Wind farm maintenance planning under forecastable weather conditions

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Wind farm maintenance planning under

forecastable weather conditions

June 30, 2019

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Master’s Thesis Econometrics, Operations Research and Actuarial Studies Supervisor: Dr. B. de Jonge

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Wind farm maintenance planning under forecastable weather

conditions

Luuk Pentinga

Abstract

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June 30, 2019 CONTENTS

List of Figures

1 Visualization of the MDP . . . 13

2 Visualization of the transition probability sub-problem structure. . . 17

3 Optimal number of units to maintain for different weather forecasts in the base case. . . 24

4 Optimal number of units to maintain for different weather forecasts in the good weather season. . . 27

5 Optimal number of units to maintain for different weather forecasts in the bad weather season. 27 6 Optimal number of units to maintain for the alternative delay-time model. . . 29

7 Optimal number of units to maintain when na= 7. . . 30

List of Tables

1 Base case wind transition probabilities for different wind states. . . 23

2 Transition probabilities for different wind states in the good weather season. . . 26

3 Transition probabilities for different wind states in the bad weather season. . . 26

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June 30, 2019 1. INTRODUCTION

1 Introduction

As industrial systems degrade over time, there is a repeating need for maintenance. This may be to improve system availability or product quality, or to prevent safety issues. Since such systems in general are becom-ing increasbecom-ingly sophisticated, it makes sense that maintenance strategies for these systems are more and more complex. Decision makers in businesses are more and more aware of the benefits of good maintenance policies. Possibly caused by this, the literature on maintenance planning is also rapidly maturing.

The problem that we consider in this research, is based on the setting of offshore wind farms. Nearly a quarter of the lifetime costs for a turbine of such a wind farm are a result of operations and management (Snyder and Kaiser, 2009). Considering that in 2015 wind energy contributed to 11.4% of the EU’s electric-ity demand (Martinez-Luengo et al., 2016), potential cost savings resulting from maintenance optimization for wind farms are enormous. Efficient maintenance strategies are expected to become even more relevant, as wind energy is supposed to account for 20% of the electricity supply in the EU in 2030 (Kang et al., 2019). Optimization of maintenance in the wind sector is complex, characterized by many relevant external factors. Expensive vessels or even helicopters must be rented to perform maintenance. Furthermore, wind speeds and weather conditions do not only influence the deterioration speeds of turbines, but they may also limit the possibility to perform maintenance. On top of this, wind farms are increasing in size, with the largest farm currently under construction counting a total of 174 turbines (Selot et al., 2019). This shows clear potential for intelligent maintenance policies, supported by scalable technical solutions. In general, maintenance optimization for the wind sector faces problems ranging from vehicle routing and inventory control to multi-unit maintenance planning.

In this study we focus on the short-term maintenance planning for offshore wind farms. Offshore wind turbines are equipped with a variety of sensors that alert the need for maintenance interventions, allowing accurate measurements of condition information. Furthermore, the effect of the weather on both the deterioration speed of the turbines and the ability to perform maintenance is taken into account. We use a Markov Decision Process (MDP) formulation to model this system, in which we aggregate the deterioration state information. This allows for great reductions in the size of the state space, and thus for optimization of relatively large amounts of units.

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June 30, 2019 2. LITERATURE

5 numerical results are given. We conclude and provide suggestions for future research in Section 6.

2 Literature

In this review, we first consider several papers on topics relevant for the operations surrounding wind tur-bines. These topics range from the logistics surrounding wind farms to condition-based maintenance mod-eling taking into account external factors. Thereafter, we review research on wind speed simulation. The third area we will review is that of multi-unit systems maintenance planning, due to potential similarities in modeling techniques used. Because maintenance planning is a diverse and increasingly well-known field of interest for researchers, we cannot provide details on the entire field. Instead, for a broader literature review we refer the reader to Ding and Kamaruddin (2015). They classify a large portion of the available literature, and include details about different areas of application. Furthermore, they stress relevant findings and future research topics. Another, more recent literature review is given by De Jonge and Scarf (2019), with a focus on the various modeling techniques used.

There are a wide variety of papers that explore details in maintenance that may be applicable to off-shore wind energy, such as external influences. Cha et al. (2018) consider a preventive maintenance model with external shocks that do not only influence deterioration rates, but also result in an immediate drop in output. Montoro-Cazorla and P´erez-Oc´on (2012) also review a system with external shocks, but assume that these shocks may cause unrepairable damage, and thus lead to eventual replacements. Byon and Ding (2010) model a single wind turbine consisting of multiple components, and determine a maintenance pol-icy dependent on seasonality in weather conditions. They assume that poor weather conditions result in the unavailability of maintenance, or interruptions in ongoing maintenance actions. Unlike our study, the component degradation is not assumed to be affected by these weather conditions. Other highly relevant research subjects concern the logistics surrounding wind farm maintenance. These include the availability of components and materials in a repair-kit fashion (Teunter, 2006), or even safe routing for maintenance vessels (Dai et al., 2013). A modeling framework for the supporting organization of wind farm mainte-nance is given by Besnard et al. (2013). For a more complete review of the studies on condition-based maintenance for offshore wind, we refer the reader to Kang et al. (2019). In addition to reviewing different maintenance policies, they also review studies on condition monitoring and system diagnosis.

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June 30, 2019 2. LITERATURE

conditions realistically. To do so, statistical models such as autoregressive models are often adopted, for example by Poggi et al. (2003) or Kavasseri and Seetharaman (2009). Although useful, these models are most suitable for short-term wind speed forecasting. A simpler way to model weather conditions is by using Markov chains. Markov chains are shown to realistically reproduce long-term distributions for wind speed and wave heights (Scheu et al., 2012). Such models are thus often used directly (Hagen et al., 2013; Nfaoui et al., 2004), or in hybrid forms combined with machine learning techniques (Yang et al., 2015) or stochastic differential equations (Ma et al., 2018).

The existing literature on multi-unit or multi-component systems is also highly relevant for this study. This literature is reviewed by Olde Keizer et al. (2017). One of the main categorizations they make, is be-tween series and parallel systems. Series systems fail completely when a single component fails, whereas parallel systems may still function (partially) upon component failure. As a wind turbine consists of mul-tiple machine parts, turbines are often modeled as a series system (Sarker and Faiz, 2016; Lu et al., 2018). However, it is not uncommon to look at maintenance policies for multiple turbines simultaneously, result-ing in a parallel system. Dresult-ing and Tian (2011) consider maintenance policies at a wind farm level, where maintenance actions are assumed to be imperfect. Scheu et al. (2012) review maintenance policies for an entire wind park, where they include weather conditions. However, they only allow the maintenance of failed turbines.

