Recursive Selective Harmonic Elimination for Multilevel Inverters: Mathematical Formulation and Experimental Validation

A recursive method that eliminates <inline-formula> <tex-math notation="LaTeX">$n+1$ </tex-math></inline-formula> harmonics and their respective multiples from the output voltage of a cascaded H-bridge (CHB) multilevel inverters with <inline-formula> <tex-math notation="LaTeX">$s=2^{n}$ </tex-math></inline-formula> dc sources <inline-formula> <tex-math notation="LaTeX">$(n=1,2,3,\ldots)$ </tex-math></inline-formula> is proposed. It solves <inline-formula> <tex-math notation="LaTeX">$2 \times 2$ </tex-math></inline-formula> linear systems with no singular matrices and always gives an exact solution with very low computational effort. Simulated results in three-phase five-, nine-, 17-, and 33-level CHB inverters, and experimental results in a five-level inverter demonstrate the validity of the method.


I. INTRODUCTION
D UE to their capability to produce approximated sinusoidal output voltages, multilevel inverters are gaining applications in many fields including renewable energy systems, electric traction, industry, and smart grids [1], [2], [3]. The shape of their waveforms gives significant improvements over conventional two-level converters, in terms of lower total harmonics distortion (THD), higher efficiency, increased voltage/power-handling capability, and reduced component stress, hence higher reliability. The main responsible for such features is the adopted modulation technique, which is highly influential both in terms of performance and implementation complexity. Basically, a modulation technique consists of the determination of a set of switching angles, which, then, can be regarded as the unknowns of the modulation problem. Usually, modulation algorithms are classified into three distinct categories: selective harmonic elimination (SHE), selective harmonic mitigation (SHM), and pulsewidth modulation (PWM) algorithms. In general, the first two approaches operate at the fundamental frequency, the latter at high frequency, but, usually, at values significantly lower than in conventional two-level PWM inverters [4], [5], [6], [7], [8], [9], [10]. Since the lower the frequency, the higher the overall efficiency, when possible, that is, in low dynamic systems, it is desirable to adopt SHE or SHM, while multilevel PWM is usually adopted in those applications requiring high bandwidth control. Among the numerous papers on the topic, Sajadi et al. [11] proposed an SHE technique for high-power asymmetrical cascaded H-bridge (CHB) inverters employed for static VAR compensation (STATCOM) application, which is capable to eliminate low-order harmonics, to reduce converter losses, and to regulate the voltage of each dc-link capacitor. To further improve the quality of the output voltages, Buccella et al. [10] proposed to vary the amplitudes of the dc sources, introducing the pulse amplitude width modulation (PAWM), which increases the number of deleted harmonics at This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ the fundamental switching frequency. Haw et al. [12] proposed a low-switching frequency multilevel SHE-PWM technique for CHB inverters used in transformerless STATCOM systems, which provides constant switching angles and linear pattern dc voltage levels over the full range of the modulation index by employing dc/dc buck converters that operate at a relatively low switching frequency of 2 kHz. Padmanaban et al. [13] proposed an SHE technique for PV systems equipped with a CHB multilevel inverter and boost dc/dc converters, based on an integrated artificial neural network (ANN) and a Newton-Raphson (NR) numerical approach. As discussed in the review paper [14], SHE and SHM modulation techniques are widely used in large electrical drives for traction applications. For instance, Ni et al. [15] proposed a faulttolerant seven-level CHB motor drive with a nonsymmetrical SHE algorithm that provides a better voltage profile and higher output line voltage amplitude with respect to a conventional frequency phase-shifted compensation approach (FPSC). Steczek et al. [16] proposed to combine SHE and SHM-PWM for railway applications. Schettino et al. [17] used an innovative SHM algorithm in an interior permanent magnet synchronous (IPMSM) drive, obtaining better current THD and drive efficiency than with a conventional multicarrier PWM technique.
This article deals with SHE at a fundamental frequency. In this framework, it is necessary to identify and implement a method capable to find the solutions of the SHE equations or of a nonlinear equations system, corresponding to converter switching angles. In this regard, numerous techniques have been proposed, including iterative approaches [18], optimization methods [19], [20], and mathematical solutions [21], [22]. The first topology presents a set of angles that depends on the initial values, which initially are unknowns and the solution could be divergent. The second topology includes genetic algorithms (GAs), bee algorithms (BAs), and in general artificial intelligence-based algorithms (AGAs) that are not significantly affected by initial values but are complex to implement and present high computational costs [23], [24], [25]. The third typology solves the transcendental equations by algebraic algorithms that are generally computationally efficient. In general, the main drawback is related to the difficulty of solving in real-time the transcendental equations but accurate harmonics cancellation is guaranteed [26], [27]. For instance, Yang et al. [28] proposed an offline computing stage in which the nonlinear equations are linearized by using the theories of symmetric polynomials and Gröbner, then, the linear equations are solved online and their solutions are used to construct a polynomial whose real roots are the switching angles. For CHB five-level inverters, SHE analytical solutions can be obtained in real-time as described in [7] and [8]. Such a method can be extended to a higher number of levels and ensures the elimination of n harmonics.
This work aims to propose a new SHE method for fundamental switching frequency for CHB multilevel inverters, which allows the elimination of more harmonics with respect to the traditional SHE methods. In particular, it allows the elimination of n + 1 harmonics and their multiples from the output voltage waveform of an l-level CHB inverter with s equal dc sources V depending on modulation index m. The number of voltage levels has to be l = 2s + 1, with s = 2 n and n = 1, 2, 3, . . . The innovation of this procedure is the recursive application of the Prosthaphaeresis formulas that allow obtaining the s switching angles by solving s − 1 2 × 2 linear system. The modulation index is directly correlated with the amplitude of the dc sources through a proportional coefficient that depends on the eliminated harmonics. In this way, the converter performance in terms of voltage distortion and converter efficiency does not depend on the modulation index.
The main advantages of the proposed procedure are summarized in the following.
1) In inverters with a low number of levels, it returns a higher number of deleted harmonics and a lower THD than the classical SHE technique described in [8].
2) It always returns an analytically exact solution. It is worth to notice that, by using the method proposed in [8], it is not possible to find a solution for all modulation index values and in some cases there are multiple solutions. 3) Being analytical, computational demand is very low, therefore, it could be easily implemented in real-time at low cost [29]. 4) The modulation index can be changed in a continuous range from zero to its maximum value. 5) Switching angles and THD do not depend on the modulation index m. 6) The modulation index linearly depends on the dc voltage amplitudes. This article is structured as follows. Section II presents the mathematical model of a three-phase l-level CHB inverter. Section III describes the proposed method for the elimination of n + 1 harmonics. Section IV presents the implementation. Section V discusses the computational complexity of the proposed SHE approach. Section VI presents the simulation results obtained with several distinct CHB inverter configurations. Section VII presents the experimental analysis that validates the effectiveness of the proposed method. Section VIII provides a comparison between the proposed SHE approach and other recent methods present in the literature. Finally, Section IX summarizes the key features of the proposed SHE method and reports the study conclusions. Fig. 1 shows a three-phase, l-level, CHB inverter. The following constraint is imposed on the number of levels: l = 2s + 1, where s = 2 n is the number of dc sources for each phase, with n = 1, 2, 3, . . . being an integer number. Due to the previous assumption, the proposed procedure can be only used with five-, nine-, 17-, 33-, 65-, or more-level inverters.

