Application of differential evolution for cascaded multilevel VSI with harmonics elimination PWM switching

Electrical Power and Energy Systems 64 (2015) 447–456

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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes

Application of differential evolution for cascaded multilevel VSI with harmonics elimination PWM switching Abdul Moeed Amjad, Zainal Salam ⇑, Ahmed Majed Ahmed Saif Centre of Electrical Energy Systems, Faculty of Electrical Engineering, Universiti Teknologi Malaysia, 81310 Johor Bahru, Malaysia

a r t i c l e

i n f o

Article history: Received 3 December 2013 Received in revised form 3 July 2014 Accepted 6 July 2014

Keywords: Harmonics elimination pulse width modulation (HEPWM) Differential Evolution (DE) Multilevel voltage source inverter (MVSI) Minimization method

a b s t r a c t Harmonic elimination pulse width modulation (HEPWM) method has been widely applied to multilevel voltage source inverter (MVSI) to remove low frequency harmonics from its output voltage. However, the computation of the HEPWM switching angles for MVSI is very challenging due to several constraints, namely angle sequencing, very tight angular spacing and the numerous possibilities of angles distribution ratio. Realizing the potential of Differential Evolution (DE) to handle complex problems, this work proposes its application to solve the HEPWM problem for cascaded MVSI. Its emphasis is on improving the availability of HEPWM for higher output voltage by extending the maximum range of modulation index (M). It also removes the discontinuities in the switching angles and reduces the number of distribution ratio required to obtain the required solution. Compared to the most advanced (similar) work, i.e., 7-level MVSI with seventeen switching angles, DE covers a wider range of M; the maximum achievable M is 2.80. Furthermore, it exhibits very low second order distortion factor (DF2): for the worst case, the value of DF2 is 0.0014%. To verify the viability of the proposed algorithm, simulation is carried out and hardware prototype is constructed. Both results show very good agreement with the theoretical prediction. Ó 2014 Elsevier Ltd. All rights reserved.

Introduction The potential of the multilevel voltage source inverter (MVSI) for electric power conversion has been extensively exploited for the medium and high power dc–ac and ac–dc converters. This is mainly due to its ability to generate high output voltage using much lower-rated power semiconductor switches. The stepped output nature of MVSI not only lessens the stress on the switching devices, it also improves the frequency spectra profile – resulting in lower harmonics distortion [1,2]. In addition, the high output voltage offers the possibility of removing the step-up transformer, which is always undesirable due to the additional cost, ohmic losses and space limitations [1,3]. Over the years, various MVSI topologies have been proposed: among others, the diode clamped [4,5], flying capacitor [4,6] and cascaded inverters [4,7] are the most popular. Each topology has its own merits and drawbacks, as summarized by Refs. [1,4]. However, the cascaded type has been the focus of most research due to its simpler structure [8,9]. Besides these arguments, the output voltage of MVSI (regardless ⇑ Corresponding author at: Faculty of Electrical Engineering, Universiti Teknologi Malaysia, 81310 Johor Bahru, Malaysia. Tel.: +60 7 5536187; fax: +60 7 5566272. E-mail address: [email protected] (Z. Salam). http://dx.doi.org/10.1016/j.ijepes.2014.07.023 0142-0615/Ó 2014 Elsevier Ltd. All rights reserved.

of any topology) contains substantial amount of harmonics that can deteriorate the performance and stability of the converter system. Enormous work has been carried out to mitigate the harmonics by improving their modulation strategies – for instance, sinusoidal pulse width modulation (SPWM), multicarrier SPWM, staircase modulation and multilevel space vector modulation [1,3,10]. In recent years, the harmonics elimination PWM (HEPWM) [11,12] has gained popularity in MVSI due to its superior harmonics profile [7,13–16]. For example, if the switching frequency of the switches is kept constant, the first harmonics component in HEPWM appears at twice the frequency if SPWM is utilized. This is particularly important for high power converters, in which the switching loss often dictates the maximum allowable switching frequency. Despite this crucial benefit, obtaining the solution for the equations that define the HEPWM switching angles is very challenging; this is due to the fact that these transcendental equations are simultaneous, non-linear, and their trigonometric functions are highly correlated to each other [11,12]. Furthermore, the number of equations that need to be solved depends on the number of harmonics to be eliminated [11,12]. To obtain a high quality waveform, a large number of harmonics need to be eliminated; consequently more equations need to be solved, which

