SPEEDAM 2006 International Symposium on Power Electronics, Electrical Drives, Automation and Motion
Application of the Optimal Minimization of the Total Harmonic Distortion technique to the Multilevel Symmetrical Inverters and Study of its Performance in Comparison with the Selective Harmonic Elimination technique Y. SAHALI *, and M. K. FELLAH * *
Intelligent Control & Electrical Power System Laboratory, University Djillali Liabes, Sidi-bel-Abbes, Algeria 1
2
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Abstract— Many interesting modulation strategies reported in literature are suitable to control multilevel power converters which presents several advantages over the conventional two-level counterpart. Since each of themes have its proper advantages and disadvantages, these strategies can be chosen according to certain performance aspects, which are required or desirable. The general performance specifications are: reducing harmonic distortion, reducing power losses and speeding transient response. In which, the Harmonic Elimination method and the Optimal Minimization technique are very important for reducing the harmonic distortion. In order to know which what control techniques the harmonic performance can be greatly enhanced or substantially improved, a detailed comparison between these two latest techniques with interpretation of its simulation results is the focus of this work.
Index Terms— Multilevel Converter, Multilevel Inverter (VSI), Cascade Inverter, Power Supplies, Pulse width Modulation, PWM techniques.
I. INTRODUCTION
C
URRENTLY, the majority of the electric drives used are ac current three-phase. These drives operate with variable speed where the traction constitutes a good example. To be able to consider the use of these drives with this operating mode, they should be equipped with variable voltage and variable frequency static converters. Several variable speed architectures accompanying the ac current drives exist. We will interest to the inverters. The first applications of these inverters exploited the twolevel structure. Because of the tendencies of important increasingly powers forwarded and the increasingly severe requirements on the harmonic depollution networks, this structure cannot be used in many fields such as the traction and the electricity distribution. Thus, these fields require the use of structures known as “Multilevel” which come to fill this gap and which are imposed in accentuated way. The merit to have created
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and put in the world the first multilevel structure, having the advantage of not using transformers, returns to A. NABAE and to its group in 1980 [1]. The general function of these multilevel structures is to synthesize a desired ac waveform from several levels of dc voltages. In this fact, they permit to overcome the voltage limitations of semiconductor devices in conventional two-level structure and to improve the quality of the output voltage waveform by reducing, for example, its Total Harmonic Distortion THD [1 to 7]. Numerous topologies have been founded in the published literature and widely used in many industrial applications, such as, static Var compensators, HVDC link, active filtering. The Multilevel inverter using cascaded-inverters with separated dc sources, hereafter called a “cascade multilevel inverter” appears to be superior to other multilevel structures in terms of its structure that is not only simple and modular but also requires the least number of components [6]. This modular structure makes it easily extensible for higher number of desired output voltage levels without undue increase in circuit power complexity. In addition, extra clamping diodes or voltage balancing capacitor are not necessary. II. CASCADE MULTILEVEL INVERTERSTRUCTURE AND MODELLING [8 to 10] Fig. 1 shows the single-phase structure of a cascade multilevel inverter. It consists of a series of H-bridge (single-phase full-bridge) inverter cells. Each inverter cell can generate, for the output voltage Vi ( i 1, 2, , S with S number of cells employed) three different values (levels), U i , 0 et U i by connecting the dc source to
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the ac output side by different combinations of the four devices. a U1
V1
U2
V2
V an
The maximum number of the line voltage levels can achieved 3 S , where S is the number of cells.
V S 1
US
Fig. 2 illustrates one of more generalized output voltage waveforms that can be synthesized by the cascaded multilevel inverter of fig. 1 and chosen for study.
VS
n Fig. 1 The single-phase structure of the multilevel cascade inverter
III. REVIEW OF DIFFERENT OPTIMIZATION TECHNIQUES FOR GENERALIZED MULTILEVELWAVEFORM
The output voltage Vi can be expressed as: Vi
U i f i1 f i 2
To improve the quality of the generalized output voltage waveform inverter by reducing its total harmonic distortion (THD), three possible techniques exists for its optimization. These techniques, which depend on choice of its parameters, are as follows:
(1)
f i1 , f i 2 are, respectively, the connection or switching functions of the upper switches ( K i1 , K i 2 ) of each cell, which define its states (switch on or off).
The optimization technique relative to the generalized waveform with equally widths or equally spaced steps, i.e. with constant distance between switching angles (Fig.3). In this case, the optimization is based on the step amplitudes (step heights).