II. MATHEMATICAL MODEL
The H-bridges are fed by equal dc sources, the amplitudes of which depend on the modulation index m, that is, whose values in per unit (p.u.) are where V dc is the rated voltage, C is a positive coefficient, and m is equal to with H 1 being the amplitude of the fundamental harmonic in p.u. Considering the phase A, the Fourier series of the line- where with α i being the switching angles with and H k is the amplitude of the k-order harmonic, expressed as From (2), it follows that

III. PROPOSED METHOD
With the previous assumptions, the following mathematical system can be defined to eliminate the n + 1 harmonics of order r 1 , r 2 , . . . , r n+1 from the phase voltage of the three-phase l-level CHB inverter In order to calculate the unknowns α i i = 1, 2, . . . , s and to solve (9), a recursive procedure can be implemented as described in the following steps.
Steps from 2 to n: By imposing H (q) r q = 0, q = 2, . . . , n, by applying again Prosthaphaeresis formulas and by introducing the new variables α . . , (s/2 q ), the following nonsingular systems are obtained: and where Also in these cases, H . . , that is, for all odd multiples of r q .

IV. IMPLEMENTATION
The proposed recursive method consists of the algorithm shown in the flowchart of Fig. 2. It starts from the solution of (18) and stops with the solutions of the s/2th systems in (12).