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often leads to numerical non-convergence. Furthermore, for MVSI, the transition between one level to another imposes an additional constraint on the sequencing of the switching angles that complicates the computational procedures further. Notwithstanding these difficulties, a significant number of HEPWM computational methods are proposed: they are generally categorized into two main types: (1) calculus [11,12] or (2) search/optimization approach [13,16–18]. For the former, Newton–Raphson is the widely used due to its simplicity and robustness [11,20]. However, the method requires an initial guess for the initial angles in such a way that they are sufficiently close to the global minimum. Choosing a good initial condition, particularly for large number of angles can be very difficult; failing to which can result in non-convergence [14]. Other refined calculus based methods, for example [21–23] are proposed, but the requirement of good initial guess remains. However, with the advent of low-cost, highly powerful processors, the trend has moved towards soft computing approach [13,14,16]. In general, these methods possess global optimizing and searching capability and are free from the requirement of good initial guess. Examples include genetic algorithm (GA) [14], particle swarm operation (PSO) [24] and bee algorithm (BA) [13]. Differential Evolution (DE) is an evolutionary-type soft computing algorithm that utilizes the mutation, crossover and selection processes to evolve an individual candidate into its fittest form. It has a natural ability to handle complex multimodal and nonlinear conditions and exhibits superiority over other algorithm methods in diverse fields [25–27]. Recently, DE has been used for HEPWM, but only to solve the trivial two-level inverter problem [18,19]. To date, it has not been utilized in multilevel structure. Realizing this potential, this work explores the application of DE to obtain the HEPWM angles for MVSI. In particular, it will focus on extending the maximum range of modulation index (M) for the switching angle trajectories. The wider range enhances the utilization of DC link voltages and extends the application of HEPWM MVSI for higher voltage applications [28]. However, extending the range of M (without adding a further output level) is extremely difficult because as M is approaching its maximum value, the angular spacing between consecutive switching angles decreases sharply, leading to situation in which the angle trajectories overlap into each other. In such a case, it is almost impossible to obtain the solution for the angles as the modulation has reached the physical limit. Furthermore, the proposed computation also eliminates the solution discontinuities; since they are no longer present, the inverter can operate smoothly over the entire range of M. Another contribution of this work is the reduction of the number of distribution ratio to obtain the HEPWM solutions. The need to correctly sequence the angles (i.e., the angles must be placed within the correct voltage level), imposes severe constraint on the algorithm due the limitation in the search space [29]. Varying the distribution ratio helps to cope with these constraints and increases the probability of obtaining solutions. However, due to the absence of an established procedure to determine the suitable values of distribution ratio, time consuming trial and error method is required to obtain their appropriate values. Thus, if less numbers of distribution ratios suffice the need, the computational burden is reduced. In this paper, the effectiveness of DE is demonstrated using five and seven level cascaded MVSIs to remove seven and seventeen harmonics, respectively. The seven level seventeen angles configuration is purposely chosen to enable a direct comparison to be made with the most advanced work published in literature [28]. The computed angles are analyzed in MATLAB simulation and verified using a low power prototype.

