# Estimation of Excitation Current of a Synchronous Machine Using Machine Learning Methods

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## Abstract

**:**

## 1. Introduction

- Easy maintenance and change of AC voltage for transmission and distribution;
- AC transmission plant costs (switches, transformers, etc.) are much lower than equivalent DC transmission;
- The power plant produces AC power, so it is better to use AC than DC instead of converting;
- In the case of major faults in the network, it is easier to disconnect an AC system because the sinusoidal current tends to zero at a certain moment.

- Is it possible to estimate the excitation current of SM using AI algorithms with a high precision rate and a small evaluation error?
- Is it possible to optimize the model and confirm the obtained results with 5 k-fold cross-validation using the randomized hyperparameter search?
- Which algorithm provides the best results with the possibility of implementation in a real-life situation?

## 2. Materials and Methods

#### 2.1. Potential Challenges When Modeling a Synchronous Motor

#### 2.2. General Information about SM

- T is the calculated torque;
- ${T}_{max}$ is the maximum torque for SM;
- $sin\left(\delta \right)$ is the sinus function of load angle.

#### 2.3. Operation Conditions and Dataset Collection

#### 2.4. Dataset Statistical Analysis

- x, angle of the density function;
- $\mu $ is a representation of measure location (the given cluster distribution around $\mu $);
- $\kappa $ is a representation of the measure concentration;
- ${I}_{0}\left(\kappa \right)$ is the modified Bessel function with order zero.

- x, a, and b are real scalars;
- b > 0 and x ∈ [0, 1] is the probability density function of the normal distribution;
- $\varphi $ is the cumulative distribution function of the normal distribution.

#### 2.5. Research Methodology

- Extra trees regressor (ETR);
- Elasticnet regressor (EN);
- K-nearest neighbor regressor (k-NN);
- Linear regressor (LR);
- Random forest regressor (RFR);
- Ridge regressor (RR);
- Stochastic gradient descent regressor (SGD);
- Support vector regressor (SVR);
- MLP regressor;
- Extreme gradient boosting regressor (XGBoost).

#### 2.5.1. Extra Trees Regressor

#### 2.5.2. Elasticnet Regressor

#### 2.5.3. K-nearest Neighbour Regressor

#### 2.5.4. Linear Regressor

#### 2.5.5. Random Forest Regressor

#### 2.5.6. Ridge Regressor

#### 2.5.7. Stochastic Gradient Descent Regressor

#### 2.5.8. Support Vector Regressor

#### 2.5.9. Multi-Layer Perceptron Regressor

#### 2.5.10. Extreme Gradient Boosting Regressor

## 3. Results and Discussion

## 4. Conclusions

- It is possible to estimate the excitation current of a synchronous motor using an AI algorithm with high precision and accuracy. Based on the given research it was shown that the most optimal algorithm was XGBoost.
- Using GS and cross-validation, the values were validated, and the parameters of the AI model were optimized, which provides suitable evaluation metrics for the estimation of the excitation current.
- From the larger number of presented algorithms in this paper, the best possible algorithm that provides optimal results and the smallest $\sigma $ is XGboost with a high value of R${}^{2}$ and small values of MSE and MAPE.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 6.**The scheme of work for the experiment with an SM [52].

**Figure 7.**Data distribution/histogram for Load current in the dataset, the histogram consists of an analyzed parameter as the number of inputs with the given value.

**Figure 8.**Data distribution/histogram for power factor PF and power factor error e in the dataset, the histogram consists of an analyzed parameter as the number of inputs with the given value.

**Figure 9.**Data distribution/histogram for excitation current I${}_{f}$ and changing of excitation current d${}_{f}$ of synchronous machine; the histogram consists of an analyzed parameter as the number of inputs with the given value.

Condition | |||
---|---|---|---|

Star connected motor ($\mathrm{Y}$) voltage [V] | Star connected motor ($\mathrm{Y}$) current [A] | Triangle connected motor ($\Delta $) | Star connected motor ($\mathrm{Y}$) current [A] |

400 | 5.8 | 231 | 10 |

Power factor (cos ∅) | |||

0.8 | |||

Apparent power (kVA) | |||

4 | |||

Revolutions per minute (rpm) | |||

1000 |

I${}_{\mathit{y}}$ | PF | e | d${}_{\mathit{f}}$ | I${}_{\mathit{f}}$ | |
---|---|---|---|---|---|

