Abstract
Compressors are one of the three major components of gas turbines, and their characteristic curves are used to analyze off-design performance. How to infer the characteristic curve based on different data is an important research topic. In this paper, PG9351FA gas turbine is taken as the research object. Two methods, artificial neural network and parameter estimation, are used to predict its characteristic curve, and the prediction accuracy and application conditions of the two methods are discussed. This article compares the two methods from the perspectives of known speed characteristic curve regression and unknown speed characteristic curve inference, analyzes the impact of sample size and sample error on their inference results, and quantitatively analyzes the error through statistical methods such as calculating the mean square deviation of the data. The application scope and conditions of different methods are provided. The research results show that the method based on neural network to infer the characteristic curve has high accuracy and is suitable for the prediction of known and unknown speed characteristic curves under sufficient data, but not for the prediction of unknown side curves. The elliptic equation fitting method based on parameter estimation has a slightly lower accuracy in processing the nearly vertical compressor characteristic curve, but it can be used as an effective and reliable method to infer the compressor characteristic curve in the case of a small amount of data. The modulization method based on parameter estimation has high accuracy and is applicable to the estimation of complete characteristic curve from partial data of known characteristic curve. In this paper, the application scope and conditions of these two methods are determined, which can provide reference for engineering practice.
1. Introduction
In recent years, the gas turbine-combined cycle (GTCC) has become increasingly popular in the power industry for its high efficiency and low pollutant emissions [1–3]. The GTCCs are more frequently used for peak shaving applications in some countries, for example, in China, requiring GTCCs to operate for a considerable period under part-load conditions. Thus, it is important to understand the off-design performance of the gas turbine through modeling and analysis. The accurate describing of the off-design performance of a gas turbine is highly dependent on the model of the compressor characteristic [4, 5]. As a highly nonlinear component, the operating characteristics of the compressor can be expressed by the four parameter relationships of mass flow, pressure ratio, efficiency, and speed. It shows how compressor efficiency and pressure ratio change with speed and mass flow by describing the compressor characteristic curve and provides important basis for studying the performance of compressor under variable operating conditions [6, 7]. Usually, the information provided by gas turbine manufacturers is very limited, so important information such as compressor characteristic curves cannot be directly obtained. Therefore, how to infer the characteristic curve based on the limited data is an important research topic [8, 9].
At present, many scholars have developed different methods for the processing of compressor characteristic curves. These methods include general mathematical expression method [10–12], polynomial fitting method [13–15], artificial neural network fitting method [16, 17], and elliptic equation parameter estimation method [18–21]. Liu et al. [22] systematically studied the modeling theory and boundary conditions of cubic spline interpolation method. Yongtao and Yanming [23] solved the mathematical relationship of algebraic polynomial curve fitting in the least square method and provided a universal C language program called function. Xie et al. [24] used cubic spline interpolation. Zhong et al. [25] used support vector machines. Dongyang [10] taking 9FA gas turbine as the object, built different neural network prediction models and made relevant discussions and concluded that RBF (radial basis function) neural network can better match the experimental samples, while BP (back propagation) neural network has better prediction performance. Elliptic equation method is a new method for fitting and modeling compressor characteristic curves, which was first proposed by Tsoutsanis et al. [26–27] of the University of Cranfield, UK. The relevant literature is rare, and the universality of the model needs to be further verified. Another method for predicting compressor characteristic curve is the modulization method based on parameter estimation. Yang and Zhang [28] found out the relationship between compressor parameters and pressure ratio through statistical analysis of a certain number of compressor characteristic curves and proposed a new method for compressor characteristic curve modulization. In the above research, different methods are presented for predicting compressor characteristic curves, and the idea for compressor characteristic curve research is developed. However, the current research mainly verifies the feasibility of different methods on the known speed characteristic curve, and the research on the influence of data sample size and sample error is insufficient.
