Abstract: This paper presents results of research of modulation processes in power conversion system with triple voltage source inverters, supplied by photovoltaic (PV) strings, and connected to a four-winding transformer. The transformer has in this case specific connections between the secondary windings and inverters, allowing providing of multilevel voltages at secondary windings of the transformer. Specialized space-vector-based strategy of pulsewidth modulation (PWM) has been used for control of inverters, insuring quarter-wave symmetry of winding voltages for any modulation indices and switching frequency of inverters. Analysis and comparison of harmonic composition of winding voltages and output voltages of inverters has been done. The validity of the analyzed processes has been verified by simulations: three basic variants of the scheme of modulation have been applied for control of inverters

Cernat, M.

Oleschuk, V.

Pastor, M.

Sanjeevikumar, P.

English

University of Johannesburg Institutional Repository

Synchronous PWM Control of Triple Transformer-Connected Inverters for Photovoltaic System  V. Oleschuk1, M. Cernat2, M. Pastor3, P. Sanjeevikumar4 1Academy of Sciences of Moldova, Chisinau, Republic of Moldova, 2Transilvania University, Brasov, Romania, 3Technical University of Kosice, Slovak Republic, 4University of Johannesburg, Auckland Park, South Africa E-mail: oleschukv@hotmail.com, m.cernat@unitbv.ro, marek.pastor@tuke.sk, sanjeevi_12@yahoo.co.in  Abstract-This paper presents results of research of modulation processes in power conversion system with triple voltage source inverters, supplied by photovoltaic (PV) strings, and connected to a four-winding transformer. The transformer has in this case specific connections between the secondary windings and inverters, allowing providing of multilevel voltages at secondary windings of the transformer. Specialized space-vector-based strategy of pulsewidth modulation (PWM) has been used for control of inverters, insuring quarter-wave symmetry of winding voltages for any modulation indices and switching frequency of inverters. Analysis and comparison of harmonic composition of winding voltages and output voltages of inverters has been done. The validity of the analyzed processes has been verified by simulations: three basic variants of the scheme of modulation have been applied for control of inverters.  I. INTRODUCTION Development of photovoltaic-focused power conversion systems is growing rapidly. Novel transformer-less and transformer-based topologies and configurations of solar photovoltaic installations for solar farms and solar factories have been proposed and investigated [1]-[4].  Recently, modified structure of transformer-based system with triple three-phase voltage source inverters has been proposed [5],[6]. Fig. 1 shows topology of this system for the case of delta-connection of secondary windings, supplied by three insulated strings of photovoltaic panels. Fig. 2 presents the corresponding secondary windings for the case of star-connection [6]. It has been shown in [6], that simple modification of connection between outputs of standard two-level inverters and windings of the transformer allow providing multilevel winding voltages for systems with insulated dc-sources. Also, winding voltages can be expressed in this case by the phase voltages of triple inverters.  Operation of inverter-based photovoltaic systems is in dependence of the used methods and techniques of PWM. Space-vector modulation is perspective method for control of inverters, including PV application. During system operation, for some regimes of photovoltaic systems (grid frequency fluctuation, different voltages of strings of PV panels, etc.) it is necessary to ensure voltage synchronization. So, this paper presents results of research of modulation processes in PV system with triple transformer-connected inverters, controlled by specialized schemes and techniques of space-vector PWM, insuring voltage synchronization and voltage symmetries for any operating conditions of the system.    Fig. 1.  Topology of transformer-based photovoltaic system with triple inverters specifically connected to secondary windings of the transformer (case of delta-connection of windings).    Fig. 2.  Secondary windings of transformer of the system for the case of star-connection.   