Dynamic Joint maintenance and spare part policy based on continuous condition monitoring and weather predictions

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,

Dynamic Joint maintenance and spare

part policy based on continuous condition

monitoring and weather predictions

Sjors Robbert Rinus ter Braak S3452301

s.r.r.ter.braak@student.rug.nl

A thesis presented for the degree of Master of Science Supply Chain Management

First supervisor: Dr. E. Ursavas (University of Groningen) Second Supervisor: Dr. I. Bakir(university of Groningen)

Third Supervisor: Dr. M. Yildrim (Wayne State University, Detroit , USA)

Faculty of Business and Economics

University of Groningen

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Abstract

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Preface

I present this thesis to the faculty of Business and Economics at University of Groningen as the final step in completing my masters program Supply Chain Management. I would like to thank my supervisors Dr. E. Ursavas and Dr. I. Bakir for providing me with valuable feedback. Furthermore, I would like to thank Dr. M. Yildrim as my third supervisor for sharing his thoughts and knowledge on the subject of offshore wind farms, providing me with the degradation data needed for this thesis, and his support during the process of writing this thesis. Lastly, I would like to thank my friends and family for their support. Sjors ter Braak

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4 Method of research 20 4.1 Research method . . . 20 4.1.1 Parameter optimization . . . 20 4.2 Input data . . . 21 4.2.1 Degradation data . . . 21 4.2.2 Weather data . . . 21 4.2.3 Probability forecasting . . . 22 4.3 Case study . . . 23 4.3.1 Cost parameters . . . 23 5 Results 24 5.1 Numerical results . . . 24 5.2 Comparative study . . . 25

5.2.1 Availability and accessibility metrics . . . 25

5.2.2 Inventory metrics . . . 26

5.3 Maintenance metrics . . . 26

5.3.1 Cost metrics . . . 27

5.3.2 Conclusion comparative study . . . 28

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5.4.1 Holding costs . . . 29

5.4.2 Maximum allowed wave height . . . 30

5.5 Price per MW . . . 31

5.5.1 Conclusion . . . 32

6 Conclusion and Limitations 34 6.1 Conclusion . . . 34

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List of Figures

1 preventive and opportunistic maintenance scheduling process . . . 10

2 Demand during scheduling . . . 13

3 Finding the policies and evaluation . . . 21

4 Forecasting process . . . 23

5 Representation of d1 and d2 vs costs . . . 24

6 w1 threshold under fixed d1 and d2 . . . 24

7 Benchmark optimum . . . 25

8 Inventory metrics . . . 26

9 Maintenance metrics . . . 27

10 Cost metrics . . . 28

11 Effect on total costs and inventory costs . . . 29

12 Effect on average inventory . . . 30

13 Effect on stock outs . . . 30

14 Effects on downtime costs . . . 30

15 Effect on availability and accessibility rate . . . 31

16 Effect on total costs . . . 31

17 Effect on total costs . . . 32

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List of Tables

1 Evaluation of prediction methods . . . 12

2 Availability and accessibility performance metrics . . . 25

3 Inventory costs . . . 26

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1 Introduction

The EU has committed to reduce CO2 emissions by 80-95 percent by 2050 (Rodrigues, Restrepo, Kontos, Pinto, & Bauer, 2015). Their plan to achieve this includes installing 150 GW in offshore wind turbines by 2030 (Rodrigues et al., 2015). For this plan to suc-ceed, offshore wind farms should be marketable and appealing for investments. Currently 75-90% of the lifetime costs is located in the operations and maintenance process(O&M) of offshore wind farms. Moreover, for increasing the marketability of offshore wind, de-creasing the O&M costs is an important element (Ding et al., 2007).

A contributing factor to the high O&M costs of OFWs is the operational availability of OWFs, indicating the amount of time the system is operating satisfactory (Van Bussel & Zaaijer, 2001). Furthermore, the missed revenue due to operational unavailability is seen as the downtime costs, which is a large contributor to the high O&M costs. The availability of offshore wind is currently between 60-70 percent, while onshore wind farms have an availability of 95-99 percent (Besnard, Fischer, & Tjernberg, 2013). In more remote areas with harsher weather conditions these costs will increase due to the inaccessibility rate of the OFW (Obdam, Rademakers, & Savenije, 2018). The accessibility rate indicates the percentage of time an OFW can be approached due to weather restrictions (Van Bussel & Zaaijer, 2001), this is considered a big obstacle in offshore wind industry (Diamond, 2012). Waiting time during maintenance activities can even account for up to 89.4% of downtime costs, due to weather constraints (Diamond, 2012). Therefore, it is of up most importance to consider downtime, failure moments and weather constraints in determining maintenance and spare part policies.

In achieving lower O&M costs academics have researched maintenance and spare part policies. Frequent used maintenance policies for offshore wind farms are preventive and corrective maintenance. Corrective maintenance (CM) occurs after failure of a compo-nent(Swanson, 1976), while preventive maintenance (PM) is aimed at postponing the moment of failure as long as possible through maintenance activities prior to the failure of a component(Shafiee, 2015). Condition based maintenance (CBM) is used to achieve ef-fective preventive maintenance. CBM makes use of monitoring the condition of the system and performing PM before reaching a threshold value (Jin & Mechehoul, 2010). Oppor-tunistic maintenance (OM) is another variant of PM, this makes use of an the opportunity of being at a wind farm in order to conduct maintenance on an additional component or turbine (Lubing, Xiaoming, Shuai, & Xin, 2019), this decreases the fixed maintenance setup costs and increases life expectancy of the system (Lubing et al., 2019).

Maintenance activities can only be executed if spare parts are available (Driessen, Arts, van Houtum, Rustenburg, & Huisman, 2015). Furthermore, spare parts for offshore wind farms are are large and expensive (C. Zhang, Gao, Yang, & Guo, 2019), this increases the holding costs. Because of the dependence of maintenance on the availability of spare parts, and the high costs of spare parts, joint spare parts and maintenance policies are a relevant field of research for OFWs. A joint spare part and maintenance policy considers both spare parts and maintenance simultaneously in order to benefit of the trade-off between maintenance costs (e.g. setup costs, labour costs, variable maintenance costs) and inventory costs (e.g. ordering costs and holding costs)Van Horenbeek, Bur´e, Cattrysse, Pintelon, and Vansteenwegen (2013).

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on the maintenance and spare part policies. C. Zhang et al. (2019) created an OM policy accompanied by a (s,S) inventory policy, in which maintenance was scheduled during low wind periods in order to minimize revenue loss. However, they did not consider the changes in spare part demand due to uncertain weather. X. Zhang and Zeng (2017) proposed a joint condition based opportunistic maintenance and spare part safety policy. However, they did not make use of real life degradation data but modeled the the degradation process themselves.

Both X. Zhang and Zeng (2017) and C. Zhang et al. (2019) formulated a model which makes use of the current degradation state of a system, this is known as a static model. A dynamic model makes use of the expected remaining life distribution, which allows it to make decisions based on expected future state instead of the current state of a system. Erguido, M´arquez, Castellano, and Fern´andez (2017) proposed such a dynamic policy by incorporating remaining life distribution and incorporating weather forecasts in their decision making process for an OM policy. However, they neglected to incorporate spare parts in their policy. Zhou and Yin (2019) created a dynamic condition based opportunistic maintenance model, but did also neglected to incorporate spare part policies.

