*5.1. Introduction*

To investigate the association of the Vernier machine with the two types of rectifiers, we model the complete energy conversion system with Simulink. The turbine is simulated with (1). The PMVM is modeled using the classical relations of the synchronous machine, with some adaptations to take its particularities into account. The converters are represented by average models. With experiments on a test bed [13], we have shown that the overall model offers a good estimate of the energy produced by the turbine, but that it is less precise for converter and machine losses.

As a model of wind, we used a profile with a mean speed of 8 m/s and including sharp variations, so as to study the dynamic behavior of the conversion system when submitted to uneven wind gusts that commonly arise in urban areas (Figure 12).

We used the parameters given in Table 1 for the SPMVM. With an additional condition imposing a 120-V phase voltage under rated speed, we calculated the parameters of the machine's electrical model:


The rectifier associated with the generator can feed a DC link with a fixed or variable voltage (see Figure 12). The load can be an accumulator, possibly fed by a chopper, or an inverter connected to a local grid [14–17].

**Figure 12.** Wind profile used for simulation and the wind energy conversion system.

Considering the high synchronous reactance of the PMVM, it is better to use a voltage source at the output of the diode bridge rectifier. We thus obtain phase currents that are naturally sinusoidal in the machine and the total harmonic distortion of the stator voltage is lower than with a smoothing inductor.

#### *5.2. Vernier Machine PWM Rectifier Association*

With a PWM rectifier, we can drive the generator to maintain the operating point of the turbine close to the maximum power point locus (Figure 3) [18–22]. The torque vs. speed characteristic of the turbine being known, we can use the driving method shown in Figure 13.

**Figure 13.** Driving method with a PWM rectifier.

From a speed measurement, the torque set point necessary to keep the working point of the turbine on the maximum power point locus is deduced. This value is used to calculate the stator current set point in a rotating frame, and the resulting control action is transformed back to the stationary frame for execution. The working point trajectory of the turbine resulting from this control method is shown in Figure 14.

**Figure 14.** Working point trajectory of the turbine.

#### *5.3. Vernier Machine Diode Rectifier Association*

The diode bridge rectifier is an easy-to-use, low-cost, and sturdy converter, which is what makes it a very interesting choice for a domestic installation. On the other hand, as it is not a driven converter, it obviously does not make it possible to impose a working point on the turbine. Consequently, the energy conversion system must be designed as a whole to obtain a working point trajectory naturally, which will come close to the one shown in Figure 14.

## 5.3.1. Constant DC-Link Voltage

In the case of constant DC-link voltage, the voltage value must be chosen to reach a compromise. For a high voltage value, the diode bridge will conduct only for high-speed, low-power working points of the turbine. With a low voltage value, the working point of the turbine will settle at low speeds, which also correspond to low power.

Therefore, there is an optimal voltage value with which the working point will settle close to the maximum power point locus (Figure 15), thus leading to maximum energy recuperation (Figure 16).

**Figure 15.** Working points for generator power with several constant DC-link voltages.

**Figure 16.** Working points for generator energy with several constant DC-link voltages.

If we pay attention to the energy produced by this conversion system, depending on whether a diode bridge or a PWM rectifier is used, the difference is much more distinct for high wind speeds: with the diode rectifier, when the wind speed rises, the derivative of the generator torque with respect to the speed of rotation δTg/δΩ tends towards negative values. In that case, the working point becomes unstable and the turbine speed increases rapidly. The working point moves away from maximum power points and, if we refer to what is produced with a PWM rectifier, the turbine is clearly under-exploited. As a consequence, we cannot take advantage of the turbine's ability to work with strong winds.

To illustrate what happens under high wind speed conditions, we use a wind profile with the same shape as that in Figure 8, but with a mean value of 19 m/s, that is, the wind speed with which we determined the rated values. With a DC-link voltage of 48 V, which is the optimal value for an average wind speed of 8 m/s, the working point trajectory of the resulting turbine is shown in Figure 17, and the energy collected on the DC-link is divided by more than four compared to the energy produced with a PWM rectifier (Figure 18).

**Figure 17.** Working points for generator power with a diode bridge rectifier for an average wind speed of 19 m/s (DC-link voltage = 48 V).

**Figure 18.** Working points for generator energy with a diode bridge rectifier for an average wind speed of 19 m/s (DC-link voltage = 48 V).

## 5.3.2. Variable DC-Link Voltage

As a starting point, we use the power expression on the generator output. From there, we can obtain the expression of the DC bus voltage uDCM for which the generator associated with a diode bridge supplies maximum power to the DC bus. Using the notation from Figure 11 and with a unit power factor, we obtain:

$$P = 3 \cdot V \cdot I = 3 \cdot \left(\frac{\sqrt{2}}{\pi} \cdot \mathcal{U}\_{\text{DC}}\right) \cdot I \tag{16}$$

In addition, from Figure 8, diode rectifier, we can deduce the stator current (*I*):

$$I = \frac{-V \cdot R\_S}{R\_S^2 + X\_S^2} + \frac{\sqrt{\left(\frac{K\_{\text{g}} \cdot \omega}{N\_{\text{K}}}\right)^2 \cdot \left(R\_S^2 + X\_S^2\right) - \left(X\_S \cdot V\right)^2}}{R\_S^2 + X\_S^2} \tag{17}$$

*Ke* being the e.m.f. coefficient of the PMVM.

