Pump working turbine performance prediction method in variable-speed pumped storage project

Abstract

The invention discloses a method for predicting the performance of a pump turbine in a variable-speed pumped storage project, which adopts a composition analysis method to determine the basic structures of a plurality of correction models, and considers that the turbine performance is closely related to the specific speed thereof under the condition of variable speed. The proportion law is divided into two classes according to the rotating speed ratio, and basic function structures of the two classes are verified respectively. After power and power constructor functions are constructed, the power and the power constructor functions are substituted into the whale optimization algorithm one by one to determine coefficient values of the whale optimization algorithm, and after the coefficient values are determined, optimal correction models, namely a frequency constructor function and a power constructor function, are effectively screened out by comparing R2 values of all the correction models. The method is accurate and fast in calculation and high in calculation precision.

Description

Technical Field

[0001] The present invention relates to the field of pumped storage, and particularly to a method for predicting the performance of a pump as a turbine in a variable-speed pumped storage project. Background Art

[0002] In the face of the dual challenges of global energy transformation and climate change, the application of clean energy such as solar and wind energy is increasing rapidly. However, the randomness, volatility, and intermittency of these energy sources pose new challenges to the stability and reliability of power supply. To address these issues, decentralized energy storage methods such as batteries and micro-pumped storage are regarded as effective solutions, which can balance the unstable output of these renewable energy sources. Some researchers have pointed out that the production and waste treatment of batteries may have a significant impact on the environment. In contrast, pumped storage systems produce almost no harmful substances because they use water as the storage medium, which shows obvious advantages in terms of environmental impact and sustainability. Through pumped storage, excess electricity can be stored during periods of energy surplus and released during peak demand periods to ensure the stable operation of the power system. Micro-pumped storage (MPHS) systems, as an innovative energy storage method, have received extensive attention due to their potential advantages in terms of economy, environmental impact, and system design optimization. Such systems generally involve lower construction and operation costs, can provide a longer equipment life, and potential economic returns. By retrofitting existing small hydropower stations and utilizing their energy storage functions, not only can the power generation time be extended, but also the economic benefits can be improved.

[0003] However, in actual use, the pump as a turbine often cannot operate under rated conditions, and the variable-speed performance parameters are often not provided by manufacturers as labelable parameters. The lack of variable-speed performance parameters seriously affects the application of the turbine pump in engineering. Therefore, the prediction of variable-speed performance parameters has extremely important engineering significance. To make up for this deficiency, some researchers have proposed empirical formulas based on experimental tests. These formulas are mainly applied in two aspects: one is to determine the best operating point (BEP) when the pump operates in the turbine state; the other is to adjust the machine speed to maximize the energy recovery efficiency of the system when adopting variable operating strategies. The proposal of these formulas provides a research direction for us to deeply understand the performance of the turbine pump when operating as a turbine.

[0004] Researchers have made certain progress in the research on the performance prediction of pumps in turbine mode. Some researchers have proposed a general empirical equation to estimate the best efficiency point when the machine converts from a pump to a turbine. Some researchers have proposed an empirical fitting equation that has been widely used in recent years through the analysis of the internal loss relationship and flow field of the pump as a turbine. Other scholars have improved the performance of these empirical equations through various refinement analysis methods based on the above empirical equations. Some researchers have used computational fluid dynamics analysis to enhance the accuracy of these methods and attempt to predict the operating point. Some researchers have reviewed various methods, established a dataset containing 181 different pump-as-turbine performance parameters, and proposed a more accurate and effective empirical formula based on this dataset. The proposed empirical formula has higher accuracy when predicting the best operating point in turbine mode. In addition, some researchers have defined a regression equation and predicted the head and efficiency curves under changing speeds by adjusting the affinity law. This method takes the proportionality law as the core and the relationship between the pump in pumping mode and turbine mode as the correction scale, and proposes a modified proportionality law for turbine mode, opening up a new research direction for the study of variable-speed performance in turbine mode.