This thesis proposes a dynamic joint spare part and maintenance policy based on both remaining life distribution and weather forecasting. Therefore, we will add to existing literature in three ways. Firstly, this thesis will use continuous condition monitoring based on real life degradation data according to Yildirim, Gebraeel, and Sun (2017). Previous research focused on modeling the degradation data. Secondly, this thesis proposes a dynamic joint spare part and maintenance policy based on remaining life distribution of real life wind turbines, while previous dynamic policies focused on either maintenance or spare parts. Thirdly, previous studies did incorporate weather effects in their model, but few have based decision making on weather. Erguido et al. (2017) did incorporate weather prediction in their decision making. But used wind speed forecasting in determining optimal opportunistic maintenance moments. Additionally, C. Zhang et al. (2019) used a 7 hours wind speed forecast for their opportunistic maintenance model. Furthermore, no previous study has incorporate weather in a spare part policy. The third contribution of this thesis is therefore to base decision making on both remaining life distribution and weather.

The joint spare part and maintenance policy proposed in this thesis makes use of three threshold values, one for preventive maintenance, one for opportunistic maintenance, and one for weather probability. Scheduling decisions in the proposed policy are based on these three threshold values, inventory decisions are based on the opportunistic and pre-ventive maintenance thresholds. Additionally, scheduling decisions are based on a wave height probability forecasts. To incorporate weather uncertainty in the inventory model, we model the seasonality of the wave heights in terms of the expected consecutive days maintenance is not allowed. We evaluate the proposed policy by conducting a case study and comparing the outputs to a benchmark policy. The benchmark policy does not make use of weather in decision making but does use remaining life distribution of wind tur-bines. This allows us to evaluate what the effects are of incorporating weather into decision making for joint spare part and inventory models.

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2 Literature review

This section gives an overview of recent literature in the field of offshore maintenance, spare parts and joint maintenance and spare part policies. Furthermore, it discusses specific weather constraints for offshore wind farms and how policies are adapted to account for these constraints. Lastly, it will discuss the difference between dynamic and static policies and will end with the contribution made by the policy proposed in this thesis.

2.1 Maintenance policies

Historically, maintenance strategies for offshore wind farms (OWFs) are divided in two cat-egories, corrective and preventive maintenance (Shafiee, 2015). Corrective maintenance (CM) takes place after a failure has occurred(Swanson, 1976). Preventive maintenance (PM) takes place before the failure and is aimed to increase the expected life of a compo-nent(Shafiee, 2015) . The remainder of this section discusses condition based maintenance (CBM) and opportunistic maintenance (OM) as maintenance policies.

2.1.1 Condition based maintenance and condition monitoring

CBM is based on performance or parameter monitoring and subsequently performing PM or OM actions (Jardine, Lin, & Banjevic, 2006; Jin & Mechehoul, 2010). Condition based maintenance makes use of either continuous or periodic condition monitoring. In continu-ous condition monitoring the system constantly monitors the condition of the machines, in periodic condition monitoring this is done at specified periods in time (Alaswad & Xiang, 2017). Continuous monitoring is the most reliable option because it decreases the risk of missing important information, it also is the more expensive option and requires many special devices (Jardine et al., 2006).Amayri, Tian, and Jin (2011) developed an optimal CBM policy for wind farms with multiple different types of wind turbines and different lead times. Andrawus, Watson, Kishk, and Adam (2006) found which CBM activities which maximize the return on investment during the logistics life cycle of offshore wind turbines. May, McMillan, and Th¨ons (2015) reviewed the economic benefits which can be achieved from implementing CBM and found that significant cost savings can be achieved.

2.1.2 Opportunistic maintenance

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2.2 Spare part policies

Spare parts are inventories used for maintenance activities of equipment or machines (Khademi & Eksioglu, 2018). The aim of spare part management therefore is to pro-vide spare parts when they are required while simultaneously keeping inventories costs low (holding cost, procurement costs, and obsolescence) (Tracht, Westerholt, & Schuh, 2013). Inventory can be reviewed continuous and periodically. In continues review sys-tems, spare part levels are continuously checked, if they pass a set threshold value a new order placed. Frequent used continuous review systems are the (s,S) and (s,Q) policies (Van Horenbeek et al., 2013). In the (s,S) policy, units are ordered to the order up to level (S), when the inventory drops below the reorder point (s). In the (r,Q) policy the fixed order quantity (Q) is ordered when inventory drops below the reorder point (r). An example of a periodic review system is, (R,S) policy in which orders are placed up to (S) at each time interval (R) depending on, for example, forecasted demand (Van Horenbeek et al., 2013).

The key components for wind turbines (generator, rotor, and gearbox) have a long expected life and stable demands, however they are large and expensive (C. Zhang et al., 2019). Furthermore, inadequate levels of spare part inventory can result in unnecessary downtime at offshore wind farms(Shafiee, 2015). Academics have focused strongly on research in the field of spare part policies for offshore wind farms due to the high costs associated with them. Tracht et al. (2013) developed a spare parts planning model subject to vessel availability limits and meteorological conditions in the environment. K. Zhang, Feng, Cui, Wang, and Yang (2018) proposed an optimization method for the number of spare parts needed based on component updating. Rausch and Liao (2010) developed an inventory control model driven by condition based maintenance.

2.3 Joint spare part and maintenance policies

The joint spare parts and maintenance models consider the trade off between maintenance and inventory (Van Horenbeek et al., 2013). Furthermore, the joint models showed sub-stantial cost improvement compared to their counterpart, the sequential models (Kabir & Al-Olayan, 1996).

C. Zhang et al. (2019) developed a model for opportunistic maintenance and spare part inventory management, considering weather conditions. They used an (s,S) spare part policy to reflect OFW spare part characteristics . X. Zhang and Zeng (2017) investigated the joint optimization for CBOM and a safety stock policy for spare parts in a multi component system. The model they proposed uses CBM and OM based on periodic condition monitoring. Furthermore, they used safety stock level inventory system which calculates the order quantity based on system status. They modeled the deterioration process stochastically and did not use any weather or accessibility constraints for the maintenance activities. Wang, Chu, and Mao (2009) presented a policy for deteriorating systems. They developed a simulation-based optimization model for the joint policy. Their model combined a CBM with periodical inspections and an s, S inventory policy. Keizer et al. (2017) used the s, S inventory policy as a benchmark for their joint policy based on JIT (just in time) in combination with condition monitoring and found that spare part policy based on system condition yielded significant savings. Dieterman (2019) proposed a joint policy with an (r, Q) spare part policy and applied CM, PM, and OM. He used two maintenance thresholds (d1 and d2) for triggering opportunistic and preventive

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2.4 Weather conditions and maintenance constraints

O&M of OWFs operate under unique condition like restricted accessibility due to high wave heights, these restrict maintenance activities (Tracht et al., 2013). A factor in the low availability of OFWs is the interruption of maintenance activities during changes in weather conditions (Byon, Ntaimo, & Ding, 2010). Therefore, it is appropriate to take these restrictions into consideration when developing a model for offshore wind farm maintenance and spare parts. Van Bussel and Bierbooms (2003) investigated the effects of wave heights on OFW accessibility. Dalgic, Lazakis, Dinwoodie, McMillan, and Revie (2015) suggested that different limits should be set based on the type of vessels used. They found for example the limit of 1.5 M and 25 m/s for crew transfer vessels. For the purpose of this study, a limit of 1.5M (wh) is used in assessing the accessibility of the wind farm.

