2.2 Simulated fault scenario

2.1 Wind turbine model

Fault Detection, Diagnosis and Prognosis

the squared wind speed v<sup>2</sup>

Eq. (2):

The turbine system consists of three submodels motivated by the power transmission flow. First, the blade and pitch block represents how the blades capture

<sup>w</sup>ð Þt

<sup>w</sup>, the air density ρ and the rotor radius R. The coefficient

n

Bdt Ng ωg

þ Bg !

Pg ¼ η<sup>g</sup> ω<sup>g</sup> τ<sup>g</sup> (5)

ω<sup>g</sup> � τ<sup>g</sup>

<sup>ω</sup><sup>r</sup> � <sup>η</sup>dtBdt N2 g

<sup>2</sup> (1)

(2)

(3)

(4)

<sup>τ</sup>rðÞ¼ <sup>t</sup> ρπR<sup>3</sup>Cqð Þ <sup>λ</sup>ð Þ<sup>t</sup> ; <sup>β</sup>ð Þ<sup>t</sup> <sup>v</sup><sup>2</sup>

Cq is usually defined using a two-dimensional map depending on the blade pitch angle β and the tip-speed ratio λ, that is, the ratio between the linear velocity of the blade tip and the wind speed. This map is represented by means of a look-up table. The blade and pitch system includes the dynamics of the pitch angle hydraulic piston servo system, which is approximated as a second-order transfer function of

<sup>β</sup>refð Þ<sup>s</sup> <sup>¼</sup> <sup>ω</sup><sup>2</sup>

For each blade, Eq. (1) describes the torque acting on the rotor τr, depending on

n s<sup>2</sup> þ 2ζω<sup>n</sup> s þ ω<sup>2</sup>

where βref is the reference pitch angle computed by the turbine controller, while

The drive-train system determines the power flow through the gear box from the rotor toward the electric generator, whose dynamics are described as in Eq. (3):

wind energy, which is based on the following aerodynamic law:

βð Þs

Jrω\_ <sup>r</sup> ¼ τ<sup>r</sup> � Kdt θ<sup>Δ</sup> � ð Þ Bdt þ Br ω<sup>r</sup> þ

<sup>θ</sup><sup>Δ</sup> <sup>þ</sup> <sup>η</sup>dtBdt Ng

shaft, Ng is the gear ratio, ηdt is the efficiency and θ<sup>Δ</sup> is the torsion angle.

τgð Þs <sup>τ</sup>g,refð Þ<sup>s</sup> <sup>¼</sup> <sup>α</sup><sup>g</sup>

torque by its speed, decreased by the efficiency coefficient ηg:

where Jr and Jg are the inertia moments of the rotor and generator shafts, respectively. Kdt is the torsion stiffness, Bdt is the torsion damping factor, Bg is the viscous friction of the generator shaft, Br is the viscous friction of the low-speed

Finally, the generator submodel represents the converter dynamics by means of

where τg,ref is the reference torque defined by the controller and α<sup>g</sup> is the transfer

Finally, the generated power Pg is computed as the product of the generator

As sketched in Figure 1, the signals generated by the wind turbine system are assumed to be acquired through the measurement block, whose objective is to

s þ α<sup>g</sup>

ζ and ω<sup>n</sup> are the transfer function parameters.

Jg <sup>ω</sup>\_ <sup>g</sup> <sup>¼</sup> <sup>η</sup>dtKdt Ng

<sup>θ</sup><sup>Δ</sup> <sup>¼</sup> <sup>ω</sup><sup>r</sup> � <sup>ω</sup><sup>g</sup>

Ng

\_

first-order transfer function of Eq. (4):

function parameter.

48

8

>>>>>>>><

>>>>>>>>:

The wind turbine simulator includes the generation of three different typical fault cases, that is, sensor, actuator and system faults, see Odgaard and Stoustrup [4] and Odgaard et al. [12].

For the case of the sensor faults, they are generated as additive signals on the affected measurements. As an example, the faulty sensor of faulty pitch angle β<sup>m</sup> provides wrong measurements on blade orientation; thus, if not handled, the controller cannot fully track the power reference signal.

On the other hand, actuator faults lead to the alteration of pitch angle or the generator torque transfer functions of Eqs. (2) and (4), by modifying their dynamics. They simulate a pressure drop in the hydraulic circuit of the pitch actuator or an electronic break down in the converter device.

Finally, a system fault affects the drive train of the turbine, which is described as a slow variation in time of the friction coefficient. This can be due to the effect of wear and tear along time of the mechanical parts.

These nine fault cases are summarised in Table 1, which also highlights which measured signals are affected by them, as shown in Figure 1.

With these assumptions, the overall model of the wind turbine process can be represented as a non-linear continuous-time function fwt describing the evolution of the turbine state vector xwt excited by the input vector u:

$$\begin{cases} \dot{\mathbf{x}}\_{wt}(t) = \mathbf{f}\_{wt}(\mathbf{x}\_{wt}, \mathbf{u}(t)) \\ \mathbf{y}(t) = \mathbf{x}\_{wt}(t) \end{cases} \tag{6}$$

where in this case, the state of the system is considered equal to the monitored system output, that is, the rotor speed, the generator speed and the generated power:


$$\mathbf{x}\_{wt}(t) = \mathbf{y}(t) = \begin{bmatrix} \alpha\_{\mathbf{g},m1}, \alpha\_{\mathbf{g},m2}, \alpha\_{r,m1}, \alpha\_{r,m2}, P\_{\mathbf{g},m} \end{bmatrix}$$

Table 1. Wind turbine simulator fault scenario. On the other hand, the input vector:

$$\mathbf{u}(t) = \left[\beta\_{1,m1}, \beta\_{1,m2}, \beta\_{2,m1}, \beta\_{2,m2}, \beta\_{3,m1}, \beta\_{3,m2}, \tau\_{\mathbf{g},m}\right]$$

consists of the measurements of the pitch angles from the three redundant sensors, as well as the measured torque. These signals are sampled with sample time T in order to acquire a number N of data uð Þk , yð Þk with k ¼ 1, …, N, in order to implement the data-driven fault diagnosis solutions proposed in this chapter.

