Method for on-line prediction of the residual life of a solid oxide fuel cell stack

Abstract

The application discloses a kind of solid oxide fuel cell stack residual life online prediction method, belong to fuel cell life prediction technical field.The prediction method of the application includes the following steps: obtaining the voltage or power data of the served stage of the stack as historical operation data and drawing the corresponding attenuation curve;According to the attenuation rate of the historical operation data at different operation time points, the attenuation trend of the stack is analyzed according to the attenuation rate, the attenuation trend change node is determined, and the voltage or power attenuation curve is divided into stages, the voltage or power attenuation trend is predicted according to the stage division, and the failure threshold is defined as the life termination value to predict the residual life.The prediction method of the application can realize the residual life online prediction of SOFC, and the prediction method is simple, reliable and practical, and has important significance for the health management of the running SOFC.

Description

Technical Field

[0001] This invention relates to the field of solid oxide fuel cell technology, and more specifically to an online prediction method for the remaining life of a solid oxide fuel cell stack. Background Technology

[0002] Solid oxide fuel cells (SOFCs) can directly convert the chemical energy stored in fuel and oxidant into electrical energy at high temperatures. They offer advantages such as wide fuel applicability, high power generation efficiency, no pollution, and low noise, making them promising for distributed power generation, backup power, and military equipment. Despite these advantages and promising prospects, SOFC technology has not yet seen widespread application, primarily due to issues such as short lifespan and poor durability. After prolonged power generation, the electrode materials of SOFCs inevitably undergo microstructural changes, reducing oxygen vacancy concentration, hindering ion / electron conduction, disrupting the three-phase reaction interface, and causing a decrease in electrochemical reaction rate and performance degradation. Remaining lifespan is a crucial component of SOFC management systems, referring to the time remaining from the current capacity to the end of its lifespan. Predicting remaining lifespan reflects the battery's health status, helps avoid serious consequences caused by degradation, and is a vital basis for developing SOFC maintenance strategies. However, current methods for predicting the remaining lifespan of SOFCs typically rely on creep fatigue damage and electrochemical performance degradation to establish prediction models. This method is relatively complex and not suitable for rapid prediction of the remaining lifespan of SOFCs in the field. Summary of the Invention

[0003] To address the aforementioned technical problems, this invention provides an online prediction method for the remaining lifetime of a solid oxide fuel cell stack.

[0004] The technical solution adopted in this invention is as follows:

[0005] This invention provides an online prediction method for the remaining lifetime of a solid oxide fuel cell stack, comprising the following steps:

[0006] (1) Obtain the current, voltage, and power data of the solid oxide fuel cell stack from the initial operation to the current time of the service stage, as historical operation data, and plot the corresponding voltage or power decay curves of the service stage.

[0007] (2) Calculate the attenuation rate every Δt hours based on the historical operating data collected in step (1) to obtain the attenuation rate at different operating time points. Analyze the attenuation trend of the fuel cell stack based on the calculated attenuation rate, determine the nodes of attenuation trend change, and divide the voltage or power attenuation curve of step (1) into stages according to the nodes. Take the attenuation rate η > 2% / kh as the first stage, which is the rapid attenuation stage in the early stage of fuel cell stack operation. Take the attenuation rate 0.1 < η ≤ 2% / kh as the second stage, which is the slow attenuation stage in the middle stage of fuel cell stack operation. Take the attenuation rate η ≤ 0.1% / kh as the third stage, which is the stable operation stage in the later stage of fuel cell stack operation.

[0008] (3) Calculate the total attenuation rate of the fuel cell stack during its service phase and define the failure threshold as the end-of-life value. Determine whether the total attenuation rate of the service phase exceeds the failure threshold. If it exceeds the threshold, determine that the fuel cell stack has no remaining service life. If it does not exceed the threshold, calculate the service life of the fuel cell stack according to step (4).

[0009] (4) Determine whether the voltage or power decay curve of the fuel cell stack during its service phase includes the first and second and / or third phases. If the voltage or power decay curve of the fuel cell stack during its service phase includes the first, second and third phases, then predict the future voltage or power decay trend based on the decay rate of the third phase and calculate the remaining lifetime. If the voltage or power decay curve of the fuel cell stack during its service phase only includes the first and second phases, then predict the future voltage or power decay trend based on the decay rates of the first and second phases at two different times combined with the fuel utilization rate and calculate the remaining lifetime.