II. TWO-LEVEL INVERTERS WITH SYNCHRONOUS PWM  To assure voltage synchronization and waveform symmetries of inverters, specialized method of space-vector modulation can be used for control of triple inverters of PV system. Table I presents peculiarities and the main control dependences of this method of synchronous modulation, which are compared here with the corresponding parameters of conventional space-vector PWM [7]-[9].  978-1-5090-1797-3/16/$31.00 ©2016 IEEE 215TABLE I. BASIC PARAMETERS OF PWM METHODS  III. SYNCHRONOUS PWM OF INVERTERS IN TRIPLE-INVERTER-BASED SYSTEM  Phase voltages Vas1, Vbs1 and Vcs1 of the first standard three-phase inverter (Inverter 1 in Fig. 1) of the system are calculated in accordance with (1)-(3) [7]:   Vas1 = Va10 + (Va10 + Vb10  + Vc10)/3                 (1)  Vbs1 = Vb10 + (Va10 + Vb10  + Vc10)/3                 (2)  Vcs1 = Vc10 + (Va10 + Vb10  + Vc10)/3,                (3)  where Va10, Vb10 and Vc10 are the corresponding pole voltages of the first inverter.  Winding voltages of secondary windings of the system (Vw1, Vw2, Vw3 in Fig. 1) can be expressed by the phase voltages of triple inverters in accordance with (4)-(9) [6]. Dependences (4)-(6) correspond to the case of delta-connection of windings, presented in Fig. 1, dependences (7)-(9) correspond to the case of star-connection, presented in Fig. 2.   Vw1 = Vas3 - Vbs1                                     (4)  Vw2 = Vbs1 - Vcs2                                (5)  Vw3 = Vcs2 - Vas3                                (6) Vw1 = 0.667Vas1  – 0.333Vbs2 – 0.333Vcs3                (7)  Vw2 = -0.333Vas1 + 0.667Vbs2 – 0.333Vcs3               (8)  Vw3 = -0.333Vas1 – 0.333Vbs2 + 0.667Vcs3               (9)  Control and output signals of triple inverters of the system are shifted by 1200 (1200 interleaving in accordance with some definitions). Also, small additional shift between control signals of three inverters, equal to 1/3 of duration of the switching period (switching sub-cycle) has been provided by the used scheme of control and modulation.  Fig. 3 – Fig. 13 .present results of simulation of processes in the system on the base of triple synchronously modulated inverters, and show basic voltage waveforms (pole voltages Va10, Vb10, Vc10 of the first inverter, phase Vas1 and line-to-line Va1b1 voltages of the first inverter, and also winding voltages Vw1phase and Vw1line, corresponding to the cases of star-connection and delta-connection of windings). It present also spectra of the phase Vas1 and line Va1b1 voltages of inverters, and also spectra of winding voltages Vw1phase and Vw1line.  Figs. 3-4 show voltage waveforms and voltage spectra of the system controlled in accordance with scheme of continuous synchronous modulation [7]. The fundamental frequency of the system is equal to F=50Hz, and switching frequency is equal to Fs=1.13kHz. Modulation index is equal to m=0.65 in this case.  Figs. 5-7 present voltage waveforms and the corresponding voltage spectra of system controlled by the scheme of synchronous discontinuous space-vector PWM with the 300-non-switching intervals ([7], DPWM30, F=50Hz, average Fs=1.13kHz, m=0.75).  Figs. 8-9 illustrate modulation processes in system controlled by scheme of synchronous discontinuous space-vector PWM with the 600-non-switching intervals ([7], DPWM60, F=50Hz, average Fs=1.13kHz, m=0.85).    Fig. 3.  Pole voltages Va10, Vb10, Vc10, phase voltage Vas1, line voltage Va1b1 of the first inverter, and the corresponding winding voltages Vw1phase and Vw1line of the system with continuous synchronous PWM (CPWM, F=50Hz, Fs=1.13kHz, Fs/F=24.6, m=0.65). Control (modulation) parameter Conventional schemes of spa-ce-vector PWM Specialized method of modulationOperating and max parameter Operating & max voltage V and Vm Operating & maximum fundamental frequency F and Fm Modulation index m mVV /  mFF /  Duration of sub-cycles T τ  Center of the k-signal kα (angles/degr.) )1( −kτ  (sec)    Switch-on durations   060[sin(1.1 mTTak =]sin) kk αα +−  kak mTt αsin1.1=   ×= mTtbk 1.1  )60sin( 0 kα−  Algebraic PWM ×−= Ak 1[1ββ])1( 1ovFKk τ− 21])(65.0[ovkikKFki τβγ−−= +− kk γβ −  Trigonomet-ric PWM ×= 1ββk])1cos[( 1ovKk τ− 21])(9.05.0[ovkikKkitn τβγ−−= +− kk γβ −  Switch-off states (zero voltage) bkakk ttTt −−=0  kk βτλ −=  Special parameters providing synchronization of the process of PWM  ×−= A1[" 1ββsov KFKk ])1( 1τ− sov KK 1)"(' ×−= βτλ cos" 1 ×= ββsov KKk ])1[( 1τ− sov KK 1)"(' ×−= βτλ 216       Fig. 4.  