## 3. Fault diagnosis techniques

This chapter considers two data-driven approaches, relying on fuzzy system and neural network structures, which are used to design the fault diagnosis schemes. Therefore, this section briefly introduces the general scheme of the fault diagnosis strategy, by recalling the basic features of the fuzzy systems and neural networks, as addressed in Sections 3.1 and 3.2, respectively. Moreover, these architectures, which are represented by NARX structures, are exploited residual generators for solving the problem of fault diagnosis, according to the analytical redundancy principle, see Chen and Patton [13].

In order to solve the fault diagnosis problem, this work assumes that the wind turbine system is affected by equivalent additive faults on the input and the output measurements, as well as measurement errors, as described by Eq. (7):

$$\begin{cases} \mathbf{u}(k) = \mathbf{u}^\*(k) + \tilde{\mathbf{u}}(k) + \mathbf{f}\_u(k) \\ \mathbf{y}(k) = \mathbf{y}^\*(k) + \tilde{\mathbf{y}}(k) + \mathbf{f}\_\mathcal{y}(k) \end{cases} \tag{7}$$

Figure 2 shows that in general the residual generators use the acquired input and output measurements uð Þk and yð Þk . As first step, the fault diagnosis scheme consists of the fault detection task. In this case, as the residual is equal to the estimated fault signal, it is easily performed via a proper thresholding logic directly operating on the residual itself, without requiring complex elaboration with proper evaluation functions, as shown in Chen and Patton [13]. Therefore, the occurrence of the ith fault can be simply detected via the threshold logic of Eq. (9) applied to the ith

ri � δσri <sup>≤</sup> ri <sup>≤</sup>ri <sup>þ</sup> δσ<sup>i</sup> fault‐free case

with rið Þk representing the ith component of the vector rð Þk . If it is considered as

Note that the parameter δ≥2 represents a tolerance variable, which has to be properly tuned in order to effectively separate the fault-free from the faulty conditions. A common choice of δ can rely on the three-sigma rule, otherwise extensive simulations can be exploited for optimising this δ value, see Chen and Patton [13]. Once the fault detection phase is accomplished, the fault isolation task is directly obtained by means of a bank of estimators. As described by Eq. (7), the faults are considered as equivalent signals that are injected and affect the input measurements

According to the scheme depicted in Figure 3, in order to uniquely isolate one of the input or output faults, under the assumption that multiple faults cannot occur, a bank of multi-input single-output (MISO) fault estimators is designed. In general, the number of this estimators is equal to the number of faults that have to be diagnosed, that is, which coincides with the number of input and output measurements, r þ m. Therefore, the ith estimator providing the reconstruction of the fault ^f kð Þ¼ rið Þ<sup>k</sup> is driven by the components of the input and output signals <sup>u</sup>ð Þ<sup>k</sup> and yð Þk , respectively. These components are selected in order to be sensitive to the

ð Þ rið Þ� k ri

2

(9)

(10)

ri values can be estimated in fault-free

or ri <sup>&</sup>gt; ri <sup>þ</sup> δσri faulty case �

condition, after the acquisition of N samples, according to Eq. (10):

ri <sup>¼</sup> <sup>1</sup> <sup>N</sup> <sup>∑</sup> N k¼1 rið Þk

σ2 ri <sup>¼</sup> <sup>1</sup> <sup>N</sup> <sup>∑</sup> N k¼1

8 >>><

>>>:

via the signal fu, or the output measurements by means of f <sup>y</sup>.

ri < ri � δσri

a random variable, its means ri and variance σ<sup>2</sup>

Fault Diagnosis Techniques for a Wind Turbine System DOI: http://dx.doi.org/10.5772/intechopen.83810

residual rið Þk :

51

Figure 2.

Fault detectors for fault diagnosis.

where <sup>u</sup><sup>∗</sup>ð Þ<sup>k</sup> and <sup>y</sup><sup>∗</sup>ð Þ<sup>k</sup> represent the actual process variables, <sup>u</sup>ð Þ<sup>k</sup> and <sup>y</sup>ð Þ<sup>k</sup> are the measurements acquired from the sensors, while u~ð Þk and y~ð Þk describe the measurement errors. According to the description of Eq. (7), signals of the faults fuð Þk and f <sup>y</sup>ð Þk also have equivalent additive effects. Obviously, these functions are different from zero in faulty cases. In general, the vector uð Þk has r components, that is, the number of process inputs, while yð Þk has m elements, that is, the number of process outputs.

Among the possible approaches exploited for residual generation, and based on the analytical redundancy principle, this work proposes to exploit fuzzy system and neural network structures, which provide an on-line estimation ^fð Þ<sup>k</sup> of the fault signals fuð Þk and f <sup>y</sup>ð Þk . Hence, as shown in Figure 1, the so-called diagnostic residuals <sup>r</sup>ð Þ<sup>k</sup> are equal to the estimated fault signals, ^fð Þ<sup>k</sup> , which are computed by the general fault estimator, as highlighted by Eq. (8):

$$\mathbf{r}(k) = \hat{\mathbf{f}}(k) \tag{8}$$

The variable ^fð Þ<sup>k</sup> is the generic fault vector, that is, ^fð Þ¼ <sup>k</sup> ^<sup>f</sup> <sup>1</sup>ð Þ<sup>k</sup> ; …; ^<sup>f</sup> <sup>r</sup>þ<sup>m</sup>ð Þ<sup>k</sup> n o. Therefore, the general fault estimate ^<sup>f</sup> <sup>i</sup> ð Þk can be equal to one of the i components of the fault vectors fuð Þk or f <sup>y</sup>ð Þk in Eqs. (7), with i ¼ 1, …, r þ m.

