Integrated condition-based opportunistic maintenance and
dynamic inventory models for offshore power systems
Niels Lobbes
A thesis presented for the degree of
MSc. Technology and Operations Management
Department of Operations
University of Groningen
Integrated condition-based opportunistic maintenance and dynamic inventory models for offshore power systems
Niels Lobbes
Abstract
1 Introduction 1
2 Literature Review 3
2.1 Maintenance Strategies . . . 3
2.2 Spare Part Control . . . 4
2.3 Modelling Techniques . . . 5
2.4 Joint Inventory Control and Maintenance Policies . . . 6
3 Maintenance- and inventory models 7 3.1 Notation . . . 7
3.2 Economic Model . . . 8
3.3 Maintenance and Availability Model . . . 10
3.3.1 Scheduling Maintenance Actions . . . 10
3.3.2 Performing Maintenance Actions . . . 10
3.4 Inventory Model . . . 12 3.5 Evaluation Method . . . 15 3.6 Decision-making framework . . . 16 4 Case Study 17 4.1 System Description . . . 17 4.2 Weather Parameters . . . 17 4.3 Cost Parameters . . . 17
5 Results and Discussion 18 5.1 Results on Risk Strategies . . . 18
5.2 Benchmark Study . . . 21
5.3 Sensitivity Analysis . . . 22
5.3.1 Impact of Holding Costs . . . 22
5.3.2 Impact on Stockout Costs . . . 24
5.3.3 Variability in Lead Time Demand . . . 26
1 Operational Decision-Making per Risk Strategy. . . 19
2 Relative Performance of Offshore Wind Operations per Risk Strategy. . . . 20
3 Maintenance Performance of Offshore Wind Operations per Risk Strategy. . 20
4 Economic Performance of Wind Turbines per Risk Strategy. . . 21
5 The Impact of Holding Spare Parts on Daily Costs. . . 23
6 The Impact of Holding Costs on Operational Parameters. . . 23
7 The Impact of Rising Electricity Prices on Daily Costs. . . 24
8 The Impact of Stockout Costs on Operational Parameters. . . 25
1 Base Case Parameters and Numerical Values. . . 18
2 Availability Rate per Risk Strategy. . . 19
3 Benchmarking Analysis. . . 22
4 Maintenance Actions Under Varying Electricity Prices. . . 25
1
Introduction
In recent years, the share of offshore wind power generation has increased significantly and is becoming the main focus for the world’s renewable energy problems (L. Wang, Wei, et al., 2009). Countries and governmental bodies such as the UK and the Netherlands set ambitious targets to curb carbon emissions and increase the share of renewable energy sources by introducing offshore wind farm projects (Kling et al., 2008; Feng et al., 2010). Offshore wind power systems have several advantages, for example the mean wind speed in sea is 25% faster than onshore wind farms. Offshore wind farms also generate substan-tial quantities of energy more cheaply comparing to other renewable energy sources, but are more expensive than its counterpart onshore mainly due to weather conditions and logistical constraints (X. Yang and Bai, 2010; Boyle, 2006)
The application of large scale offshore power systems for a sustainable energy supply increases the economic feasibility if a decent degree of operational performance can be en-sured. Researchers argue that operations and maintenance (O&M) costs contribute with 25% to 30% to the total life cycle costs per wind turbine (Lau et al., 2012). Since O&M costs comprehend approximately a quarter of total energy generation expenditures, cost re-duction strategies for the optimization of O&M are relevant for future offshore wind power operations. Various factors influence maintenance planning and wind farm operations, in-cluding the occurrence of failures, spare parts control, weather conditions, electricity prices and the chosen maintenance strategies (Seyr and Muskulus, 2019; Besnard, Patriksson, et al., 2009; Sahnoun et al., 2015).
periodic review- and continuous review systems (Kian et al., 2019; Panagiotidou, 2014). Wilson (1934) breaks the concept of inventory control into two main elements, namely determining the order quantity and determining the reorder level. These elements help to ensure the inherent uncertain expected demand of spare parts and lower the operating expenditures such as shortage- and holding costs.
2
Literature Review
Research in offshore wind operations can be distinguished as maintenance decision strate-gies, spare part control, and the impact of weather conditions. The most relevant studies and papers are summarized in this section.
