of the Maritime University of Szczecin
Akademii Morskiej w Szczecinie
2019, 57 (129), 79–87
ISSN 1733-8670 (Printed) Received: 24.01.2019
ISSN 2392-0378 (Online) Accepted: 20.02.2019
DOI: 10.17402/329 Published: 22.03.2019
Predictive regulation of the output voltage of a three-phase
two-level voltage inverter with an LC filter
Zbigniew Behrendt
Gdynia Maritime University, Faculty of Electrical Engineering, Institute of Ship Automation 81-87 Morska St., 81-225 Gdynia, Poland,
e-mail: z.behrendt@we.umg.edu.pl
Key words: uninterruptible power supply (UPS), model predictive control (MPC), predictive voltage
regula-tor, load current prediction in the time domain, load current prediction in the frequency domain, total harmonic distortion (THD)
Abstract
The article has presented a predictive output voltage regulator of a three-phase two-level voltage inverter with an LC filter with a prediction of the receiver current in the time domain and the frequency domain. Simulations of the operation of the regulator were conducted for a three-phase receiver. The circuit model and simulations of the operation of the predictive voltage regulator were performed in the PSIM program. The algorithm of the predictive regulator was written in the C programming language in the Microsoft Vi-sual Studio compiler and attached to the model performed in the PSIM program by a block that supports the Dynamic-Link Library.
Introduction
The continuous growth of the expectations that are set for voltage converters has contributed to the development of the control methods for these con-verters. Scientific research on the applications of voltage regulators is being continually conducted, in fields such as electricity distribution and energy stor-age systems (Vasquez, 2010), as well as for power systems based on renewable energy sources (Exposto et al., 2015), for parallel power supply compensation systems in distribution networks (Wojciechowski, 2013), for uninterruptible UPS systems (Cortés et al., 2009; 2010) and others (Łebkowski, 2017). Due to the increase in the processing power of signal pro-cessors it is now possible to use MPC (model predic-tive control) regulators to determine a given output voltage in three-phase voltage inverters using an LC filter.
The subject matter of the control that uses MPC regulators has been the subject of scientific studies all over the world and in Poland as well (Vasquez
et al., 2009; 2011; Rodriguez et al., 2013; Falkow-ski, 2016). The performance of the predictive con-trol method was compared with both hysteresis and PWM control for linear and nonlinear receivers in the literature (Mohamed et al., 2013). The results showed that the classical methods failed to control the output voltage for nonlinear receivers, however good results were achieved with the predictive con-trol method. The predictive voltage regulation algo-rithm determines the set voltage for each sampling period which is generated in the next sampling peri-od using the SVPWM (Space Vector PWM) mperi-odula- modula-tor. In order to calculate the set voltage of the invert-er with an LC filtinvert-er, it is necessary to predict the current of the receiver. Studies (Cortés et al., 2009) (Mohamed et al., 2013) have estimated the receiver currant and assumed that it does not change in a short period of time: this is a simplification. This paper has analyzed the operation of the predictive output voltage regulator of a three-phase two-level voltage inverter, using the prediction of the receiver current in the time domain and the frequency domain.
Diagram of the simulated system
The model of the system that was used in the sim-ulations has been shown in Figure 1. A description of the inverter, LC filter, non-linear receiver and invert-er opinvert-eration control algorithm has been presented further on in this paper.
Voltage inverter model
A three-phase two-level voltage inverter was used for the purpose of this study. It is common knowledge that six combinations of the state func-tions of an inverter switch can be distinguished, cor-responding to six active vectors of the generated out-put voltage U with the absolute value of 2/3·Udc, and
two combinations corresponding to the zero vector of the output voltage. The voltage vector U can be described using equations (1) and (2):
3 for ,16 π 1 k e U U jk (1) 8 or 7 for 0 k k U (2)
The set of all the combined voltage vectors on the
α-β plane has been presented in Figure 2.
The selection of the proper output vector U and the time of its occurrence depends on the strategy for the inverter output voltage formation in the SVPWM modulator.
LC filter model
The application of the LC filter to the output of the voltage inverter allows a sinusoidal output voltage to be produced across the capacitor and filters the high-er harmonics of the invhigh-erthigh-er’s output voltage that is connected with the frequency of the transistor con-nections. The selection of the LC filter parameters that are used to form the sinusoidal output voltage of the voltage inverters has been presented, inter alia, in the literature (Kim, Kwon & Choi, 2007; Porada, 2015). The general forms of the state equation and output equation that were adopted in the studies of the system have been presented in equations (3) and (4): Bu Ax x dt d (3) y = C x (4)
where: x is a vector of the time variables, u is the input vector, y is the output vector, A is the state vari-able matrix and C is the output matrix.
