A novel method for reliability and risk evaluation of wind energy conversion systems considering wind speed correlation

Abstract

This paper investigates an analytical approach for the reliability modeling of doubly fed induction generator (DFIG) wind turbines. At present, to the best of the authors’ knowledge, wind speed and wind turbine generator outage have not been addressed simultaneously. In this paper, a novel methodology based on the Weibull-Markov method is proposed for evaluating the probabilistic reliability of the bulk electric power systems, including DFIG wind turbines, considering wind speed and wind turbine generator outage. The proposed model is presented in terms of appropriate wind speed modeling as well as capacity outage probability table (COPT), considering component failures of the wind turbine generators. Based on the proposed method, the COPT of the wind farm has been developed and utilized on the IEEE RBTS to estimate the well-known reliability and sensitive indices. The simulation results reveal the importance of inclusion of wind turbine generator outage as well as wind speed in the reliability assessment of the wind farms. Moreover, the proposed method reduces the complexity of using analytical methods and provides an accurate reliability model for the wind turbines. Furthermore, several case studies are considered to demonstrate the effectiveness of the proposed method in practical applications.

Graphical abstract

Keywords

doubly-fed induction generator (DFIG) / composite system adequacy assessment / wind speed correlation

1 Introduction

Wind power generation in electric power systems has been widely recognized as a significant energy sources and will increase considerably throughout the world in the near future. Many countries (e.g., US, Germany, China and Denmark) have implemented or are in the process of implementing policies to integrate large scale wind power generation into the power system in the near future [1]. Nowadays, the most widely used generator types for units above 1 MW are based on the doubly fed induction generator (DFIG) technology [2−5]. In recent years, many research efforts have been devoted to reliability assessment of power systems containing wind energy. For instance, the mathematical models of the wind turbines in order to evaluate the reliability of the system have been investigated [6−9]. The reliability of power systems considering the effect of wind site correlation has been estimated [10−12]. Several factors which affect the reliability of wind generation systems have been studied in Ref. [13]. The importance of reliability studies in long-term operation of wind generation systems has been investigated [14]. Probabilistic reliability evaluation of wind generation systems has been studied [15]. The failure and repair rates of the wind turbines have been discussed [16]. A method based on the sequential Monte Carlo simulation for reliability evaluation of wind generation system has been proposed [17]. However, adequacy assessment of composite generation system incorporating large-scale wind power generation considering wind turbine outage and wind speed have not been addressed simultaneously. As a complement to the previous research works, the objectives of this paper are to extend a comprehensive model and in-depth investigation which simultaneously considers the wind speed and wind turbine outage. The Weibull-Markov method is used to investigate the effects of incorporating large-scale wind generation systems in bulk power system adequacy evaluation. This paper applies a probabilistic model of wind farm generation to actual time series of wind speed. Several case studies are conducted by adding large-scale wind generation system to the Roy Billinton test system (RBTS). The reliability indices such as loss of load expectation (LOLE), energy not supplied (EENS), probability of load curtailment (PLC), expected demand not supplied (EDNS), expected damage cost (EDC), bulk power/energy curtailment index (BPECI), modified bulk energy curtailment index (MBECI), and severity index (SI) are used for quantitative reliability analysis.

2 Wind turbine structure

The structure of the Vestas V90-2MW system is shown in Fig. 1. The majority of wind turbines are based on the DFIG technology because of its efficiency. This type of turbine was first installed in 2004. The Vestas V90-2MW system is an upwind turbine with electrically driven yaw and three blades. Its rotor has a weight of 38 tons, a nominal rotational speed of 14.9 r/min, and a diameter of 90 m. To perform the optimum power output, the pitch control system with individual pitching capability for each blade consistently adapts the blade angle to the wind direction. Additionally, it serves to control speed, turbine stops and starts-up by aerodynamic braking. A disk brake is also installed on the high-speed shaft [18−21]. All Vestas V90-2MW systems apply hybrid gearbox with two parallel-shaft stages and one planetary. The torque is then transmitted to the turbine generator through a composite coupling. A converter is used to control the current in the rotor circuit of the turbine generator, making it possible to control the reactive power which serves for smooth connection to the electric power system. In particular, the rotor speed of the turbines based on the DFIG technology varies by 30% above and below synchronous speed. The Vestas V90-2MW systems can provide up to 2 MW of electric power at 690 V and 50 Hz to the grid based on the wind speeds of 4 to 25 m/s and in a standard operating temperature ranges of − 20°C, and+ 30°C. The main components of the Vestas V90-2MW systems consist of the tower structure, rotor (blades and pitch control), mechanical gear, electrical generator, yaw mechanism, sensors and control, brake system, and transformer.

