**3.1 Wind speed data modeling**

The relation between wind speed and a WT rotor power output is expressed as [33]

$$P\_{ro} = \mathbf{0}.\mathsf{5}\rho A\_{\mathsf{SA}} \mathbf{C}\_{p}(\lambda,\beta)v^{\mathsf{3}} \tag{2}$$

*vw*<sup>2</sup> ¼ *vw*<sup>1</sup>

the shear exponent that is expressed as

*DOI: http://dx.doi.org/10.5772/intechopen.86357*

*Microgrid System Reliability*

where *Z*<sup>0</sup> is the surface roughness.

as normal, log-normal, exponential, and Weibull.

and is performed according to the statistic for MANN'S test:

*k*2∑*<sup>k</sup>*<sup>1</sup>

*<sup>M</sup>* <sup>¼</sup> *<sup>k</sup>*1∑*<sup>r</sup>*�<sup>1</sup>

*3.1.3 Distribution parameter estimation*

a linear regression line as in Eq. (7):

where

and (9).

follows:

**175**

data.

*3.1.2 Goodness-of-fit test*

*h*2 *h*1 *<sup>α</sup>*

*<sup>α</sup>* <sup>¼</sup> ð Þ <sup>0</sup>*:*096 log ð Þþ *Zo* <sup>0</sup>*:*016 log ð Þ *Zo* <sup>2</sup> <sup>þ</sup> <sup>0</sup>*:*<sup>24</sup> (5)

where *h*<sup>1</sup> and *h*<sup>2</sup> are the height of anemometer and hub, respectively; *vw*<sup>1</sup> and *vw*<sup>2</sup> are the wind velocity at anemometer height and at hub height, respectively; and *α* is

• Use Matlab distribution fitting tool to obtain probability plot of the scaled wind

• Fit the probability plot of the scaled wind data for different distributions such

• Identify the distribution corresponding to the best fit of the probability plots.

The best-fit distribution of the site wind data is tested for the goodness-of-fit

To determine the Weibull distribution parameters, the least-squares technique is

The values of *a* and *b* are determined from the least-squares fit using Eqs. (8)

By knowing the values *a* and *b*, the Weibull parameters are determined as

*<sup>θ</sup>ws* <sup>¼</sup> exp � *<sup>a</sup>*

*b*

 � ln *vwi* ð Þ *<sup>=</sup>Mi* 

(6)

*yi* ¼ *a* þ *bxi* (7)

*xi* ¼ ln *vwi* (8) *yi* ¼ *Zi* (9) *a* ¼ �*βws*ln *θ* (10) *b* ¼ *βws* (11)

(12)

*βws* ¼ *b* (13)

� ln *vwi* ð Þ *<sup>=</sup>Mi*

*<sup>i</sup>*¼*k*1þ<sup>1</sup> ln *vwi*þ<sup>1</sup>

*<sup>i</sup>*¼<sup>1</sup> ln *vwi*þ<sup>1</sup>

used because of its accuracy to fit a straight line in a given data points. In this approach, the wind speed field data are transformed to Weibull distribution to fit to

(4)

where:


The relation in Eq. (2) can be expressed as

$$P\_{vv} \propto v\_w^3 \tag{3}$$

Since wind speed is the main factor that creates uncertainty at the power output of a wind energy conversion, wind speed is considered here as the key factor to estimate the MSR. In order to relate the effects of wind speed in calculating the system's overall reliability, wind speed field data modeling is gathered. This is essential because the data itself varies not only from site to site but also according to the hub heights of the wind turbine. Wind speed data modeling for a wind turbine system includes:


#### *3.1.1 Identification of best-fit distribution*

The probability plot method is used to identify the best-fit distribution of the available wind data for a given site and for a given wind turbine hub height. The following steps are taken to accomplish the fitting of the wind data to a distribution:


*Microgrid System Reliability DOI: http://dx.doi.org/10.5772/intechopen.86357*

speed data, some of the wind speeds are below the cut-in speed of the wind turbine and, as such, will not produce power at the wind turbine output. Such wind speed data can be considered as failure events, which occur randomly. In addition, this research focuses on assessing the reliability of the microgrid system in power generation and supply, considering the wind speed as the primary uncertainty of the

The relation between wind speed and a WT rotor power output is expressed as [33]

• *ρ* is the air density. Note that for a given WT, *ASA*, *Cp*, *β*, *λ*, and *ρ* are constant.

