A joint spare part and opportunistic
condition-based maintenance policy for
offshore wind farms under continuous
condition monitoring
Bart Dieterman
Master’s thesis
MSc. Technology and Operations Management
MSc. Supply Chain Management
Supervisors: Dr. E. Ursavas
Dr. I. Bakir Dr. M. Yildirim
Student number: S2546116 Faculty of Economics and Business
A joint spare part and opportunistic condition-based
maintenance policy for offshore wind farms under
continuous condition monitoring
Abstract
Opportunistic and condition-based maintenance methods are known to effectively reduce mainte-nance costs for offshore wind farms. The maintemainte-nance policy together with the spare parts inventory policy determines the overall maintenance strategy’s effectiveness. As the maintenance and spare part policies influence each other, consideration of both polices combined could improve perfor-mance. This thesis develops a joint spare part and opportunistic condition-based maintenance policy for offshore wind farms. The policy includes continuous condition monitoring based on real-life degradation data. We use Monte Carlo simulation to assess policy performance. We use a local search heuristic to determine the best decision variables: two deterioration thresholds for triggering maintenance actions, and a reorder level and order quantity to control spare parts. In two extensions, we introduce spare part ordering based on deterioration thresholds and changing maintenance thresholds depending on weather conditions. Our results show a reduction in average daily total costs of 8.64% for our joint policy compared to the sequentially-determined policy, and of 43.39% compared to a benchmark policy with age-based maintenance. This cost reduction is attributed to reduced failure and holding costs. In the specific cases of reduced system size and increased downtime costs, the extensions outperform the joint policy in terms of costs.
Preface
This thesis is my final project for the MSc. Technology and Operations Management and MSc. Supply Chain Management at the University of Groningen. I would like to thank several people for their contributions. First, Dr. E. Ursavas and Dr. I. Bakir for their supervision and useful feedback throughout the project. I would also like to thank Dr. M. Yildirim for providing access to the degradation data and the very valuable discussions we had. Finally, I thank my family and friends for their support, proofreading and the occasional much needed distraction.
Bart Dieterman
Contents
1 Introduction 5
2 Literature review 7
2.1 Maintenance strategies . . . 7
2.2 Spare part management . . . 8
2.3 Joint spare part and maintenance policies . . . 9
2.4 Maintenance and spare part management for OWFs . . . 10
3 Methodology 10 3.1 Setting . . . 10
3.2 Model . . . 12
3.2.1 Notation . . . 13
3.2.2 Scheduling maintenance actions . . . 14
3.2.3 Spare part and resource control . . . 14
3.2.4 Performing maintenance actions . . . 15
3.2.5 Output . . . 16
3.2.6 Flow diagram simulation . . . 18
3.3 Data and parameters . . . 19
3.3.1 Degradation data . . . 19
3.3.2 Weather data . . . 19
3.3.3 Data processing for simulation . . . 20
3.3.4 Base case values . . . 20
4 Results 20
4.1 Maintenance policy . . . 21
4.2 Spare part policy . . . 22
4.3 Joint policy . . . 23
4.4 Benchmark policy . . . 24
4.5 Sensitivity analysis . . . 24
4.5.1 Impact of maintenance costs . . . 24
4.5.2 Impact of spare part costs . . . 27
4.5.3 Impact of resource constraint . . . 29
4.5.4 Impact of downtime costs . . . 30
4.6 Extension: maintenance thresholds depending on weather . . . 31
4.7 Extension: just-in-time ordering . . . 32
5 Conclusion 33
1
Introduction
A safe, reliable, affordable and sustainable energy supply is crucial for industrial competitiveness, people’s well-being and overall functioning of today’s society (European Comission, 2012). To ensure this for future generations, a radical shift in the upcoming decades from the current largely fossil-based energy supply to renewable energy sources is required. Wind power is expected to play a major role in this shift, with projections of becoming the largest power source in 2040 in Germany and United Kingdom (McKinsey, 2017). Wind farms can be classified into two categories: onshore wind farms located on land and offshore wind farms (OWFs) located at sea. Offshore wind energy is an energy source with large and unrealized potential (Lacal-Ar´antegui, Yusta, & Dom´ınguez-Navarro, 2018). Offshore wind benefits from higher wind speeds with more electricity generation potential compared to onshore wind farms, for example, and they do not affect the landscape or cause noise pollution when located far enough from coastline (Snyder & Kaiser, 2009). However, both investment and operating costs for OWFs are much higher (Irawan, Ouelhadj, Jones, St˚alhane, & Sperstad, 2017) compared to onshore farms because of more expensive foundations and higher transportation costs.
Because of these higher costs, cost reductions could increase the potential of offshore wind farm development. The total expenditures of a typical OWF are as follows: 38% turbines, 23% operations and maintenance (O&M) costs, 19% cable and network connection, 12% foundations and 8% capital costs (Karyotakis & Bucknall, 2010). As O&M is the second largest cost-driver, developing possible maintenance cost reduction strategies is important. This necessity is strengthened by comparing the costs to onshore wind farms, where O&M only accounts to 5-10% of the costs (Karyotakis & Bucknall, 2010). The difficult accessibility of OWFs partly contributes to high O&M costs. As new wind farms are placed further from shore, weather conditions are harsher (Obdam, Rademakers, & Savenije, 2014). These weather conditions constrain transportation and can lead to long downtimes in case of failures, driving up maintenance costs and reducing the availability of wind turbines.
To reduce the maintenance costs of OWFs, attention should be paid to improve the maintenance strategy. Because of their ease of management, corrective and time-based preventive maintenance actions are traditionally widespread in wind farm operations (F. Ding & Tian, 2012). Condition-based maintenance (CBM), a more recent and less widespread alternative, has become essential for operators in order to reduce maintenance costs (Byon & Ding, 2010). Thanks to advancements in sensor technology, manufacturers have started to install condition-monitoring equipment in wind turbines (Byon, P´erez, Ding, & Ntaimo, 2011). Condition-monitoring data are used to assess the state of the turbine and, based on this information, maintenance actions are performed. When multiple turbines or components are considered, maintenance actions can also be grouped together. By using a maintenance action as an opportunity to also conduct maintenance on other turbines or components, the amount of setups and costs can be reduced. This grouping of maintenance actions is called opportunistic maintenance and can lead to substantial cost savings (F. Ding & Tian, 2012). Other researchers have developed models for opportunistic maintenance with the use of condition monitoring and sensor data for wind farms (Tian, Jin, Wu, & Ding, 2011; Yildirim, Gebraeel, & Sun, 2017). Our approach to maintenance is similar, with opportunistic maintenance and continuous condition monitoring based on data from real-life rotating machinery.
Maintenance actions often require spare parts. Wind turbines consist of multiple large and expensive components, such as the rotor, generator and gearbox. Inventory costs for the spare parts are high because of the storage space and capital investments required, and expensive trans-portation resources are needed to transport the parts to the offshore location. Transtrans-portation in marine environments is constrained by weather conditions. These restrictive conditions influence the timing of spare part demand. Improvements in the supply and logistics of spare parts that consider these constraints offer high potential for cost reduction (Tracht, Westerholt, & Schuh, 2013). Although inventory control is often treated as a separate issue, it can be integrated into the maintenance strategy. This is called a joint spare part and maintenance policy. Compared to a separate or sequentially-determined spare part and maintenance strategy, joint policies can lead to significant cost reductions (Van Horenbeek, Bur´e, Cattrysse, Pintelon, & Vansteenwegen, 2013). Several studies have researched the potential to combine spare part management and maintenance strategy specifically for offshore wind farms (e.g. Dewan, 2014; Jin, Tian, Huerta, & Piechota, 2012; Tracht et al., 2013). However, none of these papers consider opportunistic or condition-based maintenance.
