*3.1. Coarsely Quantized Pulse Amplitude Modulation (CQ-PAM)*

Basically, the CQ-PAM involves applying a succession of only *M* different basic vectors per fundamental period, with all vectors having the same magnitude and being applied for equal-length time intervals. The selection of basic vectors relies on the criterion of proximity between the reference vector and the basic vectors. As seen in Figure 5a, for *M* = 12 and *l* = 2 only four different non-zero magnitudes are available, meaning very limited resolution. However, as can be noticed in Figure 5b,c, as the number of inverter modules or inverter voltage levels increases, the resolution of the CQ-PAM significantly rises—for an 18-pulse modular VSI it is 16 magnitudes and for a 12-pulse modular VSI with three-level modules it is 24 magnitudes.

For increased *M* and/or *l*, some neighboring magnitudes are close to each other, and thus it can be beneficial to use their combinations rather than stick to the same-magnitude principle. This option is illustrated in Figure 5b for the 12-pulse VSI with three-level modules: the sequence of vectors leading to the waveform shown in the lower graph is a succession of 24 (that is, 2 · *M*) basic vectors indicated in the upper graph; the magnitude of the shorter vectors is cos( *<sup>π</sup> <sup>M</sup>* ) =∼ 0.97 times that of the longer vectors. What is more, some magnitudes can be represented by more than *M* vectors. For instance, the 12-pulse VSI with three-level modules has seven magnitudes represented by 2 · *M* = 24 basic vectors. Again, this fact can be exploited in the CQ-PAM algorithm.

#### *3.2. Space Vector Pulse Width Modulation (SVPWM)*

To ensure virtually unlimited amplitude resolution of output voltage control, the PWM can be used whenever necessary or desirable. An economic solution might be the PWM discussed in [17,30] for 12-pulse rectifiers. However, this PWM technique offers only limited voltage control range and does not take advantage of the natural elimination of low harmonics in multipulse systems. Another candidate solution might be selective harmonic elimination PWM (SHE-PWM), which allows to decrease the switching frequency and remove a set of selected harmonics. Such a method was applied for parallel inverters driving a single load separated by line reactors [31]. However, this method requires precomputation of the appropriate PWM patterns, which is hardly possible in real-time [32,33]. This paper proposes an innovative space-vector PWM (SVPWM) based on barycentric coordinates.

In most cases considered in the literature, the output vectors are synthesized using the nearest three vectors (NTV) approach. A similar approach is adopted in this paper. In order to select the nearest basic vectors, the magnitude of reference vector (*V*ref) is compared with the available magnitudes of inverter basic vectors. The comparisons permit determining two neighboring magnitudes between which the reference vector is located. The closest two basic vectors of either magnitude are then searched for using a simple sorting algorithm and the following vector distance relationship

$$
\Delta V = \sqrt{(V\_{\text{refa}} - v\_a)^2 + (V\_{\text{ref}\beta} - v\_\beta)^2} \tag{16}
$$

The so obtained closest vectors can be considered vertices of a quadrangle ABCD, as illustrated in Figure 6. The quadrangle can be divided into four triangles. The tasks of the modulation include selecting the most appropriate of the triangles and computing the duty cycles of the three corresponding basic vectors.

**Figure 6.** Example reference vector and its representation by means of barycentric coordinates.

Both of the above indicated tasks can be achieved with the aid of barycentric coordinates [25]. Unlike the most popular methods of PWM computations, which are based on trigonometric functions, the use of barycentric coordinates avoids the related inconvenience. Consider the reference vector in Figure 6. It can be considered a point inside triangle ABC or triangle ACD. To fix attention, let us focus on the former triangle. The computation of duty cycles of the corresponding basic vectors is effectively means expressing the position of the reference vector as a linear combination of basic vectors. This is equivalent to expressing the Cartesian coordinates of a point inside a triangle by the barycentric

coordinates of that point. Thus, the coordinates of vector (*V*ref) in Figure 6 can be expressed by the coordinates of points A, B, and C:

$$
\begin{bmatrix} V\_{\text{refa}} \\ V\_{\text{refa}} \end{bmatrix} = \begin{bmatrix} A\_{\text{a}} & B\_{\text{a}} & C\_{\text{a}} \\ A\_{\text{f}} & B\_{\text{f}} & C\_{\text{f}} \end{bmatrix} \begin{bmatrix} N\_1 \\ N\_2 \\ N\_3 \end{bmatrix} \tag{17}
$$