2.4.1 Weather in joint maintenance and spare part models

Because of the lower operational availability and accessibility issues for offshore wind farms, several joint maintenance and spare part models are proposed in recent literature which incorporate weather restrictions. Byon et al. (2010) searched for an optimal preventive maintenance policy under stochastic weather conditions. They introduced weather limits under which preventive maintenance can be executed. Besnard, Patrikssont, Strombergt, Wojciechowskit, and Bertling (2009) used wind speed forecasting in determining the op-timal moments to perform preventive and corrective maintenance. By using wind speed forecasting their model also decreased downtime due to wind speed and wave height cor-relation. Erguido et al. (2017) used wind speed threshold in altering their maintenance threshold. This allowed the model to make scheduling decisions based on wind speed forecasts. C. Zhang et al. (2019) created a dynamic opportunistic maintenance model considering wind conditions, which were programmed stochastically. Furthermore, they used the wind condition in order to generate the maintenance wait time and incorporated weather threshold for opportunistic maintenance activities. To change this threshold, they used a forecasting period of 7 hours

2.5 Dynamic models

Static models in CBM make use of current degradation levels in making decisions. A model becomes dynamic ones it uses remaining life distributions and predictive analytics in order to make maintenance and spare part decisions into the futureVan Horenbeek and Pintelon (2013); Zhou and Yin (2019). The benefits of using predictive analytics over traditional PM and OM thresholds is shown in (Byon & Ding, 2010; Camci, 2009). Zhou and Yin (2019) proposed a dynamic opportunistic condition based maintenance policy which was able to decrease maintenance costs with up to 39.24% compared to static models. Erguido et al. (2017) created a dynamic opportunistic model which aimed at maximizing availability by incorporating wind speed thresholds.

2.6 Contribution

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3 Proposed dynamic joint maintenance and spare part model

This section will introduce the functionalities of the proposed dynamic joint maintenance and spare part model. The model uses three threshold values d1, d2, and w1. The d1

threshold indicates when opportunistic maintenance is performed. The d2 threshold

indi-cates when preventive maintenance is performed. The w1 threshold determines at which

days the probability of good weather is sufficient to schedule maintenance.

Section 3.1 will give the notations used in this sections and the remainder of this thesis. Section 3.2 gives the process of scheduling corrective, preventive, and opportunistic main-tenance. Section 3.4 illustrates the inventory model. Section 3.4.4 shows the process of performing maintenance. Section3.7 gives simplified a schematic overview of the entire decision making framework.

3.1 Notations

Binary variables

bt Variable indicating a canceled

main-tenance run at time t

df_n,t Indicates failure of wind turbine n in time period t

et Variable to indicate CM can be

scheduled

gn,t Variable indicating spare part

de-mand for n in t Mc

n,t Performed CM for wind turbine n at

time t

mc_n,t Scheduled CM for wind turbine n at time t

mo_n,t+t_f Scheduled OM for wind turbine n at time t + tf

M_n,to Performed OM for wind turbine n at time t

mp_n,t+t

f Scheduled PM for wind turbine n at time t + tf

M_n,tp Performed PM for wind turbine n at time t

ms Variable to indicate a setup is needed un,t Variable indicating spare part

re-quirement for wind turbine n in time period t

Input

cc Cost of conducting corrective main-tenance

cd Downtime costs per MW not pro-duced

ch Holding costs of spare parts ci Cost of ordering

co Costs of conducting opportunistic maintenance

cp Cost of conducting preventive main-tenance

cs Setup costs

N Total number of wind turbines n Current wind turbine

pn_t+t_f Failure probability of wind turbine n at time period t + tf

R Total number of available resources T Total simulation time

t Current time period

Tf Total forecasting time at time t

tf Time period ahead in the total

fore-casting time Tf at time t

Vt MW produced for running wind

tur-bine n at time t w1 Wave height threshold

wt Wave height at time t

Maintenance model

ao _{Indicator of available day for OM}

scheduling in Tf

ao_i Available days for scheduling oppor-tunistic maintenance

ap Indicator of available days for PM scheduling in Tf

ap_i Available days for scheduling preven-tive maintenance

Fn Failure age of wind turbine

Gn Current age of wind turbine

Ms Total number of maintenance setups in the simulation

Ra Currently available resources Outputs

A Availability of the wind farm Ac Accessibility of the wind farm

B Total number of canceled mainte-nance runs in simulation

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Ci

a Average daily inventory costs

Cm Total maintenance costs

C_am Average daily maintenance costs Cmc Total costs of canceled maintenance

run in simulation

C_amc Average daily costs of canceled main-tenance runs

Ca Average daily total costs

d1 Opportunistic maintenance

thresh-old

d2 Preventive maintenance threshold

oc Total number of CM activities in simulation run

oo Total number of OM activities in simulation run

op Total number of PM activities in simulation run

Inventory model

αDx Minimum expected lead time de-mand

αDy Minimum expected cycle time de-mand

βDx Maximum expected lead time de-mand

βDy Maximum expected cycle time de-mand

Ds Expected demand during scheduling

period

Dx Expected demand during lead time x

Dy Expected demand during cycle time

y

G(r) Critical value which minimizes hold-ing and downtime costs

Gt Total expected spare part demand

during lead time

Ht Spare part inventory at the end of

time t Ii Inventory

Io Inventory on hand at current time

period t

oi Average daily amount of inventory os Total amount of orders during

simu-lation time Q Order quantity r Reorder point

St Spare part arrivals at time t

Ut Total required spare parts required

during cycle time at time t x Lead time

Weather

At Average number of consecutive days

maintenance is not allowed at time period t

EMw Expected number of maintenance in

weather scenario w of W pi_σ,w Error associated with pi_m,w

pi_m,w Mean prediction for time i ahead in weather scenario w

W Set of weather scenarios w_tm_f Point forecast for time tf in Tf

w_tp

f Weather probability at time i ahead w_tσ_f Error value associated with point

forecast wm_t_f

wh Maximum wave height for

conduct-ing maintenance

wtf Used point forecast at time tf in Tf

3.2 Dynamic scheduling

There are three types of maintenance for which the condition of the wind turbine and the condition of the weather are evaluated in order to schedule a wind turbine for maintenance. These are corrective maintenance, preventive maintenance and opportunistic maintenance.

