Wind Turbine and Synchronous Reluctance Modeling for Wind Energy Application

Abstract

The Chapter discusses the turbine characteristics to design low-power rating generators. The low-power machine results in small wind turbines, hence distribution of power generators have attracted a growing interest from the demand, for remote and rural electrification. In renewable energy generation the design of the generator from the wind turbine is the most challenging part of the design. The generator specifications have been obtained from wind turbine models such as torque, speed and power. Based on these specifications the design of the generator with rating of 1 kW has been achieved. The turbine characteristics have been studied and various parameters of the designed machine are analyzed through analytical model and finite element analysis.

1. Introduction

In recent years, the evolution of renewable energy sources such as solar, hydroelectric, wind energy, biogas and geothermal energies have gained global attention. Rahim et al. [1] reported that an isolated self-stand generating strategy is desired to achieve electrification in distant areas. The conventional stand-alone energy generation system utilizes a synchronous generator that requires a direct current field excitation. Nagria et al. [2] identified that the convectional energy generating methods are not suitable for rural electrification. Instead, a self-stand and self-excited generating system will be more convenient for such applications, as depicted in Figure 1(b). For machines like synchronous reluctance machines and induction machine, self-excitation can be realized by the interconnection of excitation capacitors in star or delta across the stator end terminals, allowing them to be used as a self-stand generator [3, 4, 5]. In self excited induction generator, SEIG offers certain merits over a synchronous generator (convectional) as a source of separated to supply electric power, such as rugged structure, reduced size, low cost, low maintenance requirements and lack of DC source for excitation [5, 6, 7]. In spite of that, in SEIG the generated voltage frequency is influced by the loads and the capacitor bank. A self-excited or self-stand synchronous reluctance generator (SRG) has almost all the advantages of a self-excited induction generator. In addition, its rotor copper losses and the output frequency are not much influenced by the load, i.e., the load variations do not significantly affect the rotor speed and frequency of output voltage [8, 9].

Figure 1.

(a) Dq axes reference frame of the SRG. (b) Power generating system.

In literature, only few mathematical models have been built to determine the performance of a self-excited synchronous reluctance generator. Abdel-Kader et al. [10] attempted to develop an equivalent circuit for the SRG in the same manner as the SEIG. Yawei Wang et al. [5] also attempted analysis and modeling of self-excited synchronous reluctance generators. But, the load is limited to no load and resistive load conditions. Moreover, the effects of core losses and saturation effects are neglected, although the losses and saturation have significant effect on the performance of the machine. Rahim H. et al. [11, 12, 13, 14], investigated dq axes transformation based model and demonstrated its validity. T. F. Chan. et al. [12], develops a two-axis theory to model and analyze a three-phase self-excited reluctance generator which supplies to an isolated inductive load. However, in this survey, simple method to estimate the excitation capacitor is not included. Nevertheless, these research papers are mostly based on conventional salient rotor synchronous reluctance generators, i.e., no magnetic material or bridges in the rotor.

In this Chapter, wind turbine modeling, design and an analytical model in the dq rotor reference frame is developed as shown in Figure 1(a). In addition, the resistive load is considered to estimate the performance of the SRG. A new and simple method to estimate the minimum capacitance requirement for the resistive load is applied.

2. Wind turbine model

Among renewable energy sources in the world, the wind energy is more environmental and economical to generate electricity. In recent days, there is remarkable expansion in the use of wind energy generation. This result in the importance in developing in turbine and generators of maga power rating. For this reason the recent work is more based on the development of high efficiency, and low cost generators for remote area applications. The rating of the turbine is in the range of kilo watt. The turbine performance coefficient in terms of blade pitch angle and tip speed ratio (TSR) is given in Eq. (1) [15]:

$$C_p\left(\lambda ,\alpha \right)=C_1\left(\frac{C_2}{{\lambda }_1}-C_3\alpha -C_4\right)exp\frac{-C_5}{{\lambda }_1}+C_6\lambda$$E1

