A method for modeling the main steam temperature delay characteristics of dynamic operation of power plant boilers
By constructing an LSTM network model and using discrete particle swarm optimization, the delay time of the boiler main steam temperature to each input variable was determined, solving the problem of delay and lag in the control of the boiler main steam temperature, achieving precise control, and improving the safety and economy of the unit.
Patent Information
- Application Number
- CN202111148713.9
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2021-09-29
- Publication Date
- 2025-10-28
- Estimated Expiration
- 2041-09-29
AI Technical Summary
Under frequently changing operating conditions, the main steam temperature control object of the boiler has the characteristics of large delay and long lag time, resulting in poor PID control effect and affecting the safety and economy of the unit.
By collecting dynamic operation data of power plant boilers, the set of input variables affecting the boiler main steam temperature prediction model is determined, and an LSTM network model is constructed. The length of the historical sequence is optimized using discrete particle swarm optimization to characterize the delay time of the boiler main steam temperature on each input variable.
It achieves precise control of the boiler main steam temperature, improves the safety and economy of unit operation, and solves the control problem caused by the large delay characteristics.
Smart Images

Figure CN113887116B_ABST
Abstract
Description
Technical Field
[0001] This invention relates to a method for modeling the dynamic operation main steam temperature delay characteristics of a power plant boiler, belonging to the technical field of ammonia injection control strategy for power plant boilers. Background Technology
[0002] Increasing the main steam temperature can improve power generation efficiency, but while the main steam temperature is increased, the excessively high superheated steam temperature can lead to overheating of the unit's pipe walls, increased creep, reduced pipe life, and in severe cases, even pipe rupture due to overheating. Currently, the main steam temperature of 1000MW units in China is basically controlled at 605±5°C. The control scheme relies on the automatic regulation of the two-stage desuperheating water in the superheater, which is based on PID cascade control.
[0003] Routine operation reveals that the PID control of desuperheating water is less than satisfactory under frequently changing operating conditions. This is because the integration of renewable energy sources with poor stability necessitates frequent peak-shaving commands from the unit, while the main steam temperature control object exhibits significant delay and long lag time. Therefore, determining the delay characteristics of main steam temperature control is crucial for designing a main steam temperature control strategy for thermal power plant boilers.
[0004] Long Short-Term Memory (LSTM) neural networks are one of the commonly used deep learning techniques. Unlike ordinary neural networks, LSTM neural networks not only establish weight connections between different layers, but also establish weight connections at different times within the same layer, allowing them to recursively follow the sequence and fully reflect the close relationship of time series. Therefore, LSTM can learn the dynamic behavior of time series. Summary of the Invention
[0005] The technical problem this invention aims to solve is to provide a method for modeling the delay characteristics of main steam temperature during dynamic operation of power plant boilers. By collecting dynamic operating data of the power plant boiler, the method determines the set of input variables affecting the prediction model of the boiler's main steam temperature. Considering the historical sequence length of the input variables, an LSTM network model is constructed. The historical sequence length is determined using discrete particle swarm optimization, thereby characterizing the delay time of the boiler's main steam temperature with respect to each input variable. This invention is of significant importance for the design of control strategies for controlling the main steam temperature of thermal power plant boilers.
[0006] To solve the above problems, the technical solution adopted by the present invention is as follows:
[0007] A method for modeling the delay characteristics of main steam temperature during dynamic operation of a power plant boiler is proposed. By collecting dynamic operation data of the power plant boiler, the set of input variables affecting the prediction model of the main steam temperature of the boiler is determined. The historical sequence length of the input variables is considered, and an LSTM network model is constructed. The historical sequence length is determined by discrete particle swarm optimization, thereby characterizing the delay time of the main steam temperature of the boiler on each input variable.