3.2.1 Corrective maintenance scheduling

In order to schedule corrective maintenance, the current age Gn of all wind turbines n

in N are reviewed based on the failure age of the wind turbine Fn. However, corrective

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always spans for multiple days which makes scheduling CM based on this value more accurate than not considering the past weather days.

et=

(

1, if wt−1+wt−2

2 ≤ wh

0, otherwise (1)

If et= 1 we continue with scheduling corrective by looking at the failure age, Eq. 2. This

is done for all wind turbines no matter if they are already scheduled or not.

mc_n,t= (

1, if Gn≥ Fnand et= 1

0, otherwise (2)

Additionally, if a wind turbine has failed the downtime must be registered. A new binary variable registers downtime of all wind turbines which are in a failed state. In Eq. 3 df_n,t will have a value of 1 if a wind turbines is failed and it is scheduled and if a wind turbine is failed but is not scheduled due to weather.

df_n,t=      1, if Gn≥ Fnand et= 1 1, if Gn≥ Fnand et= 0 0, otherwise (3)

Furthermore, due to the stochastic degradation path each wind turbine n follows the situation can occur that a wind turbine which has failed in the time period t is already scheduled for PM or OM in future time periods. Therefore, these schedules should be evaluated and if wind turbine n is present, it should be deleted from the schedule for PM by Eq. 4 and for om by Eq. 5

mp_n,t+t

f = 0, ∀tf ∈ Tf, if m

c

n,t= 1 (4)

mo_n,t+t_f = 0, ∀tf ∈ Tf, if mcn,t= 1 (5)

3.3 Preventive and opportunistic maintenance

Figure 1 displays the process of scheduling both preventive and opportunistic maintenance. The process only starts if wind turbine n is not in a failed state and if wind turbine n is not already scheduled for either PM or OM in any tf ∈ Tf. Additionally, preventive

maintenance has priority and is scheduled first.

The process of preventive maintenance started by reviewing the failure probability for each n in N and for all tf in Tf, by Eq. 6.

pn,t+tf ≥ d2 (6) Additionally, each wind turbine is scheduled for preventive maintenance by setting a binary variable mp_n,t+t_f to 1. Eq. 7 shows that the binary variable is set to 1 if the failure probability exceeds the d2 threshold and the weather probability wptf is lower than w1, the weather thresholds.

mp_n,t+t

f = 1 if pn,tf ≥ d2 and w

p

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Figure 1: preventive and opportunistic maintenance scheduling process

Furthermore, if wp_t

f > w1 we expect that maintenance will be cancelled. Therefore we review all w_tp_f ∈ Tf to find the closest point in time where preventive maintenance is

allowed relative to where wind turbines failure probability pn passes d2 and denote this

point as ap_.

E.g. wind turbine n passes d2 at time 4 ahead of the current time period t, tf is

therefore 4. Now suppose that wp_t

f ≤ w1 at time periods 2, 3, 7, and 9 ahead of the current time period t. ap will now take the value of 3 because tf = 3 is the closest

value to 4.

As a result we schedule PM according to Eq. 8

mp_n,t+ap = 1 if p_n,t_f ≥ d₂ and ap 6= 0 (8) In all other cases PM is not scheduled. This occurs in situations where the failure proba-bility of wind turbine n does not pas d2, or all wptf > w1, as denoted by

mp_n,t+t

f = 0 if pn,tf ≤ d2 or w

p

tf > w1 (9)

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d1 ≤ pn,t+tf < d2 (10) The process of scheduling opportunistic maintenance is not based on weather but is based on scheduled PM actions. Reasoning is that opportunistic maintenance can only be exe-cuted if there is an opportunity to conduct OM, the opportunity is the presence of a PM action. Eq. 11 represents this method in which OM scheduling is only possible if there is at least 1 wind turbine scheduled for preventive maintenance at time t + tf.

mo_n,t+t f = 1, if d1 ≤ p n t+tf < d2 and N X n=1 mp_n,t+t f > 0 (11)

Additionally, the situation occurs that no preventive maintenance is scheduled at that spe-cific time period. In this situation we determine the available time period ao _{by reviewing}

all time periods tf in Tf to see if there are PM actions scheduled. ao will take the value

of the closest time period to the time period where the failure probability of wind turbine n passes d1. Eq. 12 represents this method of scheduling.

mo_n,t+ao = 1, if d₁≤ pn_t+t

f < d2 and a

o_{6= 0} ₍₁₂₎

Lastly, if the failure probability of wind turbine n does not pass d2 or there are no PM

actions scheduled, no OM maintenance is scheduled and the binary variable is set to 0 according to Eq.

mo_n,t+t_f = 0, if d1> pnt+tf or a

o_{= 0} ₍₁₃₎

3.4 Dynamic spare parts management

The dynamic spare part policy is characterized as an (r, Q) policy. In which r is the reorder point and Q is the order quantity. The (r, Q) policy is controlled by three decision variables, r, Q, and At, and one threshold value d1. Decision variable At determines the

effects weather seasonality has on the (r, Q) policy. At is computed at every time period

t and denotes the expected amount of consecutive days maintenance is not allowed in the current season, section 3.4.1 describes the process of determining At. At is used to

update the lead time and cycle time values at every time period. Decision variables r, Q are discussed in section 3.4.2 and section 3.4.3.

3.4.1 Determining lead time and cycle time

To account for uncertainty in scheduling and performing maintenance due to weather uncertainty, we adjust the length of the lead time x and cycle time y for determining the expected demand during lead time and cycle time.The dynamic spare part policy uses a decision variable in determining the total duration for the lead time and cycle time. This decision variable takes on the value of the amount of consecutive days maintenance cannot be performed due to weather in the current season At.

E.g. At any time period t the failure probability of all wind turbines are evaluated for all tf in Tf. If a wind turbine is expected to pass the d2 threshold in time period

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period tf = 1 which is t + 1, the turbine will be scheduled at time tf = 1 . If this

situation occurs at the end of cycle time - 1 day (t + y − 1) time periods ahead, a spare part is required but has not been ordered because the wind turbine did not pass the d2 threshold within t + y (see section 3.4.3), and therefore no spare part

will be available. If At is 4, y = y + At the wind turbine is accounted for in t + y,

which results in less stock outs.

Two methods are evaluated to find the best model to predict the seasonality. Modeling t−40− t, to evaluate the current season and t−365− t−365+ct to evaluate the exact cycle

time period 1 year earlier. The rolling average for both methods is calculated and are evaluated based on the actual data, which gave the MSE (mean squared error) and MAD (meand absolute deviation) scores in table 1.

Evaluation method t−40− t t−365− t−365+ct

MSE 10.49600531 10.70016737

MAD 2.381600783 2.439162327

Table 1: Evaluation of prediction methods

We will use t−40 until t to determine At at each time period t. Eq. 14 calculates At, in

which |v| represent the number of times v consecutive days maintenance is not allowed in in t − t−40, where v varies from 2 until 10 since the scheduling horizon is initially set to

10 days.

At=

P |v|v

P |v| (14)

At each time period, the lead time and cycle time are recalculated for further use in the remaining spare part model according to Eq. 15 and Eq. 16.

x = x + At (15)

y = y + At (16)

3.4.2 Determining r based on the expected demand during lead time

Expected demand during lead time is determined in two steps. First, the expected demand during scheduling (Ds) is determined, after which Dx is determined.

Demand during scheduling

Demand during scheduling (Ds) is determined through a Monte Carlo simulation as in

figure 2. Each time period the spare parts demand is generated by evaluating the required maintenance actions for all wind turbines. However, we evaluate the spare parts require-ments by using 10 different weather scenario’s, each scenario is accompanied by a different spare part demand EMw. We create a total of 10 weather scenarios W through Eq. 17.

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Figure 2: Demand during scheduling wtf,w= w m tf + w σ tfα ∀w ∈ W, ∀tf ∈ Tf (17)

Figure 18 shows that we store each expected amount of spare parts demand as EMw.