The ac output voltage V an ( U c ) is, therefore, the sum of all the individual inverter outputs: S
V an
¦ Vi
V1 V 2 V S
(2)
i 1
Using becomes:
the
connection
functions,
equation
The optimisation technique relative to the
(2)
generalized waveform with the steps of equal height, i.e. equal amplitudes ( U1 U 2 U S U ). Optimization of this waveform, shown in Fig.4 and known as “regular staircase waveform”, is based on the step spaces or switching angles.
Uc S2
The optimisation technique relative to an arbitrary generalized waveform illustrates in Fig. 5. In this case, the optimisation is based on both heights and widths (spaces) of steps.
U S U S 1
U1
3S 2
ʌ-Į2 ʌ-Į1
ʌ-ĮS ʌ-Į (S-1)
Į2
Į1
Į (S-1) ĮS
U 2
S
… U S 1 U S Fig. 2 Chosen Generalized multilevel output voltage waveform.
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Į (4S)
… U 2 U1
(3)
For a three-phase system, the output of three identical structure of single-phase cascaded inverter can be connected in either wye or delta configuration. In this case, line voltage can be expressed in term of two phase voltages. For example, the line voltage V ab is the potential between phase a and phase b which can be expressed as: Vab Van Vbn .
U c V an U S 1
U 1 ( f11 f12 ) U S ( f S1 f S 2 )
Ȧt 2S
Uc
V. HARMONIC ELIMINATION TECHNIQUE The harmonic elimination modulation strategy for multilevel inverter is also referred in the literature to as “Optimized Harmonic Stepped Waveform OHSW” method. His objective is to reduce the total harmonic THD in the output voltage.
Zt
The basic concept of this reduction is to eliminate the specific harmonics, which are generally the lowest orders, with an appropriate choice of switching angles, by combining skilfully the idea of the Selective Harmonic Eliminated PWM (SHE PWM) for 3-level inverter control and based on the unipolar PWM switching scheme [15] to a generalized multilevel waveform synthesized from several level of dc voltages.
Fig. 3 A generalized waveform with equally width steps
Uc
Zt
Because of the symmetries of the generalized waveform shown in Fig. 2 only the odd harmonics exist. For this reason, the Fourier coefficients for the chosen generalized waveform, which are calculated as the simple sum of the coefficients of all its rectangular waves, are given by the following equation:
Fig. 4 A regular staircase generalized waveform
Uc
4 nS
an Zt
S
¦ U k cos(nD k )
(4)
k 1
Assuming a regular staircase waveform, this equation becomes:
an
Fig. 5 An arbitrary generalized waveform
IV. DIFFRENTS APPROACHES FOR IMPROVEMENT QUALITY OF THE GENERALIZED MULTILEVELVOLTAGE SPECTRUM Once the generalized waveform has been chosen, two options or two approaches are then possible to improve his quality: Either the switching angles are chosen to eliminate a certain number of harmonics. This approach, referred as “ Harmonic Elimination ” technique, results in THD reduction [11-12]; Or prefers determines an appropriate switching angles to minimize, most effectively possible, this total harmonic distortion. This causes to reduce, generally, the rate of each harmonic, without eliminating it inevitably. This approach is called “Optimal Minimization of the Total Harmonic Distortion technique OMTHD” [13-14].
4U nS
S
¦ cos(nD k )
(5)
k 1
where U is the amplitude of the dc source (dc voltage supply); n is an odd harmonic order; S is the number of dc sources or H-bridge cells; D k are the optimized harmonic switching angles. Evidently, these angles must satisfy the basic constraint: S D1 D 2 D 3 D S (6) 2 Amplitude of any harmonic can be obtained by setting equation (5), with respect harmonic, equal to prespecified value. But based on the performance criteria, this equation can be solved for S variables, D1 to D S , by: Either, equating S predominant lower frequency harmonics to zero in order to cancel it.
These two approaches will be briefly defined in the following sections.
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Or provide for voltage control with simultaneous harmonics elimination, i.e. equating (S-1) lowerorder harmonics to zero and assigning a specific value to the fundamental component.
Basically, the lowest odd harmonic components should be removed from a single-phase system, whereas in the three-phase system, they are the lowest non-triplen harmonic components that need to be eliminated. Thus, to eliminate S harmonics from the output voltage inverter, S switching angles need to be known. It implies, mathematically, that S equations formed from equation (5) are necessary. These equations can be written as:
After development and simplification, the THD of our periodic multilevel waveform (chosen generalized waveform), which present the odd quarter-wave symmetric characteristic, is given by this general formula:
¦
THD
¦
¦
¦
For the single-phase system: cos (D 1 ) cos (D 2 ) cos (D S )
2 ª j 1 S ª § ·º º · § S « ¨U 2 2U ¸» » ¨ U ¸ 2 «D U 2 U D k¸ j¨ j j i¸ » « 2 ¨ « 1 1 » S j 2 i 1 ©k 1 ¹ © ¹¼ » «S ¬ » 1 « 8 u 2 § S · » « ¨ U cos D ¸ » « k k¸ ¨ »¼ «¬ k 1 © ¹
(10)
0
cos (3D 1 ) cos (3D 2 ) cos (3D S )
Proof of this expression is given in [13].
