An Improved Modulation Strategy Combining Phase Shifted ... - MDPI

energies Article

An Improved Modulation Strategy Combining Phase Shifted PWM and Phase Disposition PWM for Cascaded H-Bridge Inverters Muxuan Xiao, Qianming Xu *

ID

and Honglin Ouyang

College of Electrical and Information Engineering, Hunan University, Changsha 410082, China; [email protected] (M.X.); [email protected] (H.O.) * Correspondence: [email protected]; Tel.: +86-731-8882-3964 Academic Editor: Robert Griñó Received: 10 July 2017; Accepted: 30 August 2017; Published: 2 September 2017

Abstract: Multilevel modulation strategy is an important factor affecting the output performance of multilevel converters. In this paper, the relationship between phase-shifted pulse width modulation (PWM) and phase disposition PWM is analyzed, and then an improved hybrid modulation strategy is proposed for cascaded H-bridge. In addition, the implementation method of multilevel discontinuous modulation for an improved modulation strategy is described. The new modulation strategy is optimized to increase the DC link voltage utilization and further improvement in output harmonics by the injection of a common voltage into the reference. Simulation and experiment verify the effectiveness of the proposed modulation strategy. Keywords: cascaded H-bridge inverter; pulse width modulation; power conversion harmonics

1. Introduction Cascaded H-bridge (CHB) inverters, as one kind of existing commercial topologies of multilevel voltage-source inverters, have received a lot of attention in recent years. The obvious advantages are easily realized higher power, more outstanding harmonics performance, and modularity [1,2]. Many efforts have been made to adapt the original modulation techniques applied to the two-level converter for multilevel inverters [3,4]. Two well-known modulation techniques, phase-shifted PWM (PS-PWM) and phase disposition PWM (PD-PWM), are referred in this paper. The phase-shifted PWM is the standard modulation strategy for cascaded multilevel inverters due to the fact that it can yield a good result in reducing output harmonics and maintaining a power distribution balance among power cells [5,6]. Researchers analyzed the generalized theory of PS-PWM, which obtains the law of full-range phase-shift angle on harmonic minimization for a cascaded H-bridge [7]. Considered the operation condition, an adaptive modulation method in [8] is proposed to improve the operating characteristics under unbalanced conditions. The phase shift angle of the method is changed with the operating conditions. Currently, another research hotspot is based on the carrier phase shift of the capacitor voltage control, by implementing a balanced control algorithm in each carrier cycle to achieve voltage balance [9]. Phase disposition PWM is superior to phase-shifted PWM modulation strategies in getting the lowest harmonic distortion for line voltage, but performs poorly for balancing the power distribution among the cells, which makes it unsuitable for CHB inverters. Much work has been done to modify the phase disposition PWM for CHB inverters to achieve an improved output voltage. In [6], the author offered an equivalent PD-PWM for cascaded inverters. But this strategy can work under discontinuous modulation and a general strategy for balancing the power distribution among the cells is not presented either. Mauricio Angelo proposed a method that applies the PD-PWM to a five-level CHB inverter Energies 2017, 10, 1327; doi:10.3390/en10091327

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successfully in [10]. However, the extension of the method to a higher number of levels is not addressed. In [11], a method is presented to achieve an equivalent PD-PWM for CHB inverters. However, the author ignored the actual working situation of power semiconductors in practice and did not point out the constraint condition needed to be noticed when the strategy is applied to a three-phase system. In [12], a sorting algorithm is proposed to obtain better disposition of gates signals and distribution of switch losses when compared with traditional PD modulation, but is lacking in relevant experimental support. Some researchers have studied the loss and dissipation of the two modulation methods. The power loss and thermal model of the power device in several different modulation strategies [13] are analyzed for the half bridge, providing a reference for the full bridge of this paper, which is mainly from the perspective of total harmonic distortion analysis. A new modulation strategy combining the advantages of PS-PWM and PD-PWM for three-phase CHB inverters is introduced in this paper. The new modulation strategy not only solves the power distribution unbalancing problem in PD-PWM in virtue of features of PS-PWM but maintains the excellent performance in reducing output harmonics in PD-PWM. The new modulation strategy is optimized to increase the DC link voltage utilization and further improvement in output harmonics by the injection of a common voltage into the reference. A concise and effective multilevel discontinuous modulation strategy based on the new modulation strategy for N-level CHB inverters is also proposed to decrease the switch times and reduce the switching loss. Simulation and experimental results are given to prove the effectiveness of the new modulation strategy and discontinuous modulation strategy. 2. Principle of the Proposed Strategy The object of the new strategy is to attain a similar output performance achieved by phase disposition PWM and to take the advantage of phase-shifted PWM in effective utilizing switching frequency and balancing the power distribution among cells. 2.1. New Strategy for Single Phase System A concept of the equivalent carrier is introduced to derive the relationship between both modulation strategies in an intuitive way and the method proposed in [11] is simplified and refined to be more suitable for the practical system. Figure 1a,b illustrates PS-PWM and PD-PWM strategies with an identical reference for a five-level CHB converter, which consists of two H-bridge power cells. In both modulation strategies, four triangular carriers are necessary on account of the number of switch bridges to be driven [10]. The average frequency or the total number of switch transitions is same for both strategies. These two group carriers are subdivided into four intervals equally in the vertical direction by the dotted lines (boundary lines). The boundary lines subdivide the long line segments of carriers to short line segments. When just focusing on short line segments (the black bold lines shown in Figure 1a–c) in the interval where the reference belongs and regard these line segments as integrity, we obtain the equivalent carrier. It can be seen from the Figure 1a–c that the equivalent carrier can be considered as a series of triangle waves distributed in the different interval with the same frequency and the same amplitude. The equivalent carrier is a simplification of the original real carriers and consists of the triangle waves intersecting the reference in each interval. The introduction of equivalent carrier provides an intuitive method to analyze the relationship between the reference and the total output voltage. It can be seen from the figure that the time when the output waveforms change is just consistent with the time when the reference crosses the equivalent carrier. Additionally, it is a good choice to use the equivalent carrier to identify the similarities and differences in total output voltages between different modulation strategies because the equivalent carrier only cares about the total output voltage without focusing on the output of each power cell. Figure 1a,b show that the equivalent carriers for both modulation strategies have the same frequency and the same peak-to-peak amplitude, but differ only

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in phase from θ 1 to θ 2 . It is clearly shown that the output voltages for both strategies also differ only Energies 1327 3 of 14 from θ 12017, to θ10, Figure 1d,e. 2 in

Figure Figure 1. 1. Modulation Modulation strategies strategies and and the the corresponding corresponding output output voltages: voltages: (a) (a) phase-shifted phase-shifted pulse pulse width width modulation modulation (PS-PWM) (PS-PWM) and and phase phase disposition disposition pulse pulse width width modulation modulation (PS-PWM); (PS-PWM); (b) (b) PD-PWM; PD-PWM; (c) (c) New strategy; (d) Output voltage for PS-PWM; (e) Output voltage for PD-PWM; (f) Output voltage for New strategy; (d) Output voltage for PS-PWM; (e) Output voltage for PD-PWM; (f) Output voltage new strategy. for new strategy.

