Harmonic Elimination in Multilevel Converters

Transcription

1 1 Haronic Eliination in Multilevel Converters John Chiasson, Leon Tolbert, Keith McKenzie and Zhong Du ECE Departent The University of Tennessee Knoxville, TN Abstract A ethod is presented to copute the switching angles in a ultilevel converter so as to produce the required fundaental voltage while at the sae tie not generate higher order haronics. Using a fundaental switching schee, previous work has shown that this is possible only for specific ranges of the odulation index. Here it is shown for a three DC source ultilevel inverter that, by odifying the switching schee, one can extend the range of odulation indices for which the switching angles exist to achieve the fundaental while eliinating the 5 th and 7 th haronics. In contrast to nuerical techniques, the approach here produces all possible solutions. Keywords Multilevel Converters, Haronic Eliination, Resultants, Syetric Polynoials I. Introduction Electric power production in the 1st Century will see draatic changes in both the physical infrastructure and the control and inforation infrastructure. A shift will take place fro a relatively few large, concentrated generation centers and the transission of electricity over ostly a high voltage ac grid to a ore diverse and dispersed generation infrastructure that also has a higher percentage of dc transission lines [1]. The general function of the ultilevel inverter is to synthesize a desired ac voltage fro several levels of dc voltages. For this reason, ultilevel inverters are ideal for connecting either in series or in parallel an ac grid with distributed energy resources such as photovoltaics or fuel cells or with energy storage devices such as capacitors or batteries[]. Additional applications of ultilevel converters include such uses as ediu voltage adjustable speed otor drives, static var copensation, dynaic voltage restoration, haronic filtering, or for a high voltage dc backtoback intertie[3]. Transforerless ultilevel inverters are uniquely suited for this application because of the high VA ratings possible with these inverters [4]. The ultilevel voltage source inverter s unique structure allows it to reach high voltages with low haronics without the use of transforers or seriesconnected, synchronizedswitching devices. A fundaental issue for a ultilevel converter is to find the switching angles (ties) so that the converter produces the required fundaental voltage and does not generate specific lower order doinant haronics. In this work, a ethod is presented to copute the switching angles in a ultilevel converter so as to achieve this goal. Using a fundaental switching schee (see Figure ), previous work in [5][6] has shown that this is possible only for specific ranges of the odulation index. Here it is shown that, by odifying the switching schee, one can extend the lower range of odulation indices for which the switching angles exist. Further, in contrast with the PWM technique proposed in [7], the switching schees proposed here are only slightly above the fundaental frequency. In contrast to nuerical techniques such as used in [8], the approach here produces all possible solutions. II. Cascaded Hbridges The cascade ultilevel inverter consists of a series of H bridge (singlephase fullbridge) inverter units. The general function of this ultilevel inverter is to synthesize a desired voltage fro several separate dc sources (s), which ay be obtained fro solar cells, fuel cells, or ultracapacitors. Figure 1 shows a singlephase structure of a cascade inverter with s [4]. v a n v [(1)/] v [(1)/1] v v 1 Fig. 1. Singlephase structure of a ultilevel cascaded Hbridges inverter. Each is connected to a singlephase fullbridge inverter. Each inverter level can generate three different voltage outputs,, and by connecting the dc source to the ac output side by different cobinations of the four switches,,, and. The ac output of each level s fullbridge inverter is connected in series such that the synthesized voltage wavefor is the su of all of the individual inverter outputs. The nuber of output phase (lineneutral) voltage levels in a cascade ulitilevel inverter is then s 1,wheres isthenuberofdcsources.