Additional to differentiating between series and parallel systems, Olde Keizer et al. (2017) characterize multiple types of dependencies between units, namely economic, structural, stochastic and resource depen-dency. For multi-component system maintenance, an economic dependency is often assumed, generally in terms of setup costs necessary for maintenance actions. This allows for opportunistic maintenance, where units may be maintained preemptively, such that their maintenance can be combined with other maintenance actions. Similar to our study, Olde Keizer et al. (2018) model a parallel system, and derive a condition-based maintenance strategy for it. They find that economic dependencies may cause the postponement of maintenance actions, in order to cluster future maintenance actions.

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June 30, 2019 3. PROBLEM DESCRIPTION

Pham (2014) consider such a system, where the stochastic dependency is a result of external shocks, and the economic dependency results from lower costs for simultaneous replacements than individual ones. They require periodic inspections, and conclude that qualitative results drawn from single-component systems cannot easily be adapted to multi-component systems. Li et al. (2016) also combine these two dependen-cies, using L´evy copulas to model the correlation between unit degradation. They find that both depen-dencies significantly influence optimized maintenance strategies. However, both Mercier and Pham (2014) and Li et al. (2016) limit their studies to systems with 2 components. In general, Kang et al. (2019) stress a significant lack of research on the economic and stochastic dependency for multiple turbines at a wind farm, despite its relevance.

With this study, we aim to make the following contributions to the literature. Firstly, we address the clearly limited research on the combination of economic and stochastic dependencies, by constructing a flexible modeling framework. Our Markov model allows for the optimization of maintenance strategies for systems subject to external factors which directly influence degradation processes. Secondly, the introduced model uses degradation information more efficiently than previous models, allowing for the optimization of large multi-unit systems. Finally, in our study we let external factors influence the ability to perform maintenance, and we review the effects of such limitations on maintenance policies.

3 Problem description

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June 30, 2019 3. PROBLEM DESCRIPTION

weather type and a bad weather type. The bad weather type is assumed to result in a higher deterioration rate than the good weather type. Additionally, we assume that the bad weather type does not allow the execution of maintenance actions. At any time, the current weather state is characterized by two components. Firstly, we have the current weather conditions, given by ω1. Additionally, we have a weather forecast, denoted by

ω2. Because we distinguish two weather types, we have that ω1∈ {good, bad}, corresponding to the good

and bad weather type respectively. The number of different weather forecasts is given by nω∈ N+, and we

have ω2 = {1, 2, . . . , nω}. Here, a higher value of ω2 implies a higher probability that the weather state

in the next period is bad. However, different weather forecasts may also yield different probabilities for next period weather forecasts. Because of this, the current weather forecast also influences the likelihood of weather types more than a period from now. We denote the current weather state as ω = (ω1, ω2).

Transitions in weather states are modeled by using a discrete-time Markov chain, in which different weather forecasts result in different transition probabilities.

The deterioration of units is also modeled by a discrete-time Markov chain. As the deterioration rate is dependent on the weather type, we consider two different (m + 1) × (m + 1) transition probability matrices for this. For good weather, we have the matrix Pgood; this matrix is named Pbadfor the bad weather type.

These matrices are such that

m+1 X k=j pgood[i, k] ≤ m+1 X k=j pbad[i, k],

for all i, j ∈ {1, . . . , m + 1}. Here pgood[i, k] denotes the k-th element in the i-th row of Pgood. Due to this

condition, turbines are more likely to transition to higher deterioration states in bad weather, than when the weather is good.

Maintenance on these units can be done preventively and correctively. We assume that all maintenance actions that will be carried out in a period have their effect at the start of this period. Thus, maintenance on a unit i in period t is only corrective if this unit is in state m + 1 at the start of this period, and units that are maintained in a period also deteriorate during this same period. The costs of maintaining a single unit correctively are given by ccm, and are given by cpmwhen this maintenance action is preventive. We have

ccm≥ cpm, penalizing the failure of units. This assumption is commonly made in literature, for example by

Olde Keizer et al. (2018) and Cha et al. (2018). Furthermore, we assume an economic dependency in the sense that a setup cost csetup is incurred if maintenance is carried out, independent of the number of units

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June 30, 2019 4. METHODOLOGY

in the failed state at the beginning of a period, and it is not maintained in that period. As downtime costs depend on the production rate of a unit, we assume these costs differ for the two weather types. In the good weather state, the downtime cost is cgood_downper unit per unit time, whereas this is cbad

downper unit per unit time

if the current weather type is bad. We assume that cbad

down≥ c

good

down, as bad weather generally implies stronger

wind, and thus the ability to produce more energy.

We let na denote the maximum number of units that can be maintained in good weather. We assume

that in the bad weather type, no units can be maintained. In any period, if we choose to maintain a total of a units, we maintain the a units that are the most deteriorated. Maintenance actions are assumed to be perfect, thus the maintained units return to deterioration state 1.

4 Methodology

In this section we describe the methods used to model and analyze the problem that we consider. In Section 4.1, we model the problem that we consider as a Markov decision process. Thereafter, in Section 4.2, we explain the derivations of our probability transition matrices, which are vital to the model, in extensive detail. Lastly, in Section 4.3, we briefly describe the value iteration algorithm, which we use to determine optimal maintenance policies.

4.1 Markov decision process

We continue to model the problem that we consider as an MDP. In this formulation, we describe the system state by the deterioration states of all n units, and by the current weather type and forecast. Studies that consider multi-unit systems typically denote the deterioration state as a vector of deterioration states of the separate units. Examples of this are Zhang et al. (2012) and Olde Keizer et al. (2018). In this notation, a variable xidenotes the current deterioration level of unit i, and the complete system deterioration state is

given by x = (x1, . . . , xn). Although this notation allows for easy calculations of transition probabilities,

the number of deterioration states increases exponentially in the number of units. However, because we consider systems consisting of identical components, we can use an aggregated notation instead. In this notation we indicate the deterioration state of the system by the number of units per deterioration level.

