Multi-level condition based opportunistic maintenance optimization for oﬀshore wind farms

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Multi-level condition based opportunistic maintenance

optimization for offshore wind farms

Rijksuniversiteit Groningen

Master Thesis Technology & Operations Management

Constantijn Lennart Kruize

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Abstract

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1 Introduction

For European countries with access to open water bodies offshore wind farms have gained popularity due to the abundance of wind potential and open space for installation (Sarker & Faiz, 2016). The underlying reason for this increased popularity is to be able to fulfill the requirements posed in the EU Renewable Energy Directive, which requires all members of the EU to fulfill at least 20% of total energy consumption with renewables by 2020. However, due to the characteristics of the operating environment in which these Wind Turbines (WTs) are installed there are some serious challenges in operating and maintaining these WTs.

In contrast to the stationary conditions under which most machines operate WTs suffer from the harsh maritime environment (Y. Ding, Ntaimo, & Byon, 2010). This causes unpredictability in power output, the deterioration of components, as well as in the feasibility of maintenance actions (Byon, 2013). Under the current Operating and Maintenance (O&M) strategies costs are high with O&M costs ranging up to 20-35% of total costs (Sarker & Faiz, 2016). In contrast, the O&M costs for onshore wind energy lay between 5-10% (Shafiee & Sørensen, 2017). To be competitive with other renewable energy sources there is an increasing pressure to reduce O&M costs (Shafiee & Sørensen, 2017). To reduce these expenditures, sound O&M policies should be developed for offshore Wind Farms (WFs).

Despite their importance, O&M policies are not optimized in practice (Erguido, Crespo Márquez, Castellano, & Gómez Fernández, 2017). Currently the most dominant forms of maintenance poli-cies applied within wind farm maintenance are strictly corrective polipoli-cies, in which action is only undertaken in case of a failure, and time-based policies, in which preventive maintenance actions are performed at fixed time intervals.(F. Ding & Tian, 2011). Relatively few studies addressed other maintenance strategies such as opportunistic maintenance as a suitable maintenance policy for WFs.

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The benefits of an opportunistic maintenance strategy in an offshore wind farm setting have been demonstrated within multiple studies. These studies show clearly that when maintenance is sched-uled opportunistically significant O&M cost reductions can be achieved by reducing production losses, maintenance costs and travelling expenses in comparison to failure- and time-based policies (Abdollahzadeh, Atashgar, & Abbasi, 2016; Besnard, Patriksson, Str¨omberg, Wojciechowski, & Bertling, 2009; Besnard, Patriksson, Str¨omberg, Fischer, & Bertling, 2011; F. Ding & Tian, 2011, 2012; Erguido et al., 2017; Sarker & Faiz, 2016). However, all mentioned studies are aimed at optimizing time-based strategies, neglecting the negative influences of the operating environment on the deterioration of components (Bachant, Goude, & Wosnik, 2016; Wilson & McMillan, 2014). This leads to maintenance actions that are executed either too early, not exhausting the full life of components, or too late which in turn increases downtime and maintenance costs. In contrast to time-based maintenance policies, within Condition Based Maintenance (CBM) policies mainte-nance decisions are taken based on the actual condition of WTs components. This ensures that maintenance actions can be scheduled at the ideal moment.

Multiple studies have demonstrated the effectiveness of CBM for on- and offshore WFs in com-parison to corrective or time-based alternatives (Amayri, 2011; Besnard & Bertling, 2010; Byon & Ding, 2010; McMillan & Ault, 2008; Nilsson & Bertling, 2007; Tian, Jin, Wu, & Ding, 2011). However, these works do not consider opportunistic maintenance as a possible beneficial addition to CBM. Finally there are three works that consider both opportunistic maintenance and CBM however, Perez (2015) and Tian et al. (2011) are both focused on the maintenance of a single WT and pay no attention to dependencies between turbines, whereas Yildirim et al. (2017) in turn did not consider dependencies on component level within WTs. We expect that when the dependencies that are present within and between WTs are considered jointly that; 1) maintenance activities can be targeted more precise at the turbines that oppose the highest risk of failure, which will reduce unexpected failures and downtime, and 2) economic dependencies can be exploited more thoroughly due to the joint consideration of CBM and opportunistic maintenance on a currently not discovered level, reducing O&M costs.

Therefore, this study will try to extend the current wind farm maintenance literature by devel-oping a Mixed Integer Programming model and framework to optimize the O&M of large-scale wind farms in a Condition Based and Opportunistic manner. This will be done on component level for an entire WF to consider the economic dependencies within the WT and between WTs. The objective of the model will be to maximize the operational revenue by jointly optimizing O&M subject to operational aspects such as power output and energy price, as well as maintenance aspects such as the degradation of components based on actual sensor data, limited maintenance resources and feasibility of maintenance activities due to weather conditions.

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2 Literature review

To position this study relevant literature will be discussed with the goal of indicating what has been studied, and where the gap in literature lies. The following section will first discuss the relevant literature regarding opportunistic maintenance of WTs and its limitations. Secondly, the same will be done for relevant literature regarding CBM. Hereupon, this will be repeated for works that jointly consider CBM and opportunistic maintenance. At the end of the this section the contribution of this study to the current stream of literature will be addressed.

2.1 Opportunistic Maintenance

Only a limited number of studies are centered around opportunistic maintenance of WFs. Accord-ingly, within these works it has been demonstrated in that opportunistic maintenance can be a more effective maintenance policy in comparison to corrective and time-based alternatives. The most important works and their findings will be discussed within this section.

One of the first works that considered OM in a wind farm setting is Besnard et al. (2009). In this study preventive tasks are scheduled opportunistically over a short time horizon in case of failure. Moreover, the tasks are scheduled when weather conditions are favorable, which ensures low production losses high and turbine accessibility. In Besnard et al. (2011) this model is extended by incorporating a long term planning horizon. The results for a small near-shore WF demon-strated that indirect maintenance costs could be decreased significantly with 32% in comparison to a strictly time-based maintenance policy. However, the major shortcoming in both works is that all maintenance actions bring the turbine back into an “As good as new” condition, however, in reality preventive maintenance does not always return components into this state (Spinato, Tavner, van Bussel, & Koutoulakos, 2009). To model the effectiveness of maintenance action more realistically Ding & Tian (2011) considered imperfect preventative maintenance actions. In Ding & Tian (2012) this work is extended by developing an opportunistic maintenance approach based on component’s age threshold values, and several different preventive maintenance thresholds depending on the operating conditions. However, within this study component failures and the hereupon following corrective maintenance actions act as a prerequisite to schedule preventive maintenance actions opportunistically. Since the optimal frequency to conduct preventive maintenance actions is much higher than the frequency of corrective maintenance actions (Besnard et al., 2009; Ribrant, 2006) execution of preventive maintenance actions will deviate from its optimal frequency.

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their study adapts the decision-making process to the operating conditions fostering maintenance effectiveness whilst minimizing worker risk. Moreover, in doing so power output becomes less affected by maintenance activities.

All previously mentioned works have one aspect in common, namely they all schedule mainte-nance in a time-based fashion. Deterioration in these instances is modeled as a function of time and maintenance activities are triggered by time thresholds. However, for WTs components the actual condition is significantly influenced by the variable load due to the stochastic weather conditions (Bachant et al., 2016; Wilson & McMillan, 2014). Moreover, identical machinery also exhibits con-siderable differences in the exact moment of failure (Yildirim, Gebraeel, & Sun, 2017). This causes that reducing failures in a time-based fashion comes at the cost of performing maintenance tasks more often than necessary, and not exhausting the full life of a large share of the components within the WF. Moreover, since the actual condition and the possible differences in the exact moment of failure are not considered also more failures will occur (Garc´ıa M´arquez et al., 2012).

