Study of the operation of the output filter of a single-phase series active power filter

Mihail Antchev, Vanjo Gourgoulitsov, and Hristo Antchev Section Power Electronics, Faculty of Electronic Engineering and Technologies, Technical University, Sofia, Bulgaria College of Energy and Electronics, Technical University, Sofia, Bulgaria Section Electrical Engineering and Electronics, Department of Metallurgical Technologies, Electrical Engineering and Electronics, Faculty of Metallurgy and Mathematical Science, University of Chemical Technologies and Metallurgy, Sofia, Bulgaria


INTRODUCTION
The shunt and series active power filters are an effective and perspective means for solving the problems with the quality of electrical power [1]- [3]. Most commonly, series active power filters (SAPF) are used to improve the total harmonic distortion (THD) factor of the supply network voltage, used to power sensitive consumers [4]- [11]. In their control systems are used: sliding mode control [12], fuzzy logic control [13], [14] deadbeat control [15], monitoring of filter capacitor current [16], hysteresis control of the filter capacitor voltage or the load voltage [17], [18]. Newer control methods are also used, such as: based on perphase current calculation [19] and neural network control [20], [21]. Particular attention is paid to the accurate synchronization with the source voltage, as an overview of the main methods is done in [22].
The literature study regarding the features of the output filter shows that there are comparatively fewer studies dealing with the design features of the output passive filter. In [23] is analyzed a new seriesparallel-resonant filter as a part of a shunt active power filter. In [24] there is an analysis of the operation of filter in the configuration also of a shunt active power filter. Taking into account the high frequency modulation, the values of the current through the filter capacitor as well as through the DC capacitor are defined.

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There is not a complete study of the operation of the passive filter as part of a SAPF with hysteresis control of the filter capacitor voltage. This would have allowed the presentation of a methodology for its design. Figure 1 shows the block diagram of a SFSAPF. Its purpose is to produce such a voltage of the secondary winding of the transformer Tr, to compensate the higher harmonics of the voltage of the power source AC, so that the load voltage has a sinusoidal shape.
There are two features of the output capacitor work: 1. The voltage on the capacitor contains higher harmonics, which must be in antiphase to their respective ones, contained in the supply network voltage curve. These harmonics determine the only one component of the root mean square (RMS) value of the current through the capacitor. In addition, the current also contains a high-frequency component as a result of hysteresis control. 2. In hysteresis control, the value of the capacitor and the output inductance depend on the hysteresis parameters and the switching frequency of the power devices.
The problem is that there is no comprehensive study of the specified features of the operation of the capacitor of the output filter. Tthere is no comprehensive metology for filter design. The purpose of this article is to present a study of these features, which is lacking in the specialized literature. The novelty in the work is in determining the relationship of the value of the capacitor with the RMS value of the higher harmonic current and voltage ripple. As a result, a new methodology for the design of the output filter has been proposed.
In the second part, a mathematical description is made, and are offered expressions in accordance with the two described features. The results from the computer simulation are presented in the second part. The third part shows the experimental results of the work of the SPSAPF with linear and nonlinear load.