[0008] Specifically, this method includes the following steps:
[0009] Step S1: Select operating data with a time span of 6 months from the power plant boiler system operation database, and denote it as dataset D;
[0010] Step S2: Use the boiler main steam temperature y as the output variable of the model, the unit load as the state variable x1, and the total fuel quantity u1, total secondary air quantity u2, and the opening degrees of secondary dampers a, b, c, d, and e u7 as the input variables of the boiler main steam temperature emission model; Step S3: Use the actual operating load of the unit and the total coal feed as feature variables to filter and classify the dynamic data; Step S4: Use SD i For the training sample set, SD 6+i To test the sample set, six LSTM neural network models were constructed. During construction, the root mean square error of the predictions for the test sample set was used as the objective function. Discrete particle swarm optimization was employed to optimize the sequence length of each variable, thus obtaining the time delay τ for each variable in each group. i1 , τ i2 ,…,τ i7 That is, how much data from previous times is needed for the prediction result at the current moment;
[0011] Step S5: Calculate the average length of the historical sequences of each of the 6 groups of variables to obtain the delay time of the corresponding variable, i.e.:
[0012]
[0013] This method is specific,
[0014] The sampling frequency of dataset D is 1 data sample per minute.
[0015] This method is specific,
[0016] The boiler combustion system experienced no faults or shutdowns within the 6-month data collection period.
[0017] This method is specific,
[0018] Step S1 involves collecting data on boiler combustion operating parameters, including unit load, total fuel quantity, secondary air volume, secondary damper opening, feedwater flow rate, flue gas temperature, and boiler main steam temperature.
[0019] This method is specific,
[0020] Step S3: Select dynamic operating data of the boiler combustion process. Using the unit load x1 as the state variable, select two sets of dynamic operating data in each month's operating data in a sliding window format, resulting in a total of 12 sets of data, denoted as SD1, SD2, ..., SD1. 12 .
[0021] This method is specific,
[0022] Step S4: The input variables for each LSTM model are total fuel quantity u1, total secondary air quantity u2, total fuel quantity u1, total secondary air quantity u2, and secondary air damper a opening u3, secondary air damper b opening u4, secondary air damper c opening u5, secondary air damper d opening u6, and secondary air damper e opening u7. The output variable is boiler main steam temperature y.
[0023] This method is specific,
[0024] In step 3, the specific steps for data selection and partitioning are as follows:
[0025] Step S31: Based on the dataset D collected in Step 1, divide it by month to obtain MDi, where the value range of i is 1, 2, 3, 4, 5, 6.
[0026] Step S32, let i = 1, initialize MDi;
[0027] Step S33: Filter the dataset MDi using a sliding window;
[0028] Step S34: Let i = i + 1, repeat step S33 until i = 6, finally obtaining the partitioned data sets SD1, SD2, ..., SD 12 .
[0029] This method is specific,
[0030] Step S33, the MDi dataset filtering steps are as follows:
[0031] Step S331: Select the data at time t=1 and initialize the sliding window;
[0032] Step S322: Select the sample set from the tth to the (t+599th)th sample and calculate the dynamic trend index λt of the feature variables. The calculation formula is as follows:
[0033]
[0034] Step S333: Let t = t + 1, repeat step S322 until all data in MDi has been overwritten and the set λt is obtained;
[0035] Step S334: Sort the obtained trend index set λt, and select the sample sets corresponding to the two λt with the largest values, denoted as SD respectively. i SD 6+i .
[0036] This method is specific,
[0037] Step S4, the time delay τ of each variable in each group i1 , τ i2 ,…,τ i7 The specific steps to obtain it are as follows:
[0038] Step S41, let i = 1, initialize SDi;
[0039] Step S42, in the dataset SDi, with the total fuel quantity u1, the total secondary air quantity u2, and the opening degrees of secondary air dampers a, b, c, d, and e as input variables, and the sequence lengths of each variable being τ, respectively. i1 , τ i2 ,…,τ i7 Using the boiler main steam temperature y as the output variable, a model is constructed using LSTM theory:
[0040] y(k+1)=f i (u1(k), …, u1(k-τ) i1 ), u2(k), …, u2(k-τ) i2 ), …, u7(k) …, u7(k-τ) i7 ))
[0041] In the formula, y(k+1) represents the predicted value at the next time step, u1(k),…,u1(k-τ) i1 ) represents the input sequence of variable u1, u2(k),…,u2(k-τ) i2 ) represents the input sequence of variable u2, u7(k),…,u7(k-τ) i7 ) represents the input sequence of variable u7;
[0042] Step S43, SD i+6 For the test set, the root mean square error of the predictions on the test set is used as the fitness function:
[0043] In the formula, Yt is SD 6+i China Boiler Main The t-th actual measurement value of the steam temperature sequence This is the predicted boiler main steam temperature corresponding to the LSTM model;
[0044] Step S44, Discrete PSO is used as the optimization method, with τi1 , τ i2 ,…,τ i7 Using [1, 20] as the optimization range for each variable, we solve for the fitness function Fit. i Minimum τ i1 , τ i2 ,…,τ i7 value;
[0045] Step S45: Let i = i + 1, and repeat steps S42 to S45 until i = 6; finally, τ is obtained. i1 , τ i2 ,…,τ i7 The value of , where i = 1 to 6.