On the basis of the above equations, the state equations for the choke current (5) and the capacitor voltage (6) can be introduced:
u u L L i i L L i i dt d L L L L / 1 / 1 / 1 0 / 1 0 (5) o o o c c c c i i i C C u u C C u u dt d / 1 / 1 0 / 1 0 / 1 (6) Three-phase two-level
voltage inverter LC filter
Load
SVPWM
Figure 1. Model of the simulated system
Figure 3 shows an ideal voltage inverter model with an output LC filter in the operator domain in αβ coordinates. The input is the inverter’s input voltage
u, the output is the voltage uc across the capacitor.
The receiver current io is distorted.
Figure 3. Simplified voltage inverter model with an output LC filter in the operator domain in αβ coordinates
From the transformation of equations (5) and (6) into the operator form, the transmittance equations (7) and (8) can be derived for an ideal LC filter as well as the transmittance (9), taking into account the losses in the choke and the resistance load.
2 2 2 o o c c s u u s u u u u G c (7) 2 2 1 o c c s i u s C s u u u u G o (8) 2 2 2 2 oo o s i u s s k G o (9) where: , , . 2 2 0 R0 0 R R k C L R R
The resonant pulsation of the adopted LC filter is described by the equation ωr = (LC)−1/2. The choke’s
inductance and capacitor’s capacitance values are selected in such a manner that the basic harmonic of the output voltage f1 = 50 Hz is lower than the resonant frequency fr of the filter and the pulsation
connected with the frequency of the transistor’s con-nections fimp is higher than fr. There is an indefinite
number of combinations of the choke’s inductance
L and the capacitor’s capacitance C, thus allowing
the assumed resonant frequency to be found. In the model that was adopted for this study and using the 50 kVA receiver, an induction value of L = 400 µH and a capacitance of C = 100 µF were selected.
Receiver models
The simulations were carried out for the three-phase receivers and have been shown in Figure 4. According to the results in the literature (Akagi, Watanabe & Aredes, 2017), the assessment of the voltage susceptibility of the receivers allows for the current-type and voltage-type of the receivers to be distinguished. An example of a voltage receiver that can be used is a three-phase full-wave rectifier with a capacitor in a direct current circuit. An exam-ple of a current receiver is a three-phase full-wave rectifier with a choke in a direct current circuit. For current-type receivers and voltage-type receivers connected to the voltage inverter output with an LC filter, network chokes with an inductance of Lac were
applied. The current-type receivers presented in Fig-ures 4a and 4b can be a source of both current and voltage harmonics.
Figure 4. a) Current-type receiver, b) Voltage-type receiver, c) Resistance receiver
Measurements of electrical quantities
In each sampling period, the capacitor voltage uc,
as well as the choke iL, capacitance ic and receiver
currents io are measured. The measured quantities
are transformed into the orthogonal system α-β using the Clarke transformation with power invariance, according to equation (10).
T C B A T T C B A T u u u M u u u i i i M i i i 0 0 and (10)where: 2 2 2 2 2 2 2 3 2 3 0 2 1 2 1 1 3 2 M
The transformations of the measured currents and voltages according to equation (10) ensure that all the distortions contained in the measured electrical quantities are maintained and that the zero sequence can be separated.
Timings of the digital implementation of the control system with a predictive voltage regulator
Figure 5 has presented the digital implemen-tation of the control system. The control system algorithm is calculated for every time interval Tsampl which, with a constant frequency of the sampling of the measured currents and voltages that are equal to 16 kHz, is 0.0625 ms. The applied pulse generator operates with the period Timp = 2Tsampl. The assumed time organisation is connected with a time delay between the couplings and the control.
Predictive voltage regulator
For the time dependences of the digital imple-mentation of the control system presented in Figure 5, as well as in accordance with equations (5) and (6), the dependences of the digital predictive voltage regulator, for which there is a need for the predic-tion of the receiver current io(k+1), were derived. The
predictive voltage regulator has been presented in the literature (Wojciechowski, 2013). Figure 6 has shown the implementation of the predictive output voltage regulation of the inverter with an LC filter with receiver current prediction in the frequency domain. The operation of the predictive output volt-age regulation for the receiver current prediction, implemented in the time domain and in the frequen-cy domain, was tested.
Receiver current prediction in the time domain
By assuming the periodicity of the receiver cur-rent, it is possible to realise its prediction in the time domain by applying a cyclic buffer in accordance with the method in the literature (Wojciechowski, 2013). The length of the buffer l results from the
Figure 5. Timings of the digital implementation of the control system
Figure 6. Block diagram of the predictive output voltage regulation of the inverter with an LC filter with receiver current pre-diction in the frequency domain
quotient of the period of the receiver current T and the sampling period Tsampl. This algorithm ensures the prediction and maintains the distortions con-tained in the current samples io(k−1) that are applied to
the predictor input.
Figure 7. Cyclic buffer that realises the prediction of the receiver current
Receiver current prediction in the frequency domain
The prediction algorithm of the receiver cur-rent determined in the frequency domain relies on the computing modules of all the harmonics of the measured current io(k−1), and modifying them with an
increment of the angle Δφn,i,pred which is expressed in
the equation: 1 pred , , ni n r (11)
where: r – prediction horizon, n – harmonic number, Δφ1 – increase of the angle of the basic harmonic of the predicted receiver current.