Spectra of voltage waveforms of the system with continuous synchronous PWM (F=50Hz, Fs=1.13kHz, m=0.65). Figs. 10-13 illustrate regimes of operation of inverters with discontinuous PWM in two parts of the zone of overmodulation (Figs. 10-11: the first part of the zone of modulation – DPWM60, F=50Hz, average Fs=1.13kHz, m=0.95; Figs. 12-13: the second part of the overmodulation zone – DPWM30, F=50Hz, average Fs=1.13kHz, m=0.975).   Fig. 5.  Pole voltages Va10, Vb10, Vc10, phase voltage Vas1, line voltage Va1b1 of the first inverter, and the corresponding winding voltages Vw1phase and Vw1line of the system with discontinuous synchronous PWM (DPWM30, F=50Hz, Fs=1.13kHz, Fs/F=24.6, m=0.75).     Fig. 6.  Spectra of voltage waveforms of the system with discontinuous PWM (DPWM30, F=50Hz, Fs=1.13kHz, m=0.75). To emphasize properties of the used specialized scheme of PWM, fractional frequency ratio between the switching and fundamental frequencies (equal to 1130Hz/50Hz=22.6) has been used. Analysis of harmonic composition of symmetrical (with quarter-wave symmetry) voltage waveforms of the system shows (see Figs. 6, 7, 9, 11, 13), that spectra of the all presented waveforms do not include even harmonics and subharmonics for any values of modulation indices of inverters operating at both linear and overmodulation zones.  217   Fig. 7.  Spectra of voltage waveforms of the system with discontinuous PWM (DPWM30, F=50Hz, Fs=1.13kHz, m=0.75).   Fig. 8.  Pole voltages Va10, Vb10, Vc10, phase voltage Vas1, line voltage Va1b1 of the first inverter, and the corresponding winding voltages Vw1phase and Vw1line of the system with discontinuous synchronous PWM (DPWM60, F=50Hz, Fs=1.13kHz, Fs/F=24.6, m=0.85).  Total Harmonic Distortion factor is an important criterion for analysis and comparison of voltage waveforms in photovoltaic systems. In particular, for 50-Hz systems it has been recommended to take into consideration harmonics up to the 40th voltage harmonic for determination of total voltage harmonic distortion factor [10]. Figs. 14-15-present results of calculation of Total Harmonic Distortion factor (THD) for basic  voltage    waveforms   of  the   system  as   function   of         Fig. 9.  Spectra of voltage waveforms of the system with discontinuous PWM (DPWM60, F=50Hz, Fs=1.13kHz, m=0.85). modulation index m of triple inverters, controlled by synchronous techniques of continuous (CPWM) and disconti-nuous (DPWM30 and DPWM60) pulsewidth modulation. 218THD factor has been calculated until the 40-th low-order (k-th) voltage harmonic (Fig. 14, ==402211 )/1( 1kww kVVTHD ), and also until the 1000-th harmonic (Fig. 15, ==10002211 )/1( 1kww kVVTHD ). The fundamental frequency of the system was equal to 50Hz, and the average switching frequency of triple inverters was equal to 1.4kHz.     Fig. 10.  Pole voltages Va10, Vb10, Vc10, phase voltage Vas1, line voltage Va1b1 of the first inverter, and the corresponding winding voltages Vw1phase and Vw1line of the system with discontinuous synchronous PWM in the overmodulation zone (DPWM60, F=50Hz, Fs=1.13kHz, Fs/F=24.6, m=0.95).     Fig. 11.  Spectra of voltage waveforms of the system with discontinuous PWM (DPWM30, F=50Hz, Fs=1.13kHz, m=0.95).  Fig. 12.  Pole voltages Va10, Vb10, Vc10, phase voltage Vas1, line voltage Va1b1 of the first inverter, and the corresponding winding voltages Vw1phase and Vw1line of the system with discontinuous synchronous PWM in the overmodulation zone (DPWM30, F=50Hz, Fs=1.13kHz, Fs/F=24.6, m=0.975).     Fig. 13.  Spectra of voltage waveforms of the system with discontinuous synchronous PWM (DPWM30, F=50Hz, Fs=1.13kHz, m=0.975). Analysis of the presented in Fig. 14 and Fig. 15 results of calculation of THD factor shows, that due to the proposed in [6] connection between triple inverters and the secondary windings of the transformer, there is appreciable improvement of integral spectral characteristics of winding voltages. As an example, in accordance with diagram of Fig. 14 (for the case, if m=0.7), THD factor for winding voltages is reduced by 14% (for system with algorithm of CPWM), by 8% (for system controlled  by  algorithm  of  DPWM30),  and  219 Fig. 14.  THD factor of voltages of the system versus modulation index m (k=40).   Fig. 15.  THD factor of voltages of the system versus modulation index m (k=1000).  