[0046] The beneficial effects of adopting the above technical solution are as follows:
[0047] Precise control of boiler main steam temperature is crucial for the safety and economy of unit operation. Boiler main steam temperature is influenced by multiple factors and exhibits significant time delays, posing considerable challenges to the design of control strategies for boiler spray desuperheating systems. This method proposes to collect dynamic operating data of the power plant boiler to determine the set of input variables affecting the boiler main steam temperature prediction model. Considering the historical sequence length of these input variables, an LSTM network model is constructed. Discrete Particle Swarm Optimization (DPSO) is used to determine the historical sequence length, thus characterizing the delay time of boiler main steam temperature to each input variable. This approach is of significant importance for the design of boiler main steam temperature control strategies. Attached Figure Description
[0048] To more clearly illustrate the technical solutions in the embodiments of the present invention, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.
[0049] Figure 1 This is a flowchart of a method for modeling the delay characteristics of dynamic operation combustion parameters in power plant boilers;
[0050] Figure 2 It is a flowchart of the steps for data selection and partitioning, as well as the time delay acquisition of variables;
[0051] Figure 3 This is a schematic diagram of the system structure for obtaining the main steam temperature emission delay characteristics of boiler dynamic operation using an LSTM deep neural network. Detailed Implementation
[0052] The technical solutions of the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of this application, and not all embodiments. The following description of at least one exemplary embodiment is merely illustrative and is in no way intended to limit this application or its application or use. All other embodiments obtained by those skilled in the art based on the embodiments of this application without creative effort are within the scope of protection of this application.
[0053] It should be noted that the terminology used herein is for the purpose of describing particular embodiments only and is not intended to limit the exemplary embodiments 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.
[0054] Unless otherwise specifically stated, the relative arrangement, numerical expressions, and values of components and steps set forth in these embodiments do not limit the scope of this application. Techniques, methods, and devices known to those skilled in the art may not be discussed in detail, but where appropriate, such techniques, methods, and devices should be considered part of the specification. In all examples shown and discussed herein, any specific values should be interpreted as merely exemplary and not as limitations.
[0055] Therefore, other examples of exemplary embodiments may have different values. It should be noted that similar reference numerals and letters in the following figures denote similar items; therefore, once an item is defined in one figure, it need not be discussed further in subsequent figures.
[0056] Example 1
[0057] This embodiment provides a method for modeling the delay characteristics of main steam temperature during dynamic operation of a power plant boiler. By collecting dynamic operation data of the power plant boiler, the set of input variables affecting the prediction model of the main steam temperature of the boiler is determined. Considering the historical sequence length of the input variables, an LSTM network model is constructed. The historical sequence length is determined by discrete particle swarm optimization, thereby characterizing the delay time of the main steam temperature of the boiler on each input variable.
[0058] Specifically, in this embodiment,
[0059] like Figure 1 and 2 As shown, this method includes the following steps:
[0060] Step S1: Select operating data with a time span of 6 months from the power plant boiler system operation database, and denote it as dataset D;
[0061] In step S2, the boiler main steam temperature y is used as the output variable of the model, the unit load is used as the state variable x1, and the total fuel quantity u1, the total secondary air quantity u2, and the opening degree of secondary air damper a u3, secondary air damper b u4, secondary air damper c u5, secondary air damper d u6, and secondary air damper e u7 are used as the input variables of the boiler main steam temperature emission model.