2.2 Condition Based Maintenance

Within CBM systems are constantly assessed on their actual condition using condition information collected through condition monitoring. The common goal of CBM is to provide information to the operator to make more informed maintenance decisions. The popularity of CBM is increasing due to the rapid technological developments in the field of equipment and increased computer capability. However, not all systems can be condition-monitored. Firstly, the conditions must correlate to the moment of failure and secondly, they must be measurable (De Jonge, Klingenberg, Teunter, & Tinga, 2015). Fortunately, due to the relative simple mechanical construction of WTs, failures are easy to monitor through the use of integrated sensors (Hameed, Hong, Cho, Ahn, & Song, 2009).

The first works that considered CBM in the context of WT maintenance were mainly aimed at justifying the additional condition monitoring equipment expenses for individual components. For example, Nilsson & Bertling (2007) and McMillan & Ault (2008) both demonstrate the feasibility of a CBM strategy compared to a time-based alternative. In Besnard & Bertling (2010) CBM is not compared as in previous works against time-based maintenance, but instead against three different methods of CBM, namely; visual inspection, periodic condition monitoring and continuous condition monitoring. Here continuous condition monitoring proved to be the most beneficial in terms of the obtainable maintenance cost reductions.

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or WF level. This is understandable since it can be regarded as contradictory to schedule main-tenance activities before or after their ideal moment in time. However, it is expected that, due to the presence of the strong economical dependencies, that the negative effects of deviating from the optimal time interval are outweighed by the cost reductions that can be obtained by conducting maintenance opportunistically. The main expected benefit of combining CBM and opportunistic maintenance is instead of selecting components that have the highest age and failure risk, that components which have the highest actual priority to undergo maintenance from an economic per-spective are selected. Therefore, it is expected that the number of corrective maintenance actions can be reduced drastically since CBM will provide the ideal replacement interval and eliminates the disadvantages time-based strategies as mentioned in the previous section, whereas scheduling activities opportunistically will enable the exploitation of economic dependencies.

2.3 Condition Based Opportunistic Maintenance

The works that consider CBM as well as opportunistic maintenance are very scarce. The first to consider maintenance decisions based on the actual condition of multiple components were Tian et al. (2011). In their study the proposed maintenance policy is defined by two failure thresholds, one for preventive maintenance and a lower threshold that for opportunistic preventive maintenance. The superiority of the suggested policy in comparison to a time-based policy in terms of O&M cost reduction (44%) is clearly demonstrated. However, is this study is only aimed at opportunistically conducting maintenance tasks within the WT that has failed or needs to undergo preventive mainte-nance. This causes that opportunistic maintenance activities will only be considered in addition to corrective- or scheduled preventive actions. The opportunistic maintenance here can be seen as an addition to the dominant CBM policy. Here only opportunistic maintenance actions are considered within a single WT, no attention is devoted towards the possible economic dependencies between WTs. This limits the number of possible opportunistic maintenance actions that can be executed drastically as only possible opportunistic maintenance actions are considered in turbine a and not in turbines [b, z]. In a more recent work of Perez, Ntaimo & Ding (2015) an entire WF of 100 WTs is considered on component level. However, the developed CBOM policy again only considers opportunistic maintenance of multiple components within the same WT, neglecting possible depen-dencies between WTs. The results show that in terms of cost reduction the CBOM policy is most beneficial due to a reduction in the number of corrective maintenance actions. The dependencies between WTs are considered in Yildirim et al. (2017) who proposed an optimization framework for the O&M scheduling for an entire WF and moreover, even for multiple WFs. Although this work is the first to pay attention to dependencies between WTs, the dependencies between components within wind turbines are neglected.

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for an entire WF whilst considering failures and maintenance actions at component level within the WT but also between different WTs.

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3 Problem definition

The following section will provide a brief description of the problem that will be represented by the MIP-model. The model will be formally defined in the hereafter following section.

Consider one or multiple offshore WFs on different geographical locations, but within reason-able distance of each other so that a dedicated maintenance crew is reason-able to conduct preventive and corrective maintenance activities on all locations. Within these WFs n identical turbines are installed with each m components that need to be maintained preventively on a regular basis or correctively after the occurrence of a failure. The condition of all components is measured each time period. Based on this component level information the failure probability for the entire turbine is computed together with the associated costs of performing preventive maintenance. Additionally, the failure probability of all possible maintenance scenarios will be computed. These maintenance scenarios can be regarded as a what if analysis for all possible combinations of preventive and cor-rective maintenance actions. For example; a turbine consists of two components. This results in four possible maintenance scenarios: none of the components are maintained, only component one is maintained, only component two is maintained and finally both components are maintained. On basis of these probabilities per scenario and their associated costs a decision will be made which scenario to execute, and thus which components to maintain. For each time period all the possible scenarios are evaluated based on their corresponding condition- and cost data. Within this evalu-ation it is assessed whether it is beneficial to schedule maintenance activities opportunistically to exploit economic dependencies or to conduct maintenance right away.

When the decision is made regarding which components to maintain a maintenance crew is dispatched. The maintenance crew is dedicated to the WFs so there is no lead-time for assembly of the crew. Furthermore it is assumed that the needed equipment and spare-parts are readily available. The maintenance crew has a capacity limit on the number of tasks it can conduct per maintenance epoch.

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4 Methodology

To be able to assess the effectiveness of the proposed CBOM policy for offshore turbine mainte-nance our study will formulate a Mixed Integer Programming (MIP) model. The model will jointly optimize the execution of O&M activities subject to operational- and maintenance aspects such as; weather conditions, power generation,electricity, real-time sensor-driven failure probability and corresponding maintenance costs for turbine components. The model will be able to consider the inter-dependencies that are present such as; the cost reductions that can be obtained from schedul-ing maintenance activities opportunistically, the limited capacity of the maintenance crew, travel time between different WF locations, influences of weather conditions on the feasibility of mainte-nance activities and on power output of turbines. These inter-dependencies and the characteristics of the operating environment will be represented mathematically by a objective function and a corresponding set of constraints. To analyze the performance of the model over a long time horizon a rolling horizon framework has been developed. This framework will update the parameters and deterioration data of operational and failed WTs according to the results of each instance.

This chapter is structured as follows. First details regarding the used methodology to com-pute the sensor-driven Remaining Lifetime Distributions (RLD’s) and their corresponding Dynamic Maintenance Costs (DMC’s) will be discussed. Hereafter a problem definition will be provided after which the optimization model will be introduced followed by the set of constraints. Hereafter, a description of the rolling horizon framework, that will be used to test the performance of the CBOM maintenance policy will be provided. This section will conclude with a description of the solution procedure and the choices made regarding accuracy of the results.

