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  1. 1. Anil Kumar Baddipudi, Chintapally Archana / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 Vol. 2, Issue 5, September- October 2012, pp.308-314 A Novel Fast Acting PI based DSTATCOM 1 Anil Kumar Baddipudi, 2Chintapally Archana 1 Assistant Professor Department of Electrical and Electronics Engineering, S.V.C.E, Tirupathi, Chitoor (Dt), A.P, India. 2 Assistant Professor Department of Electrical & Electronics Engineering, Viswa Bharathi Institute of Technology and Sciences, Nadargul,Rangareddy(Dt), A.P, India.Abstract- The instantaneous response of the The shunt-connected custom power device,distribution static compensator (DSTATCOM) is called the DSTATCOM, injects current at the pointvery important while compensating rapidly of common coupling (PCC) so that harmonicvarying balanced, unbalanced and nonlinear filtering, power factor correction, and loadloads. Any change in the load affects the dc-link balancing can be achieved. The DSTATCOMvoltage directly. The proper operation of consists of a current-controlled voltage-sourceDSTATCOM requires variation of the dc-link inverter (VSI) which injects current at the PCCvoltage within the prescribed limits. through the interface inductor. The operation of VSIConventionally, a proportional-integral (PI) is supported by a dc storage capacitor with proper dccontroller is used to maintain the dc-link voltage voltage across it. One important aspect of theto the reference value. It uses deviation of the compensation is the extraction of reference currents.capacitor voltage from its reference value as its Various control algorithms are available to computeinput. However, the transient response of the the reference compensator currents. However,conventional PI dc-link voltage controller is slow. instantaneous symmetrical component theory isIn this paper, a fast-acting dc-link voltage preferred. Based on this algorithm, the compensatorcontroller based on the energy of a dc-link reference currents are given as follows:capacitor is proposed. Mathematical equations 𝑉𝑠𝑎 + 𝛾 𝑉𝑠𝑏 − 𝑉𝑠𝑐 𝑖 ∗ = 𝑖 𝑙𝑎 − 𝑓𝑎 2 𝑃𝑙𝑎𝑣𝑔 + 𝑃 𝑑𝑐are given to compute the gains of the 𝑖=𝑎,𝑏,𝑐 𝑉𝑠𝑖conventional controller based on fast-acting dc-link voltage controllers to achieve similar fast 𝑉𝑠𝑏 + 𝛾 𝑉𝑠𝑐 − 𝑉𝑠𝑎transient response. The detailed simulations are 𝑖 ∗ = 𝑖 𝑙𝑏 − 𝑓𝑏 2 𝑃𝑙𝑎𝑣𝑔 + 𝑃 𝑑𝑐 𝑖=𝑎,𝑏,𝑐 𝑉𝑠𝑖carried out on MATLAB environment to validatethe proposed controller. 𝑉𝑠𝑐 + 𝛾 𝑉𝑠𝑎 − 𝑉𝑠𝑏 𝑖 ∗ = 𝑖 𝑙𝑐 − 𝑓𝑐 2 (𝑃𝑙𝑎𝑣𝑔 + 𝑃 𝑑𝑐 )Keywords- DC-link voltage controller, 𝑖=𝑎,𝑏,𝑐 𝑉𝑠𝑖DSTATCOM, fast transient response, THD, loadcompensation, power factor, power quality (PQ), where γ = tan φ / √3, and φ is the desiredunbalance, voltage-source inverter (VSI), phase angle between the supply voltages andProportional-Integral (PI) control, CHB compensated source currents in the respectivemultilevel inverter, D-Q reference frame theory. phases. The term Plavg is the dc or average value of the load power. The term Pdc accounts for the losses I. INTRODUCTION in the VSI without any dc loads in its dc link. To The proliferation of power-electronics- generate Pdc, a suitable closed-loop dc-link voltagebased equipment, nonlinear and unbalanced loads, controller should be used, which will regulate the dchas aggravated the power-quality (PQ) problems in voltage to the reference value. The transientthe power distribution network. They cause performance of the compensator is decided by theexcessive neutral currents, overheating of electrical computation time of average load power and lossesapparatus, poor power factor, voltage distortion, in the compensator. In most DSTATCOMhigh levels of neutral-to-ground voltage, and applications, losses in the VSI are a fraction of theinterference with communication systems [1], [2]. average load power. Therefore, the transientThe literature says that the evolution of different performance of the compensator mostly depends oncustom power devices to mitigate the above power- the computation of Plavg. In this paper, Plavg isquality problems by injecting voltages/currents or computed by using a moving average filter (MAF)both into the system [3]–[6]. to ensure fast dynamic response. The settling time of the MAF is a half-cycle period in case of odd 308 | P a g e
