978-1-4244-4547-9/09/$26.00 ©2009 IEEE TENCON 2009

A Markov Model for a Wind Energy Conversion System with Condition Monitoring

Saranga K. Abeygunawardane, Panida Jirutitijaroen Department of Electrical and Computer Engineering

National University of Singapore Singapore

Abstract— Wind power is one of the most economical sources of energy. Wind Energy Conversion System (WECS) comprises of several components with diverse characteristics that affect system reliability at different level. In order to operate WECS reliably and economically, accurate modeling of a WECS is needed to study the effects of, and possibly identify, components that lead the system to fail. This paper proposes a suitable Markov model for a WECS with condition monitoring (CM). A sensitivity analysis is carried out using the developed model to identify the characteristics of components that are likely to affect system reliability the most. Results show that the components with high mean down times and high failure frequencies are more critical to the availability of a WECS than the others.

Keywords- Markov process, Reliability modeling, Wind energy

I. INTRODUCTION Wind power has drawn the attention from all over the world

due to its contribution to carbon emission reductions as one of the cheapest and cleanest sources of energy. At present, researchers and wind farm operators are more interested in effectively utilizing wind energy for a reliable power supply. Reliability of a WECS highly depends on its architecture and individual characteristics of its sub-components. A typical wind turbine (WT) system consists of a rotor, a generator, a gearbox and other mechanical and electrical sub-systems. Based on the presence of the gearbox, two main architectures can be identified; a geared WT with a gearbox and a direct drive WT without a gearbox. Even though the direct drive WT is found to have lower cumulative failure frequency of a generator and a converter than the geared WT [1, 2], it has not yet become a viable and competitive alternative to the geared WT. The main focus of WECS in this paper is therefore the geared WTs.

WECS with geared WTs is complicated in construction and consists of several sub components. Failure of one of its component will lead the system to fail and cause unavailability of wind power generation. CM is thus important to predict failures and schedule maintenance to maximize the annual output power generated. Frequent maintenance can improve reliability but at the same time may cause catastrophic failure if not properly done. It is also not an optimal or cost effective solution. Preventive maintenance, which is highly associated with CM, has now become a popular choice as a cost effective maintenance strategy.

Attempts have been made to identify crucial components of a WECS based on historical data. A recently conducted survey on failures of WECS in Sweden stated that the gearbox is the most crucial component due to its high down time per failure [3]. In [1], electrical system, rotor and converter were found as the most unreliable sub assemblies due to their high failure frequencies. In order to analytically investigate effects of sensitive components to the WECS reliability, a proper model for a WECS with CM is needed.

A probabilistic model was developed for a WECS with CM in [4, 5] considering published data, reliability data and opinions of industry experts. This model accounts for failures related to generator, gearbox, blades and electronics of a WECS. Sub-component selection criteria were downtime distributions, failure rates and the presence of CM [4, 5].

In this paper, a new model for a WECS is proposed to incorporate CM effects and failure data of all sub-components. In section I main components and failure statistics of WECSs are presented. The proposed Markov model is described in section II together with the design considerations and techniques. The developed model is incorporated into a test system in section III and it is then used to conduct a sensitivity analysis to assess which characteristics of the components are critical to the availability of a WECS. In section IV, the results of this sensitivity analysis were briefly discussed together with mathematical equations. Concluding remarks are presented in section V.

II. WIND ENERGY CONVERSION SYSTEM

A. Components of a WECS In a typical horizontal-axis geared wind turbine, three

composite blades jointed to the rotor hub drive a rotor. A main bearing is positioned to absorb static and dynamic loads and also to support a rotor shaft. The rotor drives a gearbox and a generator coupling couples the gearbox to an induction generator. The gearbox converts a low speed of the blades to a rated speed of the generator. A safety brake is located between the gearbox and the generator. The gearbox and generator cooler, a control unit and a hydraulic system are also placed in a nacelle. After positioning all these components in the nacelle frame, the complete nacelle covered by its cover, is mounted on top of a tower with the aid of a yaw bearing. The yaw drive

This work is supported by Singapore Ministry of Education-Academic Research Fund, Grant No. WBS R-263-000-487-112.

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aligns the nacelle according to the direction of the wind, sensed by wind sensors, in order to harvest maximum energy.

B. Reliability Block Diagram of a WECS When all of the above mentioned sub-components are

taken into consideration, a reliability block diagram of the complete WECS [7] can be rather complex. The reliability block diagram of a typical WECS, shown in Fig. 1, has six major sub-systems that significantly affect the system reliability. The block diagram is connected in series even though all sub-components of a WECS are not physically connected in series. This is due to the fact that failure of each sub-component causes the WECS to fail. Failure statistics of each sub-component are given in the next sub section.

