The circuit shows the capability of the ARTEMiS-SSN solver to make the real-time simulation of a WideBand line model within Simulink and SimPowerSystems.

### Demonstration

In the demo, a 25 km 3-ph cable (equivalent to 6-phase line considering the sheath) is energized with an open end. The sheath is grounded with 10 Ohms resistors at both ends. See the **Demo Validation against EMTP-RV** section at the end of this document for more info on this test case.

The WD-line model was fitted with a total of 14 poles for the characteristic admittance Yc(w) and 36 for the 6 propagation functions H1(w) to H6(w).

Real-time simulation with 2 cores of a 3.3 MHz i7 Intel target (one core for each end of the line) shows that this line can be simulated at a 25 microseconds time-step.

### More about the WB-line model and SSN

The WB-line model is a power line model that includes the frequency dependence of its parameters. For example, a line has more attenuation, described by its propagation function H(w), at high frequencies than at lower ones and this is taken into account in the WB-line model, as opposed to the Bergeron-type model with losses which assumes constant losses across all frequencies.

The characteristic impedance of the line typically diminishes also at a higher frequency while it is considered constant in non-FD models.

In addition to this frequency dependence, real power lines sometimes exhibit propagation effects that cannot be accurately described by fixed-frequency modal transformations, a mathematical method used by the FD-line model. This is especially true for cables. In that case, only full phase-domain fitting of the line impedance and propagation functions can provide an accurate simulation model.

The WideBand model makes such phase domain approximations of the line functions[1]. Both propagation function H(w) and characteristic admittance functions Yc(w) (Yc is the reverse of the impedance) are computed in the Laplace domain and obtained with a fitting routine of EMTP-RV. These Laplace functions can include real and complex poles and are listed accordingly.

The WideBand line is a line model similar to the Bergeron type in that it uses explicitly the propagation delay of the line inside the model. This has the advantage of allowing the decoupling of the simulation of a network located at both ends, a feature that is fundamental to achieve real-time simulation of large power grids in RT-LAB.

**The line parameter are specified in a MATLAB structure with the following format:**

Nph | number of phase |
---|---|

Ng | Number of propagation groups |

tau | Propagation delays |

NYc | Yc fitting order |

YcNpR | Number of real Yc poles |

YcNpC | Number of complex Yc poles |

YcR | Real Yc poles |

YcCR | Real part of complex Yc poles |

YcCI | Imaginary part of complex Yc poles |

YcstD | Yc constant residues |

YcRres | Residues of real Yc poles |

YcCRres | Real Part of Residues of complex Yc poles |

YcCIres | Imaginary part of Residues of complex Yc poles |

NH | H fitting order |

HNpR | Number of real H poles |

HNpC | Number of complex H poles |

HR | Real H poles |

HCR | Real part of complex H poles |

HCI | Imaginary part of complex H poles |

HRres | Residues of real H poles |

HCRres | Real part of Residues of complex H poles |

HCIres | Imaginary part of Residues of complex H poles |

The document entitled ** 'Obtaining WB-line model parameters from EMTP-RV'** explains how to get these parameters from the fitting routines of EMTP-RV.

A particularity of the WB-line model is that it has a very efficient formulation in the nodal method but a very poor one with the state-space method. This is caused by the numerous poles of the characteristic impedance of the WB-line model that would have to be included in the complete system state-space matrices (in ABCD formulation). SSN permits to keep the WB-line nodal formulations and couples it to the rest of the network, modeled using the state-space method of SPS.

### Demo Validation against EMTP-RV

The section presents the validation of a 25 km underground cable model in SSN against EMTP-RV. The simulation was done with a time step of 10 µs for a total simulation time of 15 ms. Fig. 1 presents the cable configuration, the table below contains the cable electrical and physical characteristics and fig. 2 shows the simulation circuit.

The simulation consists of a three-phase closure energizing an underground cable with a 138 kV voltage source. The receiving end of all three cores is left open, while both sides of the sheaths are grounded through a 10 ohms resistance.

*Fig. 1 Cable layout.*

*Fig. 2 Simulation case representation*

Fig. 2 presents the connections made for a single-phase (core and sheath) be aware that all three phases have similar connections.

In the following figures, colored lines are EMTP values while black thinner lines are WB-RT values. Where applicable, notice that both solutions are superimposed given the appearance of a darkened color line.

Fig. 3 shows the core voltages and fig. 4 shows the sheath voltages at the sending end of the cable. Fig. 5 presents the error between EMTP and WB-RT values for all sending end voltages.

* Fig. 3. Core voltages at the sending end. Color lines EMTP values, dark lines WB-RT values.*

* Fig. 4. Sheath voltages at the sending end. Color lines EMTP values, dark lines WB-RT values.*

* Fig. 5. Sending end voltage errors.*

Fig. 6 shows the core voltages while Fig. 7 shows the sheath voltages at the receiving end of the cable. Fig. 8 presents the error between EMTP and WB-RT values for all receiving end voltages.

* Fig. 6. Core voltages at the receiving end. Color lines EMTP values, dark lines WB-RT values.*

* Fig. 7. Sheath voltages at the receiving end. Color lines EMTP values, dark lines WB-RT values.*

* Fig. 8. Receiving end voltage errors.*

Fig. 9 shows the core currents and Fig. 10 shows the sheath currents at the sending end of the cable. Fig. 11 presents the error between EMTP and WB-RT values for all sending end currents.

* Fig. 9. Core currents at the sending end. Color lines EMTP values, dark lines WB-RT values.*

* Fig. 10. Sheath currents at the sending end. Color lines EMTP values, dark lines WB-RT values.*

* Fig. 11. Sending end current errors.*

Fig. 12 shows the core currents and Fig. 13 shows the sheath currents at the receiving end of the cable. Fig. 14 presents the error between EMTP and WB-RT values for all receiving end currents.

* Fig. 12. Core currents at the receiving end. Color lines EMTP values, dark lines WB-RT values.*

* Fig. 13. Sheath currents at the receiving end. Color lines EMTP values, dark lines WB-RT values.*

*Fig. 14. Receiving end currents error.*

### References

*[1] A. Morched, B. Gustavsen, M. Tartibi, “A universal model for accurate calculation of electromagnetic transients on overhead lines and underground cables”, IEEE Trans. on Power Delivery, Vol. 14, No. 3, pp. 1032-1038, July 1999.*