Generalized Multi-Carrier PWM Technique for Two-level Voltage Source Inverters

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.2022.3190872 “© 2022 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 Generalized Multi-Carrier PWM Technique for Two-level Voltage Source Inverters Abraham M. Alcaide, Member,IEEE, Vito G. Monopoli,Senior Member,IEEE, Eduardo Zafra, Member,IEEE, G. Buticchi, Senior Member,IEEE, Jose I. Leon, Fellow Member,IEEE, S. Vazquez, Fellow Member,IEEE, M. Liserre, Fellow Member,IEEE and Leopoldo G. Franquelo, Life Fellow Member,IEEE. Index Terms—Harmonic analysis, Pulse width modulation, Multilevel converters. Abstract—Two-level voltage source inverters are the mainstream solution for many power applications. The conventional modulation method of such inverters is the single-carrier pulse-width modulation (PWM) technique. In this paper, the usage of a generalized PWM method where three independent carriers is analyzed. The carrier phasedisplacement angles are evaluated in order to reduce the common-mode voltage (CMV) total harmonic distortion (THD), the dc-link capacitor current THD and the phase currents THD. These targets are difficult to be met at the same time and a cost function is deployed in order to look for a suitable solution that develops a trade-off between all the targets. The proposed method has been tested in a laboratory prototype demonstrating its superior performance. I. INTRODUCTION The evolution in the last decades of power electronics, renewable energy sources and energy storage systems helps to understand the current global energy scenario looking for avoiding the use of fossil fuels and the associated green house emissions. In this sense, the ecosystem of power converters in the industry is very wide, from motor-drive applications (including fans, pumps, conveyor belts and electrics vehicles) to grid-tied applications such as the integration of renewable energies and energy storage systems, among others [1]–[4]. Solar photo-voltaic (PV) and wind energy conversion systems are two of the most important renewable energy sources The authors gratefully acknowledge financial support provided Ministerio de Ciencia e Innovaci´ on under the project PID2020-115561RB- C31, Junta de Andaluc´ ıa under the project P18-RT-1340. Eduardo Zafra gratefully acknowledges the financial support provided by the Spanish Ministry of Universities under grant FPU18/02704. Abraham Marquez 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 and E. Zafra are with the Electronics Department, Universidad de Sevilla, Sevilla, C.P. 41092 Spain (e-mail: [email protected]; [email protected]). V. G. Monopoli is with the Department of Electrical and Information Engineering, Politecnico di Bari, 70126 Bari, Italy (e-mail: [email protected]). G. Buticchi are with the Zhejiang Key Laboratory for More Electric Aircraft Technologies, University of Nottingham Ningbo China. S. Vazquez, L. G. Franquelo and J.I. Leon are with the Laboratory of Engineering for Energy and Environmental Sustainability, Universidad de Sevilla, Sevilla, C.P. 41092 Spain (e-mail: [email protected]; lgfr[email protected]; [email protected]). M. Liserre is with the Chair of Power Electronics, Christian-Albrechts- University of Kiel, Kiel 24143, Germany (e-mail: [email protected]). and their world-wide power installation pace is exponentially growing year by year. This effect is especially pronounced in the case of PV because of the cost reduction of the PV panels in the recent years, and the possibility to install PV panels almost everywhere. In this way, in the past decade a lot of attention has been paid in the development of specific DC/AC topologies for PV applications based on modified full-bridge structures operated with carrier-based PWM methods [3], [5]– [8]. In Fig. 1 a conventional three-phase IGBT-based 2L-VSI is used in the solar photo-voltaic application. Additionally, in Fig. 2 a generalized pulse width modulation (PWM) scheme is introduced considering three independent triangular carriers (where ϕxis the phase displacement of the carrier associated to phase x, with x=a, b, c) to generate the switching signals. It is worth to be mentioned that this generalized PWM method also reproduces the