by 63% for system with DPWM60, in comparison with the corresponding line voltages. Accordingly, data of the diagram, presented in Fig. 15, show (as an example, for the case, if m=0.7), that THD factor for winding voltages of system with new connection between inverters and transformer is decreased by 23% (for system with DPWM30 or DPWM60), and by 22% (for system controlled by algorithm of CPWM), in comparison with the corresponding line voltages (case of conventional connection between inverters and transformer).  So, improved spectral composition of winding voltages of transformer-based system assures to reduce copper loss in windings of the transformer, and also to decrease switching and conduction losses at triple inverters.  IV. CONCLUSION Specialized schemes and techniques of feedforward space-vector modulation, insuring voltage synchronization and voltage symmetries of inverters, can be used effectively in transformer-based photovoltaic systems on the base of triple inverters with novel connection between inverters and windings of the transformer.  The presented voltage spectrograms illustrate qualitatively and quantitatively attenuation of the corresponding harmonics of winding voltages due to new connections inside the system. To emphasize properties of the used specialized scheme of PWM, fractional frequency ratio between the switching frequency and fundamental frequency (equal to 1130Hz/50Hz=22.6) has been chosen for analysis of modulation processes in system. Research of harmonic composition of symmetrical (with quarter-wave symmetry) voltage waveforms of the system shows (see Figs. 6, 7, 9, 11, 13), that spectra of the all presented waveforms do not include even harmonics and subharmonics for any values of modulation indices of inverters operating at both linear control zone and zone of overmodulation.. It has been shown, that in general case THD factor for winding voltages of secondary windings of the system with new connection between inverters and transformer is decreased (in average) by about 20% for systems with both continuous and discontinuous synchronous PWM, in comparison with the case of conventional connection between inverters and transformer.  ACKNOWLEDGMENT This work was supported by the Slovak Research and Development Agency under the contract no. APVV-15-0750 and by the project no. FEI-2015-3. REFERENCES [1] G. Grandi, C. Rossi, D. Ostojic, and D. Casadei, “A new multilevel conversion structure for grid-connected PV applications,” IEEE Trans. Ind. Electron., vol. 56, no. 11, pp. 4416–4426, 2009. [2] G. Spagnuolo, et al., "Renewable Energy Operation and Conversion Schemes", IEEE Ind. Electr. Magazine, vol. 4, no. 1, pp. 38-51, 2010.  [3] V. Oleschuk and G. Griva, “Simulation of processes in synchronized cascaded inverters for photovoltaic application,” International Review of Electrical Engineering, vol. 4, no. 5, pp. 975-982, 2009.  [4] Renewable Energy, Selected Issues, vol. 2. Cambridge Scholars Publishing, pp. 166-247, 2016.  [5] Yongsoon Park, Sungjae Ohn, and Seung-Ki Sul, “Multi-level operation with two-level converters through a double-delta source connected transformer,” Journal of Power Electronics, vol. 14, no. 6, pp. 1093-1099, 2014.  [6] Sungjae Ohn, Yongsoon Park, and Seung-Ki Sul, “Multi-level operation of triple two-level PWM converters,” Proc. of IEEE Energy Conversion Congress and Exposition (ECCE’2015), pp. 4283-4289, 2015. [7] F. Blaabjerg, V. Oleschuk, and F. Lungeanu, “Synchronization of output voltage waveforms in three-phase inverters for induction motor drives,” Proc. of IEEE-IEEJ Power Conversion Conf. (PCC’2002), pp. 528-533, 2002.  [8] V. Oleschuk, G. Grandi, and P. Sanjeevikumar, “Simulation of processes in dual three-phase system on the base of four inverters with synchronized modulation,” Advances in Power Electronics, vol. 2011, pp. 1-9, 2011.  [9] V. Oleschuk and F. Barrero, “Standard and non-standard approaches for voltage synchronization of drive inverters with space-vector PWM: A survey,” International Review of Electrical Engineering, vol. 9, no. 4, pp. 688-707, 2014.  [10] M. Aiello, A. Cataliotti, S. Favuzza, and G. Graditi, “Theoretical and experimental comparison of Total Harmonic Distortion factors for the evaluation of harmonic and interharmonic pollution of grid-connected photovoltaic systems,” IEEE Trans. Power Delivery, vol. 21, no. 3, pp. 1390-1397, 2006.  220Powered by TCPDF (www.tcpdf.org)

