An Integrated Optimization Framework for Multi-Component Predictive Analytics in Wind Farm Operations & Maintenance
AAn Integrated Optimization Framework for Multi-ComponentPredictive Analytics in Wind Farm Operations & Maintenance
Bakir, I. a , Yildirim, M. b,1, ∗ , Ursavas, E. a a Department of Operations, Faculty of Economics and Business,University of Groningen, Groningen, Netherlands b Industrial and Systems Engineering, Wayne State University, Detroit, MI, USA
Abstract
Recent years have seen an unprecedented growth in the use of sensor data to guide wind farm opera-tions and maintenance. Emerging sensor-driven approaches typically focus on optimal maintenanceprocedures for single turbine systems, or model multiple turbines in wind farms as single compo-nent entities. In reality, turbines are composed of multiple components that dynamically interactthroughout their lifetime. These interactions are central for realistic assessment and control of tur-bine failure risks. In this paper, an integrated framework that combines i) real-time degradationmodels used for predicting remaining life distribution of each component, with ii) mixed integer op-timization models and solution algorithms used for identifying optimal wind farm maintenance andoperations is proposed. Maintenance decisions identify optimal times to repair every component,which in turn, determine the failure risk of the turbines. More specifically, optimization modelsthat characterize a turbine’s failure time as the first time that one of its constituent componentsfail - a systems reliability concept called competing risk is developed. The resulting turbine failuresimpact the optimization of wind farm operations and revenue. Extensive experiments conducted formultiple wind farms with 300 wind turbines - 1200 components - showcases the performance of theproposed framework over conventional methods.
Highlights • A condition-based maintenance and operations model is proposed for wind farms. • Component and turbine dependencies on failure risks and maintenance are modeled. • A tailored solution algorithm is proposed to ensure computational scalability. ∗ Corresponding author
Email addresses: [email protected] (Bakir, I.), [email protected] (Yildirim, M. ), [email protected] (Ursavas, E.) The first two authors contributed equally to this work. a r X i v : . [ ee ss . S Y ] J a n A comprehensive experimental framework is developed via degradation and wind data. • The proposed approach provides significant improvements over benchmark models.
Keywords:
Degradation models, multi-component reliability, condition based maintenance, windfarm operations and maintenance, large scale mixed integer optimization
1. Introduction
Maintenance scheduling is a fundamental component of wind farm operations with far reachingimplications for equipment costs, market revenue, and maintenance crew logistics. In typical appli-cations, maintenance costs constitute 25% to 35% percent of the running costs of wind farms [1, 2, 3].To mitigate this cost, wind farm operators are continuously looking into methods to improve theeffectiveness of maintenance policies through use of real-time sensor data [4]. Condition monitoring(CM) systems identify indicators of component degradation within sensor data to develop accurateestimates for equipment condition. In traditional applications, CM systems alert the operators whenthe level of degradation in turbine components reach a certain level. These alerts are used to initiateimmediate maintenance actions. Such policies rely solely on the current degradation state of thecomponents. In reality, the complex relationships between component repair schedules, maintenancecrew logistics and revenue opportunities, require a proactive organization of wind farm operationsand maintenance (O&M) [5]. Prognostics based methods model the remaining life distribution ofthe components as they degrade, and provide the ability to plan maintenance and related activitiesbefore failure risks become eminent. This paper shows that when component-level prognostic predic-tions are integrated into wind farm operations and maintenance, the operators can sense early signsof degradation in every component, and jointly optimize O&M activities by carefully consideringthe complex stochastic and economic inter-dependencies among multiple components, turbines andwind farms.Despite their importance, wind farm O&M policies typically rely on ad-hoc decisions in practice[5, 6]. Currently the typical form of maintenance policies for wind farm maintenance is a combina-tion of strictly corrective policies that initiate maintenance actions upon the observation of failure,and time-based policies that perform maintenance at fixed time intervals [7]. In wind farm O&Mliterature, these policies are often augmented by the use of opportunistic maintenance , which groupsmaintenance actions together to reduce crew visits, production losses, traveling and setup expenses[3]. Within opportunistic maintenance, scheduled corrective and preventive maintenance tasks areused as an opportunity to perform additional maintenance tasks on other components within a singleturbine, or in multiple turbines within a farm.Conventional approaches to opportunistic maintenance in wind farms can be categorized under twoapproaches. The first approach uses optimization models to capture different aspects of operations2nd maintenance [8, 9, 10]. The second approach [11] provides structured policies to establishprocedural maintenance decisions. These approaches establish multiple threshold values in termsof age or operational conditions that identify optimal times to initiate the first and the subsequentmaintenance actions. A drawback of these conventional policies is that they do not use sensor data,and assume that the failure of identical components follow the same distribution. In reality, everycomponent exhibit significant variation in terms of how they degrade and fail, due to many factorsincluding manufacturing variations/defects, operational loading, and material imperfections.Recently, there has been a growing literature on using sensor data to infer accurate estimates forfailure likelihood of turbines and their constituent components. An extensive survey of sensor-driven failure prediction models can be found in [12]. These models rely on obtaining sensor-basedindicators called degradation signals that exhibit a strong correlation to the evolving degradationprocesses.