traditional PWM technique (where a single triangular carrier is considered) if ϕa=ϕb=ϕc. In this paper, a deep study of the influence of applying a generalized PWM method with three triangular carriers is addressed. The performance related to the reduction of the commonmode voltage (CMV), the extension of the power capacitor lifespan, and the quality of the currents injected to the grid are evaluated. The main contribution of this work is to introduce a methodology to determine a suitable solution for the carrier phase-displacement angles of the multi-carrier PWM method in order to develop a trade-off between the reduction of the CMV total harmonic distortion (THD), the capacitor current THD, and the phase currents THD. II. COMMON-MODE VOLTAGE HARMONIC AND ITS RELATION WITH THE SOLAR PV LEAKAGE CURRENT PHENOMENON In the solar PV application, the PV cells feature a semiconductor layer (where the junction is) covered by thin sheets of metal (to collect the generated current) and finally covered by the protective glass. The PV cells are then connected in series/parallel to reach the required voltage/current characteristics and enclosed into a metal frame. This structure (semiconductor/metal, glass, metal frame) is characterized by a non-negligible capacitance between the active material and the metal frame. Considering the placement of the PV panels and the environmental conditions, a thin film of moisture can be formed on top of the glass. Taking this into account, in some 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.2022.3190872 © 2022 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 L ia ib ic N vaN vbN vcN Sa Sa Sb Sb Sc Sc Vdc O a b c ix(t) iCx(t) iC(t) Fig. 1: Grid-connected three-phase 2L-VSI acting as an active front-end (AFE) for PV applications 0 1 not Sa Sa + - va * 0 1 not Sb Sb + - vb * 0 1 not Sc Sc + - vc * Áb Ác Fig. 2: Generalized PWM method of the three-phase 2L-VSI considering three triangular carriers. The conventional singlecarrier PWM method is applied if ϕa=ϕb=ϕc conditions, the distributed capacitance between the frame and PV cells can reach values of 100-160 nF/kW p. The presence of the parasitic capacitance per se does not constitute a problem, but its presence can cause the circulation of high values of ground leakage current due to the characteristics of the PV system. In fact, the metal frame of the PV panels should be grounded for safety reasons. Additionally, the lowvoltage distribution systems are usually implemented through TT or TN connections, meaning that there exists a path for the ground leakage current to flow. The path goes through the parasitic capacitance, the power electronics, the electric grid and the neutral connection of the distribution transformer [5], [9]. Several papers have analyzed the ground leakage circuit, which shows resonant characteristics caused by the grid inductors and the parasitic capacitance. Several authors have agreed that a possible way to solve the problem of the ground leakage current is to reduce the common-mode excitation of this resonant circuit. In this way, several methods have been proposed aiming to reduce the common mode voltage (CMV) variations at the output of the PV converter [10]. Although state-of-the art solutions are present for the singlephase electronics, given the complexity of the common-mode problems for three-phase power converters, the research is still active [11], [12]. In general, these solutions feature additional switches to bypass the dc and ac sides during the diode freewheeling, resulting in additional complexity and non-simple modulation method. Considering the phase output voltage in the 2L-VSI shown in Fig. 1, the resulting common-mode voltage (CMV) is the voltage between the neutral point in the grid (N) and the middle point in the dc-link (o). It is possible that the neutral point in the grid is not accessible to measure the CMV directly. In this case, as an alternative, the CMV can be obtained as follows: CMV (t) = VNo(t) = vao +vbo +vco 3(1) where vxo are the phase voltages measured between terminals x(x=a, b, c) and the point o. To analyze the CMV, the double Fourier series mathematical tool can be used, that is: x(t) = A00 2+ ∞ X n=1 hA0ncos(nω0t) + B0nsin(nω0t)i + ∞ X m=1 ∞ X n=1 hAmn cos(mωc+nω0t) + Bmn sin(mωc+nω0t)i (2) where the coefficient A00 represents the dc-component value, and the coefficients A0nand B0nrepresent the base-bands harmonic content. Additionally, the coefficients Amn,Bmn (m= 0) are representing the side-bands harmonic content. All these coefficients are calculated using the double Fourier integral [13]. Considering the carrier-based PWM method shown in Fig. 2, the phase voltage vxo(t)is determined using the double Fourier series as: vxo(t) = MVdc 2cos(ω0t+θx) + ∞ X m=1 ∞ X n=1 Vxo mn cos mωct+ϕx+nω0t+θx# (3) Vxo mn =2Vdc mπ Jnmπ 2Msin (m+n)π 2=λmnVdc (4) where, as an approximation without loss of generality, it can be considered that all the modulation indexes are equal (Ma= Mb=Mc=M) and θxis the phase angle of phase x(x= a, b, c), respectively. Vxo mn represents the magnitude of each side-band harmonic of the phase voltage vxo(t).Jn(z)is the first kind Bessel function of zand order nand the term λmn has been introduced for convenience. It is possible to calculate each harmonic component of the resulting CMV by simply evaluating (1) with the corresponding components of the phase voltages Vxo mn calculated in (4). Each single harmonic order kis defined as k=mR +n, where Ris ωc/ω0. 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.2022.3190872 © 2022 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 0 10 20 30 40 50 Harmonic order (k) 10-1 100 10-2 10-3 10-4 Spectrum magnitude (pu) m = 1 k = mR +nm = 2 n = 0 n = 2 n = 4 n = 6 n = -2 n = -4 n = -3 n = -6 n = 1n = -1 n = 3 n = -5 n = 5 Fig. 3: Typical harmonic spectrum of the phase voltage considering the definition of variables mand n CMVmn(ϕb, ϕc) = VNo mn (ϕb, ϕc) =Vxo mn"ejδa+ejδb+ejδcej(mωc+nω0)t +e−jδa+e−jδb+e−jδce−j(mωc+nω0)t#(5) where coefficients δxare introduced for convenience as: δa=mϕa+nθa δb=mϕb+nθb δc=mϕc+nθc(6) From (5), it is clear the resulting CMVmn harmonic component is highly dependent on the carrier phase displacements (ϕa,ϕb,ϕc) to be adopted in the modulation stage. Additionally, it is worth to be mentioned that base-band harmonics are eliminated and only the fundamental component and the sidebands frequency are transferred to the resulting CMV voltage. In order to mitigate the negative collateral effects of the resulting CMV on the PV array, it is possible to define the total harmonic distortion (THD) of the CMV with respect to the dc as a figure of merit. In this sense, the CMVT HD can be written as: CMVT HD =2 Vdc v u u t N−1 X m=1 j X n=−j CMVmn(ϕb, ϕc)2(7) In order to introduce the typical notation of a harmonic spectrum, the corresponding harmonic spectrum of the phase voltage is shown in Fig. 3, where the magnitude of each harmonic component follows the notation given in (5). This example has been illustrated considering that R= 21. In this sense, in the first harmonic group (m= 1) (and all odd harmonics groups) there are only side-bands components with even values of n. Similarly, in the second harmonic group m= 2 (and all even harmonic groups) the side-bands components present non-zero values with odd values of n. As an example, the resulting CMV has been evaluated for the operational condition summarized in the Table I for all 0 90 180 270 360 0 90 180 270 360 Ác (º) Áb (º) 70 70 70 70 60 60 60 60 50 50 50 50 50 45 45 45 45 44 44 44 44 43 43 43 41 41 42 42 43 50 Fig. 4: Resulting CMVT HD as function of ϕband ϕcconsidering the operational conditions summarized in Table I in the harmonic range m <= 3 and n <= 6 the possible combinations of the angles ϕband ϕcin the harmonic range m <= 3 and n <= 6. As can be observed in Fig. 4, the CMV presents two minimum values located at [ϕb, ϕc]=[120,240]◦and [240,120]◦marked with red triangles. III. RELATION OF THE POWER CONVERTER OPERATION WITH THE CAPACITOR LIFETIME ESTIMATION Capacitors and power semiconductors are key elements in power converters that are chosen and rated to maintain the cost of the system under an acceptable value. Additionally, from reliability point of view, some works have identified the capacitors among the most important sources of failure in the power conversion systems [14]–[16]. In this sense, the election of the capacitors not only comprises the minimum capacitance, but