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RESEARCH PAPER International Journal of Recent Trends in Engineering, Vol 1, No. 3, May 2009

Harmonic Reduction Technique for a Cascade Multilevel Inverter Jagdish Kumar1, Biswarup Das2, and Pramod Agarwal2 1

Indian Institute of Technology Roorkee, Roorkee, India Email: [email protected] 2 Indian Institute of Technology Roorkee, Roorkee, India Email: {biswafee, pramgfee}@iitr.ernet.in

Abstract—In this paper, an optimization technique is proposed to compute switching angles at fundamental frequency switching scheme by solving non linear transcendental equations (known as selective harmonic elimination equations), thereby eliminating certain predominating lower order harmonics, and simultaneously, control over magnitude of output voltage of a multilevel inverter is achieved. As these equations are nonlinear transcendental in nature, there may exist simple, multiple or even no solution for a particular value of a modulation index. The proposed scheme is implemented in such a way that all possible solutions are obtained without knowing proper initial guess at the solutions. Moreover, this technique is suitable for higher level of multilevel inverters where other existing methods fail to compute the switching angles due to more computational burden. For the values of modulation indices where multiple solutions exist, the solutions which produce least THD in the output voltage is chosen. A significant decrease in THD is obtained by considering multiple solution sets instead of taking a single set of solution. The computational results are shown graphically for better understanding and to prove the effectiveness of the method. An experimental 11-level cascade multilevel inverter is employed to validate the computational results. Index Terms—Cascade multilevel inverter, modulation index, objective function, total harmonic distortion (THD), sequential quadratic programming (SQP).

I. INTRODUCTION Multilevel inverters are more advanced and latest type of power electronic converters that synthesize a desired output voltage from several levels of dc voltages as inputs. By taking sufficient number of dc sources, a nearly sinusoidal voltage waveform can be synthesized. In comparison with the hard-switched two-level pulse width modulation inverters, multilevel inverters offer several advantages including their capabilities to operate at high voltage with lower voltage stress per switching, high efficiency and low electromagnetic interferences [1], [3] etc. To synthesize multilevel output ac voltage using different levels of dc inputs, semiconductor devices must be switched on and off in such a way that desired fundamental is obtained with minimum harmonic 1 Corresponding author: Tel. No. +91-9410747840, Email Addresses: [email protected], [email protected]

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distortion. There are different approaches for the selection of switching techniques for the multilevel inverters [4]-[9], one of the important techniques is selective harmonic elimination (SHE) method. In SHE technique, certain predominating lower order harmonics are eliminated whereas higher order harmonics are filtered using suitable filter. Switching angles are computed by solving the SHE equations, but it is difficult to solve SHE equations because of their nonlinear characteristics. Due to nonlinear nature, solution of these equations may be simple, multiple or even no solution for a particular value of modulation index (m). A big task is how to get all possible solutions when they exist using simple and less computationally complex method. Once these solutions are obtained, the solutions having least THD are selected for switching purpose. In [4], [5], iterative numerical techniques such as Newton- Raphson method have been implemented to solve the SHE equations producing only one solution set, and even for this a proper initial guess and starting value of m for which solutions exist are required. Some techniques as discussed in [6], [7], here SHE equations are first converted into polynomial equations, and then the resulting polynomial equations are solved using theory of resultants of polynomials and the theory of symmetrical polynomials, producing all possible solutions. A difficulty with these approaches is that for higher levels of multilevel inverter the order of polynomials becomes very high, thereby making the computations of solutions of these polynomial equations very complex. Optimization techniques based on Genetic Algorithm (GA) and Particle Swarm Optimizations (PSO) have been discussed in [8], [9] for computing switching angles only for 7-level multilevel inverters. The implementation of these approaches requires proper selection of certain parameters such as population size, mutation rate, initial weight etc. It becomes difficult to select these parameters for higher level multilevel inverters. To circumvent above mentioned problems, a simple optimization technique based on sequential quadratic programming (SQP) is proposed in this paper to solve SHE equations which produces all possible solutions. The proposed technique is implemented in such a way that all possible solutions for any number of Hbridges connected in series are computed by using any arbitrary initial guess with negligible computational effort. A complete analysis for an 11-level inverter using 181