The residual generation scheme exploiting the fault estimators as residual generator is depicted in Figure 2. Note that this strategy is able to provide both the fault detection and isolation tasks, that is, the fault diagnosis function, see Chen and Patton [13].

Fault Diagnosis Techniques for a Wind Turbine System DOI: http://dx.doi.org/10.5772/intechopen.83810

Figure 2. Fault detectors for fault diagnosis.

On the other hand, the input vector:

Fault Detection, Diagnosis and Prognosis

3. Fault diagnosis techniques

Chen and Patton [13].

of process outputs.

Patton [13].

50

uðÞ¼ t β1,m1; β1,m2; β2,m1; β2,m2; β3,m1; β3,m2; τg,m � �

consists of the measurements of the pitch angles from the three redundant sensors, as well as the measured torque. These signals are sampled with sample time T in order to acquire a number N of data uð Þk , yð Þk with k ¼ 1, …, N, in order to implement the data-driven fault diagnosis solutions proposed in this chapter.

This chapter considers two data-driven approaches, relying on fuzzy system and neural network structures, which are used to design the fault diagnosis schemes. Therefore, this section briefly introduces the general scheme of the fault diagnosis strategy, by recalling the basic features of the fuzzy systems and neural networks, as addressed in Sections 3.1 and 3.2, respectively. Moreover, these architectures, which are represented by NARX structures, are exploited residual generators for solving the problem of fault diagnosis, according to the analytical redundancy principle, see

In order to solve the fault diagnosis problem, this work assumes that the wind turbine system is affected by equivalent additive faults on the input and the output

> <sup>u</sup>ð Þ¼ <sup>k</sup> <sup>u</sup><sup>∗</sup>ð Þþ <sup>k</sup> <sup>u</sup>~ð Þþ <sup>k</sup> <sup>f</sup>uð Þ<sup>k</sup> <sup>y</sup>ð Þ¼ <sup>k</sup> <sup>y</sup><sup>∗</sup>ð Þþ <sup>k</sup> <sup>y</sup>~ð Þþ <sup>k</sup> <sup>f</sup> <sup>y</sup>ð Þ<sup>k</sup>

where <sup>u</sup><sup>∗</sup>ð Þ<sup>k</sup> and <sup>y</sup><sup>∗</sup>ð Þ<sup>k</sup> represent the actual process variables, <sup>u</sup>ð Þ<sup>k</sup> and <sup>y</sup>ð Þ<sup>k</sup> are the measurements acquired from the sensors, while u~ð Þk and y~ð Þk describe the measurement errors. According to the description of Eq. (7), signals of the faults fuð Þk and f <sup>y</sup>ð Þk also have equivalent additive effects. Obviously, these functions are different from zero in faulty cases. In general, the vector uð Þk has r components, that is, the number of process inputs, while yð Þk has m elements, that is, the number

Among the possible approaches exploited for residual generation, and based on the analytical redundancy principle, this work proposes to exploit fuzzy system and neural network structures, which provide an on-line estimation ^fð Þ<sup>k</sup> of the fault signals fuð Þk and f <sup>y</sup>ð Þk . Hence, as shown in Figure 1, the so-called diagnostic residuals <sup>r</sup>ð Þ<sup>k</sup> are equal to the estimated fault signals, ^fð Þ<sup>k</sup> , which are computed by the

The variable ^fð Þ<sup>k</sup> is the generic fault vector, that is, ^fð Þ¼ <sup>k</sup> ^<sup>f</sup> <sup>1</sup>ð Þ<sup>k</sup> ; …; ^<sup>f</sup> <sup>r</sup>þ<sup>m</sup>ð Þ<sup>k</sup>

The residual generation scheme exploiting the fault estimators as residual generator is depicted in Figure 2. Note that this strategy is able to provide both the fault detection and isolation tasks, that is, the fault diagnosis function, see Chen and

of the fault vectors fuð Þk or f <sup>y</sup>ð Þk in Eqs. (7), with i ¼ 1, …, r þ m.

<sup>r</sup>ð Þ¼ <sup>k</sup> ^fð Þ<sup>k</sup> (8)

ð Þk can be equal to one of the i components

n o

.

(7)

measurements, as well as measurement errors, as described by Eq. (7):

(

general fault estimator, as highlighted by Eq. (8):

Therefore, the general fault estimate ^<sup>f</sup> <sup>i</sup>

Figure 2 shows that in general the residual generators use the acquired input and output measurements uð Þk and yð Þk . As first step, the fault diagnosis scheme consists of the fault detection task. In this case, as the residual is equal to the estimated fault signal, it is easily performed via a proper thresholding logic directly operating on the residual itself, without requiring complex elaboration with proper evaluation functions, as shown in Chen and Patton [13]. Therefore, the occurrence of the ith fault can be simply detected via the threshold logic of Eq. (9) applied to the ith residual rið Þk :

$$\begin{cases} \overline{r}\_i - \delta \sigma\_{r\_i} \le r\_i \le \overline{r}\_i + \delta \sigma\_i & \text{fault-free case} \\ r\_i < \overline{r}\_i - \delta \sigma\_{r\_i} \text{or } r\_i > \overline{r}\_i + \delta \sigma\_{r\_i} & \text{fault case} \end{cases} \tag{9}$$

with rið Þk representing the ith component of the vector rð Þk . If it is considered as a random variable, its means ri and variance σ<sup>2</sup> ri values can be estimated in fault-free condition, after the acquisition of N samples, according to Eq. (10):

$$\begin{cases} \overline{r}\_i = \frac{1}{N} \sum\_{k=1}^{N} r\_i(k) \\ \sigma\_{r\_i}^2 = \frac{1}{N} \sum\_{k=1}^{N} \left( r\_i(k) - \overline{r}\_i \right)^2 \end{cases} \tag{10}$$

Note that the parameter δ≥2 represents a tolerance variable, which has to be properly tuned in order to effectively separate the fault-free from the faulty conditions. A common choice of δ can rely on the three-sigma rule, otherwise extensive simulations can be exploited for optimising this δ value, see Chen and Patton [13].