[0005] These existing methods only use one function for full-condition prediction, and the prediction effect in some specific speed ratio intervals is not good. Because the problem that the prediction errors of different pumps are relatively large in the lower or higher speed ratio intervals is not considered, the prediction error of using only one function for full-condition prediction is relatively large. Summary of the Invention

[0006] In view of the deficiencies of the existing technology, the present invention proposes a method for predicting the performance of a pump as a turbine in a variable-speed pumped storage project. The method uses a component analysis method to determine the basic structure of multiple correction models. Considering that the performance of the turbine is closely related to its specific speed under variable-speed conditions, this method uses a segmented analysis method. The proportionality law is divided into two categories according to the speed ratio, and the basic function structures are respectively verified. After constructing the power and power structure functions, they are substituted into the whale optimization algorithm one by one to determine their respective coefficient values. After determining the coefficient values, by comparing the R 2 values of each correction model, the best correction models, namely the frequency structure function and the power structure function, can be effectively selected.

[0007] The object of the present invention is achieved by the following technical solutions:

[0008] A method for predicting the performance of a pump as a turbine in a variable-speed pumped storage project, comprising the following steps:

[0009] Step 1: When collecting multiple centrifugal pumps with specific speed within a set range for turbine mode, obtain the head and power data corresponding to different flows at different speeds. Calculate the ratios of each speed, power, and head to their respective rated values to obtain speed ratio, head ratio, and power ratio. Then construct Dataset 1 consisting of three types of data: speed ratio, specific speed, and head ratio, and Dataset 2 consisting of three types of data: speed ratio, specific speed, and power ratio.

[0010] Step 2: Respectively use Dataset 1 and Dataset 2. Take the speed ratio and specific speed as the inputs of the polynomial regression - random forest algorithm, use multiple basic functions as the features of the polynomial regression - random forest algorithm, take the head ratio and power ratio as the outputs of the polynomial regression - random forest algorithm respectively, and train the polynomial regression - random forest algorithm respectively to obtain the importance of each basic function in the head - forming function output by the random forest algorithm and the importance of each basic function in the power - forming function. In the above two training processes, divide Dataset 1 and Dataset 2 into left and right subsets, and split the nodes according to the node splitting point position calculation method.

[0011] Step 3: Divide the speed ratio into three intervals: low, medium, and high. Construct piece - wise head - forming functions and power - forming functions on the three intervals respectively. During the construction process, sort the importance of each basic function in the head - forming function and power - forming function obtained in Step 2 from large to small, and select the basic functions with higher importance rankings to construct multiple head - forming functions and power - forming functions respectively. The head - forming function is a function with specific speed and speed ratio as independent variables and head ratio as the dependent variable; the power - forming function is a function with specific speed and speed ratio as independent variables and power ratio as the dependent variable.

[0012] Step 4: Respectively use the whale optimization algorithm to optimize the coefficients of the piece - wise function formulas of the multiple head - forming functions and power - forming functions obtained in Step 3 to obtain the coefficients corresponding to the best fitness value, and then obtain multiple head - forming functions and power - forming functions with determined coefficients.

[0013] Step 5: Calculate the head prediction value according to the head - forming function, calculate the power prediction value according to the power - forming function, and calculate the determination coefficient R 2 ; In the three speed - ratio intervals of low, medium, and high, select the head - forming function with the largest determination coefficient R 2 as the final head - forming function and power - forming function to predict the performance of the pump - as - turbine condition in the pumped - storage project.

[0014] Furthermore, the basic functions include linear function, inverse function, quadratic function, cubic function, exponential function, logarithmic function, power function, and growth function.

[0015] Furthermore, the calculation formula for the position of the node splitting point is as follows:

[0016]

[0017] where y train,L represents the left subset of the training dataset, and y train,R represents the right subset of the training dataset, and y train represents the training dataset, MSE() represents the mean squared error, and n train represents the number of samples in the entire training dataset;

[0018] When the node splitting point sc > 0, the node is considered a leaf node and needs to be split.

[0019] Furthermore, in the first step, after constructing the first dataset composed of three types of data: speed ratio, specific speed, and head ratio, divide the first dataset into a training dataset one and a test dataset one according to a ratio of 7:3. Also divide the second dataset composed of three types of data: speed ratio, specific speed, and power ratio into a training dataset two and a test dataset two according to a ratio of 7:3. In the second step, use the training dataset one and the training dataset two to train the polynomial regression-random forest algorithm respectively; in the fifth step, use the true value of the head in the test dataset one and the true value of the power in the test dataset two to calculate the coefficient of determination R 2 .