3 Reliability evaluation framework

3.1 Definitions

Reliability is defined as the probability of a component performing its function properly over a time period of t. Reliability function of a system can be obtained from Refs. [22,23].

(1) R(t)=Pr\{\mathit{T}>\mathit{t}\},

where T is the time to failure of the system (T≥0). Following this, cumulative probability distribution function can be given as

(2)

Also, the failure rate function can be expressed as

(3) \lambda (t)=\underset{\mathrm{\Delta }t\to \mathrm{0}}{\lim }\left\{{\displaystyle \frac{-[R(t+\mathrm{\Delta }t)-R(t)]}{\mathrm{\Delta }t}}\right\}{\displaystyle \frac{\mathrm{1}}{R(t)}}=-\left[{\displaystyle \frac{dR(t)}{dt}}\right]{\displaystyle \frac{\mathrm{1}}{R(t)}},

where

(4) \lambda (t)dt=-\left[{\displaystyle \frac{dR(t)}{R(t)}}\right].

Integrating Eq. (4), it can be concluded that

(5) {\displaystyle {\int }_{\mathrm{0}}^t\lambda (t^{\prime })d{t}^{\prime }={\displaystyle {\int }_t^{R(t)}{\displaystyle \frac{-dR(t^{\prime })}{R(t^{\prime })}}}.}

In terms of R(0) = 1, there is

(6) -{\displaystyle {\int }_{\mathrm{0}}^t\lambda (t^{\prime })d{t}^{\prime }=\ln R(t)},

or

(7) R(t)=\exp \left[-{\displaystyle {\int }_{\mathrm{0}}^t\lambda (t^{\prime })d{t}^{\prime }}\right].

The cumulative failure rate over the time t can be defined as

(8) L(t)={\displaystyle {\int }_{\mathrm{0}}^t\lambda (t^{\prime })d{t}^{\prime }}.

3.2 Weibull distribution

The Weibull distribution serves as a useful tool for a small and discrete set of failure rate of wind turbine systems. The shape factor in the Weibull distribution performs a wide range of shapes. The Weibull distribution is defined by Ref. [24] as

(9) f_D(\tau )={\displaystyle \frac{k}{c^k}}{\tau }^{k-\mathrm{1}}\exp \left[-{\left({\displaystyle \frac{\tau }{c}}\right)}^k\right],

where k and c denote the shape factor and the scale factor of the Weibull distribution, respectively.

Also, f _D(τ) can be written as

(10) f_D(\tau )=\mathrm{1}-\exp \left[-{\left({\displaystyle \frac{\tau }{c}}\right)}^k\right].

The Weibull probability distribution function (PDF) is similar to the negative exponential distribution when the shape parameter is k = 1.0. The probability density for a component to fail in a certain time of τ is defined as

(11) h_D(\tau )={\displaystyle \frac{k}{c^k}}{\tau }^{k-\mathrm{1}}.

The expected value of the stochastic quantity D can be obtained as

(12) E_D=c\mathrm{\Gamma }\left(\mathrm{1}+{\displaystyle \frac{\mathrm{1}}{k}}\right).

The second central moment is defined as

(13) V_D=c^{\mathrm{2}}\left\{\mathrm{\Gamma }\left(\mathrm{1}+{\displaystyle \frac{\mathrm{2}}{k}}\right)-{\left[\mathrm{\Gamma }\left(\mathrm{1}+{\displaystyle \frac{\mathrm{1}}{k}}\right)\right]}^{\mathrm{2}}\right\},

where Г(x) is the normal gamma function which is defined as

(14) \mathrm{\Gamma }(x)={\displaystyle {\int }_{\mathrm{0}}^{\infty }{t}^{x-\mathrm{1}}{e}^{-t}dt}.