4.1 Sensor driven deterioration methodology

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4.2 Formal problem description

The set of maintenance epochs is denoted by T , the set of wind farms by L, turbines by G, the set of components by K and lastly the set of maintenance scenarios is denoted by M. The sets of components and turbines are than further partitioned into Ko, Go and Kf, Gf which respectively denote the components and turbines that are operational and the components and turbines that are failed during each maintenance epoch within the planning horizon. When all components within the turbine are operational the turbine the belongs to subset Go, when one or more components have failed the turbine belongs to subset Gf. Components that are operational can undergo maintenance actions. However during the execution of these maintenance actions it is assumed that the turbine is not able to produce electricity. When preventive maintenance is executed binary variable z_t`,i,k= 1 if component k of turbine i in WF ` is maintained during maintenance epoch t. In this case the corresponding DMC of component k in turbine i of WF ` in maintenance epoch t is incurred. Failed turbines can only undergo corrective maintenance. Corrective maintenance actions will be represented by binary variable v. Here v_t`,i,k = 1 if component k of turbine i in WF ` is maintained correctively during maintenance epoch t. Additionally, binary variable u`,i,k_t = 1 if component k in turbine i in WF ` is maintained, either preventively or correctively before the end of the planning horizon. This variable will be used to ensure that components are only maintained once during the planning horizon. When a component has been maintained, either preventively or correctively it is assumed that it will not fail during the remainder of the planning horizon. Therefore, the failure probability after maintenance will be 0 for the remainder of the planning horizon. Binary variable x`

t = 1 will be used to indicate whether a maintenance crew visits WF ` during maintenance epoch t. Every time a maintenance crew is dispatched a corresponding crew deployment cost Ctv,` is incurred. Variable ρ`,i_t will be used to indicate the failure probability of turbine i in wind farm ` during maintenance epoch t. This variable will change as a function of the maintenance decisions for its constituent components. To obtain the expected turbine failure costs failure penalty C_if is multiplied by the failure probability of the turbine. Lastly variable y_t`,i represents the generated power of turbine i in WF ` during maintenance epoch t in mW h where πt indicates the price of electricity per mW h during that maintenance epoch.

max z,υ,x,y,u,ρ X `∈L X i∈G X t∈T y_t`,i· πt | {z } operational revenue −X `∈L X t∈T x`_t· C_tv,` | {z }

crew deployment cost

−X `∈L X i∈Go X k∈Ko X t∈T z`,i,k_t · C`,i,k_to k,t | {z }

component maintenance cost

−X `∈L X i∈Go X t∈T ρ`,i_t · C_`f, i | {z }

turbine maintenance cost

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Evaluation of the first two terms within the objective function is quite straightforward. Op-erational turbines generate power, which is in turn multiplied by the energy price πt during that maintenance epoch. The second term evaluates the crew deployment cost, where every time a maintenance crew is dispatched to a WF a corresponding deployment cost Ctv,` is incurred. The third term evaluates the costs for preventive maintenance actions during each maintenance epoch and is coupled to the associated DMC data. Here C_t`,i,ko

k,t represents the costs of maintenance for

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the failure probability of turbine i in WF ` during time period t. In the objective functionP t∈T ρ

`,i t gives the probability that turbine i in WF ` fails before the end of the planning horizon. This is in turn multiplied with a failure penalty C_if which represents the associated costs of failure.

4.3 Constraints

1.) Linking turbine failure to component failure probabilities

Binary variable u`,i,k_t = 1 if component k within turbine i in WF ` has been put under maintenance before the end of the planning horizon. This ensures that the failure probability is equal to zero for the remainder of the planning horizon and that components are only maintained once during the planning horizon. This variable is linked to the variable z_t`,i,k or v`,i,k_t , depending on the type of maintenance action conducted. Together constraint 2 & 3 ensure that when either z_t`,i,k= 1 or v`,i,k_t = 1 , u`,i,k_t = 1 for the rest of the remaining maintenance epochs in the planning horizon.

t−1 X

s=1

z`,i,k_s = u`,i,k_t , ∀` ∈ L∀i ∈ G, ∀k ∈ Ki

o. (2)

t−1 X

s=1

ν_s`,i,k= u`,i,k_t , ∀` ∈ L∀i ∈ G, ∀k ∈ Kif (3)

Hereafter, the component maintenance decisions are linked to all possible maintenance scenarios by making use of a so called ‘no-good-cut’ (constraints 4 & 5). The maintenance scenarios can be regarded as a set of all possible scenarios, denoted by M. When a turbine consists of n components to be maintained 2n _{maintenance scenarios are possible. For example, when a turbine consists of} n = 2 components 22 _{= 4 maintenance scenarios are possible; 1.) No maintenance is conducted} on any of the components, 2.) Maintenance is conducted on component 1, 3.) Maintenance is conducted on component 2 and 4.) Both components are maintained. For maintenance status h at maintenance epoch t binary variable ηt

his defined. For all turbines, during all maintenance epochs all scenarios in which at most n–1 components are maintained this variable is subject to:

ηt_h≥   X k∈K( ˆut h) u`,i,k_t − X k∈F( ˆut h) u`,i,k_t − _{K( ˆ}u t h) + 1  , (4)

For all turbines, during all maintenance epochs all the scenarios in which all n components are maintained binary variable ηt

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Here the index sets k ∈ K(ˆut_h_{) and k ∈ F(ˆ}ut_h) respectively represent the components that have been maintained, and those which are left unattended. This constraint ensures that the binary variable ηt

h ≥ 1 when ut= ˆu t

h, and ηht is not bounded otherwise. In turn the failure probability φt

h of the turbine can be computed for each scenario by making use of the following reliability of series systems equation (Moskowitz & McLean, 1956):

φth= 1 −(1 − φti,1)(1 − φti,2)

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ρ`,it ≥ ηthφth, ∀` ∈ L, ∀i ∈ G, ∀t ∈ T , ∀h ∈ M. (7) 2.) Maintenance coordination

Constraint 8 ensures that preventive maintenance is executed on operational components before a deterioration threshold η is reached. This threshold can be defined as the last maintenance epoch (t) before the deterioration threshold η will be exceeded, the maintenance epoch corresponding to this moment is defined by ζ`,i,k:

ζ`,i,k

X

t=1

z`,i,k_t = 1, ∀ ` ∈ L, ∀ i ∈ G, ∀k ∈ Ki_o (8)

Corrective maintenance actions are limited to at most one per component during the planning horizon by the following constraint (9):

X

t∈T

υ_t`,i,k≤ 1, ∀ ` ∈ L, ∀ i ∈ G, ∀k ∈ Kif (9)

Maintenance crew visits to WFs in case of preventive- and/or corrective maintenance actions are ensured by the following constraints(10 & 11):

z_t`,i,k≤ x`t, ∀ ` ∈ L, ∀ i ∈ G

`_{, k ∈ K}i

o, t ∈ T (10)

υt`,i,k≤ x`t, ∀ ` ∈ L, ∀ i ∈ G`, k ∈ Kif, t ∈ T (11) 3.) Maintenance crew coordination The following constraints govern the actions of the mainte-nance crew in respect to its characteristics and the weather conditions.

