Operations and Maintenance Scheduling Optimization in Offshore Wind Farms: Incorporating Failure Uncertainties in a Condition-Based Opportunistic Maintenance Model

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Operations and Maintenance Scheduling Optimization

in Offshore Wind Farms: Incorporating Failure

Uncertainties in a Condition-Based Opportunistic

Maintenance Model

Master Thesis Technology and Operations Management

Author:

Thijs van Kouwen – s2748355

Supervisors:

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Abstract

As offshore wind farms are becoming more important in the world’s energy production, the need for efficient Operations & Maintenance (O&M) schedules increases. This paper presents a model that optimizes the O&M planning for offshore wind farms. Optimizing scheduling of corrective and preventive maintenance activities using sensor driven degradation data on a wind turbine level will result in an O&M schedule that maximizes total wind farm profits. This paper adds to existing literature the incorporation of failure uncertainties (i.e. the effect of a wind turbine that was operational at the time of planning failing unexpectedly during the planning horizon). The experiments in this study show that because of an increased number of preventive maintenance actions, the availability of the wind farm increases when failure uncertainties are considered. However, more accurate expected energy generation calculations and the need for more crew visits to ensure higher availability result in only small changes in wind farm profits compared to a model that does not account for failure uncertainties.

Keywords: Offshore Wind Farm, Maintenance Optimization, Condition-Based Opportunistic

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1. Introduction

Sustainable ways of energy usage are becoming considerably more important. Since conventional ways of energy production are continued to be decommissioned, many alternative sources of energy are installed. Wind energy is one of these alternatives and an important solution for reaching the European climate goals. The aim of the European Commission to have a generation capacity between 230 and 450 GW of offshore wind by 2050 calls for many new offshore wind farms, since Europe currently has 20 GW installed (WindEurope, 2019). The rising importance of offshore wind over onshore wind is due to factors as expensive land, visual pollution near residential areas, and higher energy production at sea (Bilgili, Yasar and Simsek, 2011). One of the main disadvantages of offshore wind is the high cost of Operations and Maintenance (O&M) activities. O&M costs count for one third of the total lifetime costs of offshore wind energy (Sinha & Steel, 2015; Esteban et al., 2011). Examples of O&M costs are replacement costs, crew deployment costs, and turbine downtime costs. Due to long distances from operation centres, larger scale wind farms, and downtime losses, costs are five to ten times higher than for onshore wind farms (Scheu et al., 2012; Byon, Ntaimo and Ding, 2010; Eggen et al., 2008). A well-defined maintenance model, that combines O&M activities during planning horizons, will help decrease offshore wind farm O&M costs.

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5 2018; Nilsson and Bertling, 2007; Yildirim et al., 2017). When the degradation is below a certain level, preventive maintenance will be planned for these turbines.

Because of existing economic dependencies between wind turbines, combining maintenance activities based on individual turbine’s conditions can be profitable. Conducting corrective and preventive maintenance actions together in the same trip, called opportunistic maintenance, will help decrease system down times and consequently O&M costs (Ding and Tian, 2012). It may even be more profitable to postpone corrective maintenance, so that it can be grouped together with planned preventive maintenance actions. The resulting savings in crew deployment costs may then outweigh the loss in revenue resulting from higher downtime costs. Opportunistic maintenance works best in a condition-based maintenance strategy, given its effective scheduling of preventive maintenance tasks.

In the past decade, several academics have explored modelling offshore wind farm maintenance. The advantage of modelling and optimizing maintenance schedules is the ability to assess different maintenance strategies on their performance relatively cheap. The optimization model in this paper can be seen as an extension of the work of Yildirim et al. (2017). The model presented in the paper of Yildirim et al. (2017) includes important factors for complete wind farm O&M planning in combination with sensor data. This sensor data is used to continuously measure the turbine’s degradation. Although sensor data is a useful tool in determining the condition of turbines, turbines could fail at a different time than expected (Mo and Chan, 2017). Moreover, De Jonge, Teunter and Tinga (2017) mention that uncertainty of failure thresholds and deterioration levels are of influence on condition-based maintenance strategies. In addition, Vanli (2014) says that by including uncertainties to the maintenance-scheduling model, the number of failures will decrease, and the model is thus more cost-effective. Therefore, this paper will add to the existing literature the failure uncertainty in an offshore wind farm maintenance model. Consequently, the research question that will be answered is as follows: Is it beneficial, with respect to wind farm profit and availability, to consider failure uncertainties when making the O&M schedule in a condition-based opportunistic maintenance model?

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2. Theoretical background

A wind turbine is a system that consists of multiple components. The main components are the blades, the hub, the nacelle, and the tower. Within the nacelle smaller components are present such as a gearbox, a generator, and a converter (Le and Andrews, 2015). Technical equipment is prone to degradation. In the case of wind turbines, its rotation has a negative impact on the condition of the wind turbines. Therefore, maintenance is an important step to ensure power generation.

Maintenance in offshore wind farms is an expensive operation for several reasons. Firstly, expensive materials are needed to visit a turbine. Most of the time a service operation vessel (SOV) or a helicopter is used, which makes it more expensive than maintaining an onshore wind turbine (Besnard, Fischer and Tjernberg, 2013). Furthermore, these travel methods are highly weather dependent (i.e. repairs cannot be done in extreme weather conditions). Lastly, travel times of a SOV to the wind farm are higher. Therefore, the idle times of failed turbines are higher than turbines that are situated at onshore locations. As a result, the availability of offshore wind farms can be significantly lower than the industry availability standard of 97% (Conroy, Deane and Ó Gallachóir, 2011). The aim of a wind farm operator is to minimize idle times and O&M costs in order to run a wind farm profitably. To achieve this goal, maintenance strategies need to be implemented. These strategies can be split into corrective and preventive maintenance.

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8 Preventive maintenance on the other hand, is a maintenance strategy that carries out the maintenance before a possible failure. According to Bouwer Utne (2010), preventive maintenance can be further split into time-based and condition-based maintenance. Preventive strategies are implemented in offshore wind farms, since high costs of replacement and long idle times are prevented from occurring (Nielsen & Sørensen, 2011; Shirmohammadi, Zhang & Love, 2007)

In time-based maintenance, the choice can be to conduct maintenance based at fixed time intervals, called block-based maintenance, or to conduct maintenance based on the age of the turbine, called age-based maintenance. De Jonge and Jakobsons (2018) state that age-based maintenance results in less unnecessary maintenance actions, because after corrective maintenance the age is set to zero. In block-based maintenance on the other hand, preventive maintenance could still be performed after a failure because of the fixed times at which preventive maintenance is scheduled. This would result in unnecessary maintenance actions and higher costs. The advantage of time-based maintenance strategies is that failures are prevented from occurring and it is relatively cheap to implement, because there is no need for condition monitoring systems (Ahmad and Kamaruddin, 2012).