Once the fault detection phase is accomplished, the fault isolation task is directly obtained by means of a bank of estimators. As described by Eq. (7), the faults are considered as equivalent signals that are injected and affect the input measurements via the signal fu, or the output measurements by means of f <sup>y</sup>.

According to the scheme depicted in Figure 3, in order to uniquely isolate one of the input or output faults, under the assumption that multiple faults cannot occur, a bank of multi-input single-output (MISO) fault estimators is designed. In general, the number of this estimators is equal to the number of faults that have to be diagnosed, that is, which coincides with the number of input and output measurements, r þ m. Therefore, the ith estimator providing the reconstruction of the fault ^f kð Þ¼ rið Þ<sup>k</sup> is driven by the components of the input and output signals <sup>u</sup>ð Þ<sup>k</sup> and yð Þk , respectively. These components are selected in order to be sensitive to the

According to this approach, the approximation of non-linear multi-input single-output (MISO) systems can be achieved by the Takagi-Sugeno (TS) fuzzy reasoning, as described in Babuška [9]. The TS modelling approach proposed here, as addressed in Takagi and Sugeno [14], describes the consequents as deterministic

The fuzzy rule of the FIS has the form of Eq. (11):

Fault Diagnosis Techniques for a Wind Turbine System DOI: http://dx.doi.org/10.5772/intechopen.83810

parametric function in the affine form of Eq. (12):

partitioning the data into regions where the relations gi

is a weighted average of affine functions gi

xð Þ¼ k …; yl

where ulð Þ� and yj

are collected into the vector:

where the αð Þ<sup>i</sup>

the input ones.

53

Ri : IF fuzzy combination of inputs � � THEN output <sup>¼</sup> gi

expressed by linguistic propositions. The rule consequent function gi

gi

where a<sup>i</sup> is the parameter vector, and bi is a scalar offset, while gi

The TS prototype takes the form of the expression of Eq. (13):

well as delayed samples of the system input and output signals.

ð Þ k � 1 ; …; yl

<sup>a</sup><sup>i</sup> <sup>¼</sup> <sup>α</sup>ð Þ<sup>i</sup>

^<sup>f</sup> <sup>¼</sup> <sup>∑</sup>nC

<sup>i</sup>¼<sup>1</sup>λið Þ <sup>x</sup> gi

∑nC <sup>i</sup>¼<sup>1</sup>λið Þ <sup>x</sup>

Using this fuzzy approach, in general, the fault ^f can be reconstructed from suitable data acquired from the system under diagnosis. In other words, the fault ^f

where the weights are the combined degree of fulfilment λið Þ x of the system inputs. It is worth noting that the system under investigation corresponds to the wind turbine process described in Section 2, which has a dynamic behaviour. Therefore, the considered input vector x of the TS model of Eq. (13) contains the current as

Therefore, in order to include dynamics into the static relation of Eq. (11), the consequents are described as discrete-time linear AutoRegressive models with eXogenous input (ARX) of order o, in which the regressor vector has the form of

vectors uð Þk and yð Þk is selected via the fault sensitivity analysis tool of Section 3.3, and exploited in the scheme of Figure 3. The variable k represents the time step, with k ¼ 1, 2, …, N. The affine parameters associated to the ith model of the Eq. (12)

h i<sup>T</sup>

<sup>1</sup> ; …; αð Þ<sup>i</sup> <sup>o</sup> ; δ ð Þi <sup>1</sup> ; …; δð Þ<sup>i</sup> o

<sup>j</sup> coefficients refer to the output samples, while δ

ð Þ x

ð Þ <sup>k</sup> � <sup>o</sup> ; …ujð Þ<sup>k</sup> ; …; ujð Þ <sup>k</sup> � <sup>o</sup> ; … � �<sup>T</sup> (14)

ð Þ� are the components of the actual system input and output

ð Þ� of the inputs, while the antecedents remain fuzzy propositions.

where i refers to the number of rules. The antecedents are combined by means

of membership functions λið Þ x that take into account the logical connectives

ð Þ¼ <sup>x</sup> <sup>a</sup><sup>T</sup>

output. The number of rules is supposed equal the number of clusters nC used for

Furthermore, the antecedent of each rule defines the degree of fulfilment for the corresponding consequent model, defined by the membership function λið Þ x . Therefore, the global model is expressed as a fuzzy composition of parametric

ð Þ inputs (11)

<sup>i</sup> x þ bi (12)

ð Þ x of the input-output measurements,

ð Þ� hold, see Babuška [9].

ð Þ� is defined as

ð Þ x is the ith rule

(13)

(15)

<sup>j</sup> are associated to

ð Þi

functions gi

models gi

Eq. (14):

ð Þ x .

Figure 3. Residual generators bank with rið Þ¼ k fi ð Þk .

specific fault fi ð Þk . In fact, the design of these fault estimators is enhanced by the fault sensitivity analysis described in Section 3.3. For each case, the fault modes and their resulting effects on the rest of the system are analysed, and in particular the most sensitive input ujð Þk and output yl ð Þk measurements to that specific fault situation are selected. In this way, by means of the fuzzy system and neural network tools, it will be possible to derive the dynamic relationships between the inputoutput measurements, ujð Þk and yl ð Þk , and the faults fi ð Þk , as highlighted by Figure 3.