2.1 Maintenance Strategies
Different maintenance policies have been introduced to offshore wind farm systems. These strategies include corrective maintenance, preventive maintenance, condition monitoring and condition-based maintenance (S. Song et al., 2018).
The traditional maintenance strategies such as corrective- and preventive maintenance illustrate the basic approaches to preserve the conditions of offshore wind turbines. Scheu et al. (2012) presented a model to simulate failures and repairs by exclusively considering corrective maintenance. Hofmann and Sperstad (2013) considered time-based preventive maintenance in which components were restored to the original quality and value, while Shafiee and Finkelstein (2015) studied age-based preventive maintenance to propose an optimal group policy for multi-unit degrading systems. Sim and Endrenyi (1988) proposed a minimal maintenance model to optimize preventive maintenance for repairable devices. Byon et al. (2010) examined repair strategies for wind turbines under stochastic weather conditions to derive an optimal preventive maintenance policy.
et al. (2017) considered an imperfect maintenance schedule based on reliability for an opportunistic maintenance approach and discussed the economic advantages of corrective-and opportunistic maintenance. Ding corrective-and Tian (2011) proposed several opportunistic maintenance optimization models to realize a reduction in maintenance costs by studying three maintenance scenarios.
2.2 Spare Part Control
To maintain the conditions of offshore wind turbines, spare parts should be available in stock (Shafiee, 2015). Different inventory control policies (i.e. periodic review- and contin-uous review inventory policy) are introduced to enhance the performance of maintenance. Basic inventory policies aim to provide spare parts that are required in maintenance pro-cesses and keep inventory costs low (Tracht et al., 2013). Several types of inventory policies can be applied. Wang et al. (2008) introduced a simulation model for optimizing the joint policy of condition-based maintenance and (s, S) inventory policy for systems with identical units. Van Horenbeek et al. (2013) reviewed various joint maintenance and inventory optimization systems, including (s, Q) and (R, S) policies (L. Wang, Chu, et al., 2008; Keizer et al., 2017b). Various inventory policies also have been introduced in offshore power systems. Tracht et al. (2013), for example, studied the impact of restrictive maintenance conditions like the limited availability of vessels on spare part strategies for offshore wind turbines. Dahane et al. (2017) investigated the impact of spare parts re-manufacturing on offshore operations & maintenance by introducing a multi-agent based approach.
Several researchers applied periodic review systems in which inventory is replenished per fixed cycle time. Mjirda et al. (2016) studied the joint optimization of periodic preventive maintenance and the spare parts inventory problem. They proposed a model to find an optimal maintenance schedule by taking spare part control into account. Zhang and Zeng (2017) introduced joint optimization for periodic condition-based opportunistic preventive maintenance by considering spare parts provisioning policy in multi-unit systems.
2.3 Modelling Techniques
Different modelling techniques have been introduced to model the different factors for offshore wind power systems and capture uncertainties such as the occurrence of failures and electricity prices (Seyr and Muskulus, 2019). These techniques include discrete event simulation and Markov decision models.
Discrete event simulation models attempt to understand the behavior of systems and evaluate various strategies. Different studies have applied this modelling technique in the field of offshore wind farms (Shanoun et al., 2015; Dinwoodie et al., 2013). Kim, Singh and Spintson (2012) considered the wake effect in a wind farm and proposed an improved method for the transition rate matrix by using Monte Carlo simulation. Byon et al. (2010) proposed a simulation model to study a broad array of critical aspects of wind farm operations by using discrete event system specification formalism. Zhang and Wang (2009) studied the wake effect in power systems at one wind region in China by modelling and simulating the studied power system.
Other modelling technique is the use of Markov models in which next states only depend on the current state and thus the process is memoryless (Seyr and Muskulus, 2019). Yang, Kwan and Chang (2008) studied the optimization of substation costs and reliability by using decision-varying Markov models. Byon and Ding (2010) applied the Markov decision process to wind turbines and proposed different models to devise season-dependent condition-based maintenance and reduce operating costs. Other researchers also proposed a time series study in which they present a multivariate Markov chain model to represent uncertain weather conditions of offshore wind operations (Hagen et al., 2013).