[0020] The beneficial effects of the present invention are as follows:

[0021] The present invention corrects the method of using a centrifugal pump as a turbine, so as to more conveniently and quickly obtain the turbine performance curve of the same pump at different speeds in actual engineering applications. This method is accurate and fast in calculation and has a high calculation accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0022] Figure 1 is a schematic diagram of the method for predicting the performance of a pump as a turbine in a variable-speed pumped storage project of this embodiment.

[0023] Figure 2 is a pie chart of the importance analysis results of each component (basis function) of the head construction function.

[0024] Figure 3 is a pie chart of the importance analysis results of each component (basis function) of the power construction function.

[0025] Figure 4 is a schematic diagram of the overall prediction error of the head.

[0026] Figure 5 is a schematic diagram of the overall prediction error of the power.

[0027] Figure 6 The prediction errors obtained from the general applicability test using the test set under different specific speed pumps. Among them, Figure (a) shows the prediction error of the head construction function, and Figure (b) shows the prediction error of the power construction function. Specific implementation mode

[0028] The present invention will be described in detail below according to the attached drawings and preferred embodiments. The purpose and effects of the present invention will become more apparent. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention.

[0029] In the method for predicting the performance of a pump as a turbine in a variable-speed pumped storage project of the present invention, it is proposed to divide the speed ratio into three intervals: low, medium, and high. Correction functions are established respectively in each interval, and a specific speed correction term is added, so as to make the performance prediction of the pump as a turbine under variable speed more accurate and with higher precision.

[0030] As Figure 1 shown, the method for predicting the performance of a pump as a turbine in a variable-speed pumped storage project of the present invention includes the following steps:

[0031] Step 1: When collecting multiple centrifugal pumps with specific speeds within a set interval for turbine mode, collect the head and power data corresponding to different flows at different speeds, and calculate the ratio of each speed to the rated speed, that is, the speed ratio; calculate the ratio of each head to the head corresponding to each working condition of the rated speed, that is, the head ratio; calculate the ratio of each power to the power corresponding to each working condition of the rated speed, that is, the power ratio; then construct a data set one composed of three types of data: speed ratio, specific speed, and head ratio, and a data set two composed of three types of data: speed ratio, specific speed, and power ratio.

[0032] Among them, the calculation method of the head corresponding to each working condition of the rated speed is as follows: Each speed has a Q-H curve. Map the flow rate and head (or power) at each speed to the Q-H curve of the rated speed to obtain the "head corresponding to each working condition of the rated speed".

[0033] In this embodiment, data (including speed, flow rate, head, and power) of multiple centrifugal pumps with specific speeds within the given interval of 0.6 to 1.4 for turbine operation are collected, and the constructed data set one and data set two are evenly divided into a training set and a test set according to a ratio of 7:3, obtaining a training set one, a test set one, a training set two, and a test set two.

[0034] Step 2: Construct a polynomial regression - random forest algorithm. Using Dataset 1, take the speed ratio and specific speed as the input of the polynomial regression - random forest algorithm, take multiple basic functions as the features of the polynomial regression - random forest algorithm, and take the head ratio as the output of the polynomial regression - random forest algorithm. Train the polynomial regression - random forest algorithm to obtain the importance of each basic function in the head composition function output by the random forest algorithm. Using Dataset 2, take the speed ratio and specific speed as the input of the polynomial regression - random forest algorithm, take multiple basic functions as the features of the polynomial regression - random forest algorithm, and take the power ratio as the output of the polynomial regression - random forest algorithm. Train the polynomial regression - random forest algorithm to obtain the importance of each basic function in the power composition function output by the random forest algorithm.

[0035] In the above two training processes, divide Dataset 1 and Dataset 2 into left and right subsets, and split the nodes according to the node splitting point position calculation method.

[0036] In the random forest algorithm, divide Training Set 1, Test Set 1, Training Set 2, and Test Set 2 into left and right subsets respectively. Subsequently, compare the fitting results of multiple commonly used basic functions, and adopt the optimal method to construct regression models for the four parts of Strain,L, Strain,R, Stest,L, and Stest,R separated at the random forest nodes respectively.

[0037] Among them, multiple commonly used basic functions are shown in Table 1.

