Selective Harmonic Elimination for Multilevel Inverters with Unbalanced DC Inputs

Transcription

1 Selective Haronic Eliination for Multilevel Inverters with Unbalanced DC Inputs Abstract- Selective haronics eliination for the staircase voltage wavefor generated by ultilevel inverters has been widely studied in the last decade for ediu and high voltage applications. Most published ethods on this topic are used for balanced ultilevel inverters with the sae dc voltage agnitude. In the proposed ethod, equal area criteria and haronics injection are used for totally unbalanced ultilevel inverters. Regardless of how any voltage levels are involved, only four odified equations are proposed in the basic ethod. To eliinate the haronic coponents in unbalanced conditions, the probles of the basic ethod are discussed for a wide range of odulation indices. Then, a full set of solutions is proposed based on the cobination of optial PWM and the four-equation based ethod for a wide range of odulation indices in unbalanced conditions. Keywords: Equal Area Criteria, Modulation Index, Pulse Width Modulation, Total Haronic Distortion, Unbalanced Multilevel Inverters. I. ITRODUCTIO The recent developents of Hybrid Electric ehicles (HEs) has provided great opportunities for the ipleentation of ediu and high power inverter/converter circuits and associated control strategies []. In large electric drives with high power deands, such as heavy duty ilitary trucs, ultilevel inverters are the natural choices because of higher efficiency, lower dv/dt, and the utilization of lower voltage rated devices [],[ 3]. A typical ultilevel inverter utilizes voltage levels fro ultiple dc sources of the sae agnitude. These dc sources can be isolated as in cascade ultilevel structures or interconnected as in diode claped structures. In ost published ultilevel inverter circuit topologies, the dc sources in the circuits need to be aintained to supply identical voltage levels. Based on these identical voltage levels, with proper control of the switching angles, the switches only need to switch on and off once during a fundaental cycle to achieve the desired staircase wavefor; thus, the switching loss of the device is reduced to iniu. However, with reduced switching frequencies, even with additional voltage levels, low frequency haronics are generated in the staircase voltage [], []. Based on Fourier expansion of the staircase wavefor generated by ulti-level inverters, several ethods have been proposed to eliinate selected haronic coponents [6], [7]. In ost of these ethods, for a higher level of ultilevel Daoun Ahadi and Jin Wang* Departent of Electrical and Coputer Engineering The Ohio State University 0 Dreese Labs; 0 eil Avenue Colubus, OH 30 Phone: , Fax: Eail: wang@ece.osu.edu* inverters, the order of the equation and the nuber of the variables both increase with the voltage levels. Therefore, for a high nuber of switching angles, finding solutions for these equations would becoe difficult and often involve advanced atheatical algoriths on existing coputer algebra software tools [8]. The haronic injection and equal area criteria based fourequation ethod was proposed in 00 [9]. Regardless of how any voltage levels are involved in ultilevel inverters, four siple equations are used in the basic ethod. A full study of this ethod with an equal dc voltage level has been presented recently [0]. This paper shows that the sae ethod can be adopted to be used for haronics eliination in ultilevel inverter converters with unbalanced dc inputs. In this paper, first, the basic four-equation ethod is adopted for ultilevel inverters with unbalanced dc voltage levels. Then, the proble of the proposed ethod and proposed solutions for low and high odulation indices are discussed. Finally, for low odulation indices where haronic eliination is liited due to the sall nuber of voltage levels, a new ethod cobining a odified version of the fourequation based ethod and the basic idea of optial PWM is proposed. The full analysis of the cobined ethod and extended results will be shown in a follow up paper. II. THE PROPOSED METHOD FOR UBALACED DC IPUTS The basic idea of the four-equation ethod for haronics eliination in a ultilevel inverter with unbalanced dc levels is shown in Fig.. The haronics eliination is realized with haronics injection to the odulation wavefor and switching angle calculation based on equal area criteria. The associated algorith is suarized by the following: Equation : Based on ewton-raphson ethod, this equation is used to find nuerical solutions for the junction points of reference wavefor and voltage level: + h sin(δ )... h sin( δ ) δ = arctan( ) () cos( δ ) F /09/$ IEEE 773

