**3.1 Analytical modeling of SRG**

For the developed analytical model, the following simplifying assumptions are considered:


Pole pitch τ, as given in the equation below, is the prime parameter to obtain the outer rotor diameter of the generator.

**Figure 5.** *Flow chart of SRG design process.*

*Wind Turbine and Synchronous Reluctance Modeling for Wind Energy Application DOI: http://dx.doi.org/10.5772/intechopen.103775*


The outer rotor diameter, Dr and stack length, L could be:

$$\begin{cases} D\_r = \frac{2P\tau}{\pi} \\ L = \tau \left(\frac{L}{\tau}\right) \end{cases} \tag{8}$$

where,*Te, L/τ, kc = 1.05, ks = 1.4, P, kd*, and kq are generator torque from the wind turbine, stack aspect ratio, Carter factor, saturation factor, number of pole pairs, the ratio of *d* � and *q* � axes inductance to magnetizing inductance, respectively. The parameters are defined in **Table 2**, while the saliency ratio defined as *Ld/ Lq = (kd* � *kq)/2kq*.

The design of the stator core geometry, i.e., the stator slot dimensions and rotor design in details the structure of rotor ribs and stator have been determined. The distance of the rotor air gap ribs from the shaft radius is designed. It also shows the separation of the edge of the ribs along the inner core radius of the rotor with an angle of ∂m.

The segments/points are selected and are interpolated to get the structure of the 6 poles rotor [16].

Every flux, and air barrier consists of trapezoid shape segment with a radial thickness and tangential thickness, and the end point angle other air gap, ∂m. The parameter is designed in such a way that to ensure required electromagnetic and the structure stability at high speed.

The maximum rotor tips mechanical end point angle, ∂m expressed in terms of number of flux barriers (qi), poles pair (P) and floating angle (β) is given as (9):

$$
\partial\_m = \left(\frac{\pi}{2P} - \beta\right) / \left(q\_i + \mathbf{1}/2\right) \tag{9}
$$

Here, the floating angle β assumed to be in between 0 to 10° (0 to π/18 rad). The total slot, *d* � and *q* � axes components of ampere turns can be expressed as:

$$n\mathbf{I}\_m = \sqrt{\left(n\mathbf{I}\_d\right)^2 + \left(n\mathbf{I}\_q\right)^2} \tag{10}$$

Here,

$$nI\_d = \frac{B\_m \pi (1 + k\_s) k\_c l\_g}{3\sqrt{2} (\mathbf{q} k\_w k\_d \mu\_o)}\tag{11}$$

and,

$$
\mathfrak{n}I\_q = \mathfrak{n}I\_d \sqrt{\mathcal{L}\_d/\mathcal{L}\_q} \tag{12}
$$

where, Im = Imax, and kw is stator slot winding factor, kw = 0.955. The conductor per slot turns n is one of the key parameter in the design such as stator resistance, leakage inductance and machine inductances. The resistance per phase R, is:

$$R = \frac{2(L + L\_{\epsilon}) \text{PJnq} \rho\_{\text{r}}}{\text{I}\_{\text{m}}} \tag{13}$$

where, *Le* is end winding length, *Le* = πτ/2, *J* is current density, and ρ*<sup>r</sup>* is copper resistivity at temperature of 120°*C*. The leakage inductance *Lls* is given as:

$$L\_{ls} = (c\_s + c\_e + c\_a)2\text{LPqn}^2\mu\_o \tag{14}$$

Here, the calculated slot permeance, *cs = 2.12535*, air gap coefficient, *ca = 0.2* and the end winding length coefficient, *ce = 0.00071817*. The magnetizing inductance *Lm*, for uniform air gap can be presented as:

$$L\_m = \frac{\mathsf{G}\mu\_o \pi LP \left(qk\_w n\right)^2}{\pi^2 \mathsf{l}\_\emptyset k\_c \left(\mathbf{1} + k\_r\right)}\tag{15}$$

Therefore, for double layer winding, the stator resistance, leakage inductance, magnetizing inductance, *d*�, and *q* � *axes* inductances are the function of *n*. By simplifying the calculation, they are expressed as in **Table 3**:

Using the d-q � axes rotor reference frame in **Figure 1(a)**, equations of the SRG in the transient state are written as below.

$$\begin{cases} V\_d = \mathcal{R}\_i I\_d + \rho \lambda\_d - aP\lambda\_q \\ V\_q = \mathcal{R}\_i I\_q + \rho \lambda\_q + aP\lambda\_d \\ V\_{ph} = \sqrt{{V\_d}^2 + {V\_q}^2} \end{cases} \tag{16}$$