2 Switching Angles (degrees) An exaple phase voltage wavefor for a 7level cascaded ultilevel inverter with three SDSCs (s = 3) is shown in Figure. The output phase voltage is given by v an = v a1 v a v a Three DC Source Multilevel With Fundaental Switching p/ p 3 Fig.. Output wavefor of an 7level (3 DC source) cascade ultilevel inverter. Each of the active devices of the Hbridges switch only at the fundaental frequency, and consequently this is referred to as the fundaental switching schee. Also, each Hbridge unit generates a quasisquare wavefor by phaseshifting its positive and negative phase legs switching tiings. Each switching device always conducts for 18 (or 1 cycle) regardless of the pulse width of the quasisquare wave so that this switching ethod results in equalizing the current stress in each active device. Using the fundaental switching schee of Figure, it has been shown in [6] that achieving the fundaental while eliinating specified lower order haronics is only achievable for certain ranges of the odulation index. For exaple, Figure 3 is a plot of the switching solution angles in the case of three DC sources where the fundaental is achieved while the 5 th and 7 th haronics are eliinated. Here the paraeter is related to the odulation index by = s a where s isthenuberofdcsources(s =3 in Figure 3). Note that for in the interval [1.15,.5] there is a solution (with two different solutions in the subinterval [1.49, 1.85]). On the other hand, for [,.8] ( a [,.7]), [.83, 1.14] ( a [.8,.383]) and [.53,.76] ( a [.84,.9]) there are no solutions. The objective is to show how the range of values of the odulation index a = /s can be extended for which the fundaental is still achieved and the 5 th and 7 th haronics are also eliinated. This is done by having ore switchings per cycle. At very low odulation indices, one would surise that only one of the DC sources would be used with ultiple switchings. This is siply the unipolar prograed PWM switching schee of Patel and Hoft [9][] (see Figure 4). At slightly higher odulation indices one would surise that two DC sources would be used with a schee such as in [8] (see Figure 5) or a cobination of the unipolar schee and that of DC source ultilevel schee (see Figure 6). In the following, it is shown how the transcendental equations that characterize the haronic content for each of these switching schees canbesolvedtofind all solutions that eliinate the 5 th and 7 th haronics while achieving the fundaental Fig. 3. Schee : Fundaental switching schee. The switching angles,, in degrees vs. There are s = 3 DC sources and the odulation index is given by a = /s III. Matheatical Model of Switching The three switching schees illustrated in Figures 4, 5 and 6 are considered to eliinate the 5 th and 7 th haronics at lower odulation indices while still achieving the fundaental voltage. These schees use 4 switching angles in contrast to the 3 switching angles used by the fundaental switching schee of Figure. Consequently, their switches turn on and off at an overall frequency just above the fundaental frequency. To proceed, note that each of the wavefors of Figures 4, 5 and 6 have a Fourier series expansion of the for V (ωt) = 4 π sin(nωt) X 1 ³ 1 cos(n ) n=1,3,5,... n cos(n ) 3 cos(n ) 4 cos(n ) (1) where π/and i = ±1depending on the switching schee. Specifically, =( 1,, 3, 4 )= (1, 1, 1, 1) in the case of schee 1 (unipolar switching), =(1, 1, 1, 1) in the case of schee ( virtual stage ), and =(1, 1, 1, 1) for schee 3. As the Fourier series is sued over only the odd haronics and cos(n(π θ i )) = cos(nθ i )forn odd, equation (1) ay be rewritten in the for V (ωt) = 4 sin(nωt) () π X 1 ³ cos(nθ n=1,3,5,... n 1)cos(nθ )cos(nθ 3)cos(nθ 4) where θ i = θ i if i =1andθ i = π θ i if i = 1. The conditions becoe Inequality Conditions Schee 1 θ 1 π θ θ 3 π θ 4 π/ Schee θ 1 θ π θ 3 θ 4 π/ Schee 3 θ 1 π θ θ 3 θ 4 π/. (3)