where *N*1, *N*2, and *N*<sup>3</sup> are the barycentric coordinates of *V*ref, which can be calculated from

$$
\begin{bmatrix} N\_1 & N\_2 & N\_3 \end{bmatrix} = \begin{bmatrix} \frac{\triangle\_{V\_{\text{ref}} \text{AC}}}{\triangle\_{ABC}} & \frac{\triangle\_{V\_{\text{ref}} \text{AC}}}{\triangle\_{ABC}} & \frac{\triangle\_{V\_{\text{ref}} \text{AB}}}{\triangle\_{ABC}} \end{bmatrix} \tag{18}
$$

with the *ijk* symbols representing the areas of the small triangles defined inside triangle ABC by the vertices of the latter and *V*ref. Stated differently, the barycentric coordinates are equal to the normalized areas of their corresponding small triangles. These areas can be computed direct from the *αβ* coordinates of the appropriate basic vectors by

$$
\triangle\_{ijk} = \frac{1}{2} \cdot \left| \begin{bmatrix} v\_{i\alpha} & v\_{i\beta} & 1 \\ v\_{j\alpha} & v\_{j\beta} & 1 \\ v\_{k\alpha} & v\_{k\beta} & 1 \end{bmatrix} \right| \tag{19}
$$

A useful property of the barycentric coordinates is that their sum is equal to unity if they are calculated for a point inside a triangle (e.g., *V*ref in the triangle ABC or ACD in Figure 6), but is greater if the point lies outside the triangle (e.g., *V*ref in the triangle ABD or BCD in Figure 6). Thus, a uniform and effective method of finding a triangle or triangles containing a reference vector may be to calculate some candidate barycentric coordinates and then select the smallest (ideally 1).

As can be seen in Figure 6, the reference vector can be located inside two different triangles, and so some additional selection criterion is necessary to make the choice unique. The proposed algorithm selects the triangle for which the distance between the reference vector and the centroid (C ) is smaller. This criterion was adopted in order to minimize the occurrence of narrow pulses. The coordinates of the centroids are calculated as arithmetic means of the coordinates of vertices:

$$\mathbf{C}\_{\triangle} = \begin{bmatrix} \mathbf{C}\_{\triangle a} \\ \mathbf{C}\_{\triangle \beta} \end{bmatrix} = \begin{bmatrix} \frac{1}{3} \left( \boldsymbol{v}\_{i\alpha} + \boldsymbol{v}\_{j\alpha} + \boldsymbol{v}\_{k\alpha} \right) \\\ \frac{1}{3} \left( \boldsymbol{v}\_{i\beta} + \boldsymbol{v}\_{j\beta} + \boldsymbol{v}\_{k\beta} \right) \end{bmatrix} \tag{20}$$

The distance between *V*ref and the centroids is determined by Equation (16).

Although the proposed computational approach may seem rather complex for a system with only 64 space vectors, the practical target for the proposed method is modular VSI inverters with higher number of pulses *M* (notably, 18- and 24-pulse circuits) and using multilevel component inverters [1]. For such systems, the number of basic space vectors increases rapidly with *M* and the number of inverter levels (see Figure 5), as shown in Table 1.

**Table 1.** Number of space vectors for *M*-pulse modular VSI with *l*-level modules.


It is also important to note that with the increasing number of pulses and levels, the number of available magnitudes of inverter basic vectors also increases rapidly (e.g., 18-pulse and 24-pulse inverters with two-level modules have 16 and 67 output voltage magnitudes respectively and 12-pulse modular VSI with three-level modules has 23 output voltage magnitudes), rendering the CQ-PAM mode a true alternative to the PWM for steady-state operation, with the proposed SVPWM becoming a method for increasing the voltage control resolution, especially during transients.

### *3.3. Selection of Modulation Method*

The selection between CQ-PAM and SVPWM can be based on a variety of criteria, depending on the application (for instance the frequency criterion: SVPWM to be selected for lower fundamental frequencies, e.g., during the start-up, and CQ-PAM for higher fundamental frequencies). For the purpose of this study, the following magnitude proximity criterion is used; CQ-PAM is selected if the reference vector lies inside an annulus *Ai* defined by two circles—one inscribed in and the other described on a certain *M*-gon made of vectors of the *i*th magnitude (*Vi*); otherwise, SVPWM is chosen.

$$A\_i = \left\langle \cos\left(\frac{\pi}{M}\right) \cdot V\_i \,\, \, V\_i \right\rangle \tag{21}$$

The selection of the modulation method can be based on a decision diagram shown in Figure 7.