[0062] Step S3: Using the actual operating load of the unit and the total coal feed as characteristic variables, the dynamic data is filtered and classified; Step S4: Using SD i For the training sample set, SD i+6 To test the sample set, six LSTM neural network models were constructed. During construction, the root mean square error of the predictions for the test sample set was used as the objective function. Discrete particle swarm optimization was employed to optimize the sequence length of each variable, thus obtaining the time delay τ for each variable in each group. i1 , τ i2 ,…,τ i7 That is, how much data from previous times is needed for the prediction result at the current moment;
[0063] Step S5: Calculate the average length of the historical sequences of each of the 6 groups of variables to obtain the delay time of the corresponding variable, i.e.:
[0064]
[0065] Specifically, in this embodiment,
[0066] The sampling frequency of dataset D is 1 data sample per minute.
[0067] Specifically, in this embodiment,
[0068] The boiler combustion system experienced no faults or shutdowns within the 6-month data collection period.
[0069] Specifically, in step S1, the variables in the collected data involve boiler combustion operation parameters, including unit load, total fuel quantity, secondary air volume, secondary damper opening, feedwater flow rate, flue gas temperature, and boiler main steam temperature.
[0070] In this specific embodiment, step S3 involves selecting dynamic operating data of the boiler combustion process. Using the unit load x1 as the characteristic variable, two sets of dynamic operating data are selected from the monthly operating data in a sliding window format, resulting in a total of 12 data sets, denoted as SD1, SD2, ..., SD1. 12 .
[0071] Specifically, in step S4, the input variables for each LSTM model are the total fuel quantity u1, the total secondary air quantity u2, and the opening degrees of secondary damper a u3, secondary damper b u4, secondary damper c u5, secondary damper d u6, and secondary damper e u7. The output variable is the boiler main steam temperature y.
[0072] Specifically, in this embodiment,
[0073] In step 3, the specific steps for data selection and partitioning are as follows:
[0074] Step S31: Based on the dataset D collected in Step 1, divide it by month to obtain MDi, where the value range of i is 1, 2, 3, 4, 5, 6.
[0075] Step S32, let i = 1, initialize MDi;
[0076] Step S33: Filter the dataset MDi using a sliding window;
[0077] Step S34: Let i = i + 1, repeat step S33 until i = 6, finally obtaining the partitioned data sets SD1, SD2, ..., SD 12 .
[0078] Specifically, in this embodiment,
[0079] Step S4, the time delay τ of each variable in each group i1 , τ i2 ,…,τ i7 The specific steps to obtain it are as follows:
[0080] Step S41, let i = 1, initialize SDi;
[0081] Step S42, in the dataset SDi, with the total fuel quantity u1, the total secondary air quantity u2, and the opening degrees of secondary air dampers a, b, c, d, and e as input variables, and the sequence lengths of each variable being τ, respectively. i1 , τ i2 ,…,τ i7 Using the boiler main steam temperature y as the output variable, a model is constructed using LSTM theory:
[0082] y(k+1)=f i (u1(k), …,u1(k-τ) i1 ), u2(k-τ i2 ), …, u7(k) …, u7(k-τ) i7 ))
[0083] In the formula, y(k+1) represents the predicted value at the next time step, u1(k),…,u1(k-τ) i1 ) represents the input sequence of variable u1, u2(k),…,u2(k-τ) i2 ) represents the input sequence of variable u2, u7(k),…,u7(k-τ) i7 ) represents the input sequence of variable u7;
[0084] Step S43, SD i+6 For the test set, the root mean square error of the predictions on the test set is used as the fitness function:
[0085]
[0086] In the formula, Yt is SD 6+i The t-th actual measured value of the main steam temperature sequence of the boiler. This is the predicted boiler main steam temperature corresponding to the LSTM model;
[0087] Step S44, Discrete PSO is used as the optimization method, with τ i1 , τ i2 ,…,τ i7 Using [1, 20] as the optimization range for each variable, we solve for the fitness function Fit. i Minimum τ i1 , τ i2 ,…,τ i7 value;
[0088] Step S45: Let i = i + 1, and repeat steps S42 to S45 until i = 6; finally, τ is obtained. i1 , τ i2 , ...,
[0089] τ i7 The value of , where i = 1 to 6.