[0010] Furthermore, the formula for calculating the attenuation rate in step (2) is as follows:

[0011]

[0012] Where, η n Pn is the attenuation rate per thousand hours at time n, where n is the index and represents time n. P0 is the output power at the time when the normal operating voltage is reached. n P represents the output power at time n. n-△t This represents the output power at time n-Δt, where Δt is the time interval.

[0013] Furthermore, the formula for calculating the total attenuation rate of the fuel cell stack during its service life in step (3) is as follows:

[0014]

[0015] Where, η s P0 is the total degradation rate of the fuel cell stack during its service life, and P0 is the output power at the point of reaching normal operating voltage. s This represents the current output power of the fuel cell stack.

[0016] Furthermore, in step (3), the failure threshold is set to 10% to 40%.

[0017] Furthermore, in step (3), when the voltage or power decay curve of the fuel cell stack during its service life includes the first, second, and third stages, the future voltage or power decay trend is predicted using the decay rate of the third stage. f The output power after the time is:

[0018] P f =P s -(P0×a3×t f )

[0019] Among them, P f To extend the service life of the fuel cell stack beyond its current service time f The output power corresponding to the time, P0 is the output power at the moment of reaching the normal operating voltage, a3 is the attenuation rate of the third stage, taken as 0.1% / kh, t f For the time remaining in service in the future, kh; P s This represents the current output power of the fuel cell stack.

[0020] Furthermore, in step (3), when the voltage or power decay curve of the fuel cell stack during its service phase includes the first, second, and third phases, the formula for calculating the remaining lifetime when the output power reaches the failure threshold is as follows:

[0021]

[0022] Among them, t RUL Kh represents the remaining service life from the current moment until the failure threshold is reached; P0 represents the output power at the moment the normal operating voltage is reached. s P represents the current output power of the fuel cell stack. m To achieve the output power at the failure threshold, P m = (1-m)P0, where m is the failure threshold and a3 is the decay rate of the third stage, which is 0.1% / kh.

[0023] Furthermore, in step (3), if the voltage or power decay curve of the fuel cell stack during its service life only includes the first and second stages, then the future voltage or power decay trend is predicted by combining the decay rates at two different times in the first and second stages with the fuel utilization rate. f The output power after the time is:

[0024]

[0025] A and B are obtained using the following formula:

[0026]

[0027] Among them, P f To extend the service life of the fuel cell stack beyond its current service time f The output power corresponding to the time, P0 is the output power at the time when the normal operating voltage is reached, and t is the output power at the time when the normal operating voltage is reached. f The remaining service life is kh; A and B are the decay rate coefficients, φ is the fuel utilization rate, T is the operating temperature of the stack during stable operation, j is the current density, and C is the current density. N Let ΔP be the relative attenuation during a certain time interval in stage N, and ΔP be the power attenuation during a certain time interval in stage N. Let N be 1 or 2, and t be... N The selected relative attenuation C N The corresponding runtime.

[0028] Furthermore, in step (3), when the voltage or power decay curve of the fuel cell stack during its service phase only includes the first and second phases, the formula for calculating the remaining lifetime when the output power reaches the failure threshold is as follows:

[0029]

[0030] Among them, t RUL Let kh be the remaining service life from the current moment until the failure threshold is reached. m To reach the output power at the failure threshold, i.e., the end of life, P m = (1-m)P0, where m is the failure threshold.

[0031] Furthermore, step (2) also includes filtering and smoothing the historical operating data collected in step (1) to obtain stable historical operating data, and then using the filtered and smoothed historical operating data to calculate the attenuation rate. The calculation formula for filtering and smoothing is as follows:

[0032]

[0033] Where n is the index, representing time n, h(n) is the impulse response at time n, and τ n Let n be the analysis time point.

