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Direct Control Strategy for a Three Phase Eight-Level Flying-Capacitor Inverter
Proceedings of the 20th World Congress Proceedings of 20th The International Federation of Congress Automatic Control Proceedings of the the 20th World World Congress Proceedings of the 20th World Congress The International of Automatic Control Toulouse, France,Federation July 9-14, 2017 Available online at www.sciencedirect.com The International Federation of The International Federation of Automatic Automatic Control Control Toulouse, France, July 9-14, 2017 Toulouse, France, July 9-14, 2017 Toulouse, France, July 9-14, 2017
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IFAC PapersOnLine 50-1 (2017) 15786–15791
Direct Control Strategy for a Three Phase Direct Control Strategy Phase Direct ControlFlying-Capacitor Strategy for for a a Three Three Phase Eight-Level Inverter Eight-Level Flying-Capacitor Inverter Eight-Level Flying-Capacitor Inverter∗∗∗ ∗,∗∗∗ ∗ ∗∗
S. Laamiri ∗,∗∗∗ M. Ghanes ∗ L. Amet ∗∗ G. Santomenna ∗∗∗ ∗,∗∗∗ M. Ghanes ∗ L. Amet ∗∗ G. Santomenna ∗∗∗ S. Laamiri ∗,∗∗∗ S. M. S. Laamiri Laamiri M. Ghanes Ghanes ∗ L. L. Amet Amet ∗∗ G. G. Santomenna Santomenna ∗∗∗ ∗ LS2N/Ecole Centrale de Nantes, 44321 Nantes Cedex 3, France ∗ ∗ LS2N/Ecole Centrale de Nantes, 44321 Nantes Cedex 3, France ∗ LS2N/Ecole Centrale de Nantes, 44321 Nantes (e-mail: [email protected], [email protected]) LS2N/Ecole Centrale de Nantes, 44321 Nantes Cedex Cedex 3, 3, France France ∗∗ (e-mail: [email protected], [email protected]) (e-mail: [email protected], [email protected]) Start up company in inversores solares, Argentina (e-mail: [email protected], [email protected]) ∗∗ ∗∗ Start up company in inversores solares, Argentina ∗∗ Start up company in (e-mail: [email protected]) Start up company in inversores inversores solares, solares, Argentina Argentina ∗∗∗ (e-mail: [email protected]) (e-mail: [email protected]) GS Maintenance Company, 77000 Vaux le p´enil, France (e-mail: [email protected]) ∗∗∗ ∗∗∗ GS Maintenance Company, 77000 Vaux le p´ enil, France ∗∗∗ GS Maintenance Company, (e-mail: [email protected]) GS Maintenance Company, 77000 77000 Vaux Vaux le le p´ p´eenil, nil, France France (e-mail: [email protected]) (e-mail: [email protected]) (e-mail: [email protected]) Abstract: Flying-capacitor converters are a multilevel topology which provide several advanAbstract: Flying-capacitor converters are a multilevel topology which provide several advanAbstract: Flying-capacitor converters are multilevel topology provide several advantages over classic two level converters. a dv and offer a better harmonic Abstract: Flying-capacitor convertersThey are aareduce multilevel topology which which provide several advandt phenomena dv dv tages over classic two level converters. They reduce a phenomena and offer a better harmonic dv tages overOn classic two level level converters. They reduce phenomena and offer offer a better better harmonica content. the other, thanks to theThey seriesreduce connection of commutation cells, they provide dt phenomena tages over classic two converters. aa dt and a harmonic dt content. On the other, thanks to the series connection of commutation cells, they provide content. the thanks to connection of cells, they means increasing power and voltage. Furthermore, for certain failures, their can beaaa content.of On On the other, other, thanks to the the series series connection of commutation commutation cells,structure they provide provide means of increasing power and voltage. Furthermore, for certain failures, their structure can be means of increasing power and voltage. Furthermore, for certain failures, their structure be reconfigured in order to work in degraded mode. However, all these advantages come at the price means of increasing power and voltage. Furthermore, for certain failures, their structure can can be reconfigured in order to work in degraded mode. However, all these advantages come at the price reconfigured in order to work in degraded mode. However, all these advantages come at the price of a more complex control since it is necessary to maintain the flying capacitors voltages at their reconfigured in order to work in degraded mode. However, all these advantages come at the price of a more complex control it is necessary to maintain the flying capacitors voltages at their of a complex control since it to the capacitors voltages at target operating levels. To since overcome this problem and motivated by an industrial application in of a more more complex control since it is is necessary necessary to maintain maintain the flying flying capacitors voltages at their their target operating levels. To overcome this problem and motivated by an industrial application in target operating levels. To overcome this problem and motivated by an industrial application collaboration with GS Maintenance company, a direct control strategy is proposed for a threetarget operating levels. To overcome this problem and motivated by an industrial application in in collaboration with GS Maintenance company, aashowed direct control strategy is proposed for aa threecollaboration with GS Maintenance company, direct control strategy is proposed for threephase eight-level flying capacitor inverter. It is that the control strategy is simpler than collaboration with GS Maintenance company, a direct control strategy is proposed for a threephase eight-level flying capacitor It is showed that control strategy is simpler than phase eight-level flying inverter. It that the control strategy is than other algorithms proposedinverter. in the literature, like forthe example Width Modulation phase control eight-level flying capacitor capacitor inverter. It is is showed showed that the controlPulse strategy is simpler simpler than other control algorithms proposed in the literature, like for example Pulse Width Modulation other control algorithms proposed in the literature, like for example Pulse Width Modulation (PWM) control, and leads less expensive hardware implementations. Moreover itsModulation stability is other control algorithms proposed in the literature, like for example Pulse Width (PWM) control, leads less theory. expensive hardware implementations. its stability is (PWM) control, and leads hardware implementations. Moreover its is proved meansand of Lyapunov This approach is compared to Moreover the conventional PWM (PWM)by control, and leads less less expensive expensive hardware implementations. Moreover its stability stability is proved by means of Lyapunov theory. This approach is compared to the conventional PWM proved by means of Lyapunov theory. This approach is compared to the conventional PWM control to show its good performances during disturbances through simulation results. proved by means of Lyapunov theory. This approach is compared to the conventional PWM control to show its good performances during disturbances through simulation results. control to its during through simulation results. control to show show its good good performances performances during disturbances disturbances through simulation © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. Allresults. rights reserved. Keywords: Eight-level flying capacitor inverter, direct control, PWM control, robustness. Keywords: Eight-level flying capacitor inverter, direct control, PWM control, robustness. Keywords: Keywords: Eight-level Eight-level flying flying capacitor capacitor inverter, inverter, direct direct control, control, PWM PWM control, control, robustness. robustness. 1. INTRODUCTION the inverter power output Ladoux (2009), Stala (2009), 1. INTRODUCTION INTRODUCTION the inverter inverter power output output Ladoux (2009), (2009), Stala (2009), (2009), 1. the power Djondin´ e (2015). 1. INTRODUCTION the inverter power output Ladoux Ladoux (2009), Stala Stala (2009), Djondin´ee (2015). (2015). Djondin´ However, it is necessary to balance the floating voltages e (2015). Numerous industrial applications have begun to require Djondin´ However, it evenly is necessary necessary to balance thevoltage floating voltages However, it the voltages in order to share to thebalance DC bus across the industrial applications have begun to require require However, it is is necessary to balance the floating floating voltages Numerous industrial applications have begun to aNumerous high power level in recent years. Motor drives, active Numerous industrial applications have begun to require in in order to evenly share the DC bus voltage across the order to evenly share the DC bus voltage across the switching cells. Many control algorithms are proposed in a high power level in recent years. Motor drives, active in order to evenly share the DC bus voltage across the apower power level Motor active power years. compensation and Flexible a high high filters, power reactive level in in recent recent years. Motor drives, drives, active switching switching cells. Many control algorithms are proposed in cells. Many control algorithms are proposed in the literature such as PWM control Hemici (2015), Ghias power filters, filters,Current reactiveTransmission power compensation compensation and Flexible Flexible switching cells. Many control algorithms are proposed in power reactive power and Alternating System (FACTS) depower filters, reactive power compensation and Flexible the the literature such as PWM control Hemici (2015), Ghias literature such as PWM control Hemici (2015), Ghias (2016), sliding mode control Ben Said (2014). Alternating Current Transmission System (FACTS) (FACTS) de- the literature such as PWM control Hemici (2015), Ghias Alternating Current Transmission System devices require medium voltage and megawatt power level Alternating Current Transmission System (FACTS) de- (2016), (2016), slidingmotivated mode control control Ben Said (2014). (2014). sliding Ben Said this work, by an industrial application with vices require require medium voltage and megawatt power level level In (2016), sliding mode mode control Ben Said (2014). vices medium voltage and megawatt power Meynard (2002), Lezana (2008), Brychc´ ın (2016). vices require medium voltage and megawatt power level In In this work, motivated by an industrial application with this work, motivated by an industrial application with GS Maintenance company, a direct control law combinMeynard (2002), Lezana (2008), Brychc´ ın (2016). In this work, motivated by an industrial application with Meynard (2002), Lezana (2008), Brychc´ (2016). Lower voltage stress can be achieved by ın connecting Meynard (2002), Lezana (2008), Brychc´ ın (2016). power GS GS Maintenance company, a direct control law combinMaintenance company, a direct control law combining the projection strategy Amet (2011) with a classical Lower voltage voltage stress can be achieved by connecting connecting power GS Maintenance company, a direct control law combinLower stress can be achieved by power switchs in series such as the High Voltage Direct Current Lower voltage stress can be achieved by connecting power ing ing the the projection projection strategy Amet (2008) (2011) is with classical strategy Amet aa technique Bethoux proposed for switchs in in series seriestransmission such as as the High High Voltageon Direct Current ing the mode projection strategy Amet (2011) (2011) with with a classical classical switchs Voltage Current (HVDC) in Sweden March 1997 sliding switchs inlight series such such as the the High Voltage Direct Direct Current sliding mode technique Bethoux (2008) is proposed for sliding mode technique Bethoux (2008) is proposed for a three-phase eight-level flying-capacitor inverter. To the (HVDC) light transmission in Sweden on March 1997 sliding mode technique Bethoux (2008) is proposed for (HVDC) light Eriksson (HVDC) (1998). light transmission transmission in in Sweden Sweden on on March March 1997 1997 abest a three-phase eight-level flying-capacitor inverter. To the three-phase eight-level flying-capacitor inverter. To the of our knowledge, type of control is not yet Eriksson (1998). three-phase eight-level this flying-capacitor inverter. To the Eriksson The main(1998). problem for this method is how to guarantee the abest Eriksson (1998). of our our knowledge, this type type of ofinverter control with is not not yet of knowledge, this control is yet designed in the case of three-phase eightThe main main problem for this this method isboth howat to the guarantee the best best of our knowledge, this type of control is not yet The problem for method is how to guarantee the voltage balance among the devices static and The main problem for this method is how to guarantee the designed designed inpresent the case case of isthree-phase three-phase inverter with eightin the of inverter with eightlevel. The work organized as follows: section 2 voltage balance among the devices both at the static and designed in the case of three-phase inverter with eightvoltage balance among the devices both at the static and the dynamic transient states Nguyen (2010). voltage balance among the devices both at the static and level. level. The Theanpresent present work is organized organized as follows: follows: section 2 work is as section 2 presents operating principle of three phase seven cells the dynamic transient states Nguyen (2010). The present work is organized as follows: section 2 the Nguyen As result, atransient multilevelstates converter has(2010). been introduced as level. the adynamic dynamic transient states Nguyen (2010). presents an operating principle of three phase seven cells presents an operating principle of three phase seven cells flying-capacitor converters and describes its mathematical As aaalternative result, aa multilevel multilevel converter has been voltage introduced as presents an operating principle of three phase seven cells As converter been as an in high power and has medium situaAs a result, result, a multilevel converter has been introduced introduced as model. flying-capacitor converters and strategy describesfor itsbalance mathematical and In sectionconverters 3, the control capacan alternative alternative in(2002), high power power and (2002), mediumFazel voltage situa- flying-capacitor flying-capacitor converters and describes describes its its mathematical mathematical an in high and medium voltage situations Rodriguez Lienhardt (2007). an alternative in high power and medium voltage situa- model. model. In section section 3, the the control strategy for balance balance capacIn 3, control strategy for capacitor voltages around their reference values is detailed. In tions Rodriguez (2002), Lienhardt (2002), Fazel (2007). In section 3, the control strategy for balance capactions Rodriguez (2002), Lienhardt (2002), Fazel In particular, flying capacitors converter appeared at 1990s model. tions Rodriguez (2002), Lienhardt (2002), Fazel (2007). (2007). itor voltages around their reference values is detailed. In itor voltages around their reference values is detailed. In section 4, simulation results are given in order to compare In particular, flying capacitors converter appeared at 1990s itor voltages around their reference values is detailed. In In flying capacitors converter Meynard (1992), Djondin´ e (2015). In particular, particular, flying capacitors converter appeared appeared at at 1990s 1990s section section 4, simulation results are given in order to compare 4, simulation results are given in order to compare the performance of the proposed strategy with a classical Meynard (1992), Djondin´ e (2015). section 4, simulation results are given in order to compare Meynard (1992), Djondin´ e (2015). It consists(1992), on theDjondin´ cascaded connexion of switching cells the performance of the proposed strategy with a classical Meynard e (2015). of strategy with PWM control mainly the presence of disturbances (DC It consists consists on the the cascaded connexion of switching switching cells the performance performance of the theinproposed proposed strategy with aa classical classical It cascaded of (complementary switchs) andconnexion floating voltage sources cells (ca- the It consists on on the cascaded connexion of switching cells PWM control control mainly in the the presenceremarks of disturbances disturbances (DC PWM mainly in presence of (DC source). At last, several concluding are given in (complementary switchs) and floating voltage sources (caPWM control mainly in the presence of disturbances (DC (complementary pacitors). (complementary switchs) switchs) and and floating floating voltage voltage sources sources (ca(ca- source). source). At last, several concluding remarks are given in At last, several concluding remarks are given in section 5. pacitors). source). At last, several concluding remarks are given in pacitors). This allowed the distribution of high voltages over more section 5. pacitors). section 5. This allowed allowed the distribution distribution of high high voltages voltages over to more section 5. This the of over more switching elements and,as a consequence, the ability use This allowed the distribution of high voltages over more switching elements and,as and,as consequence, the abilityFlorea to use use switching elements aa ability of semiconductors with smaller blockingthe voltages switching elements and,as a consequence, consequence, the ability to to use of semiconductors semiconductors with(2014). smaller blocking blocking voltages voltages Florea Florea of with smaller (2012), Ponnambalam of semiconductors with smaller blocking voltages Florea (2012), Ponnambalam (2014). (2012), One of Ponnambalam the significant(2014). advantages of multilevel config(2012), Ponnambalam (2014). One of of is thethe significant advantages of the multilevel configOne the significant of uration harmonic reduction in output configwaveOne of the significant advantages advantages of multilevel multilevel configuration is the theincreasing harmonicswitching reductionfrequency in the the output output waveuration is harmonic reduction in waveform without or decreasing uration is the harmonic reduction in the output waveform without without increasing increasing switching switching frequency frequency or or decreasing decreasing form form without increasing switching frequency or decreasing Copyright 16356Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © © 2017 2017, IFAC IFAC (International Federation of Automatic Control) Copyright © 2017 IFAC 16356 Copyright © 2017 IFAC 16356 Peer review under responsibility of International Federation of Automatic Copyright © 2017 IFAC 16356Control. 10.1016/j.ifacol.2017.08.2315
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2.2 State space representation
2. THREE PHASE SEVEN CELLS CONVERTER 2.1 Operating principle The general structure of three phase serial multicellular converter with a midpoint of the DC source is presented in Fig. 1. Each leg is composed by seven cells, separated by six capacitors. The series connection between these cells, linked by capacitors, and a RL load can be considered as continuous sources Meynard (2002), Benmansour (2007), Bethoux (2008), Sadigh (2010), Amet (2011).