In addition, the expandability of the above methods to predict the unknown speed characteristic curve needs to be further explored and studied. The research of compressor characteristic curve needs to be combined with engineering practice. When considering different working conditions, different data conditions, etc., it is of great significance to analyze and compare the conventional research methods so as to clarify the used conditions of each method. In this paper, two methods for predicting compressor characteristic curves are studied: neural network method and parameter estimation method. The two methods are compared from the regression of known speed characteristic curve and the extrapolation of unknown speed characteristic curve, and the influence of sample size and sample error on the extrapolation result is analyzed. Then, the error is quantitatively analyzed by calculating the mean square deviation of the data with statistical methods, and the application scope and conditions of the different methods are given.
2. Research on the Method of Predicting the Characteristic Curve of Compressor Based on Neural Network
A neural network’s problem is to train itself repeatedly through known data so as to obtain the weights and biases. By using a neural network, the calculated output signal should be as close to the actual output signal as possible [29–31]. Based on known data, the neural network can make accurate predictions about the whole system’s operation process, which can reduce the sum of squares of the final error to the minimum. This tool is useful for analyzing compressor characteristic curves [32, 33].
2.1. Research on the Method of Neural Network Fitting Known Speed Characteristic Curve
2.1.1. Establishment of Neural Network Model
GateCycle provided the experimental data for this paper from the compressor data of PG9351FA gas turbines. So as to study and verify the feasibility of these methods, this paper takes the characteristic curve data provided by commercial software as the real characteristic curve data and selects some points as the operation data to speculate the characteristic curve and compare it with the real characteristic curve. The sampling error is considered to represent the sampling error of real operation data. Simulate the sampling error to represent the sampling error of real operation data.
In this paper, seven operating conditions with corrected speed of 81, 87, 94.6, 97.1, 100, 102, and 106.2 are extracted, and the BP neural network model is constructed by taking 15 data points from each curve. Input parameters for this model are corrected speed (CS) and normalized corrected flow (NCF); through BP neural network fitting, we obtain the normalized pressure ratio (NPR) and the normalized efficiency (NEF).
To facilitate the calculation, the corresponding parameters of the compressor are represented by the corrected speed, corrected flow, corrected pressure ratio, and corrected efficiency. The formulas are as follows:
is the corrected flow; is the corrected flow under design conditions.
is the corrected speed. is the operating speed. is the speed under design conditions. is the gas constant. is the temperature under design conditions. is the operating temperature.
is the corrected pressure ratio. is the operating pressure ratio. is the pressure ratio under design conditions.
is the corrected efficiency. is the standard efficiency at the original point. is the IGV angle (inlet guide vane angle). is the correction factor.
In the BP network, is selected as the training function, and the number of hidden layer neuron nodes is set to 20 to establish the BP neural network. The number of hidden layer neurons here cannot be too small; otherwise, the network cannot learn well and requires a lot of training, and the training accuracy is not high. The more neurons in the hidden layer, the more powerful the functions that can be achieved, but it should not be too many; otherwise, the training time will be greatly extended [34]. Secondly, in order to improve accuracy, the training frequency is set to 10000, the training error is set to 0.0001, and the learning efficiency is set to 0.01. Finally, in order to facilitate data collection work, a data reading module is added on the basis of the BP neural network, achieving the effect of inputting the corresponding reduced speed and reduced flow rate to obtain the corresponding reduced pressure ratio. At the same time, in order to prevent the collected data from being below the stall boundary or above the surge boundary, it is necessary to limit the working area of the compressor.
In order to improve the accuracy of network training, it is necessary to select appropriate training functions, the number of hidden layer nodes, normalization methods, etc. Through repeated experimental comparisons, the transfer function , the output layer transfer function , the training function , and the number of hidden layer neurons set to 20 are selected to establish a BP neural network model. In the model, the fitting of multiple curves can be completed by modifying the program input/output, and the complete fitting diagram is drawn using the plot statement. The BP neural network training flowchart is shown in Figure 1.
The BP network fitting result is shown in Figure 2. It can be seen from the figure that the fitting result is basically consistent with the actual data. Through error calculation, the mean absolute percentage error (MAPE) of the fitting result is controlled within 2%. It can be seen that the results of BP neural network training are of high accuracy and provide a high-precision method for compressor characteristic curve simulation.
2.1.2. Influence of Sample Size on Neural Network Fitting
There are many factors that affect the result of neural network fitting. This section will systematically study the influence of data volume on the result of neural network fitting.
