IEEE Electrification Magazine - December 2019 - 109

Case A
Case B
Case C

Amplitude (p.u.)

2
1.5
1
0.5
0

105

106

107

108

Frequency (Hz)
Figure 17. The A-value frequency-dependent magnitude trend.

Amplitude (p.u.)

Case A
Case B
Case C

40
30
20
10
0

104

105

106

107

108

Frequency (Hz)

Figure 18. The B-value frequency-dependent magnitude trend.

0.07
0.06
Amplitude (p.u.)

accelerate the components' thermal and electrical aging,
deteriorate the system power quality, and may cause resonances and dissymmetry. The thermal aging is mostly
related to harmonics in the current waveform, while the
electrical aging is due to voltage spikes, and they both
affect the insulation integrity.
The common models, based on low-frequency (or
even dc) parameters, can be unsuitable for studying the
disturbance propagation in the system, thus leading to
the need of frequency-dependent modeling approaches.
In this regard, the model presented in Figure 15 is used to
develop a two-port network equation system in accordance with the traveling wave theory. The input voltage
( V1) and current ( I 1) are linked to the output voltage ( V2)
and current ( I 2) by a 2 # 2 complex value matrix (i.e., the
ABCD matrix). The ABCD matrix values are analyzed in
the frequency domain to assess differences between a
frequency-dependent model compared to a fixed parameter one. A cross-linked, polyethylene-insulated MVdc
aluminum cable, 20 m in length, has been used to evaluate differences in the ABCD matrix complex values. The
electrical and geometric specification used for the reference cable are depicted in Table 2. In particular, the ABCD
matrix is calculated in three different model scenarios:
xx
Case A: The cable parameters are computed at the
dc value.
xx
Case B: The cable parameters are assumed to be frequency-dependent. The skin effect phenomenon impacts the
calculation of the cable resistance [ R( f )] and selfinductance [ L( f )], while the frequency-dependent
relative permittivity [ f( f )] affects the conductance [G( f )] and capacitance [C( f )] values. Moreover, this case needs a parameters computation for
each frequency considered to be impacting the
required computational load;
xx
Case C: To reduce the model computational load, the
RLGC cable parameters are calculated only at their
resonance frequency value.
Therefore, the trends of the A, B, and C magnitude values are shown in Figures 17, 18, and 19, respectively, in
the frequency domain. Since the ABCD matrix is symmetric, the D magnitude value has the same trend reported in Figure 17 (A-value trend). First, it can be noticed that
Case C has a completely wrong trend compared to Case A
(reference value for low frequencies) and Case B. Thus,
the computational load reduction cannot be achieved
through the Case C modeling approach. By comparing the
figures, Case A and Case B have the same trend up to the
kilohertz frequency range for all of the figures, leading to
two main considerations. First, the model presented in
Case B is valid in the low-frequency range. Second, Figures 17-19 clearly show that a fixed-frequency parameter
model is not applicable for the assessment of the highfrequency disturbance propagation in the IPES.
Nevertheless, the ABCD matrix values' dependency
on cable length has been assessed. The A value

Case A
Case B
Case C

0.05
0.04
0.03
0.02
0.01
0
104

105

106
Frequency (Hz)

107

108

Figure 19. The C-value frequency-dependent magnitude trend.

IEEE Elec trific ation Magazine / D EC EM BE R 2 0 1 9

109

IEEE Electrification Magazine - December 2019

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