3. Uncertainties model

Introducing a variable δ [rad] describing the twist of the shaft, leads to the following equation

δ þ Krδ (15)

<sup>T</sup> (17)

(18)

δ þ a14β (19)

βin (22)

Tg,r (23)

v (25)

<sup>V</sup>\_ <sup>t</sup> <sup>¼</sup> Vt (24)

Tg (20)

(21)

(16)

Tsr <sup>¼</sup> Dr \_

Ng , \_

In the above equations, Dr represents the damping and Kr denotes spring coefficient, angular speed of rotor is defined by ωr, the angular speed of generator represent by ωg, Ω<sup>r</sup> and Ω<sup>g</sup> are used to define the default shaft angle at the rotor and the default shaft angle at the generator,

As it was discussed in previous section, for design a controller a linear model of the system is

input ¼ V βin Tg,r

output ¼ ω<sup>r</sup> ω<sup>g</sup> Pe

<sup>ω</sup><sup>r</sup> � <sup>a</sup> � Dr JrNg

<sup>δ</sup> <sup>¼</sup> <sup>ω</sup><sup>r</sup> � <sup>ω</sup><sup>g</sup>

Vt � <sup>p</sup><sup>1</sup> <sup>þ</sup> <sup>p</sup><sup>2</sup> p1p<sup>2</sup>

Ng

<sup>ω</sup><sup>r</sup> � Dr Ng 2 Jg ω<sup>g</sup> þ

\_

<sup>β</sup>\_ ¼ � <sup>1</sup> τβ β þ 1 τβ

<sup>T</sup>\_ <sup>g</sup> ¼ � <sup>1</sup> τT Tg þ 1 τT

p1p<sup>2</sup>

<sup>V</sup>€<sup>t</sup> ¼ � <sup>1</sup>

<sup>ω</sup><sup>g</sup> � Kr Jr

> Kr JgNg

<sup>V</sup>\_ <sup>t</sup> <sup>þ</sup>

k p1p<sup>2</sup>

<sup>δ</sup> � <sup>1</sup> Jg

<sup>δ</sup> <sup>¼</sup> <sup>ω</sup><sup>r</sup> � <sup>ω</sup><sup>g</sup>

Ng

<sup>δ</sup> <sup>¼</sup> <sup>Ω</sup><sup>r</sup> � <sup>Ω</sup><sup>g</sup>

describing the twist of the flexible shaft [16]:

64 Stability Control and Reliable Performance of Wind Turbines

2.3. Linearized models of wind turbine system

needed. The input of this model is wind [17]:

Having all the equations, system equations become: [8]:

<sup>ω</sup>\_ <sup>r</sup> <sup>¼</sup> <sup>a</sup> � Dr Jr

<sup>ω</sup>\_ <sup>g</sup> <sup>¼</sup> Dr NgJg

And outputs of the system include:

where:

respectively.

Various uncertainties have been examined in the current literature. These uncertainties derive from approximated and process parameters in a nonlinear system which changes as the operating point changes, a matter causing the electrical power production to reduce. There are always discrepancies between real system and mathematical models, which lead to uncertain models. In this work, sources of uncertainties are taken to be:


Although, all control design presented design for moderate temperature. Decreasing the temperature in winter has devastating properties on the wind turbine. Ice on the elements of wind turbine leads some serious problems. Even a few amount of ice on the blades worsens the aerodynamic performance of the wind turbine. It not only decreases the output power energy, but also raises the abrasion between the elements [11, 16]. In other word, the ice in cold places and the high density of air at cold climate have damaging properties on aerodynamics. Fluctuation of produced power and load are reason for such dysfunctionality. Masses of the ice on the turbine cause fluctuation on the frequency of the turbine's elements and also the behavior of the system's dynamic [10–12]. In addition, this condition has effect on control plant. In other word, the performance of the turbine system worsens through wrong data sending [11]. Previous article dealing with this issue have presented approaches like observation, the use of sensors and monitors, considering aerodynamic sound, etc., to recognize ice. The control schemes are then designed to eliminate the ice [9, 10, 17].