In the previous fitting, the neural network fitting results were based on 15 reference points. In order to further explore the impact of sample size on the neural network, taking the corrected speed 97.1 in PG9351FA gas turbine as an example, 6, 10, 20, 30, 50, 100, and 150 reference points were selected in order, and BP neural network was used to operate one by one and record the operation results and then calculate the error of the results. The fitting results are shown in Figures 3–9. When the data is small, especially when the sample size is only 6 or 10, the neural network fitting result is not ideal due to the influence of the data volume, which is far from the true curve. With the increase of sample size, the curve fitted by neural network gradually coincides with the real curve. In order to obtain more accurate results, continue to make statistical analysis of the error results of the sample. Among them, MSE (mean square error), MAE (mean absolute error), and MAPE (mean absolute percentage error) are counted. The formulas are as follows:
In the equation, represents the predicted output of various network models, and represents experimental values.
The results of statistical analysis of data error of different sample sizes are shown in Table 1. It can be seen from the table that when the sample size is 6, the error is the largest, and the MAPE reaches 7.65%, which is consistent with the results observed in Figure 2. With the increase of sample size, when the sample size is 50, the fitting accuracy reaches a higher level and the error is small. When the sample size continues to increase, the error changes little, and the accuracy does not significantly improve. Subsequently, this article will also conduct experiments on other speed curves, and the conclusions obtained are similar to this curve.
In summary, the accuracy of fitting the compressor characteristic curve using neural network is poor when a small amount of data is available, and when the sample size is sufficient, a more accurate compressor characteristic curve can be fitted, but the right amount of data should be selected as the sample; too much sample size will not be more helpful to the improvement of the accuracy, but too much sample size will greatly increase the workload of the experiment.
2.1.3. Influence of Sampling Error on Neural Network Fitting
Due to the influence of instrument measurement error, when measuring the flow and pressure data of the compressor working through the sensor, it is inevitable that errors will occur, and the characteristic curve fitting will also be affected. In order to study the influence of errors on the characteristic curve fitting, this paper takes the real data plus the random error that follows the normal distribution of the relative standard deviation as a constant as the simulation experimental data.
According to the characteristics of flow and pressure sensors, the error of two groups of given size is shown in Table 2. The corrected flow error adopts relative standard deviation of 0.01 and larger relative standard deviation of 0.02, and the corrected pressure ratio error adopts relative standard deviation of 0.005 and larger relative standard deviation of 0.01. Specific errors are applied to the corrected flow data and the corrected pressure ratio data, respectively. There are four error given forms, as shown in Table 3. Then, BP neural network is used to merge and record the operation results one by one, and finally, the error of the results is calculated. Taking the corrected speed 97.1 as an example, the fitting results are shown in Figures 10–13. The figure shows that when the error of corrected flow rate and corrected pressure ratio is taken as a relatively large relative standard deviation, the curve fitted by the image is less consistent with the real curve, and the impact of sampling error is more obvious.
Table 4 is the error table of neural network fitting results when sampling error is considered. It can be seen from the table that as the relative standard deviation of each sample error increases, the MSE and MAPE of the neural network fitting results increase. When the error of corrected flow and corrected pressure ratio is taken as a small relative standard deviation, the MAPE of the fitting result increases from 0.9% (no error) to 2.9%. When the error of corrected flow and corrected pressure ratio is taken as a large relative standard deviation, the MAPE of the fitting result increases to 4.53%. It can be seen that the influence of sampling error on the fitting result of neural network is still very obvious.
In order to further explore the sensitivity of different reduced speed characteristic curves affected by sampling error, the seven corrected speed characteristic curves are fitted with the error of error form 1 in Table 3 by neural network to obtain Figure 14, and the error is calculated to obtain Table 5. It can also be seen from Table 5 that with the increase of the flow rate, the error value of the fitting curve is also increasing. Especially, when the corrected speed reaches 102 and 106.2, the MAPE of the fitting results has increased significantly, both of which have exceeded 11%. The reason is that with the increase of the flow, the characteristic curve gradually tends to be vertical, and the ordinate is very sensitive to the change of the abscissa. A small change in the abscissa will lead to a significant change in the ordinate, resulting in the error of this curve being amplified in the fitting process. It can be seen that the compressor characteristic curve with higher corrected speed is more sensitive to the influence of sampling error.