The Weibull function can be transformed into the straight lines by taking the double logarithm of the reciprocal of the survival function as

(15) \ln \left({\displaystyle \frac{\mathrm{1}}{R_D(\tau )}}\right)={\left({\displaystyle \frac{\tau }{c}}\right)}^k,

(16) \ln \left(\ln \left({\displaystyle \frac{\mathrm{1}}{R_D(\tau )}}\right)\right)=k\ln (\tau )-k\ln (c),

where a plot of ln( − ln(R _D(τ))) against lnτ will deduce the straight lines. In the Weibull distribution, the failure rate function is defined as

(17) \lambda (t)={\displaystyle \frac{k}{c}}{\left({\displaystyle \frac{t}{c}}\right)}^{(k-\mathrm{1})},\hspace{1em}k>\mathrm{0},\hspace{1em}c>\mathrm{0},\hspace{1em}t\ge \mathrm{0.}

Substituting Eq. (17) into Eq. (7), the reliability function can be expressed as

(18) R(t)=\exp \left[-{{\displaystyle {\int }_{\mathrm{0}}^t\left({\displaystyle \frac{k}{c}}\right)\left({\displaystyle \frac{t}{c}}\right)}}^{(k-\mathrm{1})}d{t}^{\prime }\right]=e^{-{(t/c)}^k}.

In this paper, the velocity of wind speed is defined as a random variable following the Weibull distribution and its probability density function is expressed as

(19) f(v)=\left({\displaystyle \frac{k}{c}}\right){\left({\displaystyle \frac{v}{c}}\right)}^{k-\mathrm{1}}\exp \left[-{\left({\displaystyle \frac{v}{c}}\right)}^k\right].

In addition, the equivalent cumulative probability function can be given as

(20) F(v)={\displaystyle {\int }_{\mathrm{0}}^{v}f(v)dv=\mathrm{1}-\exp \left[-{\left({\displaystyle \frac{v}{c}}\right)}^k\right]}.

Moreover, the outage probability of k wind turbine systems can be expressed as

(21) P_k=\left(\begin{array}{l}m\\ k\end{array}\right){\lambda }^k{(\mathrm{1}-\lambda )}^{m-k},\hspace{1em}(k=\mathrm{0},\mathrm{1},\mathrm{2},\cdots ,m).

3.3 Wind power output modeling

In this paper, the adequacy studies are based on the hourly wind speed. The hourly output power of wind systems is defined by the wind speed, wind turbine outage, and the relationship between wind turbine output and wind speed. In general, the nonlinear relationship of operating wind speed and wind turbine output power can be expressed as [25]

(22) P={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{C}_p\rho A{V}^{\mathrm{3}},

where P is the output power in W, ρ is the air density (kg/m³), V is the wind speed (m/s), A is the swept area of the turbine (m²), and C_p is the power coefficient. The wind turbine operates between the cut-in speed and cut-out speed as illustrated in Fig. 2. The relationship between wind turbine output and wind speed can be obtained by

(23)

where P_outage(t) is the output power (MW) at time t; v_i(t) is the simulated wind speed (m/s) at time t; V_ci is the cut-in speed (m/s); V_R is the rated speed (m/s); V_co is the cut-out speed (m/s); and V_R is the rated power (MW).

3.4 Homogenous Markov method

In this paper, the Markov method is used to model the output power of wind farms in reliability calculation. The homogenous Markov model enables the evaluation of state probability, frequency and duration by analytic matrix operations. The homogenous Markov model has advantages over other reliability evaluation methods due to its computational elegance. The monitored stochastic behavior of the components in the system can be obtained through a set of state and epoch combinations {(x_n,t_n)}_{n=\mathrm{0}}^{\infty }. Additionally, the exact same component under the exact same conditions, the results of another set is {(x_n,t_n)}_{n=\mathrm{0}}^{\infty }, as each next state and each state duration (t _n+1− t _n) are stochastic quantities. Each set in {(x_n,t_n)}_{n=\mathrm{0}}^{\infty } is called a component stochastic history as an outcome from an infinite number of possible outcomes. The history for each component with index “c” is given as {(X_{c,n_c},T_{c,n_c})}_{n_c=\mathrm{0}}^{\infty }. Both X_{c,n_c} and T_{c,n_c} are stochastic quantities which means the probability of X_{c,\mathrm{23}}=x_{\mathrm{1}} or the probability density function of (T_{c,\mathrm{45}}-T_{c,44}). The homogenous Markov model can be defined under a few assumptions as [24]:

1)The possible states \overline{x_c}=\mathrm{1},\mathrm{2},\cdots ,N_c, where N _c is the number of states.

