# Analysis of Very Fast Transients Using Black Box Macromodels in ATP-EMTP

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Finite Element Method for Very Fast Transients

_{r}is the relative permittivity, μ

_{r}is the relative permeability, k

_{0}is the wavenumber of free space, σ is the electrical conductivity, and $\overline{E}$ is the electric field vector.

_{r}is the dielectric constant, R

_{out}is the radius of the enclosure, and R

_{in}is the radius of the main conductor.

^{®}(Burlington, MA, USA) [10]. Insulating spacers provide mechanical support to the HV conductor and can provide a gas tight partition between gas zones.

## 3. Vector Fitting and Equivalent Circuit Extraction

_{r}> 1 at very high frequencies [13]. These earth return admittances are, however, inherently considered when using field solvers that implement full Maxwell’s equations, when appropriate modelling domains are included. Several commercial software packages allow full frequency-dependent models through the use of black box macromodels, created with rational approximations of a frequency response through VF [14]. VF is a robust method of fitting measured or simulated frequency responses, with a rational function approximation. The resulting fit can be expressed as the pole residue form in Equation (9) and the state space form in Equation (10) as given in [15]:

_{n}and c

_{n}are the poles and residues either in real or complex conjugate pairs, d and e are the constant and proportional terms, A, B, C, D and E are the state space matrices.

#### 3.1. ATP-EMTP Circuit Inclusion

#### 3.2. Model Order Approximation

## 4. Simulations and Results

^{®}) was carried out to assess the accuracy of the fit. The frequency responses were further compared with the response of a circuit-based modelling approach based on a mixture of lumped components and distributed lines to identify any significant differences. For the comparison, a 13th order bus-spacer equivalent and a 15th order elbow equivalent circuit were used. For the circuit-based modelling approach, the 1 m bus section was represented by two 0.5 m distributed transmission lines (Bergeron lines intialised at 5 MHz) and the spacer at the center was represented as a 10 pF capacitance to ground. The elbow was represented by a 0.5 m section of distributed line at each end, two spacers, each having a 10 pF capacitance to ground, representing the capacitance between the conductor and the inner surface of the enclosure, along with two parallel sections of distributed line of differing lengths. To achieve a comparison, the input impedance was evaluated in EMTP using a frequency scan with a 1 A current source, with the equivalent circuit terminated by its characteristic impedance. The results are shown in Figure 5.

_{C}), a 434 Ω resistor (maximum OHL Z

_{C}) and a capacitive-resistive termination (open disconnector 5 pF capacitor and bus Z

_{C}resistor). The validity of the pole-residue model results was assessed by comparison with the results of a transient simulation computed directly with the S-parameters in ADS for the same terminations. An overview of the VF/pole-residue modelling process and ADS comparison is given in Figure 6.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**Algorithm for pole-residue model generation and determination of the minimum order of approximation.

## References

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**Figure 1.**Three dimensional finite element models: (

**a**) section of bus-spacer intersection; and (

**b**) 90° elbow.

**Figure 2.**Bus-spacer and elbow scattering parameters (S-parameters): (

**a**) S11 (reflectance); and (

**b**) S21 (transmittance).

**Figure 5.**Impedance-frequency comparison of modelling techniques: (

**a**) bus-spacer; and (

**b**) 90° elbow.

**Figure 6.**Vector fitting (VF) modelling workflow and method validation using advanced design system (ADS).

**Figure 7.**EMTP vs. ADS elbow transient responses showing good agreement for: (

**a**) matched termination; and (

**b**) OHL termination.

**Figure 10.**Voltage at gas-air bushing for circuit-based model: (

**a**) 50 μs duration; and (

**b**) magnified 5 μs.

**Figure 11.**Voltage at gas-air bushing for pole-residue model: (

**a**) 50-μs duration; and (

**b**) magnified 5-μs.

Parameter | Bus-Spacer | 90° Elbow |
---|---|---|

Materials | Aluminium ^{1}Steel ^{1}Epoxy Resin: ε _{r} = 4.1, μ_{r}= 1, σ = 1 × 10 ^{−17} (S), tan δ = 2 × 10^{−7}SF _{6}: ε_{r}= 1.002, μ_{r}= 1, σ = 0 | Aluminium ^{1}Steel ^{1}Epoxy Resin: ε _{r} = 4.1, μ_{r}= 1, σ = 1 × 10 ^{−17} (S), tan δ = 2 × 10^{−7}SF _{6}: ε_{r}= 1.002, μ_{r}= 1, σ = 0 |

