4. Simulation results

<sup>θ</sup><sup>k</sup> <sup>¼</sup> <sup>b</sup>a1;ba2;bb1;bb<sup>2</sup> h i<sup>T</sup>

Note that the subscript k for model and controller parameters will be dropped in order to

The control law corresponding to the discrete-time adaptive controller in its difference form of

Ts TI

� �

with ek representing the tracking error, with ek = rk � yk, and rk the reference (setpoint) signal. Ts is sampling time. The controller parameters Kp and TI are here time-varying and derived from the online model parameters in the vector θk. The control law can be represented also in its

where the new controller variables q<sup>0</sup> and q<sup>1</sup> (or Kp and TI) are derived from the relations of

<sup>q</sup><sup>1</sup> ¼ �Kp <sup>1</sup> � Ts

where the parameters Kp and TI are functions of the (time-varying) critical gain and the critical

that depend on the time-varying model parameters in the vector θk. In particular, when considering a second-order model described by its (time-varying) parameters <sup>b</sup>a2, <sup>b</sup>a1, <sup>b</sup>b<sup>2</sup> and bb1, the variables KPu and Tu required by the Ziegler-Nichols method can be computed at each

In this way, the adaptive discrete-time linear controllers of Eq. (30) or (31) are designed on the basis of the time-varying linear model of Eq. (9) estimated via the online identification scheme

Ts 2TI � �

> 2TI � �

, with <sup>γ</sup> <sup>¼</sup> <sup>b</sup>a2bb<sup>1</sup> � <sup>b</sup>a1bb<sup>2</sup>

q<sup>0</sup> ¼ Kp 1 þ

8 >>><

>>>:

KPu <sup>¼</sup> <sup>b</sup>a<sup>1</sup> � <sup>b</sup>a<sup>2</sup> � <sup>1</sup> bb<sup>2</sup> � bb<sup>1</sup>

Tu <sup>¼</sup> <sup>2</sup>πTs arccosγ

from the data of the nonlinear wind turbine process of Eq. (3).

Δek 2

þ uk�<sup>1</sup>

uk ¼ q<sup>0</sup> ek þ q<sup>1</sup> ek�<sup>1</sup> þ uk�<sup>1</sup> (31)

Kp ¼ 0:6KPu , TI ¼ 0:5Tu (33)

2bb<sup>2</sup>

Δek ¼ ek � ek�<sup>1</sup> uk ¼ Kp Δek þ

8 < :

feedback formulation as described by Eq. (31):

period of oscillations, respectively, KPu and Tu:

time step k from the following relations:

8 >>><

>>>:

simplify equations and formulas.

228 Adaptive Robust Control Systems

Eq. (30):

Eq. (32):

(29)

(30)

(32)

(34)

This section presents the simulation results achieved with the proposed data-driven and model methods relying on both the fuzzy modelling technique oriented to the identification of the fuzzy controller model and the adaptive control strategy using the online estimated models. The simulations achieved with these regulators are summarised in the following.

Regarding the fuzzy modelling and identification method, the GK clustering algorithm recalled in Section 3 exploits a number M = 3 of clusters and delays n = 2. These variables were applied for clustering the first data set consisting of {Pgk, ωgk, βrk}. A number of samples k = 1, 2, …, <sup>N</sup> were considered with <sup>N</sup> = 440 � 103 . The same number of clusters and shifts were exploited for clustering the second data set {Pgk, ωgk, τgk}. After this procedure, the structures of the TS prototypes were derived for each output yk equal to Pgk and ωgk. In this way, the 2 continuous-time outputs y(t)=[ωg(t), τg(t)] of the wind turbine continuous-time model of Eq. (3) are approximated by two TS fuzzy prototypes of Eq. (8).

The performances of the fuzzy models that are derived using the procedure described above can be evaluated using the so-called Variance Accounted For (VAF) parameter [5]. In particular, the TS fuzzy model reconstructing the first output has a VAF index bigger than 90%, whereas for the second one, it was higher than 99%. This means that the fuzzy prototypes are able to describe the behaviour of the controlled process with very good precision. These estimated TS fuzzy models have been used for the derivation of the fuzzy controllers and applied to the considered wind turbine benchmark.

