A simple switching on-time calculation revision in multilevel inverter-space vector modulation to achieving extended voltage boundary operation

Bharatiraja C.1, R.K. Pongiannan2, Adedayo Yusuff3, Mohd Tariq4, Telugu Maddileti5, Tharwinkumar6 1,2,6 Department of Electrical and Electronics Engineering, SRM Institute of Science and Technology, India 1,3 Department of Electrical Engineering, University of South Africa, South Africa 4 Department of Electrical Engineering, Aligarh Muslim University, India 5 Sreenidhi Institute of Science and Technology, India


INTRODUCTION
In recent years, multilevel inverters (MLIs) have been widely used in the area of high-power medium-voltage applications. They offer a set of features that are well suited to high-voltage drive systems and power system applications such as HVDC transmission, reactive power compensation equipment [1,2]. The Neutral Point Clamped (NPC) has been mainly used MLI for motor control and PV applications [3][4][5][6]. To control the NPC-MLI, amongst various modulation techniques, SVPWM is an attractive candidate due to the following merits [7][8][9]. It directly uses the control variable given by the control system and identifies each switching vector as a point in complex space. It is useful in improving DC link voltage utilization, reducing commutation losses and Total Harmonic Distortion (THD) [10][11][12][13][14][15].
The SVPWM treats sinusoidal voltage as constant amplitude vector rotating at constant frequency with reference voltage vector V * , defined by V * =|V * |*e jwt , rotates around the centre of the space vector diagram at an angular frequency ω=2πfsys.The space vector diagram of any n-level inverter consists of six sectors, n 3 switching states and again each sector consists of (n − 1) 2 triangles [16]. Based upon the value of modulation index, it is classified as linear modulation and over modulation. Over modulation enhances the proper power utilization of installed capacity of voltage source inverter. The implementation of SVPWM for multilevel inverters is considered complex. This complexity is expected to increase further in the over modulation region due to the nonlinearity of this region. In the over modulation range, the trajectory of the reference vector is not completely circular, it is a combination of circular and hexagonal trajectory. The maximum output voltage can be increased up to 2Vdc/Π [16,17]. The algorithm proposed in Beig [7] to operate the inverter in the over modulation zone. The reference samples which are closer to the medium and large vectors are moved towards their respective nearest medium and nearest large vectors. This vector selection is based up on the angle correction factor [15]. In Seo et. at. [16] proposed a scheme for a threelevel inverter based on two-level SVPWM. The 3-level SVM diagram is divided into six two-level space vector diagrams [22,33].
McGrath et al., [17] explains the behavior of the key multilevel carrier based PWM methods for diode clamped, cascaded, and flying capacitors topologies in the over modulation region. Mondal [18] performs SVPWM based over modulation on a three-level NPC inverter. The on-time calculation equations differ for every triangular section. Due to increased computational complexity, it is cumbersome to extend this scheme to a n-level inverter. Amit kumargupta [9], the scheme easily determines the location of the reference vector and calculates on-times. Saeedifard [19] uses classification algorithm in over modulation range for SVPWM of a three-level NPC inverter and similar implementation is done by bharatiraja et.al., in [21]. It is not clear, how it can be extended to a n-level inverter. In over modulation [19-21, 29, 30] modify the trajectory of reference vector by using lookup tables. The author K.M. Kwon et.al. [20] extends his operation in to the over modulation region, the difficulty is the timing calculations which involve some trigonometric functions. In this paper, a simple SVPWM scheme for a 3-Level NPC-MLI was developed to operate the inverter in the entire modulation region. Figure 1 shows NPC topology and Figure 2 shows the SVPWM diagram of a 3-level inverter.

MODE OF OPERATIONS
In SVPWM, the three-phase voltage reference is given as a voltage reference vector V * [13]. The modulation index is defined as,

M
(1) The range from 0 to 0.907 is called as linear modulation and 0.907 to 1.0 is termed as over modulation range. In linear range the maximum obtainable voltage is 90.7% of the six-step value. It can be increased further by properly utilizing the DC link capacity through over modulation.

Linear Modulation ( 0 ≤ M.I< 0.907 )
The Figure 3 (a) shows sector-1 of space vector diagram, the tip P of the reference vector can be located in any of the 4 triangles (Δ10-Δ13). The objective here is to identify the triangle in which the point P is located. In the linear modulation the trajectory of the reference vector is entirely circular and it is always lies inside the hexagon.

Over Modulation-I (0.907 ≤ M.I< 0.9535)
The maximum allowable length of the reference vector happens when it touches the boundary of hexagon. Any further increase in the M.I causes the reference vector to be partially outside the hexagon which is termed as over modulation [16].

Over Modulation-II (0.9535 ≤ M.I< 1)
Once over modulation-I has reached the upper limit, over modulation-II becomes active. Under over modulation-II the essential feature is that the particular active voltage vector that is closest to the stator voltage reference vector is used gradually longer and longer time periods [18].