*Pro* ∞ *v*<sup>3</sup>

wind turbine. Wind speed data modeling for a wind turbine system includes:

The probability plot method is used to identify the best-fit distribution of the available wind data for a given site and for a given wind turbine hub height. The following steps are taken to accomplish the fitting of the wind data to a distribution:

• Scale the wind data according to the hub height of the wind turbine using Eq. (4):

a. Identifying best-fit distribution for 1-year wind field data

• Obtain 1-year wind speed data from the site measurement.

Since wind speed is the main factor that creates uncertainty at the power output of a wind energy conversion, wind speed is considered here as the key factor to estimate the MSR. In order to relate the effects of wind speed in calculating the system's overall reliability, wind speed field data modeling is gathered. This is essential because the data itself varies not only from site to site but also according to the hub heights of the

*Pro* <sup>¼</sup> <sup>0</sup>*:*5*ρASACp*ð Þ *<sup>λ</sup>; <sup>β</sup> <sup>v</sup>*<sup>3</sup> (2)

*<sup>w</sup>* (3)

system. Hence, Monte Carlo is applied and presented herein.

• *ASA* is the swept area covered by the turbine rotor.

**3.1 Wind speed data modeling**

*Reliability and Maintenance - An Overview of Cases*

• *Cp* is the power coefficient.

• *β* is the pitch angle of rotor blades.

b.Evaluating the goodness-of-fit test

*3.1.1 Identification of best-fit distribution*

**174**

c. Estimating the distribution parameters

The relation in Eq. (2) can be expressed as

• *vw* is the wind velocity.

• *λ* is the tip speed ratio.

where:

$$\upsilon\_{w2} = \upsilon\_{w1} \left(\frac{h\_2}{h\_1}\right)^a \tag{4}$$

where *h*<sup>1</sup> and *h*<sup>2</sup> are the height of anemometer and hub, respectively; *vw*<sup>1</sup> and *vw*<sup>2</sup> are the wind velocity at anemometer height and at hub height, respectively; and *α* is the shear exponent that is expressed as

$$a = \left(0.096 \log \left(Z\_o\right) + 0.016 \log \left(Z\_o\right)\right)^2 + 0.24\tag{5}$$

where *Z*<sup>0</sup> is the surface roughness.


#### *3.1.2 Goodness-of-fit test*

The best-fit distribution of the site wind data is tested for the goodness-of-fit and is performed according to the statistic for MANN'S test:

$$\mathcal{M} = \frac{k\_1 \sum\_{i=k\_1+1}^{r-1} \left[ \left( \ln \left( v\_{w\_{i+1}} \right) - \ln \left( v\_{w\_i} \right) \right) / \mathcal{M}\_i \right]}{k\_2 \sum\_{i=1}^{k\_1} \left[ \left( \ln \left( v\_{w\_{i+1}} \right) - \ln \left( v\_{w\_i} \right) \right) / \mathcal{M}\_i \right]} \tag{6}$$

#### *3.1.3 Distribution parameter estimation*

To determine the Weibull distribution parameters, the least-squares technique is used because of its accuracy to fit a straight line in a given data points. In this approach, the wind speed field data are transformed to Weibull distribution to fit to a linear regression line as in Eq. (7):

$$y\_i = a + b\mathbf{x}\_i \tag{7}$$

where

$$\mathbf{x}\_i = \ln \boldsymbol{v}\_{wi} \tag{8}$$

$$\mathbf{y}\_i = \mathbf{Z}\_i \tag{9}$$

$$
\sigma = -\beta\_{ws} \ln \theta \tag{10}
$$

$$b = \beta\_{ws} \tag{11}$$

The values of *a* and *b* are determined from the least-squares fit using Eqs. (8) and (9).

By knowing the values *a* and *b*, the Weibull parameters are determined as follows:

$$\theta\_{\text{ov}} = \exp\left(-\frac{a}{b}\right) \tag{12}$$

$$
\boldsymbol{\beta}\_{ws} = \mathbf{b} \tag{13}
$$

where *θws* and *βws* are defined as the scale and shape parameters, respectively, for wind speed field data.

to estimate the shape parameter and the scale factor of the gearbox. Its reliability *Rgb*

*<sup>θ</sup>gb* � �*<sup>β</sup>gb* " # � exp � *<sup>ω</sup>wt,m*

• *θgb* and *βgb* are the shape parameter and scale factor for speed seen by the gearbox.