2)The history {(X_{c,n_c},T_{c,n_c})}_{n_c=\mathrm{0}}^{\infty }, where

\begin{array}{l}-{\forall }_{n_c}\left(X_{c,n_c}\in \overline{x_c},X_{c,n_c}\ne X_{c,n_c+\mathrm{1}}\right)\hfill \\ -T_{c,\mathrm{0}}=\mathrm{0}\hspace{0.17em}\mathrm{a}\mathrm{n}\mathrm{d}\hspace{0.17em}{\forall }_{{\mathit{n}}_c}(T_{c,n_c+\mathrm{1}}>T_{c,n_c}).\hfill \end{array}

3)The set of continuous probability distribution functions F _c,ij(t) for the state durations D _c,ij.

(24) F_{c,ij}(t)=\mathrm{Pr}(D_{c,ij}\le t)=\mathrm{Pr}({\mathit{X}}_{c\mathit{,}{\mathit{n}}_c}=i,(T_{c,n_c+\mathrm{1}}-T_{c,n_c})\le t|X_{c,n_c+\mathrm{1}}=j)=\mathrm{1}-\exp \left(-{\displaystyle \frac{t}{{\lambda }_{c,ij}}}\right).

From the Markov model, the stochastic process X _c(t) can be expressed as

(25) T_{c,n_c}\le t\le T_{c,n_c+\mathrm{1}}\Rightarrow X_c(t)=X_{c,n_c}.

The state durations D _c,ij are the durations for state i, where th next state is j. The probability distribution for the duration D _c,i of state x _c,i can be obtained as

(26) \begin{array}{r}\hfill F_{c,i}(t)=\mathrm{Pr}(D_{c,i}\le t)=\mathrm{Pr}\left(\underset{j=\mathrm{1}}{\overset{N_c}{\min }}(D_{c,ij})\le t\right)\\ \hfill =\mathrm{1}-{\displaystyle \prod _{j=\mathrm{1}}^{N_c}\mathrm{Pr}(D_{c,ij}>t)=\mathrm{1}-{\displaystyle \prod _{j=\mathrm{1}}^{N_c}\exp \left(-{\displaystyle \frac{t}{{\lambda }_{c,ij}}}\right)}}\\ \hfill =\mathrm{1}-\exp \left(-{\displaystyle \frac{t}{{\lambda }_{c,i}}}\right).\end{array}

(27) {\displaystyle \frac{\mathrm{1}}{{\lambda }_{c,i}}}={\displaystyle \sum _{j=\mathrm{1}}^{N_c}{\displaystyle \frac{\mathrm{1}}{{\lambda }_{c,ij}}}}.

The state duration is exponentially distributed and can be defined by the single state transition rate λ _c,i. The state transition rates are defined in “per time” units, which also can be defined as a frequency. Frequency reveals the number of transitions out of the state per time. Moreover, the expected state duration can be obtained from the state transition rate as

(28) E(D_{c,i})={\displaystyle \frac{\mathrm{1}}{{\lambda }_{c,i}}}.

For the transition probability P_c(i,j)=\mathrm{Pr}(X_{c,n_c+\mathrm{1}}=j|X_{c,n_c}=i), there are

(29) \begin{array}{r}\hfill P_c(i,j)=\mathrm{Pr}\left(D_{c,ij}=\underset{j=\mathrm{1}}{\overset{N_c}{\min }}(D_{c,ik})\right)\\ \hfill ={\displaystyle {\int }_{\mathrm{0}}^{\infty }\mathrm{Pr}\left(\underset{k\ne j}{\min }(D_{c,ik})\ge v\right){\displaystyle \frac{\mathrm{1}}{{\lambda }_{c,ij}}}{e}^{-v/{\lambda }_{c,ij}}dv}\\ \hfill ={\displaystyle {\int }_{\mathrm{0}}^{\infty }{\displaystyle \frac{\mathrm{1}}{{\lambda }_{c,ij}}}{e}^{-v/{\lambda }_{c,ij}}dv}\\ \hfill ={\displaystyle \frac{{\lambda }_{c,i}}{{\lambda }_{c,ij}}}.\end{array}

For the Markov chains with stationary transition, the probabilities can be written as