The characteristics characteristics of both both modulation modulation strategies attribute to the difference between between these two The ◦ equivalent carriers. a sudden 180° phase change of PDS PWM’s equivalent carriercarrier when equivalent carriers. There Therewill willbebe a sudden 180 phase change of PDS PWM’s equivalent the reference transits to another interval because of theofsimilarity between the phase shifted and when the reference transits to another interval because the similarity between the phase shifted alternative phase opposition disposition (APOD) where carriers in the adjacent interval are in and alternative phase opposition disposition (APOD) where carriers in the adjacent interval are in opposition [14,15]. On the contrary, the the equivalent equivalent carrier carrier of of PD-PWM PD-PWM keeps keeps its its continuity continuity in in phase phase opposition and no sudden phase change will occur due to the fact that the carriers in each interval are in phase. and sudden phase change will occur due to the fact that the carriers in each interval are in phase. There is is no no question question that that with with the the help help of of periodicity periodicity of of the the equivalent equivalent carrier, carrier, an an appropriate appropriate There phase shift shiftofofPS-PWM’s PS-PWM’s carriers (in fact, the equivalent also shifted), at the reference time the phase carriers (in fact, the equivalent carriercarrier is also is shifted), at the time reference crosses the boundary from one interval to another, can eliminate the phase discontinuity of crosses the boundary from one interval to another, can eliminate the phase discontinuity of PS-PWM’s PS-PWM’s carrier equivalent carrier and achieve the waveforms same output waveforms the phase disposition equivalent and achieve the same output with the phasewith disposition PWM strategy. PWM strategy. Figure 1c shows the operating characteristic of the proposed strategy. The positive 45◦ phase 1c shows the operating The positive 45° phase shift Figure of the carriers occurs at θ 1 andcharacteristic θ 2 . It is easy of to the findproposed that afterstrategy. the adjustment of carriers, the shift of thestrategy carriersand occurs at θ1 and θ2. ItPWM is easy to find that after the adjustment carriers, the proposed phase disposition possess the same equivalent carrierof and the same proposed strategy and phase disposition PWM possess the same equivalent carrier and the same output waveforms. output waveforms. In fact, this operation can be extended to an arbitrary N-level CHB inverter for an equivalent PD-PWM where the main feature of carriers for PS-PWM still remains. General rules are offered for getting the carriers of the new modulation strategy. (1) Divide all the carriers of PS-PWM to N − 1 intervals of the same width in the vertical direction.

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In fact, this operation can be extended to an arbitrary N-level CHB inverter for an equivalent PD-PWM where the main feature of carriers for PS-PWM still remains. General rules are offered for getting the carriers of the new modulation strategy. Energies 2017, 10, 1327

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(1) Divide all the carriers of PS-PWM to N − 1 intervals of the same width in the vertical direction. (2) Every Everytime time when when the the reference reference crosses crosses the the boundary boundary from from one one interval interval to to the adjacent adjacent one, one, we adopted a positive phase shift of the carriers that have been generated already. The amount adopted a positive phase shift of the carriers that have been generated already. The amount of phase phase in in terms terms of of the the carrier’s carrier’s period period is is given given by by: θ = =180 180 − ,1),  /◦ (/N( N 1)

(1)

The factor 1/(N − 1) represents the frequency of the equivalent carrier N − 1 times of the real carries. The factor 1/(N − 1)that represents the frequency of the equivalent carrier 1 times of theis real In addition, the situation the reference crosses over several boundary linesNat−the same time not carries. In addition, the situation that the reference crosses over several boundary lines at the same considered here because high dv/dt in the output voltage should be avoided in practice. The reference time is be notlimited considered here because dv/dtinterval in the output voltage should avoided in practice. should to only transit to thehigh adjacent in one equivalent carrierbecycle. The reference should be limited to only transit to the adjacent interval in one equivalent carrieriscycle. It should be mentioned that although in theory a negative phase shift of the carriers also It should be mentioned that although in theory a negative phase shift of the carriers is capable capable of achieving desired results which are proposed in [11], it can cause extraalso switching of achievingofdesired results which are proposed inof [11], it can(roughly cause extra switching ofand the transitions the power semiconductor in a cycle carriers illustrated in transitions Figure 2a,b) power semiconductor in a cycle of carriers (roughly illustrated in Figure 2a,b) and is therefore not is therefore not recommended in practice. recommended in practice. At this point we have obtained the new modulation strategy for single phase system. At this point we have obtained the new modulation strategy for single phase system.

Figure 2. that thethe reference crosses the carrier four four timestimes in a cycle the carries Figure 2. Illustration Illustrationofofthe thecases cases that reference crosses the carrier in a of cycle of the after a negative phase shift of shift the carriers: (a) Positive phase shift; Negative phase shift. carries after a negative phase of the carriers: (a) Positive phase(b) shift; (b) Negative phase shift.

2.2. New Strategy for Three-Phase System Since substantial thethe first carrier component in each phase voltage and substantial harmonic harmonicenergy energyisisput putinto into first carrier component in each phase voltage cancels in theinline the PD-PWM achieves a better outputoutput line voltage than that ofthat phase-shifted and cancels the voltage, line voltage, the PD-PWM achieves a better line voltage than of phasePWM [6,15]. the same way, the achievement of the desired results for the new for strategy relies on the shifted PWMIn[6,15]. In the same way, the achievement of the desired results the new strategy effective cancellation of first carrier harmonics in line voltage. relies on the effective cancellation of first carrier harmonics in line voltage. For a three-phase system, three different groups of carriers are indispensable because each reference of three phases always does not transit to another interval at the same time. Although the equivalent carrier has been the phase between each equivalent phase of ofone onecertain certain equivalent carrier has decided, been decided, thedifference phase difference between each carrier has carrier not been yet. Thus, the how to decide adjustand a phase difference equivalent hasdecided not been decided yet.question Thus, theofquestion of howand to decide adjust a phase between each equivalent carrier becomes a new problem which does not occur in not the PD-PWM, where difference between each equivalent carrier becomes a new problem which does occur in the PDthree phase arereferences compared are with a common group of carriers. PWM, wherereferences three phase compared with a common group of carriers. Simulation results results show show that that the cancellation cancellation of first carrier harmonics harmonics in each phase voltage Simulation requires equivalent equivalent carriers carriers of each phase to be in phase. On the other hand, the possible different produces a phase difference of equivalent carrier between each phase, initial position positionofofeach eachreference reference produces a phase difference of equivalent carrier between each phase, which is a contrast to PD-PWM where all the equivalent carriers of each phase are in phase naturally because three phase references are compared with a common group of carriers. Figure 3 shows phase deviation of equivalent carriers caused by the different initial position of references. The reference for phase A starts from the interval just above zero value reference, whereas the reference for phase B starts from the interval just below zero value reference. Because of the different initial position of references, the equivalent carriers for phase A and B turn out to be