**Figure 7.** Control algorithm diagram.

#### **4. Simulation Results**

The proposed concept of output voltage control for modular VSI with coupled reactors has been verified using the PSIM11 simulation software and the Matlab environment. The simulated topologies included the 12-pulse inverter with two-level modules (Figure 1) and the 12-pulse inverter with three-level modules (Figure 8). The most important circuit and control parameters are given in Table 2.

**Figure 8.** Twelve-pulse modular VSI with coupled reactors and three-level modules.


**Table 2.** Circuit and modulation parameters used in simulation.

The operation of the considered modular VSI with CQ-PAM is illustrated in Figure 9. Characteristic values for output voltages and currents for this control mode are given in Table 3. A comparison of output voltage and current waveforms for two-level and three-level inverter modules is shown in Figure 10. To demonstrate how low the switching frequency is in relation to the output voltage frequency, the above figure also visualizes the leg voltage *u*1a waveform (scaled down to 25% in Figures 9 and 10). The output voltages shown in Figure 9 are the time waveforms corresponding to 12-pulse space-vector diagrams presented in Figure 5a. Later in this section a passage is illustrated between two steady-state operating points with the aid of SVPWM invoked during the transients (Figure 11).

**Table 3.** Number of switch commutations per period of the output voltage and the THD (of voltages and currents) for the coarsely quantized pulse amplitude modulation (CQ-PAM) presented in Figure 9.


**Figure 9.** Output voltage and current waveforms of the 12-pulse modular VSI with CQ-PAM control.

Figure 10 illustrates the maximum-magnitude and minimum-magnitude output voltages of 12-pulse modular VSIs with two-level and three-level modules, and the corresponding currents and leg voltages. As can be seen, the use of multilevel modules can increase the number of output voltage steps, so that the total harmonic distortion (THD) of the output voltage decreases (from 15.58% for two-level modules to 10% for three-level modules), and so does the THD of output currents (from 8.4% to 4.4% respectively). Moreover, for multilevel modules the dynamic range and resolution of output voltage magnitudes increases compared to the two-level modules. The minimum voltage for modular VSI with three-level modules is about four times lower than for two-level modules. The number of available non-zero voltage magnitudes also increases—from four in the case of two-level modules to twenty-three for three-level modules.

**Figure 10.** Example voltage and current waveforms of the 12-pulse modular VSI with two-level (**left**) and three-level (**right**) modules.

Figure 11 illustrates combined use of the proposed SVPWM and the CQ-PAM, ensuring smooth passage between different steady states. In this particular example the passage is between steady states at *m*<sup>a</sup> = 0.345 and *m*<sup>a</sup> = 0.488.

**Figure 11.** Example output voltage and current waveforms of the 12-pulse modular VSI during the passage between two different steady states.

### **5. Laboratory Test Results**

The laboratory tests have been performed using two prototypes of modular VSI with coupled reactors: one using two-level inverter modules, and the other equipped with three-level modules. The laboratory setup is presented in Figure 12. The most important circuit and control parameters are the same in both cases and identical to those used in the simulation (cf. Table 2). Measurements were taken by a Tektronix MDO4104B–3 oscilloscope. The control board contained a the two-core Texas Instruments digital signal processor TMS320C6672 and an Intel programmable logic device CYCLONE V. The coupled reactors shown in Figure 12 were designed for 30 kW (10 kW each) and rated frequency of 2.5 kHz. The numbers of turns of reactor coils are given in Table 2.

**Figure 12.** The laboratory set-up.

Figure 13 illustrates output phase voltages and currents, component inverter leg voltages, and the voltage across coil *N*<sup>B</sup> for all output voltage magnitudes and parameters listed in Table 2 (for CQ-PAM controlled VSI with two-level modules). The waveforms correspond to the simulation results shown in Figure 9. The laboratory test results match the results of the simulation, except for the voltage spikes appearing between the steps in the laboratory waveforms. This is a consequence of the use of dead time and the fact that several inverter legs are switched simultaneously (for voltage magnitudes other than the maximum one). However, it can be noted that the amplitudes of these spikes are not significant (smaller than *U*DC) and do not noticeably affect the current waveforms. The inverter leg voltages indicate the frequency with which the power switches commute (the waveforms confirm the data in Table 3). For the maximum available output voltage the switching frequency is equal to the fundamental output voltage frequency (in this case 1 kHz), and for the smallest CQ-PAM controlled output voltage it is equal to five times the output voltage frequency (5 kHz). One of the most important features distinguishing the coupled reactors from other magnetic integrating elements is their low rated power (related to the power of the overall system). To illustrate this feature, Figure 13 shows the voltage across the *N*<sup>B</sup> coil of the coupling reactor. The voltage is not only significantly lower than *U*DC, but may even assume values close to zero for some steps (depending on the output voltage magnitude). Therefore, the power transmitted through the reactor is only a fraction of the rated power of the inverter.