Literature on joint spare part and maintenance optimization focusing on condition-based main-tenance for multi-unit systems is rather scarce. Until the most recent paper by Zhang and Zeng (2017), only four papers considered joint optimization (Olde Keizer, Teunter, & Veldman, 2017). Zhang and Zeng (2017) were the first to include opportunistic maintenance. They use a base-stock (S − 1, S) policy for inventory control, where spare parts are replaced one-on-one whenever de-mand occurs. They consider, like the other four paper (Olde Keizer et al., 2017; Wang, Chu, & Mao, 2008b, 2009; Xie & Wang, 2008), a periodic review condition monitoring system, also known as offline monitoring. In offline monitoring, the system’s condition is retrieved by inspection at certain pre-determined time intervals. In the mentioned papers, the system deterioration is mod-eled using Markov chains and probability distributions. Technology advances make it possible to conduct continuous condition monitoring where degradation data is retrieved instantaneously via sensor technology (online monitoring). This increases the accuracy of condition monitoring and can, therefore, improve the resulting maintenance strategy.
We develop a Monte Carlo simulation model to assess the performance of the proposed policy. A local search heuristic is used to find the best decision variables, which are two deterioration thresholds for maintenance actions, and a reorder level (r) and fixed order quantity (Q) for spare part control. The structure of the remainder of this thesis is as follows. In section 2, we discuss relevant literature. In section 3, we discuss the methodology. In section 4, we present the results and two extensions. In section 5, we discuss the findings and end our thesis with a conclusion
2
Literature review
We start the literature review by explaining basic maintenance strategies and condition-monitoring concepts, and how researchers apply these strategies to wind farms (§2.1). In section §2.2, we provide a short overview of relevant spare part inventory policies. This is followed by a discussion on joint spare part and maintenance literature in §2.3. In §2.4, we discuss joint spare part and maintenance literature specific for OWFs and summarize the contributions.
2.1
Maintenance strategies
Maintenance strategies can be divided into two broad categories: corrective and preventive. Cor-rective maintenance is a reactive approach also known as run-to-failure. After a system fails, maintenance is carried out to restore system functionality. The goal of preventive maintenance is to execute maintenance to prevent failure and increase availability. Preventive maintenance can be further divided into sub-strategies, such as time-based preventive maintenance (TBM), where maintenance actions are based on pre-determined time intervals, or CBM, where maintenance ac-tions are based on information about the current state of the system collected through condition monitoring (CM) (Ahmad & Kamaruddin, 2012). CM consists of collecting data on, for example, visual inspections, vibrations, acoustics or oil analysis (Tian et al., 2011), and can be categorized as either offline/periodic or online/continuous monitoring. With offline monitoring, condition feedback is often delayed because an expert’s interpretation is required (for example, in the case of a physical visual inspection), while with online monitoring, condition feedback is instantaneous (Scarf, 2007). The choice between the two types of monitoring influences the maintenance strategy. With peri-odic monitoring, inspection costs that should be controlled in the maintenance strategy by setting appropriate inspection intervals are usually included. In contrast, continuous monitoring requires a more flexible and adaptive approach, as maintenance actions are triggered continuously instead of only at inspection periods. Continuous monitoring usually requires more advanced and expensive equipment than periodic monitoring, but total cost reductions are possible when the maintenance policy improves.
High setup costs often exist for conducting maintenance on OWFs because accessing the offshore location requires expensive transportation. Opportunistic maintenance is useful for multi-unit systems with high setup costs. With opportunistic maintenance, maintenance actions on certain system parts are used as an opportunity to also conduct maintenance on other system parts with possibly reduced setup costs. This can apply to different components within a single turbine or
across multiple wind turbines in single or multiple wind farms. Besnard, Patriksson, Str¨omberg, Wojciechowski, and Bertling (2009) propose a model that uses corrective maintenance and weather forecasts with low wind levels as opportunities to conduct time-based preventive maintenance, resulting in significant cost savings. F. Ding and Tian (2012) also model opportunistic maintenance with time-based preventive maintenance actions for a wind farm and include multiple-level imperfect maintenance thresholds.
Besides time- or age-based preventive maintenance, researchers have also extensively studied condition-based maintenance. This research has ranged from condition-based maintenance strate-gies for single components of wind turbines (Besnard & Bertling, 2010; Shafiee, Finkelstein, & B´erenguer, 2015) to CBM for single turbine systems (Y. Ding, Ntaimo, & Byon, 2010). However, when multi-unit systems are considered, economic dependencies between units arise. It might, for example, be more cost efficient to conduct maintenance on one turbine when maintenance is conducted on another turbine. When only a single unit is considered, a different decision regard-ing when to conduct maintenance is made. Tian et al. (2011) therefore include this economic dependency in their model for CBM for a multiple-turbine wind farm under continuous condition monitoring. Yildirim et al. (2017) address the same issue by combining opportunistic maintenance and CBM in their O&M optimization model for wind farms. They use real-life sensor data for con-dition monitoring, which was formerly not used in multiple-turbine CBM research. This thesis will also include CBM, grouping maintenance for multiple turbines and the use of continuous condition monitoring based on real-life degradation data. Moreover, this will be extended by including spare parts into the maintenance strategy.
2.2
Spare part management
2.3
Joint spare part and maintenance policies
In addition to considering spare part control and maintenance strategy as separate issues, O&M strategies can consider them simultaneously and interactively, which is called joint spare part and maintenance policies. For a general review on joint spare part and maintenance policies, see Van Horenbeek et al. (2013). Joint spare part and maintenance policies relevant for this thesis consider based maintenance. Kawai (1983) first published on the topic of combining condition-based maintenance and spare part management by modeling a continuous time Markovian degra-dation system and developing an optimal ordering and replacement policy assuming only a single part can be kept in inventory. Wang, Chu, and Mao (2008a) propose a model that jointly optimizes the condition-based maintenance and spare order management for a single-unit system. The sys-tem is stochastically and gradually deteriorating and is under periodic inspection. Louit, Pascual, Banjevic, and Jardine (2011) also consider a single-unit system under periodic inspection but based the ordering of spare parts on the remaining useful life of the system. Rausch and Liao (2010) and Zhao and Xu (2012) use continuous monitoring instead of periodic inspection for their CBM policy for single-unit systems.
Some systems consist of multiple identical (like wind farms) or non-identical units instead of a single unit1. Literature on joint CBM and spare parts policies for multi-unit is rather scarce. Until the most recent paper by Zhang and Zeng (2017), only four papers consider this joint problem (Olde Keizer et al., 2017). Wang et al. (2009) extend their earlier work and propose a policy for a multiple identical unit system, combining the (S, s) inventory policy with condition-based maintenance. Xie and Wang (2008) also propose a multiple identical unit model and include non-deterministic lead times. Olde Keizer et al. (2017) model a joint condition-based maintenance and inventory optimization for a system with just-in-time ordering and multiple (non-identical) components. Zhang and Zeng (2017) propose a (S − 1, S) inventory policy and are the first to include opportunistic maintenance. All these multi-unit models use periodic inspection for the condition monitoring and model the degradation using Markov chains and probability distributions. In contrast, we will develop a joint spare part and condition-based maintenance policy for multi-unit systems with a continuous monitoring system and real-life degradation data. It will also include opportunistic maintenance like Zhang and Zeng (2017), but we will extend the inventory control further by considering a more advanced (r, Q) policy instead of the base-stock (S − 1, S) policy.
We furthermore propose an extension where the (r, Q) inventory policy is replaced by two dete-rioration thresholds. These thresholds will be used to control the timing and quantity of the spare part orders. This will create a just-in-time spare part policy, which uses the systems deterioration to order spare parts. Several researchers have studied and proven that ordering spare parts based on deterioration thresholds is optimal for single component systems (e.g. Kawai, 1983; Rausch & Liao, 2010; Zhao & Xu, 2012). Olde Keizer et al. (2017) were the first to combine CBM and spare part ordering based on system degradation for multi-component systems and refer to it as a ”just-in-time” (JIT) spare part ordering policy. Due to economic dependencies between units (such as fixed ordering and setup costs), they note spare part ordering with thresholds based on the
deteri-1note that both the terms multi-unit and multi-component are used for this type of research. As long as only one
of them is considered these terms are interchangeable. No literature in this section uses a multi-unit system with multi-components, although this might be interesting for wind farms (multiple turbines with multiple components inside).
oration of single units within a multi-unit system is not necessarily optimal. In single-unit systems, normally one threshold is used to order spare parts. To capture some of the economic dependencies of a multi-unit system, we will use 2 thresholds instead. We will compare the performance of this JIT policy to the joint policy with the (r, Q) inventory policy.