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X

`∈L

x`_t≤ 1, ∀ t ∈ T (12)

For the cases in which weather conditions are in-feasible to conduct maintenance activities the following constraint (13) will be enforced:

x`_t= 0, ∀ ` ∈ L, t ∈ T_w` (13)

Here Tw` is a subset of T which consists of the average wind speeds per maintenance epoch. Weather conditions are considered as unfeasible when wind-speed threshold w is exceeded:

x`_t= (

1, if vt_{< w}

0, if vt_{≥ w}, ∀t ∈ T (14)

The following constraint (15) ensures that the travel distance between the different wind-farm locations ` and `0 is taken into consideration. This constraint ensures that crew deployment cannot be initialized before the required travel θl,l0 time has passed:

xl_t0+ xl_τ ≤ 1, ∀t ∈ {θl,l0+ 1, . . . , T }, τ ∈ {t − θ_l,l0, . . . , t} (15)

Lastly the maximum number of maintenance actions, either corrective or preventive that can be conducted during a single maintenance epoch is limited to the capacity of the maintenance crew by (16): X i∈G` o zt`,i,k+ X i∈G` f υ`,i,kt ≤ M ` t, ∀ ` ∈ L, t ∈ T (16) 4.) Operating decisions

By making use of the following two constraints (17 & 18) the decision variables z_t`,i,kand v_t`,i,kare coupled with operational variable y_ti, which is concerned with whether a turbine generates electricity or is shut down due to failure, maintenance or to low wind-speed to produce electricity. The first constraint ensures that turbine i in WF ` produces electricity during maintenance epoch t. Here pt_`,i represents the amount of electricity produced in mW h per maintenance epoch.

y_`,it ≤ pt`,i·

1 − z_t`,i,k, ∀ ` ∈ L, ∀ i ∈ G, ∀k ∈ Ko, t ∈ T (17)

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y`,it ≤ p t `,i· t−1 X j=1 υ`,i,kt , ∀ ` ∈ L, ∀ i ∈ G, ∀k ∈ Kf, t ∈ T , s ∈ S (18)

As can be seen, when turbines are maintained either correctively or preventively no electricity will be generated.

The amount of electricity that is generated when a turbine is operational is dependent on the wind speed at that time. The generated power in mW h per maintenance epoch denoted by ps t will be modelled based on the average wind speed during the maintenance epoch multiplied by the length in hours of a maintenance epoch, which is denoted by s. Turbines only produce electricity between a minimum cut-in wind speed vi_{, and a maximum cut-out wind speed v}o_{. The power} output of a turbine increases non-linearly until the wind speed at which the Rated Power (RP) and corresponding wind speed vr _{of the turbine is reached. This rated power can be regarded as the} turbine’s maximum capacity in mW h. When the wind speed exceeds the cut-out speed the turbine is shut down to avoid damage caused by overload of the components. This mathematical relation between wind speed and power output has been defined by Karki Patel (2009) as follows (19, 20, 21 & 22): pt_s=          0, if 0 ≤ vt_{≤ v}r RP (a + b·vt+c·v2t) · s, if vi≤vt≤ vr RP · s, if vi _{≤ v}t_{≤ v}o 0, if vo≤vt (19)

Where parameters a, b and c are obtained as follows:

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4.4 Rolling horizon framework

To be able to assess the performance of the policy over a long time horizon a rolling horizon frame-work has been developed. In this frameframe-work the model is solved initially one time for a planning horizon of n maintenance epochs. Hereafter, a specified number of m maintenance epochs at the beginning of the planning horizon will be frozen. For the components that have been maintained within this frozen interval new deterioration data in the form of RLD’s and the corresponding DMC will be selected and updated. Next to the parameters of maintained components all parameters of the other components in the system will be updated and initialized for the next run. In these subse-quent runs the planning horizon shifts each time by m time periods. This process will repeat itself until the the moment that the desired time horizon has been covered. Additionally, the solution of the previous run forms the starting point in solving the current run. This is beneficial in terms of the computational effort needed, and thus reduces the run-time of the model. Together, this will allow to model the behavior of the policy over a long time horizon within a reasonable time-span.

4.5 Solution method

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5 Numerical experiments and results

In this section the experiments that will place the performance of the CBOM policy into perspective will be presented. To do so the performance of the CBOM policy will be compared against the current widely applied strictly corrective- (CM), time-based preventive (PM) and condition-based (CBM) policies. For each of these benchmark policies the MIP model and its constraints will be adapted. The performance of the different policies will be analyzed for a single WF which consists of 75 identical turbines with a rated capacity of 2 mW h. Within these turbines 2 components that need to undergo periodic maintenance will be considered over a time-horizon 160 maintenance epochs of 48 hours. This process will be repeated for ten times after which the average results of these runs will be used for comparison. To rule out any influences of different component ages at the initialization a warm-up period of 80 maintenance epochs will be used. First the effects of increasing the crew deployment costs C_tv,` on policy performance will be analyzed for all the previously mentioned policies. This is followed by a section that will again analyze the influence of increasing crew deployment costs on policy performance while considering imperfect preventive maintenance effectiveness. Hereafter, the influence of the maintenance crew capacity M`

t will be analyzed for the CBOM and CBM policies. This section will conclude with a section that studies the effects of increasing electricity price πton the performance of the CBOM policy. To create same circumstances for all of the policies accessibility to the WF will be unconstrained. This ensures that within all policies maintenance actions can be executed at their most ideal time, and that variability in the outcomes is reduced. By reducing the variability the dynamics of the different policies can be distinguished more easily. It is verified that relaxing the accessibility does not change the dynamics of the policies, it only amplifies the differences. In appendix 9 the results of all the policies are provided under limited accessibility. This section will start off with a provision of the used parameters and underpinning for these parameter settings. Hereafter the performance metrics that will be tracked are presented. Lastly, the details on the benchmark policies will be provided after which the numerical studies and their results will be provided and discussed.

5.1 Weather and deterioration parameters

The deterioration data as discussed in § 4.1 consists of 40 samples which represent the deterioration process of actual rotating machinery components. These samples have been transformed according to the method as described in Yildirim et al. 2017 into RLD’s and corresponding DMC’s. For all components the failure probability increases as time proceeds.

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Figure 1: Average wind speed per maintenance epoch

Since the wind data shows a clear seasonal pattern of low- and high wind speeds, as can be seen in figure 1 it is assured that both are captured within the mentioned time horizon that. Therefore, within the numerical experiments wind-data from maintenance epoch 80-240 will be used to capture both the low- and high wind season. This is important since wind-speeds are highly influential for the amount of electricity generated and the accessibility to the WF, which in turn both influence the O&M decision making process.

5.2 Other parameters

The costs to deploy the maintenance crew vary between approximately AC 7K-80K depending on the resources needed (Le & Andrews, 2016). Within the first study the effects of increasing the crew deployment costs C_tv,`over a range fromAC 0 – 112K will be studied in relation to their impact on the performance metrics for the CBOM- and the benchmark policies. For the remainder of the numerical experiments crew deployment cost C_tv,`=AC 48K. The capacity of the maintenance crew M`

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5.3 Performance metrics

To be able to compare the performance of the CBOM policy against the other policies the perfor-mance metrics presented in Table 4 will be tracked. All perforperfor-mance metrics will be provided as averages over the entire planning horizon of all the iterations. A maintenance epoch consists of a time-frame of 48 hours.

Metric Description

Revenues

Operational Revenue (AC ) The total revenue of the WF(s) based on generated power and the electricity price.

Net Profit (AC ) Revenue minus the expenditures for maintenance and crew deployment costs.

Expenditures

Maintenance Costs (AC ) The total costs incurred for turbine maintenance. Preventive maintenance costs (AC ) Total cost related to preventive maintenance actions. Failure costs (AC ) Total cost related to corrective maintenance actions. Crew deployment costs (AC ) The total costs for crew deployment.

Performance Metrics

Corrective maintenance actions (#) The total number of corrective actions conducted. Preventive maintenance actions (#) The total number of preventive actions conducted.

Crew deployment frequency (#) The total number of times the maintenance crew has been dispatched. Availability (% · 100) The average percentage of WTs that are operational.

Downtime The total time expressed in maintenance epochs during which WTs were unavailable.

Table 1: Performance metrics

5.4 Benchmark policies

This section will present the maintenance policies which will be used as a benchmark to analyze the performance of the suggested CBOM policy. To place the performance of the CBOM policy into perspective the results will be compared against the results of a Condition-Based- (CBM), a Preventive-time-based- (PM) policy and a strictly corrective- (CR) maintenance policy. The general goal here is to study how the CBOM policy exploits the economical dependencies as stated in § 1 in comparison to the currently often used benchmark policies (F. Ding & Tian, 2011). To model the aforementioned policies the initial model will be adapted. The characteristics of each policy and the needed adaptions to the model will be discussed hereafter.