Mesh | Tetrahedral-177,116 elements | Tetrahedral-241,846 elements |

Computation time | 3 h 19 min ^{2} | 4 h 28 min ^{2} |

^{1}Material selection not meshed; Impedance Boundary Condition applied. Standard COMSOL Multiphysics

^{®}library materials used [10];

^{2}Computation time using 2 x Intel(R) Xeon(R) CPU E5-2670 0 at 2.60 GHz, 16 cores each and 128 GB RAM. Frequency sweep 0.2–150 MHz.

Termination | Bus-Spacer RMS Error | Elbow RMS Error | ||||
---|---|---|---|---|---|---|

V_{out} | I_{in} | I_{out} | V_{out} | I_{in} | I_{out} | |

Matched | 0.002 | 4.7 × 10^{−6} | 2.2 × 10^{−5} | 0.002 | 1.6 × 10^{−5} | 2.6 × 10^{−5} |

Short Circuit | 0 | 0.351 | 0.351 | 0 | 0.04 | 0.04 |

Open Circuit | 0.205 | 0.003 | 1.0 × 10^{−8} | 0.20 | 0.003 | 1.0 × 10^{−8} |

Resistive 1 Ω | 0.003 | 0.003 | 0.003 | 0.002 | 0.002 | 0.002 |

OHL Z_{0} 434 Ω | 0.003 | 3.8 × 10^{−5} | 6.8 × 10^{−6} | 0.009 | 1.3 × 10^{−4} | 2.0 × 10^{−5} |

Capacitive-Resistive | 0.019 | 3.1 × 10^{−4} | 4.5 × 10^{−5} | 0.092 | 0.001 | 1.2 × 10^{−4} |

Component | Parameters (Calculated/Assumed) | |
---|---|---|

Bus | Z_{c} = 68 Ω | |

Spacer | Pole-residue or Circuit-based = 10 pF | |

Elbow | Pole-residue or Circuit-based, Z_{0} = 68 Ω | |

DS (open) | Z_{c} = 68 Ω, gap capacitance = 5 pF | |

DS (closing/closed) | Z_{c} = 68 Ω, gap capacitance = 5 pF, R(t) = exponential decay | |

CB (Open) | Z_{c} = 68 Ω, C_{gap} = 40 pF, C_{ground} = 120 pF | |

CB(Closed) | Z_{c} = 68 Ω, C_{ground} = 120 pF | |

CT | Z_{c} = 68 Ω, C_{ground} = 50 pF | |

Bushing/downleads | Distributed parameter lines, Z_{c} varies with height |

Position | Circuit-Based Model Peak Magnitude and Time of Occurrence | Pole-Residue Model Peak Magnitude and Time of Occurrence | Absolute Difference in Magnitude |
---|---|---|---|

V_{bush} | 650 kV @ 1.12 μs | 628 kV @ 0.46 μs | 3.5% |

V_{Mid-bus} | 584 kV @ 2.16 μs | 599 kV @ 1.27 μs | 2.57% |

V_{Elbow} | 633 kV @ 1.30 μs | 617 kV @ 0.55 μs | 2.59% |

V_{DS} | 547 kV @ 0.87 μs | 531 kV @ 0.76 μs | 3.01% |

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**MDPI and ACS Style**

James, J.; Albano, M.; Clark, D.; Guo, D.; Haddad, A.
Analysis of Very Fast Transients Using Black Box Macromodels in ATP-EMTP. *Energies* **2020**, *13*, 698.
https://doi.org/10.3390/en13030698

**AMA Style**

James J, Albano M, Clark D, Guo D, Haddad A.
Analysis of Very Fast Transients Using Black Box Macromodels in ATP-EMTP. *Energies*. 2020; 13(3):698.
https://doi.org/10.3390/en13030698

**Chicago/Turabian Style**

James, Jonathan, Maurizio Albano, David Clark, Dongsheng Guo, and Abderrahmane (Manu) Haddad.
2020. "Analysis of Very Fast Transients Using Black Box Macromodels in ATP-EMTP" *Energies* 13, no. 3: 698.
https://doi.org/10.3390/en13030698