2.4
Maintenance and spare part management for OWFs
Previously discussed literature on spare parts and the joint optimizing with maintenance does not focus on OWFs. Because special resources and weather conditions alter spare part demand (Jin et al., 2012) and maintenance decisions, rather limited research has focused on integrating spare part and maintenance management for the offshore wind industry. Lindqvist and Lundin (2010) analyze the results of a simulation model for different spare part policies and find that inventory pooling and an extra central storage facility are more profitable than only local storage and handling. Dewan (2014) develops a simulation model for OWF maintenance and logistic support organization in which different spare part policies (r, Q) and (S − 1, S) were compared. M¨unsterberg (2017) develops the most comprehensive simulation for spare parts and maintenance logistics for OWFs and includes specific maintenance characteristics of OWFs, as well as weather and wave conditions. The author models maintenance actions that follow different predetermined annual frequencies.
The above theses all use simulation as a tool for addressing the problem. Two other studies also focus on joint spare part and maintenance for OWFs, but the researchers use mathematical modeling instead of simulation. Jin et al. (2012) develop a mathematical model for joint spare part and maintenance optimization for OWFs using an age-based preventive maintenance policy in which a third-party provides maintenance tasks and the wind farms owners own the spare parts. Tracht et al. (2013) develop a mathematical model for repairable spare parts planning, taking into account the restrictive maintenance conditions of OWFs, such as limited resources and weather and sea conditions. None of the studies in this section consider opportunistic maintenance or condition-based maintenance. In contrast, we develop a joint spare part and opportunistic condition-condition-based maintenance policy, including weather and resource constraints that are special to offshore wind farms. This will be our first contribution, alongside the earlier mentioned contributions to joint spare part and condition-based literature by including continuous condition monitoring based on real-life degradation data and opportunistic maintenance with an (r, Q) inventory policy.
3
Methodology
3.1
Setting
turbine goes beyond level d2, maintenance is scheduled on all wind turbines with a deterioration level beyond d1. Depending on different constraints, scheduled maintenance actions are performed. Figure 1 shows a schematic overview of the process of finding the best joint policy. A Monte Carlo simulation is used to find the average costs of a specific set of (d1, d2, r, Q) parameters. To find the best parameters, the spare part and maintenance policies are first solved sequentially. This enables us to reduce the search space for determining the joint policy. Furthermore, it allows us to directly compare the results of the joint and sequential policy. We search for values of d1 and d2 that result in lowest costs, while assuming an infinite amount of spare parts and excluding spare part costs. After finding those values, d1 and d2 are fixed to the best-found values and we search for values of r and Q that result in lowest costs. The four parameters found by solving the problem sequentially are used as a starting point and for appropriate boundary setting for a local search heuristic that searches for the best joint (d1, d2, r, Q) policy.
In the following sections, we compare the best sequential and joint policy. In the sequential policy, the maintenance policy is determined first. Based on the resulting maintenance actions, the best spare part policy is determined. This is a logical sequence, as spare part policies are normally determined based on the demand for spare parts (which in this case is equal to the maintenance actions). Determining a maintenance policy based on the spare part policy is, aside from in joint policies, uncommon. The joint policy will consider the maintenance and spare part policy simul-taneously by searching for combinations of all four (d1, d2, r, Q) parameters. The spare part and maintenance policies, therefore, influence each other, while in the sequential policy only the main-tenance policy can influence the spare part policy and not vice versa. We should note that with the discussed methodology, no guarantee for optimality can be given. In light of reducing maintenance costs by jointly considering maintenance and spare parts, however, such a “near optimal” approach seems practical (Wang et al., 2009). We programmed the model in Python and ran it on a 2.4 GHz Intel Core i5 processor with 4GB RAM.
Joint consideration of maintenance decisions and spare part control is computationally complex, as the decisions around the amount of spare parts influence the maintenance decisions and vice versa. Because of this complexity, simulation is valuable for the joint optimization of maintenance and spare parts (Hu et al., 2018). Three out of the five papers on multi-unit joint policies use Monte Carlo simulation (Wang et al., 2008b, 2009; Xie & Wang, 2008). Three out of five papers also use a search heuristic to find best decision variables(Wang et al., 2008b, 2009; Zhang & Zeng, 2017). Olde Keizer et al. (2017) are the only ones to use a full numerical approach by modeling the policy as a Markov Decision Process. They solve their model for one until 6 units. However, as the condition of every unit needs to be tracked fully, larger systems are complex and computationally time consuming (Olde Keizer et al., 2017). Using this method for the large number of wind turbines in a modern wind farm would likely be unsuitable looking at the exponentially increasing state space and computational times required. Other aspects important to OWFs, such as resource constraints, weather condition dependency and opportunistic maintenance, would further complicate the use of this methodology. Therefore, although no optimality can be guaranteed, this thesis uses a combination of simulation and a search heuristic.
Fix r and Q to large number
Range inputs [lb, ub] for each paramater
[d1, d2]
Find costs for every (d1, d2) combination Run simulation run ≤ runs Average costs of runs Choose (d1, d2)
with lowest costs
Maintenance policy
Fix d1 and d2 to best found
Range inputs [lb, ub] for each paramater
(r, Q)
Find costs for every (r, Q) combination Run simulation run ≤ runs Average costs of runs Choose [r, Q]
with lowest costs
Spare part/Sequential policy
Range inputs [d1 − lbd, d1 + ubd] [d2 − lbd, d2 + ubd] [r − lbd, r + ubd] [Q − lbd, Q + ubd]
Find costs for every (d1, d2, r, Q) combination Run simulation run ≤ runs Average costs of runs Choose (d1, d2, r, Q)
with lowest costs
Joint policy yes no yes no yes no
lb = lower bound of range to test a parameter on ub = upper bound of range to test a parameter on
lbd = distance from best found value to lower bound of the range to test a parameter on ubd = distance from best found value to upper bound of the range to test a parameter on
Figure 1: Flowchart of finding the joint policy with lowest costs
3.2
Model
3.2.1 Notation
N Total number of turbines in the wind farm T Simulation time run length
t Time period t
n Wind turbine n
d1 Failure probability maintenance threshold 1 (opportunistic maintenance) d2 Failure probability maintenance threshold 2 (preventive maintenance) Fn Current failure age of turbine n
Gn Current age of turbine n
Pn,t Predicted failure probability of turbine n at time t
mo
n,t Scheduled opportunistic maintenance on turbine n at time t (binary)
mpn,t Scheduled preventive maintenance on turbine n at time t (binary) mcn,t Scheduled corrective maintenance on turbine n at time t (binary) Mo
n,t Performed opportunistic maintenance on turbine n at time t (binary)
Mn,tp Performed preventive maintenance on turbine n at time t (binary) Mn,tc Performed corrective maintenance on turbine n at time t (binary)
Ut Maintenance actions performed at time t (binary)
r Reorder inventory point for spare parts Q Order quantity for spare parts
S Currently available spare parts I Current spare part inventory position L Lead time for spare parts in time periods Ht Spare part inventory at the end of time t
At Spare part arrivals at time t
Wt Weather conditions are feasible to conduct maintenance at time t (binary)
Vt MW produced by a running wind turbine during time period t
Rc Amount of resources available each time period (resource constraint)
Ra Amount of resources currently available
co Fixed costs for an opportunistic maintenance action
cp Fixed costs for a preventive maintenance action
cc Fixed costs for a corrective maintenance action
cm Fixed setup costs for performing a group of maintenance actions
ci Holding costs per spare part per time unit cs Fixed ordering costs for each order of spare parts cd Downtime costs for not producing a MW due to failure
oo Output variable for total amount of opportunistic maintenance actions op Output variable for total amount of preventive maintenance actions oc Output variable for total amount of corrective maintenance actions om Output variable for total amount of setups
oi Output variable for average time period inventory level
os Output variable for total amount of spare part orders
od Output variable for total MW not produced because of failures
oa Output variable for percentage availability of the turbines
Cd Average daily total downtime (opportunity) costs for a policy
Cs Average daily spare part costs for a policy
Cm Average daily maintenance costs for a policy
3.2.2 Scheduling maintenance actions
Each time period, we schedule maintenance actions based on the failure probability of the wind turbines. We treat the predicted failure probability of wind turbine n at time t (Pn,t) and the
failure age of wind turbine n (Fn) as input to the model. §3.3.1 explains the procedure as to
how the predicted failure probability is obtained. Components are failed if the current age of the component Gnis equal or larger than the failure age (Fn), and corrective maintenance is scheduled.