RESEARCH PAPER International Journal of Recent Trends in Engineering, Vol 1, No. 3, May 2009 five H-bridges per phase in series is presented, and it is shown that for a range of m, switching angles can be computed to produce the desired fundamental voltage along with elimination of 5th, 7th, 11th, and 13th order harmonic components. The computational results are validated through experiments. II. CASCADE MULTILEVEL INVERTER Cascade Multilevel Inverter (CMLI) is one of the most important topology in the family of multilevel inverters. It requires least number of components with compare to diode-clamped and flying capacitors type multilevel inverters. It has modular structure with simple switching strategy and occupies less space [1] - [3]. The CMLI consists of a number of H-bridge inverter units with separate dc source for each unit and is connected in cascade or series as shown in Fig. 1. Each H-bridge can produce three different voltage levels: +Vdc, 0, and –Vdc by connecting the dc source to ac output side by different combinations of four switches S1, S2, S3, and S4. The ac output of each H-bridge is connected in series such that the synthesized output voltage waveform is the sum of all of the individual H-bridge outputs. S1 va

S2 + Vdc _

v a1 S3

S4

S1

S2

v a2

+ Vdc _ S3

S4

S1

S2 + Vdc _

v a3 S3

S4

cascade and using proper modulation scheme, a nearly sinusoidal output voltage waveform can be synthesized. The number of levels in the output phase voltage is 2s+1, where s is the number of H-bridges used per phase. Fig. 2 shows an 11-level output phase voltage waveform using five H-bridges. The magnitude of the ac output phase voltage is given by van = va1+va2+va3+va4+va5 [2]. In general, when s number of H-bridges per phase is connected in cascade, the Fourier series expansion of the staircase output voltage waveform is given by (1).

v an ( wt ) =

∞

4Vdc (cos (kα 1 ) + ... k =1, 3, 5 kπ

∑

+ cos (kα s )) sin (kwt )

III. PROBLEM FORMULATION AND SELECTIVE HARMONIC ELIMINATION EQUATIONS For a given fundamental peak voltage V1, it is required to determine the s switching angles such that the selection of one angle is used to determine V1 and remaining (s-1) angles are used to eliminate the same number of harmonics (generally lower order harmonics), and also all switching angles should be in the range 0 ≤ α1 < α2 < … < α5 ≤ π/2. In three-phase power system, triplen harmonics are canceled out automatically in line-to-line voltages as a result only non-triplen odd harmonics are present in line-to-line voltages [3], [4]. For an 11-level cascade inverter, there are five Hbridges per phase i.e. s = 5 or five degrees of freedom are available; one degree of freedom is used to control the magnitude of the fundamental output voltage and the remaining four degrees of freedom are generally used to eliminate 5th, 7th, 11th, and 13th order harmonic components as they dominate the total harmonic distortion [5]. The modulation index, m is defined as the ratio of the fundamental output voltage to the maximum obtainable voltage (maximum voltage is obtained when all switching angles are zero). From (1), the relation between m and switching angles is given as:

cos(α 1 ) + cos(α 2 ) + ... + cos(α 5 ) = 5m S1

(1)

(2)

S2

v a4

+ Vdc _ S3

S4

S1

S2

v a5

+ Vdc _

n S3

S4

Figure 2. Output voltage waveform of an 11-level CMLI.

Figure 1. Configuration of a single-phase 11-level CMLI.