We denote our current deterioration state by s = (s1, s2, . . . , sm+1), where si is the number of units

currently in deterioration level i ∈ {1, 2, . . . , m + 1}. We denote the total set of deterioration states s by S = n(s1, s2, . . . , sm+1) :Pm+1_i=1 si= n, si∈ N+, i = 1, . . . , m + 1

o

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greatly reduced number of deterioration states, as we immediately take into account that, for instance, state (x1, x2) = (i, j) is equivalent to state (x1, x2) = (j, i) for any i, j ∈ {1, 2, . . . , m + 1}. To the best of our

knowledge, this formulation for the state space has only been used by Beek (2018). He states that, given n units and m + 1 deterioration levels, the total number of deterioration states is

   n + m m   = (n + m)! n! · m! ,

as opposed to the (m + 1)n deterioration states we would have in the more standard non-aggregated no-tation. If, for example, we consider n = 15 units and m = 2 deterioration levels before failure, this would result in only 136 deterioration states, far less than the 14,348,907 states we would have other-wise. Additional to the current deterioration state of the system, the state information also consists of the current weather type and its forecast ω = (ω1, ω2), where ω1 ∈ {good, bad} and ω2 ∈ {1, 2, . . . , nω},

with nω ∈ N+. The resulting set of weather states, which we define as Ω, is thus given by Ω =

{(ω1, ω2) : ω1∈ {good, bad}, ω2∈ {1, 2, . . . , nω}}.

In general, transitions to subsequent decision epochs can be split into three steps, as shown in Figure 1. We begin with a state, given by (s, ω). We first maintain a number of units a, altering the deterioration state to sa. Here we have that a ≤ A(ω1), where we have

A(ω1) =      0, if ω1= bad, na, if ω1= good.

As stated earlier, maintenance actions a always maintain the a units that are most deteriorated. This process is formalized by the deterministic function f (s, a), which returns a state sa_{as a function of s and}

a. Thus, f (s, a) equals f (s, a) =                                s, if a = 0, (s1+ a, s2, . . . , sm, sm+1− a), if 0 < a ≤ sm+1, (s1+ a, s2, . . . ,P m+1 i=msi− a, 0), if P m+1 i=m+1si< a ≤P m+1 i=msi, .. .

(s1+ a,Pm+1_i=2 si− a, 0, . . . , 0), if Pm+1_i=3 si< a ≤Pm+1_i=2 si,

(n, 0, . . . , 0), if Pm+1

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June 30, 2019 4. METHODOLOGY (s, ω) Maintenance action (sa_{, ω)} _(s0_{, ω)} _(s0_{, ω}0₎ sa _{= f (s, a)} Unit deterioration P (s0|sa_{, ω)} Wind changes P (ω0| ω)

Figure 1: Visualization of the MDP

Corresponding to this action, we have an immediate cost C1(s, a). Given a maintenance action a, we pay

corrective maintenance costs ccmfor a units if a ≥ sm+1, and for sm+1units otherwise. Keeping in mind

the costs for preventive maintenance cpmand setup costs csetup, we thus have the cost function

C1(s, a) = csetup· Ia>0+ cpm· a + (ccm− cpm) · min {a, sm+1} ,

where the term Ia>0in this function is an indicator function which equals 1 if a > 0, and equals 0 otherwise.

After the maintenance action has been performed, we consider the deterioration from state sa_{to state s}0_.

Given a current wind state ω, units deteriorate according to a Markov chain. For good weather conditions, we have the (m + 1) × (m + 1) transition probability matrix Pgood, and for bad weather conditions this

is Pbad instead. We note that both of these matrices are assumed to be upper triangular, implying that the

deterioration level of a unit never improves without maintenance. Given these matrices we can calculate the transition probabilities from a state sato a state s0. These transition probabilities, denoted by P (s0| sa_{, ω),}

are derived in Section 4.2. At this point we calculate the costs incurred due to downtime. As we assume maintenance to occur at the beginning of a period, and units deteriorate over the remainder of the period, we only incur downtime costs for units that are in the failed state immediately after the maintenance inter-vention. Of course, these downtime costs depend on the current weather type. We thus have

C2(sa, ω) = (cgooddown· Iω1=good) · s

a m+1+ (c bad down· Iω1=bad) · s a m+1,

where sam+1is the amount of units in m + 1 after action a is completed.

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4.2 Transition probabilities

Due to the aggregated notation for the deterioration states, calculating transition probabilities turns out to be difficult. This is the case because a transition from a state s to s0can generally occur in multiple manners. For example, when m = 2, a unit i can not only deteriorate from deterioration level 1 to level 2, but also directly to level 3. Thus, when we observe a transition from the state (1, 1, 0) to (0, 1, 1), this may either be a result of a single unit deteriorating from deterioration state 1 to state 3 directly, or a unit deteriorating from state 1 to state 2 and another unit deteriorating from state 2 to state 3. Especially when we consider more than m = 2 deterioration levels before failure, state transitions become increasingly complex. Before we extensively detail our method for deriving transition probabilities, it is important to note that in the remainder of Section 4.2, we disregard the different weather states. Thus, instead of the two matrices Pgood

and Pbad, we consider a single (m + 1) × (m + 1) transition probability matrix P , with elements p[i, j].

Our deterioration state notation has been applied before by Beek (2018). He derives the transition probabilities corresponding to a transition from state s to state s0 using multiset permutations. That is, states s and s0are transformed to the states x and x0respectively, which are vectors for which the elements denote the deterioration levels of all separate units. However, due to the aggregated notation of s and s0, there may be multiple vectors x or x0 that may correspond to these states. Thus, one feasible vector x is fixed, and the complete set of feasible vectors for x0 is considered. We denote this set by X. Given the matrix P with transition probabilities per unit, any probability of a transition from a state x ∈ X to a state x0∈ X is easily calculated. The transition probability for a transition from state s to a state s0_{is then simply}

the sum of these probabilities, over every possible x0 ∈ X. For a more detailed explanation, we refer to the original thesis.

Although this derivation is conceptually simple, it has one clear drawback. For large numbers of units n, the set X becomes extremely large. If we, for instance, consider n = 15 units, with m = 2 deterioration levels before failure, the set X contains on average 105,507 distinct elements. Additionally, a different set X has to be determined for every pair of states s, s0 ∈ S. As a result, this method quickly becomes computationally expensive, nullifying the purpose of the aggregated notation. We therefore continue by introducing a more efficient method to calculate these transition probabilities.