0
(7)
cos (n D1 ) cos (n D 2 ) cos (n D S )
Suppose, for reasons of simplicity, the step of equal heights (regular staircase waveform : U1 U 2 U S U ), the THD is given by:
0
For the three-phase system: cos (D 1 ) cos (D 2 ) cos (D S )
ª § « ¨S 2 « 2 ¨ «S u © « 8 « « «¬
0
cos (5D 1 ) cos (5D 2 ) cos (5D S )
0
cos (n D1 ) cos (n D 2 ) cos (n D S )
THD
(8)
0
where
n is an odd number (for the single-phase system) and an odd number different from three and from its multiples (for the three-phase system). We can use these two systems if we need eliminate (S1) lower-order harmonics and control the fundamental component. In this case, the first equation of these systems becomes: SMS (9) cos (D 1 ) cos (D 2 ) cos (D S ) 4 h1 with M is the modulation index. SU The resolution of these two systems, which are nonlinear, is achieves by the algorithm of NewtonRaphson method uses in this paper or by any other iterative method of nonlinear systems resolution [11-12]. VI. OPTIMAL MINIMIZATION OF THE TOTAL HARMONIC DISTORTION (OMTHD) TECHNIQUE The basic idea for such a method, confirmed by recent work of [16], is to adjust and calculate switching angles in order to get the lowest output voltage THD. To minimize the THD, which is a measure of the closeness in shape of a waveform to its fundamental component, it is necessary that its partial derivative with respect to each switching angles equals zero. It is implied that the derivative partial of its square is also set to be zero.
·º (2k 1) D k ¸ » ¸» k 1 ¹» 1 (11) 2 » S · § » ¨ cos D k ¸ » ¸ ¨ »¼ ¹ ©k 1 S
2 S
¦
¦
What implies
THD 2
ª S § ·º « ¨S2 2 ( 2k 1) D k ¸ »» « 2 ¨ ¸ S S k 1 ¹ » 1 « u© 2 «8 » S § · « » ¨ ¸ D Cos ( ) k ¸ « » ¨ ©k 1 ¹ ¬« ¼»
¦
(12)
¦
Differentiating this equation to determine the partial derivatives and set these partial derivatives equal to zero, we obtains this general expression:
w THD 2 w (D C )
S
0 (2 C 1)
¦ Cos (D k ) k 1
ª «2 «¬ k
S
º (2k 1) D k S S 2 » Sin (D C ) »¼ 1
¦
(13) 0
where C 1, 2, , S Thus to minimize the output voltage THD, S switching angles, namely D1 , D 2 , , D S determined over one-quarter-cycle, need to be know. Whereas the other angles (from D S 1 until D 4 S ) result directly by symmetry (see Fig.2). Importantly, these S switching angles must, constantly, satisfy the condition (6). To have these angles, a system with S equations formed from (13) is necessary. The system obtained is a
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nonlinear of which the resolution is, also, done by the Newton-Raphson method. VII.
Table 2 Output Voltage THD as function of the number of switching angles for the OHSW 1
COMPARISON BETWEEN OMTHD AND OHSW WITH DIFFERENT ASPECTS
Number of switching angles per quarter-cycle
To measure the harmonic performance of the two techniques for purpose of comparison, several harmonic measures are possible. The total harmonic distortion THD is one of these measures, which evaluates the quantity of harmonics contents in the output waveforms and which is the popular performance index for power converters. To calculate the values of the THD which is chosen as basic criteria in this study, MATLAB is employed as programming tool.
D2
44.8428
(14)
* eliminate the third and the fifth harmonics from a five-level waveform are: D1
12.0000 ,
D2
48.0000
(15)
* eliminate the third harmonic from a five-level waveform and control its fundamental component are: D1
15.9562 ,
D2
44.0438
Table 1 Output Voltage THD as function of the number of switching angles for the OMTHD technique.
Number of switching angles per quarter-cycle
THD %
OMTHD
S
1
28,97
S
2
16,70
S
3
11,58
S
4
08,89
1
31,08
S
2
17,46
S
3
12,53
S
4
11,66
THD %
OHSW 2
S
2
17,00
S
3
14,32
S
4
09,70
For the simulation, the 5-level inverter with twoseparated dc sources and two H-bridge cells cascaded multilevel inverter (two switching angles per quartercycle) will be used as example. The principal characteristics of this inverter (Fig.6) which is considered for simulation are as follows: The total supply voltage is 400V. Thus, the amplitude supplied each H-bridge inverter is set to be 200V; The operating frequency is 50Hz.