> Expression (16) thus becomes:

$$P = 3 \cdot \frac{V}{R\_S^2 + X\_S^2} \cdot \left( -V \cdot R\_S + \sqrt{\left(\frac{K\_e \cdot \omega}{N\_R}\right)^2 \cdot \left(R\_S^2 + X\_S^2\right) - \left(X\_S \cdot V\right)^2} \right) \tag{18}$$

Differentiating this expression with respect to the DC bus voltage UDC, we deduce that power *P* is maximum for *UDCM*:

$$\chi L\_{DCM}^2 = \frac{\left(\frac{K\_{\text{C}} \cdot \omega}{N\_R}\right)^2 \cdot \sqrt{R\_{\text{S}}^2 + X\_{\text{S}}^2} \cdot \left(\sqrt{R\_{\text{S}}^2 + X\_{\text{S}}^2} - R\_{\text{S}}\right)}{\left(\frac{2}{\pi}\right)^2 \cdot X\_{\text{S}}^2} \tag{19}$$

If we note stator angular frequency ω, as *XS*<sup>2</sup> = (*LS* · ω)<sup>2</sup> >> *RS*2, this expression becomes:

$$\mathrm{LI}\_{\mathrm{DCM}}^2 = \frac{\left(\frac{\mathrm{K}\_{\mathrm{c}} \cdot \mathrm{w}}{\mathrm{N}\_{\mathrm{R}}}\right)^2 \cdot \left(\mathrm{L}\_{\mathrm{S}} \cdot \mathrm{w} - \mathrm{R}\_{\mathrm{S}}\right)}{\left(\frac{2}{\pi}\right)^2 \cdot \mathrm{L}\_{\mathrm{S}} \cdot \mathrm{w}} = \left(\frac{\mathrm{K}\_{\mathrm{c}} \cdot \mathrm{w} \cdot \pi}{2 \cdot \mathrm{N}\_{\mathrm{R}}}\right)^2 \cdot \left(1 - \frac{\mathrm{R}\_{\mathrm{S}}}{\mathrm{L}\_{\mathrm{S}} \cdot \mathrm{w}}\right) \tag{20}$$

If the wind speed is 8 m/s, maximum turbine power is reached when N = 20 rpm. At this operating point, the stator frequency is 68 Hz and it can be found with (20) that the optimal DC bus voltage is about 48 V. If the wind speed reaches 19 m/s, the turbine's maximum power is obtained at nominal rotating speed, giving a stator frequency of 199 Hz. An amount of 48 V as the DC bus value is no longer optimal: the new optimal voltage UDCM is obtained from (20) and is about 96 V.

If we may vary the DC bus voltage, we can improve the operation of the energy conversion system with a diode bridge, by adapting this voltage to the rotating speed. However, this solution reduces simplicity, sturdiness, and does not bring significant improvement in power production. If we apply (18) with a voltage *UDC* = 96 V, the power is about 470 W, which is much lower than the 1470 W that the turbine can potentially produce when the wind reaches 19 m/s.

#### 5.3.3. Machine Optimization for a Diode Bridge Rectifier

Using (18) and (20), we can plot the evolution of the maximum power versus speed produced by the association of the generator and the diode bridge rectifier. Figure 19 is obtained by using this plot and the characteristic giving the maximum power produced by the turbine versus speed.

**Figure 19.** Evolution of the maximum power of the generator and the turbine.

There is a speed limit Nlim, beyond which the generator cannot transmit all the power produced by the turbine to the DC bus.

The generator's rated values are specified for a wind speed of 19 m/s. Thus, among the solutions plotted in Figure 10, we look for a generator giving a speed Nlim as near as possible to 54 rpm (speed of the turbine maximum power point for this wind speed in Figure 3).

We can deduce Nlim or here Ωlim expression from the relation giving the maximum power of the turbine as a function of the rotating speed:

$$P\_M = K\_T \times \Omega^3 = \left(\frac{1}{2} \cdot \mathbb{C}\_{p\_{opt}} \cdot \mathbb{S}\_T \cdot \rho \cdot \frac{R^3}{\lambda\_{opt}^3}\right) \times \Omega^3 = 9.4 \times \Omega^3 \tag{21}$$

*CPopt*: maximum power coefficient of the turbine: *CPopt* = 7.62/100; *ST*: turbine area "seen" by the wind: *ST* = 4.5 m2; *ρ*: air density: *ρ* = 1.25 kg/m3; *R*: turbine rotor radius: *R* = 0.9 m; λ represents the ratio: ( *R* · Ω/*Sw*), Sw being the wind speed; <sup>λ</sup>*op<sup>t</sup>* is the value of λ for which Cp = *CPopt* either <sup>λ</sup>*op<sup>t</sup>* = 0.26.

From (18) and (20), we can then deduce the expression of the maximum power at the generator output as a function of speed, and equate it to (22). The only coherent solution is:

$$
\Omega\_{\rm lim} \left( r d.s^{-1} \right) = \frac{K\_{\rm c} \cdot \sqrt{\frac{2}{2}}}{\sqrt{K\_T \cdot L\_S \cdot N\_R}} - \frac{R\_S}{2 \cdot L\_S \cdot N\_R} = \Omega\_1(1) - \Omega\_1(2) \tag{22}
$$

From this expression, we can make the following remarks:


To obtain an expression of *Ke* and *LS*, we use the following relations:

$$K\_{t'} = \frac{\mathcal{U}\_{\bar{r}} \cdot P\_{\bar{F}}}{\Omega\_{\bar{r}}} \tag{23}$$

with *Ur*: stator voltage at rated speed (*Ur* = 120 V), *PF*: power factor, Ω*r*: rated speed (rad·s<sup>−</sup>1).

$$L\_S = \frac{\sin\left(\cos^{-1}(P\_F)\right) \cdot \mathcal{U}\_r}{2 \cdot \pi \cdot I\_r \cdot f\_r} \tag{24}$$

*fr* rated frequency:

$$f\_r = \frac{N\_R \,\Omega\_r}{2 \cdot \pi} \tag{25}$$

*Ir* rated stator current:

$$I\_r = \frac{P\_r}{3 \cdot \mathcal{U} \cdot P\_F} \tag{26}$$

*Pr*: rated power.

Using relations (23) to (26), we find that the first term in (22) can be written:

$$\Omega\_1(1) = \sqrt{\frac{P\_r \cdot P\_F}{2 \cdot K\_T \cdot \Omega\_r \cdot \sin(\cos^{-1}(P\_F))}}\tag{27}$$

The application sets the rated values for power and rotating speed. Thus only the power factor remains as a variable.

The second part of Equation (22) can be written as:

$$\Omega\_1(2) = \frac{P\_r \cdot R\_S \cdot \Omega\_r}{6 \cdot l l\_r^2 \cdot P\_F \cdot \Omega\_r \cdot \sin(\cos^{-1}(P\_F))}\tag{28}$$

Ω1(2) being small in comparison with Ω1(1), we can approximate that stator resistance does not change from one machine to another. The stator voltage at rated speed *Ur* will remain unchanged too. We can then plot Ωlim as a function of the power factor (Figure 19).

Ωlim increases far more quickly when the power factor is greater than 0.9. If we want to obtain Ωlim = 54 rpm, we need a power factor equal to 0.928. However, the curve on Figure 20 is plotted without imposing any limit on the machine stator current.

Using (18), we calculate that a machine with a power factor of 0.928, a rated power of 1.5 kW, and a rated voltage of 120 V, would have a rated current of 4.5 A, but its stator current would reach 7.5 A at the working point. So as to not oversize the machine, among the solutions calculated in Table 1, we seek the solution that represents the best compromise between efficiency and power factor while maintaining a high torque to weight ratio (Figure 21).

**Figure 20.** Speed limit (Nlim) versus power factor (PF).

**Figure 21.** Power factor of feasible solutions versus efficiency.

In Table 2, the chosen solution is compared to the PMVM presented in Table 1.

**Table 2.** Characteristics of the solutions with the highest torque-to-weight ratio (analytical sizing).


5.3.4. Improvement Brought by the Generator with a High Power Factor

To simplify the energy conversion system, we work with a constant DC voltage at the rectifier output and, by using simulations, we try to evaluate the improvements that can be expected in energy production with a machine optimized for association with a diode bridge rectifier. In these simulations, we use, for every generator, the DC voltage calculated using (20) for speed Ωlim: this voltage seems to be the most appropriate for maximum energy production over the entire wind speed variation range. For the generator optimized for the torque-to-weight ratio: UDC = 48 V, for the generator optimized for association with the diode rectifier: UDC = 96 V.

With an average wind of 8 m/s, Figure 16 shows that the generator optimized by considering the torque/weight ratio can produce the power with a PWM rectifier. In comparison with this generator, we obtain 12% less power with the generator optimized for association with the diode bridge (Figure 22).

**Figure 22.** Energy at generator output for 8-m/s average wind speed.

With a wind profile centered on 19 m/s, the generator optimized for the diode bridge makes it possible to obtain working points nearer the maximal power points than with the machine with a high torque-to-weight ratio (Figure 23).

**Figure 23.** Power results obtained for 19-m/s average wind speed.

The energy at the generator output, despite being 30% lower than with a PWM rectifier, represents three times the energy we can obtain with a high torque/weight generator associated with a diode bridge (Figure 24).

**Figure 24.** Results obtained for energy for 19-m/s average wind speed.

With a high power factor generator, we can then maintain high-performance over a wide wind speed range.
