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Three-Phases Flying-Capacitor Multilevel Inverter with Proportional Natural PWM Control
Available online at www.sciencedirect.com
ScienceDirect Energy Procedia 74 (2015) 1061 – 1070
International Conference on Technologies and Materials for Renewable Energy, Environment and Sustainability, TMREES15
Three-Phases Flying-Capacitor Multilevel Inverter with Proportional Natural PWM Control K. Hemicia, A. Zegaouib, A. Aissa Bokhtache a,b, M.O. Mahmoudia, M. Ailleriec,d a
b Nationale Polytechnique School, 016000, Algiers, Algeria LGEER Laboratory, Hassiba Benbouali University, 02000 Chlef, Algeria c Université de Lorraine, LMOPS, EA 4423, 57070 Metz, France. d CentraleSupelec, LMOPS, 57070 Metz, France.
Abstract The flying-capacitor multilevel inverter is a recently developed topology to ensure a flexible control and a modular design. However, this inverter requires a balanced DC voltage distribution. This can be realized by using a special control leading to natural balancing or by measuring voltages and selecting appropriate switching states. To solve the capacitor voltage-balancing problem a validated solution is suggested in this contribution. It is based on a series of three phases flying-capacitor multilevel inverter model and its proportional control. The proposed command technique originally uses here the Proportional Natural Pulse Width Modulation (PN-PWM) that is compared to the basic N-PWM one. Simulation results satisfy the basic idea that floating voltage can ensure a flexible control and a modular design of flying-capacitor multilevel inverter. © Published by Elsevier Ltd. Ltd. This is an open access article under the CC BY-NC-ND license © 2015 2015The TheAuthors. Authors. Published by Elsevier (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the Euro-Mediterranean Institute for Sustainable Development (EUMISD). Peer-review under responsibility of the Euro-Mediterranean Institute for Sustainable Development (EUMISD) Keywords: PWM control, PN-PWM control, Inverters, Flying capacitors, multilevel inverter, balanced voltagesr
1. Introduction Current power electronics systems dedicated to energy conversion have important developments thanks to progress in semiconductors technology and power topology systems associated to smart approach and control. Among these systems, flying capacitor inverter built on series association of elementary commutation cells are well studied for applications in renewable energy conversion or generation. This structure appeared at the end of the 20th century [1] and makes possible to share the constraints in voltage and to improve harmonic contents of the output waveforms [2]. Moreover, modelling development of new electronic system is a very important step for the elaboration of control laws and observers synthesis. In literature, several approaches or models were considered in development control and observation methods for three phases flying capacitor inverter. According to these previous
1876-6102 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4 .0/). Peer-review under responsibility of the Euro-Mediterranean Institute for Sustainable Development (EUMISD) doi:10.1016/j.egypro.2015.07.744
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K. Hemici et al. / Energy Procedia 74 (2015) 1061 – 1070
studies, three types of models could be found. Initially, models have been developed to describe their harmonic [1] or averaging [3], instantaneous [4], behaviours. These various models were used for the development of control laws in open-loop control topology [5], which is very to ensure very simply the functioning of the three phases flying capacitor inverter with pulses delayed by 1/3 to the period relative to each other. The average model consists of calculating average value of all variables during one sampling period, nevertheless, this model cannot represent the natural balancing of floating voltage. The harmonic model consists of the calculation of the harmonic voltage phases and their magnitudes by considering the load current in steady-state operation. The instantaneous model deals with time-evolution of all variables including the switch states (discrete location). The usage of this model is delicate because the design of controllers and observers is impossible since the inverter is not a continuous system but a mixture of continuous and discrete systems. Considering these existing topologies, the stability of the voltage capacitors can be improve by the use of new controller topology based on a closed-loop control taking into account the evolution of the capacitor voltages and can meet the requirement to control and maintain defined voltage levels [6]. By else, a new model must be adequately simple to allow real time control while being enough precise to achieve the desired behaviour. Nevertheless, multilevel flying-capacitor inverter is based on continuous and discrete variables and its modelling is claimed to be difficult [7, 8]. For a better exploitation of controller possibilities, hybrid model allowing multilevel flying-capacitor inverter based on the use of analysis and synthesis corresponds to powerful tools, bringing a solution to the modelling difficulties [9]. Taking into account the previous analysis of existing controllers, mainly the N-PWM controllers, as presented in literature, the major goal of the current study is to propose an original three phases multilevel flying-capacitor inverter where each inverter arm is constituted by p cells in series. In this contribution, the inverter control technique will be improved by the use of a proportional controller of the output voltages, i.e. a Proportional Natural Pulse Width Modulation, or PN-PWM, controller. 2. Series Three-Phases Flying-Capacitor multilevel inverter modeling The studied three-phases multilevel flying-capacitor inverter topology is presented in figure 1. It is composed by four (p=4) cells per phase. These cells are connected to each other’s by capacitors. Each cell contains two complementary power electronics components controlled by a binary switch. That means that if the upper switch of th the jk cell (with is phase number j={a,b,c}) is closed u jk = 1 the lower switch is open.