C. Failure Statistics The critical components of WECS are identified by

observing the failure statistics of individual sub-components [1, 3, 4, 5]. Individual failure rates of sub components shows that the most failures are related to the electric system followed by sensors and blades/pitch. The gearbox has the highest downtime which is closely followed by the control system and the electric system.

D. Condition Monitoring of WECS Most modern WECS are incorporated with CM systems.

Blades, generator and gearbox and drive train are the three main sub-components with CM techniques in most wind turbines [8]. In addition, CM of yaw system and mechanical brake is newly proposed considering their high downtime and failure frequency [9].

III. A MARKOV MODEL FOR A WECS WECS comprises of several devices that make it

challenging to incorporate all sub-components in the reliability model. Therefore selecting and modeling sub-components are major issues in modeling a WECS.

In previous reliability models, some of the sub-components (generator, gearbox, electronics and blades) have been selected to represent the entire WECS [4, 5, 10]. In addition to the failure frequency, downtime is also considered as a selection criterion. Some components with a very high downtime, although with a low failure rate, can disturb the power generation, more than those with a very short downtime but a high failure frequency.

Instead of modeling few selected sub-components, failure and repair characteristics of all sub-components are incorporated into this proposed model. To reduce the complexity of the model, sub-components with similar characteristics are grouped together. Different modeling techniques are chosen to match the failure and repair activities according to the nature of each sub-component. This is explained in detail under subsection B.

A. Reliability Modeling Techniques for a Sub-Component Two basic models are proposed, a two-state model and an

intermediate states model. The two-state model is used to model all sub components without CM activities. Intermediate states model will introduce a better representation of CM.

1) A Two-State Markov Model In this model (Fig. 4), a single component having only up

and down states is considered. Due to simplicity, this model is widely used in reliability studies [4, 5, 11].

2) An Intermediate States Model This model (Fig. 3) is ideal for a single component having

up, down and intermediate state(s). This is widely used to model the deterioration process of a component. In this work, this model is used to represent a component under continuous condition monitoring. Transition between states depends on failure characteristics of the sub-component and the maintenance activities.

B. A Proposed Markov Model for a WECS The complexity and number of states of the model depend

on the number of sub-components and the selected reliability models to represent each sub-component. In this proposed model, all sub components are grouped into four categories. Gearbox, generator and blades/pitch were considered as separate groups as they are having CM, and all other components without CM are considered as one group.

1) Gearbox: This group includes gears and drive train. 2) Generator 3) Electronics and others: This group represents all other

components that are not included in other three groups. 4) Blades/pitch Sub component 1, 2 and 4 are represented using a three-

state model in order to include the intermediate state at which the CM system detects faults. Sub-component 3 is modeled using the two-state model as it has only up and down states to consider. Once the appropriate model is selected for each sub component, WECS can be modeled by combining the states of sub components.

C. State Space Diagram The state space diagram of the proposed WECS model is

shown in Fig. 2.

Figure 1. Reliability block diagram of a typical WECS

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Figure 2. State space diagram of the proposed WECS

Theoretically, the developed Markov model for a WECS has 54 ( 33 ×2 ) states. As in most reliability models the number of system states are reduced to 28 by using three assumptions; simultaneous failures or degradations of two components are negligible, system must transit to a failure states via a de-rated state, all failure states are considered as absorbing states.

D. Transition Rate Matrix The deterioration and failure characteristics of each sub

component are incorporated into the developed model using the transition rate matrix, parameters of which can be calculated from historical data collected over a long period of time. This paper computes transition rate parameters using average number of failures and average downtimes available in the literature for the test system. The procedure is explained in detail under section III.

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iλ , 2iλ and iμ are the transition rate

from up state to de-rated state, from de-rated state to down state and from down state to up state of sub-component i, i=1,2,4 respectively. 3λ is the transition rate from up state to

down state of sub-component 3

IV. A TEST SYSTEM The above Markov model is incorporated into a test system

with failure rates and repair rates obtained from [3]. The data is based on a recent survey conducted for more than 600 Swedish wind turbines over a period of five years. Equivalent failure rates and repair rates are calculated for each group of sub components of the selected test system.

TABLE I. EQUIVALENT FAILURE RATES AND REPAIR RATES OF SUB-GROUPS

Component eqλ (per year) eqμ (per day)

Gearbox(1) 0.000134 0.0925

Generator(2) 0.000058 0.1139

Electronics and other(3) 0.000740 0.2183

Blades/pitch(4) 0.000142 0.2620

A. Calculating Equivalent Rates For a series system with frequency of failure

approximation, equivalent failure rate ( eqλ ) and equivalent

repair rate ( eqμ ) can be calculated using following equations.

ieqλ λ=∑ (1)

i

i

eq i

MTTRMTTR

λ

λ

×= ∑

∑ (2)

1 eqeqMTTR

μ = (3)

where iλ is the failure rate of sub-component i, iMTTR is the average time required to repair component i and eqMTTR is the average down time of the WECS. Since components of each group are in series, equivalent failure and repair rates in Table I are calculated using individual failure rates and mean down times (MDTs).

