# Power Plant Energy Predictions Based on Thermal Factors Using Ridge and Support Vector Regressor Algorithms

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## Abstract

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^{2}), median absolute error (MeAE), mean absolute percentage error (MAPE), and mean Poisson deviance (MPD) are assessed after their training and testing of each algorithm. From the modeling of energy output data, it is seen that SVR (RBF) is the most suitable in providing very close predictions compared to other algorithms. SVR (RBF) training R

^{2}obtained is 0.98 while all others were 0.9–0.92. The testing predictions made by SVR (RBF), Ridge, and RidgeCV are nearly the same, i.e., R

^{2}is 0.92. It is concluded that these algorithms are suitable for predicting sensitive output energy data of a CCPP depending on thermal input variables.

## 1. Introduction

^{2}for LR is about 0.99910896 (99.91%). Zaaoumi et al. [26] used ANNs and analytical models to predict a parabolic trough solar thermal power plant (PTSTPP). The study results revealed that the ANNs model achieves better results than the analytical models. The results of the ANNs model reveal that the predicted yearly electrical energy is about 42.6 GWh/year, whereas the operational energy is around 44.7 GWh/year.

## 2. CCPP System

## 3. Power Plant Energy Output Data Modeling

#### 3.1. Ridge Regression (RR)

#### 3.2. Multiple Linear Regression (MLR)

#### 3.3. Support Vector Regression (SVR)

## 4. Performance Assessment

#### 4.1. Mean Absolute Error (MAE)

#### 4.2. R-Square (R^{2})

^{2}) is an arithmetical measure of fit that specifies how much deviation of a dependent variable is described by the independent variable in a regression model. The R

^{2}is also called the coefficient of determination. This metric offers a sign of in what mANNser good a model fits a given dataset. It specifies how close the calculated values are plotted (i.e., the regression line) to the actual data values. The value of R

^{2}lies between 0 and 1, where 0 shows that this modelis not suitable for the given data, and one shows that the model hysterics seamlessly to the provided dataset. The R

^{2}is measured as:

^{2}equation, $n$ is the number of data points, ${V}_{ot}$ and ${V}_{op}$ indicate the calculated predictions from the regressors and actual output from CCPP measured from the experiment, respectively. In the statistical study, the negative (–ve) value must be more significant enough to signify a superior precise model that can go up to a maximum equal to 1.

#### 4.3. Median Absolute Error (MeAE)

^{th}sample and ${y}_{1}$ is the equivalent actual value, then the average absolute error predicted over $n$ samples are restricted as follows:

#### 4.4. Mean Absolute Percentage Error (MAPE)

#### 4.5. Mean Poisson Deviance (MPD)

## 5. Results and Discussion

^{2}, MeAE, MAPE, and MPD are analyzed comparatively. It should be noted that the regression Ridge is analyzed in detail, and RidgeCV (Ridge cross-validated) model is used only in the last part of this section. The predictor parameter here is the output of electrical energy power (PEO) from the CCPP affected by VE (exhaust vacuum), ABT (ambient temperature), REH (relative humidity), and ABP (ambient pressure). These readings of CCPP were recorded experimentally, and the entire data set is openly available in the UCI machine learning repository made available by the work reported in [50].

^{2}= 0.9283 shows that the training is successful as the value is close to unity. Upon this comfortable training of the Ridge regressor, an attempt is made to test the ability of this model to predict the testing data. From Figure 4, the predictions made by the regressor indicate the output is in an excellent match with the experimental readings. The predictions are even better than the trained output from the regressor. The compactness and closeness with the trendlines indicate this algorithm’s ability to predict power plant energy output based on thermal parameters. Though the data is highly nonlinear, the Ridge model is successful in its predictions. The R

^{2}= 0.9297 during testing is obtained from this regressor, indicating closeness to its perfect unity. For α = 0.2 to 1.0, the training and testing results are avoided for brevity purposes.