Synchronous PWM control of triple transformer-connected inverters for photovoltaic system

Name not available

Synchronous PWM Control of Triple Transformer-
Connected Inverters for Photovoltaic System  
V. Oleschuk1, M. Cernat2, M. Pastor3, P. Sanjeevikumar4 
1Academy of Sciences of Moldova, Chisinau, Republic of Moldova, 2Transilvania University, Brasov, Romania, 3Technical 
University of Kosice, Slovak Republic, 4University of Johannesburg, Auckland Park, South Africa 
E-mail: oleschukv@hotmail.com, m.cernat@unitbv.ro, marek.pastor@tuke.sk, sanjeevi_12@yahoo.co.in 
 
Abstract-This paper presents results of research of modulation 
processes in power conversion system with triple voltage source 
inverters, supplied by photovoltaic (PV) strings, and connected 
to a four-winding transformer. The transformer has in this case 
specific connections between the secondary windings and 
inverters, allowing providing of multilevel voltages at secondary 
windings of the transformer. Specialized space-vector-based 
strategy of pulsewidth modulation (PWM) has been used for 
control of inverters, insuring quarter-wave symmetry of winding 
voltages for any modulation indices and switching frequency of 
inverters. Analysis and comparison of harmonic composition of 
winding voltages and output voltages of inverters has been done. 
The validity of the analyzed processes has been verified by 
simulations: three basic variants of the scheme of modulation 
have been applied for control of inverters. 
 
I. INTRODUCTION 
Development of photovoltaic-focused power conversion 
systems is growing rapidly. Novel transformer-less and 
transformer-based topologies and configurations of solar 
photovoltaic installations for solar farms and solar factories 
have been proposed and investigated [1]-[4].  
Recently, modified structure of transformer-based system 
with triple three-phase voltage source inverters has been 
proposed [5],[6]. Fig. 1 shows topology of this system for the 
case of delta-connection of secondary windings, supplied by 
three insulated strings of photovoltaic panels. Fig. 2 presents 
the corresponding secondary windings for the case of star-
connection [6]. It has been shown in [6], that simple 
modification of connection between outputs of standard two-
level inverters and windings of the transformer allow 
providing multilevel winding voltages for systems with 
insulated dc-sources. Also, winding voltages can be expressed 
in this case by the phase voltages of triple inverters.  
Operation of inverter-based photovoltaic systems is in 
dependence of the used methods and techniques of PWM. 
Space-vector modulation is perspective method for control of 
inverters, including PV application. During system operation, 
for some regimes of photovoltaic systems (grid frequency 
fluctuation, different voltages of strings of PV panels, etc.) it 
is necessary to ensure voltage synchronization. So, this paper 
presents results of research of modulation processes in PV 
system with triple transformer-connected inverters, controlled 
by specialized schemes and techniques of space-vector PWM, 
insuring voltage synchronization and voltage symmetries for 
any operating conditions of the system.  
 
 
Fig. 1.  Topology of transformer-based photovoltaic system with triple 
inverters specifically connected to secondary windings of the transformer 
(case of delta-connection of windings).  
 
 
Fig. 2.  Secondary windings of transformer of the system for the case of star-
connection.  
 