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which is a contrast to PD-PWM where all the equivalent carriers of each phase are in phase naturally because three phase references are compared with a common group of carriers. Figure 3 shows phase deviation of equivalent carriers caused by the different initial position of references. The reference for phase A starts from the interval just above zero value reference, whereas the reference for phase B starts from the interval just below zero value reference. Because of the different initial position of references, the equivalent carriers for phase A and B turn out to be opposite Energies 2017, 55 of Energies 2017, 10, 10, 1327 1327 of 14 14 in phase.

Figure 3. Phase deviation of equivalent carriers between between (phase A, B) caused by the different initial Figure Figure 3. 3. Phase deviation of equivalent carriers between (phase A, A, B) B) caused by the different initial position of references. position position of of references. references. FFT window: 1 of 2 cycles of selected signal

FFT window: 1 of 2 cycles of selected signal

Time (s) Fundamental (50Hz) = 3.29 , THD= 29.23% Fundamental (50Hz) = 3.29 , THD= 29.23%

Time (s) Fundamental (50Hz) = 3.291 , THD= 15.06% Fundamental (50Hz) = 3.291 , THD= 15.06%

FFT window: 1 of 2 cycles of selected signal FFT window: 1 of 2 cycles of selected signalfor phase A Figure 4a where the carriers 4 4 Figure shows the the harmonics harmonicsprofile profileof ofline linevoltage voltage where theequivalent equivalent carriers phase 4a shows shows the harmonics profile of line voltage where the equivalent carriers forfor phase A 4 4 and B are just opposite in phase. Figure 4b shows the harmonics profile of line voltage where the A and are just opposite in phase. Figure 4b shows the harmonics profile of line voltage the and BB are just opposite in phase. Figure 4b shows the harmonics profile of line voltage where 2 2 2 2 equivalent carriers for for phase phase A A and and B B are are in in phase. phase. Both harmonics profiles are obtained obtained with with 11 kHz kHz equivalent carriers Both harmonics profiles are 0 0 0 switching frequency, 50 Hz frequency reference frequency; the modulation depth is M = 0.95. For the 0 switching frequency, the frequency, 50 50 Hz Hz frequency frequency reference referencefrequency; frequency; themodulation modulationdepth depthisisM M== 0.95. 0.95. For the modulation depth M /V rp, V,dcVis the DC voltage of power cells and V is peak value of -2 -2 modulation depth M == Vdc /V is the voltage of power cells is the peak value of =V V dc rp,rp Vdc is DCDC voltage of the the power cells andand Vrp rp V isrpthe the peak value of the the dc/V dc the -2 -2 the reference. It is obvious that first carrier harmonics do not cancel in line voltage sufficiently because the reference. is obvious that first carrier harmonics notcancel cancelininline linevoltage voltagesufficiently sufficiently because reference. It isItobvious that first carrier harmonics dodo not -4 -4 -4 -4 0.025 0.02 0.015 0.01 0.01 0.015 0.02 0.025 0.03 of the the phase phase deviation. 0.025 0.02 0.015 0.01 0.01 0.015 0.02 0.025 0.03 of deviation. Time (s) Time (s)

10 10 10 10

55 00 0 0

Mag(% (%ofofFundamental) Fundamental) Mag

15 15 15 15

Fundamental Mag %%ofofFundamental Mag

20 20 20 20

（ （

） ）

Fundamental Mag %of Fundamental Mag %(% Mag(% ofFundamental) Fundamental) Mag ofof

THD THD = = 29.23% 29.23% M M= = 0.95 0.95 fr fr = = 50Hz 50Hz

） ）

25 25 25 25

00 00

5000 10000 5000 10000 5000 10000 5000 10000 Frequency (Hz) Frequency (Hz) Frequency(Hz) Frequency (Hz) (a) (a)

15000 15000 15000 15000

20000 25000 20000 25000

THD THD = = 15.06% 15.06% M M= = 0.95 0.95 fr fr = = 50Hz 50Hz

4444 3333 2222 11 000 0

00 00

5000 5000 10000 10000 5000 5000 10000 10000 Frequency (Hz) Frequency Frequency(Hz) (Hz) Frequency (Hz) (b) (b)

15000 15000 15000 15000

20000 25000 20000 25000

（ （

Figure The spectrum of line voltage for aa five-level cascaded H-bridge (CHB) under Figure 4.4. 4.The Thespectrum spectrum voltage five-level cascaded H-bridge (CHB) inverter inverter under Figure of of lineline voltage for afor five-level cascaded H-bridge (CHB) inverter under different different condition. Switching frequency is 1 kHz. Modulation depth, M = 0.95. Reference frequency different condition. Switching frequency is 1 kHz. Modulation depth, M = 0.95. Reference frequency condition. Switching frequency is 1 kHz. Modulation depth, M = 0.95. Reference frequency is 50 Hz: is Hz: profile of voltage where the carriers for phase A B is 50 50 Hz: (a) (a) Harmonics Harmonics of line linewhere voltage the equivalent equivalent carriers phase A and and B are are (a) Harmonics profile of profile line voltage thewhere equivalent carriers for phasefor A and B are opposite opposite in phase; (b) Harmonics profile of line voltage where the equivalent carriers for phase A and opposite (b) Harmonics of line voltage where the equivalent for phase in phase; in (b)phase; Harmonics profile ofprofile line voltage where the equivalent carrierscarriers for phase A andABand are B in B are are in phase. phase. in phase.

Hence, Hence, it it is is necessary necessary to to take take aa measure measure to to meet meet the the constraint constraint condition condition that that the the equivalent equivalent carriers of each phase should be in phase. Two ways are offered as follow: carriers of each phase should be in phase. Two ways are offered as follow: (1) (1) Make Make three-phase three-phase references references start start from from the the same same value value at at the the beginning. beginning. For For example, example, start start from from zero. zero. (2) (2) Take Take one one phase’s phase’s equivalent equivalent carrier carrier as as aa standard standard and and properly properly shift shift the the other other two two phases phases carriers carriers at the beginning to make their equivalent carriers’ phase consistent with the standard. at the beginning to make their equivalent carriers’ phase consistent with the standard.