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Preventive maintenance policy: within this policy preventive maintenance activities are sched-uled within a certain time interval during the components lifespan. Because of the time-based nature of this policy the sensor data in the form of the DMC will not be used which means that C_t`,i,ko

k,t = 0. The components that will be considered for preventive maintenance in this setting will

be denoted by subset Kpm which consists of all the components of which their age (a l,i,k

t ) will be within the PM time-interval [ak,l_{, a}k,u_{] during the planning horizon. Here a}k,l _{and a}k,u respec-tively represent the lower- and upper-bound of the time interval in which a preventive maintenance actions needs to be executed for component k in turbine i of WF `. These two parameters will be defined based on the time threshold (ti_{) and the the Mean Time To Failure of each component} (M T T Fi_{) which is known from the RLD’s. So, before time threshold t}i _{· M T T F}i _{is reached} during the planning horizon the component is allowed to undergo preventive maintenance within time interval [ak,l_{, a}k,u_{] = [t}i _{· M T T F}i _{− T I}i_{, t}i _{· M T T F}i_{] where T I}i _{denotes the length of the} interval. To enforce this constraint (8) is changed to:

ak,u X

ak,l

z_t`,i,k≤ 1, ∀ l ∈ L, ∀ i ∈ G, ∀k ∈ Ki

o (23)

Since constraint (23) does not force components to be maintained during their specified interval also a slight modification to the objective function has been made. This has been made deliberately because forcing the preventive maintenance actions to be executed within their interval will lead to in-feasibility. For example in cases where the maintenance crew capacity is not sufficient or when the WF is not accessible for a long period of time. The outcomes of the model in terms of performance metrics will however not be influenced by this modification to the objective function. This together with a number of minor modifications to other constraints ensures that all components within set Kpm will undergo preventive maintenance within their designated interval if there is capacity and weather conditions are feasible. If it is not possible to conduct preventive maintenance within the designated time interval the components will no longer be considered for preventive maintenance and eventually fail after which they can be maintained correctively.

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Figure 2: Determination of ti

Figure 2 displays the total average maintenance costs for a WF that consists of 75 WTs with each 2 identical components, which both have a M T T F of 250 maintenance epochs. To generate reliable results the average maintenance costs corresponding to the threshold level have been computed over a time-span of 160 maintenance epochs which has been repeated for eight times for each value in the range of; 0 < ti_{< 2. As can be seen in figure 2 at t}i_{= 0.8 (t}i _{· M T T F}i_{) the minimum average} maintenance costs per planning horizon are achieved.

Corrective maintenance policy: within this policy only maintenance actions are only conducted after a WT has failed. To model the behaviour of this policy constraint (8) is changed to:

X

t∈T

z_ti,k= 0, ∀ l ∈ L, ∀ i ∈ G, ∀k ∈ K_oi (24)

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5.5 Numerical experiments parameter settings

Table 2 provides the most important parameter settings that will be used as the default setting within the numerical experiments. It will be explicitly mentioned when parameters are changed within the numerical experiments in respect to the values provided in table 2. The performance of the CBOM policy and the in § 5.4 presented benchmark policies will be analyzed over a total time horizon of 160 maintenance epochs of 48 hours by making use of the presented rolling horizon framework.

Parameter Value

Wind Farm characteristics

Wind Farms 1 Turbines 75 Components 2 Energy price AC 20 Turbine capacity (Mw/h) 2 Cut-in (m/s) 4 Rate (m/s) 12 Cut-out (m/s) 25 Maintenance characteristics PM costs AC 4K Failure costs AC 16K

Maintenance labor capacity 8 tasks per maintenance epoch

Crew deployment costs AC 0-112K

Policy characteristics

PM interval (T Ii) 4

T Ii 0.8

M T T F 250

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5.6 Effects of crew deployment costs on policy performance

Figure 3 displays the net profit of all the four policies under crew deployment costs Ctv,` ranging fromAC 0-112K. It clearly demonstrates that the performance of the CBOM policy, in respect to net profit is less affected when crew deployment costs increase in comparison to the benchmark policies. The CBOM and CBM policies, which both utilize the real time condition data clearly outperform the time-based PM and failure based CM policies with respect to net profit. As crew deployment costs increase the CBOM policy also performs significantly better than the CBM policy, of which the net profit decreases rapidly as crew deployment costs increase. The underlying dynamics that cause these performance differences between the maintenance policies will be analyzed and discussed in detail in the following sections.

Figure 3: Influence of crew deployment costs on net profit

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conducted. For example when C_tv,`= 0 the utilization of available maintenance capacity corresponds to 19.9% whereas this is 92.2% when C_tv,` = 112K. Although this has a negative effect on failures and the corresponding turbine downtime since both increase, the negative effects of this in terms of production losses are outweighed by the cost reductions that can be obtained by reducing the crew deployment frequency. So in sum the CBOM policy shows to be capable of weighing the costs for a higher number of failures with a longer duration and their associated costs against the cost advantage of dispatching the maintenance crew less often.

Cv,`_t (AC ) 0K 16K 32K 48K 64K 80K 96K 112K Revenues Op. rev. 11.3M 11.3M 11.3M 11.3M 11.3M 11.3M 11.3M 11.3M Net profit 10.6M 10.3M 10M 9.64M 9.49M 9.21M 8.93M 8.72M Expenditures Expenditures 710K 1.03M 1.3M 1.65M 1.81M 2.07M 2.35M 2.54M Maint. exp. 708K 688K 705K 751K 717K 758K 736K 749K PM costs 394K 400K 377K 402K 381K 377K 390K 368K Failure costs 314K 288K 328K 349K 336K 381K 346K 381K

Crew dep. costs 0K 344K 598K 898K 1.09M 1.31M 1.61M 1.79M

Performance Metrics Crew dep. 46.7 21.5 18.7 18.7 17.0 16.4 16.8 16.0 C. act. 19.6 18.0 20.5 21.8 21.0 23.8 21.6 23.8 P. act. 98.5 100.1 94.2 100.5 95.3 94.2 97.5 92.0 Availability 0.95 0.95 0.95 0.95 0.95 0.95 0.95 0.95 Downtime 562.1 583.1 597.7 613.8 612.6 627.5 637.2 647.3

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When we take a closer look at the performance of the CBM policy in table 4 it can be seen that net profit decreases with approximately 2-8% each time crew deployment cost C_tv,` is increased. Maintenance costs, the corresponding preventive- and corrective maintenance frequency as well as the crew deployment frequency remain practically constant over all the cases. The main cause for this is that the decision to dispatch the maintenance crew is solely based on the state of components. Although the total maintenance costs do not increase or shift from preventive to corrective actions within the CBM policy, net profit is affected much more by the increasing crew deployment cost in comparison to the CBOM policy since the crew deployment frequency does not decrease as Ctv,` increases. For example when crew deployment costs are C_tv,`=AC 64K the total crew deployment costs are 73% higher in comparison to the CBOM policy (AC 1.21 vs. AC 2.1M). When we look at the downtime for the CBM policy it can be seen that this metric does not increase as C_tv,` increases. However, there is no significant operational benefit since WT availability is 95% for both policies and operational revenue is also practically the same. Although the performance the CBOM- and CBM policy is initially equal to each other the CBOM policy outperforms the CBM policy clearly as crew deployment costs increase. For example in the case where Ctv,` = AC 64K net profit is A