The hybrid modulation discussed in Section 3 is based on a combination of the CQ-PAM specific to the considered topologies, and the universal SVPWM—using the proposed computational approach based on barycentric coordinates. Figure 14 illustrates the phase voltages as well as phase-to-phase voltages for both modulation strategies. The voltages in Figure 14 correspond to *m*<sup>a</sup> = 0.67 for CQ-PAM, and *m*<sup>a</sup> = 0.62 for SVPWM. Figure 15 shows the voltages and currents obtained by SVPWM for two operating points: *m*<sup>a</sup> = 0.61 and *m*<sup>a</sup> = 0.42. The fundamental frequency was 1000 Hz for the higher output voltage and 600 Hz for the lower.

**Figure 13.** The output phase, leg, and coil *N*<sup>B</sup> voltages and phase currents of the 12-pulse modular VSI with two-level modules for CQ-PAM control.

**Figure 14.** Output voltages for CQ-PAM (**left**) and space-vector pulse width modulation (SVPWM) (**right**) control.

The quality of output voltage and current will improve significantly with the increase in the number of inverter levels, as exemplified in Figures 16 and 17, which provide an initial comparison between the modular VSIs with two-level and three-level inverter modules. What is more, increasing the number of levels significantly increases the amplitude resolution of the CQ-PAM and reduces the voltage stresses of individual transistors. Consequently, the use of multilevel component inverters in the modular VSIs with coupled reactors will contribute to increased attractiveness of the considered topology.

**Figure 15.** Example output voltages and currents for SVPWM control.

**Figure 16.** Output voltages and their spectra for the modular VSI with two-level (**left**) and three-level (**right**) modules.

**Figure 17.** Load currents and their spectra for the modular VSI with two-level (**left**) and three-level (**right**) modules.

The modularity of the considered topology is an important advantage in itself. It allows, inter alia, even distribution of the overall power transferred by the inverter between the reduced-power modules. This feature is illustrated in Figure 18, which shows the phase currents of the component inverters (*i*1a, *i*2a) and the resultant load currents (*i*a). As can be seen, the component inverter currents have the same amplitude close to half of the load current. The modularity also improves operational reliability of the considered topology, because it can continue working in case of a failure of one component inverter.

**Figure 18.** Phase currents of component inverters and the corresponding load currents of the modular VSI with two-level (**left**) and three-level (**right**) component inverters.

#### **6. Conclusions**

A hybrid approach to the output voltage control of modular VSI with coupled reactors has been proposed and discussed, including a novel coarsely quantized PAM and space-vector PWM based on the use of barycentric coordinates. Note that the use of these coordinates makes the SVPWM computations feasible and transparent even for such complex space-vector diagrams as those of the considered inverter topologies. The feasibility of the proposed solution has been verified by simulations and laboratory tests of the 12-pulse modular VSI with two-level and three-level component inverters. The use of multilevel inverter modules significantly improves the quality of output voltages and increases the attractiveness of the considered topology and the CQ-PAM, especially for application in high-speed motor drives (research into the latter application is planned for near future). It is also worth noting that the proposed solutions in modulation, although validated for particular inverter topologies, can be equally applicable to other topologies characterized by rich space-vector diagrams, including a variety of modular multipulse inverters.

**Author Contributions:** Conceptualization, K.J.S. and R.S.; methodology, K.J.S. and R.S.; software, K.J.S. and C.S.; validation, R.S., P.S., and C.S.; formal analysis, P.S. and J.N.; investigation, R.S., K.J.S., and C.S.; resources, K.J.S. and R.S.; data curation, R.S. and P.S.; writing—original draft preparation, K.J.S. and R.S; writing—review and editing, K.J.S., R.S., P.S. J.N., and C.S.; visualization, K.J.S., R.S., and C.S.; supervision, P.S., A.U., and J.N.; funding acquisition, P.S., J.N., and A.U. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by the LINTE<sup>2</sup> Laboratory, Gda ´nsk University of Technology, Grant DS 033784, and the Government of the Russian Federation, Grant 08–08.

**Conflicts of Interest:** The authors declare no conflict of interest.

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