# Multivariate Regression Models for Predicting Pump-as-Turbine Characteristics

^{*}

## Abstract

**:**

^{2}of the head and efficiency curves were 0.997 and 0.909, respectively. Results also showed that XGB regressors and a dimensionless dataset yielded the best-fit models overall. The high accuracy of the models, combined with their lower computational cost compared to ANN, make them a robust solution for selecting PaTs in practice.

## 1. Introduction

_{st}is the specific speed of the turbine, N

_{sp}is the specific speed of the pump and ƞ

_{p}is the pump efficiency. Specific speeds are calculated based on the best efficiency points (BEPs). Thus, the theoretical Equation (1) implies the power generated at the BEP in turbine mode is lower than the power used in pump mode. Sharma [21] developed another similar theoretical equation, assuming a smaller reduction in specific speed, as shown in Equation (2):

## 2. Materials and Methods

#### 2.1. Data Collection and Preparation

_{s}is specific speed, N is rotational speed, Q

_{BEP}is flow at BEP and H

_{BEP}is head at BEP.

#### 2.2. Model Selection

#### 2.3. Modeling and Evaluation

^{2}were then selected for the next steps of tuning BEP models. For each of these eight best types of models, three BEP models (flow, head and efficiency) and two characteristic curve models (head and efficiency) were developed. In each case, a range of hyperparameters was initially set. These parameters were tuned with randomized search cross-validation over 100 iterations. If the resulting best hyperparameters were at either end of the established range, this process would be repeated with adjusted ranges in which previously best values would be the midpoint of the new range. Results for training and testing data with the default and tuned models were then compared with cross-validation to check for over or under-fitting. This check is essential given the small number of data points. For all models, fit was evaluated according to the coefficient of determination (R

^{2}), root mean squared error (RMSE) and median absolute deviation (MAD).

^{2}, RMSE and MAD.

## 3. Results

#### 3.1. BEP Results

^{2}scores from the default hyperparameter models are compared to the optimized models. The reduction in R

^{2}scores from the default to optimized parameters of the model is a consequence of the hyperparameter tuning and fitting the model better to the data. It should also be noted that the scales of the dimensioned parameters and the dimensionless parameters are different. Models applied to the dimensioned data set performed better than the dimensionless with regard to R

^{2}. This may be explained by the fact that the dimensioned dataset contains more variables. Because dimensionless variables are normalized by impeller diameter and rotational speed, these attributes were not included separately in the dimensionless dataset.

^{2}was found for flow predictions and the lowest for efficiency. However, the dimensioned dataset performed better for flow. The rectified linear unit activation function was selected through tuning for the dimensioned models, confirming the better performance of linear models.

^{2}of 0.9319, even though the majority of head values are slightly underpredicted. Efficiency results are more scattered and are identical for dimensioned (Figure 4e) and dimensionless models (Figure 4f). This is because the orthogonal matching pursuit model was applied to both. This model has no parameters which can be tuned and look for the most highly correlated attributes. In this case, the most correlated attribute to the turbine best efficiency is the pump best efficiency, which is also inherently dimensionless. Thus, choosing dimensioned or dimensionless attributes does not impact results in this case.

#### 3.2. Characteristic Curve Results

^{2}of the efficiency curves. For both datasets, the head curve was predicted with very high accuracy, with the same R

^{2}of 0.997. Hyperparameters for the best XGB Regressor models are summarized in Table 12.

^{2}scores for the head and Ψ curves are high, 0.986 and 0.954 respectively, albeit lower than the multivariate regression models. The efficiency and η model scores are lower but nevertheless strong for both the dimensioned and dimensionless predictions. Still, the multivariate regression models performed better in predicting efficiency and η curve, as well. With more datapoints and possibly more attributes, the ANN may perform better. More research would be required to collect more data on PaTs. Nevertheless, the accuracy of the multivariate regression models is already high.

## 4. Discussion

^{2}of 0.932, followed by the model proposed by Sharma [21] with a score of 0.827. While the ANN model performed well, with a score of 0.822, the multivariate regression model and Sharma’s equation still performed better. Other previous equations had slightly lower scores, but generally above 0.7. The exception is Barbarelli et al. [23] who developed their equations based on 4 PaTs with specific speeds ranging between 14 and 45. In the present dataset, most specific speeds were below 10. Thus, the Barbarelli et al. [23] equation is not applicable to this lower range.