[0038] Table 1 Basic Functions

[0039]

[0040] In the random forest, the node splitting position calculation method is shown as the following formula:

[0041]

[0042] Among them, MSE(x) is the mean square error between x and the predicted value of x; y train,L is the left subset divided from the training set; y train,R is the right subset divided from the training set; n train is the number of test set data. When SC > 0, this node is considered a leaf node. Finally, conduct importance analysis on all basic functions. When the importance of a feature is greater than 15%, record this feature, and combine these features with the target feature for regression calculation, so as to output the required relationship conversion function.

[0043] In this embodiment, the value ranges of the internal hyperparameters of the random forest are shown in Table 2.

[0044] Table 2 Range of internal hyperparameters of random forest

[0045]

[0046]

[0047] The analysis results of the head composition components finally obtained in this embodiment are as Figure 2 shown. From Figure 2 it can be seen that in the head structure function, the quadratic function accounts for the largest proportion, reaching more than 50%. The inverse function ranks second, with a proportion of 16.1%. The proportions of the linear function, logarithmic function, and growth function are relatively close, all around 5%. The proportion of the power function is the smallest, at 1.5%. Therefore, when constructing the head function, a construction method mainly based on the quadratic function is adopted, and the specific structure function is shown in Table 3.

[0048] Table 3 Model expression of the head structure function in this embodiment

[0049]

[0050] Among them, each coefficient is a parameter to be optimized in the head structure function. N s is the specific speed, n d is the speed ratio, and h s is the head ratio.

[0051] Similarly, for the power structure function, the importance of each basic function output by the polynomial regression - random forest algorithm is as Figure 3 shown. From Figure 3 it can be seen that for power, the internal correlation with the cubic function is the largest, reaching 48.95%. The quadratic function ranks second, reaching 29.67%, and the proportions of the remaining functions are all less than 10%. The proportions of the inverse function and the linear function are relatively close, being 8.17% and 7.09% respectively. The proportions of the exponential function, power function, logarithmic function, and growth function are all small and can be ignored.

[0052] Therefore, when constructing the power structure function, the high - order terms of the function are raised or lowered, and a linear function is added as a correction term. The specific construction of the power structure function is shown in Table 4 below:

[0053] Table 4 Function expression of the power structure function

[0054]

[0055]

[0056] Among them, each coefficient is a parameter to be optimized in the efficiency structure function. p s is the power ratio.

[0057] Step 4: Respectively use the whale optimization algorithm to optimize the coefficients of the piecewise function formulas of the multiple head composition functions and power composition functions obtained in Step 3, obtain the coefficients corresponding to the best fitness value, and then obtain the multiple head composition functions and power composition functions with determined coefficients.

[0058] Considering that using a single function to represent the head and power prediction over the entire speed ratio range has a relatively large error, the speed ratio is divided into three intervals: low, medium, and high, and function relationships are constructed separately for each interval to reduce the prediction error. In this embodiment, the speed ratio interval of 0.6 - 1.4 is divided into d n < 0.6, d 1.4 > n d > 0.6 and

[0059] Table 5 n d Coefficient values of each formula for the head when n < 0.6

[0060]

[0061] Table 6 1.4 > n d Coefficient values of each formula for the head when 1.4 > n > 0.6

[0062]

[0063]

[0064] Table 7 n d Coefficient values of each formula for the head when n > 1.4

[0065]

[0066] The coefficients of the obtained power construction functions are shown in Tables 8 - 10.

[0067] Table 8 n d Coefficient values of each formula for the power when n < 0.6

[0068]

[0069] Table 9 1.4 > n d Coefficient values of each formula for the power when 1.4 > n > 0.6

[0070]

[0071]

[0072] Table 10 n d Coefficient values of each formula for the power when n > 1.4

[0073]

[0074] Step 5: Calculate the predicted head value according to the head composition function, calculate the predicted power value according to the power composition function, and calculate the determination coefficient R respectively with the true head value and the true power value 2 ; In the three speed ratio intervals of low, medium, and high, select the determination coefficient R 2 of the maximum head composition function as the final head composition function and power composition function to predict the performance of the pump as a turbine in the pumped storage project.

[0075] In this embodiment, according to the expressions of F1 - F4 in the low, medium, and high speed ratios of the head composed of the coefficients in Tables 5 - 7, and use the validation set to calculate the R 2 values. The calculation results are shown in Tables 11, 12, and 13 below.