We consider a transition from the state s = (s1, s2, . . . , sm+1) to state s0 = (s01, s02, . . . , s0m+1).

Furthermore, we let ki,j ≥ 0 denote the number of units that deteriorate from state i to state j, where

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Generally, there are multiple feasible values for these ki,j that would constitute a transition from state s

to s0. However, we initially consider these values fixed. Thus, we first assume that we know some fea-sible set of values for ki,j that would constitute a transition from state s to s0. Furthermore, we define

ki= (ki,i, . . . , ki,m+1) for all i ∈ {1, . . . , m + 1}.

We can derive the probability that this specified transition occurs, by first considering the transitions from every different deterioration level separately. Let us consider all transitions from deterioration state m. Clearly, all units that are currently in state m either remain in this state, or deteriorate further to the failed state m + 1. The quantities corresponding to these options are given by km= (km,m, km,m+1). We

define Pm(km) as the probability of the transitions given by km, given by a binomial distribution:

Pm(km) =

(km,m+ km,m+1)!

km! · km+1!

p[m, m]km,m_{· p[m, m + 1]}km,m+1_, ₍₁₎

where the fraction in the equation is the binomial coefficient. Similar derivations can be made for transitions from states lower than m. However, for any state i < m, we have to consider more than two quantities ki,j

to specify all transitions coming from deterioration state i. Thus, Pi(ki), the probability of the transition

specified by kifollows a multinomial distribution instead. We let m(ki) denote the multinomial coefficient,

given by m(ki) = Pm+1 j=i ki,j ! Qm+1 j=i ki,j! . Additionally, we define p(ki) = m+1 Y j=i p[i, j]ki,j_.

Combining this, we find an expression for Pi(ki), given by

Pi(ki) = m(ki) · p(ki). (2)

Finally, the probability of the transition specified by some fixed ki,j, is given by

P (k1, k2, . . . , km+1) = m+1

Y

i=1

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where we note that Pm+1(km+1) is equal to 1, as the final deterioration state is absorbing.

However, there are multiple combinations of values for ki,j. For a set of values for ki,jto be a feasible

solution for a transition from state s to state s0, we require that

si= m+1

X

j=i

ki,j, ∀ i ∈ {1, 2, . . . , m + 1},

simply implying that all units currently in state i end up in some state j ≥ i. Similarly, we have

s0_i=

i

X

j=1

kj,i, ∀ i ∈ {1, 2, . . . , m + 1}.

Using this, the following expression for ki,i, the number of units that remain in their respective deterioration

level i, easily follows:

ki,i= si− m+1

X

j=i+1

ki,j, ∀ i ∈ {1, 2, . . . , m + 1}. (4)

Furthermore, we have that the difference between the current and next period amounts of units in a state i is given by the amount of units that transition to state i, minus the amount of units that transition from this state. Thus, s01− s1= − m+1 X j=2 k1,j, s0_i− si= i−1 X j=1 kj,i− m+1 X j=i+1 ki,j, ∀ i ∈ {2, 3, . . . , m}, s0m+1− sm+1= m X j=1 kj,m+1.

We can rewrite this, such that we have

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s1 s2 s3 s4

s0₁ s0₂ s0₃ s0₄

k1,1 k1,2 k1,3 k1,4

(a) Original problem

s2 s3 s4 y2 2= s02− k1,2 y32 y 2 4 k2,2 k2,3 k2,4

(b) Sub-problem for fixed k1

Figure 2: Visualization of the transition probability sub-problem structure.

For the remainder of this section, we assume that empty summations evaluate to 0 for more convenient notation. For example, we assume thatP1

j=2kj,i= 0. Then, if we substitute (6) and (7) in (5), we find

s0₁− s1= m X i=2  si− s0i+ i−1 X j=2 kj,i− m+1 X j=i+1 ki,j  + sm+1− s0m+1+ m X j=2 kj,m+1 = m+1 X i=2 si− s0i.

Since s and s0 are given beforehand, we thus find that (5) is linearly dependent on equations (6) and (7) and thus is superfluous. As each of the variables k1,iis present in exactly one of the equations given by (6)

and (7), we can easily see that these equations are linearly independent. Furthermore, because the variables ki,iare already uniquely defined by (4) for all i, we have m(m − 1)/2 different variables ki,j that are not

yet given. However, as we also have m linearly independent equations given by (6) and (7), we can also express m of these variables in terms of other ki,j. As a result, we find that any transition from a state s to

s0can be fully specified by fixing at most m(m − 1)/2 − m = m(m − 3)/2 variables. There are of course other constraints on these variables ki,j, which we will specify later.

We let Ki define the set of feasible values for ki. However, this set depends on the values of kj,

where j 6= i. Because of this, we continue to derive a recursive expression for the transition probabilities. We denote yi = (yi_i, y_i+1i , . . . , y_m+1i ) as the amount of units that transition from (and therefore also to) deterioration states of at least i, given some fixed set of values for k1to ki−1. That is, yji = s0j−

Pi−1

l=1kl,j

for i ≥ 2, and y_j1 = s0_j. We then define the probability Qi(yi | s) as the probability that a transition

from s to s0 occurs, given some set of values for k1, k2, . . . , ki−1. This allows us to consider a smaller

sub-problem, in which we consider a transition from a subset of s, given by (si, si+1, . . . , sm+1), to the

state yi_{. Here Q}

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structure is visualized in Figure 2.

Firstly, we consider i = m. Here, Qm(ym| s) is the probability that the units currently in states m and

m + 1 transition such that ymmof these units end up in state m, and y m

m+1end up in state m + 1. As a result,

we have that km,m = ymm, and km,m+1 = ym+1m − sm+1. Thus, given all prior values for ki,j, there is a

unique solution for km. The resulting expression for Qm(ym| s) is then

Qm(ym| s) = Pm(ymm, y m

m+1− sm+1),

where Pmis as in (3). Then, for all states i < m, Qi(yi | s) is the probability that the transition specified

by kioccurs, multiplied by Qi+1(yi+1| s) and summed over all feasible ki. We thus have

Qi(yi| s) =

X

ki∈Ki

Pi(ki)Qi+1(yi+1 | s), ∀ i ∈ {1, 2, . . . , m − 1}. (8)

Here we note that y_ji+1 = y_ji− ki,jfor all j > i. Thus, to determine the value of Qi(yi | s), we first fix a

ki∈ Ki, and then solve the smaller sub-problem given by Qi+1(yi+1| s) specific to this ki. We do this for

every ki∈ Kiseparately.