V. COMPUTATIONAL COMPLEXITY
In the flowchart in Fig. 2, it is possible to observe that a 2 × 2 linear system has been solved before entering in the double cycle in p and i . Inside the inner cycle, in the variable i , 2 p linear systems with p in the range [1, n − 1] are solved, therefore, the number of solved systems is The computational complexity T a of the proposed algorithm is (25) where T sc is a matrix-vector scalar product with a 2 × 2 matrix, that is, T sc = 2T addition + 4T multiplication where T addition and T multiplication are time factors for addition and multiplication operations, respectively.

A. Application of the Method to a Three-Phase Five-Level Inverter
Using a five-level CHB inverter, the proposed method can eliminate two harmonics. This case is particularly simple and its implementation requires very limited effort. In fact, the method is not recursive and the switching angles are computed as follows: therefore T a = 2T addition + 2T multiplication .

B. Comparison With the SHE Algorithm [8]
In [8], the analytical solution, that is, the values of the switching angles, has been found for a five-level CHB inverter, operating at the fundamental frequency, considering the equal and fixed source voltages that do not depend on the modulation index. For a fixed modulation index m, the computational complexity of the modulation algorithm in [8] is T a [8] = (T cos + T arccos + 3T multiplication + 2T division ) (27) where T cos and T arccos are time factors of cos function and arccos functions, respectively. Therefore, being s = 2, T a = T sc < T a [8] .
Other methods described in the literature are iterative, therefore, they present poorly predictable and, in general, high computational complexity [4], [30].

VI. SIMULATED RESULTS
In order to validate the effectiveness of the proposed approach, numerical analysis has been carried out using MATLAB. 1 Figs. 3 and 4 show the phase voltage harmonic analysis and the corresponding THD for a three-phase, nine-, 17-, and 33-level CHB inverter, which can be considered cases of practical interest.
The impact of missing harmonics on the THD can be various, therefore, it is interesting to consider and compare several combinations of eliminated harmonics. Since the prototype available in the Sustainable Development and Energy Saving Laboratory (SDESLAB) of the University of Palermo is a three-phase five-level CHB inverter, this detailed analysis has been restricted to this CHB topology for consequent experimental validation. Five cases have been 1 Registered trademark. considered as reported in Table I. Control angles have been evaluated for each case study, and the corresponding modulation index range, expressed in (28) by considering (3) and (5), is reported below It must be noted that the expression (28) describes the relation of m interval with the constraint regarding the number of the dc sources and control angles, and consequently with the n + 1 eliminated harmonics. Therefore, by fixing n, the value of the modulation index depends only on the dc voltage amplitude. Fig. 5 shows the THD% trend versus modulation index [see Fig. 5(a)] and the low-order harmonics spectra comparison [see Fig. 5(b)] for each case study. In Case 1, the lowest values of THD% have been obtained by eliminating both the 5th and the 11th harmonic, moreover, it can be observed that the THD% values do not depend on the modulation index. In Case 2, by eliminating the fifth and 11th harmonics (orange bar graph), the predominant harmonics are seventh and 19th with an amplitude of about 4%, as shown in Fig. 5(b), where low-order harmonics spectra are compared. Thus, this result explains the lower values of the THD% in this case.

VII. EXPERIMENTAL RESULTS
A test rig, shown in Fig. 6, has been assembled at SDESLab. It is composed of the following.
1) A three-phase five-level CHB MOSFET-based inverter obtained by assembling six distinct H-bridges controlled by a control board employing an Intel-Altera Cyclone III FPGA, programed in VHDL with 32-bit arithmetics. The system is produced by DigiPower and the main technical data are reported in Table II. The power rating of each H-Bridge module is 5 kW, obtaining a 30-kW power rating for a three-phase five-level CHB inverter prototype. 2) Six programmable dc power supplies RSP-2400 whose main technical data are reported in Table III [ Fig. 7 shows a schematic representation of the implemented system, including the measurement circuit. The experimental validation was carried out in two steps and the experimental results were compared with those obtained with the SHE algorithm [8]. In the first analysis, load tests with an RL load (R = 20 and L = 10 mH) were carried out to validate the proposed approach, and the line voltage harmonic content of the proposed algorithm was reported by using the THD% as a comparative parameter at no-load working conditions. After that, the converter efficiency was measured in the cases summarized in Table I and compared with those measured implementing the SHE algorithm discussed in [8].