RESEARCH METHOD 2.1. Mathematical description
It should be noted, that the curves in the form of the supply network voltage have different shape and character, so it is necessary to accept a certain approximation of this form. For the purposes of the analysis, connected with the first feature (determining the RMS value of the current through the capacitor), is accepted an approximation of the curve of the supply network voltage with trapezoidal shape, shown in Figure 2. It gets close to a common case in practice-load of the network with single-phase bridge uncontrolled rectifiers, which are non-linear loads and consume current around the maximums of the network voltage. As a result, voltage drops around its maximum value. The decomposition of such a trapezoidal function in Fourier series is known: Therefore, for the THD factor, we obtain: (2) Figure 3 shows the corresponding graphical dependence for the interval 0.3 ≤ φ ≤ 1.0 rad. The sum is limited to = 25 in accordance with [25], [26].  The maximum value of the current through the capacitor of the output filter for each of these harmonics will be: = .
(4) for = 2,3 … … Considering all the harmonics, the RMS value of the current through the filter capacitor will be: The ratio of the RMS value of the current to the capacitor value at the frequency of the supply voltage = 50 is: Figure 4 shows a set of features reflecting the above dependence at parameter ℎ . The sum is limited to = 25 in accordance with [25,26].  (6) Depending on (6) the value of the capacitor is indicated and therefore it is necessary to perform an analysis to determine this value. In connection with the second feature (determining the value of the capacitor at hysteresis control), the time diagrams shown in Figure 5 are used. The examination is made within one switching period = 5 − 1 of the power devices. This period is significantly shorter than the half-period of the network voltage. At moment = 1 the transistors from this diagonal of the bridge power circuit from Figure 1 are switched on, for which the voltage decreases, and at moment = 3 −for which it increases. Since the filter is a second-order unit, after the moment = 1 there is a certain increase in voltage, and after the moment = 3 -a decrease. Provided, that the voltage ripple on the capacitor is neglected, it can be assumed that the current through the inductor − changes in both intervals according to a linear law. At these intervals, voltage with approximately constant value 2 is applied to for the interval − = 3 − 1 , and 1 for the interval = 5 − 3 . For the short duration of the period, the changes of current in both directions are equal, therefore: After simplification, it follows (8): Therefore, if the reference curve for the voltage of the capacitor ( ) = ( ) is known, then the law of change of ( ) is also known.
At hysteresis control, this reference curve is monitored with a definite hysteresis, where the capacitor voltage also contains high-frequency ripples, which are result of the switching with high frequency . Based on these ripples, a way is sought to determine the value of the capacitor . If the start of the coordinate system is translated at the moment 2 , the change of the capacitor voltage in the interval 2 − 3 can be found, ie.one part of the pulsation: The rest of the pulsation can be found when translating the start of the coordinate system at the moment 3 : by summing (11) and (12), the whole pulsation can be found: . |∆ | . determine the RMS value of the higher harmonics. Based on (14), the value of the capacitor is determined by assuming values for ∆ (as a percentage of the maximum value of the load current ) and ∆ (as a percentage of the RMS value of the higher harmonics). 3. The RMS value of the capacitor current at the value of ℎ from item 2. is determined from Figure 4. To this value is added the current with the high switching frequency = 20 . 4. The value of the inductance is determined, so that the resonant frequency of the filter to be higher than the frequency of the highest harmonic being compensated, and lower than the switching frequency . 5. The value of the hysteresis as a percentage of ∆ is determined.

Computer simulation
A study was performed by computer simulation of the operation of the SFSAPF at the thus set initial data and certain values in the above example. This is done with the ORCAD-PSPICE program. Figure 6 and Figure 7 show the results when a linear load (active-inductive) is connected to the power supply network. Figure 8 shows the results for a non-linear load for the power supply network-single-phase uncontrolled bridge rectifier with active-capacitive load.

RESULTS AND DISCUSSION
The results of the computer simulation have been confirmed experimentally. The corresponding oscillograms are shown in Figure 9 and Figure 10. The current in Figure 9 and Figure 10 was monitored with a 100: 1 current probe. The results of the experimental study confirm the results obtained from the computer simulation. Figure 11 shows the oscillograms with substantially nonlinear load-AC regulator with phase control.

CONCLUSIONS
The contribution of this study is: 1) Determining the value of the current of the higher harmonic through the capacitor of the output filter-expression (6) and Figure 4. 2) Determining the voltage ripple on the capacitor depending on its value and switching frequency-expression (14). 3) On this basis offering a methodology for designing the output filter. The results presented in this article make it possible to determine the parameters of the output capacitor as well as the output inductance of a series active power filter with hysteresis control. The results were confirmed by computer simulation and experimental testing and show sufficient for the practical purposes accuracy.