Modeling of Power Cables with Arbitrary Cross Section: From Parameter Calculation to Electromagnetic Transients Simulation

Abstract

A full computer-based methodology is proposed for electromagnetic transient simulations in power cables characterized by an arbitrary cross-section geometry. The frequency-dependent parameters of the cables are calculated using finite element method, and the three-phase cable modeling is carried out using modal decoupling and fitting techniques. The multiconductor representation of a sector-shaped cable is possible from the calculation of a constant and real modal transformation matrix, resulting four independent propagation modes (three phases and cable shield), which are modeled from the inclusion of frequency effect in the classic Bergeron method. The currents and voltages are expressed as a system of differential equations, which are presented as state equations and solved using numerical integration methods. The proposed modeling technique allows the inclusion of time-variable and nonlinear elements during electromagnetic transient simulations in the time domain, which is not possible from frequency-domain models that are solved using inverse transforms. The proposed model is validated from results simulated using numerical Laplace transform and exact modal transformation matrix for calculation of phase currents and voltages.

Appendix: Impedance and Admittance Parameters

The numerical approach for calculation of the self and mutual parameters of multiconductor cables, including phase conductors and shield mesh, can be carried out applying a current of 1.0 A at one of the conductors and calculating the voltage at all conductors. In the case of a sector-shaped cable, there are four conductors: three-phase cores and shield wire (armor). The self-impedances are calculated dividing the induced voltage by the impressed current, whereas the mutual are calculated dividing the induced voltage on other elements without impressed current (Hafner and Luz 2015). This procedure is illustrated in Fig. 15.

Initially, considering a current of 1.0 A through the phase conductor a, the mutual parameters can be calculated between a and other two phase conductors and armor. The procedure is valid considering the same current imposed in phases b or c. Thus, the frequency-dependent impedances of the sector-shaped cable can be calculated using the FEM, as a function of the current density (A/m$^{2})$ and the magnetic flux (Wb/m).

Fig. 15

Cable representation for calculation of the impedance parameters

Figure 15 shows that the cable is composed of phase cores a, b and c and armor wire g. Thus, from the voltage drops along the sector-shaped cable, the impedance matrix can be expressed in (41) (Ametani et al. 2015).

$$\begin{aligned} \left[ {\begin{array}{llll} {V_{{ag}} - V_{{a^{\prime }g^{\prime }}} } \\ {V_{{bg}} - V_{{b^{\prime }g^{\prime }}} } \\ {V_{{cg}} - V_{{c^{\prime }g^{\prime }}} } \\ {V_{{gg}} - V_{{g^{\prime }g^{\prime }}} } \\ \end{array} } \right] = \left[ {\begin{array}{llll} {Z_{{aa}} } &{} {Z_{{ab}} } &{} {Z_{{ac}} } &{} {Z_{{ag}} } \\ {Z_{{ba}} } &{} {Z_{{bb}} } &{} {Z_{{bc}} } &{} {Z_{{bg}} } \\ {Z_{{ca}} } &{} {Z_{{cb}} } &{} {Z_{{cc}} } &{} {Z_{{cg}} } \\ {Z_{{ga}} } &{} {Z_{{gb}} } &{} {Z_{{gc}} } &{} {Z_{{gg}} } \\ \end{array} } \right] \left[ {\begin{array}{l} {I_{a} } \\ {I_{b} } \\ {I_{c} }\\ {I_{g} } \\ \end{array} } \right] \end{aligned}$$

(41)

Terms ${Z}_{aa}, {Z}_{bb}, Z_{cc}$ are the self-impedances of the phase conductors, whereas $Z_{ab}, Z_{ac}$ and ${Z}_{bc}$ are mutual impedances. The impedances ${Z}_{ag}, {Z}_{bg }$ and ${Z}_{cg}$ are the mutual terms between phases a, b and c and the shield/ground wire g, respectively.

The self and mutual capacitances are considered as invariable in power cables and overhead transmissions lines up to very high frequencies. Therefore, the capacitances are usually considered constant in transmission line and cable modeling for electromagnetic transient analysis (Ametani et al. 2015).

Fig. 16

Circuit representation for calculation of the capacitances

The self and mutual capacitances are calculated as a function of the electric charge and the voltage at the conductors. Thus, the capacitance matrix of the sector-shaped cable can be expressed as (Hafner and Luz 2015; Ametani et al. 2015):

$$\begin{aligned} \left[ {\begin{array}{llll} {I_{a} - I_{{a^{\prime }}} } \\ {I_{b} - I_{{b^{\prime }}} } \\ {I_{c} - I_{{c^{\prime }}} } \\ {I_{g} - I_{{g^{\prime }}} } \\ \end{array} } \right] = \left[ {\begin{array}{llll} {C_{{aa}} } &{} {C_{{ab}} } &{} {C_{{ac}} } &{} {C_{{ag}} } \\ {C_{{ba}} } &{} {C_{{bb}} } &{} {C_{{bc}} } &{} {C_{{bg}} } \\ {C_{{ca}} } &{} {C_{{cb}} } &{} {C_{{cc}} } &{} {C_{{cg}} } \\ {C_{{ga}} } &{} {C_{{gb}} } &{} {C_{{gc}} } &{} {C_{{gg}} } \\ \end{array} } \right] \left[ {\begin{array}{llll} {V_{{ag}} } \\ {V_{{bg}} } \\ {V_{{cg}} } \\ {V_{{gg}} } \\ \end{array} } \right] \end{aligned}$$

(42)

The voltage and current vectors at the left hand in (41) and (42) are implicitly represented in terms of current density, magnetic flux and electric field, which are obtained using FEM and following the proposed boundary conditions for impedance and capacitance calculation in Figs. 15 and 16, respectively.