odulation Extenion Control for ultilevel Converter Uing Triplen Harmonic Injection with ow Switching Frequency Zhong Du, eon. Tolbert, John N. Chiaon Electrical and Computer Engineering The Univerity of Tenneee Knoxville, TN 7996- E-mail: zdu@utk.edu, tolbert@utk.edu, chiaon@utk.edu Abtract Thi paper preent a modulation extenion control method for multilevel converter with low witching frequency. The diadvantage of the fundamental frequency witching control method for multilevel converter i it narrow range of modulation indice where olution exit. To addre thi problem, a triplen harmonic compenation method i propoed. Firt, the reultant method and/or Newton Climbing method are ued to find olution of the witching angle for the fundamental frequency witching cheme control. Second, a triplen harmonic i injected into the multilevel converter accompanied with the fundamental frequency control ignal to reduce the required level number of the DC voltage without changing the fundamental component of the phae voltage. The computational reult how that the triplen harmonic method indeed reduce the required DC voltage level number to reduce the hardware cot. A 7-level example wa implemented with an -level H-bridge multilevel converter and an 8µ control reolution to demontrate the triplen harmonic compenation method. The experimental reult confirmed the method with thi example. Keyword ultilevel converter; triplen harmonic compenation, modulation extenion control. I. INTRODUCTION The multilevel converter i a promiing technology to directly interconnect utility grid with different frequencie, different voltage magnitude, and different phae. For utility application, ome method mut be ued to eliminate harmonic to atify the total harmonic ditortion (THD) requirement a given by IEEE 9 []. One control method for multilevel converter i the fundamental frequency witching method [6, 6]. The benefit of the fundamental frequency witching method i it low witching frequency compared to other witching cheme, uch a traditional PW method, pace vector PW method, ub-harmonic PW method (SH-PW) [], witching frequency optimal PW (SFO-PW) [], and carrier-baed PW method to help to optimize or balance the witch utilization in multilevel inverter [, ]. To find the witching angle for the fundamental frequency witching method, often numerical method are utilized, uch a Newton method. However, there are limitation for Newton method becaue it require a good initial gue to olve the trancendental equation characterizing the harmonic content, and Newton method doe not necearily find all the olution for the equation. For thi reaon, the reultant method wa conidered in [7] wherein the trancendental equation characterizing the harmonic content are converted into polynomial equation [7]. Elimination theory [,,, ] wa then ued (along with the pecial ymmetry propertie of the equation) to determine the witching angle to eliminate pecific harmonic, namely the th, 7th, th, and th. However, a the number of DC ource increae, the degree of the polynomial in thee equation are large, and one reache the limitation of the capability of contemporary computer algebra oftware tool (e.g., athematica or aple) to olve the ytem of polynomial equation uing elimination theory with reultant [8]. Another problem related with the fundamental frequency witching control i it narrow modulation index range compared to the traditional PW control. The third harmonic injection method ha been ued in both the traditional and multilevel PW control to increae the output voltage while taying in the linear modulation region [9,,, ]. To increae the modulation index range while uing the fundamental frequency witching control, thi paper propoe a modulation extenion control method, which i referred to here a the triplen harmonic compenation method. An experimental -level H-bridge multilevel converter i employed to validate the propoed method for the modulation extenion control. The experimental reult how that the method can effectively eliminate pecified harmonic, and the triplen harmonic compenation method can increae the modulation index range and decreae the required DC level number of a multilevel converter reulting in reduced hardware cot. II. RESUTANT ETHOD FOR FUNDAENTA FREQUENCY SWITCHING SCHEE An -level multilevel converter output voltage waveform with fundamental frequency witching cheme i hown in Fig.. The Fourier erie expanion of the output voltage waveform hown in Fig. i V V ( ωt) = dc ( nθ ) + nθ ) n=,,... n + nθ ) + + nθ )) in( nωt) where i the number of DC ource. Ideally, given a deired fundamental voltage V, one want to determine the witching angle θ,, θ o that V(ωt) = V in(ωt), and pecific higher ()