#### 3.1. The impact of cold weather on the operation of wind turbine

In this research, for the first time a new approach is advanced to enhance the wind turbine performance in cold climate conditions and to stop the damage which cause about the shutdown of the turbine. Because of the structure of the wind turbine, when frozen blades' masses changes, the rotor mass will be changed and lead to the inertia of rotor. These variations will effect on equations of the wind turbine and optimal power creation. Therefore, control of turbine and optimal power output could be possible by considering the inertia of the rotor as a new component of uncertainty in the plant. The value of this uncertainty differs with decreasing the temperature and the turbine's production capacity. In [18], imbalance of the blade was simulated by scaling the density of mass of one blade, which generates an imbalance distribution of mass with respect to the rotor. Furthermore, the aerodynamic asymmetry was simulated by adjusting the pitch of one blade, which produces an imbalance torque across the rotor. In our work, the blade imbalance in blade is due to icing on the blade. Thus, the uneven of the blade is considered about 20% as uncertainty. A robust control is designed to control the turbine in the existence of uncertainty because of the blade imbalance icing [19]. The uncertainties in the wind turbine comprise the linearized model parameters which is extracted from the nonlinear plant, spring constant, and damping coefficient that alteration as the working point diverges; another uncertainty which is added to the system because of the presence of noise and disturbance in the input signal. All of these uncertainties are considered in appropriate weather conditions. In this work, cold climate condition and inertia of rotor are considered as other sources of uncertainty in the wind turbine system [20, 21].

#### 3.2. Frequency response of wind turbine

These icy turbine blades change the rotor mass. Under the frequency response analysis of the system is shown in Figure 3, the rotor inertia uncertainty can be considered between the range of 0 and 20% and The Wind turbine system is stability:

According to, the red color is nominal frequency response and blue color is system uncertainties that uncertainty is considered between the range of 0–20%. The system has positive phase margin and positive gain margin, so it is stable.

Figure 4 shows that the system is unstable by considering range of uncertainty between 20 and

Kr ¼ Kr 1 þ pKr

Dr ¼ Dr 1 þ pDr

a ¼ a 1 þ paδ<sup>a</sup>

Jr ¼ Jr 1 þ pJr

Two reasons exist which explain that spring coefficient and the damping can be considered as uncertainties. One reason is, there exist divergence in the spring variables from fabrication to fabrication, and manufacturer to manufacturer and other reason, it should be consider that the

δKr

δDr

δJr

(26)

(27)

(28)

(30)

<sup>a</sup><sup>14</sup> <sup>¼</sup> <sup>a</sup><sup>14</sup> <sup>1</sup> <sup>þ</sup> pa<sup>14</sup> <sup>δ</sup><sup>a</sup><sup>14</sup> (29)

Designing Mu Robust Controller in Wind Turbine in Cold Weather Conditions

http://dx.doi.org/10.5772/intechopen.74626

67

30%. So system uncertainties are defined between 0 and 20%.

Figure 4. Bode diagram open loop system with 30% uncertainty of inertia rotor.

The uncertainties in parameters can be shown as follows:

Figure 3. Bode diagram open loop system with 20% uncertainty of inertia rotor.

Designing Mu Robust Controller in Wind Turbine in Cold Weather Conditions http://dx.doi.org/10.5772/intechopen.74626 67

Figure 4. Bode diagram open loop system with 30% uncertainty of inertia rotor.

turbine in the existence of uncertainty because of the blade imbalance icing [19]. The uncertainties in the wind turbine comprise the linearized model parameters which is extracted from the nonlinear plant, spring constant, and damping coefficient that alteration as the working point diverges; another uncertainty which is added to the system because of the presence of noise and disturbance in the input signal. All of these uncertainties are considered in appropriate weather conditions. In this work, cold climate condition and inertia of rotor are consid-

These icy turbine blades change the rotor mass. Under the frequency response analysis of the system is shown in Figure 3, the rotor inertia uncertainty can be considered between the range

According to, the red color is nominal frequency response and blue color is system uncertainties that uncertainty is considered between the range of 0–20%. The system has positive

ered as other sources of uncertainty in the wind turbine system [20, 21].

3.2. Frequency response of wind turbine

66 Stability Control and Reliable Performance of Wind Turbines

of 0 and 20% and The Wind turbine system is stability:

phase margin and positive gain margin, so it is stable.

Figure 3. Bode diagram open loop system with 20% uncertainty of inertia rotor.