2 Fig. : Equal area criteria in ultilevel inverters with unbalanced dc levels. Equation : Based on the junction points, switching angles are calculated by this equation: = / *( δ ) h F (cos( δ ) cos( δ )) (cos(δ ) cos(δ h (cos( δ ) cos( δ ))) δ + ))... Equation 3: The equation is used to define haronic coponents by switching angles for different frequencies: h = =,,.., ) (cos( ) cos( ( π ))) π Equation : This equation is used to generate a new reference wavefor: Where, hs ref (3) = sin( ωt) h sin( ωt) () F s is the su of h calculated in every iteration: iter h s = h ( (),,... To identify possible probles of this ethod for haronic eliination, the group of equations is applied to the 6-level wavefor with different dc agnitudes as shown in Table., for odulation index changing fro 0.6 to 0.9. The odulation index is found?? by: F MI = π where, F is the reference ac voltage in the output, is the nuber of dc levels, and is the dc agnitude for each dc ( voltage level in ultilevel inverter output wavefor. The ain proble identified in this process is the aplitude difference between the desired and resulted fundaental voltages. With the direct ipleentation of the proposed ethod, the fundaental voltage of the staircase wavefor () (6) Table. : Five different dc levels for staircase wavefor. DC Levels 3 P.U often diverts fro the desired value, as shown in Table. III. THE MODIFIED METHODS FOR DIFFERET MI To solve the above entioned proble, odulation indexes are categorized in two groups for low odulation indices at which an extra dc level is available and high odulation indices, where no extra voltage level is available, as shown in Fig.. A. Low Modulation Indices with Extra oltage Levels In these conditions, fewer dc levels are utilized to synthesize the staircase wavefor. Therefore, an extra voltage is available and can be used for fundaental voltage copensation. Based on this idea, an extra switching angle in the extra voltage level is calculated to achieve desired fundaental voltage. However, this switching angle generates additional haronic coponents. So these additional haronics are also added to the reference wavefor that is used to calculate the other switching angles. Then, the generated haronics by the extra switching angle is copensated by other switching angles. In these conditions, two equations are used to calculate the additional switching angle: ) First, the fundaental voltage based on switching angles fro to is calculated with the following equation: = cos( i ), < π ) Then, additional switching angle is calculated to copensate the difference between and fundaental voltage: (7) π + = a cos( ( F )) (8) + ) This idea for low odulation indices is shown in Fig. 3. Thus, based on four-equation based ethod, and through (7) and (8), haronic coponents are eliinated by switching angles and the reference fundaental voltage is copensated adequately by. The additional process used + for voltage copensation in + level is shown in dotted line. B. High Modulation Indices with o Extra oltage Levels At this condition, all dc levels are used for staircase generation and there is no additional voltage level for fundaental voltage copensation. Therefore, in the proposed ethod, again, two equations are used for fundaental voltage copensation: 77

3 Table. Saple points fro the direct ipleentation of the basic four-equation ethod. Reference Resulted Switching Angles (rad.) Haronics (%) MI MI 3 st th 7 th th 3 th 7 th /A /A /A /A /A /A /A /A /A /A Fig. : Two groups of odulation indices in ultilevel inverters with unbalanced dc levels. Fig. 3. Modified ethod with additional switching angle in unbalanced ultilevel inverters. ) An adjustent switching angle is calculated to achieve fundaental voltage by the following equation: * π = a cos( ( F )) (9) dc ( ) where is the total fundaental voltage generated by switching angles fro to. ) This adjustent angle is used to odify the switching angle for the last voltage level: (odified) = acos(cos( ) + cos( )) (0) Therefore, based on the switching angle adjustent for the last dc level, the desired voltage agnitude in the fundaental frequency can be achieved. However, the adjustent switching angle also causes new haronic coponents used in the final odulation wavefor in equal area criteria. The total process for this adjustent is shown in Fig.. Thus, the last switching angle,, is used, both for haronic eliination and fundaental voltage copensation. The additional process for this adjustent in four-equation based ethod is shown in dotted line. * In Table 3, soe saple points are shown for low and high odulation indices in the proposed ethod. As entioned in the results, haronic coponents are eliinated successfully and fundaental voltage is generated precisely. I. OPTIMAL PWM AD FOUR-EQUATIO METHOD COMBIED SOLUTIO At low odulation indices, fewer nuber of dc levels is available for staircase wavefor generation, thus there is decreasing of switching angle nubers. Therefore, with fourequation based ethod and Fourier analysis, staircase wavefor would have ore haronic coponents. To overcoe this proble, in each dc level, the nuber of switching angles can be increased to eliinate ore haronic coponents. In [3], it has been approved that the fourequation ethod can also be adopted to calculate the switching angles of single level optial PWM. Thus, the ongoing research is to use the four-equation based optial PWM ethod for wavefor with three levels. In this ethod, based on available dc voltage levels in ultilevel inverters, the nuber of switching angles for each level can be increased. Thus, ore haronic coponents can be eliinated specially in low odulation indices. This approach for ultilevel inverters with unbalanced dc inputs is shown in Fig.. 77

