A Direct Approach to the Positioning of the Reference Vector for Space Vector Modulation

Transcription

1 IEEE PEDS 2005 A Direct Approach to the Positioning of the Reference Vector for Space Vector Modulation M. Tavakoli Bina M. Samiei Moghadam Faculty of Electrical Engineering, Department of Electrical Power Engineering, K. N. Toosi University of Technology, K. N. Toosi University of Technology, Tehran-Iran Tehran-Iran tavakoli Skntu.ac.ir samieil352(2yahoo.com Abstract- Multilevel converters introduce the possibility of achieving higher power ratings as well as AC voltage quality improvement in the expense of increasing the power circuit and control complexity. Space vector modulation is the preferred vci C PWM technique in three-phase DC/AC multilevel converters. n-2 This paper concentrates on a specific matter of the space-vector + modulation analysis for diode-clamped multilevel converters, and VC2 v C 3 a proposes a direct positioning approach for the reference vector n-evel within the corresponding triangle. For an n-level converter, the v C M b developed method provides a (n-1)2 X 1 complex matrix that Vdc C3 MPDC b locates the exact place of the end of the reference vector. This T Converter method reduces the computation time, increasing the accuracy of positioning the switching instants for digital implementation + purposes. Optimized C programs were developed to confirm the Vc(n-2) C validity of the proposed method for microprocessor 1 implementation. + C C(n-1)lj0 Keywords-Multilevel AC/AC Converters, modulation, DC-link balancing, switching states Figure 1: General diagram of an n-level midpoint diode-clampe I. INTRODUCTION onverter. Multilevel converters provide significant benefits for high power medium and high voltage applications [1]. Compared to bi-level devices, series interconnection of switches can be then introduced a formula for calculating the two duty ratios, avoided, and quality of the output voltage is improved, avoiding the necessity of on-line computation of the However, the complexity of modulation strategy as well as the trigonometrlc function sin. Thus, it needs less computation number ofswitches is increased [2]. process, leading to reduction of sampling time as well as providing higher switching frequencies compared to Consider a standard space vector modulation strategy conventional SVM implementation. It is then extended to (SVM) for an n-level converter. One common approach to three-level cases in [3] for finding the triangle number in which compose the reference vector is the linear combination of three the end of the reference vector lies. This is also done by nearest switching instants to the reference vector during each establishing a classifier neutral network for every triangle. modulation cycle [3]. Nevertheless, it is essential for digital implementation of the SVM to locate the exact position of the This paper proposes a simple direct positioning technique end of the reference vector among (n-i)2 available triangles of that can be applied to a three-level converter as well as each sextant. This is called vector classification, which should extending the method to a multilevel case. An (n-i)2 x 1 be fast enough. Actually, the required computation time characteristic vector is introduced for a general n-level including the calculation and execution time affect the converter. Applying the reference vector to this characteristic minimum possible sampling time. Also, this can limit the vector detects its corresponding triangle in the space vector. maximum possible switching frequency, influencing the Then, the three adjacent switching instants are selected to track control system as well. the reference vector, while the on-durations of the three switching states are formulated for digital implementation. In [2], a counter propagation neural network is employed to reduce the complexity of implementation of SVM for a bi-level Further, a practical case is consildered where a conventional converter. The method seeks for the two nearest adjacent DC-link balancing strategy for a bi-level converter is switching instants to track and build the reference vector. It is developed. The work was arranged using a 150 MIPS digital /05/$ IEEE 1262

2 signal processor. Experimental results are provided to show the effectiveness of the proposed direct positioning approach. vq II. POSITIONING METHOD Assume (n-i) DC-link voltages of an n-level converter are 6 balanced (see Fig. 1). The space vector diagram of a three-level 3 converter is shown by Fig. 2, and that of a four level converter 4 by Fig. 3. These first sextants are divided into four and nine triangle regions (in general, (n-1)2 triangles), each consists of a number of switching instants to select three required basic vectors. // A rotation direction is considered for every triangle, either 4 3 clockwise for downward triangles or counter-clockwise for -) 4\ / upward triangles. Also, all lines within the first sextant are o numbered as shown in Fig. 4 for a three-level converter by --- labeling six line equations I-6. Now, if the three sides of each triangle are extended, then seven regions are distinguished as it is illustrated for an upward triangle in Fig. 5. Figure 2: Numbering and rotating direction for triangles of the first sextant of a three -level converter. When an arbitrary point (x, y) locates either above or below a line i, then i(x, y) gives either a positive or a negative value respectively. et us assign positive values of i(x, y) to one and negative values to zero. Then, all regions of Fig. 2 were vq evaluated using the explained rule and gathered in Table 1. et a be the equation of horizontal side of a triangle, 2 21 labeling the other two sides as b and c according to its direction. The first column gives the corresponding regions in g 4 Fig. 5, while columns 2--4 show the evaluated signs of these sides. Columns five and six show the signs of ab and ac Both products have negative signs when the point (x, y) situates 2 1 in either region four or region zero. Region four corresponds to 7 2 points inside an upward triangle, while region zero presents / / those of a downward triangle. Note that always the horizontal side stands for a, but clockwise and counter-clockwise 222 l directions select the inside region (O or 4). This actually paves TT 3 the way for positioning the reference vector. Based on this latter concept, we can establish two different corresponding places for positioning the instantaneous reference vector, while the triangle shape (upward and downward) chooses one region out of the two possible solutions. TABE I. igure 3: Numbering and rotating direction for triangles of the first sextant of a four -level converter. A POSSIBE REGIONS WHEN THE THREE SIDES OF A TRIANGE ARE EXTENDED. Ill. CHARACTERISTIC VECTOR Region Side 1 Side 2 Side 3 ab ac No.. b Base on the above considerations, we can now develop an 1I (n-1)2 x 1 matrix for an n-level converter using the rotation 1 0 direction of + triangles illustrated by Figs When the number of level increases, the needed calculations along with the processing time are elongated sharply. In fact, this 4 contradicts with the main purpose regarding the 1 benefits of o0o - - using multilevel converters _ Elements of this matrix are complex numbers, having real and imaginary parts in the form of columns five and six in Table 1. For example, assume an arbitrary point (x, y) within a three-level space vector diagram as Fig. 4. This point gives a Ol + + complex 4 x 1 vetor corresponding to the four triangles as 3 I 1263

