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2016 ICSEE International Conference on the Science of Electrical Engineering Optimal Switching in a Three-Level Inverter: an Analytical Approach Almog Salman, Yair Cohen, Lior Simchon, Tirza Routtenberg, Raul Rabinovici Department of Electrical and Computer Engineering Ben-Gurion University of the Negev Beer-Sheva 84105, Israel Abstract—This paper addresses the problem of optimal pulse width modulation (PWM) voltage switching signals that minimizes the total harmonic distortion (THD) for a three-level inverter. Previous works have focused on the derivation of ad-hoc methods that achieve low voltage THD values by using frequency domain analysis and brute-force search. The use of optimization techniques in power systems, especially in THD minimization, can improve system performance and reduce computational complexity, thus providing a manageable method with claims to optimality. In this paper, an optimization method, the Lagrange multipliers technique, is used to solve the THD minimization problem. For a given modulation index it is shown that the minimization of the voltage THD results in a single-switch solution. Finally, the performance of the proposed optimal singleswitch solution is examined in simulations. The proposed method reduces computational complexity, compared to the brute-force methods, and increases solution accuracy by using analytical THD time domain expressions. Index Terms—Multilevel inverter, pulse width modulation (PWM), total harmonic distortion (THD), harmonics, optimization, Lagrange multipliers. and analyzed for low switching frequencies, i.e. for each quarter of the square wave there are three switches. Then, the THD expression is formulated in the time domain and the problem is described in the context of an optimization framework. Estimation and detection theory have been shown to be powerful tools in smart grid analysis (see e.g. [7], [8], [9], and [10]). Similarly, this work demonstrates the use of modern optimization tools in classical power system problems. The optimization is conducted by using Lagrange multipliers and results in a counter-intuitive solution; in this case, less is more. That is, single switching is the optimal THD minimization solution. Finally, the performance of the proposed optimal single-switch solution is examined in simulations. It is shown that the proposed method reduces the computational complexity, compared to brute-force methods, and increases solution accuracy. I. I NTRODUCTION Multilevel inverters (MLI) play an important role in industrial power applications. In recent decades, MLI applications have grown rapidly and modulation methods have gained prominence in MLI research [1]. The main advantage of MLIs is that their output voltage can be generated with low harmonics. Thus, it is found that the harmonics decrease proportionately to the inverter level. Modulation patterns of inverters are generally characterized by performance measures, such as the commonly used total harmonic distortion (THD) [2] [3], the peak common-mode voltage (CMV) [4], the harmonic distortion factor (HDF) [5], and specific harmonic-amplitude restrictions [6]. MLI power efficiency is affected by switching losses, which are directly related to the switching frequency. High efficiency can be obtained by using a low switching frequency. However, reducing the switching frequency affects the harmonic content and increases THD values. Therefore, both the inverter voltage level and the switching frequency should be selected carefully in order to achieve low THD and high efficiency. Optimal selection requires an accurate THD analysis of inverter pulse width modulation (PWM) output. THD minimization techniques for MLI have been discussed extensively in the literature and are usually based on off-line algorithms and the use of lookup tables. The well known Selective Harmonic Elimination (SHE) method [11] reduces the THD by removing low-order harmonics. The SHE method requires the solution of a set of nonlinear equations. Therefore, SHE solutions are generally obtained by using numerical approaches. The number of the canceled harmonics is a function of the number of switching angles. This technique demands offline calculations that are stored in a lookup table. The synchronous optimal pulse width modulation (SOPWM) technique [12] operates with low switching frequencies and is usually used in motor systems. Similar to the approach outlined in this paper, the SOPWM method is obtained by using optimization methods. However, the SOPWM method approximates the objective function as a finite harmonic sum, which results in a less accurate solution that requires higher computational complexity. Another method is the Criteria Based Modulation (CBM) method [13] that offers a direct search for any given objective function. The CBM method performs an offline search by mapping all possible switching permutations. In each of these methods, a switching set is stored in a lookup table for each modulation index. A. Summary of results In this paper we consider the problem of THD minimization in a three-level inverter. First, the THD problem is described c 978-1-5090-2152-9/16/$31.00 2016 IEEE B. Related works The