Figure 3 shows this fault estimator bank, where the fault estimators are driven by the input-output signals selected via the fault sensitivity analysis procedure. In this way, the residual rið Þ¼ <sup>k</sup> ^<sup>f</sup> <sup>i</sup> ð Þk is insensitive only to the fault affecting those inputs and outputs, ujð Þk and yl ð Þk , defined by the selector blocks. It is worth noting that, using this configuration, multiple faults occurring at the same time cannot be correctly isolated.

As already remarked, the sensitivity analysis, which has to be executed before the design of the fault estimators, suggests how to select the input-output signals feeding the fault estimator modules. After this selection procedure is performed, as described in Section 3.3, the design of the fuzzy or neural network models is achieved, as recalled in Sections 3.1 and 3.2, respectively. Finally, the threshold test logic of Eq. (9) allows the achievement of the fault diagnosis task.

#### 3.1 Fuzzy system modelling and identification

This section describes the design of the fault estimators described by means of the Takagi-Sugeno (TS) prototypes, see Takagi and Sugeno [14]. Therefore, the unknown relationships between the selected measurements and the faults are described by fuzzy models, which consist of a number of rules. These rules connect the measured signals acquired from the system under diagnosis to its faults, described in form of IF)THEN relations, processed by a fuzzy inference system (FIS), see Babuška [9].

Fault Diagnosis Techniques for a Wind Turbine System DOI: http://dx.doi.org/10.5772/intechopen.83810

According to this approach, the approximation of non-linear multi-input single-output (MISO) systems can be achieved by the Takagi-Sugeno (TS) fuzzy reasoning, as described in Babuška [9]. The TS modelling approach proposed here, as addressed in Takagi and Sugeno [14], describes the consequents as deterministic functions gi ð Þ� of the inputs, while the antecedents remain fuzzy propositions.

The fuzzy rule of the FIS has the form of Eq. (11):

Ri : IF fuzzy combination of inputs � � THEN output <sup>¼</sup> gi ð Þ inputs (11)

where i refers to the number of rules. The antecedents are combined by means of membership functions λið Þ x that take into account the logical connectives expressed by linguistic propositions. The rule consequent function gi ð Þ� is defined as parametric function in the affine form of Eq. (12):

$$\mathbf{g}\_i(\mathbf{x}) = \mathbf{a}\_i^T \mathbf{x} + b\_i \tag{12}$$

where a<sup>i</sup> is the parameter vector, and bi is a scalar offset, while gi ð Þ x is the ith rule output. The number of rules is supposed equal the number of clusters nC used for partitioning the data into regions where the relations gi ð Þ� hold, see Babuška [9]. Furthermore, the antecedent of each rule defines the degree of fulfilment for the corresponding consequent model, defined by the membership function λið Þ x . Therefore, the global model is expressed as a fuzzy composition of parametric models gi ð Þ x .

The TS prototype takes the form of the expression of Eq. (13):

$$\hat{f} = \frac{\sum\_{i=1}^{n\_C} \lambda\_i(\mathbf{x}) g\_i(\mathbf{x})}{\sum\_{i=1}^{n\_C} \lambda\_i(\mathbf{x})} \tag{13}$$

Using this fuzzy approach, in general, the fault ^f can be reconstructed from suitable data acquired from the system under diagnosis. In other words, the fault ^f is a weighted average of affine functions gi ð Þ x of the input-output measurements, where the weights are the combined degree of fulfilment λið Þ x of the system inputs.

It is worth noting that the system under investigation corresponds to the wind turbine process described in Section 2, which has a dynamic behaviour. Therefore, the considered input vector x of the TS model of Eq. (13) contains the current as well as delayed samples of the system input and output signals.

Therefore, in order to include dynamics into the static relation of Eq. (11), the consequents are described as discrete-time linear AutoRegressive models with eXogenous input (ARX) of order o, in which the regressor vector has the form of Eq. (14):

$$\mathbf{x}(k) = \begin{bmatrix} \dots, y\_l(k-1), \dots, y\_l(k-o), \dots \\ \end{bmatrix}, \dots, \mathbf{u}\_j(k), \dots, \mathbf{u}\_j(k-o), \dots \end{bmatrix}^T \tag{14}$$

where ulð Þ� and yj ð Þ� are the components of the actual system input and output vectors uð Þk and yð Þk is selected via the fault sensitivity analysis tool of Section 3.3, and exploited in the scheme of Figure 3. The variable k represents the time step, with k ¼ 1, 2, …, N. The affine parameters associated to the ith model of the Eq. (12) are collected into the vector:

$$\mathbf{a}\_{i} = \begin{bmatrix} a\_{1}^{(i)}, \dots, a\_{o}^{(i)}, \delta\_{1}^{(i)}, \dots, \delta\_{o}^{(i)} \end{bmatrix}^{T} \tag{15}$$

where the αð Þ<sup>i</sup> <sup>j</sup> coefficients refer to the output samples, while δ ð Þi <sup>j</sup> are associated to the input ones.

specific fault fi

Figure 3.

Figure 3.

most sensitive input ujð Þk and output yl

output measurements, ujð Þk and yl

Residual generators bank with rið Þ¼ k fi

Fault Detection, Diagnosis and Prognosis

this way, the residual rið Þ¼ <sup>k</sup> ^<sup>f</sup> <sup>i</sup>

inputs and outputs, ujð Þk and yl

correctly isolated.

(FIS), see Babuška [9].