A three phase seven cells converter connected to a RL load can be modeling by differential equations giving its state space representation with floating voltages Vckj and load current ij as state variables. Each leg of this converter can be represented by seven following differential equations (4) 1 dVckj = ∆Ukj ij Ckj dt (4) 6 Rj 1 E Vckj dij − ij + (U7j − ) −(∆Ukj ) dt = Lj Lj 2 Lj k=1
where: j : refers to legs a, b or c. k = 1, ..., 6 : represents the number of capacitor voltages. Vckj : Capacitor voltage of leg j. Ckj : Capacitor k of leg j. Ukj : Binary switch k of leg j. ∆Ukj = U(k+1)j − Ukj . ij : Output current of legs j=a,b,c. Rj : Load resistance of leg j. Lj : Load inductance of leg j. 2.3 Output voltage
Fig. 1. Three phase seven cells inverter connected to a RL load Each cell contains two complementary power electronic components controlled by a binary switch Uk , k=[1, 2, ..., 7]. Uk =1 (respectively Uk =0) when the upper switch (respectively lower switch) in cellk is conducting. In order to ensure normal operations, it is necessary to guarantee a balanced distribution of the floating voltages as following Bethoux (2008) E (1) Vck = k 7 where E is the DC source. Each cell k is located between two floating capacitor voltages, so (2) Vcellk = Vck − Vck−1 where Vc0 = 0 and Vc7 = E. Under equilibrium conditions, cell voltages take the same value. Therefore, we obtain E Vcellk = (3) 7 As a result, the electrical stresses on each switch are reduced and more equally distributed as each switch must withstand E7 volts Shukla (2008), Hosseini (2009). Generally, the output voltages of serial multicellular con−E E E E E verter has p + 1 levels are [ −E 2 , ( 2 + p ), ..., ( 2 − p ), 2 ] depending on the configuration of the switchs Meynard (2002), Benmansour (2006), Bethoux (2008), Amet (2011). This means that the dv/dt decreases compared to the ones in classical structures.
In the midpoint of the DC source inverter configuration, voltages output can take symmetrical positive and negative values. The voltage output in each leg of this converter Vj consist of eight voltage levels. It can take: ± E/2, ± E/14, ± 3E/14, ± 5E/14. 3. PROPOSED CONTROL FOR SEVEN CELLS INVERTER 3.1 Control strategy The proposed strategy allows the control of flying capacitor converters, while it is easily scalable and easy to implement. The aim of this strategy is to find the switching state Uk for each leg j that brings errors ∆Vck to zero as fastest as possible. In other words, the switching state for varying voltage capacitors (charging, discharging or not change) is determined in order to converge states Vckj to Vckjref more quickly. Let consider λ the number of upper switchs turned ON. For λ = 0, the output voltage Vj is equal to − E2 . For λ = 7, the output voltage Vj is equal to E2 . For λ = 0 and λ = 7, for each leg, we proposed a method based to computing the projection of the error vector ∆Vck = Vckjref − Vckj on each one of the six possible state evolution V˙ ckj and picking the largest one for each leg j. Let us call Ubestj = {U7j , U6j , U5j , U4j , U3j , U2j , U1j } that yields to the largest projection. This control strategy is summarized as follows Ubestj = {U/ max{∆Vckj .V˙ ckj }} (5)
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Since the goal is to find the largest projection and not its actual value, this computation can be simplified for each leg ij } max{∆Vckj .V˙ ckj } = {max(∆Vckj .∆Ukj ) Ckj |ij | (6) } = {max(∆Vckj .∆Ukj )sign(ij ) Ckj = {max(∆Vckj .∆Ukj )sign(ij )} 3.2 Simplicity of implementation This simplification has major implications on implementation. It is no longer necessary to know the exact value of the current ij , but only its sign. In practice, this is equivalent to know its direction only. This highly simplifies the sensing circuitry, which can be reduced to a single comparator. It also implies that it is no longer necessary to use an analog to digital converter (ADC) in digital systems, which is an expensive resource. It can be replaced by a simple digital input indicating whether the current is positive or negative. In order to show the simplicity of our control strategy, we present in Table 1 capacitor voltages (charging, discharging or floating) in seven cells inverter for a voltage output is − 5E 14 . Table 1. Capacitor voltages evolution in the seven cells inverter with positive load current and λ = 1 Uj [0,0,0,0,0,0,1] [0,0,0,0,0,1,0] [0,0,0,0,1,0,0] [0,0,0,1,0,0,0] [0,0,1,0,0,0,0] [0,1,0,0,0,0,0] [1,0,0,0,0,0,0]
Vc6j ∼ ∼ ∼ ∼ ∼
Vc5j ∼ ∼ ∼ ∼ ∼
Vc4j ∼ ∼ ∼ ∼ ∼
Vc3j ∼ ∼ ∼ ∼ ∼
Vc2j ∼ ∼ ∼ ∼ ∼
where V˙ ckj−ref is considered constant. Therefore, by replacing the current path (7) in (9), the above equality becomes ij V˙ = −∆Vckj sign(∆Vckj )sign(ij ) Ckj (10) |ij | = −|∆Vckj | <0 Ckj Thus the asymptotic stability of the proposed control strategy is ensured. According to Poznyak (2011), this is a sufficient condition for finite-time convergence. Note that when ∆Ukj = 0, the Lyapunov function candidate (10) is constant (V˙ = 0), which means that the control strategy is Lyapunov stable (capacitors are bypassed). 4. SIMULATION RESULTS For the validation of the proposed control, simulations have been carried-out on a three phase seven cells inverter whose parameters are • • • • • •
E = 308 V; Ck j = 470 µF, k=1,...,6; Switching frequency : fs = 2 kHz; Lj = 60 mH; Rj = 5 Ω; Inverter output current frequencyfref = 50 Hz;
4.1 Simulation results of the proposed control Fig.2 shows floating voltages Vck for leg a. Capacitor voltages converge to their reference values 3E 4E 5E 6E [ E7 , 2E 7 , 7 , 7 , 7 , 7 ]. We can remark the presence of chattering phenomenon due of use a sliding mode. We can reduce this phenomenon by decreasing the current or by increasing the switching frequency Amet (2011).
