# Data-Driven Compartmental Modeling Method for Harmonic Analysis—A Study of the Electric Arc Furnace

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Data-Driven Compartmental Modeling Method for EAF

#### 2.1. Multi-Mode EAF Harmonic Model

_{h}is the h-th harmonic current, U

_{1}denotes the fundamental voltage, U

_{m}is the m-th harmonic voltage, and C is the constant coefficient.

_{1}denotes the fundamental current.

_{h}, leads to the heavy computational cost of harmonic power-flow calculation, due to the iterative solution of the system equations and non-linear coupling model [20,21]. Therefore, a non-linear coupling model is approximated to a linear coupling model, as shown in Equation (3). In this way, the computation burden is reduced while the convergence precision is maintained.

_{1}, U

_{2}, U

_{3}, …, U

_{m}]

^{T}as W

_{m},

^{(n)}, and C

^{(n)}.

_{m}, m, is reduced to j by PCA, then W

_{m}is processed to R

_{j}as follows,

_{m}. R

_{j}can be represented as [r

_{1}, r

_{2}, r

_{3}, …, r

_{j}]

^{T}, where r

_{1}, r

_{2}, r

_{3}, …, and r

_{j}are the first j principal components of W

_{m}.

^{−1}as A

_{p},

_{p}is the simplified coupling matrix.

_{p}, C, A

_{p}, C, …, A

_{p}

^{(n)}, and C

^{(n)}, then the model parameters A, C, A, C, …, A

^{(n)}, and C

^{(n)}can be calculated according to Equation (8),

#### 2.2. Data-Driven Compartmental Modeling Method (DCMM)

_{p}, C, A

_{p}, C, …, A

_{p}

^{(n)}, and C

^{(n)}were identified. Finally, the model parameters A, C, A, C, …, A

^{(n)}, and C

^{(n)}were calculated.

_{i}is the i-th cluster, q denotes a data point belonging to Y

_{i}, and v

_{i}is the center of Y

_{i}.

_{1}, a

_{2}, …, a

_{j}, and c are constant coefficients, j denotes the number of dimensions of matrix R

_{j}, dis

^{(i)}is the distance between the i-th data point and the clustering center, and r

_{1}

^{(i)}, r

_{2}

^{(i)}, …, r

_{j}

^{(i)}, and I

_{h}

^{(i)}are coordinates of the i-th data point.

_{p}, C, A

_{p}, C, …, A

_{p}

^{(n)}, and C

^{(n)}were identified by the least square fitting, and the model parameters A, C, A, C, …, A

^{(n)}, and C

^{(n)}were calculated according to Equation (8). In this way, the multi-mode EAF harmonic model was established by the proposed DCMM.

## 3. Performance Evaluation

_{m}, is defined as follows,

_{1}, p

_{2}] and [p

_{1}′, p

_{2}′] are equal to A

_{p}and A

_{p}′ representing the true value and the estimated value of the simplified coupling matrix, respectively.

_{c}, is defined as follows,

_{m}

_{(i)}denotes the average estimation deviation of the i-th mode.

_{c}is calculated to be 0. Nonetheless, it is quite difficult to achieve identical results when the data volumes, number of clusters, and the complexity of the sample distribution are high. Thus, the parameter identification results are considered accurate if D

_{c}is less than 5%.

#### 3.1. Case 1: 2880 Data Points, 2 Modes

_{c}is calculated to be 3.27%.

#### 3.2. Case 2: 3360 Data Points, 3 Modes

_{c}is calculated to be 2.18% in this case.