mcn,t= (
1, if (Gn≤ Fn)
0, otherwise (1)
Preventive maintenance actions are based on the failure probability of the wind turbines and the two thresholds d1 and d2. If the failure probability is between d1 and d2, opportunistic maintenance is scheduled. If it is larger than d2, preventive maintenance is scheduled.
mon,t= ( 1, if (d1 ≤ Pn,t< d2) 0, otherwise (2) mpn,t ( 1, if (Pn,t ≥ d2) 0, otherwise (3)
3.2.3 Spare part and resource control
Spare part inventory is controlled by the (r, Q) policy. If the inventory position (I) drops below the reorder level r, order size Q is added to the current inventory position (I). Spare parts orders arrive after the spare part lead time (At+L= 1).
At+L= ( 1, if (I ≤ r) 0, otherwise (4) I = ( I + Q, if (I ≤ r) I, otherwise (5)
At the beginning of each time period, the current available inventory (S) is increased by the spare part arrivals at that time (At). The available resources (Ra) are reset to the total number
S = S + (At∗ Q) (6)
Ra = Rc (7)
At the end of each time period, the amount of spare parts left in inventory is registered in Ht.
This amount is needed for calculating holding costs.
Ht= S (8)
3.2.4 Performing maintenance actions
Whether a scheduled maintenance action is performed depends on the maintenance constraints. First, the weather conditions should be feasible. We treat the feasibility of the weather at time t (Wt)
as an input to the model. §3.3.2 explains the procedure as to how feasibility of weather is determined. As every maintenance action requires one spare part and one maintenance resource, there should be at least one spare part and one maintenance resource left. Opportunistic maintenance actions are only performed if at least one preventive maintenance action is also performed at the current time. To give more critical maintenance actions priority, we schedule corrective maintenance actions first, followed by preventive and then opportunistic actions.
Mn,tc = ( mcn,t, if (Wt= 1 and Ra> 0 and S > 0) 0, otherwise (9) Mn,tp = ( mpn,t, if (Wt= 1 and Ra> 0 and S > 0) 0, otherwise (10) Mn,to = ( mo
n,t, if (Wt= 1 and Ra> 0 and S > 0 and P N n=1M
p n,t> 0)
0, otherwise (11)
After each decision to perform a maintenance action, the current available inventory (S), inven-tory position (I) and available resources level (Ra) are updated.
Ra= ( Ra− 1, if (Mn,to = 1 or M p n,t or Mn,to = 1) Ra, otherwise (14) 3.2.5 Output
To assess the performance of the policy, output variables are calculated at the end of the simulation run time. We first sum the number of the three types of maintenance actions during the simulation.
oo= N X n=1 T X t=1 (Mn,to ) (15) op= N X n=1 T X t=1 (Mn,tp ) (16) oc= N X n=1 T X t=1 (Mn,tc ) (17)
To calculate the number of maintenance setups, we first determine if a maintenance action was performed or not (Ut) for all time periods. We then sum up this variable for the simulation run
time. Ut= 1, if ( N P n=1 (Mo n,t+ M p n,t+ Mn,tc ) > 0) 0, otherwise ∀t ∈ T (18) om= T X t=1 Ut (19)
We calculate the average daily end inventory in oiand later use this to calculate holding costs.
We sum up the amount of spare part orders in os.
os=
T
X
t=1
(At) (21)
If scheduled corrective maintenance actions were not performed due to a constraint, it means the turbine was in a failed stated. We treat the number of MW a running turbine can produce at time t (Vt) as input to the model (see §3.3.2). By subtracting mon,t by Mn,to , we determine if a
turbine was scheduled for corrective maintenance, but it was not performed, and total downtime can be calculated. od= N X n=1 T X t=1 ((mcn,t− Mc n,t)Vt) (22)
We calculate system availability by examining the difference between scheduled and performed corrective maintenance actions. We divide the sum of failed turbines at all time periods by the total number of turbines times the total amount of time periods. We then subtract the outcome from 1 to arrive at the percentage availability of the turbines.
oa= 1 − N P n=1 T P t=1 (mcn,t− Mc n,t) T ∗ N (23)
We divide total costs into maintenance (Cm), downtime (Cd) and spare part costs (Cs). We
3.2.6 Flow diagram simulation
The flow chart in figure 2 summarizes the working of the simulation model (explained previously in §3.2.2-§3.2.5).
Start; initialize parameters
t = t + 1 n = 0 n = n + 1 Gn≤ Fn mc n,t= 1 d1 ≤ Pn,t< d2 mpn,t= 1 Pn,t≥ d2 mo n,t= 1 n ≤ N Schedule maintenance S = S + (At∗ Q) Ra = Rc I ≤ r At+L= 1 I = I + Q Set resources, spare parts policy
n = 0 n = n + 1 (Wt= 1 Ra> 0 S > 0) Mn,tc = mcn,t Mc n,t= 1 S = S − 1 I = I − 1 Ra = Ra− 1 n ≤ N Perform corrective maintenance n = 0 n = n + 1 (Wt= 1 Ra> 0 S > 0) Mn,tp = m p n,t Mn,tp = 1 S = S − 1 I = I − 1 Ra = Ra− 1 n ≤ N (PN n=1M p n,t > 0) Perform preventive maintenance n = 0 n = n + 1 (Wt= 1 Ra> 0 S > 0) Mn,to = mon,t Mo n,t= 1 S = S − 1 I = I − 1 Ra = Ra− 1 n ≤ N Perform opportunistic maintenance Ht= I t ≤ T Output results yes no yes no yes no yes no yes no yes no yes no no yes yes no yes no yes no yes no yes no yes no yes no yes no
3.3
Data and parameters
We use different types of data and parameters as inputs to the simulation model. §3.3.1 explains degradation data and §3.3.2 describes weather data. A flowchart in §3.3.3 shows the interaction between the simulation model and these two types of data. §3.3.4 provides an overview of the base case input parameters.
3.3.1 Degradation data
To simulate the degradation of the turbines, we use real-life degradation data. Rotating machinery with rolling element bearings are run from a new state to failure, and data continuously retrieved from vibration sensors is transformed into predicted failure probabilities using methods described in Yildirim et al. (2017). For each period during the machine’s lifetime, we calculated the failure probability for the current time period using the most recent sensor output. This resulted in 40 samples with pre-processed failure probabilities with a time interval of 48 hours. At the initialization of the simulation and after each maintenance action, we selected a random sample to simulate the turbine’s failure probability predictions (Pn,t) and failure age (Fn).
3.3.2 Weather data
We use weather data for years 2004 to 2018 from the FINO 1 project2. FINO 1 is an offshore research platform located 45 km offshore of Borkum, Germany. The platform registers a wide selection of meteorological measurements, such as wind speed, wind direction, temperature, atmospheric pressure and humidity. It also records oceanographic measurements, including water level, water temperature and current and wave height. We only use wind speed measurements at a height of 90 meters and significant wave heights registered by buoys. Days with incorrect or incomplete data are removed.