C 800K higher for the CBOM policy. This indicates that scheduling maintenance actions from solely an operational point of view results in poor economical performance when crew deployment costs increase. Cv,`_t (AC ) 0K 16K 32K 48K 64K 80K 96K 112K Revenues Op. rev. 11.2M 11.2M 11.2M 11.2M 11.2M 11.2M 11.2M 11.2M Net profit 10.4M 9.9M 9.53M 8.98M 8.29M 8.16M 7.54M 7.2M Expenditures Expenditures 809K 1.3M 1.68M 2.23M 2.93M 3.04M 3.66M 4.01M Maint. exp. 808K 861K 810K 844K 823K 861K 861K 809K Failure costs 352K 413K 349K 389K 370K 411K 414K 347K PM costs 456K 448K 461K 455K 453K 449K 451K 458K

Crew dep. costs 0K 440K 869K 1.38M 2.1M 2.18M 2.8M 3.2M

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Regarding the results of the PM policy, presented in table 5 it can be clearly seen that not making use of real-time condition data in order to anticipate to future failures highly influences the incurred failure costs. First, the number of corrective maintenance actions is over all the cases more than three times as high in comparison to the CBOM policy. This clearly highlights the importance of considering identical components as unique as stated by (Yildirim et al., 2017). Furthermore, increasing crew deployment cost Ctv,` has little influence on decreasing the crew deployment frequency of the PM policy. Only in the case where C_tv,` = 0 the crew deployment frequency is significantly higher. The constant figures for the number of conducted maintenance actions and the crew deployment frequency are a result of the practically fixed maintenance schedule, which ensures that components are maintained before before the imposed time threshold is reached. The relatively low number of preventive maintenance actions can be explained by the high value of the time threshold T Ii_{. This causes that large number of components have already failed before} they are considered for preventive maintenance actions. However, within the determination process of T Ii we have seen that setting this threshold value low results in a very high number of preventive maintenance actions and corresponding costs, and thus a high crew deployment frequency which in turn leads to even poorer overall performance. It is interesting to see that maintenance crew capacity utilization is quite high within the PM policy. This is accomplished by leaving turbines in a failed state until the next preventive maintenance action that is scheduled is executed. A negative consequence of this that downtime is high. For example when C_tv,`=AC 48K downtime is respectively 74% and 81% higher in comparison to the CBOM and CBM policy.

Cv,`t (AC ) 0K 16K 32K 48K 64K 80K 96K 112K Revenues Op. rev. 10.6M 10.6M 10.5M 10.5M 10.4M 10.5M 10.4M 10.4M Net profit 8.99M 8.48M 7.96M 7.54M 7.08M 6.87M 6.38M 6.2M Expenditures Expenditures 1.61M 2.09M 2.56M 2.93M 3.28M 3.61M 4.04M 4.2M Maint. exp. 1.61M 1.63M 1.65M 1.63M 1.61M 1.58M 1.61M 1.65M Failure costs 1.38M 1.41M 1.44M 1.42M 1.38M 1.35M 1.39M 1.44M PM costs 229K 222K 214K 216K 224K 233K 220K 213K

Crew dep. costs 0K 464K 908K 1.3M 1.67M 2.03M 2.42M 2.55M

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The results of the CM policy provided in table 6 clearly demonstrate the inferiority of only acting upon failure in comparison to the condition- and time-based benchmark policies. Here failure costs account for up to 25-45% of operational revenue over all the cases. In comparison within the CBOM policy, failure costs account for approximately 3-4.4% of operational revenue. When we look at the downtime for the CM policy it can be seen that when only corrective maintenance actions are conducted that downtime increases drastically in comparison to the other policies, for example downtime is respectively 99.8%, 114% and 18% higher in comparison to the CBOM-, CBM-and PM policy when Ctv,`=AC 48K. In turn this high downtime figure causes that less operational revenue can be generated due to turbine unavailability. The causes of this are quite straightforward. First, WTs fail more frequently, and second when WTs fail they are left considerably longer in a failed state in comparison to the CBOM-, CBM-, and PM policy because they can only be grouped with other corrective maintenance actions. When the net profit of the CM policy is compared to the net profit of the CBOM policy we can see that this is on average around 25% lower in comparison to the CBOM policy. On the other hand it can be seen that the CM policy is capable of exploiting the economic dependency created by increasing crew costs as the crew deployment frequency decreases as C_tv,`increases. However, the downside of this is significantly higher downtime and WT availability and its associated effect on the generated operational revenue.

Cv,`t (AC ) 0K 16K 32K 48K 64K 80K 96K 112K Revenues Op. rev. 9.89M 9.7M 9.63M 9.62M 9.54M 9.55M 9.5M 9.49M Net profit 7.7M 7.14M 6.83M 6.56M 6.35M 6.01M 5.76M 5.45M Expenditures Expenditures 2.19M 2.56M 2.81M 3.06M 2.09M 2.14M 2.11M 2.13M Maint. exp. 2.18M 2.17M 2.17M 2.16M 2.09M 2.14M 2.11M 2.13M Failure costs 2.18M 2.17M 2.17M 2.16M 2.09M 2.14M 2.11M 2.13M PM costs 0 0 0 0 0 0 0 0

Crew dep. costs 0K 390K 634K 902K 1.1M 1.41M 1.63M 1.9M

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5.7 Effects of crew deployment costs on policy performance with

imper-fect maintenance actions

In this section the performance of the suggested CBOM policy will be compared against the CBM policy as in the previous section only under the assumption that preventive maintenance actions are imperfect. The effectiveness of PM is here assumed to lie between 30-100%. The main reason to do so is that preventive maintenance actions often are not capable of returning the component in a state that is ’as-good-as-new’ (Spinato et al., 2009). Other than the effectiveness of preventive maintenance actions the length of the total planning horizon is slightly longer than in the previous experiments with a total length of 200 maintenance epochs instead of 160. Further all the other parameter settings will be identical to those presented in table 2. The range for crew deployment cost C_tv,` will be the same as in § 5.6.

Figure 4: Influence of crew dep. cost under imperfect maintenance effectiveness

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Cv,`_t (AC ) 0K 16K 32K 48K 64K 80K 96K 112K Revenues Op. rev. 13.4M 13.3M 13.3M 13.3M 13.3M 13.3M 13.3M 13.3M Net profit 11.8M 11.2M 10.5M 10.2M 9.65M 9.11M 8.62M 8.11M Expenditures Expenditures 1.54M 2.13M 2.72M 3.1M 3.63M 4.17M 4.65M 5.15M Maint. exp. 1.53M 1.53M 1.58M 1.48M 1.48M 1.51M 1.5M 1.54M Failure costs 724K 760K 824K 708K 710K 768K 754K 782K PM costs 810K 767K 758K 771K 772K 746K 750K 755K

Crew dep. costs 0K 606K 1.14M 1.62M 2.14M 2.66M 3.14M 3.61M

Table 7: influence of crew deployment costs on CBOM policy performance with imperfect mainte-nance action Cv,`t (AC ) 0K 16K 32K 48K 64K 80K 96K 112K Revenues Op. rev. 13.4M 13.4M 13.4M 13.4M 13.4M 13.3M 13.4M 13.4M Net profit 11.8M 10.8M 9.53M 8.52M 7.56M 6.35M 5.54M 4.47M Expenditures Expenditures 1.55M 2.55M 3.82M 4.83M 5.79M 7M 7.81M 8.88M Maint. exp. 1.55M 1.55M 1.54M 1.54M 1.56M 1.56M 1.53M 1.51M PM costs 791K 733K 787K 770K 777K 773K 749K 762K Failure costs 760K 813K 757K 768K 787K 784K 776K 744K