52

ð Þk . In fact, the design of these fault estimators is enhanced by the

ð Þk measurements to that specific fault

ð Þk is insensitive only to the fault affecting those

ð Þk , defined by the selector blocks. It is worth noting

ð Þk , as highlighted by

fault sensitivity analysis described in Section 3.3. For each case, the fault modes and their resulting effects on the rest of the system are analysed, and in particular the

ð Þk .

situation are selected. In this way, by means of the fuzzy system and neural network tools, it will be possible to derive the dynamic relationships between the input-

ð Þk , and the faults fi

Figure 3 shows this fault estimator bank, where the fault estimators are driven by the input-output signals selected via the fault sensitivity analysis procedure. In

that, using this configuration, multiple faults occurring at the same time cannot be

As already remarked, the sensitivity analysis, which has to be executed before the design of the fault estimators, suggests how to select the input-output signals feeding the fault estimator modules. After this selection procedure is performed, as described in Section 3.3, the design of the fuzzy or neural network models is achieved, as recalled in Sections 3.1 and 3.2, respectively. Finally, the threshold test

This section describes the design of the fault estimators described by means of the Takagi-Sugeno (TS) prototypes, see Takagi and Sugeno [14]. Therefore, the unknown relationships between the selected measurements and the faults are described by fuzzy models, which consist of a number of rules. These rules connect

the measured signals acquired from the system under diagnosis to its faults, described in form of IF)THEN relations, processed by a fuzzy inference system

logic of Eq. (9) allows the achievement of the fault diagnosis task.

3.1 Fuzzy system modelling and identification

A powerful approach to the design of the ith FIS as approximator for the system under diagnosis begins with the partitioning of the available data uð Þk and yð Þk of Eq. (7) into subsets, known as cluster. A cluster is defined as a set of data that are more similar to each other rather than to the members of another cluster. The similarity among data can be expressed in terms of their distance from a particular item, exploited as the cluster prototype. Fuzzy clustering provides an effective tool to obtain a partitioning of data in which the transitions among subsets are smooth, rather than abrupt. Moreover, fuzzy clustering assumes that the data of each cluster are characterised by an affine behaviour, which is indeed modelled by the relation of Eq. (12). Different clustering methods have been proposed in literature, see for example, more recent works Graaff and Engelbrecht [15] and Jun et al. [16].

With reference to this work, the design of the FIS is considered as a system identification problem from the noisy data of Eqs. (7). In fact, the estimation of the consequent parameters a<sup>i</sup> and bi of Eq. (12) is required using the input-output data for designing the bank of the fault estimations reported in Figure 3. Moreover, the data are acquired from the measurements selected from the procedure suggested in Section 3.3. The identification scheme exploited in this work was proposed by the authors in Fantuzzi et al. [17]. This approach is based on the minimisation of the prediction errors of the individual TS local affine models considered as nC-independent estimation problems. Their solutions rely on the estimation of errors-in-variables models in Fantuzzi et al. [17], which is also the assumption represented by Eq. (7).

Another key aspect, which is not considered here, regards the determination of the optimal number of clusters nC, as the clustering algorithm assumes that the number of clusters nC has been fixed. These issues are considered in the development of the estimation procedure properly integrated by the authors, which also determines the antecedent degrees of fulfilment μik required by Eq. (13) and solved with curve fitting methods, see Babuška [9].

networks are multilayer networks, in which the output of some neurons is fed back to neurons belonging to previous layers, thus the information flow in forward as well as in backward directions, allowing a dynamic memory inside the network, see

ð Þ¼ k rið Þk .

A noteworthy intermediate solution is provided by the multilayer perceptron with a tapped delay line, which is a feedforward network whose inputs come from a delay line. This study proposes to use this solution, defined as quasistatic neural network, as it represents a suitable tool to predict dynamic relationships between the input-output measurements and the considered fault functions. In this way, another NARX description is obtained, since the non-linear (static) network is fed by the delayed samples of the system inputs and outputs selected by the fault sensitivity analysis tool described in Section 3.3. Indeed, if properly trained, the

ð Þk on the

ð Þ� are the

NARX network can estimate the current (and the next) fault samples fj

ð Þk , respectively, in the same way of the fuzzy systems.

ð Þ¼ k F …; ujð Þk ; …; ujð Þ k � du ; …yl

ð Þk estimator is depicted in Figure 4.

basis of the selected past measurements of system inputs and outputs ulð Þk and

Therefore, with reference to the ith residual generator of Figure 4, which is used to design the estimator bank of Figure 3, this NARX network is described by the

ð Þk is the estimation of the generic ith fault, while ujð Þ� and yl

generic jth and lth components of the measured inputs and outputs u and y, respectively, that are selected via the fault sensitivity analysis tool. k is the time step, du and dy are the number of delay of inputs and outputs, respectively, which have to be properly selected. Fð Þ� is the function realised by the static neural network, which depends on the layer architecture, the number of neurons, their weights and their activation functions. The NARX network used as generic fault

ð Þ k � 1 ; …; yl k � dy ; … (16)

Hunt et al. [20].

Neural networks as fault estimators with ^<sup>f</sup> <sup>i</sup>

Fault Diagnosis Techniques for a Wind Turbine System DOI: http://dx.doi.org/10.5772/intechopen.83810

Figure 4.

relation of Eq. (16):

^f i

where ^<sup>f</sup> <sup>i</sup>

yj

fi

55

#### 3.2 Neural network modelling and training

This study proposes a different data-driven approach, based on neural networks, which is exploited to implement the fault diagnosis block. This section briefly recalls their general structure and properties, which are used to implement the fault estimators.

Therefore, according to the scheme shown in Figure 4, a bank of neural networks is realised in order to reproduce the behaviour of the faults affecting the system under diagnosis using a proper set of input and output measurements. The neural network structure consists of different layers of neurons, also known as perceptron, see Haykin [18], modelled as a static function f. This function is described by an activation function with multiple inputs properly weighted by unknown parameters that determine the learning capabilities of the whole network.