2.4 Joint Inventory Control and Maintenance Policies
Since maintenance strategies and spare part control are closely correlated to each other, academics and practitioners have studied this joint optimization- and integrated approach. Several studies (H. Wang, 2002; W. Wang, 2011) investigated the joint optimization of order quantity, inspection interval and maintenance by applying a two stage failure process on deteriorating systems based on the delay-time concept. Other researchers extended this joint optimization by applying difference maintenance policies such as preventive-, condition-based-, and opportunistic maintenance (Rausch and Liao, 2010; X. Zhang and Zeng, 2017). For instance, Elwany and Gebraeel (2008) integrated a degradation modelling framework for computing remaining life distributions using condition-based sensor data with replacement and inventory decision models. Louit et al. (2011) presented a model directed to the determination of the ordering decision for spare part when the component in the system is subject to condition-monitoring.
3
Maintenance- and inventory models
This section introduces the mathematical model for offshore wind operations, followed by the procedure for inventory control and maintenance decision making. A single-unit, echelon system is assumed with multiple periods and fixed replenishment lead time L. The single-unit component is subject to random failure. The input data is real-time sensor data transformed into failure probabilities using methods described by Yildrim et al. (2017). A discrete event Monte Carlo simulation for testing and evaluating the proposed model is used to portray randomness and variability accurately.
3.1 Notation
Sets and Indexes Model parameters
T Simulation run length d1 Maintenance threshold 1
t ∈ T Index for simulation length d2 Maintenance threshold 2
N Total number of WTs d3 Ordering threshold 1
n ∈ N Index for the WTs Fn Failure age of WT n
Decision variables Gn Current age of WT n
rt Reorder level at time t Pn,t Failure prob. of WT n at time t
Q Replenishment order size mon,t Scheduled OM for WT n at time t
Costs mpn,t Scheduled PM for WT n at time t
Ctot Total costs per time t mcn,t Scheduled CM for WT n at time t
Ci Inventory costs x Lead time demand
Cm Maintenance costs E(D) Expected cycle time demand
Cd Downtime costs D Actual cycle time demand
ci Cost for holding components γ Critical ratio cs Setup costs per order Q σ Standard deviation
co Cost for OM α Risk parameter 1
cp Cost for PM λ Risk parameter 2
cc Cost for CM B Expected backorders per cycle
cm Cost for setup maintenance I Inventory position
Simulation parameters L Replenishment lead time (in days)
δ Availability rate in N CT Cycle time (in days)
oo Number of OM actions Y Fill rate (in %)
op Number of PM actions S On-hand inventory level (in units) oc Number of CM actions Rc Resource availability per time t
om Amount of maintenance setups Ra Resources currently available oi Average daily inventory level Binary parameters
od Amount of MW not generated Ht Inventory at the end of time t
os Number of spare parts orders At Spare part arrivals at time t
Wt Feasibility of weather at time t
3.2 Economic Model
The objective of the maintenance- and inventory models is to minimize the costs of per-forming maintenance actions, the costs of holding and ordering spare parts and the down-time costs associated to wind energy not generated due to failures expressed in Eq. (1)
Ctot= Ci+ Cm+ Cd (1)
where Ci are the inventory costs, Cm represents the maintenance costs, and downtime costs are expressed as Cd. For estimating the inventory costs, a holding cost ci and a fixed
setup cost per order cs per t day is assumed, see Eq. (2)
Ci = o
ici+ oscs
T (2)
Output variable oi represents the average daily on-hand inventory, while os describes the order frequency expressed in Eq. (3) and (4), respectively
oi = PT t=1(Ht) T (3) os = T X t=1 (At) (4)
The maintenance costs are calculated as shown in Eq. (5), where maintenance variables co, cp and cm represent the costs for opporunistic-, preventinve- and corrective maintenance, respectively. The associated amount of maintenance actions are depicted as oo, op and oc
Cm = o
oco+ opcp+ occc+ omcm
T (5)
Maintenance variable opportunistic maintenance oo, preventive maintenance op and cor-rective maintenance oc represent the sum of maintenance actions performed per type at each wind turbine n expressed in Eq. (6), (7) and (8), respectively
oc= N X n=1 T X t=1 (Mn,tc ) (8)
Maintenance costs also comprehend the amount of maintenance setups om, see Eq. (10). Maintenance setups are considered when more than zero maintenance actions are per-formed at day t, see Eq. (9)
Ut= 1, ifPN n=1(Mn,to + M p n,t+ Mn,tc > 0) 0, otherwise (9) om= T X t=1 (Ut) (10)
Downtime costs Cdare calculated by determining whether corrective maintenance actions were scheduled mcn,t, but not performed Mn,tc due to weather conditions or absence of spare parts on-hand. We assume that one wind turbine generates Vt per t and is considered as
input for the proposed model. The difference between scheduling corrective maintenance mc and performing corrective maintenance Mc indicates the MW not generated due to failures, see Eq. (11)
3.3 Maintenance and Availability Model
A joint decision-making model is proposed where first the maintenance actions are deter-mined, followed by computing the optimal reorder level r and replenishment order size Q. Condition monitoring is in place to schedule- and perform perfect maintenance actions.