[0076] Table 11 n d <0.6 Head construction function model expression

[0077]

[0078]

[0079] Table 12 1.4 > n d >0.6 Head construction function model expression

[0080]

[0081] Table 13 n d >1.4 Head construction function model expression

[0082]

[0083] It can be seen from the above table that for the head, the R 2 values of each construction function vary greatly. Among them, the maximum R 2 value in the low speed ratio appears in the F4 function, which is 0.926. The maximum R 2 value in the medium speed ratio appears in the F1 function, which is 0.956. The maximum R 2 value in the high speed ratio appears in the F2 function, which is 0.926. Select the F4 function for the low speed ratio, the F1 function for the medium speed ratio, and the F2 function for the high speed ratio. Therefore, the final head construction function obtained is as follows:

[0084]

[0085] The overall prediction error of the head is as Figure 4As shown, it can be seen that for head prediction, its prediction R 2 is 0.941, and the prediction effect is good. Each data point can fall well on the line of y = x, and the prediction average error is about 6%.

[0086] Similarly, the coefficient values of each formula for power under the three speed ratios of low, medium, and high are calculated correspondingly. The coefficient values of each power construction function are shown in Tables 14, 15, and 16.

[0087] Table 14 n d Coefficient values of each formula for power < 0.6

[0088]

[0089] Table 15 1.4 > n d Coefficient values of each formula for power > 0.6

[0090]

[0091]

[0092] Table 16 n d Coefficient values of each formula for power > 1.4

[0093]

[0094] The coefficient compositions in the above three tables form the expressions of F1 - F4 under low, medium, and high speed ratios, and the R 2 values of each expression are calculated using the validation set. The calculation results are shown in Tables 17, 18, and 19 below.

[0095] Table 17 n d Power construction function model expression for < 0.6

[0096]

[0097] Table 18 1.4 > n d Power construction function model expression for > 0.6

[0098]

[0099]

[0100] Table 19 n d Power construction function model expression for > 1.4

[0101]

[0102] It can be seen from the above table that for the power prediction part, the maximum R 2The value appears in the F1 function and is 0.943, which is the maximum R at medium speed ratio 2 The value appears in the F3 function and is 0.947, which is the maximum R at high speed ratio 2 The value appears in the F1 function and is 0.917. Therefore, the F1 function is selected for low speed ratio, the F3 function is selected for medium speed ratio, and the F1 function is selected for high speed ratio. Thus, the finally obtained power construction function is shown as follows

[0103]

[0104] The overall prediction error of power is as Figure 5 shown. It can be seen that for the power prediction part, its prediction R 2 is 0.924, and the prediction average error is about 8%, which can basically meet the accuracy requirements in actual engineering applications

[0105] For the head and power construction functions obtained above, the test set data is used for generalization testing to observe their prediction effects in the case of pumps with different specific speeds. The prediction errors are as Figure 6 shown

[0106] It can be seen that for head prediction, its prediction error basically falls within 8%. The prediction error shows periodic fluctuations, indicating that when the predicted speed is far from the base speed during prediction, the prediction error is relatively large. However, its average error is about 7%, and excluding outliers, its maximum prediction error is about 15%, with relatively high prediction accuracy, meeting the requirements of engineering practice. For power prediction, its prediction error is lower than that of head prediction. The average prediction accuracy reaches about 4%, and the prediction error is not strongly periodic, with better prediction effects. After selecting the best expression, the method of this embodiment is compared and analyzed with the turbine proportionality law correction model proposed by other scholars. Four evaluation indicators including MAE, MSE, R 2 and MedAE are used to compare and analyze the above models to observe the effectiveness of the model established in this embodiment. The evaluation results of the four indicators are shown in Table 20

[0107] Table 20 Performance prediction method of pump as turbine at variable speed

[0108]

[0109] It can be seen that compared with the methods of other scholars, for the head and power construction functions obtained in this embodiment in head and power estimation, the three values of MSE, MAE, and MedAE are all the smallest, and R 2The value reaches the maximum. Compared with the methods used by other scholars, the mean square error of the head in this embodiment decreased by 2.87, 0.06, 0.96, and 0.42 respectively, and the mean square error of the power decreased by 53.24, 51.12, 6.83, and 0.78 respectively. Among all the models proposed by other scholars, the model proposed by Perez-Sanchez et al. performs the best, with R 2 reaching 0.9202 and 0.9175, but still lower than the method obtained in this embodiment.