The expected demand during scheduling Dscan now be calculated through Eq. 18, which

represents the last proces block in figure 2

Ds=

PW

w=0Emw

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Demand during lead time

Demand during lead time is given by the lead time demand per wind turbine sn,t and the

sum of all lead time stdemand during lead time x. Eq. 19 computes the expected demand

for each wind turbine at time t. In which wind turbines who pass threshold d1 during the

scheduling horizon Tf are excluded. Eq. 20 sums the expected demand during lead time

for all wind turbines.

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As a result total demand during lead time is given by Eq. 20.

Dx= Ds+ Gt (21)

Expected lead time demand st and cycle time demand Dy follow a uniform distribution.

The probability density function is given by Eq. 22.

f (x) = 1 βDx− αDx

(22) In Eq. 22, βDx is the maximum expected lead time demand Dx and αD x is the minimum expected lead time demand Dx. The r and Q can be found after computing the critical

value which minimizes holding cost and downtime costs in Eq. 23.

G(r) = c

d

cd_{+ c}h (23)

The optimal value of r, which minimizes costs is then determined by Eq. 24.

r = αDx+ G(r) ∗ (βDx − αDx) (24) The next step in the (r, Q) model is to trigger the reorder point, which is expressed as a binary variable Zt. Eq. 25 expresses Zt, the reorder point is triggered ones the inventory

level drops below the reorder point. The inventory level is continuously monitored in the proposed policy. Inventory on hand Io is updated by Q if the inventory level drops below

the reorder point, as stated in Eq. 26

Zt= ( 1, if Io ≤ r 0, otherwise (25) Io = ( Io+ Q, if Io≤ r Io, otherwise (26)

3.4.3 Determining Q based on expected demand during cycle time

If the reorder point is triggered (Zt = 1), Q should be determined. The first step in

determining Q is determining the expected demand during cycle time Dy. Dyis determined

based on the expected number of spare parts required during cycle time, which is calculated by setting a binary variable for each wind turbine which requires a spare part during y, according to Eq. 27. We exclude all wind turbines of which the failure probability exceed the d1 threshold during lead time, these are already considered in the demand during lead

time (Eq. 19) Additionally, total demand during cycle time is calculated as the sum all wind turbines which require spare parts during y, this is represented by Eq. 28.

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Similar to the lead time demand, cycle time demand is assumed to follow a uniform distribution. Order size Q is determined by the minimum expected lead time demand, the critical value and the maximum expected lead time demand, as in Eq.

Q = αy+ G(r) ∗ (βy− αy) (29)

On-hand inventory is updated if a replenishment order was made t − x time periods ago. St+xdenotes the spare part arrivals, which takes the value of 1 if spare parts are to arrive.

St+x=

(

1, if I ≤ r

0, otherwise (30)

Furthermore, on-hand inventory is updated when spare parts arrive by Eq. 31. Moreover, inventory on hand is registered as Ht (Eq. 32) in order to calculate ch

Ioh= Ioh+ (Q ∗ At) (31)

Ht= Ioh (32)

3.4.4 Performing maintenance

Performing maintenance is subject to three constraints, weather, spare part availability, and resource availability. Furthermore, opportunistic maintenance is only performed when also CM or PM is performed. Eq. 33 shows how the binary variable M_n,tc is set to one if CM is performed on wind turbine n in time period t. This only occurs if wind turbine n is scheduled for corrective maintenance in time period t. Additionally, maintenance can only be performed if weather allows the model to conduct maintenance, enough resources are available, and there is more than 1 spare part available. Eq. 34 shows similar method of performing maintenance as Eq. 33. Eq. 35 varies because OM can only be executed if either PM or CM is also executed at the current time period t. Furthermore, in performing OM no additional resources are used, because the resource allocated to either performing CM or PM is used. The binary variables M_n,tc , M_wt,tp , M_wt,to are ultimately summed and multiplied by the accompanying maintenance costs in section 3.5.

M_n,tc = ( mc n,t, if (Wt≤ wh and Ra > 0 and Ioh > 0) 0, otherwise (33) M_wt,tp = ( mp_wt,t, if (Wt≤ wh and Ra> 0 and Ioh > 0) 0, otherwise (34) M_wt,to = ( mo_wt,t, if (Wt≤ wh and Ioh > 0) and PW Twt=1Mwt,tp > 0 0, otherwise (35)

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weather constraint by setting a binary variable for each time period t in Eq. 36. Eq. 37 calculates the total number of canceled runs during the simulation time.

bt= ( 1, ifPN n=1m p n,t+ PN n=1mcn,t> 0 and Wt≥ wh 0, otherwise (36) B = T X t=1 bt (37)

In order to calculate the amount of stock outs, Eq.38 registers hn,t if wind turbine n was

scheduled at time t and inventory was not sufficient. Additionally, Eq. 39 calculates the total amount of stock-outs during the entire simulation time.

hn,t= ( 1, if mp_n,t+ mc_n,t+ mo_n,t> 0 and Io= 0 0, otherwise (38) H = T X t=1 N X n=1 hn,t (39)

3.5 Output and evaluation of the model

This section describes the output of the model. The output is subdivided by types of costs, total costs and availability and accessibility.

3.5.1 Maintenance costs

Total number of maintenance activities is calculated Eq. 40, Eq. 41, and Eq. 42, for opportunistic, preventive, and corrective maintenance respectively. In these Eq. (40, 41, 42 all binary variables indication performing maintenance (see section 3.4.4) are summed per maintenance type.

oc= N X n=1 T X t=1 (M_n,to (40) op = N X n=1 T X t=1 (M_n,tp (41) oc= N X n=1 T X t=1 (M_n,tc (42)

The total costs of maintenance is dependent on the number of each maintenance type performed, along with the total number of maintenance setups (Ms_{). This is determined}

based on setting a binary variable ms_t at each time period t when at least either corrective or preventive maintenance is performed, since only opportunistic maintenance is not allowed.

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This results in Eq. 44 as the calculation of the total number of setups during the simulation period. Ms= T X t=1 ms_t (44)

The total costs Cm of maintenance during the the simulation period will than be given by Eq. 45

Cm= Mscs+ occc+ opcp+ ooco (45) The average daily maintenance costs will be

C_amC

m

T (46)

3.5.2 Spare part management costs

The total inventory costs are subdivided in holding costs and ordering costs. Eq. 47 represent the average daily amount of inventory. Ordering spare parts encompass the total amount of orders placed during all time periods as seen in Eq. 48.

oi= PT t=1(Ht) T (47) os= T X t=1 (At) (48)

Total inventory costs Ci is represented by Eq. 49 and C_ai is represented by 50 and gives the average daily inventory costs.

Ci = (oich)T + osci (49)

C_ai = C

i

T (50)

3.5.3 Downtime, availability, accessibility, and total costs

Other outputs which are used to evaluate the policy are the downtime costs Cd, availability A, the accessibility Acof the OFW, and the costs of canceling maintenance due to weather.

The Cdcomes from Eq. 51 and are calculated by reviewing how many of the wind turbines scheduled for corrective maintenance are actually performed, the difference is multiplied by the potential MW output (Vt). Ultimately this is summed for all wind turbines and all

time periods and multiplied by cd.