4 Fig.. Modified ethod with adjustent switching angle for the highest voltage level in unbalanced ultilevel inverters. Table 3. Saple points based on the odified four-equation ethod for different odulation indices. Reference Resulted Switching Angles (rad.) Haronics (%) MI MI 3 st th 7 th th 3 th 7 th /A /A /A /A /A /A /A /A /A /A /A /A 00 0 /A /A /A /A Fig.. Optial PWM in low odulation indices in unbalanced ultilevel inverters. In the proposed ethod, the nuber of switching angles used for each level is not liited in theory. Thus, the haronic eliination can be proposed very efficiently, even for low odulation indices. In this condition, the advantages of optial PWM for selected haronic eliination can be applied for unbalanced ultilevel inverters. However, based on ultilevel inverters applications for ediu and high voltages, the switching frequency and nuber of switching angles applied for inverters is liited based on power electronic switches and power losses. In Table, the proposed ethod is applied for two cases in low odulation indices when three and five dc levels are available. In the first case with three dc levels, for each positive or negative polarity, one level is available. Thus, five switching angles are used in sae level. In the second case, five levels are used for ultilevel inverters. Therefore, five switching angles are applied in two dc levels for haronic eliination. In both cases, five haronic coponents can be eliinated that verifies the advantage of the proposed ethod in low odulation indices for ultilevel with unbalanced dc voltages.. SIMULATIO RESULTS To verify this iproveent for haronic eliination, the ultilevel inverter with unbalanced dc levels is siulated in low odulation indices. Then the switching angles, shown in Table, are applied to switches in the inverters. Three and five voltage levels for unbalanced input dc are shown in Fig. 6(a) and (b). The output siulated voltage is decoposed in the selected frequencies for haronic coponents. These coponents are shown for two odulation indices in Table. As entioned in Table, the selected haronic coponents are iniized successfully in the siulated ultilevel inverters that verify advantages of using OPWM in fourequation ethod for low odulation indices. I. COCLUSIO In this paper, a siple ethod using equal area criteria and haronic injection is used to eliinate selected haronic coponents for ultilevel inverter with unbalanced dc inputs. To precisely generate output voltage in the fundaental frequency, for low and high odulation indices, additional angle and adjustent angle are used respectively. However, in four-equation based ethod, for each dc level just one switching angle is used. Thus, for low odulation indices that few nubers of switching angles are available, the nuber of eliinated haronic coponent is liited. In the proposed ethod, optial PWM and four-equation based ethod are cobined together to overcoe this liitation in ultilevel inverters in low odulation. Then, the nuber of switching angles and corresponded nuber of eliinated haronics can be increased significantly. 776

5 Table. Saple points based on the OPWM and four-equation based ethod for low odulation indices # Modulation Switching Angles (rad.) Haronics (% based on fundaental output) levels Index 3 st th 7 th th 3 th 7 th (a) Three level inverters (b) Five level inverters with unbalanced DC lins Fig. 6. Siulated output voltage based on OPWM and four-equation based ethod in ultilevel inverters with unbalanced DC lins. Table. Siulation results for output voltage and selected haronic coponents on different odulation indices. Modulation Switching Angles (rad.) Haronics (% based on fundaental output) Index 3 st th 7 th th 3 th 7 th REFERECE []. J. M. Miller, A. R. Gale, Hybird-electric vehicle success will depend on low cost, efficient power electronics systes, Power Conversion & Intelligent Motion (PCIM), ov. 997, pp []. Khan, F.H, Tolbert, L.M, W Multilevel DC-DC Converter for Hybrid Electric and Fuel Cell Autootive Applications, IEEE Industry Applications Conference, nd IAS Annual Meeting, Sept. 007, pp [3]. Tolbert, L.M, F. Z. Peng, T. G. Habetler, Multilevel Converters for Large Electric Drives, IEEE Trans on Industry Applications, ol. 3, o., Jan 999, pp-36-. []. J. K. Steine, Control strategy for a three phase AC traction drive with a 3-Level GTO PWM inverter, IEEE PESC 88 Conf, 988, pp []. J. assallo, J. C. Clare, P. W. Wheeler, Power-equalized haronic-eliination schee for utility-connected cascaded H-bridge ultilevel convertes, Industrial Electronics Society, 003. IECO '03. The 9th Annual Conference of the IEEE, olue:, Pages:8-90 ol., -6 ov [6]. J.. Chiasson, L. M. Tolbert, K. J. McKenzie, and D. Zhong, Eliination of haronics in a ultilevel converter using the theory of syetric polynoials and resultants, IEEE Trans. Control Syste Technology, vol. 3, no., pp. 6--3, Mar. 00. [7]. J.. Chiasson, L. M. Tolbert, Z. Du, and K. J. McKenzie, The use of power sus to solve the haronic eliination equations for ultilevel converters, Eur. Power Electron. Drives J., vol., no., pp. 9--7, Feb. 00. [8]. B. P. McGrath, D. G. Holes, Multicarrier PWM strategies for ultilevel inverters, IEEE Trans. on Industrial Electronics, olue: 9, Issue:, Aug.00, pp [9]. L. Li; D. Czarowsi, Y. G. Liu; P. Pillay, Multilevel selective haronic eliination PWM technique in series-connected voltage inverters., IEEE Transactions on Industry Applications, ol.36, Jan.- Feb. 000, pp [0]. D. Ahadi, J. Wang, Full Study of a Precise and Practical Haronic Eliination Method for Multilevel Inverters, The Applied Power Electronics Conference and Exposition (APEC), Feb009, In Proc. [].. G. Agelidis, A. Baloutsis, and M. S.A. Dahidah, A fivelevel syetrically defined selective haronic eliination PWM 777

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