Differential iteration-based proton exchange membrane fuel cell rated power calculation method

Abstract

The invention provides a differential iteration-based rated power calculation method for a proton exchange membrane fuel cell, which is used for solving the technical problem of low precision of rated power calculation of the proton exchange membrane fuel cell. According to the method, the slope of an approximate linear equation is constructed through the difference between the maximum power point and the minimum power point, a linear approximate root is solved on the basis of the slope of the initial difference, the approximate linear equation is further constructed according to the power point corresponding to the linear approximate root and the tail end of the linear equation, iteration is repeated until the error is gradually reduced, and the power point corresponding to the linear approximate root is calculated. And solving of the rated output power of the PEMFC is realized. Complex derivative calculation of temperature, voltage and current density functions is not needed, only function values need to be calculated in each iteration, other information is not needed, the method is easy to implement, the calculation complexity is effectively reduced, and the calculation efficiency is effectively improved; the function value only needs to be calculated once in each iteration process, the efficiency is high, and efficient and highly precise solution calculation of the rated current density and the rated power point can be achieved.

Description

Technical Field

[0001] The present invention relates to the technical field of fuel cell power calculation, and in particular to a method for calculating the rated power of a Proton Exchange Membrane Fuel Cell (PEMFC) based on differential iteration. Background Art

[0002] PEMFC can directly convert hydrogen into electric energy and water through electrochemical reactions, and has the advantages of high efficiency, pollution-free, fast startup speed, etc., and has great application potential in the fields of transportation, distributed power generation, etc.

[0003] The operation of PEMFC involves complex electrode kinetic processes such as internal reaction charge transfer, reaction gas transfer, charge transport, etc. Among them, the polarization curve, which describes the output voltage value of PEMFC at different current densities, is one of its important characteristics.

[0004] The polarization curve of PEMFC fuel cell is affected by multiple factors, including equilibrium potential, activation polarization loss, internal current and permeation loss, concentration polarization loss, resistance loss, etc., and can be described as,

[0005]

[0006] In the formula, V cell is the fuel cell potential; T is the temperature, with the unit of K; R is the universal gas constant (8.314 J / mol / K); F is the Faraday constant (96487 C / mol); α = 1 is the transfer coefficient; R i is the internal resistance of the fuel cell, measured by Electrochemical Impedance Spectroscopy (EIS), range: 0.05 - 0.2, with the unit of Ω·cm 2 ; i and i0 are the actual current density and the exchange current density respectively. The actual current density is the actual working state of PEMFC, determined by the load; the exchange current density reflects the catalyst activity, obtained through experimental fitting or quantum chemical calculation, with the unit of A / m 2 ; i loss and i L are the internal loss current density and the limiting current density respectively, obtained by extrapolating the saturation region of the polarization curve or calculating through a mass transfer model, range: 1 - 3, with the unit of A / m 2 ; n = 4, which is the number of electron transfers in the whole reaction; E rTP is the equilibrium potential at the corresponding reactant pressure and temperature, which is described by the Nernst equation, that is,

[0007]

[0008] In the formula, 1.482 is the theoretical potential corresponding to the high calorific value of hydrogen; They are the pressures of the reaction hydrogen and oxygen respectively, with the unit of atm.

[0009] In formula (1), part is the voltage loss caused by activation polarization, internal current and permeation loss; is the concentration polarization voltage loss; iR i is the voltage of the resistance loss. The polarization curve of a typical PEMFC and the voltage losses of each part are as Figure 2 shown, Figure 2 and the loss curves of each part and the value of the equilibrium potential can be seen.

[0010] In the design process of PEMFC, it is necessary to determine its rated output power according to the rated output voltage of the stack. The key to calculating the rated output power according to formula (1) is to determine the current density at the rated output voltage. The difficulties are as follows: ① Formula (1) is a complex nonlinear equation and is difficult to directly solve and calculate. ② Formula (1) is greatly affected by temperature and working pressure; it is difficult to efficiently and quickly solve the rated power at different temperatures.