3 ,3 I6 1 B-4 A+21/ i 24 (1) (V4-V/ 6)(A + 21/ ) (V4-11 6)(B-21/ ) -; 6)(B - -) f (V4-6)(A + 53/3) I5 -I4~~ ~~~~~~Z +/) (V X3)B2X,4,5 ~~~~~~~(V4 (Vq - Vi 3)(A + [3 / 3) (Vq - V3 3)(B -2 / Vi) This is the characteristic vector (CHV) for the positioning technique. Each row of (1) corresponds to a triangle, which the (V4-X / 3)(B - Vi / 2) + j (V -Vi / 3)A winner row is the one with both negative real and imaginary parts. Applying the same rule, matrices similar to that of (1) B-2VX/3 A+V3/3 can be further extended for higher-level converters as this is done for the simplified CHV of a four-level case. (Vq - / 6)(A + Vi / 3) (V4- X/ 6)(B - 3 /3) IV. SlMPIFYING THE CHV FOR IMPEMENTATION (Vq - X / 6)(B -2 / V3) (Vq - /3 6)A To implement with microprocessors, a method in [3] B - Vi 3 describes the on-duration of switching instants (duty ratios) (4) that is employed following the above positioning technique. This procedure is programmed by C, and is implemented by DSP. This is detailed by the next section to confirm the application of the above proposed method. Here we discuss another step towards the implementation with processors by simplifying the obtained position evaluation matrix given by (1). For every implementation purposes is necessary to optimize the time consumption by the processors. Using this simplification technique the needed time for finding the position of the instantaneous reference vector is minimized, enabling the systems to use higher switching frequencies. / 3 Assume the instantaneous reference vector is represented by (V, Vq) in the pq plane. Further, let us define two variables 4 / A and B based on the pq values of the three phase voltages as follows: {A: Vq 3-Vi'd Figure 4: VBV + Vi \2 Then, using (1), the CHV for a three-level case can be simplified as below: abeling six line equations for a three-level case. 3 ) 92 B-Vi ]fa+3/ 2 4 (V, - [3 4)(A+ F /2) + (V4-4)(B-;i2) \ (Vq -Vi/4)(B-Vi) 6(J-7i4)A 7 B-V312 ]A J W\ (3) Note that the computation time for evaluation of (3) is much Figure 5: illustration of the seven regions inside and outside of a lower than that of (1). Moreover, the simplified CHV for a triangle. four-level case was worked out as a 9 x 1 complex matrix: 1264