(30) \begin{array}{r}\hfill \mathrm{Pr}(X_{c,n_c+m}=j|X_{c,\mathrm{0}}=k,X_{c,\mathrm{1}}=l,\cdots ,X_{c,n_c}=i)\\ \hfill =\mathrm{Pr}(X_{c,n_c+m}=j|X_{c,n_c}=i)=P_c^{(m)}(i,j),\end{array}

where P_c^{(m)}(i,j) is the value on the i and j position in the mth power of P_c. For all P_c^{(m)},\forall i{\displaystyle {\sum }_{j=\mathrm{1}}^{N_c}{P}^{(m)}(i,j)=\mathrm{1.0}}. The Markov method expresses the probability of a component in a state after a number of transitions which only depends on the starting state and on the number of transitions. The homogenous Markov model with two states is illustrated in Fig. 3(a). Since the transition probabilities and the state duration can be estimated from the transition rates, the only parameter needed to completely define the homogenous Markov model is the transition rates between the states λ _ij. In the homogenous Markov model, the conditional state durations and the transition probabilities are independent of the history of the system. In other words, when a component changes to a state, the probabilities for the next states are always unknown. Estimating the state transition probabilities and the state duration rates will result an alternative representation of the homogenous Markov model results, as shown in Fig. 3(b). The transition rates can be obtained as

(31) {\lambda }_{ij}=P_c^r(i,j){\lambda }_i.

3.5 Reliability indices in composite system adequacy assessment incorporating wind turbine generation

In this paper, several well-know reliability indices are considered such as LOLE (h/a), EENS (MWh/a), PLC, EDNS (MW), EDC (10³$/a), interrupted energy assessment rate (IEAR) ($/kWh), BPECI (MWh/MW·a), MBECI (MW/MW), and SI [26]. These indices can be obtained as follows,

(32) \mathrm{LOLE}={\displaystyle \sum _{i=\mathrm{1}}^n{P}_i{t}_i},

when the cumulative probability P _i is used, LOLE is given as

(33) \mathrm{LOLE}={\displaystyle \sum _{i=\mathrm{1}}^n{P}_i(t_i-t_{i-\mathrm{1}})},

(34) \mathrm{EENS}={\displaystyle \sum _{i=\mathrm{1}}^N{C}_i{F}_i{D}_i}={\displaystyle \sum _{i=\mathrm{1}}^{N}t{C}_i{P}_i},

(35) \mathrm{PLC}={\displaystyle \sum _{i=\mathrm{1}}^N{P}_i},

(36) \mathrm{EDNS}={\displaystyle \sum _{i=\mathrm{1}}^N{C}_i{P}_i},

(37) \mathrm{EDC}={\displaystyle \sum _{i=\mathrm{1}}^N{C}_i{F}_i{D}_{i}W},

(38) \mathrm{IEAR}={\displaystyle \sum _{k=\mathrm{1}}^{NB}{\mathrm{IEAR}}_k{q}_k},

(39) \mathrm{BPECI}={\displaystyle \frac{\mathrm{EENS}}{L}},

(40) \mathrm{MBPECI}={\displaystyle \frac{\mathrm{EDNS}}{L}},

(41) \mathrm{SI}=\mathrm{BPECI}\times \mathrm{60.}

In all equations, P _i is the probability of system state i, t _i is the duration of loss of power supply in days, L is the annual system peak load in MW, C _i is the load curtailment of system state i, N is the set of all system states associated with load curtailments, and t is the period of study. F _i and D _i are the frequency and the duration of system state i, respectively. W is the unit damage cost in $/kWh. The EDC is an important index that can be used to conduct economic analysis in composite system adequacy assessment. In this paper, this index is expressed by multiplying the EENS of the overall system by a representative system interrupted energy assessment rate (IEAR). NB is the total number of load buses in the system, IEAR _k is the interrupted energy assessment rate at load bus k, and q _k is the fraction of the system load utilized by the customers at load bus k. The representative system IEAR of the RBTS can be calculated using the data in Refs. [27]. and [28] which is 4.42 $/kWh.