B. Determining the Transition Rates for Intermediate State Model Intermediate states model for gearbox, generator and

blades/pitch is shown in Fig. 3. For gearbox, generator and blades/pitch that have an intermediate state, it is needed to calculate the transition rate from up state to de-rated state ( 1λ ) and the transition rate from de-rated state to down state ( 2λ ). The procedure of calculating 1λ and 2λ is briefly described in this section.

Figure 3. Intermediate states model

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Figure 4. Equivalent two state model

By applying frequency balancing technique to the model shown in Fig. 4, we have the following equations.

1 1 3 eqP Pλ μ= (4)

1 1 2 2P Pλ λ= (5) 2 2 3 eqP Pλ μ= (6)

1 2 3 1P P P+ + = (7) where 1P , 2 P and 3P are probabilities of being staying in

up, de-rated and down states respectively.

From (4), (5), (6) and (7),

31 2

1 1 1 1eqP μλ λ

⎛ ⎞+ + =⎜ ⎟

⎝ ⎠ (8)

Similarly, using frequency balancing technique for the equivalent two state model in Fig. 4, (9) and (10) are obtained.

4 5 eq eqP Pλ μ= (9)

4 5 1 P P+ = (10) where 4 P and 5P are probabilities of being staying in up

and down states respectively.

From (9) and (10),

51 1 1eqeq

P μλ

⎛ ⎞+ =⎜ ⎟⎜ ⎟

⎝ ⎠ (11)

Since P3=P5,

1 2

1 1 1

eqλ λ λ= + (12)

TABLE II. TRANSITION RATES FROM UP STATE TO DE-RATED STATE AND FROM DE-RATED STATE TO DOWN STATE

Component Transition Rate from Up State to De-rated State,

1λ (per day × 10-4)

Transition Rate from De-rated State to Down

State, 2λ (per day)

Gearbox 1.3444 0.0925

Generator 0.57563 0.1139

Blades/pitch 1.4254 0.2620

To determine 1λ and 2λ using (12), one of them must be known. Since, historical data is unavailable for 1λ and 2λ , considering fast transitions from de-rated state to down state,

2λ is assumed to be equal to the repair rate; eqμ . With this

approximation, 1λ can be calculated using (12) and the values for 1λ and 2λ are tabulated in Table II.

V. A SENSITIVITY ANALYSIS Maintaining high availability with minimum maintenance

cost is important to minimize operation cost of WECS. CM of sub-components can predict the failures in advance and hence maintenance scheduling can be effectively conducted. To conduct CM in a cost effective manner, identifying the more sensitive sub-components to the availability of a WECS is required.

The system availability, given by (13), shows that a high availability can be achieved either by a high mean time to failure (MTTF) or a low mean time to repair (MTTR) or a combination of both.

MTTFAvailabilityMTTF MTTR

=+

(13)

Failure statistics shows that 20 percent of downtime of a wind turbine is due to gearbox failures and hence gearbox was identified as the most critical component to the availability of a WECS [3]. Since the availability depends on both MTTF and MTTR, high downtime of a component may or may not contribute to low system availability. It is therefore important to analytically identify critical components through a sensitivity analysis. A test system in section III is used for this analysis. It is also expected that the sensitivity analysis provide insightful information on the effects of failure rates and repair rates of sub assemblies to the overall WECS reliability.

Three sensitivity analyses are carried out to observe the effects of failure rates and repair rates of each sub component on the reliability measures of the WECS namely MTTF, MTTR, and system availability. By observing the sensitivity of the sub-components to MTTF, MTTR and the availability, the components that mostly affect to the system availability can be identified.

In this analysis, failure rate and repair rate of each group of sub-components are varied from 0 to 0.001 and 0.05 to 0.3 respectively. These ranges are selected by adding a margin to the ranges of actual failure and repair rates.

A. Sensitivity to MTTF As can be seen form Fig. 5, MTTF decreases when failure

rate increases. Gearbox, generator and blades/pitch with comparatively low failure rates are not highly sensitive to MTTF of a WECS. On the other hand, electronics and other sub components with high failure rates are very sensitive to MTTF. By reducing the failure rate of this category, MTTF can be decreased. Theoretically, this trend can be well explained using (15).

For a WECS, MTTF can be computed using (14).

1

eq

MTTFλ

= (14)

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Figure 5. Variation of MTTF with failure rates

Figure 6. Variation of MTTR with failure rates

From (1) and (14), for a series system with frequency of failure approximation,

1i

MTTFλ

=∑ (15)

According to (15), MTTF is inversely proportional to the failure rates and hence, as can be seen from Fig. 5, MTTF decreases with increasing failure rates.