The real and imaginary components of the receiv-er current io(k−1) are calculated, and then the
ampli-tudes of the particular harmonics Io,αβ,n(k–1) of the
measured receiver current can be calculated from:
N n t jn k n o k n o N i e I 0 , , 1 1 , , 1 1 (12)where: N – number of measured samples per one output voltage period.
The real and imaginary components of the har-monic amplitudes of the complex receiver current are then calculated:
1
1
, 1
1
, 1 , , sin cos r n i r n i I k o k o k n o (13) 1
1
, 1
1
, 1 , , sin cos r n i r n i I k o k o k n o (14) The calculated harmonic amplitudes are modified by the angle Δφn,i,pred, according to equation (11).In order to calculate the receiver current io(k+1), the
Inverse Fourier Transforms (IFT) of the amplitudes
Io,α,n(k+1) and Io,β,n(k+1) are calculated, as well as their
superposition. 0.2 0.22 0.24 0.26Time (s) 400 200 0 –200 –400 100 50 0 –50 –100
0.34 0.36 0.38 0.4Time (s) 400 200 0 –200 –400 200 100 0 –100 –200
Figure 9. Output voltage uc waveform and receiver current io in a fixed state with receiver current prediction in the frequency
domain with a voltage receiver (R = 5 Ω and C = 165 μF)
0.34 0.36 0.38 0.4Time (s) 500 0 –500 200 100 0 –100 –200
Figure 10. Output voltage uc waveform and receiver current io in a fixed state with receiver current prediction in the time
Results of the conducted simulations
The simulations were performed in the PLECS program, for the three-phase resistive receiver, for a current-type and a voltage-type receiver. The mod-el of the circuit that was used to perform the simula-tions was designed for a 50 kVA receiver. The output voltage frequency was 50 Hz and the effective value was 230 V. Figure 8 has shown the waveforms of the output voltage and receiver current after switching on the full resistance receiver R = 5 Ω at a time of 0.22 s. The predictive voltage regulator with receiv-er current prediction was applied in the time domain. The total harmonic distortion of the ten output voltage periods in the fixed state with a resistance receiver of R = 5 Ω was THDu = 0.4%. With the application of this regulation with the receiver cur-rent prediction in the frequency domain, the value of THDu was 0.5%. Figure 9 has shown the waveforms of the output voltage and receiver current in the fixed state, with a connected voltage-type receiver with values of R = 5 Ω and C = 165 μF. A predictive voltage regulator with receiver current prediction in the frequency domain was applied; in this case the THDu was 1.75%.
The waveform of the output voltage and receiv-er current in the fixed state with a connected volt-age-type receiver and with the application of the predictive output voltage regulation with receiver current prediction in the time domain has been pre-sented in Figure 10. For the applied regulation algo-rithm and voltage-type receiver, a loss in the output voltage regulation stability was observed.
Figures 11 and 12 have shown the waveforms of the output voltage and receiver current in a fixed state, with the connected current-type receiver. The THDu in the fixed state was 1.4% for the receiver current prediction in the frequency domain, whereas for the prediction in the time domain it was 0.55%. Conclusions
The circuit model and simulations of the oper-ation of the predictive voltage regulator have been performed in the PSIM program in this paper. The algorithm of the predictive regulator was written in the C programming language in the Microsoft Visual Studio compiler and attached to the model by a block that supports the Dynamic-Link Library (DLL). The predictive output voltage regulator of the three-phase
0.34 0.36 0.38 0.4Time (s) 400 200 0 –200 –400 200 100 0 –100 –200
Figure 11. Output voltage uc waveform and receiver current io in a fixed state with receiver current prediction in the frequency
two-level voltage inverter with an LC filter that has been tested can be applied to shape the output volt-age. In the applied model, the task of the regulator was the formation of the sinusoidal phase voltage with frequency of 50 Hz and an effective value of 230 V across the LC filter capacitors, regardless of the type of connected receiver. For a voltage-type receiver and the application of the receiver current prediction in the time domain, the loss of the sta-bility of the output voltage regulation system was observed in all phases. For the same receiver and with the application of the receiver current predic-tion in the frequency domain, the voltage waveforms were distorted, but retained their sinusoidal shape. For a current-type receiver, the waveforms had a lower THDu value than the waveforms with a volt-age-type receiver for both types of receiver current predictions. After switching the resistance receiver on, the waveforms of the output voltage were quick-ly fixed and were characterised by the lowest THDu value. Further improvement of the algorithm of the predictive regulation of the inverter’s output voltage with receiver current prediction is being planned to obtain output voltage waveforms with a lower val-ue of total harmonic distortion, for voltage-type and current-type receivers in particular.
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