Vc1j ∼ ∼ ∼ ∼ ∼
Where Uj = [U7j , U6j , U5j , U4j , U3j , U2j , U1j ] is the state switching vector for leg j. From example, if the scalar product between ∆Vcj and (0,0,-1,1,0,0) is the maximum, the state switching applied for voltage capacitors balancing is Uj = [0, 0, 0, 1, 0, 0, 0]. 3.3 Stability analysis The control strategy described in Section (3.1) allows to choose the current path ∆Ukj as ∆Ukj = −sign(ij )sign(∆Vckj ) (7) which corresponds to have λ, the number of upper switchs tuned on different from 0 and 7, i.e. λ = 1 − 6. In the case where ∆Ukj = 0, which corresponds to have λ equal to 0 or 1, the capacitors of the inverter are bypassed. In order to prove the stability of the proposed control strategy, let us take the following Lyapunov candidate function 1 (8) V = ∆Vckj 2 2 The first derivative of this function is V˙ = ∆Vckj ∆V˙ ckj (9) ikj = (∆Vckj ∆Ukj ) Ckj
Fig. 2. Leg a floating voltages evolution In order to show the balance capacitor voltages for leg b and leg c, Fig.3 and Fig.4 present respectively Vc3b and Vc6c evolutions. Three phase currents output are shown in Fig.5. After a transient period, currents output are a square wave of 50 Hz with the same amplitude of 8 A and a phase difference of 2π 3 rad. Fig.6 presents a three phase voltages output having a frequency of 50 Hz and taking their eight possible values: ± 22 V, ± 66 V, ± 110 V and ± 154 V.
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Fig. 3. Evolution of the voltage capacitor Vc3 of leg b
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Fig. 6. Phase voltages output evolution waves references. The control signal of each switch is generated by the intersection between carrying triangular signals delayed by 2π 7 within frequency fs and sinusoidal modulating signals. This procedure of control is more complex and needs more computations compared to the proposed control strategy. Transient steady test In this test, the DC bus voltage will swing from 308 V to 350 V at 1 s. In Fig.7 and Fig.8 it is showed the evolution of the capacitor voltage Vc1a . Even applying this disturbance, it can be seen that our control law allows to converge capacitor voltage to its equilibrium state faster than the PWM control. In PWM control strategy, the capacitor voltage reaches its new reference value (50 V) after more than 0.2 s. However, the time convergence of the capacitor voltage to its reference is decreased to be 0.02 s by means of the proposed direct control.
Fig. 4. Evolution of the voltage capacitor Vc6 of leg c
Fig. 5. Phase currents evolution 4.2 Comparison between our control strategy and PWM control In order to test the performance of our control law, it was compared to the closed loop PWM control. In this strategy, six PI controllers are used for balancing the voltage on the capacitors. The inputs to these controllers are the error voltage capacitors ∆Vckj and the outputs are sinusoidal modulating
Fig. 7. Vc1a evolution with DC bus voltage variation in PWM control Steady state test In this test, we will compare the behavior of flying capacitor voltage in the steady state with both methods.
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Fig. 8. Vc1a evolution with DC bus voltage variation in our control strategy The evolution of Vc1a is given by Fig.9. It can be remarked that the PWM control guarantees the balance of capacitor voltages but low frequency component is well removed than our proposed control strategy.
Fig. 10. FFT harmonic spectrum of the output voltage Va using the PWM control (left) and our direct control (right) In steady state conditions, compared to the PWM control, the proposed control strategy decreased the ripple amplitude of floating capacitor voltages and a better quality of the voltage output with a lower harmonic distortion is obtained. Estimating the capacitor voltages by using a software sensor (observer) and an implementation in a real set-up which is in progress at GS Maintenance Company are the perspectives of this work. REFERENCES
Fig. 9. Vc1a evolution in steady state in two cases of control The FFT of the output voltage of leg a of the converter with its spectrum is characterized in Fig.10. We observe in the spectrum of Fig.10 that the first group of harmonics is centered around 14 KHz (7*fs ) which corresponds to the output frequency of the converter and which is a very nice property compared to the classical two level converters, since it is the frequency seen by the load. The PWM control yields T HD = 17.59%, whereas our proposed direct control gives T HD = 14.09%. 5. CONCLUSIONS AND PERSPECTIVES In this work, is studied a three phase seven cells flying capacitor inverter. To guarantee a balanced capacitor voltages, a closed loop control combining both the projection strategy with a sliding mode technique is proposed and its stability analysis is given by means of Lyapunov theory. Simulation results showed that the proposed control gives a faster dynamic response compared to the closed loop PWM control in the presence of an input variation.
Amet, L.; Ghanes, M.; Barbot, J. P. (2011). Direct control based on sliding mode techniques for multicell serial chopper. American Control Conference 2011, pp. 751– 756. Benmansour, K.; Benalia, A.; Djemai, M.; De Leon, J. (2007). Hybrid control of a multicellular converter. Nonlinear Analysis: Hybrid Systems, vol. 1, n. 1 pp. 1619. Benmansour, K.; De leon, J.; Djemai, M. (2006). Adaptive observer for multi-cell chopper. Second International Symposium on Communications, Control and Signal Processing ISCCSP,Marakesh, Morocco. Ben said, S.; Ben saad, K.; Ben rejeb, M. (2014). On two control strategies for multicellular converters. International Journal Of Control, Energy and Electrical Engineering (CEEE), Vol.1, pp.37–42. Bethoux, O.; Barbot, J. P.; Hilairet, M.(2008). Multicell actuator based on a sliding mode control. European Physical Journal - Applied Physics, 43(0), pp. 217-223. Brychc´ın, J.; Jan´ık, D.; Koˇsan, T.; Peroutka, Z. (2016). Modulator for 4-level Flying Capacitor Converter with Balancing Control in the Closed Loop. Transactions on Electrical Engineering, vol. 5, no. 3. Djondin´e, P.; Barbot, J. P.; Ghanes, M. (2015). Nonlinear phenomena study in serial multicell chopper. 4th IFAC Conference on Analysis and Control of Chaotic Systems, Tokyo, Japan. Eriksson, K.; Jonsson, T.; Tollerz, O.(1998). Small Scale Transmission To AC Networks by HVDC Light. 12th Cepsi conferencePattaya, Thailand. Fazel, S. S.; Bernet, S.; Krug, D.; Jalili, K.(2007). Design and Comparison of 4-kV Neutral-Point-Clamped,