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Hence, it is necessary to take a measure to meet the constraint condition that the equivalent carriers of each phase should be in phase. Two ways are offered as follow: (1) (2)

Make three-phase references start from the same value at the beginning. For example, start from zero. Take one phase’s equivalent carrier as a standard and properly shift the other two phases carriers at the beginning to make their equivalent carriers’ phase consistent with the standard.

A fundamental framework of the new strategy for the three-phase system has been established so far. Generally, a zero-sequence voltage is added to the three-phase references to improve DC link voltage utilization [16,17], as shown below. Vo f f = −

max(Va , Vb , Vc ) + min(Va , Vb , Vc ) , 2

(2)

It is preferable that only when the modulation depth M is more than 1, the offset voltage is injected into each phase reference to avoid introducing extra common voltage into the output phase voltage. It means that Vo f f is zero when depth M is under 1. In [18], a further adjustment of the phase reference is made to obtain an improved performance in output voltage. The author vertically shifts the reference voltages to a common carrier band and then centers the active space vectors in the switching cycle by adjusting the positions of the first and last switching transitions to optimize the harmonic profile. The new reference and the common voltage that centers the active space vectors can be described mathematically as: Vk0 = (Vk + Vo f f + Vdc )mod( Vo0 f f =

2Vdc ), k = a, b, c, N−1

max(Va0 , Vb0 , Vc0 ) + min(Va0 , Vb0 , Vc0 ) Vdc − , N−1 2

(3)

(4)

Finally, we obtain the optimized new strategy by adding Vo f f and Vo0 f f to the reference waveform Va , Vb or Vc . 2.3. Discontinuous Modulation Based on New Strategy Figure 5 shows the common carrier and the new references Va , Vb and Vc . Apparently, the easiest strategy for discontinuous modulation of multilevel inverter for PD-PWM could be achieved by eliminating the first or the last crossing (namely the first state or the last state in the space vector) with an adjustment of the new references with the offset voltage of: Vo0 f f =

Vdc − max(Va0 , Vb0 , Vc0 ), N−1

(5)

or Vo0 f f = −min(Va0 , Vb0 , Vc0 ),

(6)

Similarly, the final reference voltages for discontinuous modulation can be calculated by summing the two common voltages to Vb or Vc . However, this is not the case for the new strategy to obtain discontinuous modulation yet, considering that the new strategy is only equal to the PD-PWM in the output voltage. Clearly, the final reference to phase in which pulse is arranged to be eliminated, still crosses downwards or upwards with the real carriers, although it does not cross the equivalent carrier. Due to the intersections of the carriers and the reference, the switching transitions still exist.

or ' =  min(Va' ,Vb' ,Vc' ) , Voff

(6)

Similarly, the final reference voltages for discontinuous modulation can be calculated by Energies 2017,the 10, two 1327 common summing

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voltages to Vb or Vc .

V dc

N 1 max(Va' ,Vb' , Vc' ) 1

The first crossing

0.8

0.6

0.4

0.2

min(Va' , Vb' , Vc' )

0

The last crossing

-0.2

-0.4

-0.6

-0.8

0 -1

0.0114

0.0116

0.0118

0.012

0.0122

0.0124

0.0126

0.0128

0.013

0.0132

0, V0 references V V ' a V b'

Figure 5. The common carrier and the new Figure 5. The common carrier and the new references

a

,

b

and Vc0 . ' V and c .

In order to avoid the switching transitions, the carriers of the arranged phase should be kept However, this is not the case for the new strategy to obtain discontinuous modulation yet, unchanged in the switching cycle. Then the reference no longer crosses downwards or upwards with considering that the new strategy is only equal to the PD-PWM in the output voltage. Clearly, the the carriers, and therefore no switching transition is generated. The detailed operation of this strategy final reference to phase in which pulse is arranged to be eliminated, still crosses downwards or will be shown later in the Section 3.1. Another thing to be noted is that the efficiency to reduce the switching frequency by eliminating the first state in each switching sequence is much lower than that by eliminating the last state in practice. To eliminate the last state, we simply maintain the status that produced in the former cycle of the equivalent carrier and no switching transition is needed. However, to eliminate the first state, the output waveforms must step one level up and thus a switching transition will be introduced. Correspondingly, another switching transition is necessary to make the output waveforms step one level down when the corresponding reference do not cross the common carrier firstly. Moreover, in the multilevel application the state being the first crossing or the last crossing the common carrier rapidly rotates among three phases after the phase references are adjusted to a common carrier band, and therefore extra switching transition resulted from eliminating the first state will sharply degenerate its ability to reduce the switching frequency. 3. Simulation and Experimental Results 3.1. Simulation Results Several simulation results obtained by using MATLAB-Simulink for a three-phase 11-level CHB inverter, which consists of five power cells connected in series in each phase, are offered to verify the performance of the proposed modulation strategy in this section. Each power cell is powered by a constant DC voltage source rated at 200 V for simplicity. The switching frequency for carriers is 1 kHz and the reference voltage frequency is 50 Hz. Table 1 provides the detail parameters for simulation. Table 1. Parameters for simulation. Parameters

Value

Level of CHB Inverter DC Link Voltage Output Voltage Max Modulation Depth Switching Frequency

5,7,9,10 56 V 0–550 V (3-ph) 1.15 610 Hz

Figure 6 shows carriers and reference for the optimized new strategy under CHB inverters of various levels. It is clear that when the reference crosses the dotted line (boundary line) to enter into another interval, a sudden change of the carriers occurs and the carriers are thus shifted forward a certain phase to obtain phase continuity of the equivalent carrier. It should be noted that the carriers, after a sudden phase change for the new strategy, results from the already generated carriers but not

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from the carriers of PS-PWM. In addition, the actual carriers’ frequency is slightly higher than 1 kHz because of the forward adjustment of the original carriers. For a clear illustration, only half of the Energies 2017, 10, 1327 8 of 14 Energies 2017, 1327 of 14 carriers are10,depicted in Figure 6. The remains are just phase inverted by 180◦ of that presented 8below. Vdc Vdc

Vdc Vdc

0

0

0

0

-Vdc -Vdc

-Vdc -Vdc

t ct c

t ct c

(a) (a)

(b) (b)

Vdc Vdc

Vdc Vdc

0

0

0 -Vdc -Vdc

0 -Vdc -Vdc

ct ct(c)

ct ct(d)

(c)

(d)

Figure 6. 6. Carriers Carriers and and reference reference under under the the optimized optimized new new strategy strategy under under CHB CHB inverters inverters of of various various Figure Figure 6. Carriers and reference under the optimized new strategy under CHB inverters of various levels. (a) (a) New New strategy strategy for for five-level five-level CHB CHB inverter; inverter. (b) (b) New New strategy strategy for for seven-level seven-level CHB CHB inverter; inverter. levels. levels. (a) New strategy for five-level CHB inverter. (b) New strategy for seven-level CHB inverter. (c) New strategy for nine-level CHB inverter. (d) New strategy for 11-level CHB inverter. (c)New Newstrategy strategyfor fornine-level nine-levelCHB CHBinverter. inverter;(d) (d)New Newstrategy strategyfor for11-level 11-levelCHB CHBinverter. inverter. (c)