[0090] Example 2
[0091] This embodiment provides a method for modeling the delay characteristics of main steam temperature during dynamic operation of a power plant boiler. By collecting dynamic operation data of the power plant boiler, the set of input variables affecting the prediction model of the main steam temperature of the boiler is determined. Considering the historical sequence length of the input variables, an LSTM network model is constructed. The historical sequence length is determined by discrete particle swarm optimization, thereby characterizing the delay time of the main steam temperature of the boiler on each input variable.
[0092] Based on Example 1, this embodiment provides a specific process for filtering a dataset MDi, with the following steps:
[0093] Step S331: Select the data at time t=1 and initialize the sliding window;
[0094] Step S322: Select the sample set from the tth to the (t+599th)th sample and calculate the dynamic trend index λt of the feature variables. The calculation formula is as follows:
[0095]
[0096] Step S333: Let t = t + 1, repeat step S322 until all data in MDi has been overwritten and the set λt is obtained;
[0097] Step S334: Sort the obtained trend index set λt, and select the sample sets corresponding to the two λt with the largest values, denoted as SD respectively. i SD 6+i .
[0098] Example 3
[0099] This embodiment, based on embodiments one and two, further provides a system for obtaining the boiler main steam temperature delay characteristics using an LSTM deep neural network. The system structure includes a normal operation data acquisition subsystem, an input variable set determination subsystem, an LSTM network model construction subsystem, a model performance evaluation subsystem, and an input variable delay optimization subsystem. The system structure diagram is shown below. Figure 3 As shown.
[0100] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features. It is obvious to those skilled in the art that multiple technical solutions of the present invention can be combined. These modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.
Claims
1. A method for modeling the dynamic operation main steam temperature delay characteristics of a power plant boiler, characterized in that: By collecting dynamic operation data of the power plant boiler, the set of input variables affecting the boiler main steam temperature prediction model is determined. Considering the historical sequence length of the input variables, an LSTM network model is constructed. The historical sequence length is determined by discrete particle swarm optimization (PSO) to characterize the delay time of the boiler main steam temperature on each input variable. This includes: constructing 6 sets of LSTM neural network models, using the root mean square error of the prediction of the test sample set as the objective function during construction, and using PSO to optimize the sequence length of each variable to obtain the time delay of each variable in each set, i.e., how many previous time steps of data are needed for the prediction result at the current moment; averaging the historical sequence lengths of each variable in the 6 sets to obtain the delay time of the corresponding variable; specifically, the optimization of the sequence length of each variable using PSO is achieved by using discrete PSO as the optimization method, with the delay time of each input variable as the optimization variable, to solve for the delay time value of each input variable that minimizes the fitness function.
2. The method for modeling the main steam temperature delay characteristics of dynamic operation of a power plant boiler according to claim 1, characterized in that, Includes the following steps: Step S1: Select operating data with a time span of 6 months from the power plant boiler system operation database, and denote it as dataset D; Step S2: The boiler main steam temperature y is used as the output variable of the model, the unit load is used as the state variable x1, and the total fuel quantity u1, the total secondary air quantity u2, and the opening degree of secondary air damper a u3, secondary air damper b u4, secondary air damper c u5, secondary air damper d u6, and secondary air damper e u7 are used as the input variables of the boiler main steam temperature emission model. Step S3: Using the actual operating load of the unit, i.e., the state variable unit load, as the feature variable, and employing a sliding window approach, select two sets of dynamic operating condition datasets with the largest dynamic trend indices from the monthly operating data as the training sample sets SD. i and training / test set D i+6 i=1-6, to achieve filtering and partitioning of dynamic data; Step S4, with SD i For the training sample set, SD I+6 To test the sample set, six LSTM neural network models were constructed. During construction, the root mean square error of the predictions for the test sample set was used as the objective function. Discrete particle swarm optimization was employed to optimize the sequence length of each variable, thus obtaining the time delay τ for each variable in each group. i1 , τ i2 , ..., τ i7 i = 1-6, which means how many previous times of data are needed for the prediction result at the current time; Step S5: Calculate the average length of the historical sequences of each of the 6 groups of variables to obtain the delay time of the corresponding variable, i.e.: 。 3. The method for modeling the dynamic operation main steam temperature delay characteristics of a power plant boiler according to claim 2, characterized in that, The sampling frequency of dataset D is 1 data sample per minute.
