Sizing of a Filter (L-C) for a 180° Control Inverter Connected to a Medium Voltage Network

Kouakou Fernand Koffi^{1},
Gondo Yake^{2},
Raymond Gbegbe^{1},
Georges Loum^{1},
Olivier Asseu^{1,3}^{*}

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1. Introduction

In recent decades, the countries of sub-Saharan Africa have started to build photovoltaic plants with peak powers of around tens of megawatts [1]. These photovoltaic plants are built to reinforce the energy demand of these countries which does not cease growing.

For the transmission and distribution in MV in localities far from the national grid, of the energy produced in these photovoltaic plants, the use of three-phase voltage inverters is necessary. It is with this in mind that we propose to design and size a filter (*L*-*C*) for a 180˚ control inverter. This is because inverters do not provide sinusoidal voltage and current signals [2]. These signals contain harmonics generated by semiconductors. These harmonics hamper the proper functioning of electrical equipment [3].

The measurement parameter of these harmonics is the THD. It characterizes the quality of the voltage and current signals, and therefore the quality of the power. The lower the THD, the better the quality of the signal power, and the signal becomes closer to sinusoidal shape [3].

In order to obtain better filter efficiency (*L*-*C*) for the operation of an inverter with 180˚ control in MV, the main contribution proposed in this research article is:

- on the one hand, to discuss the applicability of existing formulas in newspapers [4];

- on the other hand, to improve them from studies so that they are applicable to the sizing of the filter (*L*-*C*) of the 180˚ control inverter connected to an MV network [5].

The document is structured as follows: section II presents the system model and the formulation of the problem; the filter (*L*-*C*) and its calculation formulas for the three-phase inverter with SPWM control are studied; in section III, the 180˚ control inverter is analysed; section IV presents the applicability of the filter formulas (*L*-*C*) of the three-phase inverter SPWM to the inverter with 180˚ control; in section V, studies to improve the parameters of the filter (*L*-*C*) for an inverter with control 180˚. Finally, section VI concludes the article.

2. System Model and Problem Formulation

The model is based on a medium voltage direct current (MVDC) electrical power transmission system. This is to improve the quality of the voltage and current signals on the AC side of the 180˚ control inverter connected to an MV network.

The scientific literature has been used to the best of our knowledge. No method of calculating a filter (*L*-*C*) for a 180˚ control inverter operating in MV. Then a method of calculating a filter (*L*-*C*) of the 180˚ control inverter operating in MV based on that of the filter (*LC*) of the SPWM control inverter is adopted, after verifying the applicability of the formulas of articles [4] [5] [6]. This is to reformulate the filter calculation method (*L*-*C*) of articles so that it can be used for a 180˚ control inverter connected to the MV network.

Filter (*L*-*C*)

Consider the diagram of a single-phase circuit of a filter (*L*-*C*) given in Figure 1.

Figure 1. Single phase filter circuit (*L*-*C*).

Figure 1 makes it possible to obtain the transfer function *G*(*P*):

$G\left(P\right)=\frac{{V}_{s}\left(P\right)}{{V}_{e}\left(P\right)}=\frac{1}{LC{P}^{2}+1}$ (1)

This filter makes an attenuation of 40 (dB)/decade [6].

Formulas of inductance *L* and capacitor *C*

According to articles [4] [5] [6], the calculation of *L* and *C* of the filter (*L*-*C*) for a three-phase inverter with SPWM control is done as follows:

$C<\frac{0.05\times {S}_{N}}{3\times \omega \times {U}_{ph}^{2}}$ and $L<\frac{3\times {U}_{ph}^{2}}{10\times \omega \times {S}_{N}}$ (2)

These formulas given by Equation (2) were obtained by minimizing two parameters: the reactive power of capacitor *C* (capped at 5% of the apparent power of the alternating load) and the ripple of the current flowing through inductor *L* (capped at 10% of the phase-to-neutral voltage supplying the alternating charge).

3. 180˚ Control Inverter (Full Wave)

Consider the diagram of a three-phase inverter below (Figure 2).

In the three-phase two-level inverter (Figure 2), there are three arms. Each arm has two controllable switches *K _{i}* = (

- Bipolar power transistor or IGBT or thyristor (controllable component);

- Diode mounted head-to-tail (antiparallel) on each controllable component.

The inverter is supplied by a DC source E.