  2. 2. Anil Kumar Baddipudi, Chintapally Archana / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 Vol. 2, Issue 5, September- October 2012, pp.308-314harmonics and one cycle period in case of even magnitude of the D-STATCOM output voltagesharmonics presence in voltages and currents. allows effective control of active and reactive powerAlthough the computation of Pdc is generally slow exchanges between the DSTATCOM and the acand updated once or twice in a cycle, being a small system. Such configuration allows the device tovalue compared Plavg to, it does not play a significant absorb or generate controllable active and reactiverole in transient performance of the compensator. In power.some applications, such as the telecommunicationsindustry uses several parallel-connected switch-mode rectifiers to support dc bus voltage. Such anarrangement draws nonlinear load currents from theutility. This causes poor power factor and, hence,more losses and less efficiency. Clearly, there arePQ issues, such as unbalance, poor power factor,and harmonics produced by telecom equipment inpower distribution networks. Therefore, thefunctionalities of the conventional DSTATCOM Figure – 1 Schematic Diagram of ashould be increased to mitigate the aforementioned DSTATCOMPQ problems and to supply the dc loads from its dclink as well. The load sharing by the ac and dc bus The VSC connected in shunt with the ac systemdepends upon the design and the rating of the VSI. provides a multifunctional topology which can beHere, there are two important issues. The first one is used for up to three quite distinct purposes:the regulation of the dc-link voltage within 1. Voltage regulation and compensation of reactiveprescribed limits under transient load conditions. power;The second one is the settling time of the dc–link 2. Correction of power factorvoltage controller. Conventionally, a PI controller is 3. Elimination of current harmonics.used to maintain the dc-link voltage. It uses the Here, such device is employed to providedeviation of the capacitor voltage from its reference continuous voltage regulation using an indirectlyvalue as its input. However, the transient response controlled converter. As shown in Figure-1 theof the conventional dc-link voltage controllers is shunt injected current Ish corrects the voltage sag byslow, especially in applications where the load adjusting the voltage drop across the systemchanges rapidly. In this paper, a fast-acting dc-link impedance Zth. The value of Ish can be controlled byvoltage controller based on the dc-link capacitor adjusting the output voltage of the converter. Theenergy is proposed. The detailed modeling and shunt injected current Ish can be written as,simulation verifications are carried out by usingMATLAB environment to prove the efficacy of this Ish = IL – IS = IL – ( Vth – VL ) / Zthfast-acting dc-link voltage controller. There is no (1)systematic procedure to design the gains of theconventional PI controller used to regulate the dc- Ish /η = IL / θ (2)link voltage of the DSTATCOM. But some,mathematical equations are given to design the gains The complex power injection of the D-STATCOMof the conventional controller based on the fast- can be expressed as,acting dc-link voltage controllers to achieve similarfast transient response. Ssh = VL Ish* (3) II.DESIGN OF MULTILEVEL BASED It may be mentioned that the effectivenessDSTATCOM of the DSTATCOM in correcting voltage sagA. Principle of DSTATCOM depends on the