By connecting the sufficient number of H-bridges in


182

Similarly from (1), expressions for 5th, 7th, 11th, and th 13 harmonic components (scaled values) are given as:


optimization toolbox [12].

cos(5α1 ) + cos(5α 2 ) + ... + cos(5α 5 ) = h5

V. COMPUTATIONAL RESULTS

cos(7α 1 ) + cos(7α 2 ) + ... + cos(7α 5) = h7 cos(11α1 ) + cos(11α 2 ) + ... + cos(11α 5 ) = h11

By implementing the proposed method, all possible solution sets for an 11-level CMLI were computed and a complete analysis is also presented. Starting with any random initial guess all solution sets were computed by incrementing m in small steps from 0 to 1. The nature of the results obtained is shown (only in the range of m where solutions occur) in Fig. 3. By using preliminary computed results from Fig. 3, complete solution sets were computed as shown in Fig. 4. It can be seen from the Fig. 4 that the solutions do not exist at lower and upper ends of the modulation indices and also for m = [0.3800 0.4400], [0.7300 0.7310], and [0.7330 0.7470]. Multiple solution sets exist for m = [0.5050 0.5800], [0.6120 0.7000]. Even some solutions existing in very narrow range of m = [0.3760 0.3790], [0.5470 0.5490], [0.7320] were also obtained by implementing the proposed method, hence this demonstrate the capability of proposed method in computing all possible solution sets.

cos(13α1 ) + cos(13α 2 ) + ... + cos(13α 5 ) = h13 (3) Equations (2) and (3) in combined form are known as SHE equations in terms of five unknowns α1, α2, α3, α4, and α5; provided that h5, h7, h11, h13 are identically zero. For the given value of m (from 0 to1), it is required to get complete and all possible solutions for the switching angle (0 to π/2) when the solutions exist, with minimum computational burden and complexity. Following objective function is formulated to solve SHE equations using optimization technique:

Φ (α1 ,α 2 , α 3 ,α 4 , α 5 ) = h5 + h7 + h11 + h13 (4) 2

2

2

2

Now, (4) is to be minimized subject to equality constraints given by (2) and 0 ≤ α1 < α2 < … < α5 ≤ π/2. Feasible solutions exist only where objective function (4) is identically zero. IV. PROPOSED OPTIMIZATION TECHNIQUE To solve the SHE equations using optimization technique, a proper initial guess at the solution is required. In general it is difficult to make initial guess i.e. what value one should start with. In the proposed technique no such specific initial guess is required i.e. one can start with any random initial guess and can obtain required information about the nature of the solutions. Once initial knowledge about the solutions is known, different solution sets (in case of multiple solutions) can be obtained by using switching angles thus obtained as initial guess. The algorithm for the optimization technique is as follows: 1) Assume any random initial guess for switching angles such that 0 ≤ α1 < α2 < … < α5 ≤ π/2. 2) Set m = 0; 3) Calculate the objective function and check equality constraints. 4) Accept the solution if Φ is zero and equality constraints are satisfied, else drop the solution. 5) Increment m by a very small value and repeat step 4). 6) Plot the switching angles thus obtained as a function of m and observe the nature of the solutions. Different solution sets appear. 7) Take one solution set at a time and compute all switching angles set for the whole range of m, where solutions exist, by taking solutions obtained in step 6 as initial guess. 8) Complete other solution sets in similar way. To implement the above algorithm, the sequential quadratic programming (SQP) method [10], [11] has been used and been implemented by using MATLAB

Figure 3. Solution sets computed with an arbitrary initial guess.

Figure 4. Exact solution sets for 11-level CMLI.

For each of the multiple solution sets as computed above, total harmonic distortion (THD) in percent is computed according to (5), the set of switching angles among multiple solutions sets which produce least THD 183


RESEARCH PAPER International Journal of Recent Trends in Engineering, Vol 1, No. 3, May 2009 is selected and termed as combined solution. The THD plots for different solution sets along with combined solution are plotted as a function of m in Fig. 5. It can be seen from the Fig. 5 that there is a significant decrease in THD if one uses all possible solution sets for determining the combined solution instead of using only one solution set as reported in [3]-[5]. For example, if one computes THD produced due to all possible solution sets at m = 0.5470 (at this value of modulation index, there are three solution sets), the difference in THD for the solution sets having highest and lowest THD is about 3%. 2

THD =

2

V17 + V19 + ... + V49 V1

2

× 100

Figure 5. THD of different solution sets.