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9 Opportunistic maintenance takes advantage of the economic dependencies between turbines. As explained before, a wind farm consists of multiple turbines. According to Tian et al. (2011), it is beneficial to take the opportunity to also repair turbines with a high risk of failure when the maintenance crew is sent to conduct maintenance on another turbine. Next to this, it might be better to wait to repair a failed turbine until the maintenance crew visits the wind farm for planned preventive maintenance (Yildirim et al., 2017). This strategy is beneficial for implementation in the O&M strategy of offshore wind farms, because the high costs of sending the maintenance crew are now divided over multiple maintenance actions. Given the effective preventive maintenance scheduling ability under condition-based maintenance and the potential cost savings of opportunistic maintenance, a combination of these two strategies, called condition-based opportunistic maintenance, is expected to give the highest profit for a wind farm.

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3. Methodology

This section presents the MIP-model for the optimization of the O&M schedule of offshore wind farms. First, the integration of degradation data will be described. Secondly, the sets and parameters used in the model will be given. Thirdly, the sensor-driven condition-based opportunistic maintenance optimization model will be introduced and elaborated on. Thereafter, the use of wind-speed data will be explained and lastly the experimental framework is presented.

3.1 Sensor-driven degradation data

Similar to the approach in the paper of Yildirim et al. (2017) we will use deterioration data of rotating machinery to emulate the degradation of wind turbines. The degradation of a turbine is determined by two types of degradation parameters, one that is equal for all turbines in a wind farm and one turbine specific parameter that accounts for unit-to-unit variability. This last element, the unit to-unit variability, results in different deterioration patterns for all turbines. In order to account for this variability in our model, we use 40 different data files with each a different so-called remaining life distribution (RLD). These RLDs represent the deterioration of all turbines in the wind farm. When the age of a turbine is higher than its lifetime (derived from the RLD), the turbine will fail. Based on this relation and taking into account the costs of maintenance and renewal reward, the dynamic maintenance cost can be calculated for each time period. The dynamic maintenance cost represents the trade-off between the cost of preventive maintenance and the risk of unexpected failures.

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11 3.2 Model description

The nomenclature that will be used in describing the model is as follows:

Sets:

𝐿: Complete set of wind farms with index 𝑙 𝐺: Complete set of wind turbines with index 𝑖 𝐺': Set of operational wind turbines

𝐺'(: Set of operational wind turbines at location 𝑙

𝐺): Set of failed wind turbines

𝐺)(: Set of failed wind turbines at location 𝑙

𝑇: Complete set of maintenance epochs with index 𝑡 Parameters:

𝑦-.: Expected energy output for turbine 𝑖 at period 𝑡

𝑝-.: Generation capacity for wind turbine 𝑖 at period 𝑡

𝜋-: Electricity price at time period 𝑡

𝑐2_: _{Preventive maintenance cost}

𝑐3_: _{Corrective maintenance cost}

𝐶-5.(: Crew deployment cost for wind farm location 𝑙 at period 𝑡

𝐶

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,-. _{: Dynamic maintenance cost for wind turbine 𝑖 at period 𝑡}

𝜉;: Maintenance criticality coefficient

𝑝<,). : Failure probability of turbine 𝑖 at the start of a sequence of time periods 𝑠

𝑀-(: Maintenance capacity for wind farm location 𝑙 at period 𝑡

Binaries:

𝑥-(: = 1 if wind farm location 𝑙 is visited by the maintenance crew at period 𝑡,

= 0 otherwise

𝑧-.: = 1 if preventive maintenance at turbine 𝑖 is performed at period 𝑡,

= 0 otherwise

𝜐-.: = 1 if corrective maintenance at turbine 𝑖 is performed at period 𝑡,

= 0 otherwise

𝜔<,-. : = 1 if wind farm location 𝑙 is visited by the maintenance crew between the start of a sequence of time

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12

Wind speed parameters:

𝑉.: wind speed

𝑉Z.: cut-in speed

𝑉[: rated speed

𝑉Z': cut-out speed

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13 3.3 Objective function

The objective function that will be used in this research is similar to that of Yildirim et al. (2017) and is as follows: max 2,3,],^ _ _ 𝑦 -._∙ 𝜋 --∈b .∈c deeefeeeg 'hi[j-.'kj( [i5ik3i − _ _ 𝑥-(∙ 𝐶-3.( -∈b (∈m deeeefeeeeg Z[in oih('^;ik- Z'<-− 𝜉;_ _ 𝑧-.∙ 𝐶-.₇8 ,--∈b .∈c8 deeeeefeeeeeg i]hiZ-io -3[p.ki ;j.k-ikjkZi Z'<-

As can be seen, the objective is to maximize wind farm revenue, while minimizing the two cost components; the crew deployment cost and the expected turbine maintenance cost. The components of the objective function are calculated as follows:

• Operational revenue is calculated by taking the product of the electricity price and energy output and summing this for all turbines and time periods.

• Crew deployment costs are only incurred when a maintenance crew visits the wind farm location. Therefore, the crew deployment costs are multiplied by the binary variable 𝑥-(. By summing this for all wind farm locations and time periods, the first cost component is calculated.

• The expected turbine maintenance cost is in place to account for the effect of real-time sensor data. The dynamic maintenance cost 𝐶

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,-. _{represents the trade-off between the cost of} preventive maintenance and the risk of unexpected failures, which are calculated as explained in section 3.1. Since this cost component determines the optimal time to schedule preventive maintenance, the dynamic maintenance cost is multiplied by the binary variable

𝑧-.. This outcome will be summed for every time period and all operational turbines to end up with the total expected turbine maintenance cost. The maintenance criticality coefficient is in place to change the importance of this last cost component.

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14 3.4 General constraints

The objective function is subject to several constraints. We start by introducing the general constraints; in section 3.5, the constraints on failure uncertainties will be presented. First, constraint (2) forces the model to schedule preventive maintenance within the time limit 𝜁. for all operational turbines. In our model, this means that preventive maintenance will be planned for an operational turbine when the turbine is over 70% of its lifetime (calculated from the RLDs).

_ 𝑧-.= 1, ∀𝑖 ∈ 𝐺' u₇

-vw

Constraint (3) limits the number of corrective maintenance actions 𝜐-. in the planning horizon to one per failed turbine. We assume perfect maintenance in our optimization model i.e. when maintenance is performed, the turbine will not fail in the same period. Note that this constraint does not force to conduct corrective maintenance on a failed turbine, enabling turbines to be idle for some time so that it can be grouped with more maintenances.