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Flying-Capacitor, and Series-Connected H-Bridge Multilevel Converters. IEEE Trans. on Industrial Applications, vol. 43, no. 4, pp. 1032– 1040. ˇ FLOREA, B. C.; STOICHESCU, D.A.; SPATARU, A.(2012). A Direct Control Method For Multicellular C onverters. U.P.B. Sci. Bull.,Series C, Vol. 74, Iss. 3, ISSN 1454-234x. Ghias, A. M. Y. M.; Pou, J.; Agelidis, V. G.(2016). An Active Voltage-Balancing Method Based on PhaseShifted PWM for Stacked Multicell Converters. IEEE Transactions On Power Electronics, vol. 31, n. 3. Hemici, K.; Zegaoui, A.; Bokhtache, A. A.; Mahmoudi, M. O.; Aillerie, M.(2015). Three-Phases Flying-Capacitor Multilevel Inverter with Proportional Natural PWM Control. Energy Procedia,Volume 74, pp. 1061–1070. Hosseini, S. H.; Sadigh, A. K.; Sharifi A.(2009). Estimation of flying capacitors voltages in multicell converters. 6th Int. ECTI Conf., Thailand, vol. 1, pp. 110-113. Ladoux, P.; Machmoum, M.; Batard, C.(2009). Harmonic currents compensation for 1.5 kV DC railway substations. International Review of Electrical Engineering,vol. 4, n. 3, pp. 380-391. Lezana, P.; Rodriguez, J.; Oyarzun, D. A. (2008). Cascaded Multilevel Inverter With Regeneration Capability and Reduced Number of Switchs. IEEE Trans. Ind. Electron, Vol. 55, No. 3, pp. 1059–1066. Lienhardt, A. M.; Gateau, J.; Meynard, T. A.(2002). Zerosteady-state-error input-current controller for regenerative multilevel converters based on single-phase cells. IEEE Trans. Ind. Electron., vol. 54, no. 2, pp. 733- 740. Meynard, T.A; Foch, H. (1992). Dispositif de conversion d’´energie ´electrique ` a semiconducteur. Brevet fran¸cais, 92, 00652(91):09582. Meynard, T. A.; Foch, H.; Forest, F.; Turpin, C.; Richardeau, F.; Delmas, L., Gateau, G.; Lefeuvre, E. (2002). Multicell converters: Derived topologies. IEEE Trans. Ind. Electron, vol. 49, no. 5, pp. 978-987. Nguyen, T. V.; Jeannin, P. O.; Vagnon, E.; Frey, F.; Crebier, J. C.(2010). Series connection of IGBT. APEC 2010, Palm Springs, United States. pp 2238–2244. Ponnambalama, P.; Praveen Kumar. M. M; Mayank Varshney; Dhanesh Choudhary; Kalyani, Ch. (2014). Control of Symmetrical Cascaded Multicell Inverter using Fuzzy Logic. Third International Conference on Advances in Control and Optimization of Dynamical Systems, Kanpur, India. Poznyak, A. S.;, Polyakov, A. Ye.; Strygin, V. V. (2011). Analysis of finite-time convergence by the method of Lyapunov functions in systems with second-order sliding modes. Journal of Applied Mathematics and Mechanics, vol. 75, n. 3, pp. 289–303. Rodriguez, J.; Jih-Sheng Lai; Fang Zheng Peng. Multilevel inverters: a survey of topologies, controls, and applications. IEEE Trans. on Industrial Electronics, 49(4). pp. 724-738. Sadigh, A. K.; Hosseini, S. H.;Sabahi, M.; and Gharehpetian, G. B. (2010). Double Flying Capacitor Multicell Converter Based on Modified Phase-Shifted Pulsewidth Modulation. IEEE Transactions On Power Electronics,VOL. 25, NO. 6. Shukla, A.; Ghosh, A.; Joshi, A. (2008). Improved multilevel hysteresis current regulation and capacitor voltage balancing schemes for flying capacitor multilevel
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Direct Control Strategy for a Three Phase Direct Control Strategy Phase Direct ControlFlying-Capacitor Strategy for for a a Three Three Phase Eight-Level Inverter Eight-Level Flying-Capacitor Inverter Eight-Level Flying-Capacitor Inverter∗∗∗ ∗,∗∗∗ ∗ ∗∗
S. Laamiri ∗,∗∗∗ M. Ghanes ∗ L. Amet ∗∗ G. Santomenna ∗∗∗ ∗,∗∗∗ M. Ghanes ∗ L. Amet ∗∗ G. Santomenna ∗∗∗ S. Laamiri ∗,∗∗∗ S. M. S. Laamiri Laamiri M. Ghanes Ghanes ∗ L. L. Amet Amet ∗∗ G. G. Santomenna Santomenna ∗∗∗ ∗ LS2N/Ecole Centrale de Nantes, 44321 Nantes Cedex 3, France ∗ ∗ LS2N/Ecole Centrale de Nantes, 44321 Nantes Cedex 3, France ∗ LS2N/Ecole Centrale de Nantes, 44321 Nantes (e-mail: [email protected], [email protected]) LS2N/Ecole Centrale de Nantes, 44321 Nantes Cedex Cedex 3, 3, France France ∗∗ (e-mail: [email protected], [email protected]) (e-mail: [email protected], [email protected]) Start up company in inversores solares, Argentina (e-mail: [email protected], [email protected]) ∗∗ ∗∗ Start up company in inversores solares, Argentina ∗∗ Start up company in (e-mail: [email protected]) Start up company in inversores inversores solares, solares, Argentina Argentina ∗∗∗ (e-mail: [email protected]) (e-mail: [email protected]) GS Maintenance Company, 77000 Vaux le p´enil, France (e-mail: [email protected]) ∗∗∗ ∗∗∗ GS Maintenance Company, 77000 Vaux le p´ enil, France ∗∗∗ GS Maintenance Company, (e-mail: [email protected]) GS Maintenance Company, 77000 77000 Vaux Vaux le le p´ p´eenil, nil, France France (e-mail: [email protected]) (e-mail: [email protected]) (e-mail: [email protected]) Abstract: Flying-capacitor converters are a multilevel topology which provide several advanAbstract: Flying-capacitor converters are a multilevel topology which provide several advanAbstract: Flying-capacitor converters are multilevel topology provide several advantages over classic two level converters. a dv and offer a better harmonic Abstract: Flying-capacitor convertersThey are aareduce multilevel topology which which provide several advandt phenomena dv dv tages over classic two level converters. They reduce a phenomena and offer a better harmonic dv tages overOn classic two level level converters. They reduce phenomena and offer offer a better better harmonica content. the other, thanks to theThey seriesreduce connection of commutation cells, they provide dt phenomena tages over classic two converters. aa dt and a harmonic dt content. On the other, thanks to the series connection of commutation cells, they provide content. the thanks to connection of cells, they means increasing power and voltage. Furthermore, for certain failures, their can beaaa content.of On On the other, other, thanks to the the series series connection of commutation commutation cells,structure they provide provide means of increasing power and voltage. Furthermore, for certain failures, their structure can be means of increasing power and voltage. Furthermore, for certain failures, their structure be reconfigured in order to work in degraded mode. However, all these advantages come at the price means of increasing power and voltage. Furthermore, for certain failures, their structure can can be reconfigured in order to work in degraded mode. However, all these advantages come at the price reconfigured in order to work in degraded mode. However, all these advantages come at the price of a more complex control since it is necessary to maintain the flying capacitors voltages at their reconfigured in order to work in degraded mode. However, all these advantages come at the price of a more complex control it is necessary to maintain the flying capacitors voltages at their of a complex control since it to the capacitors voltages at target operating levels. To since overcome this problem and motivated by an industrial application in of a more more complex control since it is is necessary necessary to maintain maintain the flying flying capacitors voltages at their their target operating levels. To overcome this problem and motivated by an industrial application in target operating levels. To overcome this problem and motivated by an industrial application collaboration with GS Maintenance company, a direct control strategy is proposed for a threetarget operating levels. To overcome this problem and motivated by an industrial application in in collaboration with GS Maintenance company, aashowed direct control strategy is proposed for aa threecollaboration with GS Maintenance company, direct control strategy is proposed for threephase eight-level flying capacitor inverter. It is that the control strategy is simpler than collaboration with GS Maintenance company, a direct control strategy is proposed for a threephase eight-level flying capacitor It is showed that control strategy is simpler than phase eight-level flying inverter. It that the control strategy is than other algorithms proposedinverter. in the literature, like forthe example Width Modulation phase control eight-level flying capacitor capacitor inverter. It is is showed showed that the controlPulse strategy is simpler simpler than other control algorithms proposed in the literature, like for example Pulse Width Modulation other control algorithms proposed in the literature, like for example Pulse Width Modulation (PWM) control, and leads less