Two (multiple input single output) MISO fuzzy controller models with two inputs and one output have been used for the compensation of the blade pitch angle β(t) and the generator torque τg(t). By using the inverse model principle, they were estimated exploiting the methodology recalled in Section 3.1. Again, the GK fuzzy clustering method has led to 2 fuzzy regulators applied to the data sets {βrk, Pgk, ωgk} and {τgk, Pgk, ωgk}, respectively, with M = 3 clusters and n = 3 lagged signals.

The controller performances were verified and validated via extensive simulations by considering different data sequences generated via the wind turbine simulator. Table 1 reports the


Table 1. Initialisation parameters of the adaptive algorithm.

values of the percent Normalised Sum of Squared tracking Error (NSSE%) index defined in Eq. (35):

$$NSSE\% = 100 \sqrt{\frac{\sum\_{k=1}^{N} \left(r\_k - y\_k\right)^2}{\sum\_{k=1}^{N} r\_k^2}}\tag{35}$$

Also in this case, with reference to the adaptive controller structure of Eq. (30) or (31), the parameters of the online controllers were tuned via the Ziegler-Nichols rules, applied to the LPV models. In this way, if both the model online parametric identification and the regulator tuning procedure are exploited, the parameter adaptation mechanisms should lead to good

The experiments with the adaptive regulators were simulated in the same situation of the fuzzy controllers. In this case, three online regulators were exploited for the compensation of both the blade pitch angle β(t) and the generator torque τg(t), in region 1 and region 2. The adaptive algorithm described above runs with initial values for its parameters reported in

With reference to the model-based adaptive approach, Figure 3 depicts the setpoint ωg(t) in bold grey line with respect to its desired value ωnom in dashed black line. By considering the full load working conditions, the adaptive regulators have replaced the wind turbine baseline

Also for the case of the adaptive regulators, Figure 3 highlights that the model-based approach

In order to analyse the performance of the proposed adaptive strategy, Table 2 reports also the

According to the simulation results summarised in Table 2, good tracking capabilities of the suggested adaptive controllers seem to be reached, and they are better than the fuzzy regulators.

<sup>2500</sup> <sup>3000</sup> <sup>3500</sup> <sup>4000</sup> <sup>161</sup>

Controller type Partial load (%) Full load (%)

Fuzzy controller 37.17 17.85 Adaptive controller 28.73 13.67

4400

*Model-based*

Set-point

Robust Control Applications to a Wind Turbine-Simulated System

http://dx.doi.org/10.5772/intechopen.71526

231

control performances.

governor at t ≥ 3300s.

leads to interesting performances.

NSSE values computed for these controllers.

(t) g [rad/s] 161.5 162 162.5 163 163.5 164

Table 2. Controllers in partial and load operations: NSSE% values.

Figure 3. ωg(t) tracking capabilities in full load conditions with adaptive controllers.

Table 1.

Noting that in partial load operation (region 1), the performance is represented by the comparison between the power produced by the generator, yk = Pgk, with respect to the theoretical maximum power output, rk = Pr. On the other hand, in full load operation (region 2), the tracking error is given by the difference between the generator speed, yk = ωgk, and its nominal value, rk = ωnom. The achieved results show the good properties of the designed fuzzy controllers, as represented also in Figure 2.

Figure 2 depicts the signal representing generator speed ωg(t) in bold grey line with respect to its desired value ωnom in dashed black line. It can be noted that in full load conditions, the fuzzy controllers derived via the data-driven approach lead to tracking errors smaller than the wind turbine baseline governor recalled in Section 2. In fact, as shown in Figure 1, the baseline regulator is working in the interval 2200s < t < 3300s. On the other hand, the fuzzy controllers are exploited during the interval 3300s < t < 4400s, when the tracking error is much lower.