^{2}, of 0.972. The next best-performing model is the Yang et al. [9] equation, at an R

^{2}of 0.965. The ANN model scored well, but the current multivariate regression model, Yang et al. [9], Sharma [21] and Stepanoff [20] was better. Efficiency results were not compared with previous studies because most authors did not develop a separate equation for efficiency. The PAT efficiency is not required to determine its BEP or create characteristic curves.

^{2}of 0.997, relatively higher than the Perez-Sanchez et al. [28] equation, with a value of 0.983. The ANN model scored very high as well, 0.986, which makes it the second best. The RMSE values confirm these results. The multivariate regression efficiency curve also had a high R

^{2}of 0.909, above the Rossi et al. [27] score of 0.869. In this case, the current ANN had the lowest score of the compared efficiency models. The results of the predicted efficiency curve values also scored highly using the multivariate regression method with a coefficient of determination of 0.901 with Rossi et al. [27] as the runner-up with a score of 0.869. The ANN method had a good score of 0.766 but the multivariate regression model and the Rossi et al. [27] model both performed better. Because some of these scores are very similar, the models are comparable, and their applicability might depend more on the range of pump values.

^{2}scores, and therefore, show no promise in predicting the attributes.

#### Limitations

^{2}of 0.98429 for the head curve, compared to 0.955 reported herein. These scores are still lower than those of the current multivariate regression model, i.e., 0.997. Furthermore, the Rossi et al. [33] scores refer to the overall training, validation and testing dataset, whereas the results presented herein are specifically for the 20 randomly selected data points.

## 5. Conclusions

^{2}for flow and head BEP were 0.972 and 0.932, respectively. On the other hand, the best characteristic curve predictions were developed with XGB Regressors, with R

^{2}of 0.994 and 0.919 for head and efficiency, respectively. Furthermore, the dimensionless dataset produced better characteristic curve and flow BEP models, whereas the dimensioned dataset provided slightly higher scores for head BEP models. Thus, a dimensionless dataset overall would be preferred.

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 4.**Predicted vs. Actual Values for Multivariate Regression Models of (

**a**) Flow—Dimensioned, (

**b**) Flow—Dimensionless, (

**c**) Head—Dimensioned, (

**d**) Head—Dimensionless, (

**e**) Efficiency—Dimensioned. (

**f**) Efficiency—Dimensionless.

**Figure 5.**Predicted vs. Actual Values for Multivariate Regression Models of (

**a**) Head Curve—Dimensioned, (

**b**) Head curve—Dimensionless, (

**c**) Efficiency curve—Dimensioned, (

**d**) Efficiency curve—Dimensionless.

Author | Flow | Head |
---|---|---|

Stepanoff [20] | $\frac{{Q}_{t}}{{Q}_{p}}=\frac{1}{\sqrt{{\eta}_{p}}}$ | $\frac{{H}_{t}}{{H}_{p}}=\frac{1}{{\eta}_{p}}$ |

Sharma [21] | $\frac{{Q}_{t}}{{Q}_{p}}=\frac{1}{{{\eta}_{p}}^{0.8}}$ | $\frac{{H}_{t}}{{H}_{p}}=\frac{1}{{{\eta}_{p}}^{1.2}}$ |

Alatorre-Frenk et al. [22] | $\frac{{Q}_{t}}{{Q}_{p}}=\frac{{{0.85\eta}_{p}}^{5}+0.385}{{{2\eta}_{p}}^{9.5}+0.205}$ | $\frac{{H}_{t}}{{H}_{p}}=\frac{1}{{{0.85\eta}_{p}}^{5}+0.385}$ |

Yang et al. [9] | $\frac{{Q}_{t}}{{Q}_{p}}=\frac{1.2}{{{\eta}_{p}}^{0.55}}$ | $\frac{{H}_{t}}{{H}_{p}}=\frac{1.2}{{{\eta}_{p}}^{1.1}}$ |

Barbarelli [23] | $\frac{{Q}_{t}}{{Q}_{p}}=0.00029{{N}_{sp}}^{2}-0.02771{N}_{sp}+2.01648$ | $\frac{{H}_{t}}{{H}_{p}}=-{3\times 10}^{-5}{{N}_{sp}}^{3}+{4.4\times 10}^{-3}{{N}_{sp}}^{2}-0.20882{N}_{sp}+4.6493$ |

Audisio [24] | $\frac{{Q}_{t}}{{Q}_{p}}=1.21{{\eta}_{p}}^{-0.25}$ | $\frac{{H}_{t}}{{H}_{p}}=1.21{{\eta}_{p}}^{-0.8}{\left[1+{\left({0.6+ln\text{}N}_{sp}\right)}^{2}\right]}^{0.3}$ |