value of Zth or fault level of the load bus. When the shunt injected current Ish is kept in A D-STATCOM (Distribution Static quadrature with VL, the desired voltage correctionCompensator), which is schematically depicted in can be achieved without injecting any active powerFigure-1, consists of a two-level Voltage Source into the system. On the other hand, when the valueConverter (VSC), a dc energy storage device, a of Ish is minimized, the same voltage correction cancoupling transformer connected in shunt to the be achieved with minimum apparent power injectiondistribution network through a coupling transformer. into the system.The VSC converts the dc voltage across the storagedevice into a set of three-phase ac output voltages. B. Control for Harmonics CompensationThese voltages are in phase and coupled with the ac The Modified Synchronous Frame methodsystem through the reactance of the coupling called the instantaneous current component (id-iq)transformer. Suitable adjustment of the phase and method that is similar to the Synchronous Reference 309 | P a g e
  3. 3. Anil Kumar Baddipudi, Chintapally Archana / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 Vol. 2, Issue 5, September- October 2012, pp.308-314Frame theory (SRF) method. The transformation control problems. The control signal from PIDangle is now obtained with the voltages of the ac controller to regulate dc link voltage is expressed asnetwork. The major difference is that, due to voltage Uc = Kp (Vdc ref – Vdc) + 𝐾 𝑖 𝑉 𝑑𝑐 𝑟𝑒𝑓 − 𝑉 𝑑𝑐 𝑑𝑡 +harmonics and imbalance, the speed of the reference 𝐾 𝑑 𝑑/𝑑𝑡(𝑉 𝑑𝑐 𝑟𝑒𝑓 − 𝑉 𝑑𝑐 ) (7)frame is no longer constant. It varies instantaneouslydepending of the waveform of the 3-phase voltage In (7), Kp, Ki and Kd are proportional,system. In this method the compensating currents integral, and derivative gains of the PID controller,are obtained from the instantaneous active and respectively. The proportional term provides overallreactive current components of the nonlinear load. control action proportional to the error signal. AnIn the same way, the mains voltages V (a,b,c) and increase in proportional controller gain (Kp) reducesthe available currents il (a,b,c) in α-β components rise time and steady-state error but increases themust be calculated as given by (4), where C is overshoot and settling time. An increase in integralClarke Transformation Matrix. However, the load gain (Ki) reduces steady state error but increasescurrent components are derived from a SRF based overshoot and settling time. Increasing derivativeon the Park transformation, where „θ‟ represents the gain (Kd) will lead to improved stability. However,instantaneous voltage vector angle (5). practitioners have often found that the derivative term can behave against anticipatory action in case 𝐼 𝑙𝑎 𝐼 𝑙𝛼 of transport delay. A cumbersome trial-and-error 𝐼 𝑙𝑏 𝐼 𝑙𝛽 = 𝐶 (4) method to tune its parameters made many 𝐼 𝑙𝑐 practitioners switch off or even exclude the derivative term. Therefore, the description of𝐼 𝑙𝑑 𝑐𝑜𝑠𝜃 𝑠𝑖𝑛𝜃 𝐼 𝑙𝛼 𝑉𝛽 −1 conventional and the proposed fast-acting dc-link𝐼 𝑙𝑞 = −𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝜃 𝐼 𝑙𝛽 , 𝜃 = 𝑡𝑎𝑛 𝑉𝛼 (5) voltage controllers using PI controllers are given in the following subsections. A. Conventional DC-Link Voltage Controller The conventional PI controller used for maintaining the dc-link voltage is shown in figure-6. To maintain the dc-link voltage at the reference value, the dc-link capacitor needs a certain amount of real power, which is proportional to the difference between the actual and reference voltages. The power required by the capacitor canFigure-2 Block diagram of SRF method be expressed as follows: Pdc = Kp (Vdc ref – Vdc) + 𝐾 𝑖 𝑉 𝑑𝑐 𝑟𝑒𝑓 − 𝑉 𝑑𝑐 𝑑𝑡 Figure-2 shows the block diagram SRF (8)method. Under balanced and sinusoidal voltageconditions angle θ is a uniformly increasing functionof time. This