Degradation models develop stochastic formulations based on these signals to predict theremaining life distribution. Degradation modeling approaches for wind turbines are straightforwarddue to the relatively simple mechanical construction of wind turbine components [13]. Integrationof these life predictions into maintenance policies, however, still remains a fundamental challenge.The earliest research efforts that incorporate sensor data to wind farm maintenance had the objectiveof justifying the additional expense of condition monitoring equipment for individual components[14, 15]. Several authors extended on these papers to develop methods for sensor-driven maintenanceand operations. Majority of the work in the area focuses on detailed modeling of sensor-driven main-tenance for single turbine systems. [16] considers the sensor-driven maintenance of gearboxes underimperfect maintenance actions. This work is extended in [17] by enabling the maintenance decisionsto adapt to season-dependent and time-varying weather conditions. These studies demonstrate thatconsidering seasonality when making maintenance decisions helps in decreasing failure rates, as wellas reducing O&M costs. [18, 16, 17] all consider a single component type, respectively blades andgearboxes. While these policies provide significant developments in the field and consider importantfactors that impact degradation, they do not necessarily capture complex degradation-based andeconomic interactions between multiple turbines and components. As suggested by [19], sensor-driven maintenance policies that focus on a single turbine, may not scale well to complex wind farmsetting where the component interactions are significant. In fact, in certain cases sensor-driven sin-gle turbine policies may perform worse than simpler periodic maintenance policies that accuratelymodel the wind farm as a whole [19].The works that consider sensor-driven policies as well as opportunistic maintenance are scarce. Oneof the first studies to consider the actual condition of multiple components when making maintenancedecisions is [20]. In this study, a maintenance policy defined by two thresholds is proposed, wherethe first threshold initiates preventive maintenance, and the second - lower - threshold is used foropportunistic preventive maintenance. In a recent paper, [21] proposed a new method to identifyoptimal two threshold policies using a detailed model of offshore maintenance cost parameters. Akey assumption in these papers is that the thresholds are static. While threshold-optimization3olicies provide an important and practical approach, they do not necessarily reflect that thesethresholds exhibit significant variation across different time periods and turbines due to multitudeof factors such as crew management, market price, and wind uncertainty. In the recent work of[22] a wind farm with 100 turbines is considered on a component level. However, the developedmaintenance policy only considers opportunistic maintenance of multiple components within thesame wind turbine, neglecting possible dependencies between turbines. The dependencies betweenturbines are considered in [19] that proposes an optimization framework for the O&M schedulingfor wind farms. Although this work models dependencies between wind turbines, the dependenciesbetween components within wind turbines are neglected.Literature on opportunistic and sensor-driven maintenance approaches clearly demonstrate thatthese two approaches provide significant benefits when applied in isolation. The benefits of theirintegration, however, can be studied in more detail: The studies considering opportunistic mainte-nance are mostly time-based, and the studies that consider sensor-driven approaches are predom-inantly aimed at component- or, single-turbine-level. The studies that consider opportunistic andsensor-driven maintenance together on a wind farm level either neglected the inter-dependenciesbetween turbines [22], or between components [19]. To the best of the authors’ knowledge, thereis no optimization framework that develops opportunistic and sensor-driven ( i.e. prognostics-based )maintenance for wind farms that accurately model the complex interactions between failures andmaintenance actions at a component level. This forms the main contribution of the framework.To bridge this gap, this work proposes a unified framework that integrates component-level prog-nostics into large-scale mixed integer optimization models used for identifying optimal wind farmmaintenance and operations decisions. The proposed model considers component-level maintenanceactions to capture the inter-dependencies