The reliability at the *ith* speed seen by the gearbox, *Rgb,wti*, can be estimated as

In order to account the effect of wind speed in estimating the wind generator's reliability of generating power, the estimation of Weibull parameters by using field data is shown herein. Such parameters are utilized to generate a set of random wind speed data. Power generated by the WT is then determined using Eq. (2). However, the power at the generator output depends on the gearbox efficiency and various losses in the generator. Efficiency of the gearbox (0.95) and generator (0.95) is considered as 90%, which is observed from the system modeling and simulation. The power at the generator output can be determined as 90% of the power at the turbine output. Thus, a power distribution at the generator output can be obtained, which also follows a Weibull distribution. This, in turn, is used to estimate Weibull distribution parameters using the least-squares parameter estimation technique. After knowing the distribution parameters of the generator output power, the reliability of generating power by the generator, *Rg*, can be evaluated as

*<sup>θ</sup>gp* � �*<sup>β</sup>gp* " # � exp � *Pg,cow*

• *θgp* and *βgp* are considered as shape parameter and scale factor for the generator

• *Pg,ciw* and *Pg,cow* are the generator power at the cut-in and cut-out wind speeds,

The reliability of generating power *Pg,i* of the generator, *RPg,i*, can be expressed as

*RPg,i* <sup>¼</sup> exp � *Pg,i*

*Rgbwt,i* <sup>¼</sup> exp � *<sup>ω</sup>wt,i*

*<sup>θ</sup>gb* � �*<sup>β</sup>gb* " # (17)

*<sup>θ</sup>gb* � �*<sup>β</sup>gb* " # (18)

*<sup>θ</sup>gp* � �*<sup>β</sup>gp* " # (19)

*<sup>θ</sup>gp* � �*<sup>β</sup>gp* " # (20)

*Rgb* <sup>¼</sup> exp � *<sup>ω</sup>wt,s*

• *ωwt,s* is the starting speed of the wind turbine.

• *ωwt,m* is the maximum operating speed of the wind turbine.

where *ωwt,I* is the *ith* speed of the WT seen by the gearbox.

*Rg* <sup>¼</sup> exp � *Pg,ciw*

can be expressed as

*Microgrid System Reliability*

*DOI: http://dx.doi.org/10.5772/intechopen.86357*

where:

*3.2.3 Generator*

where:

**177**

power distribution.

respectively.

## **3.2 Wind power generation system**

According to the microgrid configuration, all nine WTs in WPGS are connected in parallel, which are shown in the simplified RBD in **Figure 4**. In order to estimate the reliability of power generation by the WPGS, a single WT system is considered because all of them are identical both in terms of topology and subsystems context. A WT system comprising of different subsystems is shown in **Figure 6**. The different subsystems are connected in series because failure of power generation by any of the subsystems is considered as a failure of the WT system to generate power. The modeling of the reliability estimation of different subsystems in a WT system is described in the following subsections:

### *3.2.1 Wind turbine rotor*

The wind speed field data model provides information about the shape parameter and scale factor for a Weibull distribution. Such parameters are used to generate a series of random wind speed data that follow a Weibull distribution pattern. Randomly generated data are used to determine power generation by the WT using Eq. (2), which represents a Weibull distribution of power generation. Weibull parameters are determined using the parameter estimation technique described in Section 3.1. These are defined as *θtp* and *βtp*. Thus, the WT's rotor reliability *Rtp* can be expressed as

$$R\_{tp} = \exp\left[-\left(\frac{P\_{ciw}}{\theta\_{tp}}\right)^{\beta\_{tp}}\right] - \exp\left[-\left(\frac{P\_{cow}}{\theta\_{tp}}\right)^{\beta\_{tp}}\right] \tag{14}$$

where *θtp* and *βtp* are defined as shape parameter and scale factor for power distribution. *Pciw* and *Pcow* are the power at cut-in and cut-out wind speed, respectively.

The reliability of generating power at the *ith* wind speed, *RPi*, can be expressed as

$$R\_{Pi} = \exp\left[-\left(\frac{P\_i}{\theta\_{tp}}\right)^{\theta\_{tp}}\right] \tag{15}$$

where *Pi* is the power for *ith* wind speed in between cut-in and cut-out regions.

#### *3.2.2 Gearbox*

Weibull parameters obtained from field data modeling are utilized to produce a set of random wind data. Such data are used to determine the wind turbine speed using Eq. (16):

$$
\rho\_{wt} = \frac{\lambda \nu\_w}{R\_t} \tag{16}
$$

where *ωwt* is the wind turbine speed and *Rt* is the turbine radius, respectively. The wind turbine speed is also the speed seen by the gearbox's low-speed shaft. This can be represented as a Weibull distribution of speed. Such a distribution is utilized to estimate the shape parameter and the scale factor of the gearbox. Its reliability *Rgb* can be expressed as

$$R\_{\rm gb} = \exp\left[-\left(\frac{\alpha\_{\rm wt,t}}{\theta\_{\rm gb}}\right)^{\beta\_{\rm gb}}\right] - \exp\left[-\left(\frac{\alpha\_{\rm wt,m}}{\theta\_{\rm gb}}\right)^{\beta\_{\rm gb}}\right] \tag{17}$$

where:

where *θws* and *βws* are defined as the scale and shape parameters, respectively, for