A categorisation of these learning structures concerns the way in which their neurons are connected to each other, see Xu et al. [19]. This work proposes to use a feedforward network, also called multilayer perceptron, where the neurons are grouped into unidirectional layers. The first of them, the input layer, is directly fed by the network inputs; then, a hidden layer takes the inputs from the neurons of the input layer and transmits them the output to the neurons of the third layer, the output layer, which produces the final network outputs. According to this structure, neurons are connected from one layer to the next, but not within the same layer. The only constraint is the number of neurons in the output layer, that has to be equal to the number of actual network outputs. On the other hand, recurrent

Fault Diagnosis Techniques for a Wind Turbine System DOI: http://dx.doi.org/10.5772/intechopen.83810

A powerful approach to the design of the ith FIS as approximator for the system under diagnosis begins with the partitioning of the available data uð Þk and yð Þk of Eq. (7) into subsets, known as cluster. A cluster is defined as a set of data that are more similar to each other rather than to the members of another cluster. The similarity among data can be expressed in terms of their distance from a particular item, exploited as the cluster prototype. Fuzzy clustering provides an effective tool to obtain a partitioning of data in which the transitions among subsets are smooth, rather than abrupt. Moreover, fuzzy clustering assumes that the data of each cluster are characterised by an affine behaviour, which is indeed modelled by the relation of Eq. (12). Different clustering methods have been proposed in literature, see for example, more recent works Graaff and Engelbrecht [15] and Jun et al. [16]. With reference to this work, the design of the FIS is considered as a system identification problem from the noisy data of Eqs. (7). In fact, the estimation of the consequent parameters a<sup>i</sup> and bi of Eq. (12) is required using the input-output data for designing the bank of the fault estimations reported in Figure 3. Moreover, the data are acquired from the measurements selected from the procedure suggested in Section 3.3. The identification scheme exploited in this work was proposed by the authors in Fantuzzi et al. [17]. This approach is based on the minimisation of the

prediction errors of the individual TS local affine models considered as

represented by Eq. (7).

Fault Detection, Diagnosis and Prognosis

estimators.

54

with curve fitting methods, see Babuška [9].

3.2 Neural network modelling and training

nC-independent estimation problems. Their solutions rely on the estimation of errors-in-variables models in Fantuzzi et al. [17], which is also the assumption

Another key aspect, which is not considered here, regards the determination of the optimal number of clusters nC, as the clustering algorithm assumes that the number of clusters nC has been fixed. These issues are considered in the development of the estimation procedure properly integrated by the authors, which also determines the antecedent degrees of fulfilment μik required by Eq. (13) and solved

This study proposes a different data-driven approach, based on neural networks,

Therefore, according to the scheme shown in Figure 4, a bank of neural networks is realised in order to reproduce the behaviour of the faults affecting the system under diagnosis using a proper set of input and output measurements. The neural network structure consists of different layers of neurons, also known as perceptron, see Haykin [18], modelled as a static function f. This function is described by an activation function with multiple inputs properly weighted by unknown parameters that determine the learning capabilities of the whole network. A categorisation of these learning structures concerns the way in which their neurons are connected to each other, see Xu et al. [19]. This work proposes to use a feedforward network, also called multilayer perceptron, where the neurons are grouped into unidirectional layers. The first of them, the input layer, is directly fed by the network inputs; then, a hidden layer takes the inputs from the neurons of the input layer and transmits them the output to the neurons of the third layer, the output layer, which produces the final network outputs. According to this structure, neurons are connected from one layer to the next, but not within the same layer. The only constraint is the number of neurons in the output layer, that has to be equal to the number of actual network outputs. On the other hand, recurrent

which is exploited to implement the fault diagnosis block. This section briefly recalls their general structure and properties, which are used to implement the fault

Figure 4. Neural networks as fault estimators with ^<sup>f</sup> <sup>i</sup> ð Þ¼ k rið Þk .

networks are multilayer networks, in which the output of some neurons is fed back to neurons belonging to previous layers, thus the information flow in forward as well as in backward directions, allowing a dynamic memory inside the network, see Hunt et al. [20].

A noteworthy intermediate solution is provided by the multilayer perceptron with a tapped delay line, which is a feedforward network whose inputs come from a delay line. This study proposes to use this solution, defined as quasistatic neural network, as it represents a suitable tool to predict dynamic relationships between the input-output measurements and the considered fault functions. In this way, another NARX description is obtained, since the non-linear (static) network is fed by the delayed samples of the system inputs and outputs selected by the fault sensitivity analysis tool described in Section 3.3. Indeed, if properly trained, the NARX network can estimate the current (and the next) fault samples fj ð Þk on the basis of the selected past measurements of system inputs and outputs ulð Þk and yj ð Þk , respectively, in the same way of the fuzzy systems.

Therefore, with reference to the ith residual generator of Figure 4, which is used to design the estimator bank of Figure 3, this NARX network is described by the relation of Eq. (16):

$$\hat{f}\_{\text{i}}(k) = F(\ldots, u\_{\text{j}}(k), \ldots, u\_{\text{j}}(k - d\_{\text{u}}), \ldots y\_{\text{l}}(k - 1), \ldots, y\_{\text{l}}(k - d\_{\text{y}}), \ldots) \tag{16}$$

where ^<sup>f</sup> <sup>i</sup> ð Þk is the estimation of the generic ith fault, while ujð Þ� and yl ð Þ� are the generic jth and lth components of the measured inputs and outputs u and y, respectively, that are selected via the fault sensitivity analysis tool. k is the time step, du and dy are the number of delay of inputs and outputs, respectively, which have to be properly selected. Fð Þ� is the function realised by the static neural network, which depends on the layer architecture, the number of neurons, their weights and their activation functions. The NARX network used as generic fault fi ð Þk estimator is depicted in Figure 4.