The wind speeds are pre-processed into produced MW per day. This is based on the power curve of wind turbines with a cut-in wind speed of 3 m/s, a rated wind speed of 15 m/s, a cut-out wind speed of 25 m/s and rated power output of 2 MW. To do this pre-processing, we use the methods described by Giorsetto and Utsurogi (1983). To constrain the maintenance on weather conditions, we calculate the daily average significant wave height. We set the maximum average daily significant wave height at which maintenance could be performed according to a reference catamaran access vessel, which is 1.5m (Dewan & Asgarpour, 2016). If the average significant wave height of time period t is below the maximum wave height, Wt= 1 and otherwise Wt = 0. This
means weather is feasible for maintenance actions 63% of the simulation run time (5 years).
2Made available by BMWi (Bundesministerium f¨ur Wirtschaft und Energie) and PTJ (Projekttr¨ager J¨ulich) via
http://fino.bsh.de
3.3.3 Data processing for simulation
The flow chart in figure 3 gives a schematic overview of the interaction between the data and the simulation model (as previously explained in §3.3.1-§3.3.2).
Vibration data from rotating machinery
Wind speed FINO 1 project 2004-2018
Significant wave height FINO 1 project 2004-2018
Raw data
40 samples of failure probabilities at all ages until failure
MW produced per turbine per day
Feasibility of weather per day Pre-processed data Initia-lization Maintenance performed New random sample Selected samples for all turbines
Simulation Simulation
VtWt
Pn,t
Fn
Figure 3: Data processing for simulation
3.3.4 Base case values
The simulation is run for 5 years with a burn-in period of 1 year. With the burn-in period, therefore, the simulation run time for the output variables is 1,460 days (4 years). The wind farm consists of N = 80 turbines. We adapted cc = $12, 000, cp = co= $4, 000 and cs = $32, 000 from Yildirim
et al. (2017). The downtime costs per unproduced MW are cd = $100. We assume spare part
lead time L to be 3 weeks (21 time periods) and we set the resource constraint (maximum allowed maintenance actions per time period) to 4 (Rc= 4). We assume holding costs to be ci= $500 and
ordering costs cs= $20, 000 per order. In the sensitivity analysis the impact of changing most of
these values will be examined.
4
Results
policy. §4.4 compares the performance of the sequential and joint policy to a benchmark model. §4.5 describes the impact of different input parameters on the best benchmark, sequential and joint policy. §4.6 extends the model by changing maintenance thresholds based on upcoming weather conditions. Finally, §4.7 extends the model by ordering spare parts based on the deterioration of the turbines.
4.1
Maintenance policy
To find the combination of (d1, d2, r, Q) parameters that result in the lowest costs, we first consider only the maintenance decisions. We accomplish this by setting r and Q to a large number so they cannot affect any maintenance decisions. Maintenance plus downtime costs (Cm + Cd) are used
as output function, so spare part costs are excluded. We search a range of d1/d2 combinations on a logarithmic scale. The results are plotted in figure 4. Lowest costs of $5,987 are found at d1 = e−20 and d2 = e−5. Table 1 shows the output results for different d1 values with d2 fixed to the best-found value and vice versa.
For parameter d1, the best value is a trade-off between amount of maintenance setups (om) and the number of opportunistic (oo) and preventive maintenance actions (op). Increasing the threshold results in fewer opportunistic maintenance actions but increases the overall costs because of the more preventive, corrective and setup actions required. Decreasing the threshold increases the number of opportunistic actions, but the extra costs for doing more maintenance actions are higher than the small positive effect on the amount of preventive, corrective and setup actions. For parameter d2, there is a trade-off between reducing the amount of failures (oc) and increasing
the number of preventive maintenance actions. Increasing the threshold results in fewer preventive actions, but the increase in number of failures is costlier. Decreasing the threshold increases the amount of preventive maintenance actions, but this cost increase is higher than the savings from the relatively small reduction in failures.
(a) d1 vs d2 vs costs (b) d1 vs costs (c) d2 vs costs
Figure 4: Average costs for a range of d1/d2 (ex) combinations
d1 (ex) d2 (ex) oo op oc om od Cm+ Cd 0 -5 0 508 39 324 11,531 $9,704 -10 -5 269 290 29 208 9,364 $7,061 -20 -5 391 217 22 164 7,065 $5,987 -30 -5 431 224 19 171 6,174 $6,165 -40 -5 452 229 18 176 5,425 $6,298 d1 (ex) d2 (ex) oo op oc om od Cm+ Cd -20 -1 329 150 91 170 15,638 $7,091 -20 -5 391 217 22 164 7,065 $5,987 -20 -9 372 276 15 186 4,898 $6,354 -20 -13 315 359 11 228 3,743 $7,232 -20 - 17 177 516 9 307 2,951 $8,932
Table 1: Average costs for different d1 values with d2 fixed and vice versa
4.2
Spare part policy
In §4.1, we found the d1/d2 combination that resulted in the lowest maintenance costs. To find the best spare policy for this maintenance policy, d1 and d2 are fixed to the best-found values (d1 = e−20, d2 = e−5) and we search for the r/Q combination that results in lowest total costs (C). Figure 5 shows the results. Lowest costs of $9,528 are found with r = 4 and Q = 12. As the resource constraint is set to 4 per time period, the best policy has a reorder point at an inventory level enough for one maintenance setup with maximum maintenance actions. The best order quantity is a multiple of 4 — enough for three full setups. The graphs show a stuttered relation between r/Q and costs, for example, with a sharp peak in costs for Q = 13. This is also the result of the resource constraint. As there is an incentive to do maintenance setups including four maintenance actions, the extra one inventory increases costs because this can only be used for setups with fewer maintenance actions. It furthermore increases the holding costs.
(a) r vs Q vs costs (b) r vs costs (c) Q vs costs
Figure 5: Average costs for a range of r-Q combinations
maintenance actions closer to the one found in the best maintenance policy (§4.1). However, the increase in holding costs is stronger and therefore costs increase. The order quantity (Q) has the same effect on the maintenance actions. It results, furthermore, in a trade-off between holding costs and the number of spare part orders. The average inventory level in all cases might be lower than expected from the corresponding reorder level and order quantity. This indicates that all spare parts are consumed soon after their arrivals. Only after the reorder level is reached and the spare part lead time has passed, inventory is replenished and the inventory level increases temporarily.
r Q oo op oc om oi os od C 2 12 194 226 136 143 2.18 46 45,332 $10,606 4 12 281 233 75 155 3.35 49 22,622 $9,529 6 12 291 232 69 156 3.71 49 20,558 $9,557 8 12 342 226 43 163 5.65 51 13,790 $10,070 10 12 353 224 39 163 6.54 51 12,245 $10,398 r Q oo op oc om oi os od C 4 9 81 214 212 169 1.25 56 75,976 $13,444 4 12 281 233 75 155 3.35 49 22,622 $9,529 4 15 294 240 63 164 4.61 40 19,282 $9,924 4 18 303 243 56 169 5.78 33 17,811 $10,378 4 21 307 241 55 172 6.61 29 17,084 $10,728
Table 2: Output for different r values with Q fixed and vice versa
4.3
Joint policy
Until now, the maintenance and spare part policy have been considered separately. In the last step, we search for combinations of all four parameters (d1, d2, r, Q). The four parameters can now interact with each other, and both the maintenance and spare part policies are considered jointly. The values found in §4.1 and §4.2 are used as starting points, and combinations of neighbor values are tested. Table 3 shows these results. Lowest costs of $8,598 were found at d1 = e−35, d2 = e−15, r = 5 and Q = 12. This means the joint policy has a cost reduction of 8.64% compared to the best-found sequential policy. The availability of the system increases from 99.36% to 99.63%. In both cases, the low spare part inventory restricts the number of maintenance actions that can be performed and thereby influences the amount of failures. The joint policy adapts to this restriction by taking less risk on the maintenance policy. Both maintenance thresholds are decreased, and opportunistic and preventive maintenance actions are scheduled earlier and more often. With approximately the same inventory policy, this resulted in more spare part demand, lower average inventory and fewer failures. The resulting decrease in downtime, corrective maintenance and holding costs improves the joint policy performance.