[0011] Common practices in engineering: ① Combining the relevant parameters of PEMFC, traverse the output voltages at different current densities according to formula (1), and then find the current density corresponding to the rated output voltage. This method is simple and easy to implement, but the accuracy is low and it cannot achieve accurate calculation of the current density. ② Use a linear equation to approximately fit formula (1), linearly approximate the relationship between current density and voltage, and then solve the current density corresponding to the rated voltage. The accuracy of this method is even worse, especially when the linearity near the rated operating point in the actual polarization curve is poor.

[0012] The invention patent with the application number 202411272698.2 discloses a method for optimizing the power density and operating parameters of a proton exchange membrane fuel cell. Taking the operating temperature of the stack, anode pressure, cathode / anode relative humidity and current density as inputs and the power density as the output, a proton exchange membrane fuel cell surrogate model based on the random forest algorithm is constructed, and the inputs and outputs of the surrogate model are optimized in combination with the improved spectral algorithm, which can predict the maximum power density and the corresponding operating parameters. When the load demand changes, the fitness function of the improved spectral algorithm can be changed to calculate the power density corresponding to the load demand and predict its corresponding operating parameters. The above method shortens the optimization time, reduces the calculation burden in the optimization process, and improves the optimization efficiency. However, this method can only calculate the maximum power density and its corresponding parameters, depends on strict model parameters, and belongs to swarm optimization, requiring a large amount of calculation and having low calculation efficiency. Summary of the Invention

[0013] Aiming at the technical problem of low accuracy in calculating the rated power of proton exchange membrane fuel cells, the present invention proposes a method for calculating the rated power of proton exchange membrane fuel cells based on differential iteration, which is used for solving the rated current density and rated power during the design process of proton exchange membrane fuel cells, and has the advantages of small calculation amount and insensitivity to parameters.

[0014] To achieve the above object, the technical solution of the present invention is realized as follows: A method for calculating the rated power of proton exchange membrane fuel cells based on differential iteration constructs the differential slope of an approximate linear equation through the difference between the maximum power and the minimum power point. Based on the differential slope, the linear approximate root is solved, and an approximate linear equation is further constructed with the power point corresponding to the linear approximate root and the end of the initial linear equation, and iterated repeatedly until the error gradually shrinks, realizing the solution of the rated output power of PEMFC.

[0015] The steps of the present invention are as follows:

[0016] Step 1: Construct a temperature, voltage, and current density function according to the calculation equation of the fuel cell potential, and calculate the differential slope using the data points at both ends of the temperature, voltage, and current density function.

[0017] Step 2: Construct a linear straight-line equation of current density based on the data points at both ends of the temperature, voltage, and current density function and the differential slope.

[0018] Step 3: Obtain the abscissa of the intersection point with the current density axis according to the linear straight-line equation of current density, calculate the function value on the temperature, voltage, and current density function corresponding to the abscissa of the current density axis, and determine whether the function value is less than a preset threshold. If so, use the abscissa of the intersection point of the current density axis as the current density of the fuel cell potential to calculate the rated power point; otherwise, enter Step 4.

[0019] Step 4: Use the data point composed of the abscissa of the current density axis and the corresponding function value as the new endpoint and the data point at the right end of the linear straight-line equation of current density to reconstruct the linear straight-line equation of current density, and return to Step 3.

[0020] Preferably, the temperature, voltage, and current density function is:

[0021]

[0022] where, V cell is the fuel cell potential; T is the temperature; R is the gas constant; F is the Faraday constant; α = 1 is the transfer coefficient; R i is the internal resistance of the fuel cell; i and i0 are the actual current density and the exchange current density respectively; i loss and i Lare the internal loss current density and the limiting current density respectively, and n is the number of electron transfers in the whole reaction; E rTP is the equilibrium potential at the corresponding reactant pressure and temperature.

[0023] Preferably, the method for calculating the differential slope is as follows: Substitute the minimum current value i min and the maximum current value i max into the temperature, voltage, and current density function f(i) respectively to obtain the function values f(i min ), f(i max ), and obtain the data points at both ends of the temperature, voltage, and current density function f(i): [i min , f(i min )] and [i max , f(i max ); Calculate the differential slope of the two data points:

[0024] Preferably, the minimum current value i min = 0, and the maximum current value i max is the limiting current density of the PEMFC, which is determined by the electrochemical characteristics, i max = i L = nFDC B / δ, where D is the diffusion coefficient of the reactant component; C B is the total concentration of the reactant; δ is the diffusion distance.

