Extremely Low Frequency PS-PWM-based Technique for Cascaded H-Bridge Converters with Grid Voltage Compensation Capability

Depósito de investigación de la Universidad de Sevilla https://idus.us.es/ This is an Accepted Manuscript of an article published IEEE Transactions on Industrial Electronics available at: 10.1109/TIE.2023.3274870 “© 2023 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other Works” IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS 1 Extremely Low Frequency PS-PWM-based Technique for Cascaded H-Bridge Converters with Grid Voltage Compensation Capability Abraham M. Alcaide, Member,IEEE, Vito Giuseppe Monopoli, Senior Member,IEEE, Giuseppe Rendine, Lara Bruno, Jose I. Leon, Fellow Member,IEEE, Marco Liserre, Fellow Member,IEEE, and Leopoldo G. Franquelo Life Fellow,IEEE, Abstract—Cascaded H-Bridge Converters (CHB) are good candidates for many industrial applications. Among the modulation techniques that can be applied to CHB converters, the Phaseshifted PWM method (PS-PWM) provides very good harmonic quality being the most popular modulation technique at the expense of presenting high switching losses. On the contrary Block Switching technique as well as Selective Harmonic elimination/mitigation (SHE/SHM) techniques provides lower switching frequency and losses with a lower harmonic quality. Additionally with these techniques the converter operation becomes stiff. In this work a modification of PS-PWM based on an extremely low frequency carrier is proposed with the objective of achieving both low switching losses and an acceptable harmonic quality, simultaneously. The main advantage of the proposed solution is the capability of adjusting the phase-shift angle among the carriers to compensate for harmonic grid voltage disturbance. Several experimental results have been reported to demonstrate the good performance of the proposed modulation technique. Index Terms—Harmonic analysis, Pulse width modulation, Multilevel converters. I. INTRODUCTION Power energy demand is increasing every year hand by hand with the governmental policies to fight against the green house emissions as well as to limit the fossil fuels usage [1]. Power demanded by the society is not only increasing year by year, but also the requirements for grid operators and new regulations are becoming strict in terms of power quality as well as ancillary services [2], [3]. As a consequence, in The authors gratefully acknowledge financial support provided Ministerio de Ciencia e Innovaci´ on under the project PID2020-115561RB-C31 and TED2021-130613B-I00, Junta de Andaluc´ ıa under the project P18-RT-1340. Abraham M. Alcaide gratefully acknowledges the financial support provided by Contrataci´ on de Personal Investigador Doctor. (Convocatoria 2019) 43 Contratos Capital Humano L´ ınea 2. Paidi 2020, supported by the European Social Fund and Junta de Andaluc´ ıa Abraham M. Alcaide is with the Electronic Engineering Department, Universidad de Sevilla, Seville, 41092 Spain (e-mail: [email protected]g). V. G. Monopoli and Lara Bruno are with the Department of Electrical and Information Engineering, Politecnico di Bari, 70126 Bari, Italy (e-mail: [email protected]). Giuseppe Rendine is with Automation Engineer with EMS Group S.p.A. M. Liserre is with the Chair of Power Electronics, Christian-Albrechts- University of Kiel, Kiel 24143, Germany (e-mail: [email protected]). Jose I. Leon and Leopoldo G. Franquelo are with the ENGREEN Laboratory of Engineering for Energy and Environmental Sustainability, Universidad de Sevilla, Seville, 41092 Spain and also are chair professor with the School of Astronautics, Harbin Institute of Technology, Harbin (China). Cell 1 Cell 2 Cell 3 v1 v2 v3 v a n + - + - + - S 31 S 31 + S 32 S 32 V dc ,3 +- Fig. 1: Single-phase multilevel cascaded H-bridge converter many situations high power levels and/or high voltages are required [4], [5]. To fulfill these requirements, the industry and the academia are in continuous development of efficient power converter solutions to achieve high performance with limited cost [6]. In this way, multilevel converters have become a trendy solution to achieve high voltage rating with very high output waveforms quality [7]–[9]. Among the multilevel power converters portfolio, the cascaded H-bridge (CHB) power converter is a very popular solution for many industrial applications such as STATCOMs, motor-drives, pumps and fans, among many others applications [10]–[12]. CHB converter is tailored following the Power Electronics Building Block (PEBB) paradigm by the serial connection