3 3 3 3 p/ p Fig. 4. Schee 1: Output wavefor using a single DC source with a unipolar prograed PWM schee. 3 3 p/ p Fig. 5. Schee : Output wavefor using DC sources and a virtual stage schee [8]. 3 3 p/ p Fig. 6. Schee 3: Output wavefor using DC sources with a cobination of a unipolar PWM schee and a DC sources ultilevel schee. The desire here is to use these switching schees to achieve the fundaental voltage and eliinate the 5 th and 7 th haronics for those values of the odulation index a for which solutions did not exist for the switching schee of Figure (see Figure 3). That is, choose the switching angles θ 1,θ,θ 3,θ 4 to satisfy cos(θ 1)cos(θ )cos(θ 3)cos(θ 4) = cos(5θ 1)cos(5θ )cos(5θ 3)cos(5θ 4) = (4) cos(7θ 1)cos(7θ )cos(7θ 3)cos(7θ 4) = and the inequalities (3). Here, V 1 / (4 /π) and the odulation index is given by a, V 1 /V 1ax = V 1 / (s4 /π) =/s. This is a syste of 3 transcendental equations in the 4 unknowns θ 1,θ,θ 3,θ 4. In order to get a fourth constraint, consider the possibility of also eliinating the 11 th haronic using this extra switching. That is, append the condition cos(11θ 1)cos(11θ )cos(11θ 3)cos(11θ 4)= (5) to the conditions (4). The fundaental question is When does the set of transcendental equations (4), (5) have a solution? To answer this question, define x 1 =cos(θ 1),x =cos(θ ),x 3 =cos(θ 3),x 4 =cos(θ 4) and use the trigonoetric identities cos(5θ) = 5cos(θ) cos 3 (θ)16cos 5 (θ) cos(7θ) = 7cos(θ)56cos 3 (θ) 11 cos 5 (θ)64cos 7 (θ) cos(11θ) = 11 cos(θ) cos 3 (θ) 13 cos 5 (θ) 816 cos 7 (θ) 816 cos 9 (θ) 4 cos 11 (θ) to transfor the conditions (4) and (5) to p 1 (x), x 1 x x 3 x 4 = 4X ³ p 5 (x), 5x i x 3 i 16x 5 i = p 7 (x), p 11 (x), i=1 4X i=1 ³ 7x i 56x 3 i 11x 5 i 64x 7 i = 4X 11xi x 3 i 13x 5 i 816x 7 i i=1 816x 9 i 4x 11 i = (6) where x, (x 1,x,x 3,x 4 ), and the angle conditions becoe Inequality Conditions Schee 1 x 4 x 3 x x 1 1 Schee x 4 x 3 x x 1 1 Schee 3 x 4 x 3 x x 1 1. (7) This is a set of four polynoial equations in the four unknowns x 1,x,x 3,x 4. In the next section, a systeatic ethod is presented to solve these equations for all of their possible solutions. It is interesting to note that in [11] polynoial systes were also considered. IV. Solving Polynoial Equations The first equation of (6) can be solved as x 4 = (x 1 x x 3 ) to eliinate x 4 fro the reaining three equations. However, one is still left with three polynoial equations in the three unknowns (x 1,x,x 3 ). The pertinent question is then, Given two polynoial equations a(x 1,x,x 3 )=andb(x 1,x,x 3 )=,howdoesone solve the siultaneously to eliinate (say) x 3?. A systeatic procedure to do this is known as eliination theory and uses the notion of resultants [1][13]. Briefly, one considers a(x 1,x,x 3 )andb(x 1,x,x 3 ) as polynoials in x 3 whose coefficients are polynoials in (x 1,x ). Then, for

4 4 exaple, letting a(x 1,x,x 3 )andb(x 1,x,x 3 ) have degrees 3 and, respectively in x 3,theyaybewritteninthefor a(x 1,x,x 3 ) = a 3 (x 1,x )x 3 3 a (x 1,x )x 3 a 1 (x 1,x )x 3 a (x 3,x ) b(x 1,x,x 3 ) = b (x 1,x )x 3 b 1 (x 1,x )x 3 b (x 3,x ). The n n Sylvester atrix S a,b,wheren =deg x3 {a(x)} deg x3 {b(x)} =3=5,isdefined by S a,b (x 1,x ), a (x 1,x ) b (x 1,x ) a 1(x 1,x ) a (x 1,x ) b 1(x 1,x ) b (x 1,x ) a (x 1,x ) a 1(x 1,x ) b (x 1,x ) b 1(x 1,x ) b (x 1,x ) a 3(x 1,x ) a (x 1,x ) b (x 1,x ) b 1(x 1,x ) a 3(x 1,x ) b (x 1,x ) The resultant polynoial r(x 1,x )isdefined by r(x 1,x ) = Res ³a(x 1,x,x 3 ),b(x 1,x,x 3 ),x 3, det S a,b (x 1,x ) (8) and is the result of solving a(x 1,x,x 3 ) = and b(x 1,x,x 3 ) = siultaneously for (x 1,x ), i.e., eliinating x 3. In previous work [6], the ethod of resultants was used to solve three polynoial equations in three unknowns to obtain the switching angles in Figure 3. However, in the present proble, there are four polynoial equations in four unknowns as given in (6). As the nuber of equations increases, the degrees of the polynoials increase so that one has to copute sybolically the deterinant of a large n n Sylvester atrix. For exaple, after x 4 = (x 1 x x 3 ) is used in (6) to eliinate x 4, the reaining three polynoials q 5 (x 1,x,x 3 ), p 5 (x 1,x,x 3, x 1 x x 3 ) q 7 (x 1,x,x 3 ), p 7 (x 1,x,x 3, x 1 x x 3 ) q 11 (x 1,x,x 3 ), p 11 (x 1,x,x 3, x 1 x x 3 ) have degrees 4, 6,, respectively in x 3. In particular, to eliinate x 3 fro q 7 (x 1,x,x 3 ) =,q 11 (x 1,x,x 3 ) = would require the sybolic coputation of a (6 ) (6 ) = Sylvester atrix. This sybolic calculation is carried out using coputer algebra software (e.g., the Resultant coand in Matheatica [14]). However, these coputations are tie consuing, and one quickly encounters the coputational liitsofsuchsystesasthesizeofthesylvesteratrixincreases. To get around this, use is ade of the fact that the polynoials aking up the syste (6) are syetric. The theory of syetric polynoials [1] is then exploited to obtain a new set of relatively low order polynoials whose resultants can easily be coputed using existing coputer algebra software tools. In contrast to nuerical techniques, the approach here produces all possible solutions. A. Syetric Polynoials The polynoials p 1 (x),p 5 (x),p 7 (x),p 11 (x) in (6) are syetric polynoials, that is, p i (x 1,x,x 3,x 4 ) =. p i (x,x 1,x 3,x 4 ) = p i (x 3,x,x 1,x 4 ), etc. Define the eleentary syetric functions (polynoials) s 1,s,s 3,s 4 as s 1, x 1 x x 3 x 4 s, x 1 x x 1 x 3 x 1 x 4 x x 3 x x 4 x 3 x 4 s 3, x 1 x x 3 x 1 x x 4 x 1 x 3 x 4 x x 3 x 4 (9) s 4, x 1 x x 3 x 4 A basic property of syetric polynoials is that they can be rewritten in ters of the eleentary syetric functions [1] (e.g., using the SyetricReduction coand in Matheatica [14]). In the case at hand, it follows that with s =(s 1,s,s 3,s 4 ) and using (9), the polynoials (6) becoe p 1 (s) = s 1 p 5 (s) = 5s 1 s s 5 1 6s 1 s 8s 3 1s 8s 1 s 6s 3 8s 1s 3 8s s 3 () p 7 (s) = 7s 1 56s s5 1 64s s 1s 56s 3 1s 448s 5 1s 56s 1 s 896s 3 1s 448s 1 s 3 168s 3 56s 1s 3 448s 4 1s 3 56s s s 1s s 3 448s s 3 448s 1 s 3 p 11 (s) = 11s 1 s s s 7 1 (The coplete expression for p 11 (s) isgivenintheappendix). One uses p 1 (s) =s 1 = to eliinate s 1. The table below gives the degrees of the three polynoials p 5 (s),p 7 (s),p 11 (s) in the indeterinates s,s 3,s 4. degree in s degree in s 3 degree in s 4 p 5 (s) 1 1 p 7 (s) 3 1 p 11 (s) 5 3 The key point here is that the degrees of these polynoials in s,s 3,s 4 are uch less than the degrees of p 5 (x),p 7 (x),p 11 (x 1 )inx 1,x,x 3 (see (6)). In particular, the