rest of the paper is organized as follows. In Section 2 2016 ICSEE International Conference on the Science of Electrical Engineering the THD problem is formulated in the time domain, followed by an optimization solution, which proposed in Section 3. Simulations are conducted in the MATLAB environment and discussed in Section 4. Conclusions are presented in Section 5. II. THD F ORMULATION In this section we present a mathematical analysis of an MLI output voltage. The output voltage and the related THD expression are formulated in the time domain as a function of the switching angles. A general MLI output voltage signal is represented by a stepped waveform with N voltage levels and M switches. In the following we discuss a three-level inverter, i.e. N = 3, with three switching angles per quarter wave, i.e. M = 12. Therefore, the MLI output voltage is given by: θ ∈ R1 VDC −VDC θ ∈ R2 , (1) f (θ, α) = 0 otherwise where R1 = [α1 , α2 ] ∪ [α3 , π − α3 ] ∪ [π − α2 , π − α1 ], R2 = [π + α1 , π + α2 ] ∪ [π + α3 , 2π − α3 ] ∪ [2π − α2 , 2π − α3 ], and α = [α1 , α2 , α3 ]T , in which α1 , α2 , α3 are the switching angles. The inverter input DC voltage level is denoted by VDC . It should be noted that according to these definitions the switching angles should satisfy: π 0 ≤ α1 ≤ α2 ≤ α3 ≤ . (2) 2 where the associated Fourier coefficients [15] are given by: P3 4 VDC m=1 (−1)m+1 cos(nαm ) odd n πn (5) Vn = 0 even n for any n ∈ N. The output voltage THD describes the ratio between the total harmonic components and the fundamental desired harmonic, V1 . The THD is defined by [16]: T HD(α) = sP ∞ 2 n=2 Vn 2 V1 = sP ∞ 2 n=1 Vn 2 V1 −1 . In practice, evaluating the THD values from (6) is performed by using a truncated THD expression with a limited number of harmonics. Accurate THD values can be obtained by using the time domain THD expression. In particular, the Parseval identity implies: ∞ X Vn2 + Wn2 n=1 1 = π Z 2π |f (θ, α)|2 dθ . The output voltage, f (θ, α), from (1) is presented in Fig. 1. This signal is 2π periodic, and it can be seen that the signal is an odd function that has a quarter-wave symmetry. This signal can be expanded by Fourier series [14] as follows: f (θ, α) = ∞ X Wn cos(nt) + Vn sin(nt) . (3) n=1 The cosine coefficients, Wn , n = 1, . . . , ∞, are eliminated due to the odd function characteristics and the even sine coefficients, Vn , are eliminated due to quarter-wave symmetry [11]. Therefore, the Fourier series representation of the output voltage signal is given by: f (θ, α) = ∞ X n=1,3,5,... Vn sin(nt) , (4) (7) 0 By substituting (4) into (7), it can be verified that [17] ∞ X Vn2 = n=1,3... 1 2π Z 2π |f (θ, α)|2 dθ 0 (8) 4 2 = VDC (α2 − α1 − α3 + π) . π Substitution of (8) into (6) results in s 4 2 π VDC (α2 − α1 − α3 + π) T HD(α) = −1 . V12 Fig. 1. Typical output voltage for a three-level inverter, as a function of α1 , α2 , α3 . (6) (9) The modulation index describes the ratio between the inverter input DC voltage and the fundamental voltage amplitude, and is defined as m(α) = V1 . VDC (10) By substituting (10) in (9), one obtains the THD analytical expression T HD(α) = s 4(α2 − α1 − α3 ) + 2π −1 . πm2 (α) (11) Usually, the fundamental harmonic amplitude, V1 , is based on the system requirements, i.e. a given value of the modulation index, m(α). By calculating V1 using (5) with n = 1 and substituting into (10), the modulation index result is m(α) = 4 (cos(α1 ) − cos(α2 ) + cos(α3 )) . π (12) 2016 ICSEE International Conference on the Science of Electrical Engineering III. THD O PTIMIZATION for any n = 1, 2, 3. Differentiating (12) and (16) we get: The THD factor from (11) is a nonlinear function of the switching angles, where the switching angles should satisfy the condition in (2) for a given modulation index m∗ . Under this condition, this minimization problem can be formulated into a nonlinear optimization problem, where the objective function is the THD, as follows: ∂g(α) = 2(−1)n , and ∂αn ∂m(α) 4 = (−1)n sin(αn ) . ∂αn π min T HD(α) α 0 ≤ α1 ≤ α2 ≤ α3 ≤ such that m(α) = m∗ π 2 . (13) According to (11), the THD expression, which is the objective function in (13), is a positive and monotonic function. Therefore, the optimization problem from (13) can be replaced by the following equivalent optimization problem: min FT HD (α) = T HD2 (α) + 1 α 0 ≤ α1 ≤ α2 ≤ α3 ≤ π2 , such that m(α) = m∗ (14) where FT HD (α) is the modified objective function and m∗ is the desired modulation index. Constrained optimization problems can be solved by using the Lagrange multipliers method [18]. For the optimization problem from (14) the Lagrangian is defined as −L(α, λ) = FT HD (α) − λ(m(α) − m∗ ) 2g(α) − λ(m(α) − m∗ ) , = πm2 (α) (15) where g(α) = 2(α2 − α1 − α3 ) + π . (16) In order to minimize the Lagrangian, the partial derivatives of L(α, λ) w.r.t. α1 , α2 , α3 , and λ should be equated to zero. That is, (20) By using the constraint m(α) = m∗ and substituting (20) in (19) one obtains 4 3 n+1 sin(αn ) g(α) + λm∗ − m∗ = 0 . (21) 2(−1) π It can be seen that (21) implies that πm∗ sin(αn ) = n = 1, 2, 3. 