also affects to the available lifetime of the power system. Among the capacitor technologies available in the market, the aluminum electrolytic capacitors, multi-layers ceramic capacitors as well as the metalized polypropylene are the most common ones to be used in power converters [15]. The election of the capacitor technology depends on the application, rated voltage, operation condition and the required capacitance having an important impact on the cost and reliability. For instance, electrolytic capacitors are the cheapest technology but they count with the lower reliability. On the contrary, film capacitors are considered the more reliable but with a corresponding negative influence in their cost. Among the stressors for these capacitor technologies, the applied voltage V, the current that flows through the capacitor and the device hotspot temperature Thcan be found. The lifetime of the capacitor is usually modeled as (8): TABLE I: Operational conditions of the 2L-VSI Parameter Value Parameter Value Active power (P) (kW) 3Power Factor 0.948 Reactive power (Q) (kVAr) 1Modulation index 0.906 DC-link (Vdc) (V) 700 Nominal current (I) (A) 6.45 Inductance (L) (mH) 15 Switching Frequency (kHz) 10 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.2022.3190872 © 2022 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 Lc=L0 V V0!−n eEa kBT−1 h−T−1 0(8) where L0is the reference lifetime at the voltage V0and temperature T0, respectively. Eais the activation energy and kBis the Boltzmann’s constant. If the electrolytic capacitor is considered, (8) can be re-written and simplified as: Lc=L0 V V0!−n 2 T0−Th 10K(9) where it is possible to observe how an increase of the operation temperature by 10 Kleads to a reduction of the capacitor lifetime by 50%, what illustrates the importance of the hotspot temperature. The voltage dependence shown in (8) and (9) is usually provided in the capacitor datasheet, whereas the hotspot temperature of the device depends on the ambient temperature as well as the operational conditions, that is, the convolution of the capacitor loss Pdwith the thermal impedance Zth. For the sake of simplicity, only the thermal resistance Rth and the steady state operation is considered. Then, the capacitor hotspot temperature can be expressed as: Th=RthPd+Ta= ∆Th+Ta(10) where the term Pddirectly depends on the current that flows through the capacitor and its Equivalent Series Resistance (ESR), in other words, Pdis determined by the well-known Joule effect. In (11), Pdis expressed considering its frequency dependence, as follows Pd= ∞ X k=0 I2 c,kRESR,k (11) where Ic,k is the k−harmonic of the capacitor current and RESR,k is the related ESR for this frequency component. The ESR value depends on the capacitor technology as well as the frequency. The ESR can be expressed as the sum of three different components as follows (12) RESR =RESR,0+RESR,f +R T0−Th E T=R0+R T0−Th E T (12) where RESR,0is constant, RESR,f a term that depends on the frequency component and RTthat exponentially depends on the hotspot temperature [17]. In the following, the sum of RESR,0and RESR,f is defined by R0. In order to evaluate the capacitor lifespan, Fig. 1 presents the generic phase leg involved signals of the three-phase 2L- VSI. The switching state of the power devices is described by the time-variant binary switching function Sx(t). Therefore, it is easy to understand that there is a relationship between the output current delivered by one leg of a three-phase 2L-VSI and its contribution to the current to the dc link capacitor, which is expressed as: 0 90 180 270 360 0 90 180 270 360 Ác (º) Áb (º) 8.47 8.47 8.47 8.47 7.83 7.83 7.83 7.83 7.83 7.83 7.83 7.83 8.47 8.47 8.47 8.47 9.12 9.12 9.12 9.12 9.12 9.12 9.12 9.12 11.72 11.72 11.72 11.72 11.72 12.36 12.36 11.07 11.07 11.07 10.42 10.42 10.42 10.42 10.42 10.42 9.77 9.77 9.77 9.77 9.77 9.77 9.12 Fig. 5: Resulting THD of the dc-link capacitor current (IcT HD ) as function of ϕband ϕcconsidering the operational conditions summarized in Table I in the harmonic range m <= 3 and n <= 6 iCx(t) = Sx(t)ix(t)(13) It is possible to obtain the harmonic content contribution of each phase to the current in the dc capacitor by performing the following