Letting,

$${\lambda }_1=\frac{\left({\alpha }^3+1\right)\left(\lambda +0.08\alpha \right)}{{\alpha }^3-0.028\alpha -0.035\lambda +1}$$E2

For various types of wind turbine, the values of C1 ∼ C6 are different. Figure 2 depicts the group power coefficient curves and tip speed ratio for C1, C2, C3, C4, C5 and C6 as 0.200, 119, 0.4, 5.5, 12.5, and 0, respectively. The turbine output power is given in (watts)

Figure 2.

Performance coefficient at pitch angle α = 0.

$$P_m=\frac{1}{2}{R}^3{\mathit{\pi \rho }}_a{C}_p\left(\lambda ,\alpha \right){\mathrm{V}}_w^3$$E3

The value of λ has been selected for optimal point of C_p at α = 0 as shown in Figure 2. Once the values of λ and C_p are found, the turbine speed and the radius are calculated from (3) and (4). The tip speed ratio can be determine from:

$$\mathrm{\Omega }=\raisebox{1ex}{$\lambda v_w$}\!\left/ \!\raisebox{-1ex}{$R$}\right.$$E4

The torque causing the rotation of the wind turbine shafts depends on the turbine rated power output and angular velocity. It can be expressed as:

$$T_m=\raisebox{1ex}{$P_m$}\!\left/ \!\raisebox{-1ex}{$\mathrm{\Omega }$}\right.=\frac{1}{2\lambda }{R}^4{\mathit{\pi \rho }}_a{C}_p\left(\lambda ,\alpha \right){\mathrm{V}}_w^2$$E5

The gearbox is utilized to transfer torque to the generator shaft rotating at a higher speed in the wind turbine. The wind turbine side to generator side gear ratio can be expressed as:

$$R_G=\raisebox{1ex}{$\omega $}\!\left/ \!\raisebox{-1ex}{$\mathrm{\Omega }$}\right.$$E6

Here, Ω and ω are the turbine and rotational speed of the generator, respectively. Eqs. (1) to (6) are used to design a wind turbine that can produce an output shaft power of 1 kW at a rated mean wind speed of 8 meter per second.

The designed parameters of the wind turbine is summarized in Table 1. The variation of power coefficient of the wind turbine for different blade pitch angle α is shown in Figure 2. Figure 3, shows the generated mechanical power at different wind velocity. From Figure 3, it is observed that as the wind speed increases, the rotational speed of the turbine also needs to be increased to extract maximum power out of the turbine. It is also observed that for the designed wind turbine, the maximum power attains near 32 rad/s for the average wind speed of 8 m/s.

Parameters	Values (Units)
TSR(λ)	6.4
Mean wind speed	8 m/s
Gearing ratio	4.9
Blade speed	32 rad/s
Performance coefficient limit	0.414
Mechanical transmission efficiency	90%
Mechanical power	1 kW
Mechanical torque of turbine rotor	31.36 Nm
Mechanical torque of generator shaft	6.4 Nm
Generator rated speed	157.08 rad/s
Turbine blade radius	1.6 m

Table 1.

Initial data and calculated values of the wind turbine parameters.

Figure 3.

Turbine power characteristics at pitch angle α = 0.

Figure 4, shows the torque causes mechanical rotation at different wind velocity. From Figure 4, it can be observed that as the wind speed increases, the rotational speed of the turbine also needs to be increased to extract maximum rotational torque of the turbine.

Figure 4.

Turbine torque characteristics at pitch angle α = 0.

3. SRG design algorithm

The sizing procedure of the proposed generator starts with assigning the initial key parameters of wind turbine such as, speed and maximum torque. These assigned parameters are used in the calculations of magnetic, geometric and electric parameters in together with the analytical model of the generator.