[0025] Preferably, the method for constructing the linear straight-line equation of the current density is as follows:

[0026] Construct the data points at both ends: [i min , f(i min )] and [i max , f(i max )] and the straight line corresponding to the differential slope κ:

[0027] y = κ(i - i max ) + f(i max );

[0028] where y is the dependent variable and the current density i is the independent variable.

[0029] Preferably, let y = 0, solve the straight-line equation y = κ(i - i max ) + f(i max ), and obtain the intersection point i t of the t-th iteration and the current density axis i; Substitute the current density i t into the temperature, voltage, and current density function f(i) to obtain the point [i t of the intersection point it , f(i t )], check if the function value f(i t ) ≤ ε, where the error precision ε is a preset threshold.

[0030] Preferably, the implementation method of step four is as follows: using the endpoint [i max , f(i max )] and the point [i t , f(i t )] as the new two end data points, reconstruct the linear straight-line equation, and solve for the intersection point i t+1 of the (t + 1)-th iteration with the current density axis i, then return to execute step three until the function value f(i t ) ≤ ε, then it is determined that the value of the intersection point it is the current density of the fuel cell potential V cell .

[0031] Preferably, calculate the rated power point as P cell (T) = i t * V cell .

[0032] Preferably, the error precision ε is ε = 1 * 10 -7 .

[0033] Compared with the prior art, the beneficial effects of the present invention are as follows: by taking the difference between the maximum power and the minimum power points, the slope of an approximate linear equation is constructed. Based on the initial difference slope, the linear approximate root is solved, and an approximate linear equation is further constructed with the power point corresponding to this linear approximate root and the end of the initial point linear approximate equation. Through repeated iteration until the error gradually shrinks, the efficient and high-precision solution of the rated output power of the PEMFC is realized. And it has the following advantages:

[0034] (1) It does not require the complex derivative calculation of the nonlinear equation (3) of temperature, voltage, and current density function, and only the function value needs to be calculated in each iteration without other information, which is easy to implement and effectively reduces the computational complexity and improves the computational efficiency;

[0035] (2) Only the function value needs to be calculated once in each iteration process (solving equation (5)), with high efficiency, and the efficient and high-precision solution calculation of the rated current density and the rated power point can be realized. BRIEF DESCRIPTION OF THE DRAWINGS

[0036] In order to more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required for the description of the embodiments or the prior art. Obviously, the following drawings are only some embodiments of the present invention. For those of ordinary skill in the art, other drawings can be obtained based on these drawings without creative efforts.

[0037] Figure 1 This is the flowchart of the present invention.

[0038] Figure 2 This is the graph of the typical polarization curve of PEMFC and the voltage losses of each part.

[0039] Figure 3 This is the fuel cell potential V of the present invention cell = 0.6V, T = 330K, the graph of the calculation process of the rated current.

[0040] Figure 4 This is the fuel cell potential V of the present invention cell = 0.6V, the graph of the rated current density at different temperatures.

[0041] Figure 5 This is the fuel cell potential V of the present invention cell = 0.6V, 0.7V, 0.8V, the graph of the rated current density at different temperatures. Specific embodiments

[0042] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts shall fall within the protection scope of the present invention.

[0043] As Figure 1 shown, a method for calculating the rated power of a proton exchange membrane fuel cell based on differential iteration mainly has the following creative works: by taking the difference between the maximum power and the minimum power points, the slope of an approximate linear equation is constructed. Based on the initial difference slope, the linear approximate root is solved, and with the power point corresponding to this root and the end of the linear approximate equation at the initial point, an approximate linear equation is further constructed and iterated repeatedly until the error gradually shrinks, thereby realizing the efficient and high-precision solution of the rated output power of PEMFC. The main steps of the present invention are as follows:

[0044] Step 1: Construct a temperature, voltage, and current density function according to the calculation equation of the fuel cell potential, and calculate the difference slope using the data points at both ends of the temperature, voltage, and current density function.

[0045] Construct a temperature, voltage, and current density function:

[0046]

[0047] At the same time, let the iteration number t = 1.