convolution in the frequency domain: iCx(ω) = Sx(ω)∗ix(ω)(14) Assuming a sinusoidal output current waveform, each phase current can be described as: ix(t) = Ixcos(ω0t+ Φ) (15) where Ixis the phase current amplitude and Φis the power factor present in the power system. After some algebraic manipulations it is possible to demonstrate that (14) can be rewritten as iCx(ω) = Ix 2hejΦSx(ω−ω0) + e−jΦSx(ω+ω0)i(16) From (16), the harmonic spectrum of the dc capacitor current can be determined by adding up two terms, obtained through a frequency shift ±ω0of the switching function harmonic spectrum. On the other hand, if the generalized PWM strategy shown in Fig. 2 is considered, the carrier phase displacement angles need to be introduced. In this sense, the harmonic content of the switching function for each leg can be expressed as: Sx(t) = 1 2+M 2cos ω0t+θx ∞ X m=1 ∞ X n=−∞ λmn cos m(ωct+ϕx) + n(ω0t+θx)(17) Considering (17) in (16), it is possible to obtain the expression of the dc-link current as 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.2022.3190872 © 2022 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 0 90 180 270 360 0 90 180 270 360 0 90 180 270 360 0 90 180 270 360 2.76 2.76 2.76 2.76 2.76 2.76 2.76 2.76 3.04 3.04 3.04 3.04 3.04 3.04 3.04 3.04 3.04 3.04 3.04 3.04 3.04 3.04 3.04 3.33 3.33 3.33 3.33 3.33 3.33 3.33 3.33 3.33 3.33 3.33 3.33 3.33 3.33 3.61 3.61 3.61 3.61 3.61 3.61 3.61 3.61 3.61 3.61 3.61 3.61 3.61 3.61 3.61 3.61 3.61 3.89 3.89 4.17 4.45 4.45 4.45 4.45 4.45 4.45 4.45 4.45 4.45 4.73 4.73 4.45 4.73 4.73 4.73 4.73 4.73 4.73 4.73 5.02 5.02 5.02 5.02 5.02 5.02 5.30 5.30 5.30 5.30 5.30 5.30 4.73 4.73 4.45 4.45 4.17 4.17 4.17 4.17 4.17 4.17 4.17 4.17 4.17 4.17 4.17 3.89 3.89 3.89 3.89 3.89 3.89 3.89 3.89 3.89 3.89 3.89 3.89 3.89 3.89 Ác (º)Ác (º)Ác (º) Áb (º) ac b Fig. 6: Resulting Ix T HD as function of ϕband ϕcconsidering the operational conditions summarized in Table I in the harmonic range m <= 3 and n <= 6 ic(t) = X x=a,b,c icx(t) = 3 4MI0cos(Φ) + ∞ X m=1 ∞ X n=−∞ A′ mn cos mωct+nω0t +B′ mn sin mωct+nω0t(18) where coefficients A′ mn and B′ mn are A′ mn =σ1 Amn −jBmn 2+σ2 Amn +jBmn 2 B′ mn =σ1 Bmn +jAmn 2+σ2 Bmn −jAmn 2(19) where auxiliary variables µ,σand ∆as well as the coefficients Amn and Bmn are defined as follows: σ1=ejδa+ejδb+ejδc σ2=e−jδa+e−jδb+e−jδc µ= I0cos (m+n)π 2cos(Φ) mπ Amn =µhJn+1mπ 2M−Jn+1mπ 2Mi Bmn =µhJn+1mπ 2M+Jn+1mπ 2Mi (20) From (18), it is possible to notice that the current through the dc-link capacitor contains a dc component, and the carrier harmonic and the side-bands harmonic components. From Amn and Bmn calculations in (20), it is evident that in the dc-link capacitor current there are only side-bands harmonic components featuring an even value of the sum m+n. In order to evaluate the impact over current through the dclink capacitor of the generalized PWM technique shown in Fig. 2, the THD of this current can be used as figure of merit. It can be calculated as IcT HD =2 Idc v u u t N−1 X m=1 j X n=−j Ic,mn(ϕb, ϕc)2(21) As an example, IcT HD has been evaluated for the operation point summarized in Table I considering a harmonic range determined by m <= 3 and n <= 6. In this scenario, the resulting current THD has been evaluated for all possible combinations of the phase displacement angles and the results are shown in Fig. 5. It can be observed that the minimum current THD is achieved with multiple carrier phase displacement solutions (for instance considering angles equal to (ϕb= 94.54◦,ϕc= 94.54◦), among the multiple solution found in the data exploration). Additionally, it is worth to mention that the best carrier phase displacement angles set strongly depends on the converter operational conditions mainly determined by the modulation index (M) as well as the power factor (Φ) as can be observed in (18). Remark. If a classical sinusoidal modulation method is adopted for a three-phase 2L-VSI, that is ϕa=ϕb=ϕc, the generalized analytical expression of coefficients A′ mn and B′ mn shown in (19) can be particularized leading to: A′ mn =Amn1 + 2 cos n2 3π B′ mn =Bmn1 + 2 cos n2 3π (22) IV. IMPACT OF THE GENERALIZED PWM STRATEGY OVER THE 2L-VSI VOLTAGES AND CURRENTS Apart from the impact on the dc-link capacitor current and the resulting