4 Case studies

To validate the proposed method, numerous reliability and sensitivity indices have been conducted on the Roy Billinton test system (RBTS). The single line diagram of the RBTS is illustrated in Fig. 4. The period of study in the RBTS can be assumed to be an hour, a day, a week, a month or a year. In this paper, the system load is considered by the daily peak load variation curve which is also modeled as a straight line from 100% to 40% of the peak load in each case studies [29]. The system load model which is assumed to be linear is depicted in Fig. 5. The study period is considered as hour and therefore 100% on the abscissa declares to be 8760 h [30]. This system has 6 buses, 5 load buses, 9 transmission lines, and 11 generators in buses 1 and 2 which are ranged from 5 MW to 40 MW. The total installed generation capacity is 240 MW while the peak load of the system is 185 MW. The system voltage level is 230 kV. The generating unit ratings and reliability data for the RBTS are given in Table 1. Since the demand of the existing consumption is ever-increasing, a set of different peak load from 165 MW to 235 MW with steps of 10 MW is considered in calculations. The cut-in, rated, and cut-out speeds of each 2 MW wind generation systems are 4, 10 and 22 m/s, respectively. The shape factor of Weibull distributions of wind speed in the wind farm is considered to be 2, and the scaling factors are considered to be 5.5, 6.5, and 7.5 [31]. In this paper, it is assumed that the wind farm has 20 generators with the capacity of 2 MW. The failure and the repair rates of the wind turbine’s components are listed in Table 2. The COPT of the wind farm for some identified capacities is shown in Table 3. Furthermore, based on the proposed method, Weibull distribution failure rate function and reliability function can be obtained as shown in Figs. 6 and 7, respectively. The following subsections illustrate the effects of the wind farm on the overall adequacy assessment in generating bulk electric power systems.

4.1 Case 1

In this case, the performance of the RBTS with different peak loads is evaluated in reliability indices point of view. First, the RBTS with the default generations is studied. Then, the reliability of the system is improved by adding 20 × 2 MW wind turbine generators (WTGs). Finally, the WTGs are replaced with a 40 MW conventional generation. The results are depicted in Fig. 8. As it can be clearly seen from Fig. 8, great improvement has been made in all of the reliability indices. The percentage of improvement varies depending on the type of reliability index, but in a certain index, improvement in given load peak is proportional to the base value of index. Improvements made by two compensators are almost equal; however, the performance of WTGs is 0.004% to 1.137% better than conventional generation.

4.2 Case 2

Different numbers up to 50 WTGs are added to the RBTS in this case. Each WTG can generate 2 MW. The impact of this gradual addition of WTGs on the reliability indices is depicted in Fig. 9. It can be seen that there is a sharp decrease in indices before 40 MW of WTG, and after that no significant changes in reliability indices are observed. It can be concluded that adding more WTGs would not be economically reasonable for this system.

4.3 Case 3

A contingency of unavailability of a 40 MW generation with the FOR of 2% is assumed and the percentage of system degradation in reliability point of view is calculated. Then, a 20 × 2 MW of WTG is added to the system. The system status is evaluated again and compared with the default system. The results are depicted in Fig. 10. As it can be seen, when the outage happens all of the reliability indices are degraded, varying from 15.36% to 25.68%. Replacing unavailable generator with a WTG not only compensated this degradation, but also improved the indices even compared to the default system.

4.4 Case 4

In this case, 20 × 2 MW WTGs with different FORs are added to the RBTS and changes in reliability indices are evaluated and compared to the system with default generation. The results of this case are depicted in Table 4. Column 1 in Table 4 belongs to the system with default generation, and other columns belong to the system in presence of 40 MW capacity of WTGs with FOR of 0.03, 0.015, 0.007893, 0.003, and 0.001. As a result of this evaluation, it can be concluded that smaller FORs result in slightly more improvement.

5 Conclusions

Wind generation system is characterized by output fluctuation due to the stochastic nature of wind speed. On the other hand, the failure of wind turbines’ components affects the output power of the wind farm. Hence, the correlation of wind speed and wind turbine outage has a significant impact on generation system reliability. To assess an adequate reliability model of wind farms, the effects of component failure of wind turbines should be also taken into account. This paper proposes a novel method for quantitative reliability evaluation of bulk composite generation systems incorporating large-scale wind generation systems considering wind speed correlation and wind turbine outage. The proposed method was applied on the RBTS. The Weibull-Markov method is implemented to validate the calculated reliability indices based on the proposed method. Several reliability indices considering the component failure as well as wind speed are demonstrated. The obtained results reveal the validation of the proposed method, and the importance of inclusion the FOR of wind turbines and the wind speed on estimating reliability of wind generation systems. The proposed model reduces the complexity of using analytical methods, provides a comprehensive model for reliability evaluation, and assists operators and planners to evaluate the reliability benefits brought by wind generation systems.

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