4 V. ASSESSMENT OF THE PROPOSED METHOD TABE II. EXPERIMENTA RESUTS FOR THE NON-SIMPIFIED POSITIONING TECHN1QUE Modulation index Frequency (Hz) Calculation Time (.is) et us consider a three-level DC/AC converter. The main goal is to supply a pre-determined AC voltage by this converter while the DC-link capacitor voltages need to be controlled and regulated. As the main scope of this paper is the proposed 0.5 o positioning technique, here we avoid detailing various aspects 1 of modulating techniques and control strategies Nevertheless, the application determines how the reference for each phase should be, including all the information required for frequency, phase and magnitude. Thus, the reference is seeking for the location of the instantaneous reference generation is part of the practical purposes, which here we vector. This includes the dq transformation together with the assume that, in general, sinusoidal waveforms with various regular method of the positioning technique. frequencies is needed to be built. Table II presents the summarized results taken from Hence, we assume that a voltage reference is modulated by enormous number of tests. For every modulation index, the the converter using the conventional method described in [2]. calculation time introduces the average time spent by the In brief, the reference waveform is transferred to the pq plane processor for all the undertaken tests. Four different by the well-known transformation matrix. A vector is made, modulation indexes have been considered, resulting in very which moves in the pq plane. For an arbitrary instant, this close calculation times about 1lIps. vector situates within a triangle, which can be approximated by Note that this calculation time can be compared to the DSP the two surrounding sides of the triangle. These two sides specification. The DSP speed cycle is about 6.67ns, which represent switching statuses, which are practically achievable gives 1650 cycles in average. This situation can be improved by the converter. by the simplification procedure explained by (2)-(4) to speed Moreover different techniques have been proposed to up the process. Next subsection provides the simplification calculate on-durations of these switching states (e.g. [2]). In technique in this regard. total, there exist many steps in developing control algorithm with microprocessors. These include making the needed 2 S reference waveforms, dq transformation, reference positioning, Considering (3), it can be seen that the whole positioning on-duration calculation, and generating gate pulses for the procedure is summarized to simple multiplication, plus and switches. minus operations. This is expected to reduce the calculation A. Experimental results time significantly. However, the pq transformation is still in place, providing the needed variables Vd and Vq for (2) to evaluate parameters A and B. A C program has been developed to manage the foregoing methodology. This program can be uploaded by a DSP with Another C program was developed to emulate the capability of 150 MIPS. The DSP is of fixed type, and two simplified positioning technique, described by (4). This types of experiments were arranged. program was then uploaded, and again numerous tests have been arranged. The program itself is much simpler than that of A C program was developed in which the whole stated the non-simplified algorithm of the previous subsection. steps were programmed. As we are interested specifically in the needed time for the proposed positioning method, the DSP Table III introduces the obtained results for the same tests timer measures only the amount of time spent to seek the exact perfoed by the simplified method. Note that again the location of the reference vector, positioning time together with the pq transformation time were considered for the average calculation time. 1) Direct usage ofchv First,, the without considering simplification TABE III. EXPERIMENTA RESUTS FOR THE SIMPIFIED POSITIONING Flrst, thechv wlthou conslderlg the slmplficatlontechnique algorithm was implemented. Assume a very usual reference vector, a sinusoidal waveform vector. Additionally, the Modulation index Frequency (Hz) Calculation Time (ps) modulation index as well as the frequency of the reference can be changed by the program The algorithm was then uploaded by the DSP system, and various test conditions considered. Table 2 shows the needed time for the processor to emulate the program. Note that only that part of the processing time is calculated that the processor 1265

5 The obtained results show the average calculation time for simplified methods, showing a much faster response for the each modulation index, which were gathered from many tests. proposed simplified method. It can be seen that the calculation time was shortened significantly as expected. This is about ten times shorter than REFERENCES that of the non-simplified method. Therefore, the effectiveness of the simplified CHV is confirmed. [1] P.F.Seixas, M.A. Servo Mendes, and P.D.Garcia, "A Space Vector PWM Method for Three-evel Voltage Source Inverter", IEEE Applied VI. CONCUSION Power Electronics Conference, APEC'2000, Vol. 1, pp , May A direct approach isprpoedfo[2] A. Bakhshai, H. R. Saligherad, and GJoos, "A Combined Artificial A direct approach is proposed for positioning the reference Neural Network and DSP Approach to the Implementation of Space in the dq plane. For an n-level converter, each sextant is Vector Modulation Techniques", IEEE-IAS Ann. Mtg., pp , divided into (n-1)2 x 1 triangles. Every triangle has one inside October region as well as six other outside regions. Defining some rules [3] A. Bakhshai and G.Joos, "Space Vector Modulation based on will lead to exact location of a point either inside or outside a classification in three-phase multi-level Voltage Source Inverter", IEEEtriangle. This rule is applied to all triangles, which guides us to IASAnnual Meeting, Vol. 1, pp , June a complex characteristic vector. Using the characteristic vector [4] P.F.Seixas, M.A. Servo Mendes, and P.D.Garcia, "A Space Vector makes.it which.triangle.. PWM Method for Three-evel Voltage Source Inverter", Fifth Annual makes it possible to distinguish whichn triangle have negative IEEE Conff on Applied Power Electronics Con. & Exposition 2000, real and imaginary parts as a way of positioning technique. APEC 2000,Vol.l,pp Once the desired reference is placed within a triangle, three [5] J. Pou, and et al, "Voltage Balancing Strategies for Diode-Clamped adjacent switching instants are selected and corresponding duty Multilevel Converters", IEEE PESC04, pp , June ratios are obtained. Different C programs, optimized for DSP [6] J. Pou, Broyoveich, and R. Pindado, "New Feedforward space vector implementation, are provided to confirm the validity of the PWM method to optain balanced ac output voltages in a three-level NPC proposed reference vector classification. The proposed converters", IEEE Trans. Ind. Electronics, Vol. 49,no. 5,pp , positioning technique was uploaded by the developed C October programs programs,andand various experimental tests were carfied carried out. The [7] S. Vukosavic and M. Stojic, "Rrduction of parasitic spectral componets of digital space vector modulation by real-time numerical methods" results confirm the effectiveness of both non-simplified and IEEE Trans. Power Electronics., Vol. 10, no. 1, pp , Janurary