CMV, it is well-known that the application of the use of a PWM strategy with three independent triangular carriers presents a negative impact on the 2L-VSI phase voltages and currents [18], [19]. For instance, in [18]–[20], it is demonstrated that the application of a carrier phase displacement angles set equal to ϕa= 0◦,ϕb= 120◦,ϕc= 240◦ 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.2022.3190872 © 2022 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 provokes an increase of the resulting converter phase currents THD. The magnitude of each harmonic component of the resulting vxN (x=a, b, c) in the grid-tied 2L-VSI can be obtained after some mathematical manipulation from (3) and (5) as: VxN mn =|Vxo mnejδx−Vxo mn|(23) On the other hand, it is necessary to obtain the transfer function of the passive network used to connect the power converter to the grid. Assuming a simple series RL network, the magnitude of each harmonic component is: Zmn =|R+jωmnL| ωmn =ω0(mR +n)(24) and therefore, the resulting current across the RL passive network can be calculated as: Ix mn =|VxN mn ejδx−Vx mn| Zmn ≈|VxN mn | Zmn (25) where it is assumed that the grid voltage (Vx mn) does not contain any harmonic component at each ωmn frequency. Finally, in order to define a figure of merit to evaluate the impact of the PWM strategy in the converter performance, the resulting THD of the phase currents can be evaluated as: Ix T HD =2 Ixv u u t N−1 X m=1 j X n=−j Ix mn 2(26) where Ixrepresents the fundamental value of the current in the phase x. In order to explore the best carrier phase displacement angles set to minimize the phase currents THD, Fig. 6 has been generated by an exhaustive algorithm considering all the possible carrier phase displacement combinations and the operational conditions summarized in Table I. Additionally, in Fig. 6 has been highlighted the particular region where the phase currents THD present a value superior to a certain level B(Bis equal to 5% in this example but any specific value taken from actual grid codes can be imposed without any restriction). It can be seen that the optimum carrier phase displacement angles set is [ϕb, ϕc]=[0◦,0◦], that is, the conventional solution of a single-carrier PWM method. This fact highlights that the minimization of the phase currents THD, the minimization of the CMVT HD and the minimization of the IcT HD can not be achieved at the same time leading to the necessity to develop a trade-off decision making method. V. MULTI-OBJECTIVE OPTIMIZATION OF THE CARRIER PHASE-DISPLACEMENT ANGLES SET From the previous discussion, it is clear the necessity to establish a trade-off between the benefits of the reduction of the CMVT HD and IcT HD , keeping the value of all of the phase currents THD below an acceptable level (B) in order to 0 90 180 270 360 0 90 180 270 360 Ác (º) Áb (º) 82.16 78.75 75.35 75.35 75.35 75.35 78.75 71.94 71.94 71.94 71.94 68.52 68.52 68.52 68.52 65.12 65.12 61.71 61.71 61.71 61.71 51.49 58.31 54.90 54.90 54.90 54.90 58.31 58.31 58.31 58.31 58.31 51.49 65.12 65.12 Fig. 7: Cost function ρ(27) as function of ϕband ϕc considering the operational conditions summarized in Table I in the harmonic range defined by m <= 3 and n <= 6 InitEMA() Shareholders stock Stock value Á1a Á1b Á1c ½1 ½2 ½z Á2a Á2c Á2c Áza Ázb Ázc Balanced Market Evolution High-ranked Medium-ranked Low-ranked Traded Market Evolution High-ranked Medium-ranked Low-ranked [Áa Áb Ác ]Temporary solution Shareholders re-ranking (sorting ½) Evaluate Market Shareholder population evolution calculation Evaluate Market Rank Sharedholder Y=0? Y=1 Y=0 YES NO CMVTHD + ¸IcTHD ITHD< B x Fig. 8: Flow diagram of the EMA algorithm to look for the most suitable carrier phase-displacement angles to minimize the cost function ρ meet typical grid codes. In order to face this issue, the problem can be formulated as a multi-objective searching with multiple restrictions that can be written by defining a cost function to be minimized such as ρ=CMVT HD +λIcT H D , fulfilling that Ix T HD ≤B(27) where λis a factor to weigh the importance of each term in the cost function and Bdetermines the acceptable current THD boundary. For the sake of simplicity, in this work λis equal to one giving the same