The three-phases flying-capacitor multilevel inverter cells are associated in series with a load constituted with a resistor in series with an inductor and the cells separated by capacitors that can be considered as continuous voltage sources. Thus, the inverter has p-1 floating voltage sources. In order to ensure normal operations, it is necessary to guarantee a balanced distribution of the floating voltages VCj,k = kE / p . The output voltage VCj,k can attend p voltage levels E ,...,(p1) E ,E p
p
The expressions of the vector [Vcha, Vchb, Vchc] are defined by V cha Vchb V chc
2 1 1 VAM = 1 2 1 VBM 1 1 2 V CM
(1)
with
V = V E / 2 sa AM V = V E / 2 sb BM VCM = Vsc E / 2
(2)
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The output voltage Vsi corresponds to the sum of the terminal voltages of the switches. p
p
Vs = Vs = u j,k ( VCj,k VCj,k1 ) j
j,k
k=1
(3)
k=1
E et VC = 0 j,0 2 Consequently
With VC = j,p
V cha Vchb V chc
2 1 1 1 = 1 2 1 3 1 1 2
(VCa,k VCa,k1 ) k=1 p u (V V ) b,k Cb,k Cb,k1 k=1 p u c,k (VCc,k VCc,k1 ) k=1 p
u
a,k
(4)
The evolution of the floating voltages VCj,k is related to the evolution of the current i chj , which is a function of the state of the adjacent cells Cel j,k+1 ,Cel j,k and of the load current i chj . The i Cj,k current is a function of the control signals of the switches U j,k+1 and U j,k .
i Cj,k = ( u j,k+1 u j,k ).i chj
(5)
Knowing the capacitor values C j,k we obtain the equation governing the voltages evolution VCj,k
C j,k
d V = i Cj,k = ( u j,k+1 u j,k ) i chj dt Cj,k
(6)
Then
(s j,k+1 s j,k ) i d VCj,k = chi dt C j,k
(7)
In the case of a RL load, the equation describing the evolution of the current i chi is obtained as follow: Vchi = Vsi
E d E = R chi .i chi + Lchi . i chi 2 dt 2
(8)
we can write : di chj dt
=
Vsj Lchj
R chj Lchj
i chj
(9)
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Finally, by the introduction of the expression of the output voltage according to the floating voltages VCj,k , Eq 3 in Eq. 10, describing the evolution of the current i chi we obtain:
di chj dt
=
R 1 p u (V VCj,k1 ) L chj ichj Lchj k=1 j,k Cj,k chj
(10)
and the general representation of the state space of the p-cells converter is given by equation 11 dVCj,k 1 = (u u j,k )i chj dt C j,k j,k+1 di chj R 1 1 p = = u j,k (VCj,k VCj,k1 ) chj i chj Lchj Lchj k dt Lchj
..
Fig. 1. Three-Phase Flying-Capacitor Multilevel Inverter
(11)
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3. The N-PWM system control In this controller topology, we have to generate the various commands of the different cells of the system. These control signals will have phases delayed by 2 / p from each other in order to reach the optimal operating mode. In the following, we present the simplest and easiest solution based on the N-PWM method, which can be used practically. In this command technique, the control signal of each cell is generated by the intersection between carrying triangular signal within frequency f p and the sinusoidal modulating signal within frequency f mod . The triangular signals, noted Pk are generated according to: Pk =
1 2 2 arcsin sin( t jk + ) +1 2 p 2
(12)
leading to:
1 2 arcsin sin(2f p t 1 + ) +1 2 2 1 2 P2 = arcsin sin(2f p t 2 + +1 2 2 . . . 1 2 Pp = arcsin sin(2f p t p + +1 2 2
P1 =
(13)
The angle jk will be selected with equal values for all Pk i.e.