4. The method for modeling the dynamic operation main steam temperature delay characteristics of a power plant boiler according to claim 3, characterized in that, The boiler combustion system experienced no faults or shutdowns within the 6-month data collection period.
5. The method for modeling the dynamic operation main steam temperature delay characteristics of a power plant boiler according to claim 2, characterized in that, Step S1 involves collecting data on boiler combustion operating parameters, including unit load, total fuel quantity, secondary air volume, secondary damper opening, feedwater flow rate, flue gas temperature, and boiler main steam temperature.
6. The method for modeling the dynamic operation main steam temperature delay characteristics of a power plant boiler according to claim 2, characterized in that, In step 3, the specific steps for data selection and partitioning are as follows: Step S31: Based on the dataset D collected in Step 1, divide it by month to obtain MDi, where the value range of i is 1, 2, 3, 4, 5, 6. Step S32, let i-1, initialize MD i ; Step S33, process the dataset MD i Use a sliding window for filtering; Step S34: Let i = i + 1, repeat steps 3-3 until i = 6, finally obtaining the partitioned data sets SD1, SD2, ... SD 12 .
7. The method for modeling the dynamic operation main steam temperature delay characteristics of a power plant boiler according to claim 6, characterized in that, Step S33, Dataset MD t The screening steps are as follows: Step S331: Select the data at time t=1 and initialize the sliding window; Step S322: Select the sample set from the tth to the (t+599th)th sample and calculate the dynamic trend index λ of the feature variables. t The calculation formula is as follows:
8. In step S333, let t = t + 1, and repeat step S322 until MD. t All the data has been overwritten, and λ has been obtained. t gather; Step S334, obtain the trend index set λ t Sort the data and select the two largest values (λ). t The corresponding sample sets are denoted as SD. i SD i+6i .
9. A method for modeling the dynamic operation main steam temperature delay characteristics of a power plant boiler according to claim 7, characterized in that, Step S4, the time delay τ of each variable in each group i1 , τ i2 , ..., τ i7 The specific steps to obtain it are as follows: Step S41, let i = 1, initialize SDi; Step S42: Based on the dataset SDi, with the total fuel quantity u1, total secondary air quantity u2, and the opening degrees of secondary air dampers a, b, c, d, and e as input variables, the sequence lengths of each variable are τ. i1 , τ i2 , ..., τ i7 Using the boiler main steam temperature y as the output variable, a model is constructed using LSTM theory: y(k+1)=f i (u1(k), …, u1(k-τ) i1 ), u2(k), …, u2(k-τ) i2 ), …, u7(k) …, u7(k-τ) i7 In the formula, y(k+1) represents the predicted value at the next time step, u1(k), ..., u1(k-τ) i1 ) represents the input sequence of variable u1, u2(k), ..., u2(k-τ). i2 ) represents the input sequence of variable u2, u7(k), ..., u7(k-τ). i7 ) represents the input sequence of variable u7; Step S43, SD i+6 For the test set, the root mean square error of the predictions on the test set is used as the fitness function: In the formula, Yt is SD 6+i The t-th actual measured value of the main steam temperature sequence of the boiler. This is the predicted boiler main steam temperature corresponding to the LSTM model; Step S44, Discrete PSO is used as the optimization method, with τ i1 , τ i2 ,…,τ i7 Using [1, 20] as the optimization range for each variable, we solve for the fitness function Fit. i Minimum τ i1 , τ i2 ,…,τ i7 value; Step S45: Let i = i + 1, and repeat steps S42 to S45 until i = 6; finally, τ is obtained. i1 , τ i2 ,…,τ i7 The value of , where i = 1 to 6.
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