HEPWM for cascaded MVSI HEPWM switching scheme In this work the cascaded MVSI is utilized. It is considered superior to the other MVSI topologies as (1) it requires fewer numbers of components for the same output voltage level, (2) it does not require special balancing capacitors or clamping diodes, and (3) it has lower amount of associated switching losses [8]. In addition, the construction and maintenance of cascaded MVSI is much simpler due to its modular configuration [8]. A single phase cascaded MVSI using the H-bridge inverters is shown in Fig. 1(a), while its output waveform is shown in Fig. 1(b) and (c). In Fig. 1(b), the stepped output waveform of the MVSI is not modulated within the voltage levels. i.e., HEPWM angles are only placed exactly at the transitions points from one level to the next. This is the most common type of output voltage described in literature. Although simple, this ‘‘non-notch’’ HEPWM switching exhibits several drawbacks. First, for effective elimination of harmonics, the number of levels required is high, i.e., to remove N harmonics, a MVSI with 2N + 1 levels is needed. Secondly, since the H-bridge module is switched only twice per quarter cycle, the switches (particularly IGBT) are not utilized to their full potential. A more preferable option is to use the ‘‘notch-type’’ HEPWM, in which several angles are modulated within a particular level, creating notches as depicted in Fig. 1(c). Its harmonics profile is superior due to its ability to remove more harmonics using fewer levels. For instance, to eliminate 17 harmonics, eighteen switching angles have been solved using 7-level inverter [28], while to obtain similar harmonics profile for the non-notch type, a 37-level inverter is needed. In practice, such high number of levels can be prohibitive due to the large number of components and circuit complexities. Furthermore, the notch type HEPWM utilizes more switching per quarter cycle; thus the switches can be exploited to their fullest. In the light of these benefits, this work is focused on the notch type HEPWM; the non-notch types HEPWM are covered elsewhere [13,16]. Formulation of HEPWM equations To reduce the computation burden, the HEPWM exploits the quarter wave symmetry to remove the even numbered harmonics from the line to phase voltage [11,12]. Furthermore, in a three phase system, triplen harmonics are absent from the line-to-line voltage; thus only odd numbered and non-triplens harmonics (for instance 5, 7, 11, 13, etc.) need to be removed. Besides eliminating the specific harmonics (typically the lower order ones), HEPWM has the ability to control the magnitude of the fundamental component of the output voltage by setting one of the equations to a desired modulation index, M [11,12]. Fig. 2 shows the generalized quarter cycle notch type HEPWM MVSI waveform. Variables N1, N2 and Ns represent the number of angles in first, second and last voltage levels of output waveform respectively. This waveform can be represented by a generalized Fourier series of the nth harmonics component as follows [14]:

An ¼

4V f np

V s

V1

N1 N2 X X ð1Þiþ1 cos nai  V 2 ð1Þi cos nai  . . . i¼1

N X

1

ð1Þi cos nai A

i¼N1 þ1

ð1Þ

i¼N ðs1Þ þ1

In Eq. (1), variable s represents the number of dc input sources; Vf is the fundamental voltage component while V1, V2 and Vs represent the voltage magnitudes of the first, second and final level of

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Fig. 1. (a) Circuit topology of single phase cascaded MVSI, output voltage of cascaded MVSI, (b) the non notch type, and (c) the notch type.

.. . fN ¼

N1 N2 X X ð1Þi1 cosð3N  2Þai þ ð1ÞiðN1 þ1Þ cosð3N  2Þai i¼1

i¼N 1 þ1

þ... þ

N X

ð1Þ

iðN ðs1Þ þ1Þ

1

cosð3N  2Þai A ¼ eN

ð4Þ

i¼Nðs1Þ þ1

where e1 ; e2 ; . . . ; en represent the errors in the computations of objective functions. They are decided based on the allowable harmonics in the output voltage. The modulation index, M is defined as:



M¼



pV f  4V dc

; 0:01 6 M 6 s

ð5Þ

Fig. 2. Generalized quarter cycle notch type HEPWM MVSI waveform.

Physical constraints the output voltage, respectively. Here it is assumed that all input voltage sources are of equal magnitude i.e., V1 = V2 = . . . = Vs = Vdc. To utilize optimization techniques, Eq. (1) is converted into a set of objective functions (fn, n = 1, 2, . . ., N), i.e.,

f1 ¼

a1 < a2 < a3 . . . < aN <

i¼1

Due to this sequencing, the optimization algorithm is restricted to a certain region in the search space. Furthermore, for the notch type MVSI the angles must be placed within the correct voltage level. As shown in Fig. 2, N1, N2 and Ns represent number of angles in the first, second and last levels, respectively. The angle distribution ratio is defined as the ratio of the number of switching angles that is imposed on every level of the output voltage. It is denoted as N1/N2/. . ./Ns. For example, if the first level contains 5 angles, the second has 3 and the third level has 4, the angle distribution ratio is written as 5/3/4. Due to the angle sequencing and distribution ratio constraints, the computational difficulty for HEPWM is

N X

i¼N1 þ1 iðNðs1Þ þ1Þ

ð1Þ

1

cos ai A  M ¼ e1

ð2Þ

i¼N ðs1Þ þ1

1

N1 N2 X X ð1Þi1 cos 5ai þ ð1ÞiðN1 þ1Þ cos 5ai þ . . . i¼1

þ

p

N1 N2 X X ð1Þi1 cos ai þ ð1ÞiðN1 þ1Þ cos ai þ . . .