V dc v a-n v a-n * / / V dc V dc V dc θ θ θ θ v P θ v P θ v P θ v P θ v θ P θ P P P P P Fig.. Output waveform of multilevel converter with fundamental frequency witching cheme. Fig.. Switching angle olution to -level cae of multilevel converter with fundamental frequency witching cheme. harmonic of V(nωt) are equal to zero. For a three-phae application, the triplen harmonic in each phae need not be cancelled a they automatically cancel in the line-line voltage. For example, in the cae of = DC ource, the th, 7th, th, th order harmonic can cancel. Here the reultant method i employed to find the olution when they exit. ( ) + co ( θ ) + co ( θ ) + co ( θ ) + co ( θ ) = ( θ ) + co ( θ ) + co ( θ ) + co ( θ ) + co ( θ ( 7θ ) + co ( 7θ ) + co ( 7θ ) + co ( 7θ ) + co ( 7θ ( θ ) + co ( θ ) + co ( θ ) + co ( θ ) + co ( θ ( ) + co ( θ ) + co ( θ ) + co ( θ ) + co ( θ III. NEWTON CIBING ETHOD FOR UTIEVE CONVERTER The fundamental frequency witching angle computation i approached uing Newton method. The initial gue i propoed from the reult of lower order trancendental equation by the reultant method or the Newton method. For example, if olution to the m level cae are found by the reultant method [8], thee olution could be ued a initial guee for Newton method to find olution to the m+ level cae; the olution to the m+ level cae could then be ued a initial guee to find olution to the m+ level cae. To find a many olution a poible by Newton method, a many different initial guee hould be choen a poible. For thi reaon, the olution to the highet-level cae found by the reultant method hould be ued a the firt et of initial guee for Newton method. By uing uch a trategy, olution for higher- level cae can be found tep by tep. Therefore, the Newton method with uch a trategy here i referred to a Newton Climbing method. Although the Newton Climbing method cannot find all the olution for the equation, it i till practical for application becaue the THD difference for high-order olution i low for different olution et. Kato [7] ued a imilar method by implementing homotopy method and ingle-level elective harmonic elimination PW. co θ m For practical equation olving, equation require initial witching angle a initial guee. The - initial witching co angle can be ued from the previou reult of the - co () equation. Only one initial gue i required. In thi paper, the firt olution earch i to ue a olution from the low order co equation by uing the reultant method, plu an initial gue co θ near 9 degree to find a olution for the higher order equation. If the firt olution earch i not atified, the reult of the firt earch can be ued a initial guee for the econd The -level multilevel converter angle olution v. earch by hifting the modulation index value in the equation. modulation index m = V /(V dc ) are hown in Fig.. The Newton iterative method for the fundamental frequency witching computation i: x n + = x n J where x n+ i the new value, and x n i the old value. J i the Jacobian matrix for the trancendental equation, and f i the et of trancendental function. f n = n = = n = n = h h h h f () () where h i are the odd, non-triplen harmonic number.
The Jacobian matrix i in( θ) in( hθ ) J = h in( h θ ) in( hθ ) in( θ ) in( h θ ) in( θ ) in( h θ ) in( hθ ) in( θ) () in( h θ ) in( ) hθ θ θ Original Output Voltage W aveform Injected Triplen Harmonic Voltage W aveform ot of the continuou olution can be found by the propoed Newton Climbing earch method. In thi paper, up to the -level cae ha been computed. The -level cae olution v. modulation index m = V /(V dc ) are hown in Fig.. / 6 N ew O utput V oltage W aveform Fig.. Triplen harmonic compenation. Fig.. Solution to the -level cae of multilevel converter with fundamental frequency witching cheme. IV. ODUATION EXTENSION CONTRO A diadvantage of the fundamental frequency witching control i it narrow modulation index region where olution exit. To extend the linear range of operation for traditional PW, one method i to add a triplen harmonic. Here, the triplen harmonic compenation method i propoed to increae the modulation index range for the fundamental frequency witching method in multilevel converter. Aume the witching angle θ, θ, θ, θ,, θ are ordered a θ θ θ θ θ /. The working principle i hown in Fig.. Before compenation, the original output voltage waveform i hown in the top one, which i a - level output voltage. The triplen harmonic injected into the converter