In 180˚ control, each switch *K _{i}* of Figure 2 conducts for 180˚ (π radians).

Two switches of the same arm have their control shifted by 180˚ (π radians).

Two consecutive switches have their control shifted by 120˚ (2π/3 radians).

Table 1 gives a summary of the operation of the 180˚ control inverter and Figure 3 represents the voltage signals ${V}_{1}\left(\theta \right)$, ${V}_{2}\left(\theta \right)$ and ${V}_{3}\left(\theta \right)$.

Figure 3 represents the voltage signals ${V}_{1}\left(\theta \right)$, ${V}_{2}\left(\theta \right)$ and ${V}_{3}\left(\theta \right)$.

The Fourier series decomposition based on ${V}_{1}\left(\theta \right)$ of Figure 3, gives us:

${V}_{1}\left(\theta \right)=\frac{4\times E}{\pi}\times {\displaystyle {\sum}_{p=0}^{\infty}\left(\frac{1+\mathrm{cos}\left(\frac{\pi}{3}\times \left(2p+1\right)\right)}{2p+1}\right)\times \mathrm{sin}\left(\theta \left(2p+1\right)\right)}$ (3)

Figure 2. Three-phase inverter.

Figure 3. Voltage signals ${V}_{1}\left(\theta \right)$, ${V}_{2}\left(\theta \right)$ and ${V}_{3}\left(\theta \right)$.

Table 1. Summary of the operation of the 180˚ control inverter.

By replacing the values of *p* in Equation (3), we can clearly see that the harmonics which pollute the phase-to-neutral voltage signals are of ranks: 5; 7; 11; 13; 17; 19 … 25.

The effective fundamentals of voltage and current are:

${V}_{11eff}=\frac{2\times E}{\pi \sqrt{2}}$ et ${I}_{11eff}=\frac{{V}_{11eff}}{\left|Z\right|}=\frac{2\times E}{\pi \sqrt{2}\left|Z\right|}$ ; $Z=r+j\times x$ (4)

Remember that, without the filter (*L*-*C*), the harmonic distortion rate (THD = 31.08%). This does not comply with the IEEE 519 [7] standard, which requires that in MV, this THD is 8%.

4. Applicability of the Filter Formulas (*L*-*C*) of the Three-Phase Inverter SPWM to the Inverter with 180˚ Control

This is to use the formulas for calculating the inductance *L* and capacitance *C* of the filter capacitor (*LC*) of the SPWM control inverter to apply to the filter (*LC*) of the 180˚ control inverter. The formulas in articles [4] [5] [6] are used to designate the inductance *L* and the capacitor *C*:

$C={C}_{maxi}=\frac{0.05\times {S}_{N}}{3\times \omega \times {U}_{ph}^{2}}$ and $L={L}_{maxi}=\frac{3\times {U}_{ph}^{2}}{10\times \omega \times {S}_{N}}$ (5)

The formulas of Equation (5) were applied for the calculation of *L* and *C* of the 180˚ control inverter filter in MV. The following curves show the quality of the voltage and current signals.

Figure 4 shows the voltage and current waveforms and their different THDs:

The load data are as follows:

*S _{N}* = 25 MVA

cos*φ* = 0.8

*U _{ph}* = 15 kV

*E* = 19,238.25 V

*C* = 5.895 microfarads and *L* = 8.594e-3 henrys

The results of Figure 4 clearly show that for *C* = *C _{maxi}* and

5. Studies to Improve the Filter Parameters (*L*-*C*) for a 180˚ Control Inverter

In the scientific literature, the only method of calculating the filter (*LC*) for an inverter with 180˚ control is found in article [8], where it is a question of calculating a resistance *R* related to the quality factor *Q*, after having set the resonant frequency *f*_{Res} and the capacitor *C* and deduce the inductance *L*.

The SPWM inverter filter (*L*-*C*) formula verification approach applied to the 180˚ control inverter filter (*L*-*C*) has been completed. The results are inconclusive, ie the IEEE 519 [7] standard is not met.

Figure 4. Voltage and current signals for *C* = *C _{maxi}*

For this reason, a first study is carried out in order to determine the choice ranges of the capacitor *C* and the inductance *L* of the filter (*L*-*C*) of the 180˚ control inverter. In this study, *C _{maxi}* is varied by multiplying it by

From Table 2, Figures 5-7 are shown below.

Table 2 shows that from *C* = 35 × *C _{maxi}*, the THD% obtained comply with the IEEE 519 standard [7]. But for Figure 6 and Figure 7, a resonance is observed beyond

Another study is carried out by varying *L _{maxi}*. The inductance

From Table 3, Figures 8-10 are shown below.

Figure 8 clearly shows that the THD% obtained does not comply with the IEEE 519 standard [7]. Figure 9 and Figure 10 show a continuous drop in RMS voltage and RMS current. This is due to the increased impedance of the inductance. It is therefore preferable to keep *L* = *L _{maxi}*. With this in mind, taking the value of the capacity

The filter elements (*L*-*C*) of the 180˚ MV control inverter could be:

$C=35\times {C}_{0}~100\times {C}_{0}$ with ${C}_{0}=\frac{0.05\times {S}_{N}}{3\times \omega \times {U}_{ph}^{2}}$

and $L={L}_{maxi}$ with ${L}_{maxi}=\frac{3\times {U}_{ph}^{2}}{10\times \omega \times {S}_{N}}$ (6)

Table 2. THD% and RMS voltage and current values

Table 3. THD% and RMS voltage and current values.

Figure 5. THD of voltage and current as a function of *q*× *C _{maxi}*;
$q\in \left[1;1000\right]$.

Figure 6. RMS voltage as a function of *q*× *C _{maxi}*;
$q\in \left[1;1000\right]$.

Figure 7. RMS current as a function of*q*× *C _{maxi}*;
$q\in \left[1;1000\right]$.

Figure 8. THD of voltage and current as a function of *q*× *L _{maxi}*;
$q\in \left[0.01;25\right]$.

If a damping resistor is not used in series with inductor *L*.

If a damping resistor is used in series with inductance *L*, the tuning range of capacitor *C* could reach 400 × *C*_{0} because between 100 × *C*_{0} and 400 × *C*_{0}, there is resonance.

Figures 11-14 show three operating cases with the different values of *L* and *C*:

Figure 9. RMS voltage as a function of *q* × *L _{maxi}*;
$q\in \left[0.01;25\right]$.

Figure 10. RMS current as a function of *q* × *L _{maxi}*;
$q\in \left[0.01;25\right]$.

Figure 11. Voltage and current waveforms for *L* =*L _{maxi} *and

Figure 12. Voltage and current waveforms for *L* = *L _{maxi}* and

Figure 13. Voltage and current waveforms for *L* = *L _{maxi}* and

Figure 14. Voltage and current waveforms for *L* = *L _{maxi}* and

Viewing Figures 11-14, confirms that to size a filter (*L*-*C*) for an inverter with 180˚ control in MV, articles [4] [5] [6] dealing with the case of the inverter SPWM too reduced the proportion of reactive power allowing the calculation of the capacitance of the capacitor C. On the other hand, the proportion of the inductance L suits that of our study. Figures 11-14 show that the study carried out makes it possible to calculate a filter (*L*-*C*) for a 180˚ control inverter which meets the standard of the article [7].

6. Conclusions

In this article, a method was adopted: that of using the formulas of a filter (*L*-*C*) of a three-phase inverter with SPWM control obtained by minimizing the reactive power of the capacitor *C* and the current ripple in the inductor. *L*, that is to say, limits the reactive power of the capacitor to 5% of the apparent power of the load and limits the ripple of the current flowing through the inductor *L* to 10% of the current in the load.

The formulas applied to the filter (*L*-*C*) of the 180˚ MV control inverter did not allow the voltage and current THDs to be obtained which comply with the IEEE 519 standard.

Therefore, from these formulas, studies have been carried out with a view to obtaining a good compromise, where the voltage and current THDs comply with the IEEE 519 standard. From these studies, it emerges that taking the inductance *L* = *L _{maxi}* is a valid assumption and that the limitation of the reactive power of the capacitor to 5% of the apparent power of the load has shortcomings. The results also show that setting the inductance

Nomenclature

SPWM Sinusoidal Pulse-Width-Modulation

THD Total Harmonic distortion

*S _{N}* apparent power of the alternating load

MV Medium voltage alternating voltage (1 kV - 50 kV)

*U _{ph}* phase-to-phase voltage at the ac load

MVDC medium voltage direct current

*R*, *L*, *C* Resistance, Inductance, Capacitor

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