[0048] Among them, the minimum current value i min = 0, and the maximum current value i max is the limiting current density of the PEMFC, which is determined by its electrochemical characteristics. i max = i L = nFDC B / δ, where D is the diffusion coefficient of the reactant component; C B is the total concentration of the reactants; δ is the diffusion distance. Substitute the minimum current value i min , the maximum current value i max into the temperature, voltage, and current density functions of formula (3) respectively to obtain the function values f(i min ), f(i max ), and then obtain the data points [i min , f(i min )] and [i max , f(i max )] at both ends of the temperature, voltage, and current density function f(i).

[0049] Calculate the differential slopes of the data points at both ends:

[0050]

[0051] Step 2: Construct a linear straight-line equation of the current density according to the data points at both ends of the temperature, voltage, and current density function and the differential slope.

[0052] Construct the data points [i min , f(i min )] and [i max , f(i max )] at both ends and the straight line corresponding to the differential slope κ:

[0053] y = κ(i - i max ) + f(i max )(5)

[0054] where y is the dependent variable and the current density i is the independent variable.

[0055] Step 3: Obtain the abscissa of the intersection point with the current density axis according to the linear straight-line equation of the current density, calculate the function value on the temperature, voltage, and current density function corresponding to the abscissa of the current density axis, and determine whether the function value is less than the preset threshold. If so, use the abscissa of the intersection point of the current density axis as the current density of the fuel cell potential to calculate the rated power point; otherwise, go to Step 4.

[0056] Step 4: Use the data point composed of the abscissa of the current density axis and the corresponding function value as the new endpoint to reconstruct the linear straight-line equation of the current density with the data point at the right end of the linear straight-line equation of the current density, and return to Step 3.

[0057] As shown Figure 3 in the figure, let y = 0, solve the straight-line equation (5) first, and obtain the intersection point i of the straight line with the current density axis i t ; and substitute the current density it into formula (3) to obtain the point [i t , f(i t )] on the function f(i) of temperature, voltage, and current density, and check whether the function value f(i t ) ≤ ε? ε is a set error precision. When the error is less than this number, it is considered to meet the condition. In this embodiment, the threshold value is taken as ε = 1×10 -7 .

[0058] Using the endpoints [i max , f(i max )] and [i t , f(i t )] as the new two-end data points, reconstruct the linear straight-line equation, and solve the intersection point i of the straight line with the current density axis i t+1 , repeat the above steps until the function value f(i t ) ≤ ε, then it is determined that [i t , f(i t )] is the current density corresponding to the fuel cell potential V cell .

[0059] When the fuel cell potential V cell = 0.6V and T = 330K, the above calculation process is as Figure 3 shown. It can be seen through Figure 3 that as the iteration progresses, the calculated current density is getting closer and closer to the actual current density.

[0060] Calculate the rated power point P cell (T) = i t * V cell .

[0061] Substitute different working temperatures T into formula (3), and repeat the above steps to obtain the rated power points at different temperatures T. Figure 4 Shown is the rated current density at different temperatures when V cell = 0.6V, Figure 4 verifying the applicability of the method proposed by the present invention at different temperatures.

[0062] Figure 5 Shown are the rated current density curves at different temperatures when V cell = 0.6V, 0.7V, and 0.8V, verifying the applicability of the method of the present invention at different rated voltages.

[0063] As can be seen from the embodiments of the theoretical analysis section, the difference between the obtained power point and the actual power point is determined by the error precision ε. When the error precision ε = 1*10 -7 , only about 7 iterations are required to obtain a power point with an error less than 1*10 -7 *V cell . Only 7 linear calculations are required to achieve high-precision rated power acquisition.

[0064] The above are only the preferred embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principles of the present invention shall be included within the protection scope of the present invention.

Claims

1. A method for calculating the rated power of a proton exchange membrane fuel cell based on differential iteration, characterized in that, By taking the difference between the maximum power and the minimum power point, the difference slope of an approximate linear equation is constructed. Based on the difference slope, the linear approximate root is solved, and using the power point corresponding to the linear approximate root and the end of the initial linear equation, an approximate linear equation is further constructed. Through repeated iteration until the error gradually shrinks, the solution of the rated output power of the PEMFC is achieved.