expensive hardware implementations. Moreover itsModulation stability is other control algorithms proposed in the literature, like for example Pulse Width (PWM) control, leads less theory. expensive hardware implementations. its stability is (PWM) control, and leads hardware implementations. Moreover its is proved meansand of Lyapunov This approach is compared to Moreover the conventional PWM (PWM)by control, and leads less less expensive expensive hardware implementations. Moreover its stability stability is proved by means of Lyapunov theory. This approach is compared to the conventional PWM proved by means of Lyapunov theory. This approach is compared to the conventional PWM control to show its good performances during disturbances through simulation results. proved by means of Lyapunov theory. This approach is compared to the conventional PWM control to show its good performances during disturbances through simulation results. control to its during through simulation results. control to show show its good good performances performances during disturbances disturbances through simulation © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. Allresults. rights reserved. Keywords: Eight-level flying capacitor inverter, direct control, PWM control, robustness. Keywords: Eight-level flying capacitor inverter, direct control, PWM control, robustness. Keywords: Keywords: Eight-level Eight-level flying flying capacitor capacitor inverter, inverter, direct direct control, control, PWM PWM control, control, robustness. robustness. 1. INTRODUCTION the inverter power output Ladoux (2009), Stala (2009), 1. INTRODUCTION INTRODUCTION the inverter inverter power output output Ladoux (2009), (2009), Stala (2009), (2009), 1. the power Djondin´ e (2015). 1. INTRODUCTION the inverter power output Ladoux Ladoux (2009), Stala Stala (2009), Djondin´ee (2015). (2015). Djondin´ However, it is necessary to balance the floating voltages e (2015). Numerous industrial applications have begun to require Djondin´ However, it evenly is necessary necessary to balance thevoltage floating voltages However, it the voltages in order to share to thebalance DC bus across the industrial applications have begun to require require However, it is is necessary to balance the floating floating voltages Numerous industrial applications have begun to aNumerous high power level in recent years. Motor drives, active Numerous industrial applications have begun to require in in order to evenly share the DC bus voltage across the order to evenly share the DC bus voltage across the switching cells. Many control algorithms are proposed in a high power level in recent years. Motor drives, active in order to evenly share the DC bus voltage across the apower power level Motor active power years. compensation and Flexible a high high filters, power reactive level in in recent recent years. Motor drives, drives, active switching switching cells. Many control algorithms are proposed in cells. Many control algorithms are proposed in the literature such as PWM control Hemici (2015), Ghias power filters, filters,Current reactiveTransmission power compensation compensation and Flexible Flexible switching cells. Many control algorithms are proposed in power reactive power and Alternating System (FACTS) depower filters, reactive power compensation and Flexible the the literature such as PWM control Hemici (2015), Ghias literature such as PWM control Hemici (2015), Ghias (2016), sliding mode control Ben Said (2014). Alternating Current Transmission System (FACTS) (FACTS) de- the literature such as PWM control Hemici (2015), Ghias Alternating Current Transmission System devices require medium voltage and megawatt power level Alternating Current Transmission System (FACTS) de- (2016), (2016), slidingmotivated mode control control Ben Said (2014). (2014). sliding Ben Said this work, by an industrial application with vices require require medium voltage and megawatt power level level In (2016), sliding mode mode control Ben Said (2014). vices medium voltage and megawatt power Meynard (2002), Lezana (2008), Brychc´ ın (2016). vices require medium voltage and megawatt power level In In this work, motivated by an industrial application with this work, motivated by an industrial application with GS Maintenance company, a direct control law combinMeynard (2002), Lezana (2008), Brychc´ ın (2016). In this work, motivated by an industrial application with Meynard (2002), Lezana (2008), Brychc´ (2016). Lower voltage stress can be achieved by ın connecting Meynard (2002), Lezana (2008), Brychc´ ın (2016). power GS GS Maintenance company, a direct control law combinMaintenance company, a direct control law combining the projection strategy Amet (2011) with a classical Lower voltage voltage stress can be achieved by connecting connecting power GS Maintenance company, a direct control law combinLower stress can be achieved by power switchs in series such as the High Voltage Direct Current Lower voltage stress can be achieved by connecting power ing ing the the projection projection strategy Amet (2008) (2011) is with classical strategy Amet aa technique Bethoux proposed for switchs in in series seriestransmission such as as the High High Voltageon Direct Current ing the mode projection strategy Amet (2011) (2011) with with a classical classical switchs Voltage Current (HVDC) in Sweden March 1997 sliding switchs inlight series such such as the the High Voltage Direct Direct Current sliding mode technique Bethoux (2008) is proposed for sliding mode technique Bethoux (2008) is proposed for a three-phase eight-level flying-capacitor inverter. To the (HVDC) light transmission in Sweden on March 1997 sliding mode technique Bethoux (2008) is proposed for (HVDC) light Eriksson (HVDC) (1998). light transmission transmission in in Sweden Sweden on on March March 1997 1997 abest a three-phase eight-level flying-capacitor inverter. To the three-phase eight-level flying-capacitor inverter. To the of our knowledge, type of control is not yet Eriksson (1998). three-phase eight-level this flying-capacitor inverter. To the Eriksson The main(1998). problem for this method is how to guarantee the abest Eriksson (1998). of our our knowledge, this type type of ofinverter control with is not not yet of knowledge, this control is yet designed in the case of three-phase eightThe main main problem for this this method isboth howat to the guarantee the best best of our knowledge, this type of control is not yet The problem for method is how to guarantee the voltage balance among the devices static and The main problem for this method is how to guarantee the designed designed inpresent the case case of isthree-phase three-phase inverter with eightin the of inverter with eightlevel. The work organized as follows: section 2 voltage balance among the devices both at the static and designed in the case of three-phase inverter with eightvoltage balance among the devices both at the static and the dynamic transient states Nguyen (2010). voltage balance among the devices both at the static and level. level. The Theanpresent present work is organized organized as follows: follows: section 2 work is as section 2 presents operating principle of three phase seven cells the dynamic transient states Nguyen (2010). The present work is organized as follows: section 2 the Nguyen As result, atransient multilevelstates converter has(2010). been introduced as level. the adynamic dynamic transient states Nguyen (2010). presents an operating