þ

f2 ¼

Angle sequencing and distribution ratio Although it is possible to obtain a number of solutions for Eqs. (2)–(4), it does not mean they can be directly used to construct the MVSI output voltage. For a proper HEPWM waveform generation, a correct sequencing of the angles is mandatory i.e.,

N X i¼N ðs1Þ þ1

i¼N 1 þ1 iðNðs1Þ þ1Þ

ð1Þ

1

cos 5ai A ¼ e2

ð3Þ

2

ð6Þ

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severely increased. In addition, the number of angles per level also plays a crucial role in determining the effectiveness of computation. To ensure that solutions are obtained for the required modulation range, different distribution ratios are utilized by varying the values of N1, N2, . . ., Ns. As suggested by Ref. [29], if the lower levels contains fewer angles (which means less notches), the fundamental component of the output voltage is higher. On the other hand, introduction of more number of angles in lower levels yields lower fundamental component. Such variation increases the probability of getting suitable switching angles for a wider range of M. However, in the absence of an established procedure to determine the suitable values of distribution ratio, trial and error method is required to obtain their appropriate values.

where j = 1,2, . . ., D. Note that D is the dimension value of the variables to be solved, while xjL and xjH are the lower and upper bounds respectively. Parameter encoding is done in such a way that the whole search space can be explored. Once the initialization is done, the individuals (also known as target vectors) are handed over for mutation.

Angular spacing (Da)

where i – r1 – r2 – r3 are the index numbers, G is the generation count, while F is the control factor, known as the mutation constant. The value of F is a real number between 0 and 1 and it ensures that the search process is free from stagnation [32]. Vi,G+1 is termed as the mutation or donor vector and is consigned to the crossover process in order to increase the diversity of the population.

Angular spacing (Da) is defined as the amount of angle that separates two adjacent switching angles. In practice, this spacing is required in order to create a practical HEPWM waveform, taking consideration on the switching devices non-idealities and the dead time. The idea can be explained by Fig. 3, which shows the trajectories for seven switching angles. As can be seen, the switching angles a4, a5 and a6, a7 tend to merge into each other (Da ? 0) as the value of M is increased. This phenomenon is more eminent for larger values of M, where a rapid merging effect can be observed (especially for a6 and a7). Merging of these angles diminishes the ability to create notch; as a result, the HEPWM output voltage is reduced to an ordinary square waveform due to pulse dropping. In other words, the physical modulation limit of the HEPWM is reached. An extension of M is always desirable as it increases magnitude of the output voltage without necessarily increasing the DC link voltage or adding another VSI level. However, the computational effort to extend the modulation range would be very challenging due to limitation imposed by Da. Differential Evolution (DE) Differential Evolution is a meta-heuristic optimization approach which steer a random search (in a pre-specified population of vectors) to locate the global minima [25,30,31]. It perturbs the generated population of NP vectors (also termed as individuals) by employing the process of mutation, while the diversity of the population vectors is controlled by the crossover process. Next, the selection process exploits the survival of the fittest phenomenon to ensure the selected global minimum is optimized one. Initialization DE is initiated by randomly initializing a population of NP vectors or individuals. Initialized vectors are represented as X1, X2, . . ., XNP with a parameter value xj of ith vector given as:

  xj ðiÞ ¼ xjL þ rand½0; 1  xjH  xjL

ð7Þ

Mutation Mutation is the core process which perturbs the individuals. The difference vector, i.e., (Xr2,GXr3,G) is the key to the success of DE. Mutation probes through unique parents in order to produce better individuals as follows:

V i;Gþ1 ¼ X r1;G þ FðX r2;G  X r3;G Þ

ð8Þ

Crossover This process reinforces the prior success of the search process to produce a fitter individual. It produces a trial vector Ui,G+1 by mixing the components of target and donor vectors as:

uj;i;Gþ1 ¼



v j;i;Gþ1 ;

if rand 6 CR or

xj;i;Gþ1 ;

otherwise

i¼k

ð9Þ

where j is the component number of the ith vector undergoing the crossover process, while rand is a random number in the range of (0,1]. CR is defined as the crossover rate or crossover probability and its value lies inside the range of (0, 1] [25]. CR controls the crossover process and hence the diversity of the population. It should be noted that k e D is a randomly chosen index. It is introduced to ensure that the individual undergoing crossover process is at least different from the target vector by one component. Once the crossover process is over, the selection process is carried out to find the better fit individuals. Selection DE utilizes a greedy selection approach based on the survival of the fittest phenomenon to select the best individuals for the next generation. It is an elitist strategy based selection in which the fittest individual among the trial and target vector survives after a one to one competition. It can be written as:

( X i;Gþ1 ¼

  U i;Gþ1 ; if J U i;Gþ1 < JðX i;G Þ X i;G ;

otherwise

i ¼ 1; 2; . . . ; NP

ð10Þ

where J is the fitness evaluation function which ensures that only the best fitted individual is retained for the next generation. Once selection process is completed, the same series of evolutionary processes is continued till the ending criteria are met. Computational considerations Selection of DE evolutionary parameters

Fig. 3. Switching angle trajectories of bipolar VSI (seven number of switching angles).

As the HEPWM equations are transcendental and trigonometric in nature, there is always a possibility for multiple solutions to exist. Moreover, due to the constraint imposed by Eq. (6) and the stepped nature of the output waveform, the switching angle trajectories for the whole range of M cannot be covered by a single angle distribution ratio. Hence, various distribution ratios are utilized to

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obtain continuous solutions for a wider range of M [28]. The chances for convergence are further enhanced by utilizing various values of the evolutionary parameters, namely F and CR. In the case of multiple solutions, the angles that yield the best harmonics profile are selected. Since large values (close to unity) of F and CR often result in a robust solution to multimodal problems [25,30], their initial values are set to 1.00 and 0.95, respectively. If this setting is unable to produce the desired results, the value of F is decremented by 0.0125. The decrement value for F (0.0125) is chosen after many trial-runs; it is selected as it shows no significant change in convergence if it is decreased to a smaller number. On the other hand, if this value is increased beyond 0.0125, the success rate becomes very poor. The iteration is repeated until the required results are obtained or the value of F becomes zero. As far as CR is concerned, its initial value (0.95) has shown satisfactory results. However, for the cases in which the value of F reaches zero without any solution, the value of CR is decremented by 0.05 until the required fitness value is achieved. As in the case with F, extensive simulations have shown that if the value is less than 0.05, no significant change in convergence rate is observed, and it is increased beyond 0.05, the convergence is very poor. The number of population vectors (NP) is set as 10D, where D is the number of angles to be solved. Value of NP = 10D is selected based on its recommendations by various authors, for example [25,33].

switching angles contain certain amount of error in them. To obtain a certain level of accuracy on the solution, the maximum error that can be tolerated is specified through a fitness function, i.e.,

Fitness function

Five level MVSI

Since the HEPWM is solved as an optimization problem, it is impossible for the objective functions to converge to zero; the

First, DE is utilized to solve the five level notch type HEPWM problem for seven angles. For this case, the modulation index is

JðaÞ ¼ je1 j þ je2 j þ je3 j þ . . . þ jeN j

ð11Þ

As J(a) reflects the error in the solution, accurate solution requires its value as small as possible, i.e., in the order 103 or less. However, a caution need to observed: while a small value of J(a) is desirable, a very small one may result in non-convergence. On the contrary, larger J(a) value always result in convergence but it leads to erroneous result. Furthermore, a sufficient time difference between two successive angles is required for the proper generation of switching signals. This time difference accounts for the dead, on and off times of the switching devices as reflected by the angular spacing mentioned in Section ‘HEPWM for cascaded MVSI’. In the present work, the minimum value of this time difference is set as 20 us, as suggested by Ref. [28]. The computation is repeated until a stopping condition is met, i.e., either the required fitness value is achieved or the generation count has reached its maximum value Gmax. In the present case, J(a) = 104 and Gmax = 10,000.