All that remains for a complete expression of the transition probabilities is to find expressions for Ki.

Firstly, we note that as units do not improve without maintenance actions, we have

ki,i= yii, ∀ i ∈ {1, 2, . . . , m}.

This holds, because yi_iis the amount of units that end up in state i, that previously were not in a state lower than i. In terms of the sub-problem, this can be seen as the number of units in the first deterioration level, that do not deteriorate further. All units that end up in this state, must have been in state i at the beginning of the period.

Secondly, we know thatPm+1

j=i ki,j = si for all i. By rewriting this we can also find the following

expression for ki,m+1:

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We define k_i,jas the lower bound for ki,j, and ki,jas the corresponding upper bound, for all

i ∈ {1, 2, . . . , m − 1}, and j ∈ {i + 1, i + 2, . . . , m}. Finding an expression for ki,jis relatively simple.

Firstly, we know that ki,j ≤ yji for any state j > i. This holds, as otherwise

Pj

i=1ki,j > s0j, considering

all ki,j are non-negative. Thus, more units would transition to state j than ultimately end up in this state.

Furthermore, we know thatPj

l=iki,l ≤ si for any j ≥ i, as otherwise more units would transition from

state i than previously were in this state. As a result, we have that

ki,j= min ( yi_j, si− j−1 X l=i ki,l ) = min ( yi_j, si− yii− j−1 X l=i+1 ki,l ) , (9)

for all i ∈ {1, 2, . . . , m − 1} and j ∈ {i + 1, i + 2, . . . , m}. We continue by finding a lower bound for ki,j.

As we only have non-negative values for ki,j, we immediately know ki,j ≥ 0. Furthermore, to ensure the

values for ki,jare feasible, we have that

j X l=i+1 ki,l≥ j X l=i+1 yl− sl, (10)

for all j > i. The logic behind this is as follows. Suppose the number of units in deterioration level i + 1 increases such that yi

i+1− si+1 ≥ 0. This increase can only occur if a sufficient number of units transfer

from deterioration level i to i + 1. Thus, ki,i+1 ≥ yii+1− si+1. Furthermore, we might have a similar

increase in the number of units in deterioration level i + 2, thus yi_i+2− si+2≥ 0. This increase can only be

a result of transitions from deterioration level i to i + 2, or from i + 1 to i + 2. As a larger value for ki+1,i+2

would also require a larger value for ki,i+1, we thus find that ki,i+1+ ki,i+2≥ yi+1i + y i

i+2− si+1− si+2

should hold. Equation (10) generalizes this logic for all following deterioration levels. Finally, the values for ki,jare given by

k_i,j= max ( 0, j X l=i+1 (yi_l− sl) − j−1 X l=i+1 ki,l ) , (11)

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referred to as Ki, is given by

Ki =

(ki,i, . . . , ki,m+1) : ki,i= yii, ki,m+1= si− yii− m

X

i=j+1

ki,j,

ki,j ∈ {ki,j, . . . , ki,j} ∀ j = 2, . . . , m

, (12)

for all i ∈ {1, 2, . . . , m − 1}. Finally, the probability that a transition from state s to s0occurs is given by

P (s0 | s) = Q1(s0| s).

Of course, this still ignores the current weather states. Thus, to finalize the derivation of the transition probability matrices, this procedure is used twice, once with Pgoodand once with Pbad.

Example

As an example, we consider a system withn = 4 units, and m = 2 deterioration levels before failure. Each of these units deteriorates according to a Markov chain given by the transition probability matrix

P =       0.5 0.25 0.25 0 0.4 0.6 0 0 1       .

We consider a transition from the current states = (3, 1, 0) to the state s0= (2, 1, 1). We want to calculate P (s0 | s) = Q1(s0 | s). We first need to determine K1, the set of feasible values fork1, as given by(12).

Sincey1= s0, we immediately find thatk1,1= s01= 2. Then, using (11) we have that

k1,2= max {0, 1 − 1} = 0.

Additionally, by(9) we find

k1,2= min {1, 1} = 1.

We thus have thatK1 = {(2, 0, 1), (2, 1, 0)}. We note that P1(2, 0, 1) = P1(2, 1, 0) = 0.1875, as given

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different sub-problems separately. We first considerk1= (2, 0, 1). Because of this, y22= s02− k1,2 = 1 and

y2

3= s03− k1,3 = 0. Thus, we have the sub-problem given by Q2(1, 0), of which the solution is immediately

given by(1). We find Q2(1, 0) = 0.4. Similarly, if we let k1 = (2, 1, 0) we find the sub-problem given by

Q2(0, 1), where Q2(0, 1) = 0.6. Equation (8) gives the final result:

P ((2, 1, 1) | (3, 1, 0)) = 0.1875 · 0.4 + 0.1875 · 0.6 = 0.1875.

Thus, a transition from states = (3, 1, 0) to s0 = (2, 1, 1) occurs with probability 0.1875.

4.3 Value iteration

We aim to find stationary -optimal policies using value iteration. In the value iteration algorithm, we consider a sequence of values corresponding to states, vn(s, ω), where n = 0, 1, . . .. As our MDP is modeled in three distinct steps, this sequence can be determined by

un₂(s0, ω) = X ω0_∈Ω Pwind[ω, ω0] · vn−1(s0, ω0), un₁(sa, ω) = C2(sa, ω) + X s0_∈S P (s0| sa_{, ω) · u}n 2(s0, ω), vn(s, ω) = min a∈{0,...,A(ω)} {C1(s, a) + un1(f (s, a), ω)} .