Fontanella et al. [25] | $\frac{{Q}_{t}}{{Q}_{p}}=1.3595\frac{{N}_{t}}{{N}_{p}}$ | $\frac{{H}_{t}}{{H}_{p}}={1.4568\left(\frac{{N}_{t}}{{N}_{p}}\right)}^{2}$ |

Author | Variable | Equation | Applied Range |
---|---|---|---|

Derakhshan and Nourbakhsh [28] | $\frac{{H}_{t}}{{H}_{t,BEP}}$ | $1.0283{\left(\frac{{Q}_{t}}{{Q}_{t,BEP}}\right)}^{2}-0.5468\left(\frac{{Q}_{t}}{{Q}_{t,BEP}}\right)+0.5314$ | ${N}_{st}<60$ |

$\frac{{P}_{t}}{{P}_{t,BEP}}$ | $-0.3092{\left(\frac{{Q}_{t}}{{Q}_{t,BEP}}\right)}^{3}+2.1472{\left(\frac{{Q}_{t}}{{Q}_{t,BEP}}\right)}^{2}-0.8865\left(\frac{{Q}_{t}}{{Q}_{t,BEP}}\right)+0.0452$ | ||

Rossi et al. [29] | $\frac{\psi}{{\psi}_{t,BEP}}$ | ${0.2394\left(\frac{\mathsf{\Phi}}{{\mathsf{\Phi}}_{t,BEP}}\right)}^{2}+0.769\left(\frac{\mathsf{\Phi}}{{\mathsf{\Phi}}_{t,BEP}}\right)$ | $\frac{\phi}{{\phi}_{t,BEP}}\le 1.4$ |

$\frac{\eta}{{\eta}_{t,BEP}}$ | ${{-1.9788\left(\frac{\mathsf{\Phi}}{{\mathsf{\Phi}}_{t,BEP}}\right)}^{6}+{9.0636\left(\frac{\mathsf{\Phi}}{{\mathsf{\Phi}}_{t,BEP}}\right)}^{5}-13.148\left(\frac{\mathsf{\Phi}}{{\mathsf{\Phi}}_{t,BEP}}\right)}^{4}{+3.8527\left(\frac{\mathsf{\Phi}}{{\mathsf{\Phi}}_{t,BEP}}\right)}^{3}$ ${+4.5614\left(\frac{\mathsf{\Phi}}{{\mathsf{\Phi}}_{t,BEP}}\right)}^{2}-1.3769\left(\frac{\mathsf{\Phi}}{{\mathsf{\Phi}}_{t,BEP}}\right)$ | ||

Perez-Sanchez [30] | $\frac{{H}_{t}}{{H}_{t,BEP}}$ | $0.406{\left(\frac{{Q}_{t}}{{Q}_{t,BEP}}\right)}^{2}+0.621\left(\frac{{Q}_{t}}{{Q}_{t,BEP}}\right)$ | $\frac{{Q}_{t}}{{Q}_{t,BEP}}\ge 0.4$ |

$\frac{{\eta}_{t}}{{\eta}_{t,BEP}}$ | $-1.219{\left(\frac{{Q}_{t}}{{Q}_{t,BEP}}\right)}^{4}+6.95{\left(\frac{{Q}_{t}}{{Q}_{t,BEP}}\right)}^{3}$ $-14.578{\left(\frac{{Q}_{t}}{{Q}_{t,BEP}}\right)}^{2}+13.231\left(\frac{{Q}_{t}}{{Q}_{t,BEP}}\right)-3.383$ | ||

Fontanella et al. [27] | $\frac{{H}_{t}}{{H}_{t,BEP}}$ | $1+0.9633{\left(\frac{{Q}_{t}}{{Q}_{t,BEP}}-1\right)}^{2}+1.4965\left(\frac{{Q}_{t}}{{Q}_{t,BEP}}-1\right)$ | $0.33<\frac{{Q}_{t}}{{Q}_{t,BEP}}<6.25$ |

$\frac{{P}_{t}}{{P}_{t,BEP}}$ | $1+0.03499{\left(\frac{{Q}_{t}}{{Q}_{t,BEP}}-1\right)}^{4}-{0.2405\left(\frac{{Q}_{t}}{{Q}_{t,BEP}}-1\right)}^{3}$ $+{1.4326\left(\frac{{Q}_{t}}{{Q}_{t,BEP}}-1\right)}^{2}+2.7071\left(\frac{{Q}_{t}}{{Q}_{t,BEP}}-1\right)$ |

Input Variables (Dimensioned − Dimensionless) | Output Variables (Dimensioned − Dimensionless) |
---|---|

