) (22) Equation (20) can be written in matrix form as follows: (23) Small-signal analysis of equation (21) yields: (24) Wherein, the state matrix .

3. The small-signal stability analysis method for a fuel cell hybrid system according to claim 1, characterized in that, The small-signal model of the Boost converter is as follows: The small-signal analysis method adopts the state-space averaging method. According to Kirchhoff's laws, the state equation of the Boost converter is: (25) in Given the duty cycle, the small-signal model of the Boost converter can be written according to equation (25) as follows: (26) Wherein, the state matrix in and These represent the duty cycle and the steady-state value of the inductor current when the fuel cell outputs its rated power, respectively. It is the equivalent load resistance.

4. The small-signal stability analysis method for a fuel cell hybrid system according to claim 1, characterized in that, A bidirectional DC-DC converter is selected to connect the supercapacitor to the load. The supercapacitor serves as an auxiliary energy storage device. The instantaneous power of the equivalent circuit of the supercapacitor stack is: The bidirectional DC-DC converter determines its operating mode based on the conduction status of the upper and lower transistors. When the upper transistor is completely off and the lower transistor is periodically switched on and off, the circuit operates in Boost mode, and the supercapacitor discharges. When the lower transistor is completely off and the upper transistor is periodically switched on and off, the circuit operates in Buck mode, and the supercapacitor charges. Let the duty cycle of the lower transistor be... The state-space averaging method is adopted, and its state equation is: Its small-signal model is: Its state matrix .

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