3.3.1 Scheduling Maintenance Actions
Maintenance actions are scheduled based on the deterioration of single components ex-pressed in failure probabilities and the fixed values of maintenance decision thresholds d1 and d2. Corrective maintenance is scheduled if the current age of Gn is greater or equal
to the expected failure age Fn, see Eq. (12)
mcn,t= 1, if Gn≥ Fn 0, otherwise (12)
Preventive maintenance and opportunistic maintenance, however, are scheduled based on the maintenance decision thresholds. Opportunistic maintenance is scheduled if a component’s failure probability at wind turbine n surpasses d1, while exceeding d2 means scheduling preventive maintenance, see Eq. (13) and (14)
mon,t= 1, if d1 ≤ Pn,t < d2 0, otherwise (13) mpn,t= 1, if Pn,t> d2 0, otherwise (14)
3.3.2 Performing Maintenance Actions
Mn,tc = mc n,t, if (Wt= 1 and Ra> 0 and S > 0) 0, otherwise (15) Mn,tp = mpn,t, if (Wt= 1 and Ra> 0 and S > 0) 0, otherwise (16) Mn,tc =
mcn,t, if (Wt= 1 and Ra> 0 and S > 0) and PNn=1Mn,tp > 0
0, otherwise
(17)
Performing maintenance actions require spare parts and resources. On-hand inventory S, inventory position I and available resources Ra are therefore updated in Eq. (18), (19) and (20), respectively. S = S − 1, if Mn,tc = 1 or Mn,tp = 1 or Mn,tc = 1 S, otherwise (18) I = I − 1, if Mn,tc = 1 or Mn,tp = 1 or Mn,tc = 1 I, otherwise (19) Ra = Ra− 1, if Mn,tc = 1 or Mn,tp = 1 or Mn,tc = 1 Ra, otherwise (20)
For estimating the availability rate δ of all wind turbines, we assume that the difference between scheduled and performed corrective maintenance states the amount of downtime per wind turbine n per day t. The overall availability rate is calculated as
3.4 Inventory Model
The maintenance model in Section 3.3 is followed by the inventory model proposed in Section 3.4. Inventory is controlled by the reorder point, order quantity policy, known as the (r, Q) model, in which decision variables reorder point r and replenishment order size Q are determined. The expected lead time demand per wind turbine xn,t and the sum
of lead time demand xt within replenishment lead time L per day t are computed with
support of the ordering decision threshold d3, see Eq. (22) and (23).