II. TWO-LEVEL INVERTERS WITH SYNCHRONOUS PWM  
To assure voltage synchronization and waveform 
symmetries of inverters, specialized method of space-vector 
modulation can be used for control of triple inverters of PV 
system. Table I presents peculiarities and the main control 
dependences of this method of synchronous modulation, 
which are compared here with the corresponding parameters 
of conventional space-vector PWM [7]-[9].  
978-1-5090-1797-3/16/$31.00 ©2016 IEEE 215
TABLE I. BASIC PARAMETERS OF PWM METHODS 
 
III. SYNCHRONOUS PWM OF INVERTERS IN TRIPLE-
INVERTER-BASED SYSTEM  
Phase voltages Vas1, Vbs1 and Vcs1 of the first standard three-
phase inverter (Inverter 1 in Fig. 1) of the system are 
calculated in accordance with (1)-(3) [7]:  
 
Vas1 = Va10 + (Va10 + Vb10  + Vc10)/3                 (1) 
 
Vbs1 = Vb10 + (Va10 + Vb10  + Vc10)/3                 (2) 
 
Vcs1 = Vc10 + (Va10 + Vb10  + Vc10)/3,                (3) 
 
where Va10, Vb10 and Vc10 are the corresponding pole voltages 
of the first inverter.  
Winding voltages of secondary windings of the system (Vw1, 
Vw2, Vw3 in Fig. 1) can be expressed by the phase voltages of 
triple inverters in accordance with (4)-(9) [6]. Dependences 
(4)-(6) correspond to the case of delta-connection of 
windings, presented in Fig. 1, dependences (7)-(9) correspond 
to the case of star-connection, presented in Fig. 2.  
 
Vw1 = Vas3 - Vbs1                                     (4) 
 
Vw2 = Vbs1 - Vcs2                                (5) 
 
Vw3 = Vcs2 - Vas3                                (6) 
Vw1 = 0.667Vas1  – 0.333Vbs2 – 0.333Vcs3                (7) 
 
Vw2 = -0.333Vas1 + 0.667Vbs2 – 0.333Vcs3               (8) 
 
Vw3 = -0.333Vas1 – 0.333Vbs2 + 0.667Vcs3               (9) 
 
Control and output signals of triple inverters of the system 
are shifted by 1200 (1200 interleaving in accordance with some 
definitions). Also, small additional shift between control 
signals of three inverters, equal to 1/3 of duration of the 
switching period (switching sub-cycle) has been provided by 
the used scheme of control and modulation.  
Fig. 3 – Fig. 13 .present results of simulation of processes in 
the system on the base of triple synchronously modulated 
inverters, and show basic voltage waveforms (pole voltages 
Va10, Vb10, Vc10 of the first inverter, phase Vas1 and line-to-line 
Va1b1 voltages of the first inverter, and also winding voltages 
Vw1phase and Vw1line, corresponding to the cases of star-
connection and delta-connection of windings). It present also 
spectra of the phase Vas1 and line Va1b1 voltages of inverters, 
and also spectra of winding voltages Vw1phase and Vw1line.  
Figs. 3-4 show voltage waveforms and voltage spectra of 
the system controlled in accordance with scheme of 
continuous synchronous modulation [7]. The fundamental 
frequency of the system is equal to F=50Hz, and switching 
frequency is equal to Fs=1.13kHz. Modulation index is equal 
to m=0.65 in this case.  
Figs. 5-7 present voltage waveforms and the corresponding 
voltage spectra of system controlled by the scheme of 
synchronous discontinuous space-vector PWM with the 300-
non-switching intervals ([7], DPWM30, F=50Hz, average 
Fs=1.13kHz, m=0.75).  
Figs. 8-9 illustrate modulation processes in system 
controlled by scheme of synchronous discontinuous space-
vector PWM with the 600-non-switching intervals ([7], 
DPWM60, F=50Hz, average Fs=1.13kHz, m=0.85).  
 
 
Fig. 3.  Pole voltages Va10, Vb10, Vc10, phase voltage Vas1, line voltage Va1b1 of 
the first inverter, and the corresponding winding voltages Vw1phase and Vw1line 
of the system with continuous synchronous PWM (CPWM, F=50Hz, 
Fs=1.13kHz, Fs/F=24.6, m=0.65). 
Control 
(modulation) 
parameter 
Conventional 
schemes of spa-
ce-vector PWM 
Specialized 
method 
of modulation
Operating and 
max parameter 
Operating & max 
voltage V and Vm 
Operating & maximum 
fundamental frequency F and 
Fm 
Modulation 
index m mVV /  mFF /  
Duration of sub-
cycles 
T τ  
Center 
of the k-signal kα (angles/degr.) )1( −kτ  (sec) 
 
 
 
