Coordinating Maintenance With Spares Logistics to Minimize Levelized Cost of Wind Energy

Tongdan Jin Ingram School of Engineering

Texas State University San Marcos, TX 78666, USA

tj17@txstate.edu

Zhigang Tian Institute for Information Systems Engr.

Concordia University Montreal, Quebec H3G 2W1, Canada

Miguel Huerta, Jett Piechota Ingram School of Engineering

Texas State University San Marcos, TX 78666, USA

Abstract—Wind power emerges as a sustainable energy resource to meet the increasing electricity needs in the next 20-30 years. Power volatility and maintenance costs are the key challenges in harvesting this type of renewable energy. The levelized cost of energy (LCOE) allows the utility and investors to compare the costs of various generation technologies of unequal lifetimes and capacities. In this study we propose a probabilistic-based LCOE model to assess the investment risks by taking into account four major factors: wind speed, system availability, maintenance policy, and spares stock level. Moment methods are applied to estimate the mean and the variance of the energy yield. The goal of the study is to develop a decision aid methodology guiding the wind farmers to minimize the ownership cost by jointly optimizing the maintenance and the spares inventory. We assume the maintenance and repair service is carried by a third party logistics provider. Genetic algorithm is used to search the optimality of the mixed integer non-linear decision model.

Keywords-Weibull distribution; preventive maintenance; service parts logistics; performance-based maintenance; wind power

ACRONYM

Ta total calendar hours in a year X wind speed, a random variable

c,k Weibull scale and shape parameters for wind speed, respectively

fw(X) probability density function for X Fw(X) cumulative distribution function for X Pm rated power of the wind turbine vc,vr,vs cut-in, rated, and cut-off wind speeds ρ air density Aw area covered by the turbine blades max maximum wind energy conversion rate P(X) wind turbine output power, a random variable E[P(X)] expected wind turbine power G(X) annual energy production of a wind turbine E[G(X)] expected annual energy production Ceq turbine equipment annuity Cld annual land lease Com annual operations and maintenance cost Csp annual spare parts and logistics costs Cs annual system cost r interest rate compounded annually h number of years to pay off the loan

(h, r) coefficient computing the equipment annuity n system fleet size m component types, for i=1, 2, …, m ci

(p) downtime cost for a planned replacement ci

(f) downtime cost for a failure replacement c1i unit cost for component type i c2i annual holding cost for component type i c3i cost for repairing a component for type i c4i cost for reconditioning a component for type i ts time for doing a repair-by-replacement job tr repair turn-around time p rate of occurrence of failure or replacement si base stock level for component type i i replacement interval for component type i O spare parts demand As operational availability for wind turbine Ac component operational availability , scale and shape parameters for Weibull reliability Ri(t) reliability of component type i Fi(t) CDF for component type i Nf

(t) failure replacements in [0, Ta] Np

(t) planned replacements in [0, Ta] N(t) total replacements in [0, Ta]

I. INTRODUCTION

The driving force behind the adoption of the wind technology is to seek a sustainable energy solution for mitigating the global warming. By 2011, the installed wind power capacity in the United States reaches 46 GW, and the worldwide capacity exceeds 110GW [1]. Albeit the rapid growth, the current wind generation only represents 2-3% of the utility market, yet it already avoids the emissions of 60 million tons of CO2 each year [2]. It is anticipated that the cumulative wind capacity will grow steadily in the next two decades, and will reach 20-30% market share by 2030.

Wind turbines(WT) are complex aerodynamic, electro-mechanical systems subject to reliability failures. Gearbox failures account for the largest amount of downtime duration and production losses. These costly failures can consume 15-20% of the price of the turbine itself, including an unscheduled replacement of a $100,000 gearbox and an unscheduled crane cost of up to $70,000 to access the failed components [3].