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The availability A is calculated based on the total number of failed wind turbines. A = 1 − PN n=1 PT t=1d f n,t T N (52)

The accessibility rate shows how often the wind farm is accessible at time periods t that maintenance is scheduled. Total number of times maintenance is scheduled is given by Ms+ H, where H comes from Eq. 39. Eq. 53 calculates the accessibility rate of the wind farm.

Ac=

Ms

Ms_{+ H} (53)

Furthermore, we incorporate the failed maintenance runs in the total cost function by calculating the costs of not being able to perform maintenance through Eq. 54. In which we assume that all necessary fixed costs to perform maintenance cs have already been made. Eq. 55 than gives the average daily cancellation costs.

Cmc= Hcs (54)

C_amc= C

mc

T (55)

Lastly, we calculate the total costs and average daily total costs in Eq. 56 and Eq. 57

C = Cmc+ Cm+ Ci+ Cd (56)

Ca= Camc+ Cam+ Cai+ Cad (57)

3.6 Model Assumptions

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4 Method of research

This sections motivates the research method and elaborates on the process of finding the best parameter values for the proposed model in section 3. Furthermore, the input data is discussed and the pre-processing steps of the weather data are explained. Lastly, the case study which is used to evaluate the model is discussed.

4.1 Research method

This research is focused on finding if a joint maintenance and spare part policy will benefit of having a decision making structure which incorporates weather forecasts. According to Sokolowski and Banks (2011) and Carson (2004) simulation studies are adequate methods for studying complex, expensive or time consuming systems. Furthermore, the simulation study can be used to predict outcomes of a system and compare different scenario. Sim-ulation study is an appropriate method for this thesis because of the following reasons. Firstly, both failure and weather data are inherently uncertain, which makes studying them in real life complex. Secondly, offshore wind farms are complex and very expen-sive to study in real life, testing new maintenance policies can result in enormous missed revenues.

The policy proposed in this thesis is bench-marked in order to evaluate its performance. The model is bench-marked to a dynamic (d1, d2) policy. The benchmark model uses the d1

and d2 thresholds for evaluating the failure probability of the wind turbines. Additionally,

the two thresholds are used in determining the expected demand during lead time and cycle time. In the benchmark model, the maximum allowed wave height wh is used to determine

if scheduled maintenance can be performed. The model varies from the proposed model because it does not incorporate weather forecasts in the decision making process.

4.1.1 Parameter optimization

This section will describe the Monte Carlo method used in order to find the best parameter (d1, d2, w1) for this policy. Furthermore, the same approach will be used in determining

the optimal benchmark policy.

A grid search will be used to optimize the parameters of the joint policy. Because the d1, d2, w1 thresholds are all used at the scheduling part of the simulation model, the grid

search will take place for all three thresholds simultaneously, this will emphasize the effects of combining the parameters. Figure 3 gives a schematic overview of finding the best parameters for both policies and ultimately evaluating the policy. Figure 3 shows that for both the proposed policy and the benchmark policy a range of threshold values is tested through a Monte Carlo simulation. Each set of thresholds is evaluated based on the average daily output costs. Based on the previous set of thresholds and their outcomes a new range is set and evaluated.

E.g. we set range 1,2,3 for d1, 0.4,0.6,0.8 for d2 and 0.5,0.7,0.9 for w1 if the policy

1,0.4,0.7 for d1, d2, w1 respectively yields the best costs, the optimal is probably not

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therefore adapt the ranges of both d1 and d2 to be lower than the current range and

we run the Monte Carlo simulation again.

If the optimal values of both policies is reached we determine the policy by fixing the threshold values. Lastly, we evaluate the proposed model by comparing the outputs to the benchmark model.

Figure 3: Finding the policies and evaluation

4.2 Input data

4.2.1 Degradation data

The degradation data used for the wind turbines in this thesis originates from Yildirim et al. (2017). A total of 40 degradation level samples is used, each having a unique stochastically increasing degradation level and failure probability. At initialization of the simulation process each wind turbine is randomly assigned to a degradation sample and the age is randomly determined between 0 and Fn. After maintenance is performed, the wind

turbine again is randomly assigned to a sample, and the age is set to 0. In determining the future degradation level at each t, the remaining life distribution (RLD) is reviewed.

4.2.2 Weather data

Weather data used in this thesis originates from the FINO 1 database1, which is a research platform in the North Sea where also a wind farm is located. Only wave data is used in the simulation model. The data ranged from 2011 up to and including 2018, the data was 85.6 percent complete. Missing data points are filled through two steps: firstly through linear interpolation, secondly, using the linear interpolated value as a mean and the variance of the remaining data set, the new data points are filled.

1

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Wind data from the FINO 1 database is used as input to the model for determining downtime costs (cd). Data from 2013 - 2018 is used in determining the produced MW per day, in the same way as in Dieterman (2019).

4.2.3 Probability forecasting

This section describes the pre-processing step in order to compute the probability forecasts used in the simulation model in section 3.

Based on the available weather data, a forecast is made for each tf in Tf. Tf is set to

10 days. The forecast is based on time series data by using an ARIMA (autoregressive integrated moving average) model. The data itself is described in section 4.2.2. Firstly, a Dickey Fuller test was performed in order to rule out any trends present in the data, this resulted in a p-value of 2.58825E-09. We can therefore reject the null hypothesis and assume no trend is in the data, an ARMA(autoregresive moving average) model is thus appropriate. Secondly, A grid search was performed to find the ARMA model which fits best with the weather data ,based on the AIC (akaike information criteria). This results in two parameters for the ARMA(p, q) model. In which p is the number of autoregressive legs used by the model and q is the number of moving average terms used by the mode. The grid search resulted in an ARMA(2,1) model with AIC of 7680.649. Eq. 58 gives the mathematical formulation of the model, in which p and q are equal to 2 and 1. The result is that the ARMA model will use two autoregressive lags and one moving average lags(q). This indicates that the forecasted value wm

tf is based on the previous two observations wt−1 and wt−2. In Eq. 58, wmtf is the predicted value at time t for tf in Tf, φt is the coefficient for AR terms, θj gives the coefficient for the MA term, and tgives the white noise value.

w_tm f = p X i=1 φiyt−i+ q X j=1 θjt−j+ t (58)

The ARMA forecast will be based on previous day values. The further the forecasts spans in Tf the more the forecasts will level out. This is due to the fact that the forecast for tf =

4 until 10 are based on forecasted values of the ARMA model. The model does not include variability, which indicates that it will level out. Furthermore, the ARMA model does not account for forecasting errors. In order to include the error and the variation of the original data we therefore adjust the forecasted value. The process of creating the forecast is given in Figure 4. At each time period t a new wave height value is observed, which is added to the history of the model and represents the starting point of the forecast. At tf = 0

the forecasted value for tf = 1 is based on observed values wt and wt−1. The ARMA(2,1)

model used Eq.58 in order to forecast wm

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Figure 4: Forecasting process

achieved through applying a Monte Carlo simulation, in which 1000 (N ) different predic-tions are made based on the ARMA model, through Eq. 60. In which α is a random value between -1 and 1 in order to create the N number of different predictions for tf in

Tf. Each predicted value is then changed into a binary value based on Eq.61. Eq.62 then

determined the wp_t

f, which will be used for scheduling purposes. wtf,n = wtf + w σ tfα ∀n ∈ N (60) wtf,n= ( 1, if wtf,n < wh 0, otherwise ∀n ∈ N and ∀tf ∈ Tf (61) w_tp_f = PN n=1wtf,n N ∀tf ∈ Tf (62)

The forecasts, forecast errors, and probability forecasts are pre-processed for each time period t and used in section 3.2.