#### 3.3. Summary of Case 1 and 2

## 4. Case Study

#### 4.1. Parameter Identification of Multi-Mode EAF Harmonic Model

_{1}is the system impedance resulting from the minimum short-circuit capacity of the assumed source. The three-phase EAF model shown in Figure 7 is detailed in Figure 8; each EAF function block represents an electrode at each phase, and the controlled voltage source with a resistive and inductive network was adopted to model the flicker frequency and magnitude variation. The simulation time was set to 101 s, and the current and voltage signals extracted through a PQ recorder were analyzed every 0.02 s with a fixed sampling rate of 2500 Hz by the fast-Fourier transform (FFT) algorithm. Finally, we obtained 5000 sets of the fundamental current, the 2-nd to 25-th harmonic current, and the 2-nd to 25-th harmonic voltage peaks.

#### 4.2. Comparison of Different Models

^{2}denotes the coefficient of determination, FA is the fitting accuracy, N is the number of data points of validation dataset, I

^{t}

^{(i)}and I

^{c}

^{(i)}, respectively, denote the true value and calculated value of the 5-th harmonic current of the i-th data point of the validation dataset, and I′ is the mean value of the 5-th harmonic current of the validation dataset.

^{2}and FA of the proposed model are greater than those of the other two models, thus, the fitting effect of the proposed model is optimal among the three models.

#### 4.3. Application of Multi-Mode EAF Harmonic Model

_{h}denotes the h-th harmonic system impedance calculated based on the circuit diagram shown in Figure 7, U

_{h}is the h-th harmonic voltage.

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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Case | Mode | A_{p} | C |
---|---|---|---|

Case 1 | Mode 1 | [3.7099, −1.1922] | −0.0115 |

Mode 2 | [−3.9041, −0.3421] | −0.2107 | |

Case 2 | Mode 1 | [3.7099, −1.1922] | −0.0115 |

Mode 2 | [−3.9041, −0.3421] | −0.2107 | |

Mode 3 | [0.3511, −0.8861] | −0.0904 |

Case | Mode | A_{p}′ | C′ |
---|---|---|---|

Case 1 | Mode 1 | [3.7338, −1.2106] | −0.0110 |

Mode 2 | [−3.8913, −0.3801] | −0.2142 |

Case | Mode | A_{p}′ | C′ |
---|---|---|---|

Case 2 | Mode 1 | [3.7305, −1.2161] | −0.0098 |

Mode 2 | [−3.8886, −0.790] | −0.2140 | |

Mode 3 | [0.3510, −0.8861] | −0.0904 |

Case | Mode | A_{p} | C |
---|---|---|---|

EAF | Mode 1 | [0.1111, −0.0095] | −0.0007 |

Mode 2 | [0.0283, 0.0099] | 0.0020 | |

Mode 3 | [−0.0164, 0.1193] | 0.0203 |

Model | ME | MSE | R^{2} | FA |
---|---|---|---|---|

Proposed model | 1.1369 | 2.0536 | 90.28% | 98.25% |

Constant-harmonic-ratio-type model | 1.2862 | 2.5837 | 87.78% | 96.80% |

Norton equivalent model | 1.6345 | 4.2531 | 79.88% | 95.38% |

Order | Z_{h} (ohms) | U_{h} (V) |
---|---|---|

3 | 0.039 + 0.882i | 30.4131 |

5 | 0.039 + 1.372i | 18.2623 |

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**MDPI and ACS Style**

Xu, H.; Shao, Z.; Chen, F.
Data-Driven Compartmental Modeling Method for Harmonic Analysis—A Study of the Electric Arc Furnace. *Energies* **2019**, *12*, 4378.
https://doi.org/10.3390/en12224378

**AMA Style**

Xu H, Shao Z, Chen F.
Data-Driven Compartmental Modeling Method for Harmonic Analysis—A Study of the Electric Arc Furnace. *Energies*. 2019; 12(22):4378.
https://doi.org/10.3390/en12224378

**Chicago/Turabian Style**

Xu, Haobo, Zhenguo Shao, and Feixiong Chen.
2019. "Data-Driven Compartmental Modeling Method for Harmonic Analysis—A Study of the Electric Arc Furnace" *Energies* 12, no. 22: 4378.
https://doi.org/10.3390/en12224378