Impeller Diameter | Turbine Flow BEP (Q − $\mathsf{\Phi}$) |

Pump Flow BEP (Q − $\mathsf{\Phi}$) | Turbine Head BEP (H − $\mathsf{\Psi}$) |

Pump Head BEP (H − $\mathsf{\Psi}$) | Turbine Efficiency ($\eta $) |

Pump Efficiency ($\eta $) | |

Specific Speed (N_{s}) | |

Rotational Speed |

Input Variables (Dimensioned − Dimensionless) | Output Variables (Dimensioned − Dimensionless) |
---|---|

Impeller Diameter | Turbine Head Values (H_{t}/H_{tBEP} − ${\mathsf{\Psi}}_{t}/{\mathsf{\Psi}}_{t}$_{BEP}) |

Pump Flow BEP (Q − $\mathsf{\Phi}$) | Turbine Efficiency values ($\eta $_{t}/$\eta $_{tBEP}) |

Pump Head BEP (H − $\mathsf{\Psi}$) | |

Pump Efficiency ($\eta $) | |

Specific Speed (N_{s}) | |

Rotational Speed | |

Turbine Flow Values (Q_{t}/Q_{tBEP} − ${\mathsf{\Phi}}_{t}$_{BEP}) |

Parameter | Type | Values |
---|---|---|

Learning Rate | Range | 0.0001–0.1 |

Dropout Rate | Range | 0–0.99 |

Number of Hidden Layers | Range | 1–10 |

Neurons per Layer | Range | 1–300 |

Batch Size | Choice | 2, 4, 8, 16, 29, 58 |

Activation Function | Choice | Tanh, Sigmoid, Relu |

Optimizer | Choice | Adam, RMS, SGD |

Attribute | Train/Test Split | Best Model | Default R^{2} | Optimized R^{2} | RMSE | MAD |
---|---|---|---|---|---|---|

Flow | 80/20 | Huber Regressor | 0.9728 | 0.9721 | 9.7205 | 16.9551 |

Head | 90/10 | Elastic Net | 0.9549 | 0.9319 | 7.6661 | 7.1337 |

Efficiency | 80/20 | Orthogonal Matching Pursuit | 0.8147 | 0.8147 | 0.0505 | 0.0633 |

**Table 7.**Multivariate Regression Results of Turbine Mode BEP Attributes with Dimensionless Datasets.

Attribute | Train/Test Split | Best Model | Default R^{2} | Optimized R^{2} | RMSE | MAD |
---|---|---|---|---|---|---|

$\mathsf{\Phi}$ | 75/25 | Huber Regressor | 0.9749 | 0.9729 | 0.0058 | 0.0136 |

$\mathsf{\Psi}$ | 75/25 | Huber Regressor | 0.8747 | 0.8777 | 0.2227 | 0.0155 |

$\eta $ | 80/20 | Orthogonal Matching Pursuit | 0.8147 | 0.8024 | 0.0522 | 0.0708 |

Attribute | Train/Test Split | Hidden Layers | Neurons | Learning Rate | Dropout Rate | R^{2} | RMSE | MAD |
---|---|---|---|---|---|---|---|---|

Flow | 80/20 | 2 | 176 | 1 × 10^{−5} | 0.15437 | 0.914 | 17.068 | 18.256 |

Head | 80/20 | 12 | 325 | 4.38 × 10^{−5} | 0.05319 | 0.822 | 11.954 | 13.708 |

Efficiency | 80/20 | 7 | 310 | 0.00213 | 0 | 0.761 | 0.0573 | 0.0756 |

Attribute | Train/Test Split | Hidden Layers | Neurons | Learning Rate | Dropout Rate | R^{2} | RMSE | MAD |
---|---|---|---|---|---|---|---|---|

$\mathsf{\Phi}$ | 80/20 | 9 | 124 | 5.5 × 10^{−5} | 0.03443 | 0.904 | 0.0126 | 0.012 |

$\mathsf{\Psi}$ | 80/20 | 6 | 263 | 0.0001005 | 0.4 | 0.885 | 0.0665 | 0.00822 |

$\eta $ | 80/20 | 16 | 207 | 0.000299 | 0 | 0.779 | 0.055 | 0.0607 |

**Table 10.**Multivariate Regression Results of Turbine Mode Characteristic Curves with Dimensioned Datasets.

Curve | Train/Test Split | Model | Default R^{2} | Optimized R^{2} | RMSE | MAD |
---|---|---|---|---|---|---|