Sylvester atrix of the pair {p 7 (s,s 3,s 4 ),p 11 (s,s 3,s 4 )} is a 3 3 atrix (if the variable s 4 is to be eliinated) rather than the Sylvester atrix required to eliinate x 3 inthecaseof{p 7 (x 1,x,x 3 ),p 11 (x 1,x,x 3 )} in (6). To proceed, one then eliinates s 4 by coputing r q5,q 7 (s,s 3 ) = Res ³q 5 (s,s 3,s 4 ),q 7 (s,s 3,s 4 ),s 4 r q5,q 11 (s,s 3 ) = Res ³q 5 (s,s 3,s 4 ),q 11 (s,s 3,s 4 ),s 4 and finally, coputing r(s )=Res ³r q5,q 7 (s,s 3 ),r q5,q 11 (s,s 3 ),s 3 gives a polynoial in the single variable s. For each, one solves r(s ) = for the roots {s i }. Each root s i is then used to solve r q5,q 7 (s i,s 3 )=fortheroots{s 3ji }. Each pair (s i,s 3ji )isusedtosolvep 5 (, s i,s 3ji,s 4 )=

5 Switching Angles With Lowest THD (Degrees) Switching Angles (degrees) Switching Angles With Lowest THD (Degrees) 5 to get the roots s 4ki,j. Then the set of 4tuples s (s1,s,s 3,s 4 )=(, s i,s 3ji,s )forsoei, j, kª 4ki,j are the only possible solutions to (). For each solution triple (s 1,s,s 3,s 4 ), it is the corresponding values of (x 1,x,x 3,x 4 ) which are required to obtain the switching angles. Consequently, the syste of polynoial equations (9) ust be solved for the x i.todo so, one siply uses the resultant ethod again to solve the syste of polynoials not work) to achieve the fundaental without generating the 5 th, 7 th or 11 th haronics Virtual Stage PWM with Switched Negative f 1 (x) = s 1 (x 1 x x 3 x 4 ) f (x) = s (x 1 x x 1 x 3 x 1 x 4 x x 3 x x 4 x 3 x 4 ) f 3 (x) = s (x 1 x x 3 x 1 x x 4 x 1 x 3 x 4 x x 3 x 4 ) f 4 (x) = s 4 x 1 x x 3 x 4. (11) The Appendix shows how (11) is solved using resultants. The solutions of (11) which satisfy (7) are then straightforwardly used to copute the switching angles. V. Results Using the above techniques, the switching angles for the three schees vs the paraeter (odulation index is a = /s) were coputed. The switching angles for schee 1 (unipolar prograed PWM) of Figure 4 were coputed first. Figure 7 shows these results where only the switching angle set for each value of that gave the sallest THD is plotted (see Figure 11). It turns out that for.49 there are actually three different solution sets and for [.5,.55], [.68,.87] there are two different solutions sets. This shows that for low odulation indices where only one DC source is used, this schee can achieve the fundaental without generating the 5 th, 7 th or 11 th haronics. 9 8 One DC Source Prograed PWM with Four Notches Per Half Cycle 5 θ θ θ Fig. 8. Switching angles for Schee of Figure 5. Finally, the switching angles for schee 3 are shown in Figure 9. Figure 9 shows that schee 3 can be used for.56 <<.69 as neither schee nor the fundaental switching schee (schee ) work in that range of. Of course, this schee will also not generate the 11 th haronic Fig. 7. Switching angles for schee 1 of Figure 4 The switching angles for schee are shown in Figure 8. As the fundaental switching schee does not achieve the desired result for <1.15, Figure 8 shows that schee canbeusedfor.97 <<1.15 (where schee 1 will also Fig. 9. Switching angles for schee 3 of Figure 6. None of the above schees is able to achieve the desired result for.87 <<.97 (or.9 < a <.33). In the ranges of for which ore than one schee will work, a natural choice is the one which generates the sallest distortion due to higher order haronics. Figures, 11, 1 and 13 are plots of the haronic distortion vs due to the 11 th, 13 th, 17 th and 19 th haronics for each schee. These figures show for low odulation indices ( <.87 or a <.9), the unipolar PWM (schee 1) should be used except for.55 <<.7 (.55/3 < a <.7/3) where schee 3 (see Figure 13) will produce the lowest

6 THD (V dis * %) THD (V dis * %) THD (V dis * %) THD (V dis * %) 6 haronics. As pointed out above, Schee can be used for.97 <<1.15 (where no other schee works) and Figure 1 shows that the haronic distortion will be between % and 18% in this range nd Set 15 nd Set Fig.. Schee (See Figure ) THD due to the 11 th, 13 th, 17 th and 19 th vs nd Set 3rd Set Fig. 11. Schee 1 (See Figure 4) THD due to the 11 th, 13 th, 17 th and 19 th vs nd Set Fig. 1. Schee (See Figure 5) THD due to the 11 th, 13 th, 17 th and 19 th vs Fig. 13. Schee 3 (See Figure 6) THD due to the 11 th, 13 th, 17 th and 19 th vs. Acknowledgeents We would like to thank the National Science Foundation for partially supporting this work through contract NSF ECS We would also like to thank Oak Ridge National Laboratory for partially supporting this work through the UT/Battelle contract no References [1] D. Leeper and J. T. Barich, Technology for distributed generation in a global arket place, in Proceedings of the Aerican Power Conference, pp , [] L.M.TolbertandF.Z.Peng, Multilevelconvertersasautility interface for renewable energy systes, in IEEE Power Engineering Society Suer Meeting, pp , July. Seattle, WA. [3] L.M.Tolbert,F.Z.Peng,andT.G.Habetler, Multilevelconverters for large electric drives, IEEE Transactions on Industry Applications, vol. 35, pp , Jan./Feb [4] J. S. Lai and F. Z. Peng, Multilevel converters A new breed of power converters, IEEE Transactions Industry Applications, vol. 3, pp , May/June [5] J. Chiasson, L. M. Tolbert, K. McKenzie, and Z. Du, A coplete solution to the haronic eliination proble, in Proceedings of Applied Power Electronics Conference APEC 3, February 3. Miai FL. [6] J. Chiasson, L. M. Tolbert, K. McKenzie, and Z. Du, Eliinating haronics in a ultilevel inverter using resultant theory, in Proceedings of the Power Electronics Specialists Conference, pp , June. Cairns, Australia. [7] L.M.Tolbert,F.Z.Peng,andT.G.Habetler, MultilevelPWM ethods at low odulation indexes, IEEE Transactions on Power Electronics, vol. 15, pp , July. [8] F.S. Shyu and Y.S. Lai, Virtual stage pulsewidth odulation technique for ultilevel inverter/converter, IEEE Transactions on Power Electronics, vol. 17, pp , May. [9] H. S. Patel and R. G. Hoft, Generalized haronic eliination and voltage control in thryristor inverters: Part I haronic eliination, IEEE Transactions on Industry Applications, vol. 9, pp , May/June [] H. S. Patel and R. G. Hoft, Generalized haronic eliination and voltage control in thryristor inverters: Part II voltage control technique, IEEE Transactions on Industry Applications, vol., pp , Septeber/October [11] J. Sun and I. Grotstollen, Pulsewidth odulation based on realtie solution of algebraic haronic eliination equations, in Proceedings of the th International Conference on Industrial Electronics, Control and Instruentation IECON, vol.1, pp , [1] D. Cox, J. Little, and D. O Shea, IDEALS, VARIETIES, AND ALGORITHMS An Introduction to Coputational Algebraic Geoetry and Coutative Algebra, Second Edition. Springer Verlag, [13] Joachi von zur Gathen and Jürgen Gerhard, Modern Coputer Algebra. Cabridge University Press, [14] S. Wolfra, Matheatica, A Syste for Doing Matheatics by Coputer, Second Edition. AddisonWesley, 199.