4.3 Case study

This section describes the case study which is used to demonstrate the proposed model. A case study is conducted for a total of 80 wind turbines (N ). The simulation runs for a time horizon of 5 years (T ), which makes use of a warm-up period of 1 year. Furthermore, the available resources per day is set to R = 4. The arrival of replenishment order is set to a lead time of x = 21. Additionally, the cycle time which is used to determine Q is set to y = 42.

4.3.1 Cost parameters

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5 Results

This section presents the results from the research method presented in chapter 3 and 4. Section 5.1 will present the optimal threshold values for the proposed policy and the benchmark policy. Section 5.2 will evaluate the proposed policy based on the benchmark policy. Section 5.4 will perform a sensitivity analysis to give an overview of the effects of changing fixed parameters.

5.1 Numerical results

Based on the grid search process the optimal values for the d1, d2, w1 thresholds are found

for the proposed policy. The three threshold of the proposed policy are optimized simul-taneously. For representation purposes the figures are split in showing the d1, d2 threshold

versus the total costs, and the effect of the w1 threshold under fixed d1 and d2. Figure 5

shows the all combination of d1 and d2 threshold values versus the costs. We see that ones

the lower bounds of these ranges is reached the costs increase, which indicates that this is a good range to review for the optimal policy. Additionally, the start increasing when approaching the upper bounds of both ranges, again indication that this is the appropriate range to review. The optimal d1, d2 values are 1,9 and 0.4 respectively. Figure 6 presents

the resulting total costs of the w1 threshold for fixed d1 and d2 values. The optimal found

policy was the (d1, d2, w1)(1.9,0.4,0.7) policy, with an average total costs ofe 14.258.

Figure 5: Representation of d1 and

d2 vs costs

Figure 6: w1 threshold under fixed

d1 and d2

Figure 7 gives the results from the optimal combinations of d1 and d2 values for the

benchmark model. The graph shows that the costs increase at all boundary, indicating the optimal policy is within this range. The optimal policy is the (d1, d2)(2.5,0.9) policy.

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Figure 7: Benchmark optimum

5.2 Comparative study

This section discusses the results of comparing the performance of the proposed model to the benchmark model.

5.2.1 Availability and accessibility metrics

Table 2 provides a summary of the metrics used to evaluate the performance of the pro-posed model in this thesis. The availability rate decreases in the propro-posed model oppro-posed to the benchmark model with 0.44%. Additionally we see that the average daily downtime costs increase by 221.73%. This is due to the fact that the proposed model uses expected weather in scheduling decisions for failed wind turbines. As a result failed wind turbines are not automatically scheduled for maintenance the next time period, which increases the duration the wind turbine is in a failed state. Subsequently this decreases the availability rate and increases downtime costs. The accessibility rate of proposed model increases by 32.29% apposed to the benchmark model. This is the result of the model scheduling cor-rective maintenance and preventive maintenance at moments of higher probability of good weather. The resulted consequence in the costs of the proposed policy is that the costs for canceled runs decreases by 56,88%. the trade-off that the model makes in achieving costs benefits is largely based on these two types of costs. The proposed model will allow more and longer periods of downtime in order to save on fixed setup costs for not being able to complete a maintenance run.

Metric Benchmark Proposed policy Difference

Availability 99.78% 99.34% 0.44%

Accessibility 44.01% 65.01% (32.29%)

Daily cancellation costs e 3.426 e 1.477 (56.88%)

Daily downtime costs e 359 e 1.155 221.73%

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5.2.2 Inventory metrics

The metrics used to evaluate the inventory performance of the proposed model are sum-marized in figure 8. The number of orders is 31 in the proposed model and 32 in the benchmark model. Furthermore, the average inventory decreases in the proposed model opposed to the benchmark model. Average daily inventory decreases with 11.4% from 14.11 to 12.5 spare parts per day. Additionally, the stock outs decrease with 24.46% from 75.2 to 56.8 units during the total simulation time. Additionally, table 3 gives the inven-tory costs of the proposed and benchmark models. With regard to costs the proposed model achieves a 10.85% decrease. This decrease is the result of lower inventories and lower ordering costs. These results show that the proposed model achieves a lower average inventory with lower amount of orders and achieves a lower amount of stock outs with lower costs.

Figure 8: Inventory metrics

Metric Benchmark Proposed policy Difference Daily inventory costs e 7506 e 6692 (10.85%)

Table 3: Inventory costs

5.3 Maintenance metrics

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Figure 9: Maintenance metrics

precise scheduling during because of the use of w1 threshold. This is the basis of the

ac-cessibility rate in section 5.2.1. Lastly, the total number of maintenance runs executed is almost identical with 112.3 and 111.8 for the proposed and benchmark model respectively.

5.3.1 Cost metrics

The last set of metrics that will be evaluated are the costs of both the proposed and benchmark model. Firstly, table 4 gives an overview of the costs made by each model and how each type of cost changed in the proposed policy relative to the benchmark. The total costs made by the proposed model have decreased by 10.62% opposed to the benchmark model. The cost savings are located at the decreased inventory and cancellation costs. Furthermore, the maintenance and downtime costs have increased by 5.87% and 221.73%, respectively.

Metric Benchmark Proposed policy Difference

Daily cancellation costs e 3.426 e 1.477 (56.88%)

Daily downtime costs e 359 e 1.155 221.73%

Daily inventory costs e 7.506 e 6.692 (10.85%) Daily maintenance costs e 4.661 e 4.935 5.87% Total daily costs e 15.952 e 14.258 (10.62%)

Table 4: Cost metrics

While the percentage of cost savings per cost category do not outweigh the percentage increase per cost category, at the bottom line the proposed model does outperform the benchmark model. This is due to the size of the cost categories relative to the total costs. Figure 10 shows the relative costs of each cost output to the total costs made by the models.

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Lastly, we see that the relative maintenance costs increase apposed to the benchmark model. This is due to the increased amount of corrective maintenance actions required, as discussed in section 5.3, which are more expensive.

Figure 10: Cost metrics

5.3.2 Conclusion comparative study

In this section the comparative study is presented. The total costs of the proposed model apposed to the benchmark model decrease with 10.62%. This represents a daily savings of e 1.694,05 and a decrease in total costs of e 2.473.313,00 over the entire simulation period. As seen in section 5.2.1 these costs savings originate mainly in the cost savings of avoiding unnecessary setup costs. For offshore wind farms this indicates that savings can be achieved by allocating their resources differently based on a schedule which uses weather probabilities. Offshore wind farm owners or service organization can choose to out rent their resources more often, which is common practice. This allows them to gain opportunity costs by scheduling more forward-looking. Furthermore, offshore wind farm owners who don’t own service vessels can achieve similar cost savings through the use of the proposed policy.