The design of the generator starts with the precondition design output parameters such as stator geometry, rotor outer diameter etc. A new repetition with revised assigned parameters will be done, if the geometries such as stack length and stator outer diameters cannot satisfy the design parameter requirements. Otherwise, the estimated magnetic, electric and geometrical parameters are used as the design parameters. In other words, stator geometry, inductances, winding specifications, maximum torque, saliency ratio, etc., are determined if the required conditions are satisfied. Finite element (Ansys Electronic Desktop) software is used to analyze the generator’s performance related to the output functions such as torque quality, maximum developed torque, and the magnetic properties such as, magnetic field H and magnetic field B. The process ends if the Ansys Electronic Desktop software results satisfy the design requirements. Otherwise, the process is repeated by updating the assigned parameters such as tip speed ratio, efficiency coefficient, air mass density, current and magnetic loading, pole pitch to air gap ratio and stack aspect ratio to obtain the proper size (Figure 5 and Table 2).

Figure 5.

Flow chart of SRG design process.

The design constraint need to be satisfied are:

The power rated 1 kW, with electromagnetic torque ≥6.4 Nm, maximum torque ripple ≤9%, maximum back-emf ≥ 100 V.

3.1 Analytical modeling of SRG

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

• The time harmonics in current and space harmonics in air gap flux are neglected

• The core loss resistance is assumed to be constant and has no effect on excitation.

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

$$\tau =\sqrt{\frac{T_e{k}_d^2{\mu }_o}{B_m^2{P}^2\left(k_d-k_q\right)\sqrt{L_d/L_q}{k}_c\left(1+k_s\right){\mathrm{l}}_{g}L/\tau }}$$E7

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

$$\left\{\begin{array}{l}{D}_r=\frac{2\mathit{P\tau }}{\pi }\\ L=\tau \left(\frac{L}{\tau }\right)\end{array}\right.$$E8

where, T_e, L/τ, k_c = 1.05, ks = 1.4, P, k_d, and k_q 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 L_d/L_q = (k_d − k_q)/2k_q.

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 (q_i), poles pair (P) and floating angle (β) is given as (9):

$${\partial }_m=\raisebox{1ex}{$\left(\frac{\pi }{2P}-\beta \right)$}\!\left/ \!\raisebox{-1ex}{$\left(q_i+1/2\right)$}\right.$$E9

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:

$${\mathit{nI}}_m=\sqrt{{\left(n{\mathrm{I}}_d\right)}^2+{\left(n{\mathrm{I}}_q\right)}^2}$$E10

Here,

$${\mathit{nI}}_d=\frac{B_m\pi \left(1+k_s\right)k_c{\mathrm{l}}_{\mathrm{g}}}{3\sqrt{2}\left(\mathrm{q}{\mathrm{k}}_{\mathrm{w}}{\mathrm{k}}_{\mathrm{d}}{\mu }_{\mathrm{o}}\right)}$$E11

and,

$${\mathit{nI}}_q={\mathit{nI}}_d\sqrt{L_d/L_q}$$E12

where, I_m = I_max, and k_w is stator slot winding factor, k_w = 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:

$$R=\frac{2\left(L+L_e\right)\text{PJnq}{\rho }_{\mathrm{r}}}{{\mathrm{I}}_{\mathrm{m}}}$$E13

where, L_e is end winding length, L_e = πτ/2, J is current density, and ρ_r is copper resistivity at temperature of 120°C. The leakage inductance L_ls is given as:

$$L_{\mathit{ls}}=\left(c_s+c_e+c_a\right)2\mathrm{LPq}{\mathrm{n}}^2{\mu }_o$$E14

Here, the calculated slot permeance, c_s = 2.12535, air gap coefficient, c_a = 0.2 and the end winding length coefficient, c_e = 0.00071817. The magnetizing inductance L_m, for uniform air gap can be presented as:

$$L_m=\frac{6{\mu }_o\mathit{\pi LP}{\left({\mathit{qk}}_{w}n\right)}^2}{{\pi }^2{\mathrm{l}}_{\mathrm{g}}{k}_c\left(1+k_s\right)}$$E15