principle of three phase seven cells presents an operating principle of three phase seven cells flying-capacitor converters and describes its mathematical As aaalternative result, aa multilevel multilevel converter has been voltage introduced as presents an operating principle of three phase seven cells As converter been as an in high power and has medium situaAs a result, result, a multilevel converter has been introduced introduced as model. flying-capacitor converters and strategy describesfor itsbalance mathematical and In sectionconverters 3, the control capacan alternative alternative in(2002), high power power and (2002), mediumFazel voltage situa- flying-capacitor flying-capacitor converters and describes describes its its mathematical mathematical an in high and medium voltage situations Rodriguez Lienhardt (2007). an alternative in high power and medium voltage situa- model. model. In section section 3, the the control strategy for balance balance capacIn 3, control strategy for capacitor voltages around their reference values is detailed. In tions Rodriguez (2002), Lienhardt (2002), Fazel (2007). In section 3, the control strategy for balance capactions Rodriguez (2002), Lienhardt (2002), Fazel In particular, flying capacitors converter appeared at 1990s model. tions Rodriguez (2002), Lienhardt (2002), Fazel (2007). (2007). itor voltages around their reference values is detailed. In itor voltages around their reference values is detailed. In section 4, simulation results are given in order to compare In particular, flying capacitors converter appeared at 1990s itor voltages around their reference values is detailed. In In flying capacitors converter Meynard (1992), Djondin´ e (2015). In particular, particular, flying capacitors converter appeared appeared at at 1990s 1990s section section 4, simulation results are given in order to compare 4, simulation results are given in order to compare the performance of the proposed strategy with a classical Meynard (1992), Djondin´ e (2015). section 4, simulation results are given in order to compare Meynard (1992), Djondin´ e (2015). It consists(1992), on theDjondin´ cascaded connexion of switching cells the performance of the proposed strategy with a classical Meynard e (2015). of strategy with PWM control mainly the presence of disturbances (DC It consists consists on the the cascaded connexion of switching switching cells the performance performance of the theinproposed proposed strategy with aa classical classical It cascaded of (complementary switchs) andconnexion floating voltage sources cells (ca- the It consists on on the cascaded connexion of switching cells PWM control control mainly in the the presenceremarks of disturbances disturbances (DC PWM mainly in presence of (DC source). At last, several concluding are given in (complementary switchs) and floating voltage sources (caPWM control mainly in the presence of disturbances (DC (complementary pacitors). (complementary switchs) switchs) and and floating floating voltage voltage sources sources (ca(ca- source). source). At last, several concluding remarks are given in At last, several concluding remarks are given in section 5. pacitors). source). At last, several concluding remarks are given in pacitors). This allowed the distribution of high voltages over more section 5. pacitors). section 5. This allowed allowed the distribution distribution of high high voltages voltages over to more section 5. This the of over more switching elements and,as a consequence, the ability use This allowed the distribution of high voltages over more switching elements and,as and,as consequence, the abilityFlorea to use use switching elements aa ability of semiconductors with smaller blockingthe voltages switching elements and,as a consequence, consequence, the ability to to use of semiconductors semiconductors with(2014). smaller blocking blocking voltages voltages Florea Florea of with smaller (2012), Ponnambalam of semiconductors with smaller blocking voltages Florea (2012), Ponnambalam (2014). (2012), One of Ponnambalam the significant(2014). advantages of multilevel config(2012), Ponnambalam (2014). One of of is thethe significant advantages of the multilevel configOne the significant of uration harmonic reduction in output configwaveOne of the significant advantages advantages of multilevel multilevel configuration is the theincreasing harmonicswitching reductionfrequency in the the output output waveuration is harmonic reduction in waveform without or decreasing uration is the harmonic reduction in the output waveform without without increasing increasing switching switching frequency frequency or or decreasing decreasing form form without increasing switching frequency or decreasing Copyright 16356Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © © 2017 2017, IFAC IFAC (International Federation of Automatic Control) Copyright © 2017 IFAC 16356 Copyright © 2017 IFAC 16356 Peer review under responsibility of International Federation of Automatic Copyright © 2017 IFAC 16356Control. 10.1016/j.ifacol.2017.08.2315
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2.2 State space representation
2. THREE PHASE SEVEN CELLS CONVERTER 2.1 Operating principle The general structure of three phase serial multicellular converter with a midpoint of the DC source is presented in Fig. 1. Each leg is composed by seven cells, separated by six capacitors. The series connection between these cells, linked by capacitors, and a RL load can be considered as continuous sources Meynard (2002), Benmansour (2007), Bethoux (2008), Sadigh (2010), Amet (2011).
A three phase seven cells converter connected to a RL load can be modeling by differential equations giving its state space representation with floating voltages Vckj and load current ij as state variables. Each leg of this converter can be represented by seven following differential equations (4) 1 dVckj = ∆Ukj ij Ckj dt (4) 6 Rj 1 E Vckj dij − ij + (U7j − ) −(∆Ukj ) dt = Lj Lj 2 Lj k=1
where: j : refers to legs a, b or c. k = 1, ..., 6 : represents the number of capacitor voltages. Vckj : Capacitor voltage of leg j. Ckj : Capacitor k of leg j. Ukj : Binary switch k of leg j. ∆Ukj = U(k+1)j − Ukj . ij : Output current of legs j=a,b,c. Rj : Load resistance of leg j. Lj : Load inductance of leg j. 2.3 Output voltage
Fig. 1. Three phase seven cells inverter connected to a RL load Each cell contains two complementary power electronic components controlled by a binary switch Uk , k=[1, 2, ..., 7]. Uk =1 (respectively Uk =0) when the upper switch (respectively lower switch) in cellk is conducting. In order to ensure normal operations, it is necessary to guarantee a balanced distribution of the floating voltages as following Bethoux (2008) E (1) Vck = k 7 where E is the DC source. Each cell k is located between two floating capacitor voltages, so (2) Vcellk = Vck − Vck−1 where Vc0 = 0 and Vc7 = E. Under equilibrium conditions, cell voltages take the same value. Therefore, we obtain E Vcellk = (3) 7 As a result, the electrical stresses on each switch are reduced and more equally distributed as each switch must withstand E7 volts Shukla (2008), Hosseini (2009). Generally, the output voltages of serial multicellular con−E E E E E verter has p + 1 levels are [ −E 2 , ( 2 + p ), ..., ( 2 − p ), 2 ] depending on the configuration of the switchs Meynard (2002), Benmansour (2006), Bethoux (2008), Amet (2011). This means that the dv/dt decreases compared to the ones in classical structures.