_ 𝜐-. ≤ 1, -∈b

∀𝑖 ∈ 𝐺)

Turbine maintenance cannot be conducted if the maintenance crew is not at the wind farm location. Therefore, constraint (4) ensures that the maintenance crew visits a wind farm if any of the wind turbines is scheduled for preventive maintenance. Constraint (5) ensures the same for corrective maintenance.

𝑧-. ≤𝑥-(, ∀𝑙 ∈ L, ∀𝑖 ∈ 𝐺'(, ∀𝑡 ∈ Τ

𝜐-. ≤𝑥-(, ∀𝑙 ∈ L, ∀𝑖 ∈ 𝐺)(, ∀𝑡 ∈ Τ

Constraint (6) makes sure that the maintenance crew only visits one of the wind farm locations during a single maintenance epoch.

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15 In case of bad weather conditions, maintenance at sea cannot be conducted. For this reason, constraint (7) is in place. In this constraint, 𝑇n( is the set of times that the weather conditions are too bad for the crew to visit wind farm location 𝑙.

𝑥-(= 0, ∀𝑙 ∈ L, ∀𝑡 ∈ 𝑇n(

Due to labour capacity constraints and constraints on available SOVs, helicopters, etcetera, constraint (8) puts a limit on maintenance actions at time 𝑡.

_ 𝑧-. .∈c₈}

+ _ 𝜐-. ≤ 𝑀-( .∈c₈•

, ∀𝑙 ∈ 𝐿, ∀𝑡 ∈ 𝑇

Constraint (9) and (10) couple the variables 𝑧-. and 𝜐€. to the energy output variable 𝑦-.. More specifically, constraint (9) ensures that when preventive maintenance is conducted on turbine 𝑖 at time 𝑡, no electricity is generated. 𝑝-.stands for the productive capacity and is dependent on the forecasted wind power at each time period. When the turbine is operational, this is the maximum amount of electricity that can be produced. Constraint (10) is in place to make sure a failed turbine should be scheduled for corrective maintenance before electricity can be produced. In this way and in combination with constraint (3) and the objective function, the model is able to determine whether it is better to let the turbine be idle for a longer time and maintain it when other turbines are maintained as well. The resulting decrease in crew deployment costs may outweigh the loss in revenues due to the longer idle times.

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16 3.5 Constraints on failure uncertainties

This paper will add constraints to model the failure uncertainties during the planning horizon. Even though the model uses real-time sensor data in making the O&M schedule, a turbine may fail earlier than expected. By integrating new constraints, the model will account for these uncertainties. The uncertainty in failures will be based on the failure probabilities that are calculated from the RLD’s. An important assumption to be made here is that an unplanned failure of a turbine will be repaired the first time the maintenance crew visits that specific wind farm, even if no maintenance was scheduled. In order to account for this assumption, the parameter 𝜔_<,-. will be added, which is 1 when the maintenance crew visits the turbine location between and including time s and t, where s is the start of a sequence of time periods and t is the end of this sequence.

The main constraint to be added is as follows:

𝑦-.≤ „ _ 𝑧𝜏. b 𝜏v-†w ‡ ∙ „1 − _ˆ𝑝<,). ∙ ‰1 − 𝜔<,-. Š‹ -<vw ‡ ∙ 𝑝-. deeeeeeeeeeeeefeeeeeeeeeeeeeg Œj<i w: h[.'[ -' h(jkkio ;j.k-ikjkZi + 𝑧-.∙ „_ˆ𝑝<,). ∙ 𝜔<,-ƒw. ‹ -ƒw <vw ‡ ∙ 𝑝-. deeeeeeeefeeeeeeeeg Œj<i •: o3[.kŽ h(jkkio ;j.k-ikjkZi + „_ 𝑧𝜏. -ƒw 𝜏vw ‡ ∙ 𝑝-. deeefeeeg Œj<i •: j)-i[ h(jkkio ;j.k-ikjkZi ∀𝑡 ∈ Τ, ∀𝑖 ∈ 𝐺'

In modelling constraint (11), a linearized form is used, which we explain in Appendix 1. Constraint (11) accounts for the fact that electricity production is influenced by failure uncertainties in three ways:

• For case 1, to have the highest electricity generation, when preventive maintenance is planned after time t, the turbine should either have a low probability of failing in case 𝜔_<,-. = 0 or should have been visited for corrective maintenance between (and including) time 𝑠 and 𝑡 (𝜔<,-. = 1) when the probability of failure is high.

• For case 2 it holds that if preventive maintenance is planned at time t, the turbine will only produce in full capacity if it has failed and is visited for corrective maintenance before time t (𝑝_<,). = 1 & 𝜔<,-ƒw. = 1). In case the turbine is not visited between time 𝑠 and 𝑡 − 1, it

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17 for the reason that the chance of being correctively maintained between time s and t is lower, and the planned preventive maintenance is still initiated at time t.

• For case 3, it holds that the turbine will produce at full capacity, since preventive maintenance is conducted before time t and the turbine has not failed. The failure probability is left out of this part, since we assume that directly after maintenance, the failure probability is zero.

Because energy output is coupled to preventive maintenance multiplied by components on failure uncertainties, more preventive maintenance actions will be scheduled. When there is no scheduled preventive maintenance, the constraint would lead to an expected energy output value of zero. In this way, there is more pressure to schedule preventive maintenance compared to a situation where only constraint (2) is forcing to schedule preventive maintenance. Next to that, with constraint (11) there is focus on scheduling preventive maintenance sooner, because especially in case 3 this leads to higher expected energy production.

Moreover, since the failure uncertainties are coupled to the energy output variable 𝑦_-., this results in more accurate calculations of electricity generation. Where previously energy production was slightly overestimated because it was only constrained in case of preventive maintenance by constraint (9), it now includes more factors that are of impact during maintenance epochs. This allows the energy production to be higher during planned preventive maintenance if a turbine has failed and is correctively maintained. On the other hand, it allows the energy production to be lower if the chance of failure before planned preventive maintenance is high and the turbine location is not visited for corrective maintenance. Consequently, the energy production is influenced based on sensor-driven data in two ways, by the dynamic maintenance costs in the objective function and by the failure probabilities in constraint (11).

Constraint (12) couples the variables 𝜔<,-. and 𝑥‘( as both represent location visits. The

difference is that 𝜔_<,-. represents turbine location visits between and including time s and t. Whereas 𝑥‘( represents wind farm location visits for time 𝜏. Constraint (12) is on an individual

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𝜔<,-. ≥ 𝑥𝜏( ∀𝑡 ∈ Τ, ∀𝑠 ∈ {1, … , 𝑡}, ∀𝜏∈ {𝑠, … , 𝑡}, ∀𝑖 ∈ 𝐺'

Constraint (13) is in place because the sum of all visits of the maintenance crew to the wind farm location between and including time s and t should be equal or greater than the value of 𝜔_<,-. _{. This constraint makes sure that if one of the x variables in the time between s and t is 1,}

omega can be 1 too. If more x variables are 1, omega would still be one and representing a visit. If all x variables in this time span were zero, omega is zero as well because there has been no visit to the wind farm and no visit to the turbine. This is an important addition because in constraint (12) omega could still be one if all x variables were zero.

𝜔<,-. ≤ _ 𝑥𝜏( 𝑡 𝜏=𝑠

∀𝑡 ∈ Τ, ∀𝑠 ∈ {1, … , 𝑡}, ∀𝑖 ∈ 𝐺'

The combination of constraints (12) and (13) ensures that if one of the 𝑥 variables in the time periods between s and t is one, omega is one as well. On the other hand, it ensures that if none of the 𝑥 variables in this time period is one, omega will return a value of zero. This is important for the reason that omega represents visits to a turbine location during a time period and 𝑥 represents wind farm visits for a specific point in time. The assumption that a failed turbine is visited for corrective maintenance when the wind farm location is visited is important for the difference in turbine and wind farm level between variables 𝜔<,-. and 𝑥𝜏( respectively.

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19 3.6 Wind speed calculations

Wind turbines are highly dependent on the wind speed at each time period for their energy production. If wind speeds are higher, wind turbines generate more electricity. When wind speeds are too low, operating would not be profitable enough. Therefore, the cut-in speed (the minimum wind speed for a turbine to be operational) is set at 4 m/s. On the contrary, when wind speeds are too high, operating the wind turbines would harm their condition. For this reason, the turbines are turned off for wind speeds higher than 25 m/s in our model (cut-out speed). In between the cut-in and cut-out speed a wind turbine is operational. The power generation increases linearly until the rated speed is reached (12 m/s), when this wind speed is reached the wind turbine will produce at maximum power (2 MWh). To show this effect in our model, we will use data on wind speeds and calculate power generation as explained by Karki and Patel (2009). They have defined this relationship as follows:

𝑝-.= – 0, h—‰˜†™š7†Œš7›Š, h—, 0, given – 0 ≤ 𝑉. ≤ 𝑉Z. š_œ7 • š₇ žš_— š_— • š₇ Ÿ š_œ8 𝑉Z' ≤ 𝑉.

The parameters a, b, and c are calculated according to the formulas below:

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20 3.7 Experimental framework

Our optimization model consists of two different modules. The planning module and the execution module. The first module, the planning module, solves the O&M optimization model for a period of 70 days (35 maintenance epochs of 2 days). This planning will be used in the execution module, where the schedule that is made in the planning module is fixed for a freeze period of 20 days. During the freeze period, data will be recorded on when the maintenance crew is sent to the wind farm, which turbines have failed, which turbines are idle, which turbines experience maintenance, etcetera. After these 20 days, the planning module will generate a new optimal O&M schedule, taking into account the current situation (e.g. failed and operational turbines and the dynamic maintenance costs of operational turbines). This process will be repeated 20 times; as a result, the model can relatively easily show the performance of the different maintenance strategies for a period of 400 days. The performance is measured by recording the following data for the complete execution period:

• The number of crew visits, preventive and corrective maintenance actions, failures, idle days, and the availability.

• The operational revenue, which is calculated by multiplying the total wind farm electricity generation by the energy price (initially this energy price is fixed at $30 per MWh). • The crew deployment costs are incurred only when the maintenance crew visits the wind

farm location. Therefore, we sum all the incurred crew deployment costs.

• The preventive maintenance costs are calculated by multiplying the number of preventive actions times the preventive maintenance cost (𝑐2= 4000).

• The corrective maintenance costs are calculated by multiplying the number of corrective actions with the corrective maintenance cost (initially we let 𝑐3 = 4 ∗ 𝑐2).

With this information the wind farm profit can be calculated. The wind farm profit is equal to the operational revenue minus the three cost components (crew deployment costs, preventive maintenance costs, and corrective maintenance costs).

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4. Results and Discussion

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22 4.1 Benchmark analysis

To check the general model characteristics, the first study will be to compare the results of CBOM with CBM, TBM, and CM. For now, the objective function is only subject to the general constraints of section 3.4. No maintenance strategies in this benchmark analysis are subject to constraints 11-13. Section 4.2 presents the impact of the constraints on failure uncertainties.

4.1.1 Condition-Based Opportunistic Maintenance (excluding failure uncertainties)

The first maintenance strategy that we consider is CBOM without failure uncertainties. Table 1 shows the outcomes of the first experiment. The results show that when the crew deployment costs are increased, the number of deployments decreases. When crew deployment costs are low, the number of preventive maintenances is high. With every increase of the crew deployment cost, this number decreases. This is due to the increased cost of visiting the wind farm, making it less profitable to perform preventive maintenance (i.e. the higher availability when more preventive maintenance is performed does not outweigh the higher costs due to more crew deployments). Because of less preventive maintenance actions, the number of failures increases, resulting in more corrective actions and more idle days. Another observation is that for the higher crew deployment costs, the maintenance schedule is more efficient in order to minimize crew deployment costs (i.e. the crew conducts more maintenances per trip).

Table 1. Results CBOM (excluding failure uncertainties)

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23 4.1.2 Condition-Based Maintenance

CBM is slightly different from CBOM, because it does not benefit from combining maintenance trips. To see the difference between the two strategies, we set the crew deployment costs in the CBM objective function equal to zero:

𝐶-5.(= 0 ∀𝑙 ∈ 𝐿, ∀𝑡 ∈ 𝑇

Table 2 shows the performance of CBM under changing crew deployment costs. Not scheduling maintenance actions together in CBM is of great influence on the wind farm profit. When the crew deployment costs are zero CBOM and CBM behave the same. However, when the crew deployment costs increase, CBM has significantly higher costs because it is not able to schedule maintenances opportunistically and therefore visits the wind farm more often than under CBOM. Because of the large number of crew visits and the higher number of preventive actions, the availability under CBM is higher. Nonetheless, the related revenue increase does not make up for the extremely high costs of maintenance. This results in lower profits than for CBOM when the crew deployment costs are more than zero.

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24 4.1.3 Time-based Maintenance

For TBM, the preventive maintenance planning is based on time intervals, rather than on the condition of turbines. We use the characteristics of age-based maintenance (i.e. preventive maintenance periods are determined by the age of the turbine) and force the model to schedule maintenance for turbines that have reached the age between 140 and 150 days. We determined this age by taking the average turbine age at which preventive maintenance is performed under CBM (after 155 days). Because TBM does not have information on turbine conditions, we are more cautious by scheduling preventive maintenance between 140 and 150 days.

Additionally, TBM does not use sensor-driven deterioration data to schedule preventive maintenance. This is represented in the objective function by setting the dynamic maintenance cost equal to zero:

𝐶

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,-. _{= 0 ∀𝑖 ∈ 𝐺}

𝑜, ∀𝑡 ∈ 𝑇

Comparing the results of TBM in table 3 to CBM, the number of preventive maintenances is similar, whereas there are more failures under TBM. This can be explained by the fact that TBM does not take the actual condition of a turbine into consideration. Therefore, it might conduct preventive maintenance on machines that are not in need of a repair and fail to do maintenance on machines that have degraded faster than expected. On the other hand, the profits under TBM are higher than under CBM for crew deployment costs higher than € 16,000, since TBM schedules maintenance opportunistically (i.e. combining maintenance actions in a crew visit), leading to less crew visits and lower expenditures.

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25 4.1.4 Corrective Maintenance

With CM, the only difference to CBOM is that there will be no preventive maintenance actions. For this reason, constraint (2) is replaced by:

𝑧-.= 0, ∀𝑙 ∈ L, ∀𝑖 ∈ 𝐺(, ∀𝑡 ∈ Τ

The results in table 4 show that under this strategy and due to no preventive actions, the number of failures and total idle days is the highest for every value of the crew deployment cost. The relatively low revenues in combination with high maintenance costs (because of the high cost of corrective maintenance) results in the lowest profit of all maintenance strategies for a crew deployment cost lower than 48,000. When the crew deployment costs have reached a value of 48,000 CM outperforms CBM, since it benefits from opportunistic maintenance scheduling (i.e. letting failed turbines be idle to group them with other maintenances).

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26 4.1.5 Comparative study

To assess the performance of the discussed maintenance strategies, figure 1 graphically shows the total profit for the four different maintenance strategies under different crew deployment costs. As can be seen, CBM has similar profits compared to CBOM when the crew deployment cost is set to zero. When the crew deployment costs increase the difference between CBOM and CBM increases too because CBM does not benefit from combining maintenance trips with high expenditures as a result. For the lower crew deployment costs CBM is outperforming TBM and CM but as crew deployment costs become higher it is the worst choice of all maintenance strategies in terms of profit. In line with the literature, CBOM gives the highest profit for all crew deployment costs, making it the preferred maintenance strategy with respect to profits. The higher profits are explained by the combination of using sensor-driven deterioration data in scheduling preventive maintenance and by opportunistically scheduling maintenance actions.

When 𝐶-5.(= 48,000, CBOM has 13% higher profits than CBM, 4% higher profits than TBM,

and 7% higher profits than under CM. Next to that, it has 18% less idle days than under TBM and 32% less idle days than CM. In the next section (4.3), we will include failure uncertainties to this optimal condition-based opportunistic maintenance strategy.

Figure 1. Impact crew deployment cost on profit

25 27 29 31 33 35 37 39 41 43 0 16,000 48,000 80,000 112,000 Pr of it (€ M )

Crew deployment cost

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27 4.2 Impact of failure uncertainties

In this section, we consider constraints 11-13 of section 3.5 and research their impact in different environments. In the first experiment, we change the crew deployment costs. Secondly, the effect of fluctuating electricity prices will be investigated. Afterwards, we look into the effect of changing preventive maintenance costs. Lastly, the impact failure uncertainties have in different sized wind farms will be elaborated on.

4.2.1 Changing crew deployment costs

To find out the effect of including failure uncertainties in the planning phase, we run the same test as in section 4.2. The difference with CBOM in the previous section is that the objective function now is subjected to constraints 11-13. All other input parameters are the same and the results of this experiment under different crew deployment costs are shown in table 5.

Table 5. Results CBOM (including failure uncertainties)

𝐶-5.(: 0 16,000 48,000 80,000 112,000 Revenue: _{€ 41.99 M} _{€ 41.44 M} _{€ 41.13 M} _{€ 40.83 M} _{€ 40.64 M} Profit: _{€ 41.04 M} _{€ 39.97 M} _{€ 39.03 M} _{€ 38.76 M} _{€ 38.47 M} Expenditures: _{€ 0.95 M} _{€ 1.47 M} _{€ 2.10 M} _{€ 2.07 M} _{€ 2.18 M} - Maintenance _{€ 0.95 M} _{€ 1.07 M} _{€ 1.10 M} _{€ 1.11 M} _{€ 1.06 M} - Crew deployment _{€ 0.00 M} _{€ 0.40 M} _{€ 1.01 M} _{€ 0.96 M} _{€ 1.12 M} # Crew visits ₁₂₉ ₂₅ ₂₁ ₁₂ ₁₀ # Preventive actions ₁₇₈ ₁₅₉ ₁₄₆ ₁₄₂ ₁₃₆ # Corrective actions ₁₅ ₂₇ ₃₂ ₃₄ ₃₂ # Failures ₁₄ ₂₇ ₃₁ ₃₃ ₃₄ Idle days ₇₀₀ ₉₁₂ _1,088 _1,278 _1,446 Availability _0.9825 _0.9772 _0.9728 _0.9680 _0.9638

From comparing these results with the results of table 1 can be concluded that the number of preventive maintenance actions is higher when failure uncertainties are considered (19% more when 𝐶_-5.( = 48,000). This is what we would expect in case of uncertain times of failure. The new model couples failure probabilities to expected energy output variable 𝑦_-.. As a result, it will tend to schedule preventive maintenance sooner when the probability of failure is high in order to minimize failures. The increase in preventive maintenance actions leads to a decrease of failure instances of 39% when 𝐶-5.(= 48,000. Moreover, the maintenance costs (cost of

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28 As a result of considering failure uncertainties, the idle days under the newly proposed model are significantly lower than under CBOM without failure uncertainties. Figure 2 shows how the idle days differ in the different models under changing crew deployment costs. For all crew deployment costs higher than zero the newly proposed model is outperforming CBOM without failure uncertainties.

Figure 2. Impact failure uncertainties on availability

Figure 3 shows the impact of including failure uncertainties on wind farm profit. For all cases where the crew deployment cost is higher than zero, the new model outperforms the old one by generating more profits. The advantage of including failure uncertainties is highest when 𝐶_-5.( = 80,000.

Figure 3. Impact failure uncertainties on profit

0 500 1,000 1,500 2,000 2,500 0 16,000 48,000 80,000 112,000 Id le d ay s

CBOM (including failure uncertainties) CBOM (excluding failure uncertainties)

38 38.5 39 39.5 40 40.5 41 41.5 0 16,000 48,000 80,000 112,000 Pr of it (€ M )

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29 Despite the significantly higher availability, this does not result in a one-to-one increase in wind farm revenue and profit. This can be explained by:

• More crew visits in the new model that lead to higher crew deployment costs.

• Different scheduling of maintenance actions; a turbine is not able to produce electricity when maintenance is performed. During high wind speed periods, the corresponding revenue loss is higher than in times of low wind speeds. For this reason, the difference in maintenance schedules of CBOM including failure uncertainties and CBOM excluding failure uncertainties results in different energy generation numbers.

• Constraining energy output 𝑦-.in constraint 11 by failure probabilities that are between 0 and 1, leads to slightly lower energy generation in the new model. Because of the many rolls, these small differences each day lead to the revenue changes. Since more factors such as failure probabilities are taken into account, expected energy generation is calculated more reasonable than in the old model.

These last two reasons become especially clear in the situation in which there are no crew deployment costs (𝐶-5.(= 0). For these instances, the availability is the same for both models

but the revenue is lower in the model that accounts for failure uncertainties.

Another observation that stands out when 𝐶-5.( = 0 is that, contrarily to the nonzero crew

deployment costs, the number of crew visits is lower in the new model. We start the explanation by noticing that for both models the number of crew visits is significantly higher when no crew deployment costs are incurred. Normally, crew deployment costs are constraining the 𝑥 variables to be one too often because of the higher crew deployment cost factor in the objective function. This is not true for the situation in which there are no crew deployment costs, leading to the high number of crew visits. In many of these visits, the crew goes to the wind farm while conducting no maintenance. This is because there is no constraint that forces the model to conduct maintenance when the wind farm is visited. When there is a cost related to crew visits, there are no useless trips anymore resulting in a realistic overview.

The reason that the number of crew visits is lower in the new model when 𝐶-5.(= 0 is that there

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31 4.2.2 Fluctuating electricity prices

Due to reasons such as rising electricity demand, prices of offshore wind energy may increase in the future. Therefore, the impact of changing electricity prices is investigated, the results are shown in table 6. In this experiment, the crew deployment costs (𝐶_-5.() will be fixed at €48,000 and all other input parameters will remain at their initial settings.

Table 6. Impact fluctuating electricity price on CBOM (including failure uncertainties)

𝜋-: 15 30 45 60 75 90 Revenue: _{€ 20.36 M} _{€ 41.13 M} _{€ 62.01 M} _{€ 82.73 M} _{€ 103.65 M} _{€ 124.52 M} Profit: _{€ 18.20 M} _{€ 39.03 M} _{€ 59.75 M} _{€ 80.41 M} _{€ 101.19 M} _{€ 122.11 M} Expenditures: _{€ 2.16 M} _{€ 2.10 M} _{€ 2.26 M} _{€ 2.32 M} _{€ 2.47 M} _{€ 2.42 M} - Maintenance _{€ 1.15 M} _{€ 1.10 M} _{€ 1.02 M} _{€ 1.08 M} _{€ 1.08 M} _{€ 1.07 M} - Crew deployment _{€ 1.01 M} _{€ 1.01 M} _{€ 1.25 M} _{€ 1.25 M} _{€ 1.39 M} _{€ 1.34 M} # Crew visits ₂₁ ₂₁ ₂₆ ₂₆ ₂₉ ₂₈ # Preventive actions ₁₄₀ ₁₄₆ ₁₅₄ ₁₅₇ ₁₅₇ ₁₆₀ # Corrective actions ₃₇ ₃₂ ₂₅ ₂₈ ₂₈ ₂₇ # Failures ₃₆ ₃₁ ₂₅ ₂₈ ₂₈ ₂₆ Idle days _1,318 _1,088 ₉₆₆ _1,006 ₉₃₂ ₉₁₄ Availability _0.9670 _0.9728 _0.9758 _0.9748 _0.9767 _0.9771

The results clearly indicate that revenue and profits increase linearly with the electricity price. Additionally, with higher electricity prices, the focus is more on keeping turbines available. Idle turbine days result in higher lost revenues when energy prices are higher, for this reason there is more focus on preventive maintenance (i.e. the number of preventive actions increases as prices increase). Besides that, the number of crew visits increases for higher prices enabling maintenance to be conducted closer to its optimal time.

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32 Table 7. Impact fluctuating electricity price on CBOM (excluding failure uncertainties)

𝜋-: 15 30 45 60 75 90 Revenue: _{€ 20.13 M} _{€ 40.80 M} _{€ 61.19 M} _{€ 82.45 M} _{€ 102.89 M} _{€ 123.95 M} Profit: _{€ 18.25 M} _{€ 38.95 M} _{€ 58.98 M} _{€ 80.11 M} _{€ 100.52 M} _{€ 121.65 M} Expenditures: _{€ 1.89 M} _{€ 1.85 M} _{€ 2.21 M} _{€ 2.34 M} _{€ 2.38 M} _{€ 2.30 M} - Maintenance _{€ 1.41 M} _{€ 1.28 M} _{€ 1.39 M} _{€ 1.34 M} _{€ 1.37 M} _{€ 1.30 M} - Crew deployment _{€ 0.48 M} _{€ 0.58 M} _{€ 0.82 M} _{€ 1.01 M} _{€ 1.01 M} _{€ 1.01 M} # Crew visits ₁₀ ₁₂ ₁₇ ₂₁ ₂₁ ₂₁ # Preventive actions ₁₁₂ ₁₂₃ ₁₀₄ ₁₂₂ ₁₂₂ ₁₃₂ # Corrective actions ₆₀ ₄₉ ₆₁ ₅₃ ₅₅ ₄₈ # Failures ₅₉ ₅₁ ₆₃ ₅₁ ₅₄ ₄₆ Idle days _2,044 _1,700 _1,690 _1,332 _1,402 _1,266 Availability _0.9489 _0.9575 _0.9577 _0.9667 _0.9649 _0.9684

Figure 4 shows for both models how the number of idle days changes as the electricity price increases. The idle days show in both models a decreasing trend due to the higher focus on availability for higher prices. It is important to see that the number of idle days under the newly proposed model is significantly lower and have a more stable decrease than the old model has. The reason for fewer idle days is resulting from more preventive maintenance actions and more crew visits for every electricity price in the new model. Thereby, the stable decrease in idle days in the new model is a result of less fluctuation in the number of preventive actions as the electricity price increases. More crew visits and preventive maintenance actions confirm that the new model puts more pressure on scheduling preventive maintenance in order to maximize expected energy generation (as stated in section 3.5).

Figure 4. Idle days under fluctuating electricity prices

0 500 1,000 1,500 2,000 2,500 15 30 45 60 75 90 Id le d ay s

Electricity price (€ per MWh)

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33 Conducting more preventive maintenance decreases the number of failure instances and thereby the total turbine maintenance costs (costs of preventive and corrective maintenance together). Next to lower turbine maintenance cost and because of higher availability, the revenue is higher in the new model for all electricity prices. However, as can be seen in figure 5, this does not hold for wind farm profits. For both models, wind farm profits are linearly increasing and do not differ much. When taking a closer look at the profit numbers in table 6 and 7, we observe that the profit is slightly lower in the new model for an electricity price of €15. For all other prices, the new model outperforms the old one concerning profits. The explanation for lower profits in the new model with an electricity price of €15 is explained by the higher crew deployment costs that are not outweighed by the created revenue increase. For the higher electricity prices the higher availability is resulting in revenues that are outweighing the crew deployment cost (i.e. for every MWh produced more revenue is generated). Because of the fact that both models have improving availability at higher electricity prices, the relative performance on availability of CBOM including failure uncertainties to CBOM without failure uncertainties does not improve. Therefore, the profit increase is more or less the same in both models.

Figure 5. Profit under fluctuating electricity prices

€ 0 € 20 € 40 € 60 € 80 € 100 € 120 € 140 15 30 45 60 75 90 Pr of it (€M )

Electricity price (€ per MWh)

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34 4.2.3 Increasing preventive maintenance costs

To research how CBOM including failure uncertainties behaves compared to the old model when the preventive maintenance costs 𝑐2 change, we change 𝑐2 while keeping the other cost factors fixed. We fix 𝑐3 = 16,000 and 𝐶_-5.( = 48,000. The other parameters remain at their initial settings. In this way we can see how the O&M schedule changes as the cost of preventive maintenance increases relative to other cost components. The results in table 8 and 9 show that increasing preventive maintenance costs does not influence the maintenance schedules at all. Consequently, the profit is only decreased by the higher preventive maintenance costs.

Table 8. Impact increasing preventive maintenance cost on CBOM (excluding failure uncertainties) 𝑐2_: ₂₀₀₀ ₄₀₀₀ ₈₀₀₀ ₁₂₀₀₀ ₁₆₀₀₀ Revenue: _{€ 40.80 M} _{€ 40.80 M} _{€ 40.80 M} _{€ 40.80 M} _{€ 40.80 M} Profit: _{€ 39.20 M} _{€ 38.95 M} _{€ 38.46 M} _{€ 37.97 M} _{€ 37.48 M} Expenditures: _{€ 1.61 M} _{€ 1.85 M} _{€ 2.34 M} _{€ 2.84 M} _{€ 3.33 M} - Maintenance _{€ 1.03 M} _{€ 1.28 M} _{€ 1.77 M} _{€ 2.26 M} _{€ 2.75 M} - Crew deployment _{€ 0.58 M} _{€ 0.58 M} _{€ 0.58 M} _{€ 0.58 M} _{€ 0.58 M} # Crew visits ₁₂ ₁₂ ₁₂ ₁₂ ₁₂ # Preventive actions ₁₂₃ ₁₂₃ ₁₂₃ ₁₂₃ ₁₂₃ # Corrective actions ₄₉ ₄₉ ₄₉ ₄₉ ₄₉ # Failures ₅₁ ₅₁ ₅₁ ₅₁ ₅₁ Idle days _1,700 _1,700 _1,700 _1,700 _1,700 Availability _0.9575 _0.9575 _0.9575 _0.9575 _0.9575

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35 Figure 6 shows that the number of idle days is stable for both models when 𝑐2 increases. This is explained by the fact that preventive maintenance is necessary to ensure power generation. Less preventive maintenance would lead to a higher number of idle days (as can be seen in a corrective maintenance strategy). The resulting downtime losses would be higher than the cost of conducting preventive maintenance. Therefore, increasing 𝑐2 to a level where it is similar to 𝑐3_{does not lead to more corrective maintenance actions compared to the number of preventive}

maintenances. The effect of incorporating failure uncertainties is that for every value of 𝑐2 the number of idle days is lower, because of more crew visits and more preventive maintenance. This is presented graphically in figure 6.

Figure 6. Idle days under increasing preventive maintenance costs

The higher number of preventive actions when failure uncertainties are accounted for not only results in fewer idle days, it also makes sure the revenue is higher under the newly proposed model for all values of 𝑐2. However, increasing preventive maintenance costs negatively influence wind farm profits. Figure 7 displays the profit for the old CBOM model and the new CBOM model when preventive maintenance costs increase. The higher number of preventive maintenance actions when failure uncertainties are considered, result in lower profits when 𝑐2_{≥ 0.5𝑐}3_{. For instances where this is true the higher revenues in the new model do not offset}

the higher preventive maintenance cost component.

800 1,000 1,200 1,400 1,600 1,800 2000 4000 8000 12000 16000 Id le d ay s

Preventive maintenance cost

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36 Figure 7. Profit under increasing preventive maintenance costs

€ 36 € 37 € 37 € 38 € 38 € 39 € 39 € 40 2000 4000 8000 12000 16000 Pr of it (€ M)

Preventive maintenance cost

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37 4.2.4 Different wind farm sizes

This experiment looks at how including failure uncertainties in CBOM affects the O&M schedule and wind farm performance for different size wind farms, while having a crew deployment cost (𝐶_-5.() of 48,000 and all other parameters at their initial settings. First, we run the experiments on CBOM excluding failure uncertainties for wind farm sizes between 75 and 150 turbines. The results are given in table 10. The same experiment is conducted for CBOM that accounts for failure uncertainties, these results are presented in table 11.

Table 10. Performance for different wind farm sizes CBOM (excluding failure uncertainties) # turbines: 75 100 125 150 Revenue: _{€ 30.49 M} _{€ 40.80 M} _{€ 51.30 M} _{€ 61.90 M} Profit: _{€ 28.87 M} _{€ 38.95 M} _{€ 49.10 M} _{€ 59.32 M} Expenditures: _{€ 1.62 M} _{€ 1.85 M} _{€ 2.20 M} _{€ 2.58 M} - Maintenance _{€ 1.09 M} _{€ 1.28 M} _{€ 1.58 M} _{€ 1.81 M} - Crew deployment _{€ 0.53 M} _{€ 0.58 M} _{€ 0.62 M} _{€ 0.77 M} # Crew visits ₁₁ ₁₂ ₁₃ ₁₆ # Preventive actions ₈₁ ₁₂₃ ₁₇₈ ₂₂₈

# Preventive actions per turbine _1.08 _1.23 _1.42 _1.52

# Corrective actions ₄₈ ₄₉ ₅₄ ₅₆

# Failures ₄₉ ₅₁ ₅₂ ₅₆

Idle days _1,362 _1,700 _1,942 _2,014

Availability _0.9546 _0.9575 _0.9612 _0.9664

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38 Table 11. Performance for different wind farm sizes CBOM (including failure

uncertainties) # turbines: 75 100 125 150 Revenue: _{€ 30.64 M} _{€ 41.13 M} _{€ 51.46 M} _{€ 62.01 M} Profit: _{€ 29.19 M} _{€ 39.03 M} _{€ 48.45 M} _{€ 58.54 M} Expenditures: _{€ 1.45 M} _{€ 2.10 M} _{€ 3.02 M} _{€ 3.47 M} - Maintenance _{€ 0.82 M} _{€ 1.10 M} _{€ 1.43 M} _{€ 1.74 M} - Crew deployment € 0.62 M € 1.01 M € 1.58 M € 1.73 M # Crew visits ₁₃ ₂₁ ₃₃ ₃₆ # Preventive actions ₁₀₂ ₁₄₆ ₂₁₀ ₂₅₅

# Preventive actions per turbine _1.36 _1.46 _1.68 _1.70

# Corrective actions ₂₆ ₃₂ ₃₇ ₄₅

# Failures ₂₅ ₃₁ ₃₆ ₄₃

Idle days ₉₂₂ _1,088 _1,356 _1,492

Availability

0.9693 0.9728 0.9729 0.9751

Figure 8 shows the availability in the different models when wind farm sizes change. It is clear that the model that accounts for failure probabilities has a significantly better availability. This is achieved by more preventive maintenance actions that have fewer failures as a result. The improved reliability in the new model decreases the need for corrective maintenance actions. For this reason, the number of corrective maintenance actions (and consequently the total maintenance costs) decreases when failure uncertainties are considered. The higher availability results in slightly higher revenues. Nevertheless, the increase in revenue is not extremely large for reasons similar to those explained in section 4.3.1. In short, the reasons are more crew visits, different maintenance schedules, and reasonable expected energy calculations in the new model.

Figure 8. Idle days for different wind farm sizes

0.9500 0.9550 0.9600 0.9650 0.9700 0.9750 0.9800 75 100 125 150 Av ai la bi lit y

# turbines in wind farm

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39 In figure 9 we observe that the profit for CBOM that includes failure uncertainties and for CBOM that does not include failure uncertainties do not differ much. For the smaller wind farms (≤ 100 turbines), the profit is higher when failure uncertainties are included. For the other wind farm sizes, the higher revenue does not result in higher profits due to the increased expenditures. In spite of the lower maintenance costs because of less corrective actions, the total expenditures are higher in the model that includes failure uncertainties as it schedules more crew visits (75% more in a 100-turbine wind farm). The difference between crew visits becomes larger when the number of turbines increases. These higher expenditures in combination with only relatively small increases in revenue lead to no large changes in profit for the model that accounts for failure uncertainties.

Figure 9. Profit for different wind farm sizes

€ 0 € 10 € 20 € 30 € 40 € 50 € 60 € 70 75 100 125 150 Pr of it (€ M )

# turbines in wind farm

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40

5. Conclusion

This paper studied the impact of including failure uncertainties in a condition-based opportunistic maintenance strategy on the maintenance schedule, profit and availability. We have integrated sensor-driven deterioration data to measure the condition of wind turbines. First, in the benchmark analysis, the performance of the condition-based opportunistic maintenance strategy is compared to the performance of other maintenance strategies. Although condition-based maintenance provides better availability, high crew deployment costs make this strategy inferior to based opportunistic maintenance. Moreover, the condition-based opportunistic maintenance strategy has proven to outperform time-condition-based maintenance and corrective maintenance with higher profits and fewer idle days.

Next, the impact of including failure uncertainties during the maintenance-planning phase is assessed by means of four experiments. In all experiments where crew deployment costs are incurred, the newly proposed model ensures more preventive maintenance actions and an increase in crew visits. This leads to higher availability and reliability with respectively higher revenues and lower maintenance costs (due to fewer corrective maintenances) as a result. Nevertheless, the higher availability results in relatively small changes in revenue as the newly proposed model determines expected energy output more accurately by taking failure uncertainties into account. This is best observed in the situation where there are no crew deployment costs. For the reason that with similar availability rates, the revenue is lower when failure uncertainties are considered.

In spite of generally lower maintenance costs and slightly higher revenues there are no significant changes to profit when failure uncertainties are considered. This is explained by more crew visits in the new model that lead to higher crew deployment costs and the different scheduling of maintenance actions resulting in different energy generation patterns.

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42

6. Limitations and further research

The previous sections have outlined the advantage of including failure uncertainties in O&M scheduling. This section discusses the limitations of the studies and the possibilities for further research.

First, the research in this paper considered the O&M scheduling on a wind turbine level. In reality, a turbine consists of multiple components with each different deterioration patterns. When more components in a turbine have degraded, the benefits of opportunistic maintenance scheduling may increase. Therefore, it could be interesting to investigate the effect of including failure uncertainties on a model that works on component level.

Secondly, repairs are often subject to the availability of to be replaced parts, spare part unavailability is an issue in wind farm O&M activities. There is a trade-off between having a large inventory of spare parts and the turbine downtime costs in case of material unavailability (Shafiee, 2015). An opening for future research is to reconsider the assumption that the maintenance crew repairs failed turbines when they visit the turbine location, no matter if they were scheduled for preventive maintenance. Researching the effect of availability of parts on the feasibility of maintenance will give valuable insights for wind farm operators.

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Appendix A

Linearizing constraint 11

For modelling purposes, we linearized constraint (11). In order to do so we introduce three auxiliary variables 𝛿. In this way, each case (before, during, and after scheduled preventive maintenance) is represented in a separate constraint. The constraints below make sure that only one of the variables 𝛿 can be nonzero (achieved by multiplying z variables for different time periods by a large integer M). Eventually, constraint (18a) makes sure that the linearized constraints give the same output as is expected based on constraint (11).

Operations and Maintenance Scheduling Optimization in Offshore Wind Farms: Incorporating Failure Uncertainties in a Condition-Based Opportunistic Maintenance Model