OPERATION OF 3-LEVEL INVERTER OVER MODULATION REGION 3.1. Over modulation-I
In the over modulation range shown in Figure 3 (b), the trajectory of the reference vector is not completely circular but a combination of circular and hexagonal trajectory. Sector identification and triangle determination are same for both the trajectories and they are differ only in on time calculation equations. The transition from circular trajectory to the hexagon trajectory is determined by the transition angle .For ≤ <Π/3 -, the vector moves on hexagon track and for remaining part of the sector on circular track, where is given by the (2) which tells that the value of is constant for a given modulation index.

3.1.1.Circular trajectory
For applying the SVPWM technique, firstly it is required to determine the sector which the voltage vector is within. For any given reference vector, the angle γ and its sector of operation Sk can be determined by using (3) and (4) respectively, After the sector identification the triangle determination is the most important one. Each sector in the 3-level inverter can be split into four triangles Δi, where i= 0,1,2,3. The four triangles can be split into two types for the easy determination of the triangle in the sector. The sub triangle can be categorized into  The triangle Δ10,  Δ11 , Δ13 , belonged to the type 1 and the triangle Δ2 , belong to type 2.Depend upon the triangle number the on-time calculations and switching pulse can be generated. The search of the triangle of the small vector (V * ) can be narrowed down by using two integers k1 and k2. They are defined by the coordinates (Vα, Vβ) as, K1 represents the part of the sector between the two lines joining the vertices, separated by distance h and inclined at120• with respect to α-axis. From the Figure4, K1=0 signifies that the point V * is below the line X1 X2. k1=1 signifies that point V * is between line X1 X2 and line X3 X5. K2 represents the part of the sector between the two lines joining the vertices, separated by distance h and parallel to α-axis. k2=0 signifies that the point P is between line X0 X3 and line X2 X4. k2=1 signifies that the point V * is above line X2 X4. Geometrically, the values of K1and K2 are an intersection of two rectangular regions which is either a triangle or rhombus as shown in Figure 4.
for k1=1 and k2=0 ,the common intersection is rhombus which is the combination of two triangles Δ1 and Δ2 . the triangle where the reference point is located can be determined by the slope comparison Vβi≤√3Vαi. If Vβi≤√3Vαiis true, then the point V * is within the triangle Δ1(type-1), otherwise it is within the triangle Δ2(type-2). However, these on-times are modified to compensate for the loss in volt seconds during the circular trajectory by introducing a compensation factor . For a given modulation index M, the value of the is constant and it is given by Modified on-time equations for type-1 triangle: Modified on-time equations for type-2 triangle:

3.1.2.Hexagonal trajectory
If the angle , satisfies the condition ≤ <Π/3 -, means that the reference vector follows the hexagonal trajectory. During hexagonal trajectory the coordinates of tip P of the vector are given in terms of angle and level n, as The sector judgment and triangle determination can be done in the similar manner of circular trajectory. The search of the triangle of the small vector (V * ) can be narrowed down by using two integers k1 and k2. They are defined as by knowing the values of k1 and k2 the value of the triangle number can be obtained by using the (20) and the coordinates of the small vector Vs are given by the (21) and (22).
on time calculation equations are similar to two-level inverter and are determined by using the (23)-(25_.

Over modulation -11
Once over modulation-I has reached the upper limit, over modulation-II becomes active. Switching in over modulation-II is characterized by a hold angle ∝ [21], defined as 10.5 1.05 1⁄ For ≤ <Π/3 -, the vector moves on hexagon track andthe on-time calculation is same as that during the hexagonal trajectory in over modulation mode Iand for remaining part of the sector i.e., ≤ <Π/3 -and 0≤ < and Π/3 -≤ <Π/3the vector is held at one of the large vector.In Figure 3 (c) the square dots at the points 4',3', 2' and 1' represent the reference vector samples in over modulation mode II. The samples 4',3'are closer to the large vector L1 and they are moved towards the large vector L1 and the samples 2',1', are closer to the large vector L2 and they are moved towards the large vector L2.
If the reference vector makes an angle, 0 6 ⁄ then the vector is held at large vector L1 as shown in Figure 3

SIMULATION AND EXPERIMENTAL RESULTS
The performance of the proposed SVM have been investigated and simulated by MATLAB 11.b for 12 switch NPC-MLI with 300V DC-link, two 100µF capacitor, 5kHZ switching frequency fed 1.5 HP squirrel cage 3-phase induction motor open loop v/f control drive. Further the simulation is extended to laboratory scale experimental power circuit is shown in Figure 6. The proposed over modulation SVPWM algorithm is programmed in Verilog Hardware Descriptive Language (VHDL) code and synthesized in minimum computational load using SPARTAN -III-3AN -XC3S400 FPGA family board [16]. The algorithm is tested on a 3-level NPC laboratory prototype inverter with 300V DC-link, two 100 µF capacitor, 5kHz switching frequency fed 1.

CONCLUSION
This paper proposes a simple SVPWM technique for calculating the on-times in the entire modulation region, the on-times calculation is based on on-time calculation for two-level SVPWM. A simple method of calculating on-times in the over modulation range is used, hence, a solution to complex equations and lookup tables are not required. The proposed algorithm can be applied to a variety of modulation values and it can be easily applicable for any level inverter and to all the types of multilevel topologies. The experimentation validating the proposed RPWM. & AI can be extended to a broad range of applications such as driverless vehicles.