In order to do this, we first initialize v0_{(S, ω) = 0 arbitrarily, for all s ∈ S, ω ∈ Ω. The value function}

vn_{(s, ω) is then iteratively updated. We continue with these iterations until the stopping criterion is met,}

which is given by max s∈S,ω∈Ωv n_{(s, ω) − v}n−1_{(s, ω) −} _min s∈S,ω∈Ωv n_{(s, ω) − v}n−1_{(s, ω) < ,}

where is an arbitrary precision level. When this desired precision level is met, we can find our decision rule, d(s, ω), using

d(s, ω) ∈ arg min

a∈{0,...,A(ω)}

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June 30, 2019 5. NUMERICAL RESULTS

5 Numerical Results

In this section we will analyze several numerical examples for the problem that we consider. In Section 5.1, a base case is considered, for which an optimal policy is given. In Section 5.2 we consider settings in which the different weather types occur with unequal probabilities. Thereafter, Section 5.3 considers the problem with alternative deterioration probability matrices. In Section 5.4 analyzes the effect of changing the maximum number of units that can be maintained in a single period. The last setting considered is in Section 5.5, in which the number of deterioration states before failure is enlarged.

5.1 Base case

We first consider a base case. In this model we have a setup cost csetup = 5, a preventive maintenance

cost cpm = 3, and a corrective maintenance cost ccm = 5. The downtime costs per unit per period are

cgood_down = 1 in the good weather type, and cbad

down= 2 in the bad weather type. Initially, we consider n = 20

units, each with m = 2 deterioration levels before failure. With this value for m, we mimic a delay-time model, a commonly used approach to model deterioration (Wang, 2012). We thus consider a state in which units function perfectly, one in which they are deteriorated, and finally the failed state. The corresponding transition probability matrices are given by

Pgood=       0.925 0.05 0.025 0 0.925 0.075 0 0 1       and Pbad=       0.75 0.225 0.025 0 0.75 0.25 0 0 1       .

Clearly, the transition probabilities to higher states are considerably larger for the bad weather type than for the good weather type. Furthermore, we consider nω= 3 different weather forecasts, implying that we

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Table 1: Base case wind transition probabilities for different wind states.

ω0

P (ω0| ω) (good, 1) (good, 2) (good, 3) (bad, 1) (bad, 2) (bad, 3)

ω (good, 1) 0.5 0.2 0.1 0.1 0.05 0.05 (good, 2) 0.1 0.3 0.1 0.1 0.3 0.1 (good, 3) 0.05 0.05 0.1 0.1 0.2 0.5 (bad, 1) 0.5 0.2 0.1 0.1 0.05 0.05 (bad, 2) 0.1 0.3 0.1 0.1 0.3 0.1 (bad, 3) 0.05 0.05 0.1 0.1 0.2 0.5

With these values, the weather transition probabilities only depend on the current weather forecast, and not on the current weather type, i.e., the first three rows of Table 1 are equal to the last three rows. Furthermore, we have that the weather forecasts given by ω2= 1 and ω2= 3 are opposites. As the weather

forecast ω2 = 2 implies that transitions to the good and bad weather types occur with equal probability,

we have that both weather types are equally likely in the long run. With these transition probabilities, the forecast given by ω2= 1 implies that the next period weather type is good with probability 0.8. However,

for this forecast, there is also a probability of 0.6 that the next period weather forecast is ω2 = 1 as well.

Thus, this forecast not only implies that the next period is likely to have a good weather type, but also that the period thereafter is more likely to be of the good weather type than the bad one. As ω2 = 3 yields the

exact opposite transition probabilities, this forecast implies that it is likely that multiple sequential periods will have bad weather.

Lastly, we assume that at most na = 10 units can be maintained if the weather is good. Because of

this, half of the total number of turbines can be maintained in a period, which allows for flexible policies. Cases with different values for naare reviewed in Section 5.4. All policies shown are found with the value

iteration algorithm as shown in Section 4.3, based on a precision level of = 10−7.

Figure 3 shows the maintenance policy for the different weather states with ω1= good. The horizontal

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June 30, 2019 5. NUMERICAL RESULTS 0 5 10 15 20 0 5 10 15 20 state 1 state 2 (a) ω = (good, 1) 0 5 10 15 20 0 5 10 15 20 state 1 state 2 (b) ω = (good, 2) 0 5 10 15 20 0 5 10 15 20 state 1 state 2 0 2 4 6 8 10 action (c) ω = (good, 3)

Figure 3: Optimal number of units to maintain for different weather forecasts in the base case.

Firstly, opportunistic maintenance is performed in all three weather states; any maintenance intervention includes as many units as possible. Thus, when any maintenance action is performed, the optimal policy is to either maintain the maximum nanumber of units, or to maintain all units that have deteriorated or failed.

Additionally, regardless of the weather forecast, no maintenance actions are planned unless units have failed. In Figure 3, this results in the set of light gray states on and just below the diagonal, for which no units are maintained. This is likely because the failure probability of deteriorated units is not too high in our transition probability matrices. As a result, preventive maintenance is only worthwhile if it can be performed opportunistically. Furthermore, maintenance is never performed on a single unit, any actions performed maintain at least 2 units. This is likely due to the relatively high setup cost.

However, the different weather forecasts do clearly affect the optimal maintenance actions. As expected, the most positive weather forecast represented in Figure 3(a) amounts to less maintenance interventions than the worse weather forecasts found in Figure 3(b) or Figure 3(c). This is mainly observable when a significant number of units are in the deteriorated state. When observing these figures, the set of states around the diagonal in which no maintenance is performed, is clearly larger for the best weather forecast, and smaller for the worst weather forecast, when compared to the neutral forecast ω2= 2. For the neutral

forecast, maintenance is only performed if there are at least two units in the failed state, whereas performing maintenance is sometimes optimal if only on unit has failed in the most negative weather forecast. Clearly, when many units have deteriorated, the risk of not being able to maintain them for several periods results in earlier scheduling of maintenance actions. For the good weather forecast this risk is far smaller, allowing the postponement of maintenance actions.

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four units are deteriorated, maintenance is only carried out if another 3 units have failed. However, when few units are in the deteriorated state, maintenance actions are already performed when only two units have failed. We thus perform maintenance earlier in the latter case, while the overall system state is better. A likely explanation for this, is that when many units have deteriorated, clustered execution of maintenance actions in future periods is likely. However, when most states are as-good-as-new, and only a few states have failed or deteriorated, it is unlikely that future maintenance actions can be clustered by not performing maintenance in the current period. This observation, where maintenance actions are sometimes delayed to allow clustered maintenance in the future, is in line with results found by Olde Keizer et al. (2018).