Crew dep. costs 0K 1M 2.28M 3.3M 4.22M 5.44M 6.29M 7.37M

Table 8: influence of crew deployment costs on CBM policy performance imperfect maintenance action

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preventive-and corrective maintenance frequency as well as the crew deployment frequency remain practically constant each time crew deployment costs are increased. Although the total maintenance costs do not increase or shift from preventive to corrective actions within the CBM policy net profit is again affected more by the increasing crew deployment cost in comparison to the CBOM policy. For example when crew deployment costs C_tv,`=AC 64K the total crew deployment costs are doubled in comparison to the CBOM policy. When we look at the downtime for the CBM policy it can be seen that this does not increase as crew deployment costs increase. However, the benefit that is brought by this results merely in a 1% increase in operational revenue in comparison to the CBOM policy. Although the performance the CBOM- and CBM policy is initially equal to each other the CBOM policy outperforms the CBM policy significantly as crew deployment costs increase.

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5.8 Effects of maintenance crew capacity on policy performance

The following section will analyze the influence of the maintenance crew capacity Mt` on the per-formance of the CBOM and CBM policies. As demonstrated in section § 5.6 & § 5.7 the CBOM policy has proven to be able to utilize maintenance crew capacity more effectively than the other benchmark policies. In this section we will analyze how this capacity utilization is affected when the maintenance crew capacity is decreased and increased in comparison to the default of M`

t = 8. To do so M`

t, which represents the number of maintenance tasks that can be conducted during a single maintenance epoch will be varied over a range from 2-16 tasks. The experimental procedure will be identical to the previous experiments. Parameter values will remain unchanged from table 2 except for the crew deployment cost C_tv,` which will be fixed toAC 48K.

Figure 5: Influence of maintenance crew capacity on crew dep. freq.

Figure 6: Influence of maintenance crew capacity on net profit. As can be seen in figure 6 and table 9 net profit increases as crew capacity M`

t is increased. When M`

t < 6 net profit and WT availability is negatively affected. The main reason for this is that capacity is insufficient. This causes that maintenance actions cannot be executed anywhere near their most ideal moment in time and thus need to be postponed by a large number of maintenance epochs. This results in more and longer failures which is supported by the high downtime figure of 1042 maintenance epochs. When M_t` ≥ 6 the benefits of increasing crew capacity are limited in respect to net profit. On the other hand as can be seen in figure 5, a strong reduction in the crew deployment frequency and its associated costs can be seen as Mt` increases. For example, the crew deployment frequency is practically halved when Mt` is increased from 4 to 12. Increasing Mt` clearly indicates that the CBOM policy is capable of considering the trade-off between expediting and postponing maintenance actions to make use of the available capacity, which in turn has a positive effect on the total crew deployment costs incurred.

When we look at table 10 and figure 5 it can be seen that the CBM policy is not capable of considering this trade-off as the crew deployment frequency decreases only slightly as M`

t is increased. In terms of net profit it can be seen in figure 6 that over all the cases the CBM policy performs less in comparison to the CBOM policy. When crew capacity is low (M`

t = 2) the performance difference between the two policies is minimal, however it increases as M`

t is increases. For example when crew capacity M`

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M`t 2 4 6 8 10 12 14 16 18 Revenues Op. rev. 11M 11.3M 11.3M 11.3M 11.3M 11.3M 11.3M 11.3M 11.3M Net profit 7.45M 9.12M 9.61M 9.72M 9.71M 9.86M 9.87M 9.89M 9.96M Expenditures Expenditures 3.59M 2.21M 1.69M 1.58M 1.59M 1.44M 1.42M 1.41M 1.32M Maint. exp. 873K 701K 678K 710K 709K 688K 727K 735K 690K Failure costs 567K 309K 299K 309K 318K 293K 334K 313K 299K PM costs 306K 392K 378K 401K 391K 395K 393K 422K 391K

Crew dep. costs 2.72M 1.51M 1.01M 871K 878K 747K 693K 672K 631K

Performance Metrics Crew dep. 56.7 31.4 21.1 18.1 18.3 15.6 14.4 14.0 13.1 C. act. 35.43 19.29 18.71 19.29 19.86 18.29 20.86 19.57 18.71 P. act. 76.43 98.0 94.57 100.29 97.71 98.86 98.29 105.43 97.71 Availability 0.91 0.95 0.95 0.95 0.95 0.95 0.95 0.95 0.95 Downtime 1042.43 589.0 591.43 606.57 605.57 610.43 618.29 618.14 607.14

Maint. Cap. Ut. 0.99 0.93 0.90 0.83 0.64 0.63 0.59 0.56 0.49

Table 9: influence maintenance crew capacity on CBOM policy performance

M` t 2 4 6 8 10 12 14 16 18 Revenues Op. rev. 11.3M 11.3M 11.4M 11.4M 11.3M 11.4M 11.4M 11.3M 11.4M Net profit 7.28M 7.9M 8.11M 8.18M 8.2M 8.28M 8.15M 8.08M 8.34M Expenditures Expenditures 4.05M 3.43M 3.25M 3.17M 3.14M 3.07M 3.2M 3.26M 3.02M Maint. exp. 703K 746K 709K 731K 736K 731K 757K 765K 714K Failure costs 331K 357K 312K 323K 328K 320K 365K 376K 315K PM costs 373K 389K 397K 409K 408K 411K 392K 389K 399K

Crew dep. costs 3.34M 2.69M 2.54M 2.44M 2.41M 2.34M 2.44M 2.5M 2.3M Performance Metrics Crew dep. 69.67 56.0 52.83 50.83 50.17 48.83 50.83 52.0 48.0 C. act. 20.7 22.3 19.5 20.2 20.5 20.0 22.8 23.5 19.7 P. act. 93.2 97.2 99.3 102.2 102.0 102.7 98.0 97.2 99.8 Availability 0.95 0.95 0.95 0.95 0.95 0.95 0.95 0.95 0.95 Downtime 571.17 571.33 563.83 563.33 565.17 559.67 559.83 565.33 556.67

Maint. Cap. Ut. 0.81 0.53 0.37 0.30 0.24 0.21 0.17 0.15 0.14

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5.9 sensitivity analysis

In this section the influences of increasing the electricity price πton the performance of the CBOM policy will be analyzed. For the future this electricity price is assumed to increase due to increased demand, export and significant penalization of CO2 emissions of non-renewable energy sources (Wind Europe, 2018). To do so the electricity price will varied over a range of AC 10 − 100 per mW h. Crew deployment costs will be fixed to C_tv,` = AC 48K. All other parameters will remain unchanged from table 2.

Table 11 provides the results of the sensitivity analysis regarding the performance of the CBOM policy when electricity prices are increased in comparison to the default setting of πt = AC 20. First it can be seen that both operational revenue and net profit increase practically linearly with each increment of electricity price πt. Regarding the maintenance actions it can be seen that the total number of preventive maintenance actions conducted decreases as the electricity price increases, whereas the number of corrective maintenance actions conducted increases. This clearly indicates that scheduling preventive maintenance actions opportunistically to exploit the econimic dependency caused by the crew deployment costs has a lower priority as πt increases. This is moreover supported by the increasing crew deployment frequency. For example when the price for electricity is AC 10 the corresponding crew deployment frequency is 17.3 compared to 41 when the electricity price isAC 100. This means that in the case where the electricity price wasAC 10 on average, 88.4% of available crew capacity is utilized per crew deployment. In comparison when the electricity price isAC 100 crew capacity utilization has dropped to 36.6%. Instead maintenance is scheduled in such a way that production losses are minimized as the costs of these production losses begins to outweigh the cost advantage of deploying the maintenance crew less often. So as the electricity price πtincreases the focus is more pointed towards minimizing production losses than minimizing maintenance expenditures since the the costs for production losses start to outweigh the cost for additional crew deployments.

πt (AC ) 10,- 20,- 30,- 40,- 50,- 60,- 70,- 80,- 90,- 100,-Revenues Op. rev. 5.64M 11.2M 16.9M 22.6M 28.2M 33.9M 39.5M 45.2M 50.9M 56.5M Net profit 4.17M 9.61M 15.2M 20.8M 26.5M 32.1M 37.6M 43.1M 48.9M 54.3M Expenditures Expenditures 1.47M 1.58M 1.74M 1.79M 1.76M 1.8M 1.89M 2.08M 2.04M 2.19M Maint. exp. 643K 693K 793K 815K 800K 799K 841K 889K 851K 899 Failure costs 235K 306K 416K 464K 443K 425K 503K 528K 480K 546K PM costs 408K 386K 377K 351K 357K 374K 338K 361K 371K 353K

Crew dep. costs 830K 891K 946K 974K 960K 1M 1.05M 1.19M 1.19M 1.3M

Performance Metrics Crew dep. 17.3 18.6 19.7 20.3 20.0 20.9 21.9 24.7 24.7 27.0 C. act. 14.71 19.14 26.0 29.0 27.71 26.57 31.43 33.0 30.0 34.14 P. act. 102.0 96.57 94.29 87.86 89.14 93.57 84.57 90.29 92.71 88.14 Availability 0.95 0.95 0.95 0.95 0.95 0.95 0.95 0.95 0.95 0.95 Downtime 615.0 604.14 612.57 616.57 612.43 609.57 617.0 612.71 612.86 609.71

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6 Discussion

The results of the proposed CBOM policy have clearly demonstrated an economical advantage in comparison to the Condition-based, Time-based and Failure-based benchmark policies. In fact, ac-cording to the obtained results the net profit is respectively 2%, 15% and 27% higher in comparison to the CBM, PM and CM policies when Ctv,` =AC 0. Moreover, when crew deployment cost C

v,` t increases the CBOM policy demonstrated its capability of grouping maintenance activities more aggressively than the benchmark policies to minimize expenditures and to maximize the utilization of the available maintenance capacity. This is accomplished by deviating from the ideal moment, from an operational perspective to conduct maintenance activities. Although there are some nega-tive effects attached to this such as; a higher number of failures and increased downtime, the costs associated to these negative effects are outweighed by the cost benefits that are obtained by reduc-ing the needed number of crew deployments. In terms of net profit, the performance of the CBOM policy is in comparison to the CBM, PM and CM policies respectively 7%, 27% and 47% higher when C_tv,` =AC 48K and respectively 17.4%, 28.9% and 37.5% higher when C_tv,` =AC 112K. The crew deployment costs for the CBOM policy were respectively 53%, 45% and 0.5% (Ctv,`=AC 48K) lower in comparison to the CBM, PM and CM policies. The capability of expediting and postponing maintenance actions to be able to reduce the crew deployment frequency makes that the perfor-mance of the CBOM policy in terms of net profit is less affected by increasing crew deployment costs in comparison to the benchmark policies.

When crew capacity is increased the CBOM policy has proven to be capable of utilizing the available capacity more effectively than the CBM policy against which it was compared. However, when crew capacity is above 6 tasks per maintenance epoch the positive effect on net profit has proven to be minimal. On the other hand, when capacity is below 4 tasks per maintenance epoch the performance of the WF is affected negatively. However, the same holds for the CBM policy against which it was compared.

By means of a sensitivity analysis the effect of the electricity price, which is assumed to increase in the future (Wind Europe, 2018) has been studied. Here the CBOM policy demonstrated that when the price for electricity increases that the focus of the policy shifts from minimizing main-tenance expenditures, by reducing crew deployment costs towards minimizing production losses. This causes that the crew deployment frequency and its associated costs increase. However, these additional costs are outweighed by the higher energy production that is possible when failed tur-bines are maintained earlier. It is however noteworthy that the number of corrective maintenance actions also increases as the price for electricity increases since failed WTs are down for at least two maintenance epochs whereas this is only one for preventive maintenance actions. A possible explanation for this could be that the model decides to let turbines fail during periods of low wind speeds in order to still be capable of exploitting the econimic dependency caused by the crew deployment costs by grouping these corrective maintenance actions with preventive maintenance actions at their ideal moment in time.

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figures (see § 9). However these results were subject to much more variability. This makes that the dynamics that are present within the policies are harder to distinguish. We have found that the variability within the outcomes can be reduced by increasing the number of repetitions and by running the model over a longer time horizon. However, due to time and resource constraints this was not a feasible option for our study. The relaxation of WF accessibility does however not mean that the dynamics of the model are changed, they are only subject to less variability.

7 Conclusion

In this study, the applicability and performance of a Condition Based Opportunistic Maintenance (CBOM) policy has been investigated for offshore Wind Farms. The primary goal of the proposed maintenance policy is to take advantage of the econimic dependencies that are present within the operation and maintenance of these Wind Farms. Maintenance actions have been considered on component and turbine level to capture as much economical dependencies as possible. To do so a Mixed Integer Programming model has been formulated which considers various operating and maintenance aspects such as wind power, electricity price and limited maintenance capacity. The objective of the model is to jointly optimize O&M expenditures and to maximize power output to achieve the highest possible net profit. In order to model the deterioration process as realistically as possible real-world deterioration data has been used. Additionally, a rolling horizon framework has been developed to enable performance analysis over a long time-horizon.

The performance of the proposed CBOM policy has been analyzed in multiple comparative studies against the currently wide used Condition Based Maintenance-, Time-Based- and Correc-tive maintenance policies. These studies clearly demonstrated the value of scheduling maintenance actions from an economic perspective in comparison to a purely operational perspective by com-paring it to a regular Condition Based Maintenance policy. Moreover, the inadequacy of strictly time- and failure based maintenance policies for an offshore wind farm setting has been clearly demonstrated by comparing the proposed CBOM policy to a PM and CM policy. In practically all of the cases the CBOM policy outperformed the other benchmark policies clearly. Especially under increasing crew deployment costs the CBOM policy demonstrated to be capable of generating up to 47% more net profit.

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8 limitations and further research

Although this study has proven that there are some clear benefits to the suggested CBOM-policy, the results are subject to several assumptions and simplifications. These assumptions and simplifi-cations are needed to be able to model the problem mathematically and to keep the complexity of the model within reasonable bounds without resulting in excessive computation times. First, the scope of our study focused only on a limited number of components whereas in reality WTs consist of significantly more components that need to be maintained periodically. Furthermore, the future profile of inputs in the form of deterioration data and weather data were assumed to be determin-istic. In reality there is uncertainty in both the actual deterioration process of components as well as in weather predictions. Moreover, electricity prices and crew deployment costs were fixed for the entire planning horizon, in reality these values are subject to variation due to changing supply and demand patterns. Furthermore, due to time constraints and limited resources the number of conducted experiments and their scope needed to be kept quite small.

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Multi-level condition based opportunistic maintenance optimization for oﬀshore wind farms