transformation angle is sensitive tovoltage harmonics and unbalance; therefore dθ/dtmay not be constant over a mains period. Withtransformation given below the direct voltagecomponent .III. DC-LINK VOLTAGE CONTROLLERS Figure-6 Schematic diagram of the conventional dc- The sudden removal of load would result in link voltage controlleran increase in the dc-link voltage above thereference value, whereas a sudden increase in loadwould reduce the dc-link voltage below its referencevalue. As mentioned before, the source supplies anunbalanced nonlinear ac load directly and a dc loadthrough the dc link of the DSTATCOM, as shown infigure-1. Due to transients on the load side, the dcbus voltage is significantly affected. To regulate this Figure-7 Schematic diagram of the fast-acting dc-dc-link voltage, closed-loop controllers are used. link voltage controllerThe proportional-integral-derivative (PID) controlprovides a generic and efficient solution to many The dc-link capacitor has slow dynamics compared to the compensator, since the capacitor 310 | P a g e
  4. 4. Anil Kumar Baddipudi, Chintapally Archana / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 Vol. 2, Issue 5, September- October 2012, pp.308-314 voltage is sampled at every zero crossing of phase gain is selected, integral gain is tuned around and supply voltage. The sampling can also be performed chosen to be 0.5. It is found that if Kie is greater than at a quarter cycles depending upon the symmetry of Kpe / 2, the response tends to be oscillatory and if Kie the dc-link voltage waveform. The drawback of this is less than Kpe / 2, then response tends to be conventional controller is that its transient response sluggish. Hence, Kie is chosen to be Kpe / 2. is slow, especially for fast-changing loads. Also, the design of PI controller parameters is quite difficult IV. DESIGN OF CONVENTIONAL for a complex system and, hence, these parameters CONTROLLER BASED ON THE FAST- are chosen by trial and error. Moreover, the dynamic ACTING DC-LINK VOLTAGE response during the transients is totally dependent CONTROLLER on the values of Kp, and Ki, when Pdc is comparable The conventional dc-link voltage controller to Plavg. can be designed based on equations given for the fast-acting dc-link voltage controller as in (11) and B. Fast-Acting DC Link Voltage Controller can be written as To overcome the disadvantages of the 𝑃 𝑑𝑐 = 𝐾 𝑝𝑒 𝑉 𝑑𝑐 𝑟𝑒𝑓 + 𝑉 𝑑𝑐 𝑉 𝑑𝑐 𝑟𝑒𝑓 − 𝑉 𝑑𝑐 + aforementioned controller, an energy-based dc-link voltage controller is proposed. The energy required 𝐾 𝑖𝑒 𝑉 𝑑𝑐 𝑟𝑒𝑓 + 𝑉 𝑑𝑐 𝑉 𝑑𝑐 𝑟𝑒𝑓 − 𝑉 𝑑𝑐 𝑑𝑡 (13) by the dc-link capacitor (Wdc) to charge from actual voltage (Vdc) to the reference value (Vdc ref) can be It can also be written as computed as 𝑃 𝑑𝑐 = 𝐾 ′𝑝 𝑉 𝑑𝑐 𝑟𝑒𝑓 − 𝑉 𝑑𝑐 + 𝐾 𝑖′ 𝑉 𝑑𝑐 𝑟𝑒𝑓 − 𝑉 𝑑𝑐 𝑑𝑡 (14) Wdc = ½ Cdc (𝑉 2 𝑟𝑒𝑓 − 𝑉 2 ) 𝑑𝑐 𝑑𝑐 (9) Where 𝐾 ′𝑝 = 𝐾 𝑝𝑒 (𝑉 𝑑𝑐 𝑟𝑒𝑓 + 𝑉 𝑑𝑐 ) (15) In general, the dc-link capacitor voltage has ripples 𝐾 𝑖′ = 𝐾 𝑖𝑒 (𝑉 𝑑𝑐 𝑟𝑒𝑓 + 𝑉 𝑑𝑐 (16) with double frequency, that of the supply frequency. The dc power (Pdc) required by the dc-link capacitor It is observed from the aforementioned equations is given as that the gains of proportional and integral controllers 𝑊 1 𝑃′𝑑𝑐 = 𝑇 𝑑𝑐 = 2𝑇 𝐶 𝑑𝑐 (𝑉 2 𝑟𝑒𝑓 − 𝑉 2 ) (10) 𝑑𝑐 𝑑𝑐 vary with respect to time. However, for small 𝑐 𝑐 ripples in the dc-link voltage, Vdc ≈ Vdc ref, therefore, where Tc is the ripple period of the dc-link capacitor we can approximate the above gains to the following voltage. However, due to the lack of integral term, there is a steady-state error while compensating the 𝐾 ′𝑝 ≈ 2𝐾 𝑝𝑒 𝑉 𝑑𝑐 𝑟𝑒𝑓 combined ac and dc loads. This is eliminated by including an integral term. The input to this The relations give approximate gains for a controller is the error between the squares of conventional PI controller. This is due to the fact reference and the actual capacitor voltages. This that Vdc ref + Vdc is