Policy d1
(ex) (ed2x) r Q oo op oc om oi os od oa C ∆%
Joint -35 -15 5 12 227 395 48 169 2.27 56 12,816 99.63% $8,725
Sequential -20 -5 4 12 281 233 74 155 3.42 49 22,455 99.36% $9.551 -8.64%
Table 3: Results of the best joint and sequential policy
4.4
Benchmark policy
To compare the performance of the sequential and joint policy to a policy without condition-based maintenance, we introduce a benchmark age-based maintenance policy. In this policy, preventive maintenance is scheduled whenever a turbine reached age bp and opportunistic maintenance is scheduled on all turbines with an age higher than bp− bo. This makes the benchmark policy similar
to the sequential and joint policy but with maintenance based on turbines ages instead of failure probabilities. We search a range of bp and bo values to find the combination that results in the
lowest Cm. Based on the found maintenance policy, we search for the best (r, Q) policy, and the
benchmark policy with the lowest costs C is determined. Table 4 shows the results. The best benchmark policy with lowest costs of $15,413 is found at bo = 30, bp= 180, r = 7, q = 12. The
joint policy has a cost reduction of 43.39% compared to the benchmark policy, which can be mostly contributed to reduced failures. Because the condition-based maintenance policy uses information on the current degradation of each individual turbine, failures can be prevented more effectively, and a large cost reduction is possible.
Policy bo d1 (ex) bp d2 (ex) r Q oo op oc om oi os od oa C ∆% Bench. 30 180 7 12 68 320 283 196 2.39 56 72,763 97.81% $15,413 -43.39% Seq. -20 -5 4 12 281 233 74 155 3.42 49 22,455 99.36% $9,551 -8.64% Joint -35 -15 5 12 227 395 48 169 2.27 56 12,816 99.63% $8,725
Table 4: Results of best benchmark, sequential and joint policy
4.5
Sensitivity analysis
In the sensitivity analysis, we investigate the impact of all cost parameters on the performance of the best benchmark, sequential and joint policy. §4.5.1 examines the impact of opportunistic/preventive maintenance costs, corrective maintenance costs and maintenance setup costs. In §4.5.2, we look at the impact of holding and spare part ordering costs, and in §4.5.4, of the downtime costs. In §4.5.3, we analyze the impact of the resource constraint, which is the maximum allowed maintenance actions per time period. In the result tables, we refer to the policies as B (benchmark policy), S (sequential policy) and J (joint policy).
4.5.1 Impact of maintenance costs
The costs for performing an opportunistic or preventive maintenance action are set equal. Table 5 shows the impact of different co,cp values on the best sequential and joint policy. When the
costs increase, a decrease in the sum of opportunistic and preventive maintenance (co,cp) actions is
downtime costs that results in lowest costs. Changing the costs for only the opportunistic and preventive maintenance actions alters this best-found mix relatively little.
The relative cost reduction between the joint and sequential policy reduces for higher co,cp.
The joint policy is able to perform better than the sequential policy by increasing the maintenance thresholds while keeping low inventory levels. This results in more preventive/opportunistic main-tenance actions, and when their costs increase, the benefit of the joint policy decreases. A similar effect is seen in comparison to the benchmark policy.
P co,cp b o d1 (ex) bp d2 (ex) r Q oo op oo+ op oc om oi os od oa C ∆% B $1,000 50 180 9 11 27 423 450 253 199 1.50 64 71,514 97.88% $13,977 -47.11% $2,000 70 190 7 12 66 331 397 282 194 2.24 57 75,100 97.74% $14,922 -47.81% $4,000 30 180 7 12 68 320 388 283 196 2.39 56 72,763 97.81% $15,413 -43.39% $12,000 30 180 7 12 69 317 385 285 196 2.40 56 74,168 97.77% $17,631 -31.56% S $1,000 -25 -5 7 11 386 186 573 56 122 3.27 57 16,158 99.54% $8,375 -11.73% $2,000 -20 -5 5 12 286 230 516 73 156 3.64 49 21,925 99.38% $8,912 -12.62% $4,000 -20 -5 4 12 281 233 515 74 155 3.42 49 22,455 99.36% $9,551 -8.64% $12,000 -15 -3 4 12 256 194 450 105 156 3.63 46 29,866 99.14% $12,761 -5.44% J $1,000 -35 -17 6 12 199 428 628 47 171 2.09 56 12,876 99.63% $7,392 $2,000 -35 -15 6 12 225 398 623 46 169 2.19 56 12,651 99.65% $7,787 $4,000 -35 -15 5 12 227 395 622 48 169 2.27 56 12,816 99.63% $8,725 $12,000 -40 -15 4 12 230 389 619 49 168 2.21 56 13,148 99.62% 12,066
Table 5: Impact of co,cp on best benchmark, sequential and joint policy
We will now look at the impact of corrective maintenance actions (cc) on the performance of the
three different policies. Table 6 shows the results. It is expected that the policies would decrease the amount of corrective maintenance actions/failures (oc) with increasing corrective maintenance costs
cc). The effect on the amount of corrective maintenance actions might be smaller than expected
looking at the results for cc=$4,000, $16,000 and $32,000. This can be contributed to downtime
costs that also occur with failures alongside cc. Increasing the amount of failures by doing less preventive maintenance actions is, therefore, less beneficial. A large increase to cc= $48, 000 makes the effect clearer. The reorder level r increases with higher cc, so the risk of failures because of spare part unavailability decreases. In the joint policy, the maintenance thresholds decrease for higher cc to reduce the risk of failures further. The cost reduction of the joint policy compared to the sequential policy increases for higher cc. The joint policy is more effective in reducing corrective
maintenance actions by increasing the maintenance thresholds while maintaining lower average inventory levels. Compared to the benchmark policy, the cost reduction of the joint policy also increases because it is again more effective in reducing failures.
P cc b o d1 (ex) bp d2 (ex) r Q oo op oc om oi os od oa C ∆% B $4,000 30 180 7 12 68 318 283 195 2.39 56 74,941 97.76% $13,203 -37.30% $16,000 30 180 7 12 68 320 283 196 2.39 56 72,763 97.81% $15,413 -43.39% $32,000 70 180 11 12 136 334 257 226 4.32 61 52,921 98.44% $18,480 -50.41% $48,000 60 120 6 16 35 633 207 220 2.48 55 55,455 98.36% $19,238 -50.65% S $4,000 -20 -5 4 12 281 234 74 156 3.38 49 22,274 99.36% $8,912 -7,10% $16,000 -20 -5 4 12 281 233 74 155 3.42 49 22,455 99.36% $9,551 -8.64% $32,000 -20 -5 6 12 292 232 68 156 3.72 49 19,606 99.44% $10,236 -10.47% $48,000 -20 -5 8 8 349 227 40 165 4.59 77 13,401 99.65% $10,767 -11.83% J $4,000 -40 -15 5 12 201 425 49 168 2.21 56 13,047 99.63% $8,279 $16,000 -35 -15 5 12 227 395 48 169 2.27 56 12,816 99.63% $8,725 $32,000 -35 -17 6 12 207 422 45 172 2.15 56 12,172 99.65% $9,165 $48,000 -45 -17 9 12 315 399 22 185 2.93 61 6,421 99.82% $9,493
Table 6: Impact of cc on best benchmark, sequential and joint policy
Setup costs occur when one or more maintenance actions are performed during a time period. To see how this cost parameter impacts the policies, different maintenance setup costs (cm) are
tested. Table 7 shows the results. We expect the amount of maintenance setups (om) to reduce
with higher cm. In the sequential policy, however, we observe only a change with cm = $4, 000.
This is because the resource constraint is set to four actions per maintenance setup. The average maintenance setup size with cm = $4, 000 is 3.53, while with cm=$16,000, $32,000 and $64,000 it
is 3.78. Because opportunistic maintenance actions are only triggered with scheduled preventive actions, the corrective maintenance actions cause an average of 3.78 maintenance actions per setup to be the maximum under these circumstances. Further increasing cm shows no significant effect
on the policy.