Figure 4 shows that the system is unstable by considering range of uncertainty between 20 and 30%. So system uncertainties are defined between 0 and 20%.

The uncertainties in parameters can be shown as follows:

$$K\_r = \overline{K}\_r \left(1 + p\_{K\_r} \delta\_{K\_r}\right) \tag{26}$$

$$D\_r = \overline{D}\_r \left( 1 + p\_{D\_r} \delta\_{D\_r} \right) \tag{27}$$

$$a = \overline{a}(1 + p\_a \delta\_a) \tag{28}$$

$$a\_{14} = \overline{a}\_{14} \left( 1 + p\_{a\_{14}} \delta\_{a\_{14}} \right) \tag{29}$$

$$J\_r = \overline{J}\_r \left(1 + p\_{l\_r} \delta\_{l\_r}\right) \tag{30}$$

Two reasons exist which explain that spring coefficient and the damping can be considered as uncertainties. One reason is, there exist divergence in the spring variables from fabrication to fabrication, and manufacturer to manufacturer and other reason, it should be consider that the


Table 1. Parameter uncertainties of the wind turbine.

spring parameter and damping can be changed during a long time because of continuous working, and aging. In previous equations, Kr, Dr, a, a<sup>14</sup> are nominal parameter values, resulting from the spring constant, damping coefficient, linearization process and rotor inertia, respectively. pKr , pDr , pa, pa<sup>14</sup> indicate maximum relative uncertainties that are for uncertainty parameters shown in Table 1.

δKr , δDr , δa, δ<sup>a</sup><sup>14</sup> and δJr are relative changes in these parameters. Therefore:

$$\mathbb{1}\left|\delta\_{\mathbb{K}\_r}\right| \le \mathbf{1}, \, |\delta\_{D\_r}| \le \mathbf{1}, \, |\delta\_a| \le \mathbf{1}, \, |\delta\_{a\_{14}}| \le \mathbf{1}, \, \left|\delta\_{\mathbb{J}\_r}\right| \le \mathbf{1} \tag{31}$$

represents the performance when model is ideal, to which the designed closed-loop system wants to reach [23]. The transfer function of model is chosen in a way that the time response of the reference signal has an overshoot less than around 5%. Inside the dots rectangle is the ideal model, which shows with, Gnom of the wind turbine model and the block Δ that parameterizes uncertainties in the model. To find the wanted performance, inputs r can be obtained from the transfer function matrix, and we need to find disturbance in V and also n to outputs eu and ey. Thus the infinity norm of that transfers function can be minor for the entire existing uncertainty variable. The position noise signal is attained by moving the unit-bounded signal which is shows by n through the weighting transfer matrix which denotes Wn. The transfer matrices Wp and Wu represent the relative significance of the diverse frequency spans for which the performance is needed. So, the performance aim can be reorganize, with probable slight conservativeness, like that transfer function matrix infinity norm be less than 1. So Δ matrix is

Δ ¼ diag pKr

<sup>¼</sup> Wpð Þ <sup>S</sup>oGu<sup>K</sup> � <sup>R</sup> WuSiK

internally stable for each possible plant dynamics G ¼ Fuð Þ Gnom; Δ .

4.2. Matching transfer function and weighting transfer functions

the performance criterion should be satisfied for each G ¼ Fuð Þ Gnom; Δ [21].

Note that SoG is the transfer function between V and y.

; pDr

WpSoG<sup>v</sup> �WuKSoG<sup>v</sup>

where Si <sup>¼</sup> ð Þ <sup>I</sup> <sup>þ</sup> KG �<sup>1</sup> and So <sup>¼</sup> ð Þ <sup>I</sup> <sup>þ</sup> GK �<sup>1</sup> are the input and output sensitivities, repectively.

By definition, the closed loop system achieves robust stability if the closed loop system is

The closed loop system must remain internally stable for each G ¼ Fuð Þ Gnom; Δ and in addition

In the case of mu controller optimization design, we have to define the model transfer function which is denote with R and the weighting transfer functions that are nominated with Wn, Wp

The model transfer function is selected therefore the time response to the reference signal has an overshoot fewer than 50% and a settling time not more than 1 ms. A probable plant which

" # r

; pa; pa<sup>14</sup> ; pJr

� � (32)

V n

3 7

<sup>5</sup> (33)

2 6 4

�WpSoGuKWn

Designing Mu Robust Controller in Wind Turbine in Cold Weather Conditions

http://dx.doi.org/10.5772/intechopen.74626

69

�WuKSoWn

given in following form:

4.1.1. Robust stability

4.1.2. Robust performance

please the requirements is:

and Wu.