not really equal to 2Vdc ref until controller is shown in figure-7 and the total dc variation in Vdc is small during transients. Hence, power required by the dc-link capacitor is computed the designed conventional PI controller works only as follows: on approximation. The open-loop gains for the two 𝐾 ′Pdc = Kpe (𝑉 2 𝑟𝑒𝑓 − 𝑉 2 ) + 𝐾 𝑖𝑒 𝑑𝑐 𝑑𝑐 𝑉 2 𝑟𝑒𝑓 − 𝑉 2 𝑑𝑡 (11) 𝑑𝑐 𝑑𝑐 𝐾 ′𝑝 (𝑠+ ′ 𝑖 ) 𝐾𝑝 𝑃 𝑑𝑐 cases are given by = 𝐸𝑟 𝑠 The coefficients Kpe and Kie are the proportional and 𝐾1 𝑖 integral gains of the proposed energy-based dc-link where Er = Vdc ref - Vdc. Since 𝐾1 is the same as Kie / 𝑝 voltage controller. As an energy-based controller, it Kpe, the higher gain in the conventional PI controller gives fast response compared to the conventional PI renders less stability than that of the proposed controller. Thus, it can be called a fast acting dc-link energy-based dc-link controller. For nearly the same voltage controller. The ease in the calculation of the performance, the conventional PI controller has proportional and integral gains is an additional gains which are 364 (40/0.11 from Table-III) times advantage. The value of the proportional controller larger than that of that proposed one. Also, the gain (Kpe) can be given as amplifier units used to realize these gains need more design considerations and are likely to saturate when Kpe = Cdc / 2Tc (12) used with higher gains. For example, if the value of dc-link capacitor is 2200 μF and the capacitor voltage ripple period as 0.01 s, then Kpe is computed as 0.11 by using (12). The selection of Kie depends upon the tradeoff between the transient response and overshoot in the compensated source current. Once this proportional 311 | P a g e
  5. 5. Anil Kumar Baddipudi, Chintapally Archana / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 Vol. 2, Issue 5, September- October 2012, pp.308-314TABLE -III: SIMULATION PARAMETERS V. SELECTION OF THE DC-LINK Figure-8 Matlab/Simulink power circuitCAPACITOR model of DSTATCOM The value of the dc-link capacitor can beselected based on its ability to regulate the voltage Figure-9 shows the three phase sourceunder transient conditions. voltages, three phase source currents and loadLet us assume that the compensator in Fig. 1 is currents respectively without DSTATCOM. It isconnected to clear that without DSTATCOM load current anda system with the rating of X kilovolt amperes. The source currents are of the system is given by X x 1000 J/s. Letus further assume that the compensator deals withhalf (i.e.X/2) and twice (i.e., 2X) capacity under thetransient conditions for cycles with the systemvoltage period of Ts. Then, the change in energy tobe dealt with by the dc capacitor is given asΔE = (2X – X/2) nT (21)Now this change in energy (21) should be supportedby the energy stored in the dc capacitor. Let usallow the dc capacitor to change its total dc-linkvoltage from 1.4 Vm to 1.8 Vm during the transientconditions where Vm is the peak value of phasevoltage. Hence, we can write½ Cdc [(1.8Vm)2 – (1.4Vm)2] = (2X – X/2) nT (22) Figure-9 Source voltage, current and load currentwhich implies that without DSTATCOM 3𝑋𝑛𝑇Cdc = (1.8𝑉 )2 −(1.4𝑉 )2 (23) Figure-10 shows the three phase source 𝑚 𝑚 voltages, three phase source currents and loadFor example, consider a 10-kVA system (i.e., X = currents respectively with DSTATCOM. It is clear10 kVA), system peak voltage Vm = 325.2 V, n = that with DSTATCOM even though load current is0.5, and T = 0.02 s. The value of Cdc computed non sinusoidal source currents are sinusoidal.using (23) is 2216 μF. Practically, 2000 μF isreadily available and the same value has been takenfor simulation and experimental studies.IV MATLAB/SIMULINK MODELING ANDSIMULATION RESULTS Figure-8 shows the Matab/Simulink powercircuit model of DSTATCOM. It consists of fiveblocks named as source block, non linear loadblock, control block, APF block and measurementsblock. The system parameters for simulation studyare source voltage of 11kv, 50hz AC supply, DC bus Figure-10 Source voltage, current and load currentcapacitance 1550e-6 F, Inverter series inductance 10 with DSTATCOMmH, Source resistance of 0.1 ohm and inductance of0.9 mH. Load resistance and inductance are chosen Figure-11 shows the DC bus voltage. Theas 30mH and 60 ohms respectively. DC bus voltage is regulated to 11kv by using PI regulator. 312 | P a g e