Figure 7 shows the phase voltage and line voltage for phase-shifted PWM and the optimized new Figure Figure77shows showsthe thephase phasevoltage voltageand andline linevoltage voltagefor forphase-shifted phase-shiftedPWM PWMand andthe theoptimized optimizednew new modulation strategy. The shapes of phase voltage for both modulation strategies look similar, but the modulation strategy. The shapes of phase voltage for both modulation strategies look similar, but the modulation strategy. The shapes of phase voltage for both modulation strategies look similar, but the line voltage for the optimized new modulation strategy is neater than that for phase-shifted PWM and line linevoltage voltagefor forthe theoptimized optimizednew newmodulation modulationstrategy strategyisisneater neaterthan thanthat thatfor forphase-shifted phase-shiftedPWM PWMand and keeps favorable stepped waveforms like the phase voltage. The stepped waveforms of line voltage also keeps stepped waveforms likelike thethe phase voltage. The The stepped waveforms of line also keepsfavorable favorable stepped waveforms phase voltage. stepped waveforms ofvoltage line voltage are a clear sign that the equivalent carrier of each phase was kept in phase fairly well. are a clear equivalent carriercarrier of eachofphase kept inkept phase also are asign clearthat signthe that the equivalent each was phase was infairly phasewell. fairly well. 300 300

Phase Voltage(v) Phase Voltage(v)

Phase Voltage(v) Phase Voltage(v)

300 300

0

300 300600 600

Line Voltage(v) Line Voltage(v)

300 300600 600

Line Voltage(v) Line Voltage(v)

0

0

0

0

600 600 0 0

0

0

0

t rt r (a) (a)

3π 3π

600 600 0 0

t rt r

3π 3π

(b) (b)

Figure 7. Phase voltage and line voltage for PS-PWM and the optimized new strategy: (a) Phase voltage Figure 7. Phase voltage and line voltage for PS-PWM and the optimized new strategy: (a) Phase voltage Figure 7. Phase voltage and line (b) voltage forvoltage PS-PWM optimized new strategy: new (a) Phase voltage and line voltage for PS-PWM; Phase andand linethe voltage for the optimized strategy. and line voltage for PS-PWM; (b) Phase voltage and line voltage for the optimized new strategy. and line voltage for PS-PWM; (b) Phase voltage and line voltage for the optimized new strategy.

Figure Figure 8a,b 8a,b show show the spectrum spectrum of of phase phase voltage voltage for phase-shifted phase-shifted PWM and the optimized new Figure 8a,b show the spectrum of phase voltage for phase-shifted PWM and the optimized new modulation modulation strategy, strategy, respectively. respectively.ItItisisclear clearthat that under under the the optimized optimized new new modulation modulation strategy strategy more modulation strategy, respectively. It is clear that under the optimized new modulation strategy more harmonic energy is injected into the phase voltage than that under phase-shifted PWM, harmonic is injected into the phase voltage that phase-shifted PWM, and and most most of of harmonic energy is injected into the phase voltage than that under phase-shifted PWM, and most of the the harmonics harmonics center around the first carrier. carrier. Figure 8d shows shows that the the harmonics harmonics around the first the harmonics center around the first carrier. Figure 8d shows that the harmonics around the first carrier carrier are are almost almost canceled canceled in in line line voltage, voltage, as expected. expected. However, However, for for the line voltage of phase-shifted phase-shifted carrier are almost canceled in line voltage, as expected. However, for the line voltage of phase-shifted PWM, most harmonics are still retained. From this aspect, the optimized new modulation PWM, harmonics still retained. aspect, the optimized new modulation strategy strategy PWM, most harmonics are still retained. From this aspect, the optimized new modulation strategy achieves achieves the the anticipated anticipated results. results. achieves the anticipated results.

Energies 2017, 10, 1327 Energies2017, 2017,10, 10,1327 1327 Energies

Mag % of Fundamental Mag % of Fundamental

33

））

22

（（

11

00

11

44

33 22 4 4Hz) Frequency(10 (10 Frequency Hz) (a)(a)

44

THD =15.06% 15.06% THD =266 Fundamental(50Hz) Fundamental(50Hz) = =266 0.95 THD= =12.61% 12.61% MM= =0.95 THD 50Hz frfr= =50Hz

88 66 44 22 00

55

Fundamental(50Hz)= =460.1 460.1 Fundamental(50Hz) THD= =9.81% 9.81% THD

1010

00

11

Mag % of Fundamental Mag % of Fundamental

33 22

33 22 4 4Hz) Frequency(10 (10 Frequency Hz) (b) (b)

44

55

Fundamental(50Hz)= =460.7 460.7 Fundamental(50Hz) THD= =6.86% 6.86% THD

）） 11

0.5 0.5

（（

））

Mag % of Fundamental Mag % of Fundamental

THD= =12.58% 12.58% THD Fundamental(50Hz) =265.6 265.6 Fundamental(50Hz) 0.95 MM===0.95 THD= =12.58% 12.58% fr = 50Hz THD fr = 50Hz 256.6 256.6

44

（（

00

））

Mag % of Fundamental Mag % of Fundamental

9 of 14 9 9ofof1414

11

（（

00

00

11

33 22 4 4Hz) Frequency(10 (10 Frequency Hz) (c)(c)

44

55

00

00

11

33 22 4 4Hz) Frequency(10 (10 Frequency Hz) (d) (d)

44

55

Figure 8. 8. Spectrum ofofof phase voltage and PS-PWM and the optimized newstrategy strategy Figure 8.Spectrum Spectrum phase voltage andline linevoltage voltagefor forPS-PWM PS-PWMand andthe theoptimized optimizednew new strategy Figure phase voltage and line voltage for (Modulation depth, M = 0.95): (a) Spectrum of phase voltage for PS-PWM; (b) Spectrum of phase (Modulationdepth, depth,MM= =0.95): 0.95):(a) (a)Spectrum Spectrumofofphase phasevoltage voltagefor forPS-PWM; PS-PWM;(b) (b)Spectrum Spectrumofofphase phase (Modulation voltage forfor the optimized new strategy; (c) Spectrum voltage for PS-PWM; (d) Spectrumofof ofline line voltage for the optimized new strategy; (c) Spectrumof linevoltage voltagefor forPS-PWM; PS-PWM;(d) (d)Spectrum line voltage the optimized new strategy; (c) Spectrum ofofline line voltage forfor the optimized new strategy. voltage for the optimized new strategy. voltage the optimized new strategy.