jk = (k 1)
2 p
(14)
The N-PWM method consists to determine the control signal u jk obtained according to • If mod jk p k then u jk = 1 • if mod jk p k then u jk = 0 with sinusoidal modulation signals, mod jk described by the following set of equations for the three phases mod1k = r.sin(2f mod t) 2 ) 3 2 = r.sin(2f mod t ) 3
mod 2k = r.sin(2f mod t mod 3k
(15)
In Eqs. 15, r represents the modulation depth varying between 0 and 1. The choice of a regular phase of (2 / p) between the different cells brings a significant improvement in the outputvoltage spectrum level. We note that harmonics are gathered in families centered on a multiple frequencies p.m.f mod
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, where m = f p /f mod represents the modulation index. 4. Proportional N-PWM control of the output voltages The proportional control takes into the count only the regulation of the floating voltage. This method is based on a modulated duty cycle. During operating mode of p-cells inverter, the law of control must take into account the sign of the load current which is sinusoidal M jp = M jref M jk = M K+1 sign(i chj ).G.(
C jk Tdec .I chjmax
).(
k.E VCjk ) p
with sign(i chj ) represents the sign of load current
M jref =
(16)
and defined by M jref
p mod 1 .sin(2f mod t) + 2 2
(17)
Moreover, the maximum value i chjmax of the current load is to be considered. Figure 2 presents the control bloc diagram.
Fig 2. Control proportional of the output voltages
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The adequate choice of the constant G allows ensuring a non-saturation of the manipulated variables. In the case of the starting operating phase of the inverter with null voltages, we can obtain G as the follow:
M j1init = M jref sign(i chj ).
p(p 1) G j .C jk E . . Tdec .Ichjmax p 2
(18)
Thus from Eq. 18, G can be deduced according to M j1init = 0 with the reference modulating of M jref = 1 and the 2
load current positive corresponding to sign(i chj ) = 1 , given
Gj =
Tdec .Ichjmax
(19)
C jk .(p 1).E
5. Simulation results and discussions Simulations were performed under Matlab/ Simulink environment. The parameters of the studied series three-phases five levels flying capacitor inverter (having four cells) connected to an RL load is summarized in table 1: Table 1. Parameters of the studied three-phases filing capacitor multilevel inverter E
Lchj
Rchj
Cjk
fmod
Fdéc
m
r
400VDC
0.5mH
10
40nF
50KHZ
5MHZ
180
0.8
The chronograms in Fig 3 present the shape of the control signals of the various switch states varying between 0 and 1 in the inverter.
Fig. 3. Control signals of one inverter arm.
We clearly see in Figs. 3 the modulated duty cycle associated to the natural pulse width modulation NP-PWM. Within these control signals, the floating-capacitor voltages according to the two compared control techniques are
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shown in Fig.4. We have plotted in Fig.4 the floating capacitor voltages when inverter is controlled by the N-PWM technique. It clearly shows disturbances in the response of the system with appearance of ripples at about 10% of the mean values of the voltages recorded at terminals of the floating-capacitors. In order to suppress these fluctuations, the control was performed by the NP-PWM technique and floating capacitor voltages are represented in Fig. 4.b. We clearly observed that the implementation of such control hugely improves the voltage quality by a quasisuppression, (down to less than 1% of the mean values) of the voltage ripples.
Fig. 4.a: Floating capacitor voltages when inverter is controlled by the N-PWM technique. b: Floating capacitor voltages when inverter is controlled by proportional-N-PWM technique.
According to these results, we present in Fig. 5 the inverter output voltage and in Fig. 6 the load voltage in the case of the four levels flying-capacitor inverter. These results show the performances of our proposed control method.
K. Hemici et al. / Energy Procedia 74 (2015) 1061 – 1070
Fig. 5. The sinusoidal modulating signal.
Fig. 6. Phase voltage for four level flying capacitor inverter (load voltage)
Fig.7. FFT harmonic spectrum of the modulating voltage.