978-1-4673-0788-8/12/$31.00 ©2012 IEEE 1022

Today the U.S. has installed more than 30,000 turbines. This means everyday 72 turbines are expected to fail based on the current 10,000 hours MTBF (mean-time-between-failures) [4]. The maintenance issue becomes more pronounced as the installed base reaches 100,000-150,000 by 2030 in the U.S. Other countries such as China and European Unison are confronted with the similar issues as more turbines are connected to the grid. Therefore, it is imperative to develop effective maintenance and logistics support programs to ensure the availability of fielded systems.

Three maintenance strategies have been extensively discussed in the literature pertaining to WT systems [5-7]: corrective maintenance (CM), condition-based maintenance (CBM), and preventative maintenance (PM). CM is preferred if the equipment downtime cost is small or negligible. CBM is an enabling technology aiming to achieve just-in-time replacement by constantly monitoring failure precursors or degradation signals via in-situ sensors. This technology would gain in popularity in wind power industry if the implementation costs could be further reduced. PM is a relatively mature technology in term of maintenance scheduling and parts provisioning. In particular, age-based PM policy is a preferred scheme for WT considering the fact that an unexpected failure cost is much higher than a planned downtime cost. Therefore, this study adopts the age-based PM policy to design an optimum WT sustainment approach.

Parts repair and supply is fundamental to support WT maintenance tasks. Traditional PM policy focuses on finding the optimal inspection and replacement time such that the system downtime cost is minimized. Readers are referred to [8] for the comprehensive review on this topic. The basic assumption is that the spare part is always available upon the request. In wind power industry, spare parts such as gearboxes and blades are quite expensive. Building a large spares inventory is not a viable approach due to the excessive capital and holding costs. Efforts have been made to jointly optimize the replacement interval and the spares stock size [9-13]. These models are tailored to non-repairable parts, and the goal is to minimize the system downtime cost. Our focus is on repairable components. Repairable parts are capital-intensive and the replacement and repair activities are often carried out by external service providers. We attempt to minimize the levelized cost of energy (LCOE) by coordinating the maintenance interval with the spares stock level. Our study differs from prior literature in two aspects. First, we formulate the cost model from a lifecycle perspective taking into account the initial capital. Second, the defective components are repairable, and the repair service is performed by a third party logistics (3PL) provider in a multi-echelon supply chain network.

The rest of the paper is organized as follows. Section II reviews the spare parts logistics models currently used in wind power industry. In Section III, the moment method is used to characterize the variability of wind energy yield. Section IV formulates the mixed integer programming model to optimize the maintenance schedule and the spares stock size. In Section V numerical examples are presented to demonstrate the proposed method, and Section VI concludes the work.

II. WIND TURBINE RELIABILITY AND MAINTENANCE

A) Reliability of Wind Turbines

As shown in Fig. 1, the reliability of a wind turbine can be modeled as a series system interconnected by the blades/rotor, the main shaft/bearing (MS/B), the gearbox, the generator, power electronics, and control mechanisms, among others. Upon failure, the defective component is removed and replaced by a good part, and the system can be restored to production immediately. Throughout the paper, components and parts are used interchangeably.

Blades/Rotor

Main shaft & Bearing

Gearbox GeneratorPower

ElectronicsControl

Mechanisms Figure 1. Reliability block diagram for the wind turbine.

The reliability of a wind turbine depends on the manufacturing technology, the operating condition, and the turbines size. Actual reliability data directly from the manufacturer is very difficult to obtain. Based on the studies in [4,7], we estimate the WT reliability for three major equipment manufacturers, namely GE, Siemens, and Vestas. As shown in Fig. 2, large MTBF variations are observed between different components types. For the same component type, MTBF also varies between different manufacturers. For instance, the MS/B MTBF varies from 43,800 to 807, 174 hours across three original equipment manufacturers (OEM).

-

100

200

300

400

500

600

700

800

900

Bla

des

Mai

n S

/B

Gea

rbox

Gen

erat

or

P.E

.

MT

BF

(in

1,00

0 ho

urs)

MTBF Breakdown by Component Type

Main S/B=Main Shaft/BearingsP.E.=Power Electronics

Figure 2. MTBF breakdown by component types.