Figure 9a,b offer adetailed detailed comparisonof bothmodulation modulationstrategies strategieson onthe theperformance performanceofof Figure 9a,b offer comparison ofofboth both modulation Figure 9a,b offer a adetailed comparison strategies on the performance of output line voltages. Obviously, the new modulation strategy achieves an improved total harmonic output line voltages.Obviously, Obviously,the thenew newmodulation modulation strategy strategy achieves output line voltages. achievesan animproved improvedtotal totalharmonic harmonic distortion(THD) (THD)and andshows showsaasharp sharpadvantage advantagetotocanceling cancelingthe thefirst firstcarrier carrierharmonics harmonicscompare comparetoto distortion distortion (THD) and shows a sharp advantage to canceling the first carrier harmonics compare to phase-shiftedPWM. PWM. phase-shifted phase-shifted PWM.

Figure 9.Detailed Detailed comparison the THD andfirst firstcarrier carrierharmonics harmonicsabout aboutthe theline linevoltage voltagePS-PWM PS-PWM Figure comparison ofof the THD and first carrier Figure 9. 9. Detailed comparison of the THD and harmonics about the line voltage PS-PWM and the optimized new strategy: (a) Comparison of THD; (b) Comparison of first carrier harmonics. and optimizednew newstrategy: strategy: (a) (a) Comparison of of first carrier harmonics. and thethe optimized ComparisonofofTHD; THD;(b) (b)Comparison Comparison first carrier harmonics.

Figure 10 10 shows shows the the carriers carriers and and reference reference under under discontinuous discontinuous modulation modulation where where Figure Figure 10 shows the carriers and reference under discontinuous modulation where ' ' ' ' min( ) . Unlike the carriers shown in Figure 6, the carriers under discontinuous ' off min( VV VV ,0 V,V ,0V,V off a' a b' b c' )c 0 . Unlike the carriers shown in Figure 6, the carriers under discontinuous 0 Vo f f = −min(Va , Vb , Vc ). Unlike the carriers shown in Figure 6, the carriers under discontinuous modulationare arekept keptunchanged unchanged incertain certaintime timeintervals intervalstotoavoid avoid theintersections intersections of thecarriers carriers and modulation modulation are kept unchanged inincertain time intervals to avoid the the intersectionsofofthe the carriersand and thereference. reference.ItItisisapparent apparentthat thatno noswitching switchingtransitions transitionswill willbe begenerated generatedfor forthe thetime timethis thisphase phaseisis the the reference. It is apparent that no switching transitions will be generated for the time this phase is designatedtotoeliminate eliminateits itsstate stateininthe theswitching switchingsequence. sequence. designated designated to eliminate its state in the switching sequence.

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10 10 of of 14 14 10 of of 14 14 10

Figure 10. Carriers and reference under the discontinuous modulation. Figure 10. 10. Carriers Carriers and reference reference under under the the discontinuous discontinuous modulation. modulation. Figure Carriers and

Figure 11 shows one phase reference voltage in a cycle for discontinuous modulation where the Figure 11 shows shows one one phase reference reference voltage voltage in in aa cycle cycle for for discontinuous discontinuous modulation modulation where where the the Figure last switching are eliminated, a corresponding that indicates the state and Figure 11 11 transitions shows onephase phase referenceand voltage in a cycle forsignal discontinuous modulation where last switching switching transitions transitions are are eliminated, eliminated, and and aa corresponding corresponding signal signal that that indicates indicates the the state state and and last whether this phasetransitions is the last crossing the common carrier. When the signal turns 1, thisthe phase the the last switching are eliminated, and a corresponding signal that indicates stateisand whether this phase is the last crossing the common carrier. When the signal turns 1, this phase is the whether this phase is the last crossing the common carrier. When the signal turns 1, this phase is last crossing the common carrier and no switching should occur in power cells of this phase. It is clear whether this phase is the last crossing the common carrier. When the signal turns 1, this phase is the the last crossing the common carrier and no switching should occur in power cells of this phase. It is clear last the carrierquickly and no no from switching should occur in power power cellsthat of this this phase. is that the indicating signal alters 1 to 0should or 0 tooccur 1, which confirms the phase. state isIt first last crossing crossing the common common carrier and switching in cells of Itthe is clear clear that the indicating signal alters quickly from 1 to 0 or 0 to 1, which confirms that the state is the first that the indicating signal alters quickly from 1 to 0 or 0 to 1, which confirms that the state is the crossing the common carrier or the last crossing the common carrier and rapidly rotates among three that the indicating signal alters quickly from 1 to 0 or 0 to 1, which confirms that the state is the first first crossing the common carrier or the last crossing the common carrier and rapidly rotates among three crossing the carrier or the crossing phases, is mentioned crossingwhich the common common carrierin orSection the last last2.3. crossing the the common common carrier carrier and and rapidly rapidly rotates rotates among among three three phases, which which is is mentioned mentioned in in Section Section 2.3. 2.3. phases, phases, which is mentioned in Section 2.3.

Figure Figure 11. 11. Reference Reference and and indicating indicating signal signal under under the the discontinuous discontinuous modulation. modulation. Figure 11. 11. Reference Reference and and indicating indicating signal signal under under the the discontinuous discontinuous modulation. modulation. Figure

Figure 12 shows the phase voltage and line voltage under the discontinuous modulation. The Figure 12 12 shows showsthe thephase phasevoltage voltage and line voltage under the discontinuous modulation. and line voltage under the discontinuous discontinuous modulation. The 12and shows phasestill voltage line voltage under the modulation. The phaseFigure voltage linethe voltage keep aand fairly stair-like shape. The phase voltage and line voltage still keep a fairly stair-like shape. phase voltage voltage and and line line voltage voltage still still keep keep aa fairly fairly stair-like stair-like shape. shape. phase

Voltage(v) Voltage(v) Voltage(v)

600 600 600

0 0 0

600 6000 600 0 0

rt rtt  r

3π 3π 3π

Figure 12. Phase and line voltage under the discontinuous modulation. Figure 12. Phase and line voltage under the discontinuous modulation. Figure 12. 12. Phase Phase and and line line voltage voltage under under the the discontinuous discontinuous modulation. modulation. Figure

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Voltage BridgeVoltage Bridge HH