among components within a turbine, as well as compo-nents across turbines in multiple wind farms. A unique aspect of this approach is the integrationof opportunistic maintenance across three levels: components, turbines, and multiple wind farms.Main contributions of the paper are listed below:1. A novel wind farm operations and maintenance framework that adapts to real-time sensordata to capture dynamic interactions among turbine components, turbines and wind farmsis developed. The proposed framework offers a significant shift from existing sensor drivenpolicies that i) focus on single wind turbine systems with multiple components, or ii) modelmaintenance of multiple wind farms without considering component-level interactions.2. Component degradation models are integrated within a novel mixed integer decision opti-mization formulation used for identifying optimal wind farm operations and maintenance.This approach models dynamically evolving turbine failure risks as a function of i) real-timepredictions on component remaining life distributions, and ii) component-level maintenancedecisions. Component maintenance decisions and the resulting turbine failure risks are tightlyintegrated into O&M of multiple wind farms, and the routing for the maintenance crew.4. Modeling enhancements and algorithmic approaches to solve large instances of the resultingdecision optimization model is proposed. More specifically, a linearization that enforces a one-to-one mapping between component-level maintenance decisions and the state of the turbinesystem is developed. The model is decomposed into a two-stage form, where the masterlevel problem consists of wind farm- and turbine-level decisions, and the subproblem identifiesdecisions for the constituent components. The resulting reformulation enables a master-slavetype solution algorithm based on Benders’ and integer cuts.A comprehensive experimental platform that models the impact of different maintenance and oper-ations decisions is developed. Degradation processes in turbine components are emulated by using adatabase of vibration based degradation data collected from rotating machinery. Wind is modeledby using the database from KNMI North Sea Wind Atlas project [23]. The complex degradationbased and economic interactions between multiple components, wind turbines, and wind farms aresimulated. Extensive set of experiments suggest that the proposed policy provides significant ad-vantages over existing approaches, in terms of cost, reliability and the efficient use of equipmentlifetime.The remainder of the paper includes the following: Section 2 introduces the method for the unifiedframework, which covers degradation modeling for multi-component turbine systems, and its inte-gration into a novel large scale optimization model. Section 3 introduces the solution algorithm andhighlights its computational benefits. Section 4 presents the experimental results for benchmark androbustness studies. Finally, the conclusion is provided in Section 5.
2. Method
This section formally introduces the integrated wind farm operations and maintenance schedulingframework that is composed of two main modules: i) sensor-driven multi-component degradationmodels for quantifying failure risks, and ii) sensor-driven adaptive optimization for multi-componentopportunistic maintenance and operations. These modules are tightly integrated, i.e. sensor-drivencomponent remaining life predictions continuously update the operations and maintenance schedul-ing for the wind farm.The proposed framework assumes that condition monitoring technology exists for wind turbinecomponents to enable the predictions of component-level remaining life distributions. Conditionmonitoring, diagnostics and prognostics in wind turbines has a rich and growing literature. Formore details on this literature, we refer the reader to [24, 25, 26, 27, 28, 13].To elucidate the framework in detail, this section demonstrates how component specific low-levelsensor observations are used to update a series of modules that drive high-level decisions in wind5arm operations and maintenance.
The first task is to use real-time sensor information to derive accurate predictions on when theturbines and their constituent components are likely to fail. This requires a two step procedure.The first step is to use the component specific sensor data to predict remaining life distributionof all the components in a turbine. A mapping between component remaining life and the failureprobabilities of the turbine is then derived. This degradation framework enables the quantificationof failure risks both at a component and a turbine level.