Switch-on 
durations 
 
 
060[sin(1.1 mTTak =
]sin) kk αα +−  
kak mTt αsin1.1=  
 
×= mTtbk 1.1  
)60sin( 0 kα−  
Algebraic 
PWM 
×−= Ak 1[1ββ
])1( 1ovFKk τ−
 
2
1
])(6
5.0[
ov
kik
KFki τ
βγ
−
−= +−
 
kk γβ −  
Trigonomet-
ric PWM 
×= 1ββk
])1cos[( 1ovKk τ−
 
2
1
])(9.0
5.0[
ov
kik
Kkitn τ
βγ
−
−= +−
 
kk γβ −  
Switch-off 
states (zero 
voltage) 
bkakk ttTt −−=0  kk βτλ −=  
Special 
parameters 
providing 
synchronization 
of the process 
of PWM 
 ×−= A1[" 1ββ
sov KFKk ])1( 1τ−
 
sov KK 1
)"(' ×−= βτλ
 
cos" 1 ×= ββ
sov KKk ])1[( 1τ−
 
sov KK 1
)"(' ×−= βτλ
 
216
 
 
 
 
 
 
 
Fig. 4.  Spectra of voltage waveforms of the system with continuous 
synchronous PWM (F=50Hz, Fs=1.13kHz, m=0.65). 
Figs. 10-13 illustrate regimes of operation of inverters with 
discontinuous PWM in two parts of the zone of 
overmodulation (Figs. 10-11: the first part of the zone of 
modulation – DPWM60, F=50Hz, average Fs=1.13kHz, 
m=0.95; Figs. 12-13: the second part of the overmodulation 
zone – DPWM30, F=50Hz, average Fs=1.13kHz, m=0.975).  
 
Fig. 5.  Pole voltages Va10, Vb10, Vc10, phase voltage Vas1, line voltage Va1b1 of 
the first inverter, and the corresponding winding voltages Vw1phase and Vw1line 
of the system with discontinuous synchronous PWM (DPWM30, F=50Hz, 
Fs=1.13kHz, Fs/F=24.6, m=0.75). 
 
 
 
 
Fig. 6.  Spectra of voltage waveforms of the system with discontinuous PWM 
(DPWM30, F=50Hz, Fs=1.13kHz, m=0.75). 
To emphasize properties of the used specialized scheme of 
PWM, fractional frequency ratio between the switching and 
fundamental frequencies (equal to 1130Hz/50Hz=22.6) has 
been used. Analysis of harmonic composition of symmetrical 
(with quarter-wave symmetry) voltage waveforms of the 
system shows (see Figs. 6, 7, 9, 11, 13), that spectra of the all 
presented waveforms do not include even harmonics and 
subharmonics for any values of modulation indices of 
inverters operating at both linear and overmodulation zones.  
217
 
 
 
Fig. 7.  Spectra of voltage waveforms of the system with discontinuous PWM 
(DPWM30, F=50Hz, Fs=1.13kHz, m=0.75). 
 
 
Fig. 8.  Pole voltages Va10, Vb10, Vc10, phase voltage Vas1, line voltage Va1b1 of 
the first inverter, and the corresponding winding voltages Vw1phase and Vw1line 
of the system with discontinuous synchronous PWM (DPWM60, F=50Hz, 
Fs=1.13kHz, Fs/F=24.6, m=0.85). 
 
Total Harmonic Distortion factor is an important criterion 
for analysis and comparison of voltage waveforms in 
photovoltaic systems. In particular, for 50-Hz systems it has 
been recommended to take into consideration harmonics up to 
the 40th voltage harmonic for determination of total voltage 
harmonic distortion factor [10]. Figs. 14-15-present results of 
calculation of Total Harmonic Distortion factor (THD) for 
basic  voltage    waveforms   of  the   system  as   function   of  
 
 
 
 
 
 
 
Fig. 9.  Spectra of voltage waveforms of the system with discontinuous PWM 
(DPWM60, F=50Hz, Fs=1.13kHz, m=0.85). 
modulation index m of triple inverters, controlled by 
synchronous techniques of continuous (CPWM) and disconti-
nuous (DPWM30 and DPWM60) pulsewidth modulation. 
218
THD factor has been calculated until the 40-th low-order (k-
th) voltage harmonic (Fig. 14, 
=
=
40
2
2
11 )/1( 1
k
ww kVVTHD ), 
and also until the 1000-th harmonic (Fig. 15, 

=
=
1000
2
2
11 )/1( 1
k
ww kVVTHD ). The fundamental frequency of 
the system was equal to 50Hz, and the average switching 
frequency of triple inverters was equal to 1.4kHz.  
  