xn,t= 1, if Pn,t+L ≥ d3 0, otherwise (22) xt= N X n=1 xn,t∀t ∈ T (23)
where xt follows a Normal distribution in which standard deviation σx represents the
variability in lead time demand in set N . We consider three different risk models to compute the optimal r and Q by introducing risk parameters α and λ. Both r and Q can be found by first computing the critical ratio γ to minimize holding costs chplus downtime costs cd in relation to MW produced Vt per wind turbine n, see Eq. (24)
γ = (c d∗ V t) (cd∗ V t) + ch (24)
Since γ follows a Normal distribution, the optimal rtcould be computed. The reorder level,
however, is subject to various risk scenarios in which a averse, neutral or risk-taking strategy can be followed. A risk-averse strategy (i.e. rt> xt, λ > 0.5) maximizes
the availability wind turbine by holding a safety stock based on Eq. (24) and standard deviation σx, while a risk-taking strategy (rt < xt, λ < 0.5) minimizes inventory costs
and accept potential backorders by lowering reorder levels. The risk-neutral strategy is a more balanced approach in which the interaction between lead-time demand, critical ratio and variance is considered, see Eq. (25). The risk parameters α and λ determine which strategy is in place to compute the reorder level (Jammernegg and Kischka, 2007)
where α = 0.5, 0 < λ < 1, and F−1 denotes the inverse of the cumulative distribution function. The risk strategies aim to optimize availability (risk-averse) or minimize net inventory level (risk-taking), where carrying a spare part unit in stock is typically more costly than backorders. It is the generally the case that r is an increasing function of expected lead time demand x. Backorders occur if lead time demand xt exceeds the
optimal rt illustrated in Eq. (26)
Backorders = 0, if xt≤ rt xt− rt, if xt> rt (26)
The expected average number of backorders per cycle is then given by Eq. (27)
B = (xt− rt)[1 − Φ(z)] + σxφ(z) (27)
where z = (rt− xt)/σx and Φ and φ represent the cumulative distribution function (cdf)
and the probability density function (pdf), respectively (Hopp and Spearman, 2011). The B function computes the amount of unmet demand and is therefore referred to as the loss function. The (r, Q) inventory model continuously monitors inventory position I and places a replenishment order Q when I triggers the reorder level expressed as Zt, see Eq.
(28) and (29) Zt= 1, if I ≤ r 0, otherwise (28) I = I + Q, if I ≤ r I, otherwise (29)
where inventory position I is on-hand inventory plus replenishment order size minus back-orders. Inventory position I in the (r, Q) model should oscillate between r and r +Q. Only if Zt= 1, the expected- and actual cycle time demand (E(D) and D, respectively) should
D = E(D) − xt+ B (32)
The economic order quantity (EOQ) model is applied to determine the replenishment order size Q because it may simply not be feasible to base this decision at first on historical data. The appropriateness of the EOQ model, however, diminishes when time proceeds because it assumes that demand is deterministic, while it is in fact stochastic. Similar to the lead time demand, the expected cycle time demand D follows a Normal distribution and the standard deviation is σD. Q is computed in Eq. (33) or (34)
Q = q 2csD ci , if t = 0 Q∗, otherwise (33) Q∗ = D + (γ ∗ σD), if α < λ F−1(γ), if α = λ D − ((1 − γ) ∗ σD), if α > λ (34)
The fraction of orders filled from stock, known as the fill rate, represents the service level and is expressed in Eq. (35). This formula, known as type II service, is an approxima-tion and tends to underestimate the true fill rate, however it turns out to be an useful intermediate measure.
Y = 1 −B
Q (35)
On-hand inventory is replenished at the beginning of day t if a replenishment order Q was made t − L days ago. The spare part arrivals At are calculated as
At+L= 1, if I ≤ r 0, otherwise (36)
After performing maintenance actions (see §3.3.2), on-hand inventory S is reviewed at the beginning of each time period and increases by the spare parts that arrive at that specific day (At). The available resources Ra are reset to the total number of available resources
Rc
The on-hand inventory is registered at the end of each today as Ht to calculate total
holding costs, see Eq. (39)
Ht= S (39)
3.5 Evaluation Method
4
Case Study
4.1 System Description
The proposed model in Section 3 is demonstrated by the means of a case study with N = 80 wind turbines. Each maintenance action requires one resource, where the availability is constraint to Rc= 4 per day t. Receiving components after an order takes a replenishment lead time of L = 21 days, while cycle lead time is CT = 42 days. For the maintenance strategies, the deterioration state of the component is inspected every day t. We assume that maintenance actions provide perfect maintenance to restore the state of the wind turbine.
4.2 Weather Parameters
Wave and wind data are based on the FINO 1 offshore research project from 2004 to 2018 that is 45 km offshore of Borkum, Germany. Wind speeds are processed into MW generated per day t. Wave heights are also calculated to constraint the performance of maintenance actions. If the wave height at day t is less than or equal to the maximum wave height of 1.5m, maintenance actions can be performed Wt= 1, otherwise Wt= 0.