[0110] Those of ordinary skill in the art can understand that the above are only preferred examples of the invention and are not used to limit the invention. Although the invention has been described in detail with reference to the foregoing examples, for those skilled in the art, they can still modify the technical solutions recorded in the foregoing examples, or perform equivalent replacements for some of the technical features. Any modifications, equivalent replacements, etc. made within the spirit and principle of the invention shall be included within the protection scope of the invention.

Claims

1. A method for predicting the performance of a pump as a turbine in a variable-speed pumped storage project, characterized in that, It includes the following steps: Step 1: When collecting multiple centrifugal pumps with specific speed within a set range for turbine mode, collect the head and power data corresponding to different flows at different speeds, and calculate the ratios of each speed, power, and head to their corresponding rated values to obtain the speed ratio, head ratio, and power ratio; then construct Dataset 1 composed of three types of data: speed ratio, specific speed, and head ratio, and Dataset 2 composed of three types of data: speed ratio, specific speed, and power ratio; Step 2: Respectively use Dataset 1 and Dataset 2. Take the speed ratio and specific speed as the inputs of the polynomial regression - random forest algorithm, take multiple basic functions as the features of the polynomial regression - random forest algorithm, take the head ratio and power ratio as the outputs of the polynomial regression - random forest algorithm respectively, train the polynomial regression - random forest algorithm respectively, and obtain the importance of each basic function in the head composition function output by the random forest algorithm and the importance of each basic function in the power composition function; in the above two training processes, divide Dataset 1 and Dataset 2 into left and right subsets, and split the nodes according to the node splitting point position calculation method; Step 3: Divide the speed ratio into three intervals: low, medium, and high. Construct piecewise head composition functions and power composition functions on the three intervals respectively; during the construction process, sort the importance of each basic function in the head composition function and power composition function obtained in Step 2 from large to small, and select the basic functions with the top - ranked importance to construct multiple head composition functions and power composition functions respectively; the head composition function is a function with specific speed and speed ratio as independent variables and head ratio as the dependent variable; the power composition function is a function with specific speed and speed ratio as independent variables and power ratio as the dependent variable; Step 4: Respectively use the whale optimization algorithm to optimize the coefficients of the piecewise function formulas of the multiple head composition functions and power composition functions obtained in Step 3, obtain the coefficients corresponding to the best fitness value, and further obtain multiple head composition functions and power composition functions with determined coefficients; Step 5: Calculate the predicted head value according to the head composition function, calculate the predicted power value according to the power composition function, and calculate the determination coefficient R with the true head value and the true power value respectively 2 ; In the three speed ratio intervals of low, medium and high, select the head composition function with the largest determination coefficient R 2 as the final head composition function and power composition function to predict the performance of the pump-turbine operation in the pumped storage project 2. The method for predicting the performance of a pump as a turbine in a variable-speed pumped storage project according to claim 1, wherein The basic functions include linear function, inverse function, quadratic function, cubic function, exponential function, logarithmic function, power function, and growth function.

3. The method for predicting the performance of a pump as a turbine in a variable-speed pumped storage project according to claim 1, wherein, The calculation formula of the node splitting point position calculation method is as follows: Among them, y train,L represents the left subset of the training dataset, y train,R represents the right subset of the training dataset, y train represents the training dataset, MSE() represents the mean squared error, and n train represents the number of samples in the entire training dataset; When the node splitting point sc > 0, the node is considered a leaf node and needs to be split.

4. The method for predicting the performance of a pump as a turbine in a variable-speed pumped storage project according to claim 1, wherein In the first step, after constructing the first data set composed of three types of data: speed ratio, specific speed, and head ratio, divide the first data set into a training data set one and a test data set one according to a ratio of 7:

3. Also divide the second data set composed of three types of data: speed ratio, specific speed, and power ratio into a training data set two and a test data set two according to a ratio of 7:

3. In the second step, use the training data set one and the training data set two respectively to train the polynomial regression-random forest algorithm; in the fifth step, use the true value of the head in the test data set one and the true value of the power in the test data set two to calculate the coefficient of determination R 2 .

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