HEPWM angles computation

Fig. 4. Switching angle trajectories against respective values of modulation index and percentage residual amplitude of the harmonics w.r.t fundamental component against respective values of modulation index for the distribution ratios (a) 5/2, (b) 3/4 and (c) 1/6.

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set in the range 0.6 6 M 6 1.8 with an incremental step of 0.05. For M < 0.6, no solutions are sought as they can be conveniently solved using two or three level inverter. Furthermore, for M > 1.8, seven or higher level MVSI must be used because the five level inverter has reached its modulation limit. To ensure discontinuity is avoided within the whole range of M, three angle distribution ratios, namely 5/2, 3/4 and 1/6 are used. Fig. 4(a) depicts the trajectories of switching angles for distribution ratio of 5/2. The solutions are obtained by solving Eqs. (2)–(4) until the condition J(a) < 104 is reached. The harmonics eliminated are in the range of 0.60 6 M 6 0.90. Note that no discontinuity occurs within this range. However, since J(a) is a non-zero, residual harmonics remains, as illustrated by the spectra distribution of Fig. 4(a). Nevertheless, it can be clearly observed that the low order harmonics (5th, 7th, 11th, 13th, 17th and 19th) are minimized to a great extent, i.e., all components are less than 0.20% of the respective fundamental (with the exception of 5th harmonics for M = 0.65). Fig. 4(b) shows the trajectories for the distribution ratios of 3/4. For this distribution ratio, the achievable range is 0.90 6 M 6 1.35. Once again the switching angle trajectories are continuous with all harmonics components recording less than 0.20% of the fundamental. In general, to achieve higher fundamental magnitude the lower level of the inverter need to be switched less rapidly [29]. The trajectories for distribution ratio of 1/6 are shown in Fig. 4(c). It covers the range of 1.35 6 M 6 1.80. The residual harmonics amplitude of harmonics for any value of M is always less than 0.20% of its fundamental value (with the exception of the 13th harmonics component for M = 1.60). Once all the solutions for the desired range of M are computed, the output waveform is constructed using MATLAB Simulink. The Simulink model for the five level cascaded MVSI is shown in Fig. 5(a). The distribution ratio of 3/4 is used to illustrate the validity of the algorithm. Lossless universal H-bridge is chosen to simplify the design. Furthermore, equal DC voltages are used as

inputs. For a snapshot, line to neutral and line to line voltage of the output for a specific value of M = 1.10 is plotted in Fig. 5(b) and (c), respectively. The harmonics spectrum of the line to line voltage is shown in Fig. 5(d). It can be clearly observed that the output voltage is in accordance to the general MVSI output waveform of Fig. 1(c). Moreover, the harmonics under consideration (5th, 7th, 11th, 13th, 17th and 19th) along with the triplens are successfully eliminated. The 23rd harmonic is the first non-eliminated harmonics of the line to line voltage. The quality of a VSI output voltage is often benchmarked by the total harmonic distortion (THD) that it produces. If Vf is the fundamental voltage, the THD for line to line voltage is given as:

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi, 1 X THD ¼ ðV 6i1 Þ2 V f

ð12Þ

i¼1

In Eq. (12), the THD defines the non eliminated harmonics components present in the output voltage [34]. The recommended limit for THD is set by several organizations, for example [35]. For the five level case, THD of the picked value of M (i.e., M = 1.10) is calculated to be 24.7%. However, for HEPWM, a more realistic figure of merit is the second order distortion factor (DF2), as described in Ref. [15]. It considers the attenuation of the harmonics components through L-C filtering, i.e.,

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #2ffi u " N X 100 u V 6i1 t %DF2 ¼ 2 Vf i¼1 ð6i  1Þ

ð13Þ

To find out the effective value of harmonics in the line to line output voltage of the inverter system, the value of DF2 for the line to line output voltage at M = 1.60 (worst case) is computed. It turned out to be 0.0074%, which is much below the limit set in Ref. [35].

Fig. 5. (a) Cascaded 5-level MVSI topology for HEPWM, (b) output waveform (line to neutral voltage) of 5-level MVSI using the angle distribution ratio of 3/4, (c) output waveform (line to line voltage) of 5-level MVSI using the angle distribution ratio of 3/4, and (d) harmonics spectrum (of line to line voltage) of 5-level MVSI.