Voltage BridgeVoltage Bridge HH

Figure 13 shows the output voltage of the power cell under continuous modulation and Figure 13 shows the output voltage of the power cell under continuous modulation and discontinuous output voltage continuous modulation consists of 13 pulses in Figure 13modulation. shows the The output voltage of under the power cell under continuous modulation and discontinuous modulation. The output voltage under continuous modulation consists of 13 pulses in one cycle. The output voltage under discontinuous modulation consists of nine pulses in one cycle. discontinuous Theunder output voltage undermodulation continuousconsists modulation consists ofin 13one pulses in one cycle. The modulation. output voltage discontinuous of nine pulses cycle. It is cycle. clear that discontinuous modulation can greatly reduceconsists the frequency the switch. one Thethe output voltage under discontinuous modulation of nineof pulses in one cycle. It It is clear that the discontinuous modulation can greatly reduce the frequency of the switch. is clear that the discontinuous modulation can greatly reduce the frequency of the switch.

rt rt (a) (a)

` ` rt rt (b) (b)

Figure 13. 13. The output voltage of the power power cell under under continuous modulation modulation and discontinuous discontinuous Figure Figure 13. The The output output voltage voltage of of the the power cell cell under continuous continuous modulation and and discontinuous modulation. (a)(a) Output under modulation. Output voltage voltage under under continuous continuous modulation. modulation; (b) (b) Output Output voltage voltage modulation. (a) Output voltage under continuous modulation. (b) Output voltage under under discontinuous modulation. discontinuous modulation. discontinuous modulation.

3.2. Experimental Results 3.2. 3.2. Experimental Experimental Results Results To confirm the feasibility and the validity of the proposed modulation strategy, a three-phase, To To confirm confirm the the feasibility feasibility and and the the validity validity of the proposed modulation strategy, a three-phase, 11-level CHB inverter is established. The switching frequency for carriers is about 610 Hz and the 11-level 11-level CHB CHB inverter inverter is is established. established. The switching frequency for carriers is about 610 Hz and and the the reference voltage frequency is 50 Hz. Figure 14 shows the picture of the experimental prototype. reference Figure 14 shows the picture of the experimental prototype. reference voltage voltage frequency is 50 50 Hz. Hz. Figure Table 2 provides the detailed parameters for experiments. Table Table22provides providesthe thedetailed detailedparameters parametersfor forexperiments. experiments.

Figure 14. Picture of the experimental prototype. Figure 14. Picture of the experimental prototype.

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Table 2. Parameters for Experiment.

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Parameters Value Table 2. Parameters for Experiment. Number of Power Cells 3 × 5 unit DC Link Voltage 56 V Parameters Value Output Voltage 0–550 V (3-ph) Number of Power Cells 31.15 × 5 unit Max Modulation Depth DC Link Voltage 56 V Switching Frequency 610 Hz Output Voltage 0–550 V (3-ph) Insulated Gate 2MBI100VA-120-50 Max Modulation Depth 1.15 Bipolar Transistor(IGBT) Switching Frequency 610 Hz Drivers 2SD315-17 Insulated Gate Bipolar Transistor (IGBT) 2MBI100VA-120-50 Digital Signal Process(DSP) TMS320F2812 Drivers 2SD315-17 Field－Programmable Digital Signal Process (DSP) TMS320F2812 Altera EPF10K20T Gate Array(FPGA) Field—Programmable Gate Array (FPGA) Altera EPF10K20T

Figure 15 shows the experimental phase voltage and line voltage under the optimized new Figure 15 shows the experimental phase voltage and line voltage under the optimized new modulation strategy. It can be seen that the shapes of both output voltage are consistent with the modulation strategy. It can be seen that the shapes of both output voltage are consistent with the simulation results shown above. simulation results shown above.

Figure phase voltage and line under the optimized new strategynew (Modulation 15.Experimental Experimental phase voltage andvoltage line voltage under the optimized strategy Figure 15. depth, M = 0.95). (Modulation depth, M = 0.95).

Figure 1616shows shows experimental lineunder voltage under discontinuous Figure the the experimental phasephase voltagevoltage and lineand voltage discontinuous modulation modulation with the mode eliminating the last state in the space vector. The number of phasepulses voltage with the mode eliminating the last state in the space vector. The number of phase voltage is pulses is distinctly reduced distinctly to that 12. But there is no significant reduced comparedcompared to that shown inshown Figure in 12.Figure But there is no significant differencedifference between between line for voltages for discontinuous modulation and the optimized new strategy. line voltages discontinuous modulation and the optimized new strategy.

16. Experimental discontinuous modulation strategy Figure 16. Experimental phase phase voltage and line voltage under discontinuous depth, M M == 0.95). (Modulation depth,

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Figure 17a shows the output voltage of one power cell under the optimized new modulation Figure 17a shows the output voltage of one power cell under the optimized new modulation strategy. Figure 17b shows the output voltage of one power cell under discontinuous modulation. strategy. Figure 17b shows the output voltage of one power cell under discontinuous modulation. Carefully observing Figure 17a,b, we find that the number of output voltage pulses for discontinuous Carefully observing Figure 17a,b, we find that the number of output voltage pulses for discontinuous modulation is about two thirds of that for the optimized new strategy, which confirms that the modulation is about two thirds of that for the optimized new strategy, which confirms that the switching frequency of power semiconductor has decreased by the theoretical factor of one third. switching frequency of power semiconductor has decreased by the theoretical factor of one third. 80

H Bridge Voltage (V)

H Bridge Voltage (V)

80

40

0

-40

-80

40

0

-40

-80 0

5

10

15

Time (mSec) (a)

20

25

0

5

10

15

20

25

Time (mSec) (b)

Figure17. 17.Output Outputvoltages voltagesof ofone onepower powercell cellunder underthe theoptimized optimizednew newstrategy strategyand anddiscontinuous discontinuous Figure modulation strategy (Modulation depth, M = 0.95): (a) Output voltage under the optimized new modulation strategy (Modulation depth, M = 0.95): (a) Output voltage under the optimized new strategy; (b) Output voltage under discontinuous modulation strategy. strategy; (b) Output voltage under discontinuous modulation strategy.