Based on the above analysis, sampling has a significant impact on the fitting results of the neural network. Since there will be errors in the acquisition process of the samples, the accuracy of the fitting results will also decline. At the same time, the modified compressor characteristic curve with higher speed is more sensitive to the impact of the sampling error, resulting in a significant decline in the fitting accuracy. The error cannot be avoided and can only be reduced. Only by selecting appropriate tools and minimizing the error during sampling can the fitting result be more accurate.
2.2. Research on the Method of Inferring Unknown Speed Characteristic Curve by Neural Network
2.2.1. Inferring Method of Unknown Speed Characteristic Curve
When the amount of data on a known compressor characteristic curve is large enough, the neural network method can be considered to obtain an unknown compressor characteristic curve.
Also, data comes from GateCycle, 7 corrected speeds were extracted, and 15 data points were taken from each curve as experimental data.
The first method is to infer the unknown characteristic curve through six characteristic curves. Assuming that one of the seven curves provided by GateCycle is an unknown characteristic curve, the unknown characteristic curve can be inferred by learning the remaining six characteristic curves through neural network. The feasibility of the method is analyzed through the comparison and error calculation between the conjectured data and the real data.
The second method is to infer the unknown characteristic curve through the two adjacent curves of the unknown characteristic curve. Select three adjacent characteristic curves, assume that the middle curve is unknown characteristic curve, and infer the middle unknown characteristic curve by learning the characteristic curves on both sides through neural network. The feasibility of the method is analyzed through the comparison and error calculation between the conjectured data and the real data.
2.2.2. Analysis of the Results of Inferred Method I
Taking the corrected speed 97.1 as an example, assuming that the curve of the corrected speed 97.1 is an unknown characteristic curve, six curves of the corrected speeds 81, 87, 94.6, 100, 102, and 106.2 are taken as the learning objects of the neural network, and Figure 15 is inferred. Through the error calculation, the , the , and the are obtained, and the error is maintained at a low level. It can be seen that this method has a high prediction accuracy for the curve with corrected speed of 97.1 and a low error in the prediction result. In order to avoid the accident of the experiment, continue to speculate on the characteristic curves at other corrected speeds.
The extrapolation effect diagram of the characteristic curve at other speeds is shown in Figure 16, and Table 6 is obtained through error calculation. It can be seen from Figure 16 that the predicted curve of the corrected speed 81 is not consistent with the real curve. The calculation of the error of the predicted results in Table 6 also confirms this point. The MAPE of the predicted curve reaches 17.729%. The reason is that CS81 is a boundary curve. In its inference process, the six curves of corrected speeds 87, 94.6, 97.1, 100, 102, and 106.2 are used as the learning objects of the neural network. The corrected speed of these six curves is greater than 81, and there is no curve less than the corrected speed 81 for the neural network to study. Therefore, the training results of the neural network are not accurate, resulting in a large error in the inference of this speed curve. Excluding the boundary characteristic curve, the extrapolation results of other characteristic curves are relatively stable, with the maximum MAPE of 6.8931% and the minimum MAPE of 0.0882%. The best extrapolation effect is to correct the speed of 100. Compared with the curve with higher corrected speed, the error of the predicted curve of corrected speeds 87 and 94.6 is significantly increased. The reason is that the distribution of the corrected speed of the PG9351FA gas turbine compressor characteristic curve provided by GateCycle is not uniform, and the difference between the corrected speed 87 and the corrected speed 94.6 is far, which has a certain impact on the learning of neural network, so the final predicted result will also be affected.
2.2.3. Analysis of the Results of Inferred Method II
Similarly, taking the corrected speed 97.1 as an example, the characteristic curves CS94.6 and CS100 on both sides are selected as the learning objects of the neural network, and Figure 17 is obtained by extrapolation. By comparing Figure 15 with Figure 17, it can be seen that the predicted curve in Figure 17 is closer to the real curve; that is, the predicted accuracy of the second method is higher. The error analysis is also carried out. The , the , and the are calculated. All the error values are less than the error of the conjectured method I. Therefore, the conjectured CS97.1 curve shows that the precision of method II is higher and closer to the real curve. In order to avoid the chance of the experiment, the rest of the curves are speculated below. Since CS81 and CS106.2 are boundary curves, we will learn and speculate through the two curves before and after them.
Figure 18 is inferred from the neural network. It can be seen from the figure that there are two curves with poor prediction effect, namely, the curves of CS81 and CS106.2. Table 7 is obtained through error calculation, which can more intuitively see the prediction effect. The MAPE of CS81 conjectured curve reached 30.1290%, and the MAPE of CS106.2 conjectured curve was 8.6795%. The two curves had poor conjecture effect, and the error also increased significantly compared with the conjectured method. This is because CS81 and CS106.2, as boundary curves, did not correct the curve with speed less than or greater than it for neural network learning. When the number of curves for neural network learning is reduced to two, the prediction accuracy of neural network will be greatly reduced, so the prediction result is not ideal. The best predicted curve is the predicted curve of CS102, and the MAPE is only 0.0034%. The predicted results of the three curves CS97.1, CS100, and CS102 have lower errors than the predicted results of the first method.
2.2.4. Summary
To sum up, inferred method I has higher conjecture accuracy than inferred method II, but the latter will have larger errors in conjecture of boundary curves. For the boundary curve, the prediction result of neural network will have low accuracy. Therefore, this method is applicable to the prediction of unknown speed characteristic curve under sufficient data, but not to the prediction of side curve.
3. Research on the Method of Predicting Compressor Characteristic Curve Based on Parameter Estimation
3.1. Elliptic Equation Fitting Method Based on Parameter Estimation
Parameter estimation method is a research method that uses statistical methods to infer one or several parameter values. The elliptic equation used in this paper is a point estimation method in the parameter estimation method. The point estimation method is to obtain the unknown parameters in the estimation model through the total sample, replace the real value with the estimated value, and finally obtain the complete model results.
3.1.1. Fitting Characteristic Curve with Elliptic Parameter Method
Through observation, it can be found that the corrected speed curve is similar to an arc on an ellipse, so the elliptic parameter method can be considered to fit the corrected speed curve [35]. The first step of the elliptic parameter estimation method is to establish a suitable elliptic equation. This step is very important, because the establishment form of the elliptic equation largely determines the highest accuracy that the final elliptic parameter method can achieve.
To improve the accuracy of the elliptic parameter method, the following elliptic equation is established: where is the corrected flow rate of the compressor characteristic curve, is the corrected pressure ratio, and are the short and long half axes of the ellipse, respectively, is the elliptic center coordinate, and is the angle of the ellipse. In this way, the equation has a total of 7 parameters, of which and are known quantities that can be obtained from known compressor characteristic curves, and the other 5 parameters are unknown quantities. Use the parameter estimation function in gPROMS software to calculate five unknown parameters. gPROMS can be used to estimate parameters of complex models using dynamic and steady-state experimental data. Take the corrected flow rate and the corrected pressure ratio as an example, select fifteen groups of data as the known quantity input, obtain the reference values of five unknown quantities through the parameter estimation operation of gPROMS software, bring the reference values of these five unknown quantities back to the elliptic equation, and obtain the ellipse as shown in Figure 19.
3.1.2. Influence of Sample Size on Elliptic Parameter Estimation
In order to study the influence of sample size on the elliptic parameter estimation method, taking the corrected speed 97.1 as an example, 3, 6, 15, and 30 reference points were selected as experimental data to estimate the parameters, and the elliptic equation was obtained as shown in Figures 20–23. When the sample size is only 3, it can be seen from Figure 20 that only two points are on the elliptic curve. At this time, the fitting result error is large, and the MAPE reaches 0.2. The elliptic equation cannot accurately represent the characteristic curve. When the number of samples increases to 6, it can be observed from Figure 21 that the elliptic equation is very consistent with the characteristic curve and the fitting accuracy is high. MAPE calculated by error is only 0.0006. As the sample size increases to 15 and 30, the fitting accuracy of the elliptic equation does not significantly improve, and the error does not change much. Therefore, the elliptic parameter estimation method is applicable to the establishment of a relatively reliable compressor characteristic curve model with a small amount of compressor characteristic curve data and can be used as an effective and reliable method to predict the compressor characteristic curve with a small amount of data.