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Seven level inverter To benchmark the performance of DE, comparison is made with the work carried out in Ref. [28]. The objective is to solve seventeen angles in a 7-level cascaded MVSI, which is the most severe condition reported in literature. Fig. 6 displays the switching angle trajectories of seventeen angles along with the residual harmonics. It is worth mentioning that various angle distribution ratios have been tried to find the best solution. Fig. 6(a) shows the switching angle trajectories obtained through the angle distribution ratio of 7/5/5. It can be observed that this ratio has covered the range 1.50 6 M 6 2.00 without any discontinuity. Furthermore, the non-eliminated harmonics components are always less than 0.10% of fundamental component. Similar behavior is exhibited by the angle distribution ratios of 5/7/5 and 3/5/9 as shown in Fig. 6(b) and (c), respectively. The distribution ratio of 5/7/5 has covered the range of 2.00 6 M 6 2.45 while 2.45 6 M 6 2.80 range is covered by 3/5/9. The residual harmonics components are always less than 0.20% of fundamental component for all cases. Fig. 7(a) and (b) depicts the simulated line to neutral and line to line output voltage waveforms for M = 2.00. Harmonics spectrum of the line to line output voltage is shown in Fig. 7(c). It can be observed that all the triplens and low order harmonics under consideration are removed successfully, with 53rd

453

harmonics appears as the first non-eliminated harmonics component. The THD value turned out to be 13.21% of the fundamental component, while the DF2 is 0.0014%. The performance of DE is compared with the work presented in Ref. [28]. The key results are summarized in Table 1. The latter utilized the minimization method, as described in Ref. [20]. It can be observed that in the case of DE, no discontinuity of the solution is observed within 1.50 6 M 6 2.80. On the other hand, the work in Ref. [28] exhibits certain discontinuities (i.e., no solution is found for M = 2.59 to M = 2.63). Hence, DE provides a continuous smooth operation of HEPWM multilevel inverters. The ability to achieve wider modulation range is another reason for the preference of MVSI over the two or three level inverter. Extended modulation range increases the utilization of the DC link voltage, thus making it more attractive for high power applications [28,36]. However to obtain the solution for higher range is not easy due to the need for angle sequencing and the limitation due to angular spacing. These constraints impose a physical limit for the value of M. Hence, as the modulation reaches its limiting value, the difficulty to reach for the solutions increases tremendously. So, an algorithm that can further increase the range of M without introducing another inverter level is preferable. In this context, the range extension using DE (from 2.67 to 2.80) is crucial. Furthermore, the angle distribution ratio plays an important role in the solution of notch HEPWM problem.

Fig. 6. Switching angle trajectories against respective values of modulation index and percentage residual amplitude of the harmonics w.r.t fundamental component against respective values of modulation index for the distribution ratios (a) 7/5/5, (b) 5/7/5, and (c) 3/5/9.

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Fig. 7. (a) Output waveform (line to neutral voltage) of 7-level MVSI using the angle distribution ratio of 7/5/5, (b) line to line output voltage of 7-level MVSI using the angle distribution ratio of 7/5/5, and (c) harmonics spectrum (of line to line voltage) of 7-level MVSI.

Table 1 Summary of a comparison between the present work and the work of Ref. [28]. Comparison parameters

Technique in Ref. [28]

DE (proposed)

Modulation range covered Discontinuities in solution Number of angle distribution ratios

1.51–2.67 2.58–2.64 4

1.50–2.80 0 3

This is because to determine the suitable distribution ratio requires tedious trial and error process. Therefore, wherever possible, the least distribution ratio is desirable. In this context, DE used fewer distribution ratios than the minimization method.

Experimental validation Hardware set-up Experimental verification of the simulated results is carried out by constructing single phase five and seven level H-bridge cascaded MVSI prototypes. Since the relationship between single and three phase systems is well known, there is no significant reason to consider the latter. The power circuit are built using the IRF610 (200 V, 3.3 A) MOSFET with internal anti parallel diodes. Selected switching angles of different distribution ratios and M are used to generate gate drive signals. The switching waveform is created using the QUARTUS II software, which is designed for

Fig. 8. (a) Experimental line to neutral output voltage of with notch HEPWM problem for five level cascaded MVSI with distribution ratio of 5/2 with seven number of switching angles per quarter cycle (M = 0.85). (b) Frequency spectrum of the line to neutral output voltage of (a).