weight to each term and Bhas been set to 5%. Taking into account this cost function ρdefining the multiobjective optimization problem, although an exhaustive search 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.2022.3190872 © 2022 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 Fig. 9: Laboratory prototype method can be used, it is preferable to use an advanced mathematical search algorithm in order to accurately and quickly look for the figure of merit optimum value. In this way, the academia is developing mathematical searching algorithms such as Genetic Algorithm, Particle Swarm Optimization and Exchange Market Algorithm (EMA), just to name a few [21]– [23]. These algorithms present mechanisms to avoid the local minimum stacking as well as it is not required to reset the search when the operational conditions change because the algorithm automatically evolves to the new optimum solution [24], [25]. In this sense, these algorithms are the perfect candidate to be executed in the background task in a multicore control platform (such as the TMS320F28377S microcontroller by Texas Instruments belonging to the off-the- shelf mid-range microcontroller [26]). In this work, the EMA is adopted to solve this particular problem. The technical implementation of EMA can be found in [22]–[25]. EMA is based on an optimization of the stock in a trading market. Different shareholders participate in the trading market always looking for maximizing their stock value. In [24], [25], this economic problem was translated to apply EMA to face a power electronics problem. In this work, each shareholder is a solution set with specific carrier phase-displacement angles. These specific angles are the stock of each shareholder and the value of the stock corresponds to the value of the cost function ρto be minimized. A figure representing the EMA algorithm with the translation to apply the method to face the multi-carrier PWM method can be observed in Fig. 8. It is worth to be mentioned that the performance comparison between different searching algorithms as well as meta-heuristic techniques are out of the scope of this paper. Considering the same operational condition referred in Table I, in Fig. 7 the value of the cost function ρfor all the possible carrier phase displacement values [ϕb, ϕc]is shown considering also the boundary Bimposed in Ix T HD what creates forbidden areas (these zones are drawn using the same color scheme introduced in Fig. 6). VI. EXPERIMENTAL RESULTS In order to check the performance of the proposed generalized PWM method minimizing the cost function ρdefined in (27), the laboratory prototype shown in Fig. 9 has been used. The PV application has been emulated using a PV arrays emulator power supply as DC source. When a specific active power is imposed in the power system, it comes because the PV arrays emulator is configured to operate with temperature and solar radiation corresponding to generate this specific active power value. The reactive power to be injected in the grid is defined by the power system operator in the VOC method giving the possibility to fulfill current grid standards where it is required that PV systems inject reactive power in order to improve the grid stability. The operational conditions imposed in the converter are also those summarized in Table I and the control scheme is based on the conventional stationary dq frame [27], [28]. The EMA is executed in background looking for the minimum value of ρmodifying every iteration the carrier phase displacement angles set to be applied in the generalized PWM method. A flow diagram of the implementation of the EMA algorithm is represented in Fig. 8. The algorithm was implemented in the TMS320F28377S control platform achieving an execution time below 1 millisecond per iteration. The way to implement the proposed method in the control platform in order to reduce the computational burden was very similar to that introduced in [25]. The convergence is very fast achieving steady state values of the carrier phasedisplacement angles in less than 15 iterations. The steady state results are obtained evaluating the performance of the three individual control targets. On one hand, the obtained CMV and its corresponding