^{2}, MeAE, MAPE, and MPD) of the Ridge regressor at different α values are shown. This performance of the Ridge regressor is obtained from the training data set computed following experimental data. From the figure, the MAE value is seen rising linearly with the α values. As discussed, the increase in α values impacts the data shrinkage growth; the MAE tends to increase accordingly. The R

^{2}value also indicates that the increase in α values hampers the accuracy of the Ridge regressor. However, the best closet value of R

^{2}obtained is at α = 0. The R

^{2}looks to be a very uniform thought, but a closer look reveals that its value is decreasing, which is unacceptable. The MaAE, MAPE, and MPD metrics also increase with an increase in α values as the training ability of the regressor deteriorates. In Figure 8, the trend of all the metrics is shown with an increase in α values obtained from the testing session. As expected, the performance of the Ridge regressor has worsened with the α values, as also observed from the training data set. However, the metrics are seen to linearly increase with the α values, not in the previous metrics.

^{2}is above 0.9. This training is performed using 90% of the total data available for training purposes. The remaining 10% of data being used to test the model, which is trained using this 90% data. The testing of the Linear regressor model is carried out with the 10% of data, and a comparative plot obtained with experimental data is shown in Figure 10. Significantly few data points fall outside the clustered region, where most of the data points are close to the trendline. The testing of the Linear regressor model depicts that it is also as suitable as the Ridge model in the prediction of PEO of a CCPP. The R

^{2}is above 0.9 in testing results, clearly showing that the Linear regressor model can give a very close result. This indicates that more computational cost and time can be saved by adopting this simple model than other models. In Figure 11, the performance evaluation of the trained and tested Linear regressor is shown in a 3D bar graph for better illustration. The metrics are very close to each other from training and testing. IN BOTH CASES, the MAE, R2, MeAE, MAPE, and MPD are approximately the same, which is usually rare in most modeling processes. A very close value to zero from MAPE and MPD indicates that the error is less and the difference in predicted and the actual value is also significantly less irrespective of whether the data point numerical value involved is large or less.

^{2}for both cases is shown in the graphs at the top left corner. The densely packed data points indicate the training computations performed successfully, and the testing results also show a comfortable prediction from the SVR (LR) trained. The R

^{2}value for both is above 0.91, indicating a good regression of the power plant energy data. In Figure 13, using the SVR-based RBF—SVR (RBF) algorithm, the training of the data set is most successful in this study. The R

^{2}for SVR (RBF) based training is above 0.98, which is the closest unity obtained. The testing of this regressor is shown in Figure 13b, which also indicates a good prediction. However, these predictions during SVR (RBF) testing are not dominant to other algorithms as the R

^{2}is close to 0.93 as in other cases.

^{2}is 0.91, also obtained from previous models. The main drawback observed during the training is the excessive computational time taken by this model for 1 degree of polynomial functions. For 3 degrees polynomial functional, the time was not tolerable and did not continue with the training process. Surprisingly, during the testing of this data, the closeness with actual values predicted was slightly better than the training.

## 6. Conclusions

^{2}, MeAE, MAPE, and MPD analysis also revealed that the SVR (RBF) is the best algorithm that gives very close predictions to actual values. The following stands are RidgeCV, SVR (Poly.) Ridge, SVR (LR), and then last is the Linear regression model in predictions. This finally indicates that the selected algorithms are well suited for modeling CCPP output energy based on thermal input parameters.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Symbols | Description | Symbols | Description |