Fig. 9. (a) Experimental line to neutral output voltage of with notch HEPWM problem for five level cascaded MVSI with distribution ratio of 1/6 with seven number of switching angles per quarter cycle (M = 1.55). (b) Frequency spectrum of the line to neutral output voltage of (a).

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Fig. 10. (a) Experimental line to neutral output voltage of with notch HEPWM problem for seven level cascaded MVSI with distribution ratio of 7/5/5 (N1 = 7, N2 = 5, N3 = 5) with seventeen number of switching angles per quarter cycle (M = 1.65). (b) Frequency spectrum of the line to neutral output voltage of (a).

Fig. 11. (a) Experimental line to neutral output voltage of with notch HEPWM problem for seven level cascaded MVSI with distribution ratio of 3/5/9 (N1 = 3, N2 = 5, N3 = 9) with seventeen number of switching angles per quarter cycle (M = 2.70). (b) Frequency spectrum of the line to neutral output voltage of (a).

the ALTERA Cyclone II 2C20F484 FPGA board. The HEPWM signals (generated through ALTERA board) are passed on to the gating drivers which generate gating signals for the power circuit. The DC link voltage is kept as 20 V for each H-bridge module. Output voltages and their frequency spectra are analyzed through LeCroy 44MXs-B, 400 MHz 5 GS/s oscilloscope. Experimental waveforms For five level MVSI, the operating points of M = 0.85 and M = 1.55 are selected for the demonstration. These values of M belong to the distribution ratios of 5/2 and 1/6, respectively. Their line to neutral output voltages along with the respective frequency spectra are shown in Figs. 8 and 9. Figs. 8(b) and 9(b) illustrate that targeted harmonics (5th, 7th, 11th, 13th, 17th and 19th) are successfully removed from the output voltages. As expected, the 23rd harmonics component is the first non-eliminated component. Moreover, magnitudes of the fundamental components of output voltages are also in accordance with Eq. (5). It is re-emphasized that triplen harmonics are not considered for removal as they are eliminated in the line to line output voltage spectrum. The case of seven level cascaded MVSI with seventeen angles is also verified experimentally. Distribution ratios of 7/5/5 and 3/5/9 with the values M = 1.65 and M = 2.70 respectively are selected to generate the output line to neutral voltages. The respective output voltages and their frequency are presented in Figs. 10 and 11. It is important to mention that the value of M = 2.70 has been chosen to demonstrate the extended range of M. The minimization technique [28] is unable to solve the HEPWM switching angles for M > 2.67. Figs. 10(b) and 11(b) show that the targeted low order harmonics (seventeen in this case) are successfully removed from the frequency spectra of output voltages, with 53rd harmonics component as the first non-eliminated component. In addition, the fundamental component value is also controlled according to Eq. (5). Clearly, these results are in agreement with the simulation and are proof of the credibility of the DE. Conclusions The Differential Evolution optimization has been utilized to solve the HEPWM angles for cascaded MVSI. It has fulfill its

objectives to (1) extend the maximum range of modulation index (M) for the switching angle trajectories, (2) remove the discontinuities in the switching angle and to (3) reduce the number of distribution ratio required to obtain the solution for a wide range of M. Five and seven level cascaded inverters are selected as test cases; comparison between the minimization method and DE has also been carried out. Different parameter settings of DE and various distribution ratios of switching angles are employed to obtain optimum switching angles. Results obtained validate the successful elimination of harmonics under consideration. The solution angle trajectories are free from any discontinuity and a wider range of M has been covered. Coverage of wider range of M without a compromise to the harmonics components (1) enhances the utilization of DC link voltages and (2) extends the application of notch type HEPWM for high power applications. Selected operating points have also been utilized for the hardware verification of the obtained results which are in accordance with the simulation results, and hence are a clear evidence of the superiority of DE for the HEPWM problem of MVSI.

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