i hown in the middle waveform of Fig., and the compenated voltage waveform i hown in the bottom waveform, which i a -level output voltage. From Fig., it can be een that to generate the original output voltage, a - level multilevel converter i neceary. However, after compenation, a -level converter can generate the required output voltage waveform. Thi triplen harmonic voltage doe not change the fundamental frequency content, and i cancelled in the line-line voltage. If the witching angle atify the condition θ θ, (6) then by uing thi triplen harmonic compenation method, one can produce the deired output voltage with a converter that ha - DC ource intead of DC ource. Similarly, if the witching angle atify the condition θ θ, (7) then by uing thi triplen harmonic compenation method, a multilevel converter with - DC ource can eliminate the ame number of harmonic a a multilevel converter with DC ource and no triplen harmonic injection. Repeating thi proce, one can find all the poible compenation with the fundamental frequency witching angle. For convenience, here the normalized modulation index i defined a m a = V /(V dc )/. A comparion of the normalized modulation index range with and without the triplen harmonic compenation i hown in Table I for the cae of = ( level converter). Table I. Comparion of the modulation index range with and without triplen harmonic compenation for -level converter ( = ) evel Number m anc * m ac ** ~.9 ~.9 7 ~. ~.76 9 ~. ~.66 ~. ~.6 ~. ~. ~. ~.6 7 ~.6 ~.66 9 ~.6 N/A ~.66 N/A *m anc : odulation index range without triplen harmonic compenation. **m ac : odulation index range with triplen harmonic compenation.
Fig.. Phae voltage and line-line voltage imulation of a 7-level multilevel converter without triplen harmonic compenation (m a =.9). harmonic compenation method, a 7-level cae with m a =.9 i imulated. The output phae voltage without triplen harmonic compenation require 7 level; the phae voltage with triplen harmonic compenation require level. The output line-line voltage, which have no harmonic below the rd, are the ame with and without triplen harmonic compenation. The imulated voltage waveform with and without triplen harmonic compenation are hown in Fig. and 6, eparately. From the imulation figure, it can be derived that the phae voltage without triplen harmonic compenation ha 7 level, and the phae voltage with triplen harmonic compenation ha jut level. The two phae voltage generate the ame lineline voltage. The normalized FFT analyi of the line-line voltage hown in Fig. 7 ha no harmonic below the rd. The imulation reult confirm the witching angle computation and modulation extenion with the triplen harmonic compenation method. Fig. 6. Phae voltage and line-line voltage imulation of a 7-level multilevel converter with triplen harmonic compenation (m a =.9). V. EXPERIENTA IPEENTATION To experimentally validate the modulation extenion control method, a 7-level cae with triplen harmonic compenation i implemented on an -level H-bridge multilevel converter. The prototype three-phae -level cacaded H-bridge multilevel converter wa built at Oak Ridge National aboratory uing 6 V, 7 A OSFET a the witching device and i hown in Fig. 8. A battery bank of eparate DC ource (SDCS) of V DC each feed the converter (five SDCS per phae). A real-time controller baed on an Altera FEX K field programmable gate array (FPGA) i ued to implement the algorithm with 8 µ control reolution. Fig. 8. kw multilevel converter prototype. Fig. 7. Normalized FFT analyi of the line-line voltage hown in Fig. and Fig. 6. The triplen harmonic compenation method increae the modulation index range. For example, without the triplen harmonic compenation, a modulation index m a =.66 realization require a -level converter, but with the triplen harmonic compenation method, uch a modulation realization only need a 7-level converter. Thi can decreae the hardware cot. To confirm the witching angle computation for the fundamental frequency witching cheme and the triplen The experimental phae voltage hown in Fig. 9 confirm that a 7-level voltage cae can be implemented uing an - level multilevel converter with the triplen harmonic compenation method. The line-line voltage generated by the -level multilevel converter with the triplen harmonic compenation method i hown in Fig., and it normalized FFT analyi given in Fig. how that all the harmonic below the rd are zero. Therefore, the experiment confirm the witching angle computation uing the triplen harmonic compenation method combined with the Newton Climbing method.