2. The method for calculating the rated power of a proton exchange membrane fuel cell based on differential iteration according to claim 1, wherein The steps are as follows: Step 1: Construct a temperature, voltage, current density function according to the calculation equation of the fuel cell potential, and calculate the difference slope using the data points at both ends of the temperature, voltage, current density function; Step 2: Construct a linear straight-line equation of the current density based on the data points at both ends of the temperature, voltage, current density function and the difference slope; Step 3: Obtain the abscissa of the intersection point with the current density axis according to the linear straight-line equation of the current density, calculate the function value on the temperature, voltage, current density function corresponding to the abscissa of the current density axis, and determine whether the function value is less than a preset threshold. If so, use the abscissa of the intersection point of the current density axis as the current density of the fuel cell potential to calculate the rated power point; otherwise, go to Step 4; Step 4: Use the data point composed of the abscissa of the current density axis and the corresponding function value as the new endpoint and the data point at the right end of the linear straight-line equation of the current density to reconstruct the linear straight-line equation of the current density, and return to Step 3.

3. The method for calculating the rated power of a proton exchange membrane fuel cell based on differential iteration according to claim 2, wherein The temperature, voltage, current density function is: Among them, V cell is the fuel cell potential; T is the temperature; R is the gas constant; F is the Faraday constant; α = 1 is the transfer coefficient; R i is the internal resistance of the fuel cell; i and i0 are the actual current density and the exchange current density respectively; i loss and i L are the internal loss current density and the limiting current density respectively, and n is the number of electron transfers in the whole reaction; E rTP is the equilibrium potential at the corresponding reactant pressure and temperature.

4. The method for calculating the rated power of a proton exchange membrane fuel cell based on differential iteration according to claim 3, characterized in that, The method for calculating the differential slope is as follows: Substitute the minimum current value i min , the maximum current value i max into the temperature, voltage, and current density function f(i) respectively to obtain the function values f(i min ), f(i max ), and obtain the data points at both ends of the temperature, voltage, and current density function f(i) [i min , f(i min )] and [i max , f(i max )]; Calculate the differential slope of the two data points:

5. The method for calculating the rated power of a proton exchange membrane fuel cell based on differential iteration according to claim 4, wherein The minimum current value i min = 0, and the maximum current value i max is the limiting current density of the PEMFC, which is determined by the electrochemical characteristics. i max = i L = nFDC B / δ, where D is the diffusion coefficient of the reactant component; C B is the total concentration of the reactants; and δ is the diffusion distance.

6. The method for calculating the rated power of a proton exchange membrane fuel cell based on differential iteration according to claim 4 or 5, characterized in that The method for constructing the linear straight-line equation of the current density is: Data points at both ends of the structure [i min , f(i min )] and [i max , f(i max )], and the straight line corresponding to the differential slope κ: y = κ(i - i max ) + f(i max ); Among them, y is the dependent variable, and the current density i is the independent variable.

7. The method for calculating the rated power of a proton exchange membrane fuel cell based on differential iteration according to claim 6, wherein Let \(y = 0\), and solve the linear equation \(y=\kappa(i - i max )+f(i max ) to obtain the intersection point \(i t \) of the \(t\)-th iteration and the current density axis \(i\); substitute the current density \(i t \) into the temperature, voltage, current density function \(f(i)\) to obtain the point \([i t , f(i t )]\) of the intersection point \(i t \) on the temperature, voltage, current density function \(f(i)\), and check if the function value \(f(i t )\leq\varepsilon\), where the error precision \(\varepsilon\) is a preset threshold.

8. The method for calculating the rated power of a proton exchange membrane fuel cell based on differential iteration according to claim 7, wherein The implementation method of the fourth step is as follows: Using the endpoint [i max , f(i max )] and the point [i t , f(i t )] as the new two end data points, reconstruct the linear straight-line equation, and solve for the intersection point i t+1 of the (t + 1)-th iteration and the current density axis i, then return to execute the third step until the function value f(i t ) ≤ ε, then it is determined that the value of the intersection point it is the current density of the fuel cell potential V cell .

9. The method for calculating the rated power of a proton exchange membrane fuel cell based on differential iteration according to claim 8, characterized in that, Calculate the rated power point as P cell (T) = i t * V cell .

10. The method for calculating the rated power of a proton exchange membrane fuel cell based on differential iteration according to any one of claims 7-9, characterized in that, The error precision ε is ε = 1*10 -7 .

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