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The assessments are carried out under the assumption of converters connected to a network of infinite power through their switching reactance. In particular, considering different values of the modulation index, the voltage and current behaviors in terms of harmonic spectrum and in terms of Total Harmonic Distortion (THD) are compared. Analysis of “sensitivity” have also been conducted to vary at first the circuit parameters (for example by varying the output reactance of the two inverters interleaved) and the characteristics of modulated and carriers waveforms (for example in the case of error in synchronization during the generation of the two carriers out of phase by 180°).

In regards to the generated voltage, the analysis show that the three-level inverter presents a harmonic contribution lower than the two-level, since the voltage assumes a shape closer to the sinusoidal waveform. Considering the currents injected into the network, the interleaved solution presents an harmonic impact less than in case of a single inverter, both two or three-level one. The detailed comparisons relating to configurations analyzed are shown in the next.

Comparative impact of harmonic voltage

To operate a comparison in terms of harmonic impact of different inverter configurations the THD_V (Total Harmonic Distortion) index is calculated, as stated in §1.1:

1 2

2

_ V

V V

THD

N

n

∑

n

= =

where V_n is the RMS voltage value at the harmonic n, n is the maximum harmonic order considered and V₁ is the RMS value of the voltage at fundamental frequency.

Since the solution interleaved doesn’t provide benefit in terms of harmonic voltage impact⁶ with respect to the other configurations in examination (two and three-level), it has operated only the comparison in terms of voltage harmonic distortion generated for a single two-level and three-level three phase inverter, both with PWM modulation.

As already mentioned in §1.2.3, also the THD_V comparison confirms the output voltage of a three-level inverter has less harmonic contents than a two-three-level (Figure 1-22, 1-23).

Only for the three-level inverter, the behavior with the modulations APOD and PD has been analyzed, making sure that for the phase voltage there aren’t any differences in THD while the line to line THD_V is lower in case of PD modulation (Figure 1-24), as seen in the § 1.2.3.

6 This is due to the fact that the two inverters are connected in parallel to the network.

400 500 600

-180 -90 0 90 180

Current phase - 1st two-level inverter - fsw

[Degree]

400 500 600

-180 -90 0 90 180

Current phase - 2nd two-level inverter - fsw

[Hz]

[Degree]

850 950 1050 1150

-180 -90 0 90 180

Current phase - 1st two-level inverter - 2*fsw

[Degree]

850 950 1050 1150

-180 -90 0 90 180

Current phase - 2nd two-level inverter - 2*fsw

[Hz]

[Degree]

27 a)

b)

Figure 1-21: THD_V line to midpoint (a) and line to line (b) comparison for a two-level and three-level inverter with APOD modulation

a)

b)

Figure 1-22: THD_V line to midpoint (a) and line to line (b)comparison for a two-level and three-level inverter with PD modulation

28

Figure 1-23: THD_V line to line comparison for a three-level levels inverter with modulation APOD and PD

Comparative impact of harmonic current

To make the comparison between the different types of inverter (two, three-level and interleaved converters) it is assumed to have inverters connected to a network of infinite power through the switching reactance. In the interleaved configuration the inverter are designed to have the total power generated by the two converters interleaved the same as the single two or three-level inverter.

Three phase two-level Inverter and interleaved inverter

In Figure 1-24 it’s possible to verify that in the spectrum of the overall current generated in the network by the two interleaved inverter there aren’t the harmonics multiple of the switching frequency and the related “sidebands”, this means a reduced harmonic impact on network with respect to the single inverter.

Figure 1-24: Comparison between current harmonic spectrum for two-level and for interleaving inverter configuration

In addition, with reference to the THD_I index, already expressed in (1.2):

1 2

2

_ I

I I THD

N

n

∑

n

= =

Figure 1-25 represents its trend, considering different modulation index values, and the best performance of the interleaved configuration is confirmed.

0 500 1000 1500 2000 2500

0 0.05 0.1 0.15 0.2

Current Harmonic Spectrum - two-level inverter

0 500 1000 1500 2000 2500

0 0.05 0.1 0.15 0.2

Current Harmonic Spectrum - two-level interlaced inverters

[Hz]

29

Figure 1-25: THD_I comparison between two-level and interleaved inverter

Comparison between three phase two-level, three-level inverter and interleaved inverters

Operating a comparison between all the three configurations considered, analyzing the current harmonic spectrum⁷ there is a lack, for the interleaved inverter, of the bell-shaped distribution centered on the switching frequency and harmonic multiples of the switching frequency. In addition for the three-level configuration with APOD modulation (Figure 1-26) there is an harmonic contribution similar to that of two interleaved inverters even if starting from the bell centered on the switching frequency.

Figure 1-26: Current harmonic spectrum comparison for two-level inverter, three-level inverter with APOD modulation and interleaved inverter

Figure 1-27: Current harmonic spectrum comparison for two-level inverter, three-level inverter with PD modulation and interleaved inverter

Finally, by working the comparison with the THD_I it’s verified that the current generated by a three-level inverter (both with modulation APOD and PD) presents an index of total harmonic distortion bigger than the interleaved configuration. In Figure 1-28 the trend of the THD_I for different values of the modulation index is represented, this confirms the lower harmonic impact of the interleaved configuration which is favored in modular multi inverter architectures.