In the reliability model of a WECS, all sub components of a WECS are connected in series. For this series system MTTF is a function of failure rates of all components and not a function of repair rates (15). Therefore, MTTF remains constant with varying repair rates.

B. Sensitivity to MTTR As can be seen from Fig. 6, there are two major trends in

the variation of MTTR with failure rates. For sub components with high average down times such as gearbox and generator, MTTR increases, when failure rate increases. It is due to the fact that the system MDT increases as components with high average down times fail frequently than the others. On the other hand, for sub-components with low average down times, MTTR decreases, when failure rate increases. This implies that the quick repairable failures are more desirable than the failures that take a long time to repair.

The above trends can be theoretically explained using following mathematical expressions. For a component i,

1 i iMTTRμ

= (16)

Figure 7. Variation of MTTR with repair rates

From (2) and (16), 1

ii

eq iMTTR

λμ

λ

×=∑∑

(17)

As given in (17), MTTR of a WECS is a function of repair rates and failure rates of its sub components. To discuss the trends in Fig. 6, the first derivative of eqMTTR with respect to

iλ can be computed as given in (18).

2

( )

( )

kk

k ii k

eq k ii i k

k i

d MTTRd

λλ

μ μλ λ λ

∀ ≠

∀ ≠

∀ ≠

−=

+

∑∑∑

(18)

The numerator that decides the sign of the derivative can

be rearranged as 1 1( )ki k

k i

λμ μ∀ ≠

−∑ . For a component having a

comparatively small repair rate, this term is positive and leads to an increasing trend, as for the gearbox and generator in Fig. 6. On the other hand, for a component with a high repair rate, this term is negative and leads to a decreasing trend.

MTTR of the system is inversely proportional to the repair rates of individual sub components (17). Fig. 7 shows that MTTR decreases when repair rate increases. By increasing the repair rate of the components that fail frequently, MTTR of the system can be dramatically reduced. Therefore by increasing the repair rate of the components with high failure rates such as electronics and blades, MTTR can be decreased.

Figure 8. Variation of availability with failure rates

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Figure 9. Variation of availability with repair rates

C. Sensitivity to Availability Fig. 8 shows that the failure rates of generator and gearbox

are more sensitive to the availability than the failure rates of others. The gearbox has the most sensitive failure rate to the availability of the WECS. Although failure rate of electronics is comparatively less sensitive to the availability, its small failure rate is also important to guarantee a significantly high availability of the system.

Fig. 9 shows that the repair rate of electronics is highly sensitive to the availability of a WECS, than the repair rates of other components. However this high sensitivity decreases when repair rate increases.

D. Discussion The availability of a WECS depends on both the MTTF

and MTTR. This sensitivity analysis examines the effect of failure rates and repair rates of three major components with CM techniques to the overall reliability of a WECS. Combined effect of all other components without CM techniques is also examined. Although some components are lumped together in the model, there exist significant differences in failure and repair rates among sub-groups (Table I). For example, generator and gearbox have low failure rates and high repair rates, while electronics and other components considered as group 3 is having a high failure rate and a low repair rate. These different characteristics of sub-groups are useful to generalize the results for any sub component with known failure and repair rates, to predict the sensitivity to the availability of a WECS.

Results show that MTTF decreases rapidly with the failure rates of the components that typically have high failure rates. Therefore, a decrease in failure rates of the components with high failure frequency can significantly increase MTTF of the system. MTTR can be reduced significantly by reducing failure rates of sub-components with high MDTs or by increasing repair rate of sub-components with high failure frequency.

The results of this analysis emphasize the importance of minimizing sudden failures of the components with high MDTs such as gearbox and generator using CM techniques. Most unreliable components such as electronics are also critical and investigating new techniques to minimize failures of these components or to replace them with highly reliable components is also favorable for a higher availability of

WECSs. Another possibility is to further increase the repair rates of the components that fail frequently.

VI. CONCLUSION In this paper, a new model for a WECS with CM is

proposed to incorporate failure data of all sub-components. The model is then utilized to identify sensitive components of a WECS by conducting several sensitivity analyses. With the proposed model, the sensitivity analyses show that components with high MDTs or high failure frequencies can greatly affect the reliability of a WECS. Although all sub components are grouped into four in order to reduce complexity, the results can be generalized to identify the effect of any component with known failure and repair rates.

Results also show that the failure rates of the components with high MDT and the failure rates and the repair rates of the components with high failure frequencies are equally important to the overall reliability of a WECS. Not only the components with high MDTs but also the components that fail frequently are critical and highly sensitive to the availability of a WECS.

In future studies, this model can be useful to accurately assess the benefits of CM in monetary terms.

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