 
Fig. 10.  Pole voltages Va10, Vb10, Vc10, phase voltage Vas1, line voltage Va1b1 of 
the first inverter, and the corresponding winding voltages Vw1phase and Vw1line 
of the system with discontinuous synchronous PWM in the overmodulation 
zone (DPWM60, F=50Hz, Fs=1.13kHz, Fs/F=24.6, m=0.95). 
 
 
 
 
Fig. 11.  Spectra of voltage waveforms of the system with discontinuous 
PWM (DPWM30, F=50Hz, Fs=1.13kHz, m=0.95). 
 
Fig. 12.  Pole voltages Va10, Vb10, Vc10, phase voltage Vas1, line voltage Va1b1 of 
the first inverter, and the corresponding winding voltages Vw1phase and Vw1line 
of the system with discontinuous synchronous PWM in the overmodulation 
zone (DPWM30, F=50Hz, Fs=1.13kHz, Fs/F=24.6, m=0.975). 
 
 
 
 
Fig. 13.  Spectra of voltage waveforms of the system with discontinuous 
synchronous PWM (DPWM30, F=50Hz, Fs=1.13kHz, m=0.975). 
Analysis of the presented in Fig. 14 and Fig. 15 results of 
calculation of THD factor shows, that due to the proposed in 
[6] connection between triple inverters and the secondary 
windings of the transformer, there is appreciable 
improvement of integral spectral characteristics of winding 
voltages. As an example, in accordance with diagram of Fig. 
14 (for the case, if m=0.7), THD factor for winding voltages 
is reduced by 14% (for system with algorithm of CPWM), by 
8% (for system controlled  by  algorithm  of  DPWM30),  and  
219
 
Fig. 14.  THD factor of voltages of the system versus modulation index m 
(k=40). 
 
 
Fig. 15.  THD factor of voltages of the system versus modulation index m 
(k=1000). 
 
by 63% for system with DPWM60, in comparison with the 
corresponding line voltages. Accordingly, data of the 
diagram, presented in Fig. 15, show (as an example, for the 
case, if m=0.7), that THD factor for winding voltages of 
system with new connection between inverters and 
transformer is decreased by 23% (for system with DPWM30 
or DPWM60), and by 22% (for system controlled by 
algorithm of CPWM), in comparison with the corresponding 
line voltages (case of conventional connection between 
inverters and transformer).  
So, improved spectral composition of winding voltages of 
transformer-based system assures to reduce copper loss in 
windings of the transformer, and also to decrease switching 
and conduction losses at triple inverters.  
IV. CONCLUSION 
Specialized schemes and techniques of feedforward space-
vector modulation, insuring voltage synchronization and 
voltage symmetries of inverters, can be used effectively in 
transformer-based photovoltaic systems on the base of triple 
inverters with novel connection between inverters and 
windings of the transformer.  
The presented voltage spectrograms illustrate qualitatively 
and quantitatively attenuation of the corresponding harmonics 
of winding voltages due to new connections inside the 
system. 
To emphasize properties of the used specialized scheme of 
PWM, fractional frequency ratio between the switching 
frequency and fundamental frequency (equal to 
1130Hz/50Hz=22.6) has been chosen for analysis of 
modulation processes in system. Research of harmonic 
composition of symmetrical (with quarter-wave symmetry) 
voltage waveforms of the system shows (see Figs. 6, 7, 9, 11, 
13), that spectra of the all presented waveforms do not 
include even harmonics and subharmonics for any values of 
modulation indices of inverters operating at both linear 
control zone and zone of overmodulation.. 
It has been shown, that in general case THD factor for 
winding voltages of secondary windings of the system with 
new connection between inverters and transformer is 
decreased (in average) by about 20% for systems with both 
continuous and discontinuous synchronous PWM, in 
comparison with the case of conventional connection between 
inverters and transformer.  
ACKNOWLEDGMENT 
This work was supported by the Slovak Research and 
Development Agency under the contract no. APVV-15-0750 
and by the project no. FEI-2015-3. 
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Synchronous PWM control of triple transformer-connected inverters for photovoltaic system

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