7 In the case of interleaved inverters this is the overall current injected into the network.

THD_I % output current referred to the fundamental component

0 20 40 60 80 100 120 140 160

0 0.2 0.4 0.6 0.8 1

m

THD_I [%]

THD_I_2lev THD_I_inter

0 500 1000 1500 2000 2500

0 0.1 0.2

Current Harmonic Spectrum - two-level inverter

0 500 1000 1500 2000 2500

0 0.1

0.2 Current Harmonic Spectrum - two-level interlaced inverters

0 500 1000 1500 2000 2500

0 0.1 0.2

Current Harmonic Spectrum - three-level inverter

[Hz]

0 500 1000 1500 2000 2500

0 0.1 0.2

Current Harmonic Spectrum - two-level inverter

0 500 1000 1500 2000 2500

0 0.1

0.2 Current Harmonic Spectrum - two-level interlaced inverters

0 500 1000 1500 2000 2500

0 0.1 0.2

Current Harmonic Spectrum - three-level inverter

[Hz]

30

Figure 1-28: THD_I comparison between two-level inverter, three-level inverter with PD and APOD modulation and interleaved inverter

Even if the two-level interleaved inverters are designed with the same power and parameters, it can be useful to analyse what happens, in the current harmonic spectrum, in the hypothesis of “unbalance”

between the two interleaved converters, examining the cases with:

• a switching reactance varies in a range of ± 20 %, with respect to the nominal value;

• the angles

φ

₀ and

φ

_c from the equation (1.37), which respectively represent the phase shift of the modulating signal (with respect to the absolute reference) and of the carriers of two inverters interleaved, are variable⁸.

Switching reactance variation

In Figure 1-29 the THD_I referred to the fundamental current are shown when the reactance of an inverter varies between ± 20% of the designed value. For the same modulation index, in the range of variation positive or negative, there is an increase in the harmonic distortion injected into the network, even if the variation of the THD_I compared to the ideal case “balanced” is less than 2%

(Figure 1-30). From the comparison with the Figure 1-25 it’s possible to confirm better performance of the interleaved inverter solution, even in the presence of unbalance between the switching reactance.

8 These conditions may occur, for example, in the case of synchronization errors in the generation of the waves.

THD_I % output current referred to the fundamental component

0 10 20 30 40 50 60 70

0 0.2 0.4 0.6 0.8 1

m

THD_I [%]

THD_I_2lev THD_I_inter THD_I_3lev_PD

THD_V % line to line voltage referred to the fundamental component

0 100 200 300 400 500 600 700 800

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

m

THD_V [%]

THD_Vab_3lev_APOD THD_Vab_3lev_PD THD_V_inter

31 a)

b)

Figure 1-29: THD_I deviation from the nominal case, when the reactance of the second interleaved inverter is less (a) or bigger (b)than the first one

Figure 1-30: THD_I deviation from the ideal case

φ

0 and

φ

_c angles variation

At the end, the voltage and current Total Harmonic Distortion are also influenced by the phase shifts φ and 0 φ . The angle _c φ is the shift of the reference current from the zero cross and ₀ φ is an added _c shift between the carriers waveform of the interleaved inverters evaluated in case of a synchronization error during the generation of these signals.

THD_I in function of reactance unbalance

0 2 4 6 8 10 12 14 16

0 0.2 0.4 0.6 0.8 1

m

THD_I

-20%

-10%

-5%

-1%

THD_I_ref

THD_I in function of reactance unbalance

0 2 4 6 8 10 12 14 16 18 20

0 0.2 0.4 0.6 0.8 1

m

THD_I

THD_I_ref 1%

5%

10%

20%

Deviation from the ideal case

0.8 1 1.2 1.4 1.6 1.8 2

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

m

-20%

-10%

-5%

-1%

I_Ref 1%

5%

10%

20%

32

For instance, the THD_I variation, for all the possible values assumed by these shifts, is less than 10%

for a range of modulation index between 0.6 and 0.8 (Figure 1-31 and 1-32) ⁹.

Regarding the spectrum of the total current injected into the network, it has been reported an influence of the angle φ : assuming, for example, _c φc = 30° and a modulation index m=0.7, the harmonic spectrum presents the “first bell”, corresponding to the switching frequency, that is not present in the ideal case, due to the not perfect compensation in phase opposition of the harmonics of the two inverters. In Figure 1-31b the spectrum of the current in the event of φ0 = 90° is shown: only the sidebands harmonic components appear in correspondence of the switching frequency, since the cancellation of the fc is ensured by 180° phase shift between the carriers of two inverters.

a) b)

Figure 1-31: THD_I (a) and current harmonic spectrum (b) in case of a ^φ0 shift of one interleaved inverter

a) b)

Figure 1-32: THD_I (a) and current harmonic spectrum (b) in case of a φ_c shift between the carries waveform of the interleaved inverter

As presented in the previous results, considering the configuration of two three phase two-level interleaved inverters, the positive effects to reduce the current harmonic distortion injected into the network are confirmed in any case.

9 These considerations are valid in the hypothesis that the angle

φ

₀ is different, from the ideal case, only for one of two inverters interlaced modulating waveform (if the phase shift is identical for both converters, variations in THD_I aren’t present).

0 10 20 30 40 50 60 70 80 90

300 400 500 600 700 800 900 1000 1100 1200

0

300 400 500 600 700 800 900 1000 1100 1200

0

33

1.3 Interaction between inverter and network in the presence of variable

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