B) Service Logistics Supply Chain

Maintenance, repair and overhaul tasks are often carried out by the 3PL or the OEM based on the service agreement signed by the wind farmer. These service agreements, though varies in terms and conditions, belong to what is called material-based contracts (MBC). Under BMC, the wind farmer compensates the service provider for the labor, materials, and tools each time the service task is accomplished. Large wind farmers may have their own maintenance crew, yet spare parts and critical tools still rely on the 3PL and the OEM. This is largely due to the complexity of technology, not mentioning certain repair tools are only available from the service provider.

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Fig. 3(on next page) shows a two-echelon service supply chain network to support n turbines at the customer site. We assume the OEM designs and produces WT systems. The 3PL owns the repair center and performs maintenance service. The wind farmer owns the turbine fleet and the spares inventory. The service is executed as follows: under the age-based PM policy, the 3PL routinely inspects and replaces the components. If an unexpected failure occurs, the 3PL accesses the faulty turbine to replace the defective component (e.g. gearbox) using a spare part from the inventory. After the turbine is resorted to the operating state, the defective part is shipped to the repair center for root-cause analysis. Upon fix, the part is sent back to the stockroom for future replacement. Obviously, the turbine availability relies on the spares stock level and the time in performing a repair-by-replacement job.

Repair CenterM/G/

New System Shipping/Installation

OEM

OEM for design and prodcution

Wind Farmer

Repair by replacement

Replenish inventory

3PL

Wind TurbineFleet with n

Systems

Spares Inventory

Figure 3. An integrate service logistics supply chain network.

C) System Operational Availability

The steady-state operational availability A is defined as the ratio of the system uptime vs. the total time [8]. Assuming the uptime is equal to MTBF, then A can be estimated by

MDTMTBF

MTBFA

(1)

Where, MDT is the mean downtime. MDT can be further broken into two terms: MTTR (mean-time-to-repair) and the MLDT (mean-logistics-delay-time). MTTR is the hands-on time to restore the system given the spare part is available. MLDT is the time elapsed while waiting for the arrival of the maintenance crew, materials, and service tools.

Studies [14-16] have shown that A is affected by multiple factors, including inherent failure rate , usage rate , spare parts stock size s, the fleet size n, and the repair times {ts, tr}. Notice that ts is the MTTR, and tr is the defective repair turn-around time between the wind farm and the repair center. A unified formula synthesizing all six factors into one analytical formula is derived in [16], and it is restated as follows

s

j

tnjr

rs j

etntt

Ar

0 !

)(11

1

(2)

In this equation, tr is modeled as a M/M/ queue which is quite reasonable for critical components with low failure rates [17]. Equation (2) is derived under the premise of CM policy. We will extend this model to the PM policy in Section IV.

III. ANNUAL WIND ENERGY PRODUCTION

A) Wind Speed Distribution

The intermittency of wind speed can be characterized by probabilistic distribution functions. Normal and Weibull distributions are the two popular models to characterize the stochastic wind speed behavior [18,19]. Without loss of generality, let X be the random wind speed following the Weibull distribution, the probability density function and the cumulative distribution function (CDF) are given

kcxk

w ec

x

c

kxf )/(

1

)(

, x0 (3)

kcx

w exF )/(1)( , x0 (4)

Where c is the Weibull scale parameter and k is the shape parameter. These parameters can be estimated based on the local meteorological records.

B) Modeling Wind Power Volatility

Cubic power curves were developed to characterize the relationship between the output power and the wind speed when the turbine operates in the non-linear phase [18,20]. Mathematically the cubic power function can be expressed as

srm

rcw

sc

vxvP

vxvxA

vxvx

xP 3max5.0

,0

)( (5)

In equation (5), ρ is the air density, and Aw is the area covered by the turbine blades. max is the maximum electricity conversion rate. In theory max=0.5926, but the actual value falls in [0.3, 0.5]. Notice that vc, vr, and vs are the cut-in, the rated, and the cut-off speeds, respectively.