Turbine components degrade over time due to use and aging. Degradation models characterize thisdegradation processes in detail to improve failure risk predictions. This paper uses a generalizedparametric degradation model for component k in turbine i in the form of the function: D i,k ( t ) = φ i,k ( t ; κ, θ i,k ) + (cid:15) i,k ( t ; σ ). In this formulation, φ i,k ( t ; κ, θ i,k ) defines the underlying base degradationfunction for component k of turbine i , and (cid:15) i,k ( t ; σ ) models the uncertainty due to degradation andmeasurement errors, with an associated variance parameter σ . The parameters κ and θ i,k denotethe deterministic and stochastic degradation parameters, respectively.It is assumed that engineering knowledge and historical data yield an initial estimate for the dis-tribution of the stochastic degradation parameter θ i,k , denoted by π ( θ i,k ). Further assumption isthat the operators observe a set of degradation signals that enable the update of the distribution ofthe stochastic parameter θ i,k to its posterior distribution counterpart υ ( θ i,k ) using Bayesian update.Given the updates on the degradation parameters, remaining life distribution of component k inturbine i can be computed as the first time that the degradation signal crosses a predefined failurethreshold Λ i,k . The remaining life of component k in turbine i , namely τ i,k can be evaluated as P ( τ i,k = t ) = P ( t = min ( s ≥ | D i,k ( s | υ ( θ i,k )) ≥ Λ i,k ). See [29] for more details on this class ofdegradation models.These remaining life predictions drive two submodules that are used by the operations and main-tenance optimization model: i) dynamic component maintenance cost functions, and ii) turbineremaining life predictions. 6 .1.2. Dynamic Component Maintenance Cost Function Given the updated remaining life distributions of component k , the expected cost of conductingmaintenance on the component at time t is derived, as follows [29, 30]: C i,kt o ,t = c pi,k P ( τ i,k > t ) + c fi,k P ( τ i,k ≤ t ) (cid:82) t P ( τ i,k > z ) dz + t o , (1)where c pi,k and c fi,k are the costs for preventive maintenance, and unexpected failure for compo-nent k , respectively. The function translates the remaining life distribution of component k into adegradation-based function of expected cost over time. Predictions on the component remaining life distribution are used to derive failure likelihood ofturbines as well. Assume that K i is the set of critical components in turbine i . The failure time ofturbine i , τ i can be defined as the first time that one of its critical components fail: P ( τ i > t ) = (cid:90) P (cid:18) max k ∈K i sup t o ≤ s ≤ t o + t ϕ ( s, θ i,k ) > (cid:19) (cid:89) k ∈K i υ ( θ i,k ) dθ i,k where ϕ ( s, θ i,k ) = D i,k ( s | θ i,k ) − Λ i,k , and t o is the age of the component. This failure definition usesa well known failure modeling concept called competing risk [31, 32, 33]. Turbine failure risks are highly dependent on component maintenance schedules. There are twomain assumptions to capture the impact of maintenance on turbine failure risks: i) a componentbecomes as good as new after maintenance, ii) a maintained component does not fail again withinthe planning horizon. Second assumption is reasonable so long as the planning horizon doesn’texceed an annual span. In fact, it is trivial to augment the proposed model to eliminate the secondassumption, which is chosen to be enforced here for notational convenience.The maintenance time for component k in turbine i is denoted as χ i,k . This is an integer value:assume that the maintenances may take place at the beginning of each period. The probability thata turbine survives until time t is: P (cid:0) τ i > t (cid:1) = P ( τ i,k > min ( t, χ i,k ) ∀ k ∈ K i )A key observation from this equation is that the probability of failure during ( t, t + 1], directlydepends on whether a maintenance was conducted at or before t , i.e. ϕ i,k ≤ t . This means that the7 igure 1: Integrating Component & Turbine-Level Risks.set of possible maintenance scenarios at each time period ( t, t + 1], has a cardinality of 2 |K i | ; i.e. eachcomponent has two states: maintained or not maintained. It is not computationally demanding toprecompute the turbine failure probabilities of these maintenance scenarios