In the midpoint of the DC source inverter configuration, voltages output can take symmetrical positive and negative values. The voltage output in each leg of this converter Vj consist of eight voltage levels. It can take: ± E/2, ± E/14, ± 3E/14, ± 5E/14. 3. PROPOSED CONTROL FOR SEVEN CELLS INVERTER 3.1 Control strategy The proposed strategy allows the control of flying capacitor converters, while it is easily scalable and easy to implement. The aim of this strategy is to find the switching state Uk for each leg j that brings errors ∆Vck to zero as fastest as possible. In other words, the switching state for varying voltage capacitors (charging, discharging or not change) is determined in order to converge states Vckj to Vckjref more quickly. Let consider λ the number of upper switchs turned ON. For λ = 0, the output voltage Vj is equal to − E2 . For λ = 7, the output voltage Vj is equal to E2 . For λ = 0 and λ = 7, for each leg, we proposed a method based to computing the projection of the error vector ∆Vck = Vckjref − Vckj on each one of the six possible state evolution V˙ ckj and picking the largest one for each leg j. Let us call Ubestj = {U7j , U6j , U5j , U4j , U3j , U2j , U1j } that yields to the largest projection. This control strategy is summarized as follows Ubestj = {U/ max{∆Vckj .V˙ ckj }} (5)
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Since the goal is to find the largest projection and not its actual value, this computation can be simplified for each leg ij } max{∆Vckj .V˙ ckj } = {max(∆Vckj .∆Ukj ) Ckj |ij | (6) } = {max(∆Vckj .∆Ukj )sign(ij ) Ckj = {max(∆Vckj .∆Ukj )sign(ij )} 3.2 Simplicity of implementation This simplification has major implications on implementation. It is no longer necessary to know the exact value of the current ij , but only its sign. In practice, this is equivalent to know its direction only. This highly simplifies the sensing circuitry, which can be reduced to a single comparator. It also implies that it is no longer necessary to use an analog to digital converter (ADC) in digital systems, which is an expensive resource. It can be replaced by a simple digital input indicating whether the current is positive or negative. In order to show the simplicity of our control strategy, we present in Table 1 capacitor voltages (charging, discharging or floating) in seven cells inverter for a voltage output is − 5E 14 . Table 1. Capacitor voltages evolution in the seven cells inverter with positive load current and λ = 1 Uj [0,0,0,0,0,0,1] [0,0,0,0,0,1,0] [0,0,0,0,1,0,0] [0,0,0,1,0,0,0] [0,0,1,0,0,0,0] [0,1,0,0,0,0,0] [1,0,0,0,0,0,0]
Vc6j ∼ ∼ ∼ ∼ ∼
Vc5j ∼ ∼ ∼ ∼ ∼
Vc4j ∼ ∼ ∼ ∼ ∼
Vc3j ∼ ∼ ∼ ∼ ∼
Vc2j ∼ ∼ ∼ ∼ ∼
where V˙ ckj−ref is considered constant. Therefore, by replacing the current path (7) in (9), the above equality becomes ij V˙ = −∆Vckj sign(∆Vckj )sign(ij ) Ckj (10) |ij | = −|∆Vckj | <0 Ckj Thus the asymptotic stability of the proposed control strategy is ensured. According to Poznyak (2011), this is a sufficient condition for finite-time convergence. Note that when ∆Ukj = 0, the Lyapunov function candidate (10) is constant (V˙ = 0), which means that the control strategy is Lyapunov stable (capacitors are bypassed). 4. SIMULATION RESULTS For the validation of the proposed control, simulations have been carried-out on a three phase seven cells inverter whose parameters are • • • • • •
E = 308 V; Ck j = 470 µF, k=1,...,6; Switching frequency : fs = 2 kHz; Lj = 60 mH; Rj = 5 Ω; Inverter output current frequencyfref = 50 Hz;
4.1 Simulation results of the proposed control Fig.2 shows floating voltages Vck for leg a. Capacitor voltages converge to their reference values 3E 4E 5E 6E [ E7 , 2E 7 , 7 , 7 , 7 , 7 ]. We can remark the presence of chattering phenomenon due of use a sliding mode. We can reduce this phenomenon by decreasing the current or by increasing the switching frequency Amet (2011).
Vc1j ∼ ∼ ∼ ∼ ∼
Where Uj = [U7j , U6j , U5j , U4j , U3j , U2j , U1j ] is the state switching vector for leg j. From example, if the scalar product between ∆Vcj and (0,0,-1,1,0,0) is the maximum, the state switching applied for voltage capacitors balancing is Uj = [0, 0, 0, 1, 0, 0, 0]. 3.3 Stability analysis The control strategy described in Section (3.1) allows to choose the current path ∆Ukj as ∆Ukj = −sign(ij )sign(∆Vckj ) (7) which corresponds to have λ, the number of upper switchs tuned on different from 0 and 7, i.e. λ = 1 − 6. In the case where ∆Ukj = 0, which corresponds to have λ equal to 0 or 1, the capacitors of the inverter are bypassed. In order to prove the stability of the proposed control strategy, let us take the following Lyapunov candidate function 1 (8) V = ∆Vckj 2 2 The first derivative of this function is V˙ = ∆Vckj ∆V˙ ckj (9) ikj = (∆Vckj ∆Ukj ) Ckj
Fig. 2. Leg a floating voltages evolution In order to show the balance capacitor voltages for leg b and leg c, Fig.3 and Fig.4 present respectively Vc3b and Vc6c evolutions. Three phase currents output are shown in Fig.5. After a transient period, currents output are a square wave of 50 Hz with the same amplitude of 8 A and a phase difference of 2π 3 rad. Fig.6 presents a three phase voltages output having a frequency of 50 Hz and taking their eight possible values: ± 22 V, ± 66 V, ± 110 V and ± 154 V.
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Fig. 3. Evolution of the voltage capacitor Vc3 of leg b
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Fig. 6. Phase voltages output evolution waves references. The control signal of each switch is generated by the intersection between carrying triangular signals delayed by 2π 7 within frequency fs and sinusoidal modulating signals. This procedure of control is more complex and needs more computations compared to the proposed control strategy. Transient steady test In this test, the DC bus voltage will swing from 308 V to 350 V at 1 s. In Fig.7 and Fig.8 it is showed the evolution of the capacitor voltage Vc1a . Even applying this disturbance, it can be seen that our control law allows to converge capacitor voltage to its equilibrium state faster than the PWM control. In PWM control strategy, the capacitor voltage reaches its new reference value (50 V) after more than 0.2 s. However, the time convergence of the capacitor voltage to its reference is decreased to be 0.02 s by means of the proposed direct control.
Fig. 4. Evolution of the voltage capacitor Vc6 of leg c
Fig. 5. Phase currents evolution 4.2 Comparison between our control strategy and PWM control In order to test the performance of our control law, it was compared to the closed loop PWM control. In this strategy, six PI controllers are used for balancing the voltage on the capacitors. The inputs to these controllers are the error voltage capacitors ∆Vckj and the outputs are sinusoidal modulating
Fig. 7. Vc1a evolution with DC bus voltage variation in PWM control Steady state test In this test, we will compare the behavior of flying capacitor voltage in the steady state with both methods.
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Fig. 8. Vc1a evolution with DC bus voltage variation in our control strategy The evolution of Vc1a is given by Fig.9. It can be remarked that the PWM control guarantees the balance of capacitor voltages but low frequency component is well removed than our proposed control strategy.
Fig. 10. FFT harmonic spectrum of the output voltage Va using the PWM control (left) and our direct control (right) In steady state conditions, compared to the PWM control, the proposed control strategy decreased the ripple amplitude of floating capacitor voltages and a better quality of the voltage output with a lower harmonic distortion is obtained. Estimating the capacitor voltages by using a software sensor (observer) and an implementation in a real set-up which is in progress at GS Maintenance Company are the perspectives of this work. REFERENCES
Fig. 9. Vc1a evolution in steady state in two cases of control The FFT of the output voltage of leg a of the converter with its spectrum is characterized in Fig.10. We observe in the spectrum of Fig.10 that the first group of harmonics is centered around 14 KHz (7*fs ) which corresponds to the output frequency of the converter and which is a very nice property compared to the classical two level converters, since it is the frequency seen by the load. The PWM control yields T HD = 17.59%, whereas our proposed direct control gives T HD = 14.09%. 5. CONCLUSIONS AND PERSPECTIVES In this work, is studied a three phase seven cells flying capacitor inverter. To guarantee a balanced capacitor voltages, a closed loop control combining both the projection strategy with a sliding mode technique is proposed and its stability analysis is given by means of Lyapunov theory. Simulation results showed that the proposed control gives a faster dynamic response compared to the closed loop PWM control in the presence of an input variation.
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