[0034] The beneficial effects of this invention are as follows:

[0035] This invention provides an online prediction method for the remaining life of a solid oxide fuel cell stack. Based on historical operating data from the current service phase, a performance degradation curve for the solid oxide fuel cell stack is constructed. The degradation trend is analyzed based on the curve, and a calculation formula for future voltage or power degradation trends is established. Then, based on data from the current service phase, the future degradation trend of the stack can be predicted, and the remaining life of the stack can be predicted according to a defined failure threshold. This prediction method enables online prediction of the remaining life of a solid oxide fuel cell stack. The method is simple, reliable, and highly practical, suitable for rapid prediction of the remaining life of stacks in the field, and is of great significance for the health management of solid oxide fuel cell stacks in operation. Attached Figure Description

[0036] To clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0037] Figure 1 This is a flowchart of the lifetime prediction method of the present invention;

[0038] Figure 2 Historical operating data of output voltage obtained in embodiments of the present invention;

[0039] Figure 3 This is the historical operating data of the output voltage after filtering.

[0040] Figure 4 This is the historical operating data of the output voltage after being divided into stages. Detailed Implementation

[0041] This invention provides an online prediction method for the remaining lifetime of a solid oxide fuel cell stack. To make the objectives, technical solutions, and effects of this invention clearer and more explicit, the invention is further described in detail below. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.

[0042] The present invention will now be described in detail with reference to the accompanying drawings.

[0043] Reference Figure 1 This embodiment provides an online prediction method for the remaining lifetime of a solid oxide fuel cell stack, including the following steps:

[0044] (1) Obtain the current, voltage, and power data of the solid oxide fuel cell stack from the initial operation to the current time of the service stage, as historical operation data, and plot the corresponding voltage or power decay curves of the service stage.

[0045] (2) Based on the historical operating data collected in step (1), filter and smooth the data to obtain stable historical operating data. Then, use the filtered and smoothed historical operating data to calculate the attenuation rate every Δt hours to obtain the attenuation rate at different operating time points. Analyze the attenuation trend of the fuel cell stack based on the calculated attenuation rate, determine the nodes of attenuation trend change, and divide the voltage or power attenuation curve of step (1) into stages according to the nodes. Take the attenuation rate η > 2% / kh as the first stage, which is the rapid attenuation stage in the early stage of fuel cell stack operation. Take the attenuation rate 0.1 < η ≤ 2% / kh as the second stage, which is the slow attenuation stage in the middle stage of fuel cell stack operation. Take the attenuation rate η ≤ 0.1% / kh as the third stage, which is the stable operation stage in the later stage of fuel cell stack operation.

[0046] The calculation formula for the above filtering and smoothing process is as follows:

[0047]

[0048] Where n is the index, representing time n, h(n) is the impulse response at time n, and τ n Let n be the analysis time point;

[0049] The formula for calculating the attenuation rate is as follows:

[0050]

[0051] Where, η n Pn is the attenuation rate per thousand hours at time n, where n is the index and represents time n. P0 is the output power at the time when the normal operating voltage is reached. n P represents the output power at time n. n-△t This represents the output power at time n-Δt.

[0052] △t is the time interval;

[0053] (3) Calculate the total attenuation rate of the fuel cell stack during its service phase, and define a failure threshold as the end-of-life value. Using an attenuation rate of 10% to 40% as the lifespan threshold, determine whether the total attenuation rate during the service phase exceeds the failure threshold. If it does, determine that the fuel cell stack has no remaining service life. If it does not, calculate the service life of the fuel cell stack according to step (4). The formula for calculating the total attenuation rate of the fuel cell stack during its service phase is:

[0054]

[0055] Where, ηs P0 is the total degradation rate of the fuel cell stack during its service life, and P0 is the output power at the point of reaching normal operating voltage. s This represents the current output power of the fuel cell stack.

[0056] (4) Determine whether the voltage or power decay curve of the fuel cell stack during its service phase includes the first and second and / or third phases. If the voltage or power decay curve of the fuel cell stack during its service phase includes the first, second and third phases, then predict the future voltage or power decay trend based on the decay rate of the third phase and calculate the remaining lifetime. If the voltage or power decay curve of the fuel cell stack during its service phase only includes the first and second phases, then predict the future voltage or power decay trend based on the decay rates of the first and second phases at two different times combined with the fuel utilization rate and calculate the remaining lifetime.