for each time period andturbine. This is an important point for integrating the model to optimization; i.e. only 2 |K i | eventsfor time t are considered, as opposed to the entire history with 2 t ·|K i | unique events. Given the predictive degradation models, expectations on failure likelihoods and associated main-tenance costs, the next challenge is to develop a fully-adaptive and comprehensive, multi-windfarmoptimization model for operations and maintenance scheduling. The proposed Multi-ComponentCondition Based Opportunistic Maintenance and Operations Model (MC-CBOM) tightly integratesthe dynamic maintenance cost functions and the predicted likelihoods of failure for every componentand turbine. Figure 1 highlights the information flow across component and turbine-level analyticsthat feed into the optimization models. Information exchange occurs in two levels. At a componentlevel, the dynamic maintenance cost function (1) is discretized for each time period and integratedinto the objective function of the optimization model. Turbine failure probabilities are reconstructedwithin the optimization model as a function of maintenance decisions and component failure proba-bilities. More specifically, variables associated with maintenance decisions and turbine maintenancescenarios ( as introduced in Section 2.1.4 ) enable the optimization model to capture the impact ofmaintenance decisions on turbine failure probability. The coupling between turbine maintenancesand operations is also explicitly captured within the framework.More specifically, the proposed model considers multiple layers of dependencies across componentsand turbines. In what follows, these dependencies are motivated with practical examples:8 omponent-level dependencies:
When a maintenance team visits a turbine to fix a certain compo-nent, it may make sense to fix another component that may also be highly degraded. A simplifiedexample could be that a blade repair order may also include a fix to another blade or a yaw controlsystem that may also be highly degraded. This would be particularly important when the team hasa significant setup cost associated with setting up the crane to access the turbine components.
Component-to-turbine dependencies:
Turbine failure is continuously characterized as a function ofits constituent components. A maintenance action on a component, would have a direct impacton component failure probability, and an indirect impact on the failure likelihood of the turbine aswhole. Both of these aspects are explicitly modeled in the MC-CBOM.
Turbine-level dependencies:
Given maintenance crew routing requirements, only a subset of turbinesthat are located in the same wind farm location l can be maintained simultaneously. It wouldtherefore be important to understand the cumulative impact of failure risks from multiple turbinesbefore a crew visits a certain location within a wind farm.Before formally developing the model, the notation is introduced. Let T , L , G , and K denote thesets of time periods, wind farm locations, wind turbines, and turbine components, respectively. Set G l represents the wind turbines in wind farm l , and its two subsets G lo and G lf denote the operationaland failed turbines at the start of the planning horizon. A turbine is classified as operational if all ofits components are operational, and failed if at least one of its components is in a failed state. Foreach turbine i within wind farm l , the set of components is also partitioned into subsets K l,io and K l,if , representing operational and failed components of turbine i at the beginning of the planninghorizon, respectively.To denote the preventive maintenance decisions, the binary decision variable z is used, which isdefined only over the set of currently operational components. Variable z l,i,kt will assume the value1 if preventive maintenance on currently operational component k of turbine i in wind farm location l is initiated at period t (and 0 otherwise). The preventive maintenance decisions incur a dynamicmaintenance cost, as given in equation (1). Corrective maintenance actions are represented withbinary variable ν , which is defined only over the set of components that are currently at a failedstate. The variable ν l,i,kt will assume the value 1 if corrective maintenance on