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.

$$\left\{\begin{array}{l}{V}_d=R_s{I}_d+{\mathit{\rho \lambda }}_d-\mathit{\omega P}{\lambda }_q\\ V_q=R_s{I}_q+{\mathit{\rho \lambda }}_q+\mathit{\omega P}{\lambda }_d\\ V_{\mathit{ph}}=\sqrt{{V_d}^2+{V_q}^2}\end{array}\right.$$E16

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

$$V_{\mathit{ph}}=n\sqrt{10}$$E17

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, I_q and I_d 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 I_q and I_d < 0 or I_q and I_d > 0, T_e > 0, generating operations are in the second and fourth quadrants which means I_d > 0 and I_q < 0 or I_d < 0 and I_q > 0, T_e < 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:

$$T_e=\frac{3}{2}\mathrm{P}\left({\mathrm{L}}_{\mathrm{d}}-{\mathrm{L}}_{\mathrm{q}}\right){\mathrm{I}}_{\mathrm{d}}{\mathrm{I}}_{\mathrm{q}}=\frac{3}{2}\mathrm{P}\left({\lambda }_{\mathrm{d}}{\mathrm{I}}_{\mathrm{q}}-{\lambda }_q{\mathrm{I}}_d\right)$$E18

Figure 6.

Torque verses current curves of the machine.

Figure 7.

Flux lines and magnetic flux density distribution of the machines.

Figure 8.

Variation of SRG electromagnetic torque with power angle, for R_s ≠ 0.

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:

$$\mathit{\rho \omega }=\frac{1}{2J_i}\left({\mathrm{T}}_m-T_e-2\mathrm{B}\omega \right)$$E19

3.1.1 Excitation capacitance and load modeling

The inductive load R_L, X_L 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:

$$\left\{\begin{array}{l}{I}_{\mathit{dc}}=-I_d-I_{\mathit{dL}}\\ I_{\mathit{qc}}=-I_q-I_{\mathit{qL}}\end{array}\right.$$E20

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

$$\left[\begin{array}{l}{I}_{\mathit{dc}}\\ I_{\mathit{qc}}\end{array}\right]=\left[\begin{array}{cc}\mathit{\rho C}& \mathit{\omega PC}\\ -\mathit{\omega PC}& \mathit{\rho C}\end{array}\right]\left[\begin{array}{c}{V}_{\mathit{dL}}\\ V_{\mathit{qL}}\end{array}\right]$$E21

Whereas the voltages in rotor reference frame are given as:

$$\left\{\begin{array}{l}\rho V_{\mathit{qL}}=\raisebox{1ex}{$\omega {\mathit{PV}}_{\mathit{dL}}+\left(-I_q-I_{\mathit{qL}}\right)$}\!\left/ \!\raisebox{-1ex}{$C$}\right.\\ \rho V_{\mathit{dL}}=\raisebox{1ex}{$-\omega {\mathit{PV}}_{\mathit{qL}}+\left(-I_d-I_{\mathit{dL}}\right)$}\!\left/ \!\raisebox{-1ex}{$C$}\right.\end{array}\right.$$E22

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

$$\left\{\begin{array}{l}{I}_{\mathit{qL}}=\int \frac{1}{L}\left(V_{\mathit{qL}}-I_{\mathit{qL}}{R}_L+\omega {\mathit{PLI}}_{\mathit{dL}}\right)\mathit{dt}\\ I_{\mathit{dL}}=\int \frac{1}{L}\left(V_{\mathit{dL}}-I_{\mathit{dL}}{R}_L-\omega {\mathit{PLI}}_{\mathit{qL}}\right)\mathit{dt}\end{array}\right.$$E23

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

Dimension of the Generator	Values
Bin	[0.5, 1] T
kd	[0.6, 1]
kq	[0, 0.4]
L/τ	[0.3, 6]
vw	[5, 25]m/s
ku	[0.4, 0.7]
J	[4, 9] A/mm^2
S	36
P	3

Table 2.