(16)
The output voltage THD of different multilevel inverter waveforms calculated by MATLAB is obtained by substituting the switching angle values into equation (10). Its evolution, as function of the number of switching angles, is tabulated for the OMTHD technique in summary table 1, for the OHSW technique without voltage control (OHSW 1) in summary table 2 and for the OHSW with voltage control (OHSW 2) in summary table 3.
S
Number of switching angles per quarter-cycle
* optimize the THD of a five-level waveform are: 13.7610 ,
OHSW 1
Table 3 Output Voltage THD as function of the number of switching angles for the OHSW 2
After running our programs, some analytical results giving the appropriated switching angles are obtained. For instance, the values of switching angles (in degrees) to
D1
THD %
The simulation results are shown in Fig. 7, Fig. 8 and Fig. 9. They present frequency spectrums of the output voltage waveforms of multilevel inverter. a. SUMMARY-COMMENTS For the two techniques, the effect of the increase of the number of switching angles, i.e. of the number of inverter levels, on the output voltage THD has been foreseeable. First, we see clearly from the three summary tables (1, 2 and 3) that as the number of switching angles increases, the output inverter THD decreases. This improvement obtained in the output voltage quality result from the notable reduction of their THD.
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Reduction obtained in term of THD OMTHD comparatively with OHSW.
Table.4
by the
Fundamental : 427.9 V Number of switching angles per quarter-cycle
The 3rd harmonic : 04. 3 V The 5rth harmonic : 18.1 V The 7rth harmonic : 21.2 V
Fig. 7 Frequency spectrum of the output voltage of the five-level inverter controlled by OMTHD technique
*
Reduction obtained (%) OHSW 1 OHSW 2 to to OMTHD OMTHD
S
1
06,78*
/
S
2
04,35
01,76
S
3
07,58
19,13
S
4
23,76
08,35
this validates, very well, the result found in [17].
Nevertheless, the THD is not enough because it doesn’t constitute the only comparison criterion. The results must be confronted with those concerning the fundamental component and the remaining harmonics (especially the lowest).
Fundamental : 419.5 V
The examination of the different frequency spectrums obtained shows clearly that the first harmonics which are eliminated by the OHSW technique, have increasingly reduced amplitudes with the increase in the number of switching angles per quarter-cycle if the inverter is controlled by the OMTHD technique. The calculation of these amplitudes confirms that, these harmonics have relative amplitudes lower than 3 % of the fundamental. By more increasing the number of switching angles (increasing the output voltage inverter levels), these harmonics become increasingly negligible and approaching zero. Then, concerning the first harmonic remaining in the output voltage frequency spectrum of the inverter controlled by OHSW, it presents in the case of OMTHD technique the lower amplitude than those obtained by OHSW.
The 7rth harmonic : 37 V
Fig. 8 Frequency spectrum of the output voltage of the five-level inverter controlled by OHSW 1 technique
Fundamental : 427.9 V
In addition, the amplitude of the fundamental component relating to OMTHD technique is higher than those relating to OHSW 1 technique. The reduction of this component obtained while passing from OMTHD to OHSW 1 is given in table 5.
The 5rth harmonic : 29.9 V
Table 5 Fig. 9 Frequency spectrum of the output voltage of the five-level inverter controlled by OHSW 2 technique
Reduction of the fundamental component obtained while passing from OMTHD to OHSW 1
Number of switching angles per quarter-cycle
Table 4 specifies, for some switching angles, the rate of this reduction. From this last table, it is preferable for minimizing the output voltage THD of multilevel inverters to privilege the OMTHD technique over the OHSW technique.
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OHSW 1
S
1
05,74
S
2
01,95
S
3
02,76
S
4
02,53
Static Var Generator Using Cascade Multilevel Inverters . IEEE Transactions on Industry Applications, Vol. 33, No. 3, May/ June 1997.
VIII. CONCLUSION The comparison between the two modulation techniques, based on the output THD revealed the superiority of Optimal Minimization of the Total Harmonic Distortion OMTHD technique over Optimized Harmonic Stepped-Waveform OHSW technique. The study carried out shows that this approach is particularly interesting. It appeared very useful since it makes possible to act on the total harmonic distortion with an increased precision, i.e. with a notable reduction of the output signals harmonic contents as well as the number of levels increases, so that the resulting staircase output voltage which has the minimum harmonic content approaches a desired waveform. The advantage of the improvement obtained in terms of the total harmonic distortion, is unfortunately obtained to the detriment of the lowest harmonics disappearance. In spite of their presences, these harmonics present a very low amplitudes (