4.3 Cost Parameters
The maintenance costs are defined as the costs for performing the different types of main-tenance in simulation length T = 5 years, including a warm-up period of Tint = 1 year.
It is assumed that the fixed setup costs for performing a group of maintenance actions at day t is cm = $32, 000. The costs for performing corrective maintenance, preventive maintenance and opportunistic maintenance is cc = $16, 000 and cp = co = $4000 per action, respectively.
If a wind turbine failure occurs, a new component is ordered to arrive at destination after replenishment lead time. Ordering components have a fixed setup cost of cs = $20, 000 per order, while the costs for holding a single component at day t is ci = $500 per day. Downtime costs associated to MW not generation due to wind turbine failures - or electricity prices - has a cost of cd= $100 per MW
Table 1: Base Case Parameters and Numerical Values. Parameter Numerical Value
T [yr] 5 Tint [yr] 1 N [#] 80 L [d] 21 CT [d] 42 Rc [#] 4 cm [$] 32,000 cc [$] 16,000 cp, co [$] 4,000 cs [$] 20,000 ci [$/d] 500 [25 - 6,400]1 cd[$/MW] 100 [25 - 6,400]1
1 range for sensitivity analysis.
5
Results and Discussion
In this section, several case study simulations are presented to provide insights into the capabilities of the methodology described in Section 3. The proposed model is validated by providing a benchmark study based on the paper of Dieterman (2019). This baseline study has been chosen for several reasons. The author introduced a similar methodology for the analysis of offshore wind operations by considering static decision-making. The ap-plication of the proposed inventory model exposes the differences between the operational performances of the two policies. For the purpose of this paper, the risk-neutral scenario is used as reference material to determine the optimal solutions. This section also demon-strates how the operational metrics evolve under the different scenarios by providing a sensitivity analysis on the proposed model.
5.1 Results on Risk Strategies
Table 2: Availability Rate per Risk Strategy. Strategy Availability (%) Risk-neutral 99.93%
Risk-averse 99.95%
Risk-taking 99.77%
are associated with an increase in holding costs. The results on the risk-taking strategy do not only illustrate the effect of a lower reorder level and order size on average inventory level (see Fig. 1), but also show the impact on the availability rate of the wind turbines (see Tab. 2). For example, offshore wind operations is not always able to deliver the required spare parts on time because they are simply not in stock which lowers the availability rate.
Figure 1: Operational Decision-Making per Risk Strategy.
Figure 2: Relative Performance of Offshore Wind Operations per Risk Strategy.
Scheduling and performing maintenance actions also has a direct impact on offshore wind operations. Fig. 3 illustrates the number of maintenance actions per type per risk strat-egy. Similar to the operational performance, the risk-neutral and risk-averse strategy perform comparable in which opportunistic maintenance is performed frequently and cor-rective maintenance is uncommon. Performing opportunistic maintenance clearly reduces the chance of component failures, downtime and increases inventory turnover, however a significant share of the remaining useful life of the wind turbines stays unexploited. Main-tenance actions under are differently distributed among the types of mainMain-tenance when the risk-taking strategy is followed. We observe that a balanced combination between preventive- and opportunistic maintenance is in place, but the amount of corrective main-tenance actions also increases. This inflates downtime and is therefore one example why several wind turbines missed out on the generation of MW (see Fig. 2).
The economic performance illustrated in Fig. 4 shows that the daily O&M costs per risk strategy oscillate between $7,900 and $8,300. The risk-taking strategy presents the lowest daily costs comparing to the other risk strategies. This result supports the view that dy-namic inventory decision-making considers realistic uncertainties and shows that inventory minimization provide cost-benefits to offshore wind systems. Academics and practitioners, however, should also consider the role of other indicators in offshore wind operations such as reliability and availability. The risk-taking strategy, for example, does not take the volatility of demand into account and is therefore more vulnerable for randomness.
Figure 4: Economic Performance of Wind Turbines per Risk Strategy.
5.2 Benchmark Study
Table 3: Benchmarking Analysis.