We use the moment methods to characterize the wind power volatility. This approach is relatively simple, yet quite accurate to predict the long-term energy yield [21]. By defining q=0.5maxA, the mean power can be obtained by taking the expectation with respect to P(x) as follows:

)()()()()()]([ 3

0rwswm

v

vww vFvFPdxxfxqdxxfxPXPE

r

c

(6)

The annual energy yield depends on the wind speed and the system availability. Let G(X) be the annual energy yield, then we have

)()( XPTAXG as . (7)

Where As is the WT operational availability, and Ta is the hours in a calendar year (i.e. Ta=8,760 hours). Similarly, we can also estimate the mean value of G(X) as

)]([)]([ XPETAXGE as , (8)

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IV. MINIMIZING LEVELIZED COST OF ENERGY

A) Wind Turbine Lifecycle Cost

Wind turbines are often installed as a group in an area where a considerable wind speed profile exits during the course of the year. The cost of an entire fleet is the sum of individual system costs. Hence, the problem of assessing the fleet cost can be translated into the lifecycle cost analysis of individual turbines. The lifecycle costs of a single turbine comprise the initial capital, land lease, operations and maintenance (O&M) costs, and spare parts inventory. By aggregating all the cost items, the annual system cost, denoted as Cs, can be expressed as

spomldeqs CCCCrhC ),( , (9)

with

1)1(

)1(),(

h

h

r

rrrh . (10)

Notice that (h, r) is the coefficient computing the equipment annularity. h is the number of years to pay off the equipment loan, and r is the interest rate compounded annually. Ceq is the initial capital cost. Cld is the annual land lease. Com is the annual O&M cost, and Csp is the annual spare parts logistics cost. Since Ceq and Cld are fixed costs, we focus our attention on Com and Csp in the following paragraphs.

B) Operating and Maintenance Costs

Under an age-based PM policy, the O&M costs include the expenses for parts transportations, the labor, the crane rental, and the production losses, either in a failure or in a planned replacement. In this study, the spare parts cost is considered as part of the inventory cost, not the O&M costs. The goal is to find the component replacement interval such that the O&M costs is minimized during [0, Ta]. Such an optimum policy can be determined by minimizing the following well-known equation

m

i i

iip

iiif

iaom

i dttR

RcFcTC

10

)()(

)(

)()((

τ) , (11)

where, ci(f) is the cost incurred due to a failure of component

type i, and ci(p) is the cost incurred for a planned replacement.

In general, ci(f)>> ci

(p). Notice that Fi() and Ri() are the CDF and reliability for component type i, respectively, and m is the number of component types in the system.

C) Spare Parts Logistics Cost

Instead of using MTBF, the component reliability under the age-based PM policy is often quantified by MTBR (mean-time-between-replacements) which can be estimated as

0

)( dttRMTBR (12)

Two cases are observed under the age-based replacement policy: the component either fails before or survives by .

By synthesizing both scenarios, the MDT under the PM policy can be estimated as

)(}Pr{)( FsOttRtMDT rss (13)

with

s

j

tnjrp

j

etnsO

rp

0 !

)(1}Pr{

(14)

0)(

1

dttRp . (15)

In equation (13), the first term is the downtime if the component survives through for a planned replacement. Similar to [12], we assume no parts backorder occurs for a planned replacement. The second term is the downtime in the circumstance that the component fails prior to . The random variable O represents the spare parts demand, and s the base inventory stock level for the component. Let N(t)(, s), N(f)(, s), and N(p)(, s) be the total replacements, failure replacements, and planned replacements in [0, Ta], then we have

apt Tτ,sN )()( , (16)

),()(),( )()( sNFsτN tf , (17)

),()(),( )()( sNRsN tp . (18)

It is worth mentioning that equations (12)-(18) are derived for one particular component type. The annual spare parts logistics costs comprise the inventory capital, the holding cost, and the cost for repairing or reconditioning the replaced components. The following model enveloping the costs of all component types is normalized over a single WT system,

m

i

pii

fii

m

iii

m

iiisp NcNcscscrh

nC

1

)(4

)(3

12

11),(

1),( sτ . (19)

Where n is the number of turbines in the fleet. The first item in equation (19) is the annual spare parts capital cost, and c1i is the part unit cost. The second term represents the spare parts holding cost, and c2i is the annual holding cost. The third summation is the annual costs for repairing and reconditioning field returned components. Note that c3i and c4i are the costs for repairing and reconditioning a component, respectively. In general c3i>c4i because a planned replacement part technically is still a working unit, hence the cost for reconditioning a working unit is relatively lower than repairing a failed part. All shipping costs are included in c3i and c4i.