5.2 Weather transition probabilities

Of course, weather conditions are subject to seasonality. In the base case, we assumed good and bad weather to be equally likely. It makes sense to also look at different settings. We thus consider a setting in which good weather states are more likely than bad ones, and vice versa. We introduce a good weather season, which could be considered to be summer, in which good weather states occur most often. Additionally, we consider a bad weather season, winter, in which bad weather states are more likely. Table 2 shows the transition probabilities corresponding to the good season, Table 3 those corresponding to the bad season. Given these transition probabilities, the good season will have good weather approximately 64.4% of all periods. These transition probabilities are such, that when ω2 = 3, there is still a 80% chance that the

weather in the next period will be bad. However, as the prediction ω2 = 1 is far more likely than ω2 = 3,

the weather generally does not remain bad for many subsequent periods. As the bad season is simply the opposite of the good season, it has bad weather nearly 64.4% of the periods. For both weather seasons, the single period weather forecast given by ω2= 2 is equal to that of the base model, implying that both weather

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Table 2: Transition probabilities for different wind states in the good weather season.

ω0

Table 3: Transition probabilities for different wind states in the bad weather season.

ω0

Figure 4 shows the policies for the good weather season. It is immediately visible that the optimal maintenance actions are largely different from those in the base case. Most interesting is that with the good weather forecast, no preventive maintenance is performed. Any maintenance action targets only the units in the failed state. A probable explanation for this, is that with the current transition probabilities, an alert that a unit has deteriorated is given quite early. Thus, if it is highly likely that we can perform maintenance in following periods, preventive maintenance on deteriorated units would target units earlier than would be worthwhile.

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June 30, 2019 5. NUMERICAL RESULTS 0 5 10 15 20 0 5 10 15 20 state 1 state 2 (a) ω = (good, 1) 0 5 10 15 20 0 5 10 15 20 state 1 state 2 (b) ω = (good, 2) 0 5 10 15 20 0 5 10 15 20 state 1 state 2 0 2 4 6 8 10 action (c) ω = (good, 3)

Figure 4: Optimal number of units to maintain for different weather forecasts in the good weather season.

0 5 10 15 20 0 5 10 15 20 state 1 state 2 (a) ω = (good, 1) 0 5 10 15 20 0 5 10 15 20 state 1 state 2 (b) ω = (good, 2) 0 5 10 15 20 0 5 10 15 20 state 1 state 2 0 2 4 6 8 10 action (c) ω = (good, 3)

Figure 5: Optimal number of units to maintain for different weather forecasts in the bad weather season.

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visible by the diagonal set of states in the figures, for which no maintenance is performed. This set of states is larger in the good forecast than in the weather forecasts that are worse.

Figure 5 shows the policy for the bad weather season. Unlike the policy for the good season, this maintenance policy is similar to the base case. Preventive maintenance is clearly worthwhile, as units are expected to deteriorate faster, and future maintenance actions are more likely to be unavailable. When comparing to the base case, we see that maintenance actions are performed earlier in general, again visible by the smaller diagonal set of states in which no maintenance in performed. In the worst weather forecast, ω2 = 3, for which multiple sequential periods with the bad weather type are likely, maintenance is even

performed when no units have failed. In any state for which at least 8 units have deteriorated or failed, maintenance is performed.

The differences between the weather forecasts are relatively small in this weather season. In general, the number of units that should have deteriorated or failed before maintenance is viable, is smaller for worse weather forecasts. However, even when the next period is also highly likely to have good weather, maintenance is performed, early, sometimes even when only a single unit has failed.

The different cases amount to large differences in the average cost per period. In the good case, the long-run average cost per period is only 10.71, which is a decrease of 18.2% compared to the base case. Also, despite the small differences in the policy compared to the base case, the bad case results in a far greater average cost per period of 15.84, an increase of 21.0%. Clearly, different likelihoods of certain weather types greatly influence the costs for maintaining wind farms.

5.3 Alternative deterioration probabilities

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June 30, 2019 5. NUMERICAL RESULTS 0 2 4 6 8 10 10 12 14 16 18 20 state 1 state 2 (a) ω = (good, 1) 0 2 4 6 8 10 10 12 14 16 18 20 state 1 state 2 (b) ω = (good, 2) 0 2 4 6 8 10 10 12 14 16 18 20 state 1 state 2 0 2 4 6 8 10 action (c) ω = (good, 3)

Figure 6: Optimal number of units to maintain for the alternative delay-time model.

consider the matrices

Pgood=       0.9 0.075 0.025 0 0.7 0.3 0 0 1       and Pbad=       0.8 0.175 0.025 0 0.6 0.4 0 0 1       .

In Figure 6, a part of the optimal policy corresponding to these alternative probability matrices is shown. Only optimal actions for the states for which at least 10 units are as-good-as-new are presented. We note that for all system states that are not visible in the figure, the optimal decision is to maintain a = 10 units. When we compare this case to the base case, one main difference is observable. In this case, maintenance actions are performed more often, already when only a few units have reached the deteriorated state. Due to the large probability that deteriorated units fail quickly, the benefits of preventive maintenance are far greater than in the base case. As of such, when enough units have either deteriorated or failed, we always maintain as many units as possible.

Due to this, the differences between the forecasts are also less visible. However, we can see that a bad weather forecast may lead to earlier maintenance than the good forecast. For example, if one unit has failed and one unit is deteriorated, maintenance is only carried out for the worst weather forecast.

5.4 Maximum number of maintainable units

So far, we have considered the maximum number of units that can be maintained in good weather to be na = 10. Depending on the length of a time period, this may be an unrealistic number, as it allows for the

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na. Note that all other parameters are such as in the base case in Section 5.1.

We first consider the maximum number of units that can be maintained to be na= 7. The corresponding

maintenance policy is shown in Figure 7. We note that in this figure, the states that are coloured black correspond to an action a = 7, whereas this was a = 10 in previous sections. Firstly, we note that the

0 5 10 15 20 0 5 10 15 20 state 1 state 2 (a) ω = (good, 1) 0 5 10 15 20 0 5 10 15 20 state 1 state 2 (b) ω = (good, 2) 0 5 10 15 20 0 5 10 15 20 state 1 state 2 0 2 4 6 action (c) ω = (good, 3)

Figure 7: Optimal number of units to maintain when na= 7.

0 5 10 15 20 0 5 10 15 20 state 1 state 2 (a) ω = (good, 1) 0 5 10 15 20 0 5 10 15 20 state 1 state 2 (b) ω = (good, 2) 0 5 10 15 20 0 5 10 15 20 state 1 state 2 1 2 3 4 5 action (c) ω = (good, 3)

Figure 8: Optimal number of units to maintain when na= 5.