Dynamic Inventory Models Metric Joint Risk-neutral Risk-averse Risk-taking
r 5 5.49 5.65 3.67 Q 12 11,48 11.89 8.64 B - 0.17 0.28 1.24 Y - 98.45% 97.51% 81.25% oi 2.27 2.4 2.86 0.57 os 56 58 55 75 oo 227 369 381 256 op 395 259 255 299 oc 48 18 15 54 om 169 186 187 182 od 12,816 2,174 1,595 7,551 δ 99.63% 99.93% 99,95% 99,77% C $8,725 $8,117 $8,271 $7,922 ∆ - 6.97% 5.2% 9.2% 5.3 Sensitivity Analysis
Section 5.3 provides insights into the relationship between several parameters and vari-ables on the performance of the risk strategies. We examine the impact of holding costs, stockout costs and variance in demand on cost- and operational performance, followed by a discussion about the consequences for offshore wind operations.
5.3.1 Impact of Holding Costs
Figure 5: The Impact of Holding Spare Parts on Daily Costs.
Figs. 6a and 6b illustrate that both the reorder level and replenishment order size are a decreasing function of the holding costs. We argue that holding spare parts in stock be-comes less feasible for offshore wind operations because the critical ratio - or the point the minimize holding- plus backorder costs - decreases if holding costs increase. Offshore wind operations reduces the inventory level to counter the rising holding costs by lowering the reorder level and replenishment order size. This would increase the order frequency and the lack of spare parts in stock would harm the availability of the wind turbines because components are sometimes unavailable when they are required for performing mainte-nance. Comparing the risk strategies, we observe that the risk-taking strategy presents lower reorder levels and order quantities in contrast to the risk-averse and risk-neutral strategy. It is important to note that the risk-neutral strategy considers underestimations with regard to the replenishment order size based on the inverse of the cumulative distri-bution, but still performs at higher reorder levels and order size. We argue that offshore wind operations under the risk-neutral strategy balances its inventory- and purchasing management based on the interaction between variation in demand, the critical ratio and expected demand. This generally results in over-estimations of expected demand and explains the relative higher operational decisions for spare parts control.
(a) Reorder Level (b) Replenishment Order Size
The position of the maintenance- and inventory decision thresholds determine when main-tenance actions per type should be performed and replenishment orders should be made, respectively. Increasing expected maintenance actions performed at offshore wind turbines tend to increase both reorder level and replenishment order size. Offshore wind operations should raise replenishment order size, inventory- and reorder level to satisfy demand with spare parts, while maintaining a high degree of availability. An increasing holdings cost, however, decrease optimal values r and Q∗. The rule of thumb is that the more expensive it is to stock spare parts, the less we should actually hold. This effect is expressed in Eq. 24 for computing the critical ratio. Volatility in the holdings cost is therefore crucial for decision-making in offshore wind operations, where a trade-off between low on-hand inventory levels and high availability rates should be made.
5.3.2 Impact on Stockout Costs
Fig. 7 presents the stacked daily costs per risk scenario as a function of the stockout costs expressed in MW. The daily costs of offshore wind operations for each risk strategy increase due to rising prices per MW. The behaviour among the risk strategies, however, differs significantly in which the risk-neutral strategy only shows a slight, linear increase, while the averse and taking strategy grow exponentially. We argue that the risk-neutral strategy follows a balanced approach in which rising downtime costs of offshore wind turbines are taken into account in operational decision-making. Preventive- and opportunistic maintenance actions are performed earlier, while a buffer of spare parts is acquired as well as enlarged. Nonetheless, offshore wind operations are clearly vulnerable for increasing electricity prices if it aims for inventory minimization expressed as the risk-taking strategy.
Rising electricity prices should trigger offshore wind operations to enhance the generation of wind energy because inventory minimization becomes less feasible comparing to wind power generation. Tab. 4 shows that the number of preventive- and opporunistic mainte-nance actions increase, while corrective maintemainte-nance actions decrease. This behaviour in maintenance actions is supported by the expectation that reorder level and replenishment order size increase. Figs. 8a and 8b back this view by showing that both reorder level and order size increase as the price for MW increases. We observe that wind farm opera-tions under the risk-neutral strategy increase safety stock to cover for expensive downtime by increasing reorder level and replenishment order size. In this approach, they consider higher inventory levels and -costs, but compensate for them by extremely low downtime and high availability rates for the offshore wind turbines.