Assigned parameters.

Parameters	Magnitude
Rs	0.92x10^−3 (n)^2 Ω
Lls	1.4x10^−6 (n)^2 H
Lm	2.99x10^−5 (n)^2 H
Ld	2.93x10^−5 (n)^2 H
Lq	2.993x10^−6 (n)^2 H

Table 3.

Parameters expression.

Parameters	Quantity	Parameters	Quantity
Lm	17.24 mH	Dr	104 mm
Ld	16.9 mH	Din	105 mm
Lq	1.73 mH	Do	180 mm
Lls	0.80 mH	h	13 mm
Im	20.6 A	n	24
L	40 mm	nIq	453.6 AT
lg	0.5 mm	nId	195.2 AT
ku	0.5	Eph	100 V
Rs	0.53 Ω	nIm	493.8 AT

Table 4.

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

3.1.2 Resistive load condition

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

$$Z=-{\mathit{BjR}}_L+\mathrm{B}{\mathrm{X}}_{\mathrm{c}}$$E24

Where,

$$B=\frac{X_c{R}_L}{{R_L}^2+{X_c}^2}$$E25

Which provides voltage at terminal

$$\begin{array}{l}V=-\mathit{IZ}=\left(-{\mathit{BX}}_c+{\mathit{jBR}}_L\right)\left(I_d+{\mathit{jI}}_q\right)\\ =-B\left(X_c{I}_d+R_L{I}_q\right)+\mathit{jB}\left(R_L{I}_d‐X_c{I}_q\right)\end{array}$$E26

The voltage equations can be reduced to:

$$\begin{array}{l}{V}_d=R_s{I}_d-X_q{I}_q=-B\left(X_c{I}_d+R_L{I}_q\right)\\ V_q=R_s{I}_q+X_d{I}_d=B\left(R_L{I}_d-X_c{I}_q\right)\end{array}$$E27

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 I_d and I_q (Figure 9).

Figure 9.

Synchronous reluctance generator equivalent circuit with capacitor C, resistive R_L and inductive X_L loads (V = V_d + jV_q, I = I_d + jI_q, Λ = λ_d + jλ_q).

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.

Figure 10.

Machine emf at given frequency.

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 11.

Peak emf curves as a function of current.

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 (L_d-L_q) 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.

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.

Parameters	Analytical	FEA
Te	6.4 Nm	6.493 Nm
Lls	0.80 mH	0.78 mH
Lm	17.42 mH	18 m H
Ld	16.9 mH	17.4 mH
Lq	1.73 mH	1.65 mH
Tr	—	9%

Table 5.

Analytical, and FEA results of the machines.

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.

Appendices and nomenclature

ω	Generator rated speed (mechanical)
ωe	Generator rated speed (electrical)
ku	Fill factor
Bm	Air gap flux density
kd	Ratio of Ld/Lm
kq	Ratio of Lq/Lm
Ls/τ	Stack aspect ratio
Vw	Mean wind speed
J	Current density
ρa	Air density
P	Number of pole pairs
Ω	Turbine tip speed
Eph	Emf induced per phase
Gr	Gear ratio
λ	Tip speed ratio
Do	Stator air gap diameter
Din	Stator yoke diameter
Dr	Rotor outer diameter
Rs	Phase resistance
Rt	Wind turbine radius
Lls	Stator leakage inductance
Id	d-axis current
Iq	q-axis current
Xd	d-axis reactance
Xq	q-axis reactance
n	Number of conductor turns per phase
lg	Air gap length
λq	q-axis flux
λd	d-axis flux
Lq	q-axis inductance
Ld	d-axis inductance
L	Active stack length