CCPP | Combined cycle power plant | α | Alpha |

LR | Linear regressor | SHRGs | Steam heat recovery generators |

SVR | Support vector regressor/regression | EP | Electrical power |

RBF | Radial basis function | HE | Heat exchanger |

MAE | Mean absolute error | MW | Mega watt |

R^{2} | R-squared | AP | Atmospheric pressure |

MeAE | Median absolute error | RH | Relative humidity |

MAPE | Mean absolute percentage error | AT | Ambient temperature |

MPD | Mean poison deviance | ESP | Exhaust steam pressure |

CV | Cross-validated | I/P | Input |

ML | Machine learning | O/P | Output |

ANNs | Artificial neural networks | RR | Ridge regression |

GT | Gas turbine | MLR | Multiple linear regression |

MLP | Multi-layer perception | VE | Exhaust vacuum |

ST | Steam turbine | RidgeCV | Ridge cross-validated |

P_{E} | Electrical power | ABT | Ambient temperature |

FFNNs | Feed forward neural networks | REH | Relative humidity |

NNs | Neural networks | MLP | Multi-layer perception |

IoT | Internet of things | ELM | Extreme learning machine |

GA | Genetic algorithm | SVM | Support vector machines |

EEP | Electrical energy power | ||

Symbols | |||

X | Independent variable | $\left|{x}_{a}-x\right|$ | Absolute errors |

Y | Dependent variable | n | Number of data points |

e | Errors | ${V}_{ot}$ | Calculated predictions from regressors |

B | Regression coefficient | ${V}_{ot}$ | Calculated actual O/P |

x | Input value | $\widehat{y}$ | Expected value of i^{th} sample |

y | Output value | ${y}_{1}$ | Equivalent actual value |

A_{0}, A_{1} | Scale factors | ${A}_{t}$ | Actual value |

σ | Radial basis function kernel | F_{t} | Forecast value |

A_{1}, A_{2} | Linear datasets | ${\theta}_{u}$ | Calculated mean |

p | Number of errors | ${A}_{u}$ | Observed value |

$\sum $ | Summation of all absolute errors | P_{D} | Poisson deviance |

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**Figure 1.**The simple layout of CCPP (reprinted with permission from Elsevier 2017 [44]).

**Figure 5.**The trend of experimental readings and predictions made by Ridge regressor for the first 30 data points during training with different α values ranging from 0 to 1.0.

**Figure 7.**Performance of Ridge regressor at different α values from the training computations (R

^{2}is shown as R2 in the graph legends).

**Figure 8.**Ridge regressor performance metrics change linearly with α values in the tested dataset (R

^{2}is shown as R2 in the graph legends).

**Figure 9.**Training output from Linear regressor and experimental output compared along a line where the R

^{2}is 0.904 from the trendline.

**Figure 10.**Testing predictions from Linear regressor and experimental output plotted along a line where the R

^{2}is 0.907 from the trendline.

**Figure 11.**The different forms of errors were analyzed in the Linear regressor obtained from the training and testing points (R

^{2}is shown as R2 in the horizontal x-axis).

**Figure 12.**(

**a**) Training and (

**b**) testing of PEO using support vector regressor (SVR) based LR—SVR (LR) algorithm.

**Figure 13.**SVR-based radial basis function—SVR (RBF) algorithm (

**a**) Training and (

**b**) testing results for the CCPP energy output.

**Figure 14.**Polynomial regression-based SVR (SVR Poly.) algorithm used for (

**a**) Training and (

**b**) testing of PEO data.

**Figure 15.**Comparison between the actual readings of CCPP energy output (PEO) and trained values produced from computations of SVR (LR), SVR (RBF), and SVR (Poly.) algorithms during the training indicating first 20 data points only.

**Figure 16.**The predicted data of CCPP using SVR (LR), SVR (RBF), and SVR (Poly.) algorithms were obtained after testing in comparison with experimental readings.

**Figure 17.**Assessment of SVR (LR), SVR (RBF), and SVR (Poly.) algorithms compared with RidgeCV used for (

**a**) training and (

**b**) testing of energy output data. (R

^{2}is shown as R2 in the graph legend).

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## Share and Cite

**MDPI and ACS Style**

Afzal, A.; Alshahrani, S.; Alrobaian, A.; Buradi, A.; Khan, S.A.
Power Plant Energy Predictions Based on Thermal Factors Using Ridge and Support Vector Regressor Algorithms. *Energies* **2021**, *14*, 7254.
https://doi.org/10.3390/en14217254

**AMA Style**

Afzal A, Alshahrani S, Alrobaian A, Buradi A, Khan SA.
Power Plant Energy Predictions Based on Thermal Factors Using Ridge and Support Vector Regressor Algorithms. *Energies*. 2021; 14(21):7254.
https://doi.org/10.3390/en14217254

**Chicago/Turabian Style**

Afzal, Asif, Saad Alshahrani, Abdulrahman Alrobaian, Abdulrajak Buradi, and Sher Afghan Khan.
2021. "Power Plant Energy Predictions Based on Thermal Factors Using Ridge and Support Vector Regressor Algorithms" *Energies* 14, no. 21: 7254.
https://doi.org/10.3390/en14217254