  6. 6. Anil Kumar Baddipudi, Chintapally Archana / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 Vol. 2, Issue 5, September- October 2012, pp.308-314 Figure-14 Harmonic spectrum of Phase-A Source current without DSTATCOM Figure-15 shows the harmonic spectrum of Phase – A Source current with DSTATCOM. The THD of Figure-11 DC Bus Voltage with PI source current without DSTACOM is 5.05%controller Figure-15 Harmonic spectrum of Phase-A Source current with DSTATCOM VI CONCLUSION Figure-12 DC Bus Voltage with fast PI A new control scheme which maintains thecontroller power balance at the PCC to regulate the DC capacitor voltages is proposed. A new PI control Figure-12 shows the DC bus voltage with scheme has been implemented to control the VSIfast PI controller. Form the figure it is clear that used in the D-STATCOM. The DC link voltage ofwith fast PI controller DC link voltage controlled the DSTATCOM has been regulated to the referencewith less time. DC link voltage under varying loads. Matlab/Simulink base model is developed and Figure-13 shows the phase-A source simulation results are presented. From the results,voltage and current, even though the load is non the transient response by the conventional controllerlinear RL load the source power factor is unity. and ts= 0.24sec for DC link voltage, where the transient response by using fast acting DC link controller is ts=0.12sec for voltage. Finally, it is concluded that by fast acting DC-link control the settling time for DC link capacitor is reduced. References [1] K.A Corzine, and Y.L Familiant, “A New Cascaded Multi-level H-Bridge Drive,” Figure-13 Phase-A source voltage and IEEE Trans. Power.Electron., vol.17, no.1,current pp.125-131. Jan 2002. [2] J.S.Lai, and F.Z.Peng “MultilevelFigure-14 shows the harmonic spectrum of Phase – converters – A new bread of converters,A Source current without DSTATCOM. The THD ”IEEE Trans. Ind.Appli., vol.32, no.3,of source current without DSTACOM is 36.89%. pp.509-517. May/ Jun. 1996. [3] T.A.Maynard, M.Fadel and N.Aouda, “Modelling of multilevel converter,” IEEE Trans. Ind.Electron., vol.44, pp.356-364. Jun.1997. [4] P.Bhagwat, and V.R.Stefanovic, “Generalized structure of a multilevel 313 | P a g e
  7. 7. Anil Kumar Baddipudi, Chintapally Archana / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 Vol. 2, Issue 5, September- October 2012, pp.308-314 PWM Inverter,” IEEE Trans. Ind. Appln, Vol.1A-19, no.6, pp.1057-1069, Nov./Dec..1983.[5] J.Rodriguez, Jih-sheng Lai, and F Zheng peng, “Multilevel Inverters; A Survey of Topologies, Controls, and Applications,” IEEE Trans. Ind. Electron., vol.49 , no4., pp.724-738. Aug.2002.[6] Roozbeh Naderi, and Abdolreza rahmati, “Phase-shifted carrier PWM technique for general cascaded inverters,” IEEE Trans. Power.Electron., vol.23, no.3, pp.1257- 1269. May.2008.[7] Bhim Singh, Kamal AlHaddad & Ambrish Chandra, 1999, A Review of Active Filter for Power Quality Improvements, IEEE Trans on Industrial Electronics, 46(5), pp.960970[8] Mauricio Angulo, Pablo Lezana, Samir Kouro, Jos´e Rodr´ıguez and Bin Wu,”Level-shifted PWM for Cascaded Multilevel Inverters with Even Power Distribution” IEEE Power Electronics specialist conference, 17-21 june 2007, pp.2373-2378.[9] B. P. McGrath and D. G. Holmes, “Multicarrier PWM strategies for multilevel inverters,” IEEE Trans. Ind. Electron., vol. 49, no. 4, pp. 858–867, August 2002. 314 | P a g e