*Wind Turbine and Synchronous Reluctance Modeling for Wind Energy Application DOI: http://dx.doi.org/10.5772/intechopen.103775*

At steady state, the voltage equation can be represented as:

$$V\_{ph} = n\sqrt{10} \tag{17}$$

The above Eqs. (16) are written utilizing the motor convention for the reluctance machine. When synchronous reluctance machine work in generating mode, it converts mechanical energy in to electrical energy, however, it requires capacitor over reactive p to magnetize their magnetic field paths for self-excitation. In motor mode, Iq and Id are of the same sign, while in generator mode they are of opposite sign. Referring to the current q and d-axes frame, motor operations are in the first and third quadrants, in other words the motoring mode is when Iq and Id < 0 or Iq and Id > 0, Te > 0, generating operations are in the second and fourth quadrants which means Id > 0 and Iq < 0 or Id < 0 and Iq > 0, Te < 0. **Figures 6** and **8**, show the reluctance torque developed by the SRG. Neglecting the effect of stator resistance, the torque equation is given as:

$$T\_{\epsilon} = \frac{3}{2} \mathbf{P} (\mathbf{L\_{d}} - \mathbf{L\_{q}}) \mathbf{I\_{d}} \mathbf{I\_{q}} = \frac{3}{2} \mathbf{P} (\lambda\_{\mathbf{d}} \mathbf{I\_{q}} - \lambda\_{q} \mathbf{I\_{d}}) \tag{18}$$

Since, the wind turbine rotates at low speed, gearbox are utilized to increase the speed of rotation of generator shaft. The swing-equation corresponding to the combined turbine generator system is given as:

$$
\rho o = \frac{1}{2f\_i} (T\_m - T\_e - 2\text{Bo}o) \tag{19}
$$

#### *3.1.1 Excitation capacitance and load modeling*

The inductive load RL, XL is connected to a capacitor bank in shunt at the stator terminals.

The equation which relates the stator current, load current, and terminal voltages are presented as follow:

**Figure 6.** *Torque verses current curves of the machine.*

**Figure 7.** *Flux lines and magnetic flux density distribution of the machines.*

$$\begin{cases} I\_{dc} = -I\_d - I\_{dL} \\ I\_{qc} = -I\_q - I\_{qL} \end{cases} \tag{20}$$

The excitation capacitance in rotor reference frame is as follows (21)

$$
\begin{bmatrix} I\_{dc} \\ I\_{qc} \end{bmatrix} = \begin{bmatrix} \rho \mathbf{C} & \rho \mathbf{P} \mathbf{C} \\ -\alpha \mathbf{P} \mathbf{C} & \rho \mathbf{C} \end{bmatrix} \begin{bmatrix} V\_{dL} \\ V\_{qL} \end{bmatrix} \tag{21}
$$

Whereas the voltages in rotor reference frame are given as:

$$\begin{cases} \rho \mathbf{V}\_{qL} = {}^{aPV\_{dL}} + \left( {}^{-I\_{q} - I\_{qL}} \right) \mathbf{\varprojlimits} \\ \rho \mathbf{V}\_{dL} = {}^{-aPV\_{qL} + (-I\_{d} - I\_{dL})} \mathbf{\varprojlim} \end{cases} \tag{22}$$

The R-L load model are obtained as (23)

$$\begin{cases} I\_{qL} = \int \frac{\mathbf{1}}{L} \left( V\_{qL} - I\_{qL} \mathbf{R}\_L + a \text{PLI}\_{dL} \right) dt \\\ I\_{dL} = \int \frac{\mathbf{1}}{L} \left( V\_{dL} - I\_{dL} \mathbf{R}\_L - a \text{PLI}\_{qL} \right) dt \end{cases} \tag{23}$$

Eq. (23) is obtained using a general balanced RL load model (V = RLI + LdI/dt). **Table 4** summarizes the calculated parameters, the designed parameters of the generator, and their performance.

