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This paper studies the design and sizing of a filter (
*L-C*) for an inverter with 180
° control in Medium voltage (MV), based on formulas of the capacitance of the capacitor
*C* and the inductance
*L* of the filter (
*L-C*) of an SPWM inverter. These formulas were obtained by minimizing two parameters: the reactive power of the capacitor (capped at 5% of the apparent power of the load) and the ripple of the current flowing through inductance
*L* (capped at 10% of the current supplying the load). The application of these formulas for the calculation of the filter (
*L-C*) of the 180
° control inverter in MV is not conclusive. Studies have been carried out to make them applicable. The results show that limiting the current ripple in the inductor to 10% of the load current is a valid assumption and that limiting the reactive power of the capacitor to 5% of the apparent power of the load presents shortcomings. The results also show that setting the inductance
*L* of the filter to
*L _{maxi }*and the capacitor

*C*from 35 ×

*C*to 400 ×

_{maxi}*C*

*gives voltage and current THDs that meet the 519 IEEE-2014 standards.*

_{maxi }In recent decades, the countries of sub-Saharan Africa have started to build photovoltaic plants with peak powers of around tens of megawatts [

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 [

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 [

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 [

- 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 [

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.

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 [

Filter (L-C)

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

G ( P ) = V s ( P ) V e ( P ) = 1 L C P 2 + 1 (1)

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

Formulas of inductance L and capacitor C

According to articles [

C < 0.05 × S N 3 × ω × U p h 2 and L < 3 × U p h 2 10 × ω × 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).

Consider the diagram of a three-phase inverter below (

In the three-phase two-level inverter (_{i} = (T_{i}; D_{i}) (with i = 1, 2, 3, 4, 5, 6). Each controllable switch consists of:

- 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

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).

The Fourier series decomposition based on V 1 ( θ ) of

V 1 ( θ ) = 4 × E π × ∑ p = 0 ∞ ( 1 + cos ( π 3 × ( 2 p + 1 ) ) 2 p + 1 ) × sin ( θ ( 2 p + 1 ) ) (3)

θ | 0 | π 3 | 2 π 3 | 3 π 3 | 4 π 3 | 5 π 3 2 π |
---|---|---|---|---|---|---|

K_{1} | 1 | 1 | 1 | 0 | 0 | 0 |

K_{2} | 0 | 0 | 1 | 1 | 1 | 0 |

K_{3} | 1 | 0 | 0 | 0 | 1 | 1 |

K_{4} | 0 | 0 | 0 | 1 | 1 | 1 |

K_{5} | 1 | 1 | 0 | 0 | 0 | 1 |

K_{6} | 0 | 1 | 1 | 1 | 0 | 0 |

V 1 ( θ ) | E/3 | (2/3) × E | E/3 | −E/3 | −2/3E | −E/3 |

V 2 ( θ ) | −(2/3) × E | −E/3 | E/3 | (2/3) × E | E/3 | −E/3 |

V 3 ( θ ) | E/3 | −E/3 | −(2/3) × E | −E/3 | E/3 | (2/3) × E |

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 11 e f f = 2 × E π 2 et I 11 e f f = V 11 e f f | Z | = 2 × E π 2 | Z | ; Z = r + j × x (4)

Remember that, without the filter (L-C), the harmonic distortion rate (THD = 31.08%). This does not comply with the IEEE 519 [

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 [

C = C m a x i = 0.05 × S N 3 × ω × U p h 2 and L = L m a x i = 3 × U p h 2 10 × ω × 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.

The load data are as follows:

S_{N} = 25 MVA

cosφ = 0.8

U_{ph} = 15 kV

E = 19,238.25 V

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

The results of _{maxi} and L = L_{maxi} of the three-phase inverter with SPWM control is not suitable for obtaining voltage and current THDs meeting the IEEE 519 standard [

In the scientific literature, the only method of calculating the filter (LC) for an inverter with 180˚ control is found in article [_{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 [

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 q varying from 1 to 1000 and the value of L_{maxi} is kept unchanged.

From

_{maxi}, the THD% obtained comply with the IEEE 519 standard [_{maxi}. Between C = 35 × C_{maxi} and C = 100 × C_{maxi}, the THD% obtained in _{maxi}; 100 × C_{maxi}] because of the resonance phenomenon observed in

Another study is carried out by varying L_{maxi}. The inductance L_{maxi} is multiplied by q which varies from 0.01 to 25 while keeping the value of C = C_{maxi} unchanged.

From

_{maxi}. With this in mind, taking the value of the capacity C in the range of 35 × C_{maxi} to 100 × C_{maxi} would not be a bad choice for the inverter with 180˚ control in MV, if the resonance phenomenon is avoided.

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

C = 35 × C 0 ~ 100 × C 0 with C 0 = 0.05 × S N 3 × ω × U p h 2

and L = L m a x i with L m a x i = 3 × U p h 2 10 × ω × S N (6)

Voltage THD% | 114.36 | 68.09 | 93.95 | 53.48 | 21.77 | 13.45 | 9.62 | 7.42 | 5.98 |
---|---|---|---|---|---|---|---|---|---|

Current THD% | 12.88 | 17 | 30.02 | 17.15 | 6.93 | 4.26 | 3.04 | 2.34 | 1.88 |

RMS VOLTAGE (V) | 12,510 | 12,720 | 12,990 | 13,270 | 13,560 | 13,860 | 14,180 | 14,520 | 14,870 |

RMS CURRENT (A) | 802.4 | 815.7 | 833 | 851 | 869.8 | 889.4 | 909.8 | 931.3 | 953.7 |

C = q × C_{maxi}; q | 1 | 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 |

Voltage THD% | 4.97 | 4.23 | 2.25 | 1.39 | 0.27 | 0.24 | 0.38 | 0.47 | 0.58 | 0.66 |
---|---|---|---|---|---|---|---|---|---|---|

Current THD% | 1.56 | 1.33 | 0.7 | 0.43 | 0.08 | 0.07 | 0.12 | 0.15 | 0.18 | 0.21 |

RMS VOLTAGE (V) | 15,230 | 15,620 | 17,860 | 20,800 | 49,960 | 37,480 | 17,560 | 11,180 | 6432 | 3919 |

RMS CURRENT (A) | 977.2 | 1002 | 1146 | 1334 | 3205 | 2404 | 1126 | 717.2 | 412.6 | 251.4 |

C = q × C_{maxi}; q | 45 | 50 | 75 | 100 | 200 | 300 | 400 | 500 | 700 | 1000 |

Voltage THD% | 57.17 | 79.62 | 64.95 | 125.19 | 80.95 | 57.91 | 49.72 | 51.83 |
---|---|---|---|---|---|---|---|---|

Current THD% | 7.52 | 7.62 | 7.62 | 8.39 | 7.89 | 7.66 | 7.66 | 7.76 |

RMS VOLTAGE (V) | 14,970 | 14,930 | 14,870 | 14,800 | 14,760 | 14,740 | 14,480 | 14,220 |

RMS CURRENT (A) | 960.5 | 958 | 953.8 | 949.6 | 947.1 | 945.4 | 928.7 | 912.1 |

L = q × L_{maxi}; q | 0.01 | 0.025 | 0.05 | 0.075 | 0.09 | 0.1 | 0.2 | 0.3 |

Voltage THD% | 95.4 | 73.71 | 45.79 | 76.96 | 120.3 | 64.6 | 114.36 |
---|---|---|---|---|---|---|---|

Current THD% | 9.5 | 8.75 | 7.75 | 9.48 | 12.48 | 8.91 | 12.88 |

RMS VOLTAGE (V) | 13,960 | 13,710 | 13,460 | 13,220 | 12,980 | 12,740 | 12,510 |

RMS CURRENT (A) | 835.7 | 879.5 | 863.5 | 847.8 | 832.4 | 817.2 | 802.4 |

L = q × L_{maxi}; q | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1 |

Voltage THD% | 45.63 | 86.9 | 49.74 | 48.94 | 59.66 | 47.67 | 39.34 |
---|---|---|---|---|---|---|---|

Current THD% | 8.17 | 12.51 | 8.72 | 8.85 | 10.36 | 8.89 | 7.96 |

RMS VOLTAGE (V) | 10,470 | 8896 | 7684 | 6739 | 4110 | 2937 | 1862 |

RMS CURRENT (A) | 671.6 | 570.6 | 492.9 | 432.3 | 263.7 | 188.4 | 119.6 |

L = q × L_{maxi}; q | 2 | 3 | 4 | 5 | 10 | 15 | 25 |

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:

Viewing Figures 11-14, confirms that to size a filter (L-C) for an inverter with 180˚ control in MV, articles [

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 L of the filter to L_{maxi} and the capacitor C from 35 × C_{maxi} to 400 × C_{maxi} gives voltage and current THDs that meet the IEEE 519 standard. The simulations on the MATLAB/Simulink software made it possible to justify this method. Also for the range, 100 × C_{0} to 400 × C_{0}, the sizing of a resonance damping resistor is to be expected.

The authors declare no conflicts of interest regarding the publication of this paper.

Koffi, K.F., Yake, G., Gbegbe, R., Loum, G. and Asseu, O. (2021) Sizing of a Filter (L-C) for a 180˚ Control Inverter Connected to a Medium Voltage Network. Open Journal of Applied Sciences, 11, 565-576. https://doi.org/10.4236/ojapps.2021.115040

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