According to the microgrid configuration, all nine WTs in WPGS are connected in parallel, which are shown in the simplified RBD in **Figure 4**. In order to estimate the reliability of power generation by the WPGS, a single WT system is considered because all of them are identical both in terms of topology and subsystems context. A WT system comprising of different subsystems is shown in **Figure 6**. The different subsystems are connected in series because failure of power generation by any of the subsystems is considered as a failure of the WT system to generate power. The modeling of the reliability estimation of different subsystems in a WT system is

The wind speed field data model provides information about the shape parameter and scale factor for a Weibull distribution. Such parameters are used to generate a series of random wind speed data that follow a Weibull distribution pattern. Randomly generated data are used to determine power generation by the WT using Eq. (2), which represents a Weibull distribution of power generation. Weibull parameters are determined using the parameter estimation technique described in Section 3.1. These are defined as *θtp* and *βtp*. Thus, the WT's rotor reliability *Rtp* can

� exp � *Pcow*

*θtp* � �*<sup>β</sup>tp* " #

(14)

(15)

(16)

wind speed field data.

*3.2.1 Wind turbine rotor*

be expressed as

respectively.

*3.2.2 Gearbox*

using Eq. (16):

**176**

**3.2 Wind power generation system**

*Reliability and Maintenance - An Overview of Cases*

described in the following subsections:

*Rtp* <sup>¼</sup> exp � *Pciw*

*θtp* � �*<sup>β</sup>tp* " #

distribution. *Pciw* and *Pcow* are the power at cut-in and cut-out wind speed,

*RPi* <sup>¼</sup> exp � *Pi*

where *θtp* and *βtp* are defined as shape parameter and scale factor for power

The reliability of generating power at the *ith* wind speed, *RPi*, can be expressed as

where *Pi* is the power for *ith* wind speed in between cut-in and cut-out regions.

Weibull parameters obtained from field data modeling are utilized to produce a set of random wind data. Such data are used to determine the wind turbine speed

> *<sup>ω</sup>wt* <sup>¼</sup> *<sup>λ</sup>vw Rt*

where *ωwt* is the wind turbine speed and *Rt* is the turbine radius, respectively. The wind turbine speed is also the speed seen by the gearbox's low-speed shaft. This can be represented as a Weibull distribution of speed. Such a distribution is utilized

*θtp* � �*<sup>β</sup>tp* " #


The reliability at the *ith* speed seen by the gearbox, *Rgb,wti*, can be estimated as

$$R\_{\mathcal{g}b\_{wt,i}} = \exp\left[-\left(\frac{o\_{wt,i}}{\theta\_{\mathcal{g}b}}\right)^{\beta\_{\mathcal{g}b}}\right] \tag{18}$$

where *ωwt,I* is the *ith* speed of the WT seen by the gearbox.

### *3.2.3 Generator*

In order to account the effect of wind speed in estimating the wind generator's reliability of generating power, the estimation of Weibull parameters by using field data is shown herein. Such parameters are utilized to generate a set of random wind speed data. Power generated by the WT is then determined using Eq. (2). However, the power at the generator output depends on the gearbox efficiency and various losses in the generator. Efficiency of the gearbox (0.95) and generator (0.95) is considered as 90%, which is observed from the system modeling and simulation. The power at the generator output can be determined as 90% of the power at the turbine output. Thus, a power distribution at the generator output can be obtained, which also follows a Weibull distribution. This, in turn, is used to estimate Weibull distribution parameters using the least-squares parameter estimation technique. After knowing the distribution parameters of the generator output power, the reliability of generating power by the generator, *Rg*, can be evaluated as

$$R\_{\rm g} = \exp\left[-\left(\frac{P\_{\rm g,ciw}}{\theta\_{\rm gp}}\right)^{\beta\_{\rm gp}}\right] - \exp\left[-\left(\frac{P\_{\rm g,cow}}{\theta\_{\rm gp}}\right)^{\beta\_{\rm gp}}\right] \tag{19}$$

where:


The reliability of generating power *Pg,i* of the generator, *RPg,i*, can be expressed as

$$R\_{\mathbf{f},i} = \exp\left[-\left(\frac{P\_{\mathbf{g},i}}{\theta\_{\mathbf{g}p}}\right)^{\beta\_{\mathbf{g}}}\right] \tag{20}$$

where *Pg,I* is the generator power at the *ith* wind speed in between cut-in and cut-out regions.