The design parameters are represented by the number of neurons and the number of delays of the network inputs and outputs, while the value of the weights of each neuron are derived from the network training from the data acquired from the system under diagnosis, see Hunt et al. [20].

#### 3.3 Fault sensitivity analysis

The design of the fault diagnosis schemes proposed for the application example considered in this chapter have been summarised in Section 4. However, the tool addressed in this chapter enhances the design of the banks of these fault estimators depicted in Figure 3.

This tool consists of a fault sensitivity analysis that has to be performed on the wind turbine simulator. It is aimed at defining the most sensitive measurements ujð Þk and yl ð Þk with respect to the fault conditions fi ð Þk considered in Section 2.2. In practice, the considered fault signals have been injected into the wind turbine simulator, assuming that only a single fault may occur. Then, the relative mean square errors (RMSE) between the fault-free and faulty measured signals are evaluated, so that, for each fault, the most sensitive signal ujð Þk and yl ð Þk can be selected. The results of the fault sensitivity analysis are summarised in Table 2 for the wind turbine system.

In particular, the fault sensitivity analysis is conducted on the basis of a selection algorithm that is performed by introducing the normalised sensitivity function Nx, defined in Eq. 17:

$$N\_{\mathfrak{x}} = \frac{\mathbb{S}\_{\mathfrak{x}}}{\mathbb{S}\_{\mathfrak{x}}^{\*}}\tag{17}$$

measurements that are most affected by the considered fault lead to a value of Nx equal to 1. Otherwise, a smaller value of Nx, that is, close to zero, represents a signal x kð Þ not affected by the fault. Those signals characterised by high value of Nx are thus selected as the most sensitive measurements, and they will be considered in the

Fault case fi Most sensitive inputs uj Most sensitive outputs yl

 β1,m1, β1,m<sup>2</sup> ωg,m<sup>2</sup> β1,m2, β2,m<sup>2</sup> ωg,m<sup>2</sup> β1,m2, β3,m<sup>1</sup> ωg,m<sup>2</sup> β1,m<sup>2</sup> ωg,m2, ωr,m<sup>1</sup> β1,m<sup>2</sup> ωg,m2, ωr,m<sup>2</sup> β1,m2, β2,m<sup>1</sup> ωg,m<sup>2</sup> β1,m2, β3,m<sup>2</sup> ωg,m<sup>2</sup> β1,m2, τg,m ωg,m<sup>2</sup> β1,m<sup>2</sup> ωg,m1, ωg,m<sup>2</sup>

Fault Diagnosis Techniques for a Wind Turbine System DOI: http://dx.doi.org/10.5772/intechopen.83810

The complete results of the fault sensitivity analysis are summarised in Table 3. For each fault case, the selected signals of the wind turbine benchmark are marked

This method represents a key feature of the proposed approach to fault diagno-

exploiting a reduced number of signals, thus leading to a noteworthy simplification of the overall complexity, and a decrease in the computational cost of the training

This section summarises the simulations performed with the considered wind turbine benchmark, and the performances of the proposed fault diagnosis solutions. Due to the presence of the uncertainty and disturbance effects included in the benchmark, the robustness features of the developed fault diagnosis techniques are

With reference to the wind turbine benchmark of Section 2, all simulations are driven by the same wind mean speed sequence. It was acquired from a real measurement of wind speed, which represents a good coverage of typical operating conditions, as it ranges from 5 to 20 m/s, with a few spikes at 25 m/s, see Odgaard et al. [12]. The simulations last for 4400 s, with single fault occurrences. The discrete-time simulator runs at a sampling frequency of 100 Hz, so that

N = 440,000 samples are acquired during each simulation. With reference to the different fault cases reported in Section 2.2, Table 4 shows the shape and the timing of the fault modes affecting the process. They model input (actuator) or output (sensor) additive faults, which are used for sensitivity analysis of Section 3.3.

As an example, in order to highlight the actual fault effect on the wind turbine measurements, Figure 5 shows the fault sensitivity test. In particular, the cases of

the faults 1, 2, 3 and 8 in fault-free and faulty conditions are depicted.

design of the fault diagnosis modules of the bank sketched in Figure 3.

sis. In fact, the fault estimators of the bank of Figure 3 can be designed by

as inputs or outputs.

Table 3.

Fault sensitivity test.

and identification phases.

4. Simulation results

also verified in simulation.

57

with

$$\mathcal{S}\_{\mathbf{x}} = \frac{||\boldsymbol{\omega}\_{f}(\boldsymbol{k}) - \boldsymbol{\omega}\_{n}(\boldsymbol{k})||\_{2}}{||\boldsymbol{\omega}\_{n}(\boldsymbol{k})||\_{2}}\tag{18}$$

and

$$S\_{\mathbf{x}}^{\*} = \max \frac{\left\| \boldsymbol{\omega}\_{f}(\boldsymbol{k}) - \boldsymbol{\omega}\_{n}(\boldsymbol{k}) \right\|\_{2}}{\left\| \boldsymbol{\omega}\_{n}(\boldsymbol{k}) \right\|\_{2}} \tag{19}$$

The value of Nx indicates the effect of the considered fault case with respect to the general measured signal x kð Þ, with k ¼ 1, 2, …, N. The subscripts 'f' and 'n' indicate the faulty and the fault-free case, respectively. Therefore, the


Table 2.

Fault sensitivity fi ð Þk with respect ujð Þk and yl ð Þk . Fault Diagnosis Techniques for a Wind Turbine System DOI: http://dx.doi.org/10.5772/intechopen.83810