[0057] The specific steps are as follows:

[0058] When the voltage or power degradation curve of the fuel cell stack during its service life includes the first, second, and third stages, the future voltage or power degradation trend is predicted using the degradation rate of the third stage. The future service life t f The output power after the time is:

[0059] P f =P s -(P0×a3×t f (4)

[0060] Among them, P f To extend the service life of the fuel cell stack beyond its current service time f The output power corresponding to the time, P0 is the output power at the moment of reaching the normal operating voltage, a3 is the attenuation rate of the third stage, taken as 0.1% / kh, t f For the time remaining in service in the future, kh; P s This represents the current output power of the fuel cell stack.

[0061] In the above scenario, the formula for calculating the remaining lifetime when the predicted output power reaches the failure threshold is:

[0062]

[0063] Among them, t RUL Kh represents the remaining service life from the current moment until the failure threshold is reached; P0 represents the output power at the moment the normal operating voltage is reached. s P represents the current output power of the fuel cell stack. m To achieve the output power at the failure threshold, P m = (1-m)P0, where m is the failure threshold and a3 is the decay rate of the third stage, taken as 0.1% / kh;

[0064] If the voltage or power degradation curve of the fuel cell stack during its service life only includes the first and second stages, then the future voltage or power degradation trend is predicted by combining the degradation rates at two different times in the first and second stages with the fuel utilization rate. f The output power after the time is:

[0065]

[0066] A and B are obtained using the following formula:

[0067]

[0068] Among them, P f To extend the service life of the fuel cell stack beyond its current service time f The output power corresponding to the time, P0 is the output power at the time when the normal operating voltage is reached, and t is the output power at the time when the normal operating voltage is reached. f The remaining service life is kh; A and B are the decay rate coefficients, φ is the fuel utilization rate, T is the operating temperature of the stack during stable operation, j is the current density, and C is the current density. N Let ΔP be the relative attenuation during a certain time interval in stage N, and ΔP be the power attenuation during a certain time interval in stage N. Let N be 1 or 2, and t be... N The selected relative attenuation C N The corresponding running time, kh;

[0069] In the above scenario, the formula for calculating the remaining lifetime when the predicted output power reaches the failure threshold is:

[0070]

[0071] Among them, t RUL Let kh be the remaining service life from the current moment until the failure threshold is reached. m To reach the output power at the failure threshold, i.e., the end of life, P m = (1-m)P0, where m is the failure threshold.

[0072] Furthermore, in the actual operation of a fuel cell stack, the first stage is relatively short. When making predictions about the fuel cell stack, the stack has usually already completed the first stage. Therefore, this invention mainly studies the case where the service stage includes the first and second stages or the first, second, and third stages. If, when predicting the remaining life of a fuel cell stack, calculations show that the voltage or power decay curve of the service stage only includes the first stage, and predictions based solely on the data from the first stage are inaccurate, then the fuel cell stack can continue to operate for a period of time, allowing it to reach the second or even third stage.

[0073] Additionally, it should be noted that the output power in the above calculation process is obtained by multiplying the output voltage by the current. Since the current remains constant during the operation of the fuel cell stack, the output power and the output voltage exhibit the same attenuation trend.

[0074] Example

[0075] This embodiment uses 40 anode-supported SOFCs (10×10cm) 2 Taking a fuel cell stack as an example, the prediction method of the invention will be explained. The operating temperature is 750℃, the fuel gas is hydrogen with a flow rate of 1800 mL / min, and the cathode gas is air with a flow rate of 9000 mL / min. The specific steps are as follows:

[0076] (1) Conduct continuous operation tests on the solid oxide fuel cell stack to obtain historical operating data; specifically:

[0077] (11) An activation test was conducted on the fuel cell stack, namely, nitrogen gas was introduced into the anode of the fuel cell stack and air was introduced into the cathode. The fuel cell stack was heated from room temperature to 800°C at a heating rate of 1°C / min. Then, hydrogen gas was introduced from the anode to reduce it for 4 hours and then cooled down to the working temperature of 750°C.

[0078] (12) A loading test was conducted on the fuel cell stack. While maintaining a constant operating temperature, the current was increased stepwise from 0, increasing by 5A each time, and held for 1 minute after each increase, until the actual output voltage (32V) of the fuel cell stack exceeded the normal operating output voltage (30V). Then, the current was decreased stepwise, decreasing by 1A each time, until the actual output voltage of the fuel cell stack reached the normal operating output voltage (30V). The corresponding output current (35A), output voltage (30V), and output power (1.05kW) were recorded. During normal operation, the current density was 350 mA·cm². -2 ;

[0079] (13) Conduct a continuous operation test on the fuel cell stack, i.e., keep the output current constant at 35A and allow the fuel cell stack to operate continuously and stably for at least 2000 hours. Simultaneously, record the output voltage and output power of the fuel cell stack during continuous and stable operation every hour as historical operating data. The resulting output voltage-time curve is shown below. Figure 2 As shown, a total of 2026 hours of historical data were obtained.