A. Validation and Converter THD Analysis
In order to demonstrate the effectiveness of the proposed approach, Fig. 8 shows the line voltage (yellow curve), phase current (cyan curve), and corresponding voltage harmonic distribution in the frequency domain (red bar graph) for each analyzed case at dc input voltages equal to 40 V. The voltage, current, and time scales were set to 50 V/div, 2 A/div, and 5 ms/div, respectively; the harmonic spectra in the frequency domain present a voltage scale equal to 20 V/div and a frequency scale equal to 200 Hz/div. As shown in Fig. 8, it should be noted that the selected harmonics were eliminated in each case demonstrating the effectiveness of the proposed algorithm. In order to evaluate dynamic performances during the elimination of pairs of harmonics described in Table I, several tests were carried out for dc voltage steps, corresponding to the modulation index step inside the interval shown in Table I. In this analysis, it is necessary to highlight that the switching angles are constants within all modulation index ranges, resulting in a negligible impact on CHB inverter dynamic performances. Therefore, the CHB inverter transient behavior depends only on the time variation of dc voltage power supplies. In order to show the benefit described impact on CHB inverter dynamic performances, the high-side gate signals of each H-bridge module of the same phase, input dc voltages of the same phase, load phase voltage, and phase current were acquired and analyzed. By way of example, Fig. 9 shows high-side gate signals of H-bridge modules (white, purple, orange, and red curves), the load phase voltage (cyan curve), phase current (green curve), and relative input dc voltages (yellow and violet curves) of the CHB inverter when the dc voltage input values change from 20 to 40 V. The corresponding modulation index values were calculated by (28). In detail, Fig. 9 Fig. 10(a) shows the comparison between experimental and simulation results of THD% versus modulation index for each case analyzed at no load conditions. Moreover, the experimental values of THD% were evaluated for several values of the fundamental frequency. The experimental data confirm the simulation results. In fact, the lowest THD% values were detected in the second case, where the fifth and 11th harmonics were eliminated. Moreover, as expected, no significant THD% variations were detected as the fundamental frequency varies.  The proposed approach is suitable in several applications where different fundamental working frequencies are required. In order to demonstrate the improved performance of the proposed algorithm with respect to other traditional SHE algorithms, the experimental THD% values were compared with those obtained by using the SHE algorithm discussed in [8]. In detail, Fig. 10(b) shows that the proposed approach reduces THD% in almost the whole modulation range considered in Table I, with respect to the SHE algorithm proposed in [8] that individually eliminates the fifth, seventh, 11th, and 13th harmonic, respectively. It is worth to notice that comparable results were detected only when the modulation index was around 0.9. Therefore, it is possible to assert that lower values of the THD% were obtained with the proposed algorithm in all cases as shown in Fig. 10(b).

B. Converter Efficiency Analysis
Efficiency is of paramount importance. This section aims to characterize the proposed algorithm in terms of converter  efficiency for several values of the modulation index and the fundamental frequency. Moreover, it discusses the input power distribution provided by each dc source, which supplies each H-bridge module. The same working points, analyzed in the no-load tests, were considered for load tests by connecting an electric load with R = 20 and L = 10 mH. Fig. 11 shows the experimental converter efficiency versus modulation index evaluated for different values of the fundamental frequency from 100 to 400 Hz with 100-Hz steps.
By varying the fundamental frequency, different load displacement power factor (DPF) values-0.95, 0.85, 0.73, and 0.62-were obtained. The efficiency increases as the fundamental frequency increase. This phenomenon can be attributed to the RL load filter effect on the current harmonics. Fig. 12(a) shows converter efficiency versus the modulation  Table I, and obtained with traditional SHE proposed in [8] eliminating one by one the fifth, the seven7th, the 11th, and the 13th harmonics. Instead, Fig. 12(b) shows the same comparison of converter efficiency as in Fig. 12(a) but at a fundamental frequency equal to 400 Hz (DPF = 0.62), typical of avionics. It is worth to observe that, in all considered cases, converter efficiency is higher than with method [8], in particular, at lower values of the modulation index, which means better control characteristics, especially for the 13th harmonic case and for all considered modulation index values.
Another interesting consideration is the dc power distribution between the H-Bridge modules P HBi as a function of the modulation index. Fig. 13 shows P HBi between two H-Bridge modules of the same phase, in percent with respect to the total input power P dc j ( j = A, B, C) of the same phase for the proposed approach [see Fig. 13(a)] and for SHE proposed in [8] [see Fig. 13(b)]: the proposed algorithm returns a uniform dc power distribution, which is not the case of [8]. In summary, the analysis carried out and results obtained validate the effectiveness of the proposed SHE approach and highlight better performances with respect to the SHE approach proposed in [8].

VIII. COMPARISON WITH OTHER SHE ALGORITHMS
In order to highlight the main benefits of the proposed SHE approach, a comparative analysis with other recent SHE approaches described in [13], [20], [21], and [29] has been carried out by considering as evaluation parameters the number and the order of eliminated harmonics, the implementation complexity, the capability to operate with continuity in a wide modulation index range, the computational cost, and the solution employed. The results of such analysis are summarized in Table IV. Ahmed et al. [29] proposed a real-time solution for the determination of switching angles by using a proportional-integral (PI) controller which is based on the definition of an initial guess. The determination of the switching angles, which are suitable only for fundamental low-frequency applications, guarantees low-order harmonics elimination within the whole modulation index range and the elimination of s − 1 harmonics. Nevertheless, this approach requires a significant amount of computational time and it does not consider high fundamental frequency analyses. Memon et al. [20] proposed an SHE approach based on hybrid asynchronous particle swarm optimization (APSO) NR algorithm. According to the authors, it provides efficient and accurate identification of the switching angles and fewer iterations than other genetic, Bee and PSO algorithms. However, although it requires just few iterations, its implementation is complex, the number of harmonics eliminated is equal to s − 1, and it does not guarantee results for the whole modulation index range. Padmanaban et al. [13] proposed a hybrid ANN-NR based on an SHE procedure and suitable for CHB inverters for PV applications. Such an approach employs an ANN for the offline estimation of the switching angles that are used as initial guesses for the NR method. Since it requires just few iterations for optimal identification of the switching angles, it ensures quick convergence, however, it eliminates s −1 low-order harmonics but only within a modulation index range [0.68, 0.72]. Although the authors estimate its power losses and efficiency, no comparative analysis with other methods is reported. Finally, Ahmed et al. [21] proposed an interesting general mathematical solution for SHE purposes valid for both symmetrical and asymmetrical CHB inverter configurations. The approach eliminates the targeted harmonics in a wide range of modulation indices and, in asymmetrical CHB configurations, guarantees the cancellation of a larger number of harmonics, but at the cost of higher implementation complexity. Its computation cost increases with the number of eliminated harmonics, and its THD is a function of the modulation index value. For symmetrical multilevel inverter configurations, eliminated harmonics are equal to n and multiples. It is worth noticing that all previously considered papers do not include either a detailed efficiency analysis or the dc power distribution among the H-Bridge modules as well.
To compare THD, the methods proposed in [32] and [10] were implemented and analyzed. The first method, the so-called middle-level SHE pulse amplitude modulation (ML-SHE-PAM), does not delete the harmonics having order 2(l − 1)z ± 1, the second one, the SHE PAWM (SHE-PAWM), does not delete the harmonics having order 2lz ± 1 with z = 1, 2, . . . Fig. 14 shows a comparison among the phase voltage THD% obtained with the proposed approach and ML-SHE-PAM and SHE-PAWM approaches considering up to 301th harmonic. The proposed method offers better performances with CHB inverters with few levels, whereas its performances are comparable in the case of 33-level CHB inverters. Hence, the proposed method outperforms the other methods in cases of more practical interest.

IX. CONCLUSION
In this article, a new SHE algorithm based on a recursive application of the Prosthaphaeresis formulas has been proposed for the elimination of n +1 harmonics and their multiples from the output voltage waveform of an l-level CHB inverter with s equal dc sources V (m). The proposed approach has been numerically evaluated for different CHB inverter configurations and experimentally validated with a three-phase five-level CHB inverter. Several tests with or without load and considering several case studies confirm that the proposed approach is very effective. Comparisons with the modulation described in [8] highlight much improved performance and lower computational effort. Overall efficiency improves increasing frequency, which makes it suitable for variable speed electrical drives. Moreover, a comparison with some recent SHE approaches described in the literature is reported to highlight the main benefits of the proposed approach. The algorithm can be useful in multilevel photovoltaic inverters, too, taking benefit from dc-dc stages used in maximum power point tracking. After graduation, she attended the postgraduate Inter-University School, Perugia, Italy, and the CNR Computational Mathematics School, Naples, Italy. Since 1994, she has been a Researcher in numerical analysis with the University of L'Aquila. Her research activities deal with new spline operators for the approximation of functions, for numerical evaluation of Cauchy principal value integrals, and for the numerical solution of integro-differential equations, and since some years with analytical and numerical methods for modulation algorithms for multilevel converters.
Prof. Cimoroni has been a reviewer for several international conferences and IEEE TRANSACTIONS. She was the Publication Chair for the IEEE IECON 2016 and the Publicity Chair for the 5th International Symposium on Environment Friendly Energies and Application 2018 in Rome.