Additionally, the inventory metrics in section 5.2.2 shows that additional savings can be achieved by using weather uncertainty in the inventory model. The cost savings originate in more effectively using the inventory because more accurate prediction can be made of the expected demand. Furthermore, less stock-outs appear in the proposed policy because of the more accurate prediction of expected demand. The benefits of the savings originating in the stock outs are not quantified in this thesis, but do show additional potential for the proposed policy. The decrease in stock outs means that the proposed policy makes more effective use of the setup costs allocated to conducting maintenance. Because corrective maintenance and preventive maintenance get priority in using spare parts in this policy, this indicates that the stock outs are primarily located at opportunistic maintenance actions. The savings achieved by lower stock outs therefore result in a higher life expectancy of turbine components.

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avoiding additional setup costs and incurring additional downtime costs. This trade-off will shift if energy prices would rise. This means that offshore wind farm owners should continuously make this trade off in their decision making policy.

5.4 Sensitivity analysis

This section will elaborate on the effects of changing fixed parameters on the optimal proposed policy. This will test the robustness of the model and will give additional insights. The parameters that are discussed are the holding costs, the maximum allowed wave height for maintenance, and MW price.

5.4.1 Holding costs

Figure 11 shows the effect of increasing the holding costs on the total costs and inventory costs of the model. The expectations are that the that the costs increase because of higher costs of holding inventory. Figure 11 shows that both the total costs and inventory costs show a similar growth path, indicating that the total increased costs are primarily located in the inventory costs, which corresponds to our expectations

Figure 11: Effect on total costs and inventory costs

Figure 12 shows the negative relation between an increase in holding costs and the average inventory the model holds. Because the holding costs are used in the calculation of the critical ratio of the inventory model. The effects of an increasing holding costs are therefore a decrease of the reorder level r and a decrease in the order quantity Q, resulting in lower average inventories. Furthermore, the decrease of average inventory levels also results in a higher level of stock outs, as seen in figure 13.

Ultimately, figure 14 shows the effects that an increase of holding costs has on the downtime costs. Due to the increased amount of stock-outs more wind turbines will fail and an increase in downtime costs is the result.

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Figure 12: Effect on average

inven-tory Figure 13: Effect on stock outs

Figure 14: Effects on downtime costs 5.4.2 Maximum allowed wave height

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Figure 15: Effect on availability and accessibility rate

Figure 16 shows the negative relation between the maximum allowed wave height and the total costs. The costs naturally decreases because of the decrease in cancelled runs costs and a decrease in downtime costs, because more scheduled maintenance runs are also executed and corrective maintenance is more frequently scheduled. The slope in figure 16 flattens at the point of wh = 2.5 and shows a steep decrease in the early steps. This

indicates that at a certain point, it is not beneficial anymore to increase the accuracy of the forecast.

Figure 16: Effect on total costs

In figure15 the accessibility rate is at 0.87 if the wh = 2.5M . This represents the point

where the slope flattens in figure 16. In this scenario it will not be beneficial to invest in increasing the accessibility rate because achieving the last percentage increase is hard and the added benefits in terms of costs are minimal. For offshore wind farms this indicates that there is a limit of achieving cost benefits by increasing forecasting accuracy.

5.5 Price per MW

The expected effects of the price per MW on the proposed policy is an increase in total costs and downtime costs. The downtime costs increase while the same level of failed wind turbines is maintained. Figure 17 shows the effects of increasing the MW price on the total costs. As expected the costs increase with an increase in MW price.

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Figure 17: Effect on total costs

increase is achieved because all other costs stay on the same level if the MW price is increased.

Figure 18: Effect on the relative downtime costs

The effect of the relative downtime costs show that the importance of downtime costs and failed wind turbines increases if the MW price increases. For offshore wind turbines this shows that variation in energy prices should be considered in the decision making framework. Additionally, this highlights the importance of an accurate scheduling pro-cess. By increasing the accuracy of a forecast the number of failed wind turbines can be decreased. Especially when variation exists in energy prices, this is a tool which can be used to counter the negative effects increased MW price.

5.5.1 Conclusion

This section describes the sensitivity analysis in order to evaluate if the model behaved as expected and to review additional outcomes. The results of the sensitivity analysis showed that the model behaved as expected, which indicates that model is robust. Additionally, increasing holding costs ultimately will increase the downtime of the model. This shows that the inventory element of the proposed policy must be altered for different types of spare parts.

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shows that the positive effects of increasing the accuracy of forecasting diminished ones the forecasting accuracy reaches 86%.

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6 Conclusion and Limitations

The goal of this thesis was to research whether or not a joint maintenance and spare part policy could benefit of the use of weather forecasts in its decision making. This sections will give a short summary of the outcomes and their meaning, followed by the limitations present in this thesis.

6.1 Conclusion

The model proposed in this thesis is a dynamic joint spare part and maintenance model which uses three threshold values. The d1 threshold is used for reviewing opportunistic

maintenance. The d2 thresholds is used for reviewing preventive maintenance. The w1

threshold indicates the wave height probability threshold, which determines the weather probability is good enough to schedule preventive maintenance. The model makes use of remaining life distributions and a wave height probability forecast to schedule wind turbines for preventive and opportunistic maintenance. Furthermore, the model uses the wave heights of the past two days to determine if corrective maintenance is scheduled. Furthermore, the (r, Q) inventory model within the proposed policy uses the d1 and d2

thresholds to determine the expected demand during lead time and demand during cycle time. This allows it to recalculate the reorder level r and the order quantity Q at each time period. Additionally, the spare part model incorporates uncertainty which originates from the uncertain weather conditions in which offshore wind farms operate. It incorporates this uncertainty in the reorder level r by evaluating multiple weather scenarios and calculate the average demand during the scheduling horizon. Furthermore the policy models seasonality in terms of consecutive days maintenance is not allowed, and increases the length of the lead time and cycle time based on this seasonality.

A comparative study is used in order to evaluate the proposed policy. In evaluating the policy, a benchmark model is used. The benchmark policy is a dynamic policy in which decision making is solely based on the remaining life distribution of the wind turbines and the d1(opportunistic maintenance) and d2(preventive maintenance) thresholds. In

determining the optimal threshold values a Monte Carlo simulation in combination with a grid search is conducted.

The results of the comparative study show that the proposed model outperforms the benchmark model. The proposed model achieves a total of 10.62% cost savings, which resulted in a total cost savings ofe 2.473.313 over the total simulation time. Additionally, the results show that the cost savings are primarily based on the avoidance of unnecessary maintenance setups. In striving to decrease these unnecessary costs and increasing the accessibility of the wind farm, the model allows for more turbines to fail for a longer period of time. This is due to a more selective scheduling process by including wave height probability forecasts. Additionally, incorporating uncertainty due to weather in the inventory model, allows for a more effective inventory model. The inventory element achieved lower costs due to lower average inventories along sides a lower amount of stock-outs.

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6.2 Limitations

The model proposed here based all scheduling decisions on a probability forecast which was made using time series data. The ARMA model used has the negative side effect of not being a good predictor of variability in the data. These effects are partly diminished by changing the actual point forecast to a probability forecast using the forecast error associated with the point forecast. While using a Monte Carlo simulation for this process should provide us with a near accurate density forecast of the wave height, the accuracy of the forecast can be improved. Furthermore, the results show that even a slight increase in prediction accuracy showed an increase in overall performance of the policy.

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Dynamic Joint maintenance and spare part policy based on continuous condition monitoring and weather predictions