frequency witching method can eliminate the harmonic a expected, and with the triplen harmonic compenation method, it ha wider modulation index range than that of the fundamental frequency witching method. ACKNOWEDGENTS We would like to thank the National Science Foundation for partially upporting thi work through contract NSF ECS- 988. We would alo like to thank Oak Ridge National aboratory for partially upporting thi work through UT/Battelle Contract No. 7. Fig. 9. Experimental phae voltage of a 7-level output uing triplen harmonic compenation with only an -level converter (m a =.9). Fig.. Experimental line-line voltage of a 7-level multilevel converter (m a =.9). Fig.. Normalized FFT analyi of the line-line voltage hown in Fig.. VI. CONCUSIONS The fundamental frequency witching angle for multilevel converter have been computed by the reultant method and the Newton Climbing method. The triplen harmonic compenation method ha been propoed to extend the modulation index range for the fundamental frequency witching control. The imulation and experiment validated that the fundamental REFERENCES [] H. S. Patel and R. G. Hoft, Generalized harmonic elimination and voltage control in thyritor inverter: Part I harmonic elimination, IEEE Tran. Indutry Application, vol. 9, ay/june 97, pp. -7. [] H. S. Patel and R. G. Hoft, Generalized harmonic elimination and voltage control in thyritor inverter: Part II voltage control technique, IEEE Tran. Ind. Application, vol., Sept./Oct. 97, pp. 666-67. [] J. K. Steinke, Control trategy for a three phae AC traction drive with a -level GTO PW inverter, IEEE PESC, 988, pp. -8. [] P. N. Enjeti, P. D. Zioga, J. F. inday, Programmed PW technique to eliminate harmonic: A critical evaluation, IEEE Tran. Indutry Application, vol. 6, no., arch/april, 99. pp. 6. [] C. K. Duffey, R. P. Stratford, Update of harmonic tandard IEEE-9: IEEE recommended practice and requirement for harmonic control in electric power ytem, IEEE Tran. Indutry Application, vol., no. 6, Nov./Dec. 989, pp. -. [6].. Tolbert, F. Z. Peng, T. G. Habetler, ultilevel converter for large electric drive, IEEE Tran. Indutry Application, vol., no., Jan./Feb. 999, pp. 6-. [7] J. N. Chiaon,.. Tolbert, K. J. ckenzie, Z. Du, Control of a multilevel converter uing reultant theory, IEEE Tran. Control Sytem Technology, vol., no., ay, pp. -. [8] J. N. Chiaon,.. Tolbert, K. J. ckenzie, Z. Du, A unified approach to olving the harmonic elimination equation in multilevel converter, IEEE Tran. Power Electronic, vol. 9, no., arch, pp. 78-9. [9] P. Hammond, A new approach to enhance power quality for medium voltage ac drive, IEEE Tran. Indutry Application, vol., Jan./Feb. 997, pp. 8. [] W. A. Hill and C. D. Harbourt, Performance of medium voltage multilevel inverter, IEEE Indutry Application Society Annual eeting, October 999, Phoenix, AZ, pp. 86 9. [] G. Carrara, S. Gardella,. archeoni, R. Salutari, G. Sciutto, A new multilevel PW method: A theoretical analyi, IEEE Tran. Power Electronic, vol. 7, no., July 99, pp. 97-. [].. Tolbert, F. Z. Peng, T. G. Habetler, ultilevel PW method at low modulation indice, IEEE Tran. Power Electronic, vol., no., July, pp. 79-7. [].. Tolbert, T. G. Habetler, Novel multilevel inverter carrier-baed PW method, IEEE Tran. Indutry Application, vol., no., Sept./Oct. 999, pp. 98-7. [] D. G. Holme, The ignificance of zero pace vector placement for carrier baed PW cheme, IEEE IAS Annual eeting, 99, pp. -8. [] J. Vaallo, J. C. Clare, P. W. Wheeler, A power-equalized harmonicelimination cheme for utility-connected cacaded H-bridge multilevel converter, IEEE Indutrial Electronic Society Annual Conference, -6 Nov., pp. 8 9. [6] S. Siriukpraert, J.-S. ai, T.-H. iu, Optimum harmonic reduction with a wide range of modulation indexe for multilevel converter, IEEE Tran. Ind. Electronic, vol. 9, no., Aug., pp. 87-88. [7] T. Kato, Sequential homotopy-baed computation of multiple olution for elected harmonic elimination in PW inverter, IEEE Tran. Circuit and Sytem I, vol. 6, no., ay 999, pp. 86-9.