Head | 80/20 | XGB Regressor | 0.993 | 0.997 | 0.0186 | 0.2369 |

Efficiency | 80/20 | XGB Regressor | 0.908 | 0.901 | 0.0539 | 0.0394 |

**Table 11.**Multivariate Regression Results of Turbine Mode Characteristic Curves with Dimensionless Datasets.

Curve | Train/Test Split | Model | Default R^{2} | Optimized R^{2} | RMSE | MAD |
---|---|---|---|---|---|---|

$\mathsf{\Psi}$ | 80/20 | XGB Regressor | 0.994 | 0.997 | 0.0179 | 0.1940 |

$\eta $ | 80/20 | XGB Regressor | 0.919 | 0.897 | 0.0516 | 0.0364 |

**Table 12.**Hyperparameters for best XGB Regressor models selected to predict head and efficiency curve.

Hyperparameter | Head Curve | Efficiency Curve |
---|---|---|

Subsample | 0.4 | 0.8 |

n_estimators | 2500 | 1300 |

Min_child_weight | 1 | 1 |

Max_depth | 4 | 9 |

Max_delta_step | 10 | 6 |

Learning_rate | 0.15 | 0.75 |

eta | 0.8 | 0 |

Curve | Train/Test Split | Hidden Layers | Neurons | Learning Rate | Dropout Rate | R^{2} | RMSE | MAD |
---|---|---|---|---|---|---|---|---|

Head | 80/20 | 7 | 145 | 3.86 × 10^{−5} | 0.005 | 0.986 | 0.03848 | 0.20587 |

Efficiency | 80/20 | 15 | 180 | 5.34 × 10^{−6} | 0 | 0.766 | 0.0776 | 0.0339 |

Curve | Train/Test Split | Hidden Layers | Neurons | Learning Rate | Dropout Rate | R^{2} | RMSE | MAD |
---|---|---|---|---|---|---|---|---|

$\mathsf{\Psi}$ | 80/20 | 8 | 141 | 7.27 × 10^{−5} | 0 | 0.980 | 0.0455 | 0.2159 |

$\eta $ | 80/20 | 19 | 100 | 5.73 × 10^{−6} | 0 | 0.816 | 0.0687 | 0.0299 |

Method | R^{2} Head | RMSE Head | R^{2} Flow | RMSE Flow |
---|---|---|---|---|

Current study multivariate regression | 0.932 | 7.666 | 0.972 | 9.720 |

Current study ANN | 0.822 | 11.954 | 0.914 | 17.068 |

Stepanoff [20] | 0.798 | 8.833 | 0.915 | 16.163 |

Sharma [21] | 0.827 | 13.582 | 0.927 | 17.904 |

Alatorre-Frenk et al. [22] | 0.750 | 16.359 | 0.819 | 17.052 |

Yang at al. [9] | 0.744 | 15.738 | 0.965 | 19.946 |

Barbarelli et al. [23] | −6.807 | 32.252 | 0.739 | 23.387 |

Audisio [24] | −4.97 | 68.161 | 0.971 | 10.685 |

Fontanella et al. [25] | 0.391 | 48.772 | 0.967 | 11.34 |

Method | R^{2} Head Curve | RMSE Head Curve | R^{2} Efficiency Curve | RMSE Efficiency Curve |
---|---|---|---|---|

Current study multivariate regression | 0.997 | 0.019 | 0.909 | 0.054 |

Current study ANN | 0.986 | 0.038 | 0.766 | 0.078 |

Derakhshan and Nourbakhsh [26] | 0.545 | 0.239 | 0.297 | 0.158 |

Rossi et al. [27] | 0.983 | 0.047 | 0.777 | 0.089 |

Perez-Sanchez et al. [28] | 0.955 | 0.076 | 0.868 | 0.068 |

Fontanella et al. [25] | 0.874 | 0.126 | 0.869 | 0.068 |

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## Share and Cite

**MDPI and ACS Style**

Brisbois, A.; Dziedzic, R.
Multivariate Regression Models for Predicting Pump-as-Turbine Characteristics. *Water* **2023**, *15*, 3290.
https://doi.org/10.3390/w15183290

**AMA Style**

Brisbois A, Dziedzic R.
Multivariate Regression Models for Predicting Pump-as-Turbine Characteristics. *Water*. 2023; 15(18):3290.
https://doi.org/10.3390/w15183290

**Chicago/Turabian Style**

Brisbois, Alex, and Rebecca Dziedzic.
2023. "Multivariate Regression Models for Predicting Pump-as-Turbine Characteristics" *Water* 15, no. 18: 3290.
https://doi.org/10.3390/w15183290