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We have characterized the FFT of the sinusoidal modulating voltage and we present the spectrum in Fig. 7. The output voltage obtained presents a sinusoidal shape and is closest to a pure sine wave with a total harmonic distortion, THD equal to 4.02%. We observe in the spectrum of Fig. 7 that harmonics are grouped into two families of multiple frequencies centered around pmfmod and at the double, respectively, corresponding in our case m = 180, p = 4 therefore mp = 720. 6. Conclusion In this paper, we have presented a detail analysis of a flying capacitor multilevel inverter for regulation of capacitor voltages controlled by natural pulse width modulation NP-PWM and associated to a modulated duty cycle. A threephases flying capacitor multilevel inverter is implemented instead to have a balanced distribution of the voltage. We have presented the theoretical analysis, design, and simulations under Simulink/Matlab environment of the inverters built with four levels. This four-levels flying capacitor inverter generates a sinusoidal waveform within a minimum of distortion increasing the overall efficiency of existing inverters based on N-PWM control allowing, now a THD achieving a low value equal to 4.02%. Compared to the N-PWM control technique, the originally developed PNPWM technique shows satisfactory results with a huge increase of the quality of the output signal with a decreasing of the amplitude of the ripples of the floating capacitor voltages. References [1] [2] [3] [4] [5] [6] [7] [8]
[9]
R. Stala, S. Pirog, A. Mondzik, M. Baszynski, A. Penczek, J. Czekonski and S. Gasiorek, “Results of investigation of multicell converters with balancing circuit,” IEEE Trans. Industrial Electronics,vol. 56, no. 7, pp. 2620–2628, July 2009. A. Ajami and M. Armaghan, “Vector control of induction motor drive based on mixed multi-cell cascaded inverter,” International Review on Mod- elling and Simulations vol. 3, no. 5, pp. 767-774, October 2010. A.K Sadigh, and G.B. Gharehpetian, “Double flying capacitor multicell converter based on modified phase-shifted pulse width modulation,” IEEE Transactions on Power Elec- tronics, vol. 259, no. 6, pp. 1517-1526, June 2010 P. Ladoux, M. Machmoum, C. Batard, “Harmonic currents compensation for 1.5 kV DC railway substations,” International Review of Electrical Engi- neering, vol. 4, no 3, 380-391, June 2009. Bethoux, O., and J.P. Barbot, “Multi-cell chopper direct control law preserving optimal limit cycles”,in Proceedings of CCA, Glasgow,2002 M. Ghanes,F. Bejarano and J.P. Barbot, “On sliding mode and adaptive observers design for multicell converter” in Proceedings of the 2009 IEEE ACC,, pp. 2134-2139, June 2009 D. Patino, P. Riedinger and C. Iung, “Predictive control approach for multicellular converters,” in Proceedings of the 2008 IEEE IECON, pp. 3309- 3314, November 2008 T.A. Meynard, H. Foch, P. Thomas, J. Courault, R. Jakob and M. Nahrstaedt, “Multicell converters: basic concepts and industry applications,” IEEE Trans. Industrial Electronics, vol. 49, no. 5, pp. 955–964, October 2002M. Young, The Technical Writer’s Handbook. Mill Valley, CA: University Science, 1989. G. Gateau, M. Fadel, P. Maussion, R. Bensaid and T.A. Meynard, “Multicell converters: active control and observation of flying-Capacitor voltages, ” IEEE Trans. Industrial Electronics, vol. 49, no. 5, pp. 998–1008 October2002.
ScienceDirect Energy Procedia 74 (2015) 1061 – 1070
International Conference on Technologies and Materials for Renewable Energy, Environment and Sustainability, TMREES15
Three-Phases Flying-Capacitor Multilevel Inverter with Proportional Natural PWM Control K. Hemicia, A. Zegaouib, A. Aissa Bokhtache a,b, M.O. Mahmoudia, M. Ailleriec,d a
b Nationale Polytechnique School, 016000, Algiers, Algeria LGEER Laboratory, Hassiba Benbouali University, 02000 Chlef, Algeria c Université de Lorraine, LMOPS, EA 4423, 57070 Metz, France. d CentraleSupelec, LMOPS, 57070 Metz, France.
Abstract The flying-capacitor multilevel inverter is a recently developed topology to ensure a flexible control and a modular design. However, this inverter requires a balanced DC voltage distribution. This can be realized by using a special control leading to natural balancing or by measuring voltages and selecting appropriate switching states. To solve the capacitor voltage-balancing problem a validated solution is suggested in this contribution. It is based on a series of three phases flying-capacitor multilevel inverter model and its proportional control. The proposed command technique originally uses here the Proportional Natural Pulse Width Modulation (PN-PWM) that is compared to the basic N-PWM one. Simulation results satisfy the basic idea that floating voltage can ensure a flexible control and a modular design of flying-capacitor multilevel inverter. © Published by Elsevier Ltd. Ltd. This is an open access article under the CC BY-NC-ND license © 2015 2015The TheAuthors. Authors. Published by Elsevier (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the Euro-Mediterranean Institute for Sustainable Development (EUMISD). Peer-review under responsibility of the Euro-Mediterranean Institute for Sustainable Development (EUMISD) Keywords: PWM control, PN-PWM control, Inverters, Flying capacitors, multilevel inverter, balanced voltagesr
1. Introduction Current power electronics systems dedicated to energy conversion have important developments thanks to progress in semiconductors technology and power topology systems associated to smart approach and control. Among these systems, flying capacitor inverter built on series association of elementary commutation cells are well studied for applications in renewable energy conversion or generation. This structure appeared at the end of the 20th century [1] and makes possible to share the constraints in voltage and to improve harmonic contents of the output waveforms [2]. Moreover, modelling development of new electronic system is a very important step for the elaboration of control laws and observers synthesis. In literature, several approaches or models were considered in development control and observation methods for three phases flying capacitor inverter. According to these previous
1876-6102 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4 .0/). Peer-review under responsibility of the Euro-Mediterranean Institute for Sustainable Development (EUMISD) doi:10.1016/j.egypro.2015.07.744
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K. Hemici et al. / Energy Procedia 74 (2015) 1061 – 1070
studies, three types of models could be found. Initially, models have been developed to describe their harmonic [1] or averaging [3], instantaneous [4], behaviours. These various models were used for the development of control laws in open-loop control topology [5], which is very to ensure very simply the functioning of the three phases flying capacitor inverter with pulses delayed by 1/3 to the period relative to each other. The average model consists of calculating average value of all variables during one sampling period, nevertheless, this model cannot represent the natural balancing of floating voltage. The harmonic model consists of the calculation of the harmonic voltage phases and their magnitudes by considering the load current in steady-state operation. The instantaneous model deals with time-evolution of all variables including the switch states (discrete location). The usage of this model is delicate because the design of controllers and observers is impossible since the inverter is not a continuous system but a mixture of continuous and discrete systems. Considering these existing topologies, the stability of the voltage capacitors can be improve by the use of new controller topology based on a closed-loop control taking into account the evolution of the capacitor voltages and can meet the requirement to control and maintain defined voltage levels [6]. By else, a new model must be adequately simple to allow real time control while being enough precise to achieve the desired behaviour. Nevertheless, multilevel flying-capacitor inverter is based on continuous and discrete variables and its modelling is claimed to be difficult [7, 8]. For a better exploitation of controller possibilities, hybrid model allowing multilevel flying-capacitor inverter based on the use of analysis and synthesis corresponds to powerful tools, bringing a solution to the modelling difficulties [9]. Taking into account the previous analysis of existing controllers, mainly the N-PWM controllers, as presented in literature, the major goal of the current study is to propose an original three phases multilevel flying-capacitor inverter where each inverter arm is constituted by p cells in series. In this contribution, the inverter control technique will be improved by the use of a proportional controller of the output voltages, i.e. a Proportional Natural Pulse Width Modulation, or PN-PWM, controller. 2. Series Three-Phases Flying-Capacitor multilevel inverter modeling The studied three-phases multilevel flying-capacitor inverter topology is presented in figure 1. It is composed by four (p=4) cells per phase. These cells are connected to each other’s by capacitors. Each cell contains two complementary power electronics components controlled by a binary switch. That means that if the upper switch of th the jk cell (with is phase number j={a,b,c}) is closed u jk = 1 the lower switch is open.