Mean value | 4.499 | 0.825 | 0.174 | 0.350 | 1.530 |

Minimum value | 3.0 | 0.650 | 0.0 | 0.037 | 1.217 |

Standard deviation | 0.896 | 0.103 | 0.103 | 0.180 | 0.180 |

Maximum value | 6.0 | 1.0 | 0.350 | 0.769 | 1.949 |

Parameter Name | Minimum Value | Maximum Value |
---|---|---|

n_estimators | 1000 | 10,000 |

criterion | squared error, friedman_mse | |

max_depth | None | |

min_samples_split | 2 | 10 |

max_features | auto, sqrt, log2 | |

random state | 0 | 63 |

Parameter Name | Minimum Value | Maximum Value |
---|---|---|

alpha | 0.1 | 10 |

l1_ratio | 0.1 | 10 |

max_iter | 1000 | 10,000 |

selection | random, cyclic |

Parameter Name | Minimum Value | Maximum Value |
---|---|---|

n_neighbours | 1 | 1000 |

weights | uniform, distance | |

leaf_size | 1 | 1000 |

algorithm | auto, ball_tree, kd_tree, brute | |

p | 2 | 50 |

Parameter Name | Minimum Value | Maximum Value |
---|---|---|

fit_intercept | True, False | |

normalize | True, False | |

positive | True, False |

Parameter Name | Minimum Value | Maximum Value |
---|---|---|

n_estimators | 100 | 5000 |

criterion | squared_error, absolute_error, Poisson | |

max_features | sqrt, log2 |

Parameter Name | Minimum Value | Maximum Value |
---|---|---|

alpha | 1.0 | 100.0 |

max_iter | 1000 | 50,000 |

tol | 1 × 10${}^{-5}$ | 1 × 10${}^{-1}$ |

fit_intercept | True, False | |

solver | svd, cholesky, lsqr, sparse_cg, sag, saga |

Parameter Name | Minimum Value | Maximum Value |
---|---|---|

alpha | 0.0001 | 10.0 |

max_iter | 1000 | 10,000 |

validation_fraction | 0.15 | |

power_t | 0.1 | 0.5 |

learning_rate | invscaling, optimal, constant | |

shuffle | True, False | |

l1_ratio | 0.0001 | 0.5 |

Parameter Name | Minimum Value | Maximum Value |
---|---|---|

kernel | linear, poly, rbf, sigmoid | |

degree | 1 | 10,000 |

gamma | scale, auto | |

coef0 | 0.0 | 10.0 |

tol | 1 × 10${}^{-5}$ | 1 × 10${}^{-10}$ |

C | 0.5 | 20.0 |

epsilon | 0.05 | 20.0 |

max_iter | 100 | 10,000 |

Parameter Name | Minimum Value | Maximum Value |
---|---|---|

hidden_layer_sizes (2,3 and 4 hidden layers) | 5 | 600 |

activation | tahn, relu, identity, logistic | |

solver | adam, lbfgs | |

alpha | 0.02 | 0.5 |

power_t | 0.5 | 2.0 |

max_iter | 1000 | 10,000 |

tol | 1 × 10${}^{-5}$ | 1 × 10${}^{-1}$ |

max_iter | 1000 | 10,000 |

Parameter Name | Minimum Value | Maximum Value |
---|---|---|

learning_rate | 0.2 | 0.02 |

max_depth | 6 | 64 |

min_child_weight | 1 | 10 |

gamma | 0.0 | 1.5 |

colsample_bytree | 0.1 | 1.0 |

max_delta_step | 0.0 | 1.5 |

**Table 13.**Table of the results and evaluation metrics of all algorithms used in this research for default hyperparameters.

Regressor Name | ${\mathit{R}}^{2}$ | MSE | MAPE |
---|---|---|---|

Extra Trees | 0.9795 | 0.0006 | 0.0166 |

ElasticNet | 0 | 0.0297 | 0.1442 |

k-Nearest Neighbour | 0.9517 | 0.0015 | 0.0294 |

Linear | 0.8875 | 0.0035 | 0.0465 |

Random Forest | 0.9909 | 0.0003 | 0.0121 |

Ridge | 0.9012 | 0.0028 | 0.0423 |

Stochastic Gradient Descent | 0 | 0.0402 | 0.1611 |

Support Vector Machines | 0.8662 | 0.0041 | 0.0506 |

Multi-layer Perceptron | 0.8534 | 0.0052 | 0.0585 |

Extreme Gradient Boosting | 0.99433 | 0.0001 | 0.0074 |

**Table 14.**Table of results and evaluation metrics of all algorithms used in this research for varied hyperparameters and cross-validated models.

Regressor Name | $\overline{{\mathit{R}}^{2}}$ | $\overline{\mathbf{MSE}}$ | $\overline{\mathbf{MAPE}}$ |
---|---|---|---|

Extra Trees | 0.9784 | 0.0006 | 0.0164 |

ElasticNet | 0 | 0.0297 | 0.1442 |

k-Nearest Neighbour | 0.4484 | 0.0173 | 0.1127 |

Linear | 0.8881 | 0.0035 | 0.0462 |

Random Forest | 0.9746 | 0.0008 | 0.0223 |

Ridge | 0.4833 | 0.01487 | 0.1054 |

Stochastic Gradient Descent | 0.8731 | 0.0046 | 0.0535 |

Support Vector Machines | 0.8844 | 0.0045 | 0.0523 |

Multi-layer Perceptron | 0.9303 | 0.0025 | 0.0392 |

Extreme Gradient Boosting | 0.9963 | 0.0001 | 0.0057 |

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## Share and Cite

**MDPI and ACS Style**

Glučina, M.; Anđelić, N.; Lorencin, I.; Car, Z.
Estimation of Excitation Current of a Synchronous Machine Using Machine Learning Methods. *Computers* **2023**, *12*, 1.
https://doi.org/10.3390/computers12010001

**AMA Style**

Glučina M, Anđelić N, Lorencin I, Car Z.
Estimation of Excitation Current of a Synchronous Machine Using Machine Learning Methods. *Computers*. 2023; 12(1):1.
https://doi.org/10.3390/computers12010001

**Chicago/Turabian Style**

Glučina, Matko, Nikola Anđelić, Ivan Lorencin, and Zlatan Car.
2023. "Estimation of Excitation Current of a Synchronous Machine Using Machine Learning Methods" *Computers* 12, no. 1: 1.
https://doi.org/10.3390/computers12010001