In the joint policy, the same effect appears with no significant change in policy and ombetween cm=$32,000 and $64,000. However, because the maintenance and spare part policies are considered jointly in this policy, the amount of corrective maintenance actions are reduced further. This causes the maximum average maintenance actions per setup to be close to 4, and the policy chooses to do fewer actions with cm=$4000 and $16,000. The relative cost reduction between the sequential
P cm b o d1 (ex) bp d2 (ex) r Q oo op oc om ω∗ oi os od oa C ∆% B $4,000 30 160 8 13 81 407 250 238 3.11 3.23 57 53,970 98.35% $10,821 -52.18% $16,000 30 189 8 12 108 319 268 221 3.15 4.22 58 55,578 98.36% $13,232 -48.62% $32,000 30 180 7 12 68 320 283 196 3.41 2.39 56 72,763 97.81% $15,413 -43.39% $64,000 30 180 7 12 69 318 281 195 3.43 2.47 56 72,757 97.80% $19,653 -37.15% S $4,000 -20 -11 4 12 216 348 58 176 3.53 2.88 52 17,247 99.51% $5,987 -13.58% $16,000 -20 -5 4 12 280 234 75 156 3.78 3.40 49 22,589 99.36% $7,852 -13.39% $32,000 -20 -5 4 12 281 233 74 155 3.78 3.42 49 22,455 99.36% $9,551 -8.64% $64,000 -20 -5 4 12 280 234 76 156 3.78 3.34 49 22,440 99.37% $12,944 -4.58% J $4,000 -45 -21 8 11 158 500 39 190 3.67 1.73 63 10,078 99.71% $5,174 $16,000 -45 -17 7 11 202 449 38 188 3.66 1.95 63 10,330 99.70% $6,799 $32,000 -35 -15 5 12 227 395 48 169 3.96 2.27 56 12,816 99.63% $8,725 $64,000 -40 -15 4 12 220 401 49 168 3.99 2.15 56 13,192 99.63% $12,352
Table 7: Impact of cmon on best benchmark, sequential and joint policy ∗ω=(oo+ op+ oc)/om
4.5.2 Impact of spare part costs
We are now going to look at the impact of spare part costs. We first search for the best policies with different holding costs (ci). Table 8 shows the results. In the sequential policy, d1 and d2
are determined without considering spare parts, so the sequential policy cannot adapt these two parameters to changing spare part costs. Instead, r and Q both decrease for increased holding costs. This causes the average inventory level oito decrease. Meanwhile, this decrease in available
inventory increases the number of failures and downtime.
With relatively low holding costs of ci=$100, we do not observe a significant difference between
the sequential and joint policies. The portion of holding costs in total costs is not significant enough to change the best-found maintenance thresholds in the sequential policy anymore. For ci=$250,
$500 and $1000, the spare part policy is identical, but the average inventory is reduced for higher ci by decreasing the maintenance thresholds. This increases the amount of maintenance actions and thereby the demand for spare parts. With an unchanged spare part policy, this reduces the average inventory oi while the amount of failures remains stable. This results in a strong increasing relative cost reduction of the joint policy compared to the sequential policy with higher holding costs. At ci=$1000, this reduction increases to 12.55%. For the relative cost reduction between the joint and
benchmark policy, we observe a small decrease because the portion of holding costs in total costs increases and the average inventory level is closer for these two policy.
P ci b o d1 (ex) bp d2 (ex) r Q o o op oc om oi os od oa C ∆% B $100 30 180 13 14 151 312 254 253 8.30 51 38,420 98.94% $13,759 -44.36% $250 30 180 11 12 129 313 262 231 4.95 59 49,878 98.54% $14,598 -44.39% $500 30 180 7 12 68 320 283 196 2.39 56 72,763 97.81% $15,413 -43.39% $1000 30 180 7 12 68 317 282 195 2.41 56 74,196 97.78% $16,681 -42.01% S $100 -20 -5 9 15 354 229 33 167 7.49 41 10,506 99.73% $7,662 -0.08% $250 -20 -5 8 12 341 226 43 163 5.63 51 13,140 99.65% $8,594 -5.54% $500 -20 -5 4 12 281 233 74 155 3.42 49 22,455 99.36% $9,551 -8.64% $1000 -20 -5 5 11 236 246 93 161 2.59 52 27,459 99.20% $11,061 -12.55% J $100 -20 -5 10 16 372 223 29 165 9.06 39 9,567 99.77% $7,656 $250 -35 -13 5 12 248 365 48 167 2.36 55 13,190 99.63% $8,118 $500 -35 -15 5 12 227 395 48 169 2.27 56 12,816 99.63% $8,725 $1000 -40 -17 5 12 199 429 47 170 2.01 56 13,485 99.61% $9,673
Table 8: Impact of ci on best on best benchmark, sequential and joint policy
For every spare part order made, fixed ordering costs occur. To understand the impact of this cost parameter, we search for the best policies with different ordering costs (cs). Table 9 shows
the results. For both the sequential and joint policy, the effect of cs on the amount of spare part
orders (os) is low for cs=$5,000, $20,000 and $50,000. The incentive for ordering in multiples of
four because of the resource constraint (as discussed in §4.5.2) causes Q to stay at 12. The relative low contribution of ordering costs to total costs cause the policies to be relatively insensitive. With a large increase to cs=$100,000, both policies changed to a Q of 16. The other parameters adapt to
P cs b o d1 (ex) bp d2 (ex) r Q o o op oc om oi os od oa C ∆% B $5,000 30 180 9 4 79 322 279 202 1.90 170 63,742 98.11% $14,489 -44.08% $20,000 30 180 7 12 68 320 283 196 2.39 56 72,763 97.81% $15,413 -43.39% $50,000 30 180 7 12 70 317 196 283 2.42 56 73,532 97.79% $16,602 -41.19% $100,000 30 180 7 12 69 318 197 285 2.45 56 72,948 97.80% $18,546 -38.28% S $5,000 -20 -5 6 12 293 232 69 157 3.57 49.49 20,694 99.41% $9,009 -10.07% $20,000 -20 -5 4 12 281 233 74 155 3.42 49.05 22,455 99.36% $9,551 -8.64% $50,000 -20 -5 4 12 281 233 75 156 3.41 49.07 22,679 99.35% $10,674 -8.52% $100,000 -20 -5 4 16 320 235 52 162 5.27 37.58 15,685 99.58% $11,931 -4.07% J $5,000 -40 -15 7 12 228 391 49 168 2.23 55.61 12,894 99.63% $8,102 $20,000 -35 -15 5 12 225 397 46 169 2.21 55.58 12,059 99.66% $8,598 $50,000 -35 -15 5 12 227 395 48 169 2.27 56 12,816 99.63% $8,725 $100,000 -35 -13 3 16 256 360 52 169 3.25 41.69 14,647 99.60% $11,446
Table 9: Impact of cson best on best benchmark, sequential and joint policy
4.5.3 Impact of resource constraint
In the base case, the resource constraint (Rc), which is the maximum allowed maintenance actions
per time period, is set to 4 maintenance actions per time period. To see how this impacts the policies, we search for the policies with lowest costs for different resource constraints. The results are shown in table 10. In both the sequential and joint policy, the gap between d1 and d2 increases when the resource constraint increases. This enables the maintenance policy to schedule more opportunistic maintenance actions (oo) per preventive action (op). It thereby reduces the amount of maintenance
setups (om). The spare part policy adjusts accordingly to the changed spare part demand size.
Comparing the joint to the sequential policy, the cost reduction for the joint policy increases with higher resource levels. As seen before, the joint policy takes fewer risks in the maintenance policy by increasing the maintenance thresholds. With low resource constraint levels, these higher numbers of maintenance actions results in a high amount of maintenance setups and the benefit of the joint policy decreases. With higher resource constraint levels, we observe the opposite effect, and the cost reduction increases to 12.22% and 12.52% for Rc=6 and 8, respectively. Compared to the benchmark policy, an increase in cost reduction can be seen. With a resource constraint of 8, the policy remains on average more below the maximum allowed amount of actions per setup. This indicates that more resources are less beneficial.