Table 4: Maintenance Actions Under Varying Electricity Prices. 25 50 100 200 400 800 1,600 3,200 6,400
CM 40 27 19 12 8 6 4 3 2
PM 286 271 257 252 244 240 234 227 226
OM 296 337 369 396 411 423 432 440 440
Both the risk-neutral and risk-averse strategy explicitly consider safety stock to provide a buffer against unexpected failures of offshore wind turbines. The importance of safety stock grows further if electricity prices fluctuate and tend to increase. These two strategies force offshore wind operations to increase optimal values r and Q in order to reduce the number of stockouts and downtime costs despite rising inventory levels. Decision-makers should consider whether the offshore wind operations have the capacity to stock large amount of spare parts and find suppliers who can deliver components in bulk on infrequent basis.
(a) Reorder Level (b) Replenishment Order Size
5.3.3 Variability in Lead Time Demand
Fig. 9 presents the daily costs per risk strategy under an increasing variability in lead time demand. High levels of variability in lead time demand impede offshore wind opera-tions because it becomes more difficult to predict the amount of spare parts required for maintenance actions. We observe that an increasing variability amplifies the costs for all risk strategies, however the risk-neutral and risk-taking strategies show a steeper increase comparing to the risk-taking strategy. The effect of variability on daily costs is also mini-mal at first, but grows significantly afterwards. Offshore wind operations should therefore not only consider inventory buffers to counter variability, but also capacity buffers. For example, offshore wind turbines that exceed maintenance decision threshold d2 could lower their capacity to slow the component’s deterioration process down, which gives offshore wind operations time to respond to this development accordingly.
Figure 9: The Impact of Variability in Lead Time Demand on Daily Costs.
Table 5: Order Frequencies Under Variability in Lead Time Demand.
1 2 3 4 5 6 7 8 9 10
Risk-neutral 55 56 57 58 60 61 62 63 62 64 Risk-taking 73 74 75 76 78 79 81 80 82 84 Risk-averse 58 58 60 62 63 64 65 66 67 68
We argue that an increasing variability in lead time demand tends to increase optimal r and inventory level of spare parts. More specifically, the risk-averse and risk-neutral strategies safeguard offshore wind operations against stockouts by considering higher safety stocks compared to stable lead time demand processes. This insight, however, only holds when the critical ratio is larger than 0.5. In the case of the risk-taking strategy, the optimal reorder level actually decreases because the critical ratio will be negative. This phenomena occurs when it is optimal to set the fill rate low for the components. Decision-makers should think about the aim of the offshore wind operations, for example a risk-neutral or risk-averse strategy should be applied for reliability-centered approaches, while a risk-taking strategy is more appropriate for cost minimization.
(a) On-hand inventory (b) Reorder level
6
Conclusion
This paper presents a cost-based optimization model for offshore wind operations by exam-ining condition-based opportunistic maintenance and spare part inventory control policies. The proposed model considers a (r, Q) continuous review inventory policy to capture the effect of uncertainties in offshore wind power systems. Sensor-driven deterioration data is in place to compute optimal maintenance performance and optimal spare part inventory control for a single-unit, echelon offshore wind system. A Monte Carlo simulation method is proposed to evaluate the expected maintenance-, inventory and downtime costs as well as the operational performance under different risk strategies. A benchmark study is in-troduced to compare the expected economic- and operational performance to a baseline study.
Results show that the optimal reorder level and replenishment order size decrease down-time and increase the availability rate of offshore wind turbines under the risk-neutral strategy to 2,174 MW and 99.93%, respectively. All risk strategies (i.e. neutral, risk-averse and risk-taking) would be beneficial comparing to the joint policy with an expected annual cost reduction of $221,920, $165,710 and $293,095, respectively. A sensitivity anal-ysis was conducted to determine the influence of varying holding costs, stockout costs and variability in lead time demand. A risk-taking strategy is the optimal inventory control strategy for increasing holding costs and variability in lead time demand, while varying stockout costs require a risk-neutral strategy. If the holding costs increase, offshore wind operations require less spare parts because the feasibility of having safety stocks diminishes comparing to the downtime costs. In the case of increasing stockout costs, the offshore wind operations should consider decent inventory levels and a risk-neutral strategy is more appropriate. These results not only illustrate a trade-off between variability and on-hand inventory, but also present the decision between reliability-centered approaches and cost minimization approaches.
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