3.1.3. Influence of Sampling Error on Elliptic Parameter Estimation
The method for studying the influence of sampling error on parameter estimation is the same as that for studying the influence of sampling error on neural network. According to the characteristics of sensors, the error levels are also divided into two types, the same as Table 2. There are also four given forms of errors, as shown in Table 3. First, synthesize the data with errors from MATLAB, then import the data into gPROMS software for parameter estimation, and obtain five unknown parameters. Specific errors are added to the corrected flow data and the corrected pressure ratio data, and the elliptic equation is obtained by parameter estimation through gPROMS one by one. Taking the corrected speed 97.1 as an example, the fitting results are shown in Figures 24–27. It can be clearly seen from Figure 27 that when the corrected flow rate and the corrected pressure ratio errors are taken as larger relative standard deviations, the fitting effect is not as good as taking smaller relative standard deviations. Through error calculation, the effect of sampling error on the fitting result of elliptic equation can be observed more accurately.
By calculating the errors of the fitting results, Table 8 is obtained. From the table, it can be observed that the fitting results of the elliptic equation fitting method based on parameter estimation are slightly less accurate compared to the neural network. As the sampling error increases, the MSE and MAPE of the parameter estimation results also increase in both sets of error values, but the difference between the two fitting error values is not significant compared to the error-free fitting results. Therefore, the effect of sampling error on the fitting of the elliptic equation parameter estimation method is not significant. Analyzing the reason, the accuracy of the elliptic parameter fitting method is lower than that of the neural network method, and the fitting results are not significantly affected by the sampling error.
In order to further explore the sensitivity of different corrected speed characteristic curves affected by sampling error, the error of error form 1 in Table 3 is applied to seven corrected speed characteristic curves, and then, the elliptic equation is fitted to obtain Figure 28. When fitting the characteristic curve with large corrected flow, the elliptic equation fitting method also has the problem of neural network fitting: the fitting result error of the characteristic curve with large corrected flow is large, and the curve fitting degree is poor. The reason is analyzed. The characteristic curve with large corrected flow is almost vertical, and its curve form is not close to the elliptic equation. The elliptic equation curve obtained by parameter estimation is not consistent with the real curve. When the data has sampling error, the result will be even more unexpected.
To sum up, it is not difficult to see that the fitting accuracy of the elliptic equation fitting method based on parameter estimation is not as good as that of the neural network fitting method, but its sensitivity to sampling error is low, and the impact of small sampling error on the elliptic parameter estimation method is not very large, but if the sampling error is large, it will also affect the accuracy of parameter estimation. For the characteristic curve with large corrected flow, because its curve is close to the vertical state and is not close to the curve form of the elliptic equation, the elliptic equation curve obtained by parameter estimation is not consistent with the real curve, so the elliptic parameter estimation method is not suitable for fitting the characteristic curve with large corrected flow.
3.2. The Modulization Method Based on Parameter Estimation
The modulization method is used to predict the unknown speed characteristic curve based on parameter estimation [36]. The data basis of this method is the known characteristic curve at a certain speed and a part of the measured data on the unknown characteristic curve, which is combined with the great likelihood parameter estimation method to calculate the best modular factor to obtain the unknown compressor characteristic curve.
3.2.1. Calculation of Modular Factor
The modulization method based on parameter estimation is divided into two steps: one is to correct the characteristic curve of the design operating point, and the other is to correct the variable operating condition.
To begin with, the design working point’s characteristic curve is corrected. The original characteristic curve’s design working point (, ) is chosen as the central reference point for the modular process. Using the modular factor and the distance from the central reference point, the new characteristic curve can display the position of any point from the original characteristic curve.