With reference to the second model-based design approach using adaptive solutions, the two outputs Pg(t) and ωg(t) of the wind turbine continuous-time nonlinear model of Eq. (3) were approximated by two second-order time-varying MISO discrete-time models of Eq. (9) with two inputs and one output. Using these one LPV prototypes, the model-based approach for determining the adaptive controllers recalled in Section 3.2 was exploited and applied to the wind turbine benchmark of Section 2. Thus, according to Section 3.2, the parameters of the adaptive controllers were computed online. In particular, for each output, two second-order (na = nb = 2) time-varying MISO prototypes were identified, and the adaptive regulator parameters in Eq. (30) or (31) were computed analytically at each time step k.

Figure 2. Generator speed (bold grey line) ωg(t) and its reference (dashed black line) ωnom.

Also in this case, with reference to the adaptive controller structure of Eq. (30) or (31), the parameters of the online controllers were tuned via the Ziegler-Nichols rules, applied to the LPV models. In this way, if both the model online parametric identification and the regulator tuning procedure are exploited, the parameter adaptation mechanisms should lead to good control performances.

values of the percent Normalised Sum of Squared tracking Error (NSSE%) index defined in

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

P N k¼1 r2 k

rk � yk � �<sup>2</sup>

(35)

P N k¼1

vuuuuuut

Noting that in partial load operation (region 1), the performance is represented by the comparison between the power produced by the generator, yk = Pgk, with respect to the theoretical maximum power output, rk = Pr. On the other hand, in full load operation (region 2), the tracking error is given by the difference between the generator speed, yk = ωgk, and its nominal value, rk = ωnom. The achieved results show the good properties of the designed fuzzy control-

Figure 2 depicts the signal representing generator speed ωg(t) in bold grey line with respect to its desired value ωnom in dashed black line. It can be noted that in full load conditions, the fuzzy controllers derived via the data-driven approach lead to tracking errors smaller than the wind turbine baseline governor recalled in Section 2. In fact, as shown in Figure 1, the baseline regulator is working in the interval 2200s < t < 3300s. On the other hand, the fuzzy controllers are exploited during the interval 3300s < t < 4400s, when the tracking error is much lower.

With reference to the second model-based design approach using adaptive solutions, the two outputs Pg(t) and ωg(t) of the wind turbine continuous-time nonlinear model of Eq. (3) were approximated by two second-order time-varying MISO discrete-time models of Eq. (9) with two inputs and one output. Using these one LPV prototypes, the model-based approach for determining the adaptive controllers recalled in Section 3.2 was exploited and applied to the wind turbine benchmark of Section 2. Thus, according to Section 3.2, the parameters of the adaptive controllers were computed online. In particular, for each output, two second-order (na = nb = 2) time-varying MISO prototypes were identified, and the adaptive regulator param-

2500 3000 3500 4000

4400

Set-point

eters in Eq. (30) or (31) were computed analytically at each time step k.

Figure 2. Generator speed (bold grey line) ωg(t) and its reference (dashed black line) ωnom.

NSSE% ¼ 100

Eq. (35):

230 Adaptive Robust Control Systems

lers, as represented also in Figure 2.

The experiments with the adaptive regulators were simulated in the same situation of the fuzzy controllers. In this case, three online regulators were exploited for the compensation of both the blade pitch angle β(t) and the generator torque τg(t), in region 1 and region 2. The adaptive algorithm described above runs with initial values for its parameters reported in Table 1.

With reference to the model-based adaptive approach, Figure 3 depicts the setpoint ωg(t) in bold grey line with respect to its desired value ωnom in dashed black line. By considering the full load working conditions, the adaptive regulators have replaced the wind turbine baseline governor at t ≥ 3300s.

Also for the case of the adaptive regulators, Figure 3 highlights that the model-based approach leads to interesting performances.

In order to analyse the performance of the proposed adaptive strategy, Table 2 reports also the NSSE values computed for these controllers.

According to the simulation results summarised in Table 2, good tracking capabilities of the suggested adaptive controllers seem to be reached, and they are better than the fuzzy regulators.

Figure 3. ωg(t) tracking capabilities in full load conditions with adaptive controllers.


Table 2. Controllers in partial and load operations: NSSE% values.