component k of turbine i in wind farm l is initiated at period t (and 0 otherwise).Visits of the maintenance crew to wind farm locations and wind turbines are denoted with binarydecision variables x and α , respectively. If wind farm location l is visited by the maintenance crewat period t , then x lt will be 1; and if that visit included turbine i , then α l,it will also be 1. The crewdeployment costs for wind farm and turbine visits are denoted with V v,lF,t and V v,l,iT,t , respectively.Energy generation is represented with a continuous decision variable y , where y l,it denotes the energy(in MWh) generated by wind turbine i in period t ∈ T . There is a revenue π t associated with eachgenerated MWh of electricity. 9he remainder of this section details various components, i.e., objective function and constraints, ofthe MC-CBOM. The objective of the model is to maximize the total profit from all wind farm locations throughoutthe planning horizon. The objective function (2) calculates the net profit by subtracting crewdeployment, turbine maintenance, and expected turbine failure costs from the operational revenue.max (cid:88) l ∈L ,i ∈G l ,t ∈T (cid:16) π t y l,it − V v,lT,t α l,it (cid:17) − (cid:88) l ∈L ,t ∈T V v,lF,t x lt − (cid:88) l ∈L ,i ∈G lo ,k ∈K l,io ,t ∈T C l,i,kt ok ,t z l,i,kt − (cid:88) l ∈L ,i ∈G lo ,t ∈T C f,l,i ρ l,it (2)Note that every time the model is constructed, the dynamic maintenance costs, denoted with C l,i,kt ok ,t ,are computed using the remaining life predictions of operational wind turbine components andupdated with recent sensor readings. These costs are crucial in making preventive maintenancedecisions.The last term of the objective function evaluates the costs associated with turbine failure based onactual failure probabilities of the components within the turbine. Here, ρ l,it represents the failureprobability of turbine i of wind farm l in time period t . This variable is multiplied with a failurecost C f,l,i to obtain the expected failure cost. Constraints (3) mandate that a component undergoes preventive maintenance before its reliabilitydrops below a prespecified threshold ¯ ζ . To enforce this requirement, a time limit ζ l,i,k := min { t ∈T : P ( R l,i,kt oi > t ) < ¯ ζ } is computed for all components; and constraints (3) ensure that a preventivemaintenance is initiated on or before this time limit. Constraints (4) enforce that at most onecorrective maintenance is conducted on each failed component during the planning horizon. ζ l,i,k (cid:88) t =1 z l,i,kt = 1 , ∀ l ∈ L , ∀ i ∈ G l , ∀ k ∈ K l,io (3) (cid:88) t ∈T ν l,i,kt ≤ , ∀ l ∈ L , ∀ i ∈ G l , ∀ k ∈ K l,if (4)Turbine maintenance is denoted with binary variable α , where α l,it = 1 indicates that a maintenanceactivity (preventive or corrective) is being conducted on turbine i in period t . Constraints (5) and (6)10nsure the coupling of turbine visit and component maintenance decisions. This way, a turbine visitis scheduled if any of the turbine’s components undergoes preventive or corrective maintenance.Constraints (7) couple the wind farm and turbine visit variables by enforcing that a crew visitis planned for a wind farm location whenever a turbine in that location is to be visited by themaintenance crew. z l,i,kt ≤ α l,it , ∀ l ∈ L , ∀ i ∈ G l , ∀ k ∈ K l,io , ∀ t ∈ T (5) ν l,i,kt ≤ α l,it , ∀ l ∈ L , ∀ i ∈ G l , ∀ k ∈ K l,if , ∀ t ∈ T (6) α l,it ≤ x lt , ∀ l ∈ L , ∀ i ∈ G l , ∀ t ∈ T (7) Constraints (8) ensure that the same crew cannot visit multiple locations at the same time period.Constraints (9) limit the crew visit to location l , when the conditions do not allow a visit. (cid:88) l ∈L x lt ≤ , ∀ t ∈ T , (8) x lt = 0 , ∀ l ∈ L , ∀ t ∈ ¯ T l , (9)where ¯ T l denotes the set of time periods when a maintenance crew visit to location l is not possibledue to limiting conditions, such as weather-related or organizational restrictions. Constraints (10)enforce the required travel time for traveling between location l and l (cid:48) . x lt + x l (cid:48) t (cid:48) ≤ , ∀ l ∈ L , ∀ l (cid:48) ∈ L \ { l } , ∀ t ∈ { , . . . , T − θ l,l (cid:48) } , ∀ t (cid:48) ∈ { t, . . . , t + θ l,l (cid:48) } (10)where θ l,l (cid:48) denotes the required time to travel from location l to l (cid:48) . Constraints (11) limit the number of turbines on which maintenance