of the several power modules. In this sense, it is possible to affirm that the CHB converter presents natural fault-tolerant capability because of its inherent modularity. Both are great features from the manufacturer and maintenance cost point of view [13], [14]. As an example, a single-phase three-cell CHB converter is represented in Fig. 1. Among the modulation techniques available in the literature to operate a CHB converter, the most conventional method is the well-known phase-shift pulse-width modulation (PSPWM) technique [9], [15], [16]. PS-PWM method proposes to operate each power cell using an unipolar PWM method implemented with the modulating signal, the modulating signal with opposite phase, and a triangular carrier with fcfrequency [17]. To achieve the optimum performance, the PS-PWM method requires a specific phase displacement among the car- This article has been accepted for publication in IEEE Transactions on Industrial Electronics. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TIE.2023.3274870 © 2023 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information. IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS 2 riers of each H-bridge. This phase displacement is, in general, determined, with respect to the carrier phase displacement corresponding to the first power cell, as: ϕx= (x−1) π N(1) where xrepresents the cell number (x=1,. . .,N), and Nis the number of cells per phase of the CHB [17], [18]. The advantages and properties of the PS-PWM method are well-known. As a general rule, a converter operated with PSPWM method achieves an inherent power sharing between the power modules and also equally distributed power losses. As a consequence, the aging of the power devices is naturally equalized [8], [9], [17]. On the other hand, the converter also has a superior harmonic performance since the output voltage and current present a multiplicative effect in the switching frequency [17], [19]. In this way, although the power devices switch at a frequency equal to fc, the CHB output waveforms operated with the PS-PWM method with unipolar PWM per H- bridge present an virtual switching frequency equal to 2Nfc. However, a multilevel converter operated with PS-PWM strategy usually presents higher power loss than applying other modulation techniques such as staircase modulation, hybrid PWM methods, or even the well-known selective harmonic elimination/mitigation (SHE/SHM), which is specifically designed for high-power converters to meet the grid standards with very reduced power loss [20]. In SHE/SHM methods, the output voltage and current quality highly depend on the number of switching angles (αi)that are considered in the calculation of a pre-programmed switching pattern [21]–[23]. The SHE/SHM methods are stiff since the pre-programmed solution should be updated each fundamental period. The switching angles set should be obtained off-line and the computational cost depends on the number of considered switching angles (i.e. on the number of harmonics to be eliminated/mitigated) [24], [25]. In this work, a variation of the PS-PWM technique based on an extremely low-frequency carrier will be considered. This adopted solution allows to obtain a good trade-off between low switching losses and acceptable harmonic quality of the output waveforms. On the other hand, in grid connected applications, it is usual to deal with non-desired harmonic distortion, which limits the operation of the converter to meet the standards. It leads to the use of complex controllers to compensate them and/or filter structures which in general are expensive, heavy and bulky [26], [27]. This phenomenon is shown in Fig. 2 where a simplified equivalent circuit of a distorted grid (that includes a non-negligible third harmonic content) connected to a converter is shown. If the power converter does not compensate this distortion, the corresponding grid current also presents poor quality. This phenomenon can be observed in Fig. 3, drawn in blue. However, if the output voltage generated by the converter contains the appropriated harmonic magnitude and phase, the resulting grid current does not present this undesired harmonic distortion as it is shown in Fig. 3, drawn in red. This feature is the basis of the proposed modulation method in order to compensate the distortion present in the grid. Therefore, the innovation introduced by the proposed L i R vg1(t)=A1sin(w1 t +©1 ) vg3(t)=A3sin(w3 t +©3 ) vg(t)=v g1 (t)+v g3 (t) Grid sideConverter side + + + v(t) Fig. 2: Simplified equivalent circuit of a power converter connected to a