After completing the design condition correction, carry out the variable condition correction. A new variable condition modular factor is introduced, which can link variable condition with design condition. The corrected modular factor of the modified flow and pressure ratio under off-design conditions is defined as follows:
Among them, is the design operating point, is any point on the original characteristic curve, is the corresponding point of on the new characteristic curve, is the pressure ratio, and is the corrected flow.
Through the correction factor, all data on the new and old characteristic curves can be one-to-one corresponded. As shown in Figure 29, using the modular factor and the distance from the central reference point, the new characteristic curve can display the position of any point from the original characteristic curve. The pressure ratio and flow value of the new characteristic curve are as follows:
Different from the design condition modular method, the modular factor of the off-design condition is unknown. Through the optimization algorithm of parameter estimation, the off-design modulation program calculation is aimed at finding the best modular factor, so that the new characteristic curve is consistent with the given characteristic curve.
3.2.2. Results and Discussion
Take the curve of corrected speed 100 as the known speed curve, and the point on the speed curve is the point on the default original characteristic curve before the correction of off-design condition. Select some points of other speed curves as corresponding points after modulization of off-design conditions. The modular factors at different speeds are estimated by gPROMS parameters, as shown in Table 9.
Bring the obtained modular factor back to formulas (4) and (5), and the complete characteristic curve can be obtained through calculation. Figure 30 shows the comparison between the inferred characteristic curve inferred by the modular method and given original characteristic curve, “o” represents the original characteristic curve, and “” represents the inferred characteristic curve. It can be seen from the figure that the characteristic curve inferred by this parameter estimation-based modular method can better match the original characteristic curve. In order to obtain more accurate comparison results, Table 10 was calculated through error calculation, the MAPE of all the predicted values is not more than 2%. Based on the above analysis, this method has high accuracy and is applicable to the partial data of the known characteristic curve to infer the complete characteristic curve.
4. Conclusion
In this paper, PG9351FA gas turbine is taken as the research object. Two methods, artificial neural network and parameter estimation, are used to predict its characteristic curve, and the prediction accuracy and application conditions of the two methods are discussed. The following conclusions are drawn.
The method of inferring known speed characteristic curve based on neural network has high accuracy. The accuracy of the compressor characteristic curve obtained when the sample size is sufficient is much higher than that obtained by the elliptic parameter method using a small amount of data. The fitting result of the neural network is sensitive to the influence of sampling error. At the same time, with the increase of the corrected flow, the influence of the fitting curve on the sampling error will also increase, resulting in a significant decline in the fitting accuracy. This method is applicable to the prediction of known speed characteristic curve under sufficient data.
In the process of inferring unknown characteristic curves through neural networks, when two curves around the unknown characteristic curve are selected as the neural network learning data, the accuracy of the curve is higher than that when six characteristic curves are selected, but the former will show a large error in the prediction of the boundary curve. Therefore, this method is applicable to the prediction of unknown speed characteristic curve under sufficient data, but not to the prediction of side curve.
The elliptic equation fitting method based on parameter estimation can be used as an effective and reliable method to predict the compressor characteristic curve in the case of a small amount of data. Its fitting accuracy is not as good as that of neural network fitting method, but its sensitivity to sampling error is low, and the impact of small sampling error on elliptic parameter estimation method is not very large. For the characteristic curve with large corrected flow rate, the curve is close to the vertical state, and the elliptic parameter method has a slightly lower accuracy in processing the compressor characteristic curve with approximate vertical state, so the elliptic parameter estimation method is not suitable for fitting the characteristic curve with large corrected flow rate.
The modulization method based on the parameter estimation takes the design operating point as the center and one characteristic curve as the benchmark. When some data of other speed characteristic curves are known, the whole characteristic curve can be inferred by calculating the modular factor. This method has high accuracy and is applicable to the partial data of known characteristic curve and the complete characteristic curve.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there is no conflict of interest regarding the publication of this paper.
Acknowledgments
This work was supported by the Fundamental Research Project in Chinese National Sciences and Technology Major Project (grant number: 2017-I-0002-0002), Fundamental Research Funds for the Central Universities, National Natural Science Foundation of China (grant number: 52121003), and National Key Research and Development Program of China (grant number: 2022YFC2904105).