The three-phases flying-capacitor multilevel inverter cells are associated in series with a load constituted with a resistor in series with an inductor and the cells separated by capacitors that can be considered as continuous voltage sources. Thus, the inverter has p-1 floating voltage sources. In order to ensure normal operations, it is necessary to guarantee a balanced distribution of the floating voltages VCj,k = kE / p . The output voltage VCj,k can attend p voltage levels E ,...,(p1) E ,E p
p
The expressions of the vector [Vcha, Vchb, Vchc] are defined by V cha Vchb V chc
2 1 1 VAM = 1 2 1 VBM 1 1 2 V CM
(1)
with
V = V E / 2 sa AM V = V E / 2 sb BM VCM = Vsc E / 2
(2)
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The output voltage Vsi corresponds to the sum of the terminal voltages of the switches. p
p
Vs = Vs = u j,k ( VCj,k VCj,k1 ) j
j,k
k=1
(3)
k=1
E et VC = 0 j,0 2 Consequently
With VC = j,p
V cha Vchb V chc
2 1 1 1 = 1 2 1 3 1 1 2
(VCa,k VCa,k1 ) k=1 p u (V V ) b,k Cb,k Cb,k1 k=1 p u c,k (VCc,k VCc,k1 ) k=1 p
u
a,k
(4)
The evolution of the floating voltages VCj,k is related to the evolution of the current i chj , which is a function of the state of the adjacent cells Cel j,k+1 ,Cel j,k and of the load current i chj . The i Cj,k current is a function of the control signals of the switches U j,k+1 and U j,k .
i Cj,k = ( u j,k+1 u j,k ).i chj
(5)
Knowing the capacitor values C j,k we obtain the equation governing the voltages evolution VCj,k
C j,k
d V = i Cj,k = ( u j,k+1 u j,k ) i chj dt Cj,k
(6)
Then
(s j,k+1 s j,k ) i d VCj,k = chi dt C j,k
(7)
In the case of a RL load, the equation describing the evolution of the current i chi is obtained as follow: Vchi = Vsi
E d E = R chi .i chi + Lchi . i chi 2 dt 2
(8)
we can write : di chj dt
=
Vsj Lchj
R chj Lchj
i chj
(9)
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Finally, by the introduction of the expression of the output voltage according to the floating voltages VCj,k , Eq 3 in Eq. 10, describing the evolution of the current i chi we obtain:
di chj dt
=
R 1 p u (V VCj,k1 ) L chj ichj Lchj k=1 j,k Cj,k chj
(10)
and the general representation of the state space of the p-cells converter is given by equation 11 dVCj,k 1 = (u u j,k )i chj dt C j,k j,k+1 di chj R 1 1 p = = u j,k (VCj,k VCj,k1 ) chj i chj Lchj Lchj k dt Lchj
..
Fig. 1. Three-Phase Flying-Capacitor Multilevel Inverter
(11)
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3. The N-PWM system control In this controller topology, we have to generate the various commands of the different cells of the system. These control signals will have phases delayed by 2 / p from each other in order to reach the optimal operating mode. In the following, we present the simplest and easiest solution based on the N-PWM method, which can be used practically. In this command technique, the control signal of each cell is generated by the intersection between carrying triangular signal within frequency f p and the sinusoidal modulating signal within frequency f mod . The triangular signals, noted Pk are generated according to: Pk =
1 2 2 arcsin sin( t jk + ) +1 2 p 2
(12)
leading to:
1 2 arcsin sin(2f p t 1 + ) +1 2 2 1 2 P2 = arcsin sin(2f p t 2 + +1 2 2 . . . 1 2 Pp = arcsin sin(2f p t p + +1 2 2
P1 =
(13)
The angle jk will be selected with equal values for all Pk i.e.
jk = (k 1)
2 p
(14)
The N-PWM method consists to determine the control signal u jk obtained according to • If mod jk p k then u jk = 1 • if mod jk p k then u jk = 0 with sinusoidal modulation signals, mod jk described by the following set of equations for the three phases mod1k = r.sin(2f mod t) 2 ) 3 2 = r.sin(2f mod t ) 3
mod 2k = r.sin(2f mod t mod 3k
(15)
In Eqs. 15, r represents the modulation depth varying between 0 and 1. The choice of a regular phase of (2 / p) between the different cells brings a significant improvement in the outputvoltage spectrum level. We note that harmonics are gathered in families centered on a multiple frequencies p.m.f mod
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, where m = f p /f mod represents the modulation index. 4. Proportional N-PWM control of the output voltages The proportional control takes into the count only the regulation of the floating voltage. This method is based on a modulated duty cycle. During operating mode of p-cells inverter, the law of control must take into account the sign of the load current which is sinusoidal M jp = M jref M jk = M K+1 sign(i chj ).G.(
C jk Tdec .I chjmax
).(
k.E VCjk ) p
with sign(i chj ) represents the sign of load current
M jref =
(16)
and defined by M jref
p mod 1 .sin(2f mod t) + 2 2
(17)
Moreover, the maximum value i chjmax of the current load is to be considered. Figure 2 presents the control bloc diagram.