harmonic spectrum are represented in Fig. 10. On the other hand, the dc-link capacitor current and its corresponding harmonic spectrum are represented in Fig. 11. Finally, the phase currents and their corresponding harmonic spectra are represented in Fig. 12. A detail of the harmonic spectra of the phase currents is presented in Fig. 13, where the harmonic content at low frequency can be observed. The obtained phase currents THD applying the generalized PWM method with three independent carriers is 1.3% that is a reasonable value for any standard. Also, it can be observed that the CMV THD has been reduced from 38.39% up to 19.97%. At the same time, the capacitor current THD has been also reduced from 21.41% up to 15.5%. Considering these experimental results, the capacitor hotspot temperature has been estimated considering different Rth values. These results are summarized in Table II where it can be clearly observed the beneficial impact of applying the proposed adaptive threecarrier PWM method in the capacitor lifespan. It is important to highlight that usage of capacitor in long term provokes that Rth becomes larger what accelerates the capacitor degradation. This fact makes even more beneficial the application of the proposed PWM method. In addition, it is also important to highlight that this improvement can be achieved for free without introducing extra cost in the power system. These facts demonstrate that the proposed generalized PWM method with the application of the most suitable carrier phase displacement angles set leads to present very good performance in terms of limited distortion in both the dc-link capacitor current, the phase currents and the CMV. 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.2022.3190872 © 2022 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 0 10 20 30 40 0 10 20 30 40 0 200 400 600 800 0 10 20 30 0 10 20 30 0 10 20 30 40 Time (ms)Time (ms) Frequency (kHz)Frequency (kHz) Spectrum magnitude (%) Voltage (V) a c b d CMVTHD=37.28% CMVTHD=19.94% Fig. 10: CMV of the 2L-VSI applying the conventional single-carrier PWM strategy a) Waveform b) Corresponding harmonic Spectrum. CMV of the 2L-VSI applying the generalized PWM with three independent carriers with the final carrier phase displacement angles obtained by the EMA c) Waveform d) Corresponding harmonic spectrum 0 302010 40 0 302010 40 -20 -40 0 20 40 0 10 20 30 0 10 20 30 0 10 20 30 Time (ms)Time (ms) Frequency (kHz)Frequency (kHz) Spectrum magnitude (%) Current (A) a c b d IC THD=36.56% IC THD=30.50% Fig. 11: Capacitor current in the 2L-VSI applying the conventional single-carrier PWM strategy a) Waveform b) Corresponding harmonic Spectrum. Capacitor current in the 2L-VSI applying the generalized PWM with three independent carriers with the final carrier phase displacement angles obtained by the EMA c) Waveform d) Corresponding harmonic spectrum VII. CONCLUSIONS In this paper a generalized PWM method where an independent carrier is applied to each converter phase is presented and deeply analyzed. The corresponding carrier phase displacement angles are evaluated taking into account three targets: minimization of the CMV THD, minimization of the phase currents THD, and minimization of the dc-link capacitor THD. These targets are colliding and a trade-off method has been addressed in order to obtain a suitable solution that, improving the converter performance applying the conventional single-carrier PWM method, also meet the minimum current standards. The required mathematical descriptions of the CMV, phase currents and dc-link capacitor current have been addressed, all of them depending on the carrier phase displacement angles. A cost function has been defined in order to look for the most suitable carrier phase displacement angles solution set. The exchange market algorithm has been chosen to look for this solution set achieving good results after few iterations, what permits the possibility to obtain the most suitable solution set in real time by implementing the method in an off-the-shelf microprocessor. The method has been validated by showing 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.2022.3190872 © 2022 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information.