grid with a non-negligible third harmonic content 0 40 80 120 Magnitude (A) Current (A) 0 4 1 32 85.4A 0 1 32 4 5 6 7 8 9 10 Harmonic order (n) Time (ms) 0 20 40 60 80 100 -120 -80 -40 0 40 80 120 Uncompensated Compensated a b Fig. 3: Grid current modulation technique is the capability of compensating grid voltage harmonic disturbances and hence of producing voltage harmonics of given amplitude and phase by properly adjusting the phase-shift angles among the different carriers. The proposed solution finds the optimal set of the carrier phaseshift angles through an optimization algorithm, namely the Exchange Market Algorithm (EMA). The manuscript is organized as follows: Section II mathematically describes the proposed modulation technique whereas Section III illustrates the output voltage optimization process. Finally, Section IV and Section V show the corresponding experimental validation and the conclusions, respectively. II. MATHEMATICAL DESCRIPTION OF THE OUTPUT VOLTAGE OF THE CHB CONVERTER CHB converter is usually operated by the PS-PWM technique because of its good features. In this modulation method, each power cell is operated using unipolar PWM strategy, where each leg of the CHB is operated considering the This article has been accepted for publication in IEEE Transactions on Industrial Electronics. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TIE.2023.3274870 © 2023 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information. IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS 3 0.0 5 10 15 20 0.0 1.0 -1.0 0.0 1.0 -1.0 0.0 1.0 -1.0 0.0 1.0 -1.0 Time (ms) c Áa,1 v-v Áb,1 Áb,2 Ác,1 Ác,3 d a b Fig. 4: Implementation of the proposed modified unipolar PWM technique modulating signan and another 180◦-shifted modulating signal with just one carrier signal [17]. This fact is the responsible of the double multiplicative effect in the harmonic spectrum of each power cell output voltage. In the previous approaches that modify the implementation of the PS-PWM method applied to a CHB converter, there is only one degree of freedom to act over the resulting output voltage of the converter (the carrier phase-displacement angle between the carriers of the power cells) [16], [28]. On the contrary, the proposed modulation technique presents extra degrees of freedom because the unipolar PWM method is generalized considering the modulating signal, the 180◦- shifted modulating signal and two carrier signals with an specific carrier phase-displacement, each one associated to each H-bridge leg. Therefore, in the case of a three-cell CHB converter, there are five carrier phase-displacement angles that can be tuned in order to optimize the resulting CHB output voltage quality. In order to study the harmonic behavior of the output waveforms generated by each power cell, the double Fourier Integral can be used [17]. As a general rule, this mathematical tool allows to describe any periodical signal as follows: f(t) = f0+f1(t) + fbb(t) + fsb(t)(2) where f0and f1(t)are the dc and the fundamental harmonic components, whereas fbb(t)and fsb(t)are referred to the base-band and side-band harmonic components, respectively. Following this notation, the resulting output voltage of each individual branch in a H-brigde is determined by: 0 10 20 30 40 50 60 70 80 90 100 Harmonic order (k) 10-1 1 10-2 10-3 10-4 Spectrum magnitude (pu) m = 1 k = 2mR +(2n-1) m = 2 n = 1n = 0 n = 2n = -1 n = 3n = -2 n = 1n = 0 n = 2n = -1 n = 3 n = -2 Fig. 5: Example of notation for the harmonic spectrum of the output voltage in a CHB converter calculated by the Double Fourier Integral approach va,x(t) = H0,x +H1,x cos (ωot+ Θo) + ∞ X m=1 ∞ X n=−∞ Hmn,x cos zt + (mϕa,x +nΘo)(3) and vb,x(t) = H0,x +H1,x cos (ωot+ Θo+π) + ∞ X m=1 ∞ X n=−∞ Hmn,x cos zt + (mϕb,x +nΘo+nπ)(4) where variables ωoand Θoare referred to reference signal frequency and phase, and the index xis referred to the particular power cell of the CHB. The coefficients z,H0,x, H1,x and Hmn,x are calculated as follows: z=mωc+nωo H0,x =Vdc,x 2 H1,x =Vdc,xMx 2 Hmn,x =2Vdc,x mπ JnmπMx 2sin (m+n)π 2(5) where Jnis the first kind Bessel function of order n,m determines the harmonic group and nis referring to a specific harmonic component within the group m(see the notation example figure shown in Fig. 5). It is worth to be mentioned that the resulting output voltage in the H-Bridge does not contain base-band components is the converter is not operated in over-modulation region. Finally, Mxis referred as the desired modulation index which is defined as: VT dc = N X x=1 Vdc,x (6) M=H1 VT dc (7) This article has been accepted for publication in IEEE Transactions on Industrial Electronics. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TIE.2023.3274870 © 2023 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information. IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS 4 where Vdc,x is the dc voltage of the power cell xand H1is the desired fundamental voltage. After the corresponding coefficients calculation, the resulting output voltage of each power cell as well as the total output voltage of the CHB converter can be determined as: vab,x(t) = van,x(t)−vbn,x(t)(8) v(t) = N X x=1 vab,x(t)(9) It can be observed that harmonic components of the odd groups in (3) and (4) do not cancel each other as it happens in the conventional uni-polar PWM technique. This fact is essential for harmonic disturbance cancellation purposes. In general, a particular carrier and side-band harmonic component of the power cell xcan be expressed as: vmn,x(t) = Hmn,xcos zt + (mϕa,x +nΘo) −cos zt + (mϕb,x +nΘo+nπ)(10) After some mathematical manipulation, in (10) cosine terms can be decoupled and each particular harmonic component of the resulting output voltage can be described in magnitude and phase as: vmn,x(t) = −2Hmn,x sin m(ϕa,x −ϕb,x) 2 sin zt +m(ϕa,x +ϕb,x) + nπ 2+nΘo)(11) that is: |Hx|=|vmn,x(t)|= 2Hmn,x sin m(ϕa,x −ϕb,x)−nπ 2 ∠(Hx) = ∠(vmn,x(t)) = m(ϕa,x +ϕb,x) + nπ 2+nΘo) (12) where the resulting magnitude and phase can be linked to expression (9) by Hk= N X x=1 vmn,x(t) = N X x=1 |Hx|∠(Hx)(13) As it is shown in (12), it is possible to manipulate the magnitude and phase of the resulting CHB output voltage as a function of the phase-displacement angle ϕa,x and ϕb,x. III. PROPOSED MODULATION TECHNIQUE FOR THE CHB CONVERTER As it was mentioned above, in the conventional PS-PWM strategy applied to the CHB converter, the carrier phasedisplacement angles between the power cells are fixed. Additionally, the technical literature also provides an advanced version of this modulation technique (called Variable-angle PS-PWM, VA-PS-PWM) which is able to modify these carrier phase-displacement angles during the operation of the converter to achieve an improved output voltage harmonic performance [16], [28], [29]. In this sense, it is possible to affirm that previous VA-PS-PWM techniques count with N−1degrees of freedom for the optimization of the resulting output voltage. However, the 180◦carrier phase-displacement into each unipolar PWM strategy is still imposed in previous approaches. On the contrary, the proposed modulation technique allows to set each individual carrier phase-displacement angle respect to the carrier aof the first branch of the first power cell. This leads to use variable carrier phase-displacement angles even within each power module modifying the conventional operation of the unipolar PWM. In this sense, the proposed technique counts with 2N−1degrees of freedom (corresponding to the carrier phase displacements of each H-bridge branch of the CHB converter) for the optimization of the resulting CHB output voltage. As it was introduced previously, the equations to calculate the harmonic description of the CHB output voltage lead to deal with a complex mathematical problem. Additionally, the complexity of the mathematical problem to improve the harmonic performance of the CHB converter is also increasing when the number of power cells grows. In this sense, it is more appropriated to transform the mathematical problem into a multi-variable non-lineal mathematical minimization problem by defining a cost function as follows: e1= (H⋆ 1−H1)2 ek= (H⋆ k−Hk)2 τ=e1+ek+λWTHD0 (14) where H⋆ 1and H⋆ krepresent the desired fundamental harmonic component and the required harmonic component to compensate the distorted grid (in a complex form, that is, magnitude and phase as shown in 10), respectively. H1and Hkare the fundamental and the kth harmonic component obtained by the application of the proposed phase-displacement angle set. Therefore, cost function τis defined as the minimization of both error variables shown in (14). For practical implementation, the magnitude and phase expression shown in (12) can be used. In addition, in the cost function τ, the WTHD0 value is included in order to consider the reduction of the rest of higher-order harmonic components. A weighting factor λhas been included in the cost function in order to properly tune the cost function because the minimization of the first two terms has more importance, letting the WTHD0 reduction as a secondary target. The minimization of a cost-function usually presents a high computational burden. The use of artificial intelligence tools such as the meta-heuristic algorithms has allowed the resolution of problems not solved in the recent past. The variety of techniques is growing year by year since the academia is highly focused on this topic. For instance, the Genetic Algorithm (GA) is inspired by the natural selection process present in the biological evolution [30]. Particle Swarm Optimization This article has been accepted for publication in IEEE Transactions on Industrial Electronics. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TIE.2023.3274870 © 2023 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information. IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS 5 [Áa1 Áb1 Áa2 Áb2 Áa3 Áb3 ]Solution Shareholders Ranking (according their ¿i ) Evaluate Cost Function ¿ (14) InitEMA() Shareholders M=0 Stock Á10 Á11 Á1k-1 ¿1 ¿2 ¿p Á20 Á21 Á2k-1 Áp0 Áp1 Ápk-1 Balanced Market Evolution Trading Market Evolution M=0? Shareholder evolution Evaluate Market Rank Sharedholder M =1 M =0 YES NO Fig. 6: EMA implementation to look for the reduction of the cost function τ 0 200 400 600 Magnitude (V) 0 2010 30 5040 Harmonic order (k) -180 -90 0 90 180 Phase (º) 1 0 50 100 3 5 7 9 11 13 15 17 19 Fig. 7: Theoretical evaluation of the particular operation point of the CHB output voltage given in Table I. a) Harmonic Spectrum Magnitude b) Harmonic Spectrum Phase (PSO) is based on the leadership present in the social animal behavior [31]. As a combination of both techniques, Exchange Market Algorithm (EMA) mixes their benefits achieving a good trade-off between computational burden and convergence performance avoiding the local minima [29], [32]. EMA is inspired by shares traded on the stock market and it is the selected method for the harmonic optimization in this problem by looking for the most proper carrier phase displacement angles in order to minimize the cost function defined in (14). The block diagram of the EMA implementation in order to reduce the cost function is represented in Fig. 6. In this EMA implementation, a population of p= 30 carrier phase-displacement solutions is iteratively evaluated. Once the cost functions are determined, the solutions sets are ranked depending on the instantaneous value of the corresponding cost function. For the next iteration, the solutions set are modified accordingly 0 0.02 0.04 150 0.06 0.08 100 0.1 3 rd harmonic amplitude (V) 3rd harmonic phase (º) 50 0 WTHD (%) 0 90 180 270 360 Fig. 8: Results of the optimization cost function τfor a modulation index equals to 0.9. Resulting value of the figure of merit WTHD0 as function of the amplitude and phase of the third harmonic injected the trading markets (slight modifications for good solution sets and large or even random modifications for solutions sets that achieve a poor performance). Two trading markets are alternatively considered, one trading market (M= 1) where the modifications of the solutions sets are large, and another more stable (M= 0) where the solutions sets modifications are not so large. After modifying the solutions sets, the cost functions are again calculated in a non-stop iterative procedure in order to minimize the cost function. More details of EMA implementation are addressed in [29], [32]. However it is worth to be mentioned that any meta-heuristic method can be applied to minimize the cost function. Although the minimization of the cost function cannot be mathematically demonstrated, all these mathematical search algorithms, properly tuned, can achieve a reduction of the cost function. In Table I an application example for the theoretical evaluation of the proposed method is presented. It is important to highlight that the carrier frequency is very low (150 Hz) leading to very reduced switching losses. In Table I the carrier TABLE I: Operational conditions for the theoretical validation CHB Operation parameter DC Voltage Vdc,x (V) 200 Desired modulation index M0.9 Fundamental frequency fo(Hz) 50 PWM Carrier frequency fc(Hz) 150 Undesired Distortion H⋆ 330V∠300◦ EMA-based Solution for phase-displacement angles Power Cell 1 [ϕa,1, ϕb,1](◦)[0,289.75] Power Cell 2 [ϕa,2, ϕb,2](◦)[301.36,237.20] Power Cell 3 [ϕa,3, ϕb,3](◦)[156.82,60.33] Resulting CHB output voltage Magnitude of fundamental H1∠(H1) 540∠0 Magnitude of third harmonic H3∠(H3) 30∠60 This article has been accepted for publication in IEEE Transactions on Industrial Electronics. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TIE.2023.3274870 © 2023 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information. IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS 6 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 0.5 1.0 1.5 2.0 Modulation index WTHD0 (%) WTHD0 Fig. 9: Results of the optimization cost function τfor a specific harmonic injection equals to H⋆ 3= 30V∠30◦. TABLE II: Operational conditions of the experimental setup Operation parameters Unit Values DC Voltage Vdc,x (V) 120 Desired modulation index M0.9 Fundamental frequency fo(Hz) 50 PWM Carrier frequency fc(Hz) 150 Desired Fundamental H⋆ 1324∠0 phase-displacement angles set obtained by the execution EMA algorithm is also presented. These carrier phase displacement angles minimize the cost function τfor a desired output voltage composed by [H1, H3]equal to [540,30]with a phase [P1, P3]equal to [0,60]◦. In Fig. 7 the resulting output voltage magnitude spectrum and its phase are presented. It is important to highlight that the proposed modulation technique achieves the desired output voltage harmonic performance for the complete modulation index range in a wide range of magnitude and phase of the third harmonic. In this sense, Fig. 8 shows the result of the defined cost function for a particular harmonic injection for a modulation index equal to M= 0.9. On the other hand, in Fig. 9 the cost function minimization solution for a particular harmonic injection (H⋆ 3= 30V∠30◦) considering the complete modulation index range is shown. As it can be observed, the resulting value of the figure of merit essentially depends on the desired harmonic distortion to be compensated. However for an specific harmonic injection the resulting WTHD0 figure of merit is essentially the same value for the whole modulation index range. IV. EXPERIMENTAL RESULTS In order to validate the performance of the proposed modulation technique, several experiments have been performed in the low-voltage laboratory setup shown in Fig. 10. This experimental setup is composed by a three-cell single-phase CHB converter connected to a grid emulator. The dc voltage of each power cell is supplied by an independent isolated dc source. The operation of the whole converter is performed by the low-cost mid-range TMS320F28379D Digital Signal Processor (DSP) platform from Texas Instruments [33]. Parameter λin the cost function defined in (14) has been fixed to one obtaining good results. The rest of the operational parameters of the converter are reported in Table II. TABLE III: Specific Operational conditions of the proposed technique H⋆ 3Phase-displacement angles Values 20∠120 Power Cell 1 [ϕa,1, ϕb,1](◦)[0,289.75] Power Cell 2 [ϕa,2, ϕb,2](◦)[301.36,237.20] Power Cell 3 [ϕa,3, ϕb,3](◦)[156.82,60.33] 30∠300 Power Cell 1 [ϕa,1, ϕb,1](◦)[0,79.92] Power Cell 2 [ϕa,2, ϕb,2](◦)[55.37,310.17] Power Cell 3 [ϕa,3, ϕb,3](◦)[303.83,26.25] Fig. 10: Down-scaled single-phase three-cell CHB prototype The experiments consider two different scenarios to demonstrate the ability of the proposed modulation technique to achieve not only the desired fundamental voltage, but also the rejection of an undesirable harmonic component present in the grid. The fundamental voltage meets the requirements shown in Table II, and the particular scenarios under study are also presented in Table III. In these scenarios a non-negligible third harmonic content is present in the grid voltage. The proposed method is applied in order to generate the desired fundamental component and to compensate this negative loworder harmonic component. The particular phase-displacement angles to be applied in the proposed modulation technique are also reported in Table III. In each case, the EMA has been executed in order to determine the most suitable carrier phase-displacement angles to achieve all the objectives by minimizing the cost function τ. For each experiment, the resulting output voltage, current as well as their corresponding harmonic spectrum magnitude are obtained. First of all, the PS-PWM method applying the conventional carrier phase-displacement angles with carrier frequency equal to 150 Hz is tested generating a fundamental voltage with modulation index equal to 0.9 . The obtained results are represented in Fig. 11, where it can be observed that a non-negligible third harmonic component is present in the output current due to the distorted grid. On the other hand, the corresponding results applying the proposed method are represented in Fig. 12 and Fig. 13, where it can be observed that a superior performance is achieved even with a distorted grid. As it can been observed, the objectives of the cost function are achieved not only in the fundamental magnitude, but also in the harmonic disturbance rejection ability in the resulting current injected to the grid emulator. The achieved THD figure of merit (calculated considering up to 50th harmonic order) is also reported in the corresponding This article has been accepted for publication in IEEE Transactions on Industrial Electronics. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TIE.2023.3274870 © 2023 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information. IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS 7 0 10 20 30 40 -10 -5 0 5 10 Time (ms) Harmonic order (n) Current (A)Current (A) 0 10 20 30 40 50 0 1 2 c d -400 -200 0 200 400 Voltage (V)Voltage (V) 0 30 60 Harmonic order (n) 0 10 20 30 40 50 0 10 20 30 40 Time (ms) THD = 8.97% a b Fig. 11: Experimental results considering the PS-PWM method applying the conventional carrier phase-displacement angles a) Output voltage. b) Output voltage harmonic spectrum. c) output current d) output current harmonic spectrum. 0 10 20 30 40 -10 -5 0 5 10 Time (ms) Harmonic order (n) Current (A)Current (A) 0 10 20 30 40 50 0 1 2 c d -400 -200 0 200 400 Voltage (V)Voltage (V) 0 30 60 Harmonic order (n) 0 10 20 30 40 50 0 10 20 30 40 Time (ms) THD = 5.29% a b Fig. 12: Experimental results for proposed modulation technique considering the operational point H⋆ 3=20∠120 reported in Table II . a) Output voltage. b) Output voltage harmonic spectrum. c) output current d) output current harmonic spectrum. figures. A. Comparison with SHE/SHM and high frequency carrierbased PS-PWM methods The proposed modulation method can be compared with the fundamental switching SHE/SHM methods because both strategies are looking for very low switching losses. An experiment applying a SHE/SHM method considering a modulation index equals to 0.9, an specific 3rd harmonic component equals to 12 V and the elimination of the 5th harmonic component has been carried out. The obtained results are represented in Fig. 14, where it can be observed that, although the third and fifth harmonics contents achieve the expected values in the output voltage, a non-negligible third harmonic component is present in the output current. This demonstrate that SHE/SHM methods cannot carry out a proper harmonic compensation because the phase of the harmonic components is not considered in the switching angles determination. In addition, the SHE methods do not consider the higher order components that are not eliminated leading to obtain high THD values. This is also faced by the proposed method by introducing WTHD0 term in the cost function. This article has been accepted for publication in IEEE Transactions on Industrial Electronics. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TIE.2023.3274870 © 2023 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information. IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS 8 0 10 20 30 40 -10 -5 0 5 10 Time (ms) Harmonic order (n) Current (A)Current (A) 0 10 20 30 40 50 0 1 2 c d -400 -200 0 200 400 Voltage (V)Voltage (V) 0 30 60 Harmonic order (n) 0 10 20 30 40 50 0 10 20 30 40 Time (ms) THD = 6.48% b a Fig. 13: Experimental results for proposed modulation technique considering the operational point H⋆ 3= 30∠300 reported in Table II a) Output voltage. b) Output voltage harmonic spectrum. c) output current d) output current harmonic spectrum. Another extra experiment has been carried out in order to test the performance of the PS-PWM method considering a high frequency carrier signal (equals to 3 kHz) and conventional carrier phase-displacement angles. The harmonic compensation is enabled adding a required third harmonic component (considering magnitude and phase) to the voltage reference of one of the power modules. The obtained results are represented in Fig. 15, where it can be observed that a superior performance is achieved in the current harmonic quality at the expense of very high switching loss. As a summary, the proposed method can be considered as a trade-off solution between the SHE/SHM methods and the PS-PWM technique with high frequency carrier signals. The proposed method achieves the harmonic compensation of the undesired third harmonic component and the reduction of higher order harmonics but keeping limited the switching loss. V. CONCLUSIONS Multilevel converters are key players in the current energy scenario. Cascaded H-bridge converters operated using the PSPWM method have shown multiple benefits in the past and the expense of higher power loss. 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