#### *3.1.2 Resistive load condition*

For the resistive load case, the capacitor C, and the load, RL are connected in parallel to the stator terminals, as shown in **Figure 8**. Therefore, the impedance can be determined as:

$$Z = -B\mathbf{j}\mathbf{R}\_L + \mathbf{B}\mathbf{X}\_c \tag{24}$$

Where,

*Wind Turbine and Synchronous Reluctance Modeling for Wind Energy Application DOI: http://dx.doi.org/10.5772/intechopen.103775*


#### **Table 4.**

*Evaluated performance parameters and approximated quantity of 1 kW of SRG.*

**Figure 8.** *Variation of SRG electromagnetic torque with power angle, for Rs* 6¼ *0.*

$$B = \frac{X\_c R\_L}{R\_L^{\cdot^2} + X\_c^{\cdot^2}} \tag{25}$$

Which provides voltage at terminal

$$\begin{split} V &= -l\mathbf{Z} = \left(-\mathbf{B}\mathbf{X}\_{\boldsymbol{c}} + j\mathbf{B}\mathbf{R}\_{L}\right) \left(\mathbf{I}\_{d} + j\mathbf{I}\_{q}\right) \\ &= -\mathbf{B} \left(\mathbf{X}\_{\boldsymbol{c}}\mathbf{I}\_{d} + \mathbf{R}\_{L}\mathbf{I}\_{q}\right) + j\mathbf{B} \left(\mathbf{R}\_{L}\mathbf{I}\_{d}\mathbf{X}\_{\boldsymbol{c}}\mathbf{I}\_{q}\right) \end{split} \tag{26}$$

The voltage equations can be reduced to:

$$\begin{aligned} V\_d &= R\_s I\_d - X\_q \ I\_q = -B \left( X\_c I\_d + R\_L \ I\_q \right) \\ V\_q &= R\_s I\_q + X\_d I\_d = B \left( R\_L \ I\_d - X\_c I\_q \right) \end{aligned} \tag{27}$$

The values of minimum capacitance required can be determined for self-excited synchronous reluctance generator under resistive condition is by rearranging Eq. (27) and eliminate Id and Iq (**Figure 9**).

#### *3.1.3 Finite element analysis of SRG*

The present section includes the finite element validation of the proposed design of the SRG, as shown in **Figure 1(a)**. The finite element analysis (FEA) performance is evaluated as per the proposed design specifications. The maximum induced electromotive forces (emfs) in the stator winding of the 1 kW SRG with symmetric design are shown in **Figure 10**. The emf in the synchronous reluctance machine is given by Eqs. (16), which clearly indicates that if the effective q-axis flux is reduced, it leads to a decrease in emf. The results obtained through finite element analysis/ simulation with the excitation current of the machine is shown in **Figure 10**.

The peak values of the back emf produced at 1500 *rpm* corresponding to different magnetizing currents for the machine is shown in **Figure 11**. It is observed that, emf starts with zero voltage.

**Figure 6** shows that the performance of the machine in motor and generator modes. From **Figure 6**, it is clear the average torque is as a function of square of stator current till the current of the machine is 9A. But, after 9 A the different between (Ld-Lq) is approximately constant, hence, the variation of average torque is observed to be linear.

**Figure 8** represents the electromagnetic torque of the machine with variations of γ. Moreover, as the magnitude of average torque increases, the torque ripple also increases and viva-versa.

#### **Figure 9.**

*Synchronous reluctance generator equivalent circuit with capacitor C, resistive RL and inductive XL loads (V = Vd + jVq,I=Id + jIq, Λ = λ<sup>d</sup> + jλq).*

**Figure 10.** *Machine emf at given frequency.*

*Wind Turbine and Synchronous Reluctance Modeling for Wind Energy Application DOI: http://dx.doi.org/10.5772/intechopen.103775*

#### **Figure 11.**

*Peak emf curves as a function of current.*


#### **Table 5.**

*Analytical, and FEA results of the machines.*

**Table 5** provides the performance of self-excited SRG generator. From **Table 5** and **Figure 7**, the performance, flux line, and field density are observed. Overall, it can be said that the design of SRG is more robust and less costly. Since, cost effectiveness and robustness are major criteria for the suitability of generator for rural electrification application, SRG is more suitable for rural electrification. **Figure 7** provides magnetic field density distribution and flux lines of SRG generator.

**Table 5** provided that the performance comparison of torque, linkage inductance, magnetizing inductance, d and q-axes, and ripple torque percentage. It can be observed that the analytical and FEA similar with small deviation.

### **4. Conclusion**

The design and the modeling of synchronous reluctance generator for 6 poles, 1500 rpm, 1 kW, from wind turbine modeling specifications, are presented. The performance verses tip speed, mechanical power and torque verses turbine speed have been evaluated. The rotor design reducing q-axis inductance of this generator have been analyzed. Therefore, the torque ripple has been reduced. The relationship between generated emf voltage, and torque with the change of time are evaluated.

The effects of stator resistance on electromagnetic torque with variation of power angle have been considered. The design algorithm of reluctance generator are analyzed. Using finite element, the performances of the machine for field density, and flux line are also determined. With increase in current the performance of developed torque and generated voltage have been presented. The analytical and finite element results are evaluated and compared.