4g(α) + λm3∗ (22) Since the sine function is an injective function within [0, π2 ], (22) implies that the optimal solution is obtained where all three switching angles are equal, i.e. α1 = α2 = α3 . That is, a single switching angle provides the minimal THD value. Since all switching angles are equal, using α = αn for any n=1,2,3 will provide a single switch expression. Finally, the THD expression is obtained for a single switch from (11): r π 2 − 2πα T HD(α) = −1 . (23) 8 cos2 α Similarly to (23), the modulation index expression is obtained for a single switch from (12): 4 cos α . (24) π The single switch α is given for any fixed modulation index m(α) = 4 (25) α = cos−1 ( m∗ ) . π This section implies that for any given modulation index the switching angles for which the voltage output THD is minimal are all equal and can be directly calculated based on (25). IV. S IMULATION In this section we examine and discuss the performance of the proposed analytical method solution. All simulations were performed by using the MATLAB software with a Lenovo ∂L(α, λ) (17) G50 Intel i5 CPU with 4GB RAM. The SHE and CBM = m(α) − m∗ = 0 ∂λ methods were implemented with three switching angles per and quarter wave, where the SHE solution was obtained using the fmincon.m MATLAB function. The CBM method was ∂L(α, λ) = implemented for low and high resolution of the switching ∂αn angle steps. In Fig. 2, the THD of the different methods is ∂g(α) ∂m(α) (18) presented versus the modulation index. It can be seen that the ∂m(α) 2 ∂αn m(α) − 2g(α) ∂αn −λ = π m3 (α) ∂αn SHE method provides the highest THD values; the behavior of the CBM method can be attributed to the discrete search =0, approach, with its inherent resolution errors. for n = 1, 2, 3. It can be verified that Equation (18) can be Simulation runtime was analyzed for each method and is rewritten as presented in Table 1. In order to achieve lower THD via the SHE or CBM methods, higher resolution is required. How∂m(α) π ∂m(α) 3 ∂g(α) m(α) − 2g(α) −λ m (α) = 0 , (19) ever, this increases problem complexity, resulting in longer ∂α ∂α 2 ∂α n n n 2016 ICSEE International Conference on the Science of Electrical Engineering runtime, which precludes real-time applications or requires larger lookup table sizes. Since the proposed optimal switch solution has a significantly lower runtime it does not require any lookup table. Thus, the proposed approach is suitable for real-time applications. Fig. 2. The THD for the proposed optimal solution, CBM with low/high resolution and SHE method versus modulation index. Method Grid res.[rad] Runtime[sec] No. of values in table SHE 0.01 0.001 5.1 41.7 92 902 3 1 0.2 0.011 0.28 27.4 4960 125580 15288900 CBM proposed solution has been developed for theoretical models and is valid for pure resistor loads. One of the known uses of inverters is the electrical motor drive, which is characterized by an RL circuit. Therefore, future work to minimize THD for RL loads is required and can be done as an extension to the presented method. ACKNOWLEDGMENT TABLE I RUNTIME TABLE FOR THE Fig. 3. THD mapping with fixed modulation index m=0.9 and feasible α1 , α2 .α3 is obtained by (10) SHE AND CBM. In Fig. 3. the following scenario was implemented: the modulation index was predefined to a specific value, α1 and α2 were mapped, and α3 was evaluated from (12). Valid switching sets were selected from (2). The obtained THD value is presented versus the mapped switching angles, for each value of α1 and α2 , and presented in Fig. 3. It can be seen that the minimal THD values are obtained for α1 = α2 , which is equivalent to the proposed solution of a single switch α3 . This simulation validates the proposed analytical solution presented in this paper, and shows that the optimal solution can be calculated analytically for any given modulation index. V. C ONCLUSION In this paper, we discuss the THD problem for MLI and develop an analytical approach based on optimization methods. This method has proven more efficient for the THD problem since it provides a global minimum THD value and since switching angles are calculated online and no lookup tables are used. Optimized analytical THD has two advantages: first, it is a simple analytical continuous expression that has realtime applications; second, a single switching signal provides the lowest switching losses in a simple hardware design. The This research was partially supported by THE ISRAEL SCIENCE FOUNDATION (grant No. 1173/16) R EFERENCES [1] J. Rodriguez, J.-S. Lai, and F. Z. Peng, “Multilevel inverters: a survey of topologies, controls, and applications,” IEEE Trans. Industrial Electronics, vol. 49, no. 4, pp. 724–738, 2002. [2] H. S. Patel and R. G. Hoft, “Generalized techniques of harmonic elimination and voltage control in thyristor inverters: Part I–harmonic elimination,” IEEE Trans. Industry Applications, no. 3, pp. 310–317, 1973. [3] A. K. Rathore, J. Holtz, and T. Boller, “Synchronous optimal pulsewidth modulation for low-switching-frequency control of medium-voltage multilevel inverters,” IEEE Trans. Industrial Electronics, vol. 57, no. 7, pp. 2374–2381, 2010. [4] A. M. Hava and E. Un, “A high-performance PWM algorithm for common-mode voltage reduction in three-phase voltage source inverters,” IEEE Trans. Power Electronics, vol. 26, no. 7, pp. 1998–2008, 2011. [5] M. S. Dahidah, G. Konstantinou, N. 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