D) Operational Availability under PM Policy

Under the age-based PM policy, the operational availability for component type i can be estimated by substituting equations (12) and (13) into (1) with the replacement of MTBF with MTBR, that is

1025

)(}Pr{)()(

)(),(

,0

0

iiiirisiiiisi

iiii

FsOttRtdttR

dttRsA

i

i

(20)

with i=1, 2, …, m. For a series system comprising multiple components each failing independently, the system operational availability can be estimated as

m

iiiis sAA

1),(),( sτ . (21)

E) Minimizing Levelized Cost of Wind Energy

Levelized cost of energy (LCOE) is a fundamental concept that allows the utility and investors to compare various generation technologies of unequal lifetimes and capacities without resorting to a full-blown project finance model. In this study, we adopt the LCOE formula from the U.S. NREL to assess the cost of wind energy production [22]. The following model, denoted as Problem P1, is formulated to minimize the LCOE by incorporating the maintenance decision and the spare parts logistics planning.

Problem P1:

Min: txspomldeq cCCCCrhXGE

g ),()(),()],;([

1),( sττ

sτsτ (22)

where, =[1, 1, …, m] and s=[s1, s2, …, sm] are decision vectors representing the component maintenance intervals and the spare parts stock level, respectively. In particular, i for all i are non-negative numbers, and si for all i are non-negative integers. Note that E[G(X; , s)] is the expected annual energy yield which is a function of and s as shown in (8) and (21). ctx is the renewable energy tax credit which is constant.

Problems P1 belongs to non-linear mixed integer programming. Existing algorithms relying on the successive solutions of closely related non-linear programming problems and further applying the branch-and-bound technique is not efficient. Recently, genetic algorithm and heuristic methods [10,11] have shown to be effective in searching the optimal or near optimal solution within a reasonable computational time. Therefore, we use the genetic algorithm to solve Problem P1 in the following case study.

V. NUMERICAL STUDIES

Our numerical example concentrates on five major components: blades, the MS/B, the gearbox, the generator, and the control module. Parameters related to the design, manufacturing, spare parts, and repairs are listed in Tables I. Component reliability, cost, and inventory data are estimated based on the publications in [4,24]. In addition, we use the Weibull distribution to model the component lifetime, and the values of and are estimated based on [7]. The unit of is failures/hour.

A) Optimization Search Algorithm

The optimal program given in Problem P1 is searched using the Genetic Algorithm (GA) embedded with the greedy heuristic [14]. The GA in this example used crossover and

mutation operators. The crossover is given 50% probability while and the mutation is given 10% probability. A population size of 20 is adopted. The main deficiency of GA is the randomness involved in the search process. To compensate this shortcoming, the gradient information of the objective functions is calculated for the members of the new population. The integration of GA and the gradient method generates an intelligent or guided, instead of random, search mechanism. The offspring with the genes having the largest gradient value is prioritized to survive in the next evolution.

TABLE I. COMPONENT RELIABILITY AND COSTS (h=20 years, r=5%)