This transfer function can be written as [21]:

eu ey � �

#### 4. Robust design controller

#### 4.1. Closed loop system design specifications

Figure 5 shows the block diagram of Wind turbine closed loop system, including the feedback structure, the controller, as well as the model uncertainties and performance objectives weights.

In Figure 5, (r) shows the reference input, (V) represents the wind speed which has disturbance, (n) denotes noise and eu and ey are considered as two output costs. The system (R)

Figure 5. Block diagram of the closed-loop system with performance specifications [22].

represents the performance when model is ideal, to which the designed closed-loop system wants to reach [23]. The transfer function of model is chosen in a way that the time response of the reference signal has an overshoot less than around 5%. Inside the dots rectangle is the ideal model, which shows with, Gnom of the wind turbine model and the block Δ that parameterizes uncertainties in the model. To find the wanted performance, inputs r can be obtained from the transfer function matrix, and we need to find disturbance in V and also n to outputs eu and ey. Thus the infinity norm of that transfers function can be minor for the entire existing uncertainty variable. The position noise signal is attained by moving the unit-bounded signal which is shows by n through the weighting transfer matrix which denotes Wn. The transfer matrices Wp and Wu represent the relative significance of the diverse frequency spans for which the performance is needed. So, the performance aim can be reorganize, with probable slight conservativeness, like that transfer function matrix infinity norm be less than 1. So Δ matrix is given in following form:

$$\Delta = \text{diag}\left(p\_{K\_r}, p\_{D\_r}, p\_a, p\_{a\_{14}}, p\_{l\_r}\right) \tag{32}$$

This transfer function can be written as [21]:

$$
\begin{bmatrix} \varepsilon\_u \\ \varepsilon\_y \end{bmatrix} = \begin{bmatrix} W\_p (\mathbf{S}\_o \mathbf{G}\_u \mathbf{K} - \mathbf{R}) \ W\_p \mathbf{S}\_o \mathbf{G}\_v & -W\_p \mathbf{S}\_o \mathbf{G}\_u K W\_n \\ W\_u \mathbf{S}\_i K & -W\_u K \mathbf{S}\_o \mathbf{G}\_v - W\_u K \mathbf{S}\_o W\_n \end{bmatrix} \begin{bmatrix} r \\ V \\ n \end{bmatrix} \tag{33}
$$

where Si <sup>¼</sup> ð Þ <sup>I</sup> <sup>þ</sup> KG �<sup>1</sup> and So <sup>¼</sup> ð Þ <sup>I</sup> <sup>þ</sup> GK �<sup>1</sup> are the input and output sensitivities, repectively. Note that SoG is the transfer function between V and y.

#### 4.1.1. Robust stability

spring parameter and damping can be changed during a long time because of continuous working, and aging. In previous equations, Kr, Dr, a, a<sup>14</sup> are nominal parameter values, resulting from the spring constant, damping coefficient, linearization process and rotor inertia,

Parameter Description Unit Tolerance Kr spring coefficient Nm=rad �10%

A Linearization parameter — �20% a<sup>14</sup> Linearization parameter — �20% Jr rotor inertia kg:m<sup>2</sup> þ20%

δKr j j ≤ 1, δDr j j ≤ 1, j j δ<sup>a</sup> ≤ 1, δ<sup>a</sup><sup>14</sup> j j ≤ 1, δJr

Figure 5 shows the block diagram of Wind turbine closed loop system, including the feedback structure, the controller, as well as the model uncertainties and performance objectives weights. In Figure 5, (r) shows the reference input, (V) represents the wind speed which has disturbance, (n) denotes noise and eu and ey are considered as two output costs. The system (R)

δKr , δDr , δa, δ<sup>a</sup><sup>14</sup> and δJr are relative changes in these parameters. Therefore:

Figure 5. Block diagram of the closed-loop system with performance specifications [22].