Fig 2. Control proportional of the output voltages
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The adequate choice of the constant G allows ensuring a non-saturation of the manipulated variables. In the case of the starting operating phase of the inverter with null voltages, we can obtain G as the follow:
M j1init = M jref sign(i chj ).
p(p 1) G j .C jk E . . Tdec .Ichjmax p 2
(18)
Thus from Eq. 18, G can be deduced according to M j1init = 0 with the reference modulating of M jref = 1 and the 2
load current positive corresponding to sign(i chj ) = 1 , given
Gj =
Tdec .Ichjmax
(19)
C jk .(p 1).E
5. Simulation results and discussions Simulations were performed under Matlab/ Simulink environment. The parameters of the studied series three-phases five levels flying capacitor inverter (having four cells) connected to an RL load is summarized in table 1: Table 1. Parameters of the studied three-phases filing capacitor multilevel inverter E
Lchj
Rchj
Cjk
fmod
Fdéc
m
r
400VDC
0.5mH
10
40nF
50KHZ
5MHZ
180
0.8
The chronograms in Fig 3 present the shape of the control signals of the various switch states varying between 0 and 1 in the inverter.
Fig. 3. Control signals of one inverter arm.
We clearly see in Figs. 3 the modulated duty cycle associated to the natural pulse width modulation NP-PWM. Within these control signals, the floating-capacitor voltages according to the two compared control techniques are
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shown in Fig.4. We have plotted in Fig.4 the floating capacitor voltages when inverter is controlled by the N-PWM technique. It clearly shows disturbances in the response of the system with appearance of ripples at about 10% of the mean values of the voltages recorded at terminals of the floating-capacitors. In order to suppress these fluctuations, the control was performed by the NP-PWM technique and floating capacitor voltages are represented in Fig. 4.b. We clearly observed that the implementation of such control hugely improves the voltage quality by a quasisuppression, (down to less than 1% of the mean values) of the voltage ripples.
Fig. 4.a: Floating capacitor voltages when inverter is controlled by the N-PWM technique. b: Floating capacitor voltages when inverter is controlled by proportional-N-PWM technique.
According to these results, we present in Fig. 5 the inverter output voltage and in Fig. 6 the load voltage in the case of the four levels flying-capacitor inverter. These results show the performances of our proposed control method.
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Fig. 5. The sinusoidal modulating signal.
Fig. 6. Phase voltage for four level flying capacitor inverter (load voltage)
Fig.7. FFT harmonic spectrum of the modulating voltage.
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We have characterized the FFT of the sinusoidal modulating voltage and we present the spectrum in Fig. 7. The output voltage obtained presents a sinusoidal shape and is closest to a pure sine wave with a total harmonic distortion, THD equal to 4.02%. We observe in the spectrum of Fig. 7 that harmonics are grouped into two families of multiple frequencies centered around pmfmod and at the double, respectively, corresponding in our case m = 180, p = 4 therefore mp = 720. 6. Conclusion In this paper, we have presented a detail analysis of a flying capacitor multilevel inverter for regulation of capacitor voltages controlled by natural pulse width modulation NP-PWM and associated to a modulated duty cycle. A threephases flying capacitor multilevel inverter is implemented instead to have a balanced distribution of the voltage. We have presented the theoretical analysis, design, and simulations under Simulink/Matlab environment of the inverters built with four levels. This four-levels flying capacitor inverter generates a sinusoidal waveform within a minimum of distortion increasing the overall efficiency of existing inverters based on N-PWM control allowing, now a THD achieving a low value equal to 4.02%. Compared to the N-PWM control technique, the originally developed PNPWM technique shows satisfactory results with a huge increase of the quality of the output signal with a decreasing of the amplitude of the ripples of the floating capacitor voltages. References [1] [2] [3] [4] [5] [6] [7] [8]
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