Parts blades MS/B gearbox generator Control i 1 2 3 4 5

(10-5) 2.2593 2.0257 2.2568 1.6354 4.0567

1.8757 2.5 2.6295 3.1814 1.264 c1i ($/item) 333,000 67,500 193,650 51,600 221,850

ci(p) ($) 5,000 5,000 5,000 5,000 5,000

ci(f) ($) 20,000 20,000 20,000 20,000 20,000

c2i ($) 49,950 10,125 29,048 7,740 33,278 c3i ($/item) 99,900 20,250 58,095 15,480 66,555 c4i ($/item) 33,300 6,750 19,365 5,160 22,185

ts (days) 5 4 7 4 3 tr (days) 100 or 50 80 or 40 140 or 70 80 or 40 60 or 30

B) Results Analysis

We consider four planning cases: 1) n=50 and a regular tr; 2) n=50, and a short tr; 3) n=100 and a regular tr; and 4) n=100 and a short tr. The purpose is to examine whether the fleet size and the repair turn-around time pose significant impacts on LCOE. In all these cases, we assume c=10 m/s and k=2.5 for the Weibull wind speed. The WT system under study has Pm=1.5 MW, and vc, vr, and vs are 2.5, 12, and 25 m/s, respectively.

TABLE II. OPTIMAL DECISION FOR CASES 1 AND 2 (n=50 Systems)

Case Parts blades MS/B gearbox generator control

1

tr (days) 100 80 140 80 60 s 4 2 0 1 1

(hours) 54,023 43,675 54,358 66,090 28,442 Ac 0.9838 0.9938 0.9632 0.9970 0.9751 As 0.9155

LCOE 0.035 ($/KWh)

2

tr (days) 50 40 70 40 30 S 3 2 0 3 3

(hours) 57,532 63,113 57,954 51,901 34,062 Ac 0.9944 0.9972 0.9888 0.9977 0.9928 As 0.9712

LCOE 0.032 ($/KWh)

The optimum or near-optimum values of s and are

presented in Table 2 under a regular and shorter tr, respectively. A major finding is that the system availability is improved from 0.9155 to 0.9712 if tr is reduced by 50%. This represents more than 6% increase of the system availability. We also expect that the LCOE should decease in a similar trend. In fact, the LCOE is reduced from $0.035/kWh to $0.032/kWh, or 8% reduction. In both cases, the aggregate spare parts inventory size remains almost the same even

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though the quantity of individual spare components may change.

We also compute the LCOE assuming the fleet size increases from n=50 to 100. Table 3 presents the cost, the maintenance intervals, and the spares stocking under the regular and the smaller tr. The result shows that it is more cost-effective to keep a larger spares pool as n becomes larger. For instance, under a regular tr with n=100, we can hold more than twice of the spare parts, yet the LCOE is $0.034/kWh, lower than $0.035/kWh with n=50. Another observation is that when n increases, the impact of tr on the system availability becomes smaller.

TABLE III. OPTIMAL DECISION FOR CASES 3 AND 4 (n=100 Systems)

Case Parts blades MS/B gearbox generator Control

3

tr (days) 100 80 140 80 60 s 9 5 0 3 3

(hours) 56634 59055 59035 48002 30563 Ac 0.9896 0.9969 0.9672 0.9975 0.9815 As 0.9342

LCOE 0.034 ($/KWh)

4

tr (days) 50 40 70 40 30 s 7 4 2 1 2

(hours) 57992 56006 50457 63488 31644 Ac 0.9962 0.9976 0.9923 0.9981 0.9949 As 0.9793

LCOE 0.032 ($/KWh)

VI. CONCLUSION

This paper jointly optimizes the maintenance policy and the spare parts logistics to minimize the levelized cost of wind energy. The optimization model is formulated under the circumstance that the 3PL undertakes the maintenance tasks while the wind farmer owns the spare parts. We derived a holistic LCOE estimate enveloping multiple performance drivers, include wind speed, reliability, maintenance intervals, spare parts stocking, the repair time, and the fleet size. This analytical insight enables the wind farmer to coordinate the maintenance policy with the spare parts logistics in order to mitigate the LCOE. Two interesting observations are made from the case study. First, the repair turn-around time has a major impact on the spares stock size and the system availability. Second, a high system availability may not lead to a significant reduction of LCOE (e.g. As=0.9155 vs. 0.9712, and LCOE=0.035 vs. 0.032). Under the deregulated electricity market, it might be also interesting to propose new cost models, such as the levelized profit of energy, to assess the performance of wind technology. Another direction worth of exploring is to study performance-based service, i.e. performance-based contracting, for sustaining wind power generators.

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