(5)

Where V17, V19 …V49 are magnitudes of 17th, 19th, 49th order harmonic components (non-triplen odd harmonics only) respectively, and V1 is magnitude of fundamental voltage. One of the key features of the proposed method is that the switching angles can be obtained for those values of modulation indices for which solutions of sets of equations given by (2) and (3) do not exist, thus producing continuous solution for the complete range of m from 0 to 1. In this case, the objective function given by (4) is minimized (instead of making it zero) subjected to equality constraint given by (2) with switching angles range 0 ≤ α1 < α2 < … < α5 ≤ π/2. Variation of the objective function with m is shown in Fig. 6. The switching angles corresponding to least THD for the values of m where solutions exist, and corresponding to least objective function for the values of m where solutions do not exist are plotted in Fig. 7. The plot for switching angles in Fig. 7 is plotted only for the range of m where THD is below certain limits.

Figure 6. Variation of objective function.

VI. EXPERIMENTAL RESULTS A prototype single-phase 11-level CMLI has been built using 400V, 10A MOSFET as the switching device. Five separate dc supply of 13V each is obtained using step down transformers with rectifier unit. Pentium 80486 processor based PC with clock frequency 2MHz with timer I/O card is used for firing pulse generation. Firing pulses to the switching devices are given through a delay circuit which provides 5µsec delay to avoid any short circuit due to simultaneous conduction of devices in the same leg of H-bridges. In order to validate the analytical and simulated results, an 11-level single-phase output voltage at m = 0.5480 (for this value of m three solution sets exist) was synthesized at fundamental frequency (f = 50Hz). The fundamental voltage V1 produced was 45.34V (peak) as calculated using (1) and (2). For each of the multiple solution set, THD in line to line voltage was computed as per (5).

Figure 7. Switching angles.

The experimentally produced phase voltage along with its harmonic spectrum corresponding to lowest THD is shown in Fig. 8, and in Fig. 9 phase voltage with corresponding harmonic spectrum for the solution having highest THD is depicted. The harmonic spectrums of synthesized phase voltage show that 5th, 7th, 11th, and 13th harmonics are almost absent as predicted analytically. The THD in line to line voltage as computed analytically and experimentally are: 5.70% and 5.25% respectively for the solution set producing least THD in the output voltage. Corresponding data for the solution set having highest THD are 8.28% and 7.03%. The above data show that experimental results are in close agreement with theoretical values. The triplen harmonics are present in Figs. 8 and 9 due to the fact that the synthesized wave form is a single-phase one. 184



[5]

[6]

[7] Figure 8. Phase voltage and harmonic spectrum for the solution set having smallest THD at m = 0.5480.

[8]

[9]

[10] [11]

Figure 9. Phase voltage and harmonic spectrum for the solution set having highest THD at m = 0.5480.

[12]

VII. CONCLUSION The selective harmonic elimination method at fundamental frequency switching scheme has been implemented using the optimization technique that produces all possible solution sets when they exist. In comparison with other suggested methods, the proposed technique has many advantages such as: it can produce all possible solution sets for any numbers of multilevel inverter without much computational burden, speed of convergence is fast, it can produce continuous solutions for the complete range of modulation index thereby giving more flexibility in control action etc. The proposed technique was successfully implemented for computing the switching angles for 11-level CMLI. A complete analysis for 11-level inverter has been presented and it is shown that a significant amount of THD reduction can be attained if all possible solution sets are computed. REFERENCES [1] Jih-Sheng Lai, Fang Zheng Peng, “Multilevel ConvertersA New Breed of Power Converters”, IEEE Trans. on Industry Applications, vol. 32, no. 3, pp. 509-517, May/June 1996. [2] Fang Zheng Peng, Jih-Sheng Lai, et al, “A Multilevel Voltage-Source Inverter with Separate DC Sources for Static Var Generation”, IEEE Trans. on Industry Applications, vol. 32, no. 5, pp. 1130-1138, September/October 1996. [3] F. Z. Peng, J. W. McKeever, and D. J. Adams, “Cascade Multilevel Inverters for Utility Applications”, IECON Proceedings (Industrial Electronics Conference), vol. 2, pp. 437-442, 1997. [4] L. M. Tolbert, F. Z. Peng, and T.G. Habetler, “Multilevel converters for large electric drives”, IEEE Transactions on

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