4. Conclusions 4. Conclusions The relationship between PS-PWM and PD-PWM is well established by introducing the new The relationship between PS-PWM and PD-PWM is well established by introducing the new concept of the equivalent carrier. The concept could easily be applied to other multilevel modulation concept of the equivalent carrier. The concept could easily be applied to other multilevel modulation strategies and allow the quick establishment of the relationship between them. strategies and allow the quick establishment of the relationship between them. A modulation strategy combining the PS-PWM and PD-PWM is proposed to improve the A modulation strategy combining the PS-PWM and PD-PWM is proposed to improve the quality quality of output line voltage performance for CHB inverters. A discontinuous modulation strategy of output line voltage performance for CHB inverters. A discontinuous modulation strategy that could that could reduce the switching frequency to the theoretical value is offered for multilevel inverters reduce the switching frequency to the theoretical value is offered for multilevel inverters as well. These as well. These two strategies are the embodiments of a way of new thinking, which entails adjusting two strategies are the embodiments of a way of new thinking, which entails adjusting the carriers to the carriers to obtain improved performance of modulation strategy. Evidently, this way of thinking obtain improved performance of modulation strategy. Evidently, this way of thinking also has a role in also has a role in improving other modulation strategies. improving other modulation strategies. Detailed simulation and experimental results are offered to support the modulation strategy Detailed simulation and experimental results are offered to support the modulation strategy presented in this paper. presented in this paper. Acknowledgments:This Thiswork workwas wassupported supportedby bythe theNational NationalNatural NaturalScience ScienceFoundation Foundation of of China China (NSFC) (NSFC) Acknowledgments: underGrant GrantNo. No.51677063 51677063and andPostdoctoral PostdoctoralInnovative InnovativeTalent TalentSupport SupportProgram ProgramofofChina. China. under Author thethe authors conceived and designed the study. Qianming performed AuthorContributions: Contributions:AllAll authors conceived and designed theMuxuan study. Xiao, Muxuan Xiao, Xu Qianming Xu the simulation and the experiment and wrote the manuscript with guidance from Honglin Ouyang. Muxuan Xiao, performed the simulation and the experiment and wrote the manuscript with guidance from Honglin Ouyang. Qianming Xu and Honglin Ouyang reviewed the manuscript and provided valuable suggestions. Muxuan Xiao, Qianming Xu and Honglin Ouyang reviewed the manuscript and provided valuable suggestions. Conflicts of Interest: The authors declare no conflict of interest. Conflicts of Interest: The authors declare no conflict of interest.

References References 1. 1. 2. 2. 3. 3.

4.

Lai, J.S.; Peng, F.Z. Multilevel converters—A new breed of power converters. IEEE Trans. Ind. Appl. 1996, 32, Lai, J.S.; Peng, F.Z. Multilevel converters—A new breed of power converters. IEEE Trans. Ind. Appl. 1996, 509–517. 32, 509–517. Malinowski, M.; Gopakumar, K.; Rodriguez, J.; Pérez, M.A. A survey on cascaded multilevel inverters. Malinowski, M.; Gopakumar, K.; Rodriguez, J.; Pérez, M.A. A survey on cascaded multilevel inverters. IEEE Trans. Ind. Electron. 2010, 57, 2197–2206. [CrossRef] IEEE Trans. Ind. Electron. 2010, 57, 2197–2206. Miguel, M.; Francisco, H. A Comparison of modulation techniques for modular multilevel converters. Miguel, M.; Francisco, H. A Comparison of modulation techniques for modular multilevel converters. Energies 2016, 9, 1091. [CrossRef] Energies 2016, 9, 1091. Xiongmin, T.; Zhang, J. A switching frequency optimized space vector pulse width modulation (SVPWM) scheme for cascaded multilevel inverters. Energies 2017, 10, 725.

Energies 2017, 10, 1327

4. 5. 6. 7. 8.

9. 10.

11. 12.

13.

14. 15.

16. 17.

18.

14 of 14

Xiongmin, T.; Zhang, J. A switching frequency optimized space vector pulse width modulation (SVPWM) scheme for cascaded multilevel inverters. Energies 2017, 10, 725. [CrossRef] Liang, Y.Q.; Nwankpa, C.O. A new type of STATCOM based on cascading voltage source inverters with phase-shifted unipolar SPWM. IEEE Trans. Ind. Appl. 1998, 35, 1447–1453. McGrath, B.P.; Holmes, D.G. Multicarrier PWM strategies for multilevel inverters. IEEE Trans. Ind. Electron. 2002, 49, 858–867. [CrossRef] Li, Y.; Wang, Y.; Li, B.Q. Generalized theory of phase-shifted carrier PWM for cascaded H-bridge converters and modular multilevel converters. IEEE J. Emerg. Sel. Top. Power Electron. 2016, 4, 589–605. [CrossRef] Marquez, A.; Leon, J.I.; Portillo, R.; Vazquez, S.; Franquelo, L.G.; Kouro, S. Adaptive phase-shifted PWM for multilevel cascaded H-bridge converters for balanced or unbalanced operation. In Proceedings of the IECON 2015—41st Annual Conference of the IEEE Industrial Electronics Society, Yokohama, Japan, 9–12 November 2015; pp. 5124–5129. Deng, F.; Chen, Z. Voltage-balancing method for modular multilevel converters under phase-shifted carrier-based pulsewidth modulation. IEEE Trans. Ind. Electron. 2015, 62, 4158–4169. [CrossRef] Angulo, M.; Lezana, P.; Kouro, S.; Rodriguez, J.; Wu, B. Level-shifted PWM for cascaded multilevel inverters with even power distribution. In Proceedings of the Power Electronics Specialists Conference, Orlando, FL, USA, 17–21 June 2007; pp. 2373–2378. Naderi, R.; Rahmati, A. Phase-shifted carrier PWM technique for general cascaded inverters. IEEE Trans. Power Electron. 2008, 23, 1257–1269. [CrossRef] De Paris, J.M.; Nicolini, A.M.; Carnielutti, F.; Massing, J.; Pinheiro, H. Sorting algorithm for a PD modulation for a cascaded multilevel converter. In Proceedings of the 2015 IEEE 13th Brazilian Power Electronics Conference and 1st Southern Power Electronics Conference (COBEP/SPEC), Fortaleza, Brazil, 29 November–2 December 2015; pp. 1–6. Liu, H.; Ma, K.; Blaabjerg, F. Device loading and efficiency of modular multilevel converter under various modulation strategies. In Proceedings of the 2016 IEEE 7th International Symposium on Power Electronics for Distributed Generation Systems (PEDG), Vancouver, BC, Canada, 27–30 June 2016; pp. 1–7. Carrara, G.; Gardella, S.; Marchesoni, M.; Salutari, R.; Ciutto, G. A new multilevel PWM method: A theoretical analysis. IEEE Trans. Power Electron. 1992, 7, 497–505. [CrossRef] McGrath, B.P.; Holmes, D.G. A comparison of multicarrier PWM strategies for cascaded and neutral point clamped multilevel inverters. In Proceedings of the Power Electronics Specialists Conference, Galway, Ireland, 23 June 2000; pp. 674–679. Blasko, V. A hybrid PWM strategy combining modified space vector and triangle comparison methods. In Proceedings of the Power Electronics Specialists Conference, Baveno, Italy, 23–27 June 1996; pp. 1872–1878. Holmes, D.G. The general relationship between regular-sampled pulse-width-modulation and space vector modulation for hard switched converters. In Proceedings of the Industry Applications Society Annual Meeting, Houston, TX, USA, 4–9 October 1992; pp. 1002–1010. McGrath, B.P.; Holmes, D.G.; Lipo, T. Optimized space vector switching sequences for multilevel inverters. IEEE Trans. Power Electron. 2003, 18, 1293–1301. [CrossRef] © 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).