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every period, resulting in even earlier scheduling of maintenance. Note that, although the policies are structurally similar, the costs increase significantly when less units can be maintained per period. For na = 7 the long-term average cost per period is 13.50, and for na = 5 this is 14.31, resulting in a 3.1%

and 9.3% increase in costs respectively, when compared to the base case. Alternatively, we may consider

0 5 10 15 20 0 5 10 15 20 state 1 state 2 (a) ω = (good, 1) 0 5 10 15 20 0 5 10 15 20 state 1 state 2 (b) ω = (good, 2) 0 5 10 15 20 0 5 10 15 20 state 1 state 2 0 2 4 6 8 10 12 action (c) ω = (good, 3)

Figure 9: Optimal number of units to maintain when na = 13.

the case where up to na = 13 units can be maintained in a single period. The corresponding policy can be

found in Figure 9. As we can see from Figures 9(b) and 9(c), the policies for the two worst weather forecasts change little compared to the base case. We always maintain as many units as possible. Also, when we have a bad weather forecast, we maintain slightly less often than in the bad case. However, differences are marginal.

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a large grouped maintenance action. If instead a smaller number of units are deteriorated or failed, we are likely able to maintain all of them. This would result in most units being in the same deterioration level, and thus in a large probability of future clustered maintenance actions.

5.5 Multiple deteriorated states

In previous sections, we considered only m = 2 deterioration levels before failure. However, due to the wide variety of sensors available in wind turbines, more degradation information may be available for these units. Therefore, in this section, we will instead look at setting with m = 3 deterioration levels before failure. Here, the deterioration level 2 would imply that a unit has slightly deteriorated, while a deterioration level 3 would imply significant deterioration. We assume the transition probability matrices

Pgood=          0.8 0.15 0.0375 0.125 0 0.8 0.15 0.05 0 0 0.8 0.2 0 0 0 1          and Pbad=          0.7 0.2 0.075 0.025 0 0.7 0.2 0.1 0 0 0.7 0.3 0 0 0 1          .

We considered a case with n = 10 units, and assumed that at most Agood = 5 of these units can be

maintained in good weather types. Due to the higher dimensionality of this setting, we do not provide the entire policy. Instead, Table 4 shows the optimal decisions corresponding to some of the system states. Table 4: Optimal number of units to maintain for several states in a model with m = 3 deterioration levels before failure. ω2 s1 s2 s3 s4 1 2 3 5 0 4 1 5 5 5 5 0 5 0 0 5 5 6 0 4 0 0 0 4 3 4 2 1 0 0 3 4 3 2 1 0 3 3 8 0 0 2 2 2 2

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June 30, 2019 6. CONCLUSION AND DISCUSSION

unit has failed. Furthermore, in the bad weather forecast, it is even optimal to maintain 4 units preventively, while all other 6 units are in either deterioration level 1 or deterioration level 2.

Additionally, units in deterioration level 2 are never maintained. However, the number of units in this state may still determine the optimal action. Consider for example, the optimal action when one unit has failed, and two units are in the most deteriorated state. In the good weather forecast, we never maintain units, whereas we always do so in the bad weather forecast. However, for a neutral forecast, we only maintain units here when at most three units are only slightly deteriorated. Again, this is to group as many units in the same deterioration level as possible, to make future combined maintenance actions more likely. In general, we never maintain single units.

6 Conclusion and discussion

We have considered the optimization of a condition-based maintenance strategy for an offshore wind-farm consisting of multiple turbines, with both an economic and a stochastic dependency. The stochastic depen-dency is implemented by considering different weather conditions that have an impact on the deterioration processes of the turbines. Furthermore, these weather conditions potentially also limit the ability to main-tain units. We assume that forecasts for future weather conditions are available; these forecasts are therefore used to optimize maintenance policies.

We have formulated the problem as a Markov decision process. In this formulation we have used a compact notation for the system deterioration state, which allowed us to determine policies for far greater numbers of units than previous studies that used Markov decision processes. Furthermore, we presented a recursive framework to efficiently determine the state transition probabilities that result from this notation. The resulting maintenance policies were found using value iteration.

In our numerical results, we reviewed models with up to 20 turbines. A representative policy was given. We found that the different weather forecasts can significantly alter the planned maintenance actions. When the outlook for future weather conditions is positive, preventive maintenance is performed less often. However, when future weather is expected to restrict the opportunity to perform maintenance, combined maintenance actions may be viable even when little to no units have broken down.

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June 30, 2019 REFERENCES

operational costs rise significantly. Furthermore, when good weather is most likely, policies without pre-ventive maintenance may be optimal in settings with only a few deterioration states. Thereafter, a system in which degradation is revealed only shortly before failures is considered. We find that such a system leads to a simpler policy, which is less dependent on the different weather forecasts. Thereafter, the model is tested for different maximum amounts of units that can be maintained. Lastly, a smaller wind farm setting with a larger amount of deterioration states is reviewed.

As a result of this study, several future research topics have presented themselves. Firstly, despite the efforts made in this study, the combination of stochastic and economic dependencies has received lim-ited attention in literature. Models with more sophisticated stochastic dependencies are highly relevant for offshore wind farm maintenance planning. For instance, models with external factors that have both con-tinuous and shock-based effects on degradation processes could be used to explore the effects of wind and waves on turbines.

Furthermore, due to the complexity of wind farms and their maintenance optimization, many other additions can be made to this study. An example of this is to consider variable production rates. Uit het Broek et al. (2019) have shown that flexible production rates have the potential to reduce operational costs significantly, a topic that may be interesting to extend to multi-unit systems. A last suggestion we make, is related to the technical limitations of Markov decision processes. Although we successfully increased the number of turbines modeled, wind farms are also becoming increasingly large. As of such, increasing the size and complexity of these models is an ongoing problem. Future research may be directed towards the use of function approximators, such as multilayered perceptrons, to estimate the value functions used.

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Wind farm maintenance planning under forecastable weather conditions

Wind farm maintenance planning under

forecastable weather conditions

Wind farm maintenance planning under forecastable weather

conditions

Luuk Pentinga

Contents

List of Figures

List of Tables

1

Introduction

2

Literature

3

Problem description

4

Methodology

5

Numerical Results

6

Conclusion and discussion

References