*Parallel Operation of Converters*

A balanced three-phase electric grid has been considered as a generated supply for the validation of the proposed converter operation. Mathematical form of voltages in a three-phase system can be written as:

$$\begin{aligned} \mathcal{U}\_{\mathcal{S}^a} &= ESin(wt) \\ \mathcal{U}\_{\mathcal{S}^b} &= ESin(wt - 120) \\ \mathcal{U}\_{\mathcal{S}^c} &= ESin(wt + 120) \end{aligned} \tag{1}$$

where *Uga, Ugb*, and *Ugc* are the phase voltages of the deliberated electric grid, while '*E'* and '*w'* represent the peak voltage and angular frequency of the input source.

In the proposed configuration, NPC works as a full-scale converter for the considered three-phase power feedback system, while others interleaved Vienna rectifiers work as a unidirectional power flow from generator to the DC link. In the planned system, the total output power of the generator divided by a factor of 1/(*n* + 1) for each converter. Where '1' in the denominator represent to the NPC, while 'n' represent the number of other parallel connected Vienna rectifiers.

The uppermost NPC converter, as depicted in Figure 3, have dual nature of operation (rectifier & inverter) while all other parallel-connected circuits only work as a unidirectional rectifier mode of operation. Phase voltages of NPC converter as a rectifier mode are mentioned in Table 2 while assuming ideal power switches with their following switching states:

$$d\_{\mathbf{x}} = \begin{cases} 1 \ S\_{\mathbf{x}1'}^{+} \ S\_{\mathbf{x}2}^{+} : ON & \mathcal{S}\_{\mathbf{x}1'}^{-} \ S\_{\mathbf{x}2}^{-} : OFF \\ 0 \ S\_{\mathbf{x}1'}^{-} \ S\_{\mathbf{x}2}^{+} : ON & \mathcal{S}\_{\mathbf{x}1'}^{+} \ S\_{\mathbf{x}2}^{-} : OFF \\ -1 \ S\_{\mathbf{x}1'}^{-} \ S\_{\mathbf{x}2}^{-} : ON & \mathcal{S}\_{\mathbf{x}1'}^{+} \ S\_{\mathbf{x}2}^{+} : OFF \end{cases} \tag{2}$$

Whereas, *x* = *a*, *b*, *c* for three-phase system. There are three kinds of switching states for a single leg 3L-NPC. Therefore, overall states for three-phase converter are '3<sup>3</sup> =27', which can be expressed as:

$$dL\_{\mathcal{S}^2} = \begin{cases} V\_1 & \text{if} \quad d\_{\mathcal{X}} = 1 \\ 0 & \text{if} \quad d\_{\mathcal{X}} = 0 \\ -V\_2 & \text{if} \quad d\_{\mathcal{X}} = -1 \end{cases} \tag{3}$$

**Table 2.** Configurations and state of 3L-NPC converter.


Moreover, for 'n' number of parallel-connected unidirectional Vienna rectifiers, the most e fficient topology, like t-type inverter with six active switches, has been used. Assuming the balanced grid supply voltage at the input of three-phase Vienna rectifier the terminal voltages with switching states and polarity of the phase current can be expressed as:

$$\begin{aligned} \mathcal{U}\_{\mathcal{S}A0} &= \frac{V\_{\rm dc}}{2} \text{sgn}(i\_{\rm s2\_n}) (1 - S\_a) \\\\ \mathcal{U}\_{\mathcal{S}B0} &= \frac{V\_{\rm dc}}{2} \text{sgn}(i\_{\rm s2\_n}) (1 - S\_b) \\\\ \mathcal{U}\_{\mathcal{S}\mathcal{C}0} &= \frac{V\_{\rm dc}}{2} \text{sgn}(i\_{\rm s2\_n}) (1 - S\_c) \end{aligned} \tag{4}$$

where *Sa*, *Sb*, and *Sc* are the switching states that switched between 1 and 0. Table 3 mentions eight di fferent switching conditions [32–34].


**Table 3.** Switching conditions for three-phase Vienna Rectifier.

Vienna rectifiers have boosting ability with the inductive filter at the input side of the circuit, which can be calculated while using Equation (5) [35].

$$L\_i = \frac{V\_{bus}}{8 \ast F\_{sw} \ast \Delta I\_{pp \text{ max}}} \tag{5}$$

where *Li* is the input inductor, *Vbus* is a dc bus voltage, *FSW* is the value of switching frequency, and Δ*Ippmax* is a maximum ripple current value.

Finally, the output of the rectifier terminals has connected with the common dc-link capacitors to obtain the regulated suppressed ripple content and for power flow control problems. The dc-link consists of two equally sized capacitors that were also used to provide a low inductive path for the turned-o ff current. The dc-link capacitance can be calculated by Equation (6).

$$\mathcal{C}\_{\rm DC} = \frac{2\pi S\_N}{V\_{\rm DC}}\tag{6}$$

where *SN* is a nominal apparent power of the converter, τ is the time constant that usually considered as less than 5 ms, and *VDC* is the total dc-link voltage [36]. Equation (7) determines the individual capacitor sizes.

$$\frac{1}{\mathcal{C}\_{\rm DC}} = \frac{1}{\mathcal{C}\_{\rm DC1}} + \frac{1}{\mathcal{C}\_{\rm DC2}} \tag{7}$$

having the same magnitude of current. Moreover, the dc-link voltage is to be kept constant for the input variations due to wind or load side dynamics. The overall control strategy of the proposed structure is demonstrated in the next section.

### **4. Control Strategy for the Proposed Converter**

Wind energy produces dynamic alternating energy that is not acceptable by appliances and for long-distance transmission. Therefore, some control schemes on directly connected wind generator side converter require application to achieve a regulated energy. Some aspects of the modified hybrid control strategy for the new system were taken into account, such as good power factor, low THD of input current, and equal sharing of power in each parallel-connected module. The detail of the converter control is discussed in the following subsections:
