**2. BDFRG Dynamic Model**

The BDFRG dynamic model is based on the theory of space vector, shown inEquations (1)–(3)[4]. The process of decoupling both the torque-producing and flux-producing current components is only possible in a special kind of frame transformation. This is mainly because of the bizarre structure of the BDFRG and the fact that the primary and secondary windings electrical quantities (current, flux, etc.) both have variant pole numbers and also different frequencies. The primary winding quantities are transformed into a general reference frame that rotates at a speed of *ω* while the secondary winding quantities are transformed into another frame that rotates at speed of (*ω<sup>r</sup>* − *ω*) [25]. The general frame speed is preferred to be the same as the primary winding supply frequency. The resulting dynamic model based on the dual frame transformation is described by Equation (1). The flux equation is presented in Equation (2). Finally, the electromagnetic torque and mechanical speed expressions can be stated as in Equation (3) [26,27].

$$v\_{dp} = r\_p \ i\_{dp} + \frac{d}{dt} \ \lambda\_{dp} - \omega \ \lambda\_{qp}$$

$$v\_{qp} = r\_p \ i\_{qp} + \frac{d}{dt} \lambda\_{qp} + \omega \ \lambda\_{dp}$$

$$v\_{dc} = r\_c \ i\_{dc} + \frac{d}{dt} \ \lambda\_{dc} - (\omega\_r - \omega) \ \lambda\_{qc}$$

$$v\_{qc} = r\_c \ i\_{qc} + \frac{d}{dt} \ \lambda\_{qc} + (\omega\_r - \omega) \ \lambda\_{dc} \tag{1}$$

$$\begin{aligned} \lambda\_{dp} &= L\_p \ i\_{dp} + L\_{pc} \ i\_{dc} \\ \lambda\_{qp} &= L\_p \ i\_{qp} - L\_{pc} \ i\_{qc} \\ \lambda\_{dc} &= L\_c \ i\_{dc} + L\_{pc} \ i\_{dp} \\ \lambda\_{qc} &= L\_c \ i\_{qc} - L\_{pc} \ i\_{qp} \\ T\_c &= \frac{3}{2} \ \newline \newline \left[ \lambda\_{dp} \ i\_{qp} - \lambda\_{qp} \ i\_{dp} \right] \end{aligned} \tag{2}$$

$$\begin{aligned} \left( \left[ f\_r + n\_g^2 \; \; \; \right]\_g \right) \frac{d \; \; \omega\_{rm}}{dt} &= n\_g \ \newline \left[ \rm n\_{\!\!\!} + n\_g^2 \ \; \; \! \; T\_c \end{aligned} \tag{3}$$

As described in Figure 1, the system consists of the BDFRG driven by WT The BDFRG is tied to the network by the power and control windings. The power winding is directly connected, while the control winding is connected via two converters: Machine Side Converter (MSC) and Grid Side Converter (GSC). A capacitance is placed between two converters. Each converter has its own controller in order to assure that it functions properly. Both the MSC and GSC controllers are shown in Figure 1. The MSC controller has two branches; one of them is connected to MPPT, this branch calculates the reference quadrature secondary voltage by using indirect field-oriented control. The other branch calculates the reference direct secondary voltage. These two reference voltages are transformed from dq frame to abc frame by using (θ*s*) to obtain appropriate gate signals from Sinusoidal Pulse Width Modulation (SPWM). On the other hand, the GSC controller has also two branches as illustrated in Figure 1. The two branches calculate reference quadrature and direct grid voltages. Then, they are transformed from dq frame to abc frame by using (θ*g*) to get appropriate gate signals from SPWM to adjust the DC link voltage and secondary reactive power on its reference values. *Energies* **2022**, *15*, x FOR PEER REVIEW 4 of 29

**Figure 1.** Block diagram of the studied BDFRG. **Figure 1.** Block diagram of the studied BDFRG.

as mentioned in the paper [28–30].

**4. The Proposed Crowbar Control Strategy** 

signal which lead to complicating the crowbar system.

#### **3. The Crowbar 3. The Crowbar**

The crowbar is one of the used solutions in the cases of severe disturbances such as heavy faults. In general, the crowbar provides a safe path for the heavy fault currents by short-circuiting the original path, subsequently damping the destroying fault currents and protecting the generator from the increased (overrated) currents. Basically, crowbar con-The crowbar is one of the used solutions in the cases of severe disturbances such as heavy faults. In general, the crowbar provides a safe path for the heavy fault currents by short-circuiting the original path, subsequently damping the destroying fault currents and protecting the generator from the increased (overrated) currents. Basically, crowbar

sists of a three-phase resistance called crowbar resistance. These resistances could be connected and disconnected by means of power electronics switches. Severe disturbances

the DC link voltage. In general, the activation (connection) process of the crowbar protection system is carried out only in the case of severe disturbances (heavy faults). After clearing the disturbance (fault), if the current and the DC link voltage are returned back to its allowable values, the crowbar will be deactivated (disconnected). Moreover, the generator returned back to its normal configuration. If the current and the DC link voltage are not returned to its allowable values, the activation of the crowbar protection system can be restarted. Moreover, the crowbar has been studied in a lot of fields by researchers

The crowbar activation and deactivation processes are important issues, there are many used techniques for crowbar activation and deactivation processes. In Ref. [29], an Adaptive Neuro Fuzzy Inference System (ANFIS) is used to produce the crowbar control

This work aimed to propose a simple crowbar control strategy. As shown in Figure 2, the used crowbar technique is the outer crowbar. The main components of the proposed crowbar control system are voltage measuring units for all phases, a control program and an automatic switch. The main aim of the crowbar control program is producing the control signal which would control the automatic switch. The main idea of the proposed strategy is based on continuously monitoring and measuring the per unit rms terminal voltage for each phase individually. According to the flowchart shown in Figure 3, if the measured rms voltage value, of each phase, is within the predefined voltage constraint limits (more than or equals to 0.7 p.u.), the control scheme would behave in this case as a normal steady-state operating condition, then the control program would set the output control signal to (1). Otherwise, if any one of the measured rms voltage values is lower than the consists of a three-phase resistance called crowbar resistance. These resistances could be connected and disconnected by means of power electronics switches. Severe disturbances lead to raising the currents in the generator above the allowable values, and also disturb the DC link voltage. In general, the activation (connection) process of the crowbar protection system is carried out only in the case of severe disturbances (heavy faults). After clearing the disturbance (fault), if the current and the DC link voltage are returned back to its allowable values, the crowbar will be deactivated (disconnected). Moreover, the generator returned back to its normal configuration. If the current and the DC link voltage are not returned to its allowable values, the activation of the crowbar protection system can be restarted. Moreover, the crowbar has been studied in a lot of fields by researchers as mentioned in the paper [28–30].