[0080] (2) Using the above formula (1), the historical operating data collected in step (1) is filtered and smoothed to obtain stable historical operating voltage data, as shown in the figure. Figure 3 As shown in Table 1 below, the data from the early activation stage are then removed, and the attenuation rate of the historical running data after filtering and smoothing is calculated every 100 hours using the above formula (2).

[0081] Table 1

[0082]

[0083] Based on the attenuation rates calculated in Table 1 above, the attenuation trend of the fuel cell stack is analyzed. The nodes of attenuation trend change are determined and divided into stages according to these nodes. The first stage, i.e., the rapid attenuation stage in the initial phase of the fuel cell stack, is defined as an attenuation rate η > 2% / kh. The second stage, i.e., the slow attenuation stage in the middle phase of the fuel cell stack, is defined as an attenuation rate 0.1 < η ≤ 2% / kh. The third stage, i.e., the stable operation stage in the later phase of the fuel cell stack, is defined as an attenuation rate η ≤ 0.1% / kh. Figure 4 As shown.

[0084] (3) The total attenuation rate of the fuel cell stack during its service phase is calculated to be 11.72%. The failure threshold of 20% is defined as the end-of-life value. The total attenuation rate of the fuel cell stack during its service phase does not exceed the failure threshold, indicating that the fuel cell stack still has a remaining service life.

[0085] (4) The curves of the above-mentioned fuel cell stack in service include the first, second, and third stages. Using the third stage's attenuation rate of 0.1% / kh to predict the future voltage attenuation trend, the future output power attenuation trend can be predicted according to the above formula (4). The future service life t f The output power after a time (assuming 100,000 hours) is 893.62W; and the remaining lifespan when the output power reaches the failure threshold is calculated to be 36493h using the above formula (5).

[0086] It should be noted that any parts not mentioned in this invention can be achieved by using or referencing existing technologies.

[0087] Of course, the above description is not intended to limit the present invention, and the present invention is not limited to the examples given above. Any changes, modifications, additions or substitutions made by those skilled in the art within the scope of the present invention should also fall within the protection scope of the present invention.

Claims

1. A method for online prediction of the remaining lifetime of a solid oxide fuel cell stack, characterized in that, Including the following steps: (1) Obtain the current, voltage, and power data of the solid oxide fuel cell stack from the initial operation to the current time of the service stage, as historical operation data, and plot the corresponding voltage or power decay curves of the service stage. (2) Calculate the attenuation rate every Δt hours based on the historical operating data collected in step (1) to obtain the attenuation rate at different operating time points. Analyze the attenuation trend of the fuel cell stack based on the calculated attenuation rate, determine the nodes of attenuation trend change, and divide the voltage or power attenuation curve of step (1) into stages according to the nodes. Take the attenuation rate η > 2% / kh as the first stage, the attenuation rate 0.1 < η ≤ 2% / kh as the second stage, and the attenuation rate η ≤ 0.1% / kh as the third stage. (3) Calculate the total decay rate of the fuel cell stack during its service phase, and define the failure threshold as the end-of-life value to determine whether the total decay rate during the service phase exceeds the failure threshold. If the lifespan is exceeded, it is determined that the fuel cell has no remaining service life. If the lifespan is not exceeded, the service life of the fuel cell is calculated according to step (4). (4) Determine whether the voltage or power decay curve of the fuel cell stack during its service phase includes the first and second and / or third phases. If the voltage or power decay curve of the fuel cell stack during its service phase includes the first, second and third phases, then predict the future voltage or power decay trend based on the decay rate of the third phase and calculate the remaining lifetime. If the voltage or power decay curve of the fuel cell stack during its service phase only includes the first and second phases, then predict the future voltage or power decay trend based on the decay rates of the first and second phases at two different times combined with the fuel utilization rate and calculate the remaining lifetime.

2. The method for online prediction of the remaining lifetime of a solid oxide fuel cell stack according to claim 1, characterized in that, The formula for calculating the attenuation rate in step (2) is as follows: Where, η n Pn is the attenuation rate per thousand hours at time n, where n is the index and represents time n. P0 is the output power at the time when the normal operating voltage is reached. n P represents the output power at time n. n-△t This represents the output power at time n-Δt, where Δt is the time interval.

3. The method for online prediction of the remaining life of a solid oxide fuel cell stack according to claim 1, characterized in that, In step (3), the failure threshold is set to 10% to 40%.

4. The method for online prediction of the remaining lifetime of a solid oxide fuel cell stack according to claim 1, characterized in that, The formula for calculating the total attenuation rate of the fuel cell stack during its service life in step (3) is as follows: Where, η s P0 is the total degradation rate of the fuel cell stack during its service life, and P0 is the output power at the point of reaching normal operating voltage. s This represents the current output power of the fuel cell stack.

5. The method for online prediction of the remaining life of a solid oxide fuel cell stack according to claim 1, characterized in that, In step (4), when the voltage or power decay curve of the fuel cell stack during its service life includes the first, second, and third stages, the future voltage or power decay trend is predicted using the decay rate of the third stage. The future service life t f The output power after the time is: P f =P s -(P0×a3×t f ) Among them, P f To extend the service life of the fuel cell stack beyond its current service time f The output power corresponding to the time, P0 is the output power at the moment of reaching the normal operating voltage, a3 is the attenuation rate of the third stage, taken as 0.1% / kh, t f For the time remaining in service in the future, kh; P s This represents the current output power of the fuel cell stack.

6. The method for online prediction of the remaining lifetime of a solid oxide fuel cell stack according to claim 5, characterized in that, In step (4), when the voltage or power decay curve of the fuel cell stack during its service phase includes the first, second, and third phases, the formula for calculating the remaining lifetime when the output power reaches the failure threshold is as follows: Among them, t RUL Kh represents the remaining service life from the current moment until the failure threshold is reached; P0 represents the output power at the moment the normal operating voltage is reached. s P represents the current output power of the fuel cell stack. m To achieve the output power at the failure threshold, P m = (1-m)P0, where m is the failure threshold and a3 is the decay rate of the third stage, which is 0.1% / kh.

7. The method for online prediction of the remaining lifetime of a solid oxide fuel cell stack according to claim 1, characterized in that, In step (4), if the voltage or power decay curve of the fuel cell stack during its service life only includes the first and second stages, then the future voltage or power decay trend is predicted by combining the decay rates at two different times in the first and second stages with the fuel utilization rate. f The output power after the time is: A and B are obtained using the following formula: Among them, P f To extend the service life of the fuel cell stack beyond its current service time f The output power corresponding to the time, P0 is the output power at the time when the normal operating voltage is reached, and t is the output power at the time when the normal operating voltage is reached. f The remaining service life is kh; A and B are the decay rate coefficients, φ is the fuel utilization rate, T is the operating temperature of the stack during stable operation, j is the current density, and C is the current density. N Let ΔP be the relative attenuation during a certain time interval in stage N, and ΔP be the power attenuation during a certain time interval in stage N. Let N be 1 or 2, and t be... N The selected relative attenuation C N The corresponding runtime, kh.

8. The method for online prediction of the remaining lifetime of a solid oxide fuel cell stack according to claim 7, characterized in that, In step (4), when the voltage or power decay curve of the fuel cell stack during its service phase only includes the first and second phases, the formula for calculating the remaining lifetime when the output power reaches the failure threshold is as follows: Among them, t RUL Let kh be the remaining service life from the current moment until the failure threshold is reached. m To reach the output power at the failure threshold, i.e., the end of life, P m = (1-m)P0, where m is the failure threshold.

9. The method for online prediction of the remaining lifetime of a solid oxide fuel cell stack according to claim 1, characterized in that, Step (2) further includes filtering and smoothing the historical operating data collected in step (1) to obtain stable historical operating data, and then using the filtered and smoothed historical operating data to calculate the attenuation rate. The calculation formula for filtering and smoothing is as follows: Where n is the index, representing time n, h(n) is the impulse response at time n, and τ n Let n be the analysis time point.

Images