Dr Damping kg:m<sup>2</sup>

, pa, pa<sup>14</sup> indicate maximum relative uncertainties that are for uncertainty

  rad:s

≤ 1 (31)

�10%

respectively. pKr

, pDr

Table 1. Parameter uncertainties of the wind turbine.

68 Stability Control and Reliable Performance of Wind Turbines

4. Robust design controller

4.1. Closed loop system design specifications

parameters shown in Table 1.

By definition, the closed loop system achieves robust stability if the closed loop system is internally stable for each possible plant dynamics G ¼ Fuð Þ Gnom; Δ .

#### 4.1.2. Robust performance

The closed loop system must remain internally stable for each G ¼ Fuð Þ Gnom; Δ and in addition the performance criterion should be satisfied for each G ¼ Fuð Þ Gnom; Δ [21].

#### 4.2. Matching transfer function and weighting transfer functions

In the case of mu controller optimization design, we have to define the model transfer function which is denote with R and the weighting transfer functions that are nominated with Wn, Wp and Wu.

The model transfer function is selected therefore the time response to the reference signal has an overshoot fewer than 50% and a settling time not more than 1 ms. A probable plant which please the requirements is:

$$R(\text{s}) = \frac{1}{0.48\text{s}^2 + 0.95\text{s} + 1} \tag{34}$$

In Figure 6, the response of matching model to power electrical input is shown.

The noise shaping function Wn is determined on the basis of the spectral density of the position noise signal. In the given case it is taken as the high-pass filter. In this case output has a noteworthy spectral content more than 500 Hz. For this type of filter, the position noise signal is just 0.95 V in the low-frequency values but it is 1 V in the high-frequency values that matches to a position error of around 5% without width.

$$W\_n(s) = 95 \times 10^{-5} \frac{0.1s + 1}{0.001s + 1} \tag{35}$$

So the frequency response is shown in Figure 7.

The closed-loop system performance specifications are reflected by the weighting performance function Wp.

$$W\_p(s) = 0.95 \frac{2s + 50}{2s + 0.005} \tag{36}$$

disturbances. The weight of control function is usually selected as high-pass filters to make

According to the above figure, effort control is very low in low- frequencies that cause reduced

The objective of controller with applying the μ synthesis scheme is the stabilizing of the closed loop of the plant and pleasing all of the control demands. In the existence of measurement noise, disturbance in the wind, and uncertainties, the closed-loop plant should have the robust efficiency. Therefore, the purpose of μ controller scheme is design a controller where the wind

<sup>s</sup><sup>2</sup> <sup>þ</sup> <sup>20</sup> � <sup>10</sup>�<sup>2</sup>

Designing Mu Robust Controller in Wind Turbine in Cold Weather Conditions

http://dx.doi.org/10.5772/intechopen.74626

71

<sup>s</sup> <sup>þ</sup> <sup>2</sup> (37)

Wuð Þ¼ <sup>s</sup> <sup>0</sup>:<sup>022</sup> <sup>0</sup>:8s<sup>2</sup> <sup>þ</sup> <sup>10</sup><sup>s</sup> <sup>þ</sup> <sup>2</sup> <sup>3</sup>:<sup>5</sup> � <sup>10</sup>�<sup>4</sup>

Figure 9 shows the frequency response of this weighting function Wu.

sure that the control action will not surpass 25�.

Figure 7. Frequency response of noise filter Wn:

the control cost.

4.3. Robust μ controller design

Figure 8 shows the frequency response of the inverses of this weighting function.

It is shown that in a selection, the objective is to obtain a minor variance between the system and outputs of the model, and a minor effect of the disturbance on outputs of the system. This will ensure nice tracking of the reference input and minor error because of the low-frequency

Figure 6. Response of matching model to electrical power input.

Figure 7. Frequency response of noise filter Wn:

R sð Þ¼ <sup>1</sup>

The noise shaping function Wn is determined on the basis of the spectral density of the position noise signal. In the given case it is taken as the high-pass filter. In this case output has a noteworthy spectral content more than 500 Hz. For this type of filter, the position noise signal is just 0.95 V in the low-frequency values but it is 1 V in the high-frequency values that matches

Wnð Þ¼ <sup>s</sup> <sup>95</sup> � <sup>10</sup>�<sup>5</sup> <sup>0</sup>:1<sup>s</sup> <sup>þ</sup> <sup>1</sup>

The closed-loop system performance specifications are reflected by the weighting performance

Wpð Þ¼ <sup>s</sup> <sup>0</sup>:<sup>95</sup> <sup>2</sup><sup>s</sup> <sup>þ</sup> <sup>50</sup>

It is shown that in a selection, the objective is to obtain a minor variance between the system and outputs of the model, and a minor effect of the disturbance on outputs of the system. This will ensure nice tracking of the reference input and minor error because of the low-frequency

Figure 8 shows the frequency response of the inverses of this weighting function.

In Figure 6, the response of matching model to power electrical input is shown.

to a position error of around 5% without width.

70 Stability Control and Reliable Performance of Wind Turbines

So the frequency response is shown in Figure 7.

Figure 6. Response of matching model to electrical power input.

function Wp.

<sup>0</sup>:48s<sup>2</sup> <sup>þ</sup> <sup>0</sup>:95<sup>s</sup> <sup>þ</sup> <sup>1</sup> (34)

<sup>0</sup>:001<sup>s</sup> <sup>þ</sup> <sup>1</sup> (35)

<sup>2</sup><sup>s</sup> <sup>þ</sup> <sup>0</sup>:<sup>005</sup> (36)

disturbances. The weight of control function is usually selected as high-pass filters to make sure that the control action will not surpass 25�.

$$\mathcal{W}\_{\rm u}(\mathbf{s}) = 0.022 \frac{0.8 \mathbf{s}^2 + 10 \mathbf{s} + 2}{3.5 \times 10^{-4} \mathbf{s}^2 + 20 \times 10^{-2} \mathbf{s} + 2} \tag{37}$$

Figure 9 shows the frequency response of this weighting function Wu.

According to the above figure, effort control is very low in low- frequencies that cause reduced the control cost.

#### 4.3. Robust μ controller design

The objective of controller with applying the μ synthesis scheme is the stabilizing of the closed loop of the plant and pleasing all of the control demands. In the existence of measurement noise, disturbance in the wind, and uncertainties, the closed-loop plant should have the robust efficiency. Therefore, the purpose of μ controller scheme is design a controller where the wind

turbine can track the reference input of generated electrical energy when noise and disturbance are exist. In Figure 10, the schematic of the closed loop model which is utilized with the

From the figure, it can be seen that yc is defined as the difference between noisy output and the reference and the transfer function P(s) shows the open-loop transfer function matrix with 10 inputs and eight outputs. The upper linear fractional transformation (LFT) of the closed loop

where Pnom is the nominal transfer function matrix, Δ<sup>r</sup> includes five uncertainties in the wind turbine model. We assume Δ<sup>P</sup> is defined the structure of uncertainties block as follows:

Controller, which is attained with this scheme, is typically of high order controller that cause to challenges in a real-world implementation. So for this purpose, it is recommended to decrease the order of the control plant until it is feasible to simplify the closed loop scheme theory and operation. The results of the singular parameter of planned after repeating five iterations of

It can be seen that the maximum value of μ is 11.648 that is achieved in the first iteration. Similar to this method, next steps are done to finally value of γ is less than 1. The designed μ controller synthesis is of order 17, and it is achieved after five iterations. In the final iteration, the value of γ is reached to 0.732 and, μ reaches to 0.730 that is less than 1. In other word, the closed-loop scheme has robust performance due to the structured singular parameter is less

; Δ<sup>r</sup> ∈ ℜ<sup>5</sup>�<sup>5</sup>

( )

P ¼ Fuð Þ Pnom;Δ<sup>r</sup> (38)

Designing Mu Robust Controller in Wind Turbine in Cold Weather Conditions

http://dx.doi.org/10.5772/intechopen.74626

(39)

73

;Δ<sup>f</sup> ∈ C<sup>4</sup>�<sup>2</sup>

defined uncertainties is depicted for μ controller model design.

<sup>Δ</sup><sup>p</sup> <sup>¼</sup> <sup>Δ</sup><sup>r</sup> <sup>0</sup>

D-K procedure are demonstrated in Table 2.

Figure 10. Closed loop system model for design μ robust controller.

0 Δ<sup>f</sup> " #

system is:

Figure 8. Frequency response of weighting function <sup>1</sup> Wp .

Figure 9. Frequency response of weighting function Wu.

turbine can track the reference input of generated electrical energy when noise and disturbance are exist. In Figure 10, the schematic of the closed loop model which is utilized with the defined uncertainties is depicted for μ controller model design.

From the figure, it can be seen that yc is defined as the difference between noisy output and the reference and the transfer function P(s) shows the open-loop transfer function matrix with 10 inputs and eight outputs. The upper linear fractional transformation (LFT) of the closed loop system is:

$$P = F\_u(P\_{nom}, \Delta\_r) \tag{38}$$

where Pnom is the nominal transfer function matrix, Δ<sup>r</sup> includes five uncertainties in the wind turbine model. We assume Δ<sup>P</sup> is defined the structure of uncertainties block as follows:

$$
\Delta\_p = \left\{ \begin{bmatrix} \Delta\_r & 0 \\ 0 & \Delta\_f \end{bmatrix} ; \Delta\_r \in \mathfrak{R}^{5 \times 5}, \Delta\_f \in \mathbb{C}^{4 \times 2} \right\} \tag{39}
$$

Controller, which is attained with this scheme, is typically of high order controller that cause to challenges in a real-world implementation. So for this purpose, it is recommended to decrease the order of the control plant until it is feasible to simplify the closed loop scheme theory and operation. The results of the singular parameter of planned after repeating five iterations of D-K procedure are demonstrated in Table 2.

It can be seen that the maximum value of μ is 11.648 that is achieved in the first iteration. Similar to this method, next steps are done to finally value of γ is less than 1. The designed μ controller synthesis is of order 17, and it is achieved after five iterations. In the final iteration, the value of γ is reached to 0.732 and, μ reaches to 0.730 that is less than 1. In other word, the closed-loop scheme has robust performance due to the structured singular parameter is less

Figure 10. Closed loop system model for design μ robust controller.

Figure 8. Frequency response of weighting function <sup>1</sup>

72 Stability Control and Reliable Performance of Wind Turbines

Figure 9. Frequency response of weighting function Wu.

Wp .


speeds. This express that a sudden variation in the wind speed, the robust controller attempts to control the pitch angle for setting the electrical power at it's at most efficiency in the third

Designing Mu Robust Controller in Wind Turbine in Cold Weather Conditions

http://dx.doi.org/10.5772/intechopen.74626

75

Due to the linear dependency among the speed of the rotor and the speed of generator over the gear ratio, the results of applying the control scheme which seeks to maintain the electrical power fixed and regulate speed of the generator- can be examined after modeling of speed of

Considering Figure 12(a), the deflection of output about the nominal value which is changed by noise and disturbance in wind is not high that is acceptable. Maximum variation around the reference value is 2.1% and it is equal to 0.027 kW that expresses nice disturbance cancelation in areas in the matching scheme. In Figure 12(b), at most variations around nominal values are equivalent to 3.068 rad/s and it is equal to 9.2%. It is sates a disturbance cancelation and nice following of generator reference parameter in the existence of disturbance with wind speed in

In Figure 13, it can be found that the control effort or the adjustment of pitch angle is between of 14 and 22.5. Due to changed areas of wind turbine efficiency, aim in this work is, regulating the power and speed of the generator at the nominal parameter in the third performance part. So, in this scenario by growing the value of wind to cut-out, the pitch angle has been improved, and this scenario leads to the decrease of power coefficient and the power is at its nominal value [4]. Furthermore, by reducing the speed of the wind, the blade pitch angle is decreased and at this step, to regulate the power and generator speed at nominal value, robust controller is designed. This controller attempts to control the pitch angle for accessing the high electric power and adjust the speed of generator about its nominal value. So, by applying this kind of controller, the control effort remains fewer than 25, that is the utmost pitch angle of

Figure 12. (a) The response of electric power produced by wind turbines to track input reference electrical power using μ controller. (b) The response of speed of the wind turbine generator to track input reference speed of generator using μ

performance area.

controller.

the wind turbine and modeling the robust μ controller.

the good way and with the least variations.

the wind turbine, stays bounded in the third area.

Table 2. Obtained results of the robust controller (μ).

than 1 in any frequency. Also, the gamma parameter denotes the value that the function Flð Þ P;K infinity norm is fewer than that value.