P Rc b o d1 (ex) bp d2 (ex) r Q oo op oc om ω∗ oi os od oa C ∆% B 2 10 180 9 12 35 345 285 353 1.89 3.97 55 64,233 98.13% $19,047 -31.32% 4 30 180 7 12 68 320 283 196 3.41 2.39 56 72,763 97.81% $15,413 -43.39% 6 70 180 11 6 157 313 258 156 4.65 1.76 121 60,942 98.17% $14,264 -48.46% 8 70 180 11 12 200 293 251 162 4.59 2.94 62 56,236 98.31% $13,838 -50.21% S 2 -15 -5 8 6 130 345 79 281 1.97 3.81 92 24,607 99.33% $13,165 -0.63% 4 -20 -5 4 12 281 233 74 155 3.78 3.79 49 22,455 99.36% $9,551 -8.64% 6 -25 -5 7 11 386 186 56 122 5.18 3.27 57 16,158 99.54% $8,375 -12.22% 8 -30 -5 5 15 460 152 52 99 6.73 3.40 44 16,849 99.53% $7,875 -12.52% J 2 -25 -15 9 6 93 482 60 319 1.99 2.62 106 16,034 99.55% $13,082 4 -35 -15 5 12 227 395 48 169 3.96 2.27 56 12,816 99.63% $8,725 6 -40 -15 4 12 253 356 57 113 5.90 1.49 55 15,811 99.55% $7,351 8 -50 -17 3 14 387 336 31 109 6.89 1.68 54 8,727 99.76% $6,889
Table 10: Impact of Rc on best benchmark, sequential and joint policy ∗ω=(oo+ op+ oc)/om
4.5.4 Impact of downtime costs
When a turbine fails, the MW it would have produced during that time period is registered and multiplied by cdto include costs for downtime based on the weather. In the base case, the downtime costs are cd = $100. We expect a reduction in the amount of corrective maintenance actions (oc) and downtime (od) for higher cd. We search for the best policies with different levels of cd to better understand the impact of this cost parameter. Table 11 shows the results. The maintenance thresholds in the sequential policy could be changed, but this seems unbeneficial. Instead, inventory levels are increased. This causes fewer failures and, therefore, less downtime. In the joint policy, the maintenance thresholds change slightly cd=$25, $50 and $100, but for these
values, the largest difference also comes from changing the spare part policy. With cd=$200, the
P cd b o d1 (ex) bp d2 (ex) r Q o o op oc om oi os od oa C ∆% B $25 30 180 3 12 26 291 313 164 1.61 52 112,770 96.59% $11,348 -30.31% $50 30 180 4 12 52 318 289 188 2.12 55 80,960 97.58% $12,882 -36.21% $100 30 180 7 12 68 320 283 196 2.39 56 72,763 97.81% $15,413 -43.39% $200 40 180 11 12 99 413 242 221 3.74 63 53,337 98.40% $18,935 -50.24% S $25 -20 -5 3 12 196 225 134 144 2.20 46 44,131 98.70% $8,257 -4.22% $50 -20 -5 4 12 280 233 75 156 3.42 49 22,655 99.35% $8,789 -6.50% $100 -20 -5 4 12 281 233 74 155 3.42 49 22,455 99.36% $9,551 -8.64% $200 -20 -5 8 8 345 228 39 164 4.68 76 12,734 99.67% $10,729 -14,19% J $25 -35 -17 3 12 119 417 93 158 1.55 52 27,667 99.19% $7,908 $50 -40 -17 4 12 198 424 50 169 2.06 56 13,314 99.61% $8,217 $100 -35 -15 5 12 227 395 48 169 2.27 56 12,816 99.63% $8,725 $200 -60 -21 8 12 249 482 23 189 2.17 63 6,285 99.83% $9,207
Table 11: Impact of cd on best benchmark, sequential and joint policy
4.6
Extension: maintenance thresholds depending on weather
Weather conditions can increase the number of corrective maintenance actions because preventive maintenance actions cannot be performed on time to prevent failure. Because failures are likely to last longer when caused by bad weather conditions, weather conditions can further increase downtime. While previously the policies already could adapt to weather conditions by deciding to perform preventive maintenance earlier, the maintenance thresholds were fixed. To take upcoming weather conditions into account while deciding on maintenance actions, two new thresholds are introduced: d3 and d4. When upcoming weather conditions are bad, we use thresholds d3 and d4 instead of d1 and d2. To decide on if upcoming weather conditions are bad, we search for the first time period after the current time period that weather conditions are feasible (Wt= 1, see §3.3.2).
If this time period is four or more time periods away, d1 and d2 are replaced with d3 and d4. We search for combinations of d1, d2, d3 and d4 with r and Q fixed to the best-found values in the joint policy. In the base case, the extension has no effect on the policy (d3=d1 and d4=d2). Therefore, we increase the downtime costs to $200 and $300 and compare the extension and best joint policy. Table 4.6 shows the results. We see a small reduction in total downtime (od) and
costs. d3 and d4 are set lower than d1 and d2, which means that maintenance is scheduled earlier when upcoming weather is bad. The cost reduction, however, is rather low, especially taking into account that there is some variance in the output. Because the joint policy already reduces failures effectively with increased downtime costs, and only a part of the total failures are due to bad weather (others, for example, results from stock-outs), a large cost reduction is also unlikely. When a (long) maintenance lead time would have been used, the impact of constraining maintenance on weather would likely have been greater. When preventive maintenance is not carried out on time to prevent failure due to bad weather conditions, a long maintenance lead time would increase the impact of the failure on the downtime. This could also increase the cost reduction potential of this
extension but is, however, out of the scope of the current model. P cd d1 (ex) d2 (ex) d3 (ex) d4 (ex) r Q oo op oc om oi os od oa C ∆% Joint $200 -60 -21 8 12 249 482 23 189 2.17 63 6,285 99.83% $9,207 Ext. $200 -65 -21 -70 -28 8 12 240 497 23 190 2.03 63 5,978 99.84% $9,150 -0.61% Joint $300 -65 -21 8 12 242 490 24 189 2.11 63 6,154 99.83% $9,644 Ext. $300 -50 -21 -75 -28 8 12 259 477 22 190 2.19 63 5,574 99.84% $9,596 -0.49%
Table 12: Results of joint policy and weather extension for different cd
4.7
Extension: just-in-time ordering
In the main policy, the (r, Q) policy is used to control the spare part inventory. However, as deterioration levels are used to trigger maintenance actions, this could also be used to order spare parts. This could create a ”just-in-time” (JIT) spare part ordering policy. As mentioned earlier in the literature review, ordering spare parts based on deterioration thresholds is not necessarily optimal in multi-unit systems. The one paper that proposes a JIT policy for a multi-unit system instead uses a Markov Decision Process that makes decisions based on the state of all units, available inventory and outstanding spare part orders (Olde Keizer et al., 2017). However, the complexity and size of the system considered here would make the usage of such a method difficult.
To model a JIT policy with spare part ordering based on deterioration within the current methodology and capture some of the economic dependencies between multiple units, we introduce a novel two-thresholds spare part ordering policy. We replace r and Q parameters with two new deterioration thresholds: s1 and s2. When the first turbine reaches level s2, spare parts are ordered for all turbines with a failure probability higher than s1. When a spare part is ordered for a specific turbine, no new spare parts can be ordered until a maintenance action is performed on that turbine. This causes s2 to determine the ordering time and the gap between s1 and s2 to determine the order size. To ensure a stable system and to enable setting s1 and s2 independent of d1 and d2, spare parts are backlogged and added to the soonest spare part order when a maintenance action on a turbine is performed without an ordered spare part during its current life cycle. We search for the best (d1, d2, s1, s2) in the same way as the joint policy.
The joint policy outperforms the JIT policy for every cost parameter configuration tested. Only when we reduce the number of turbines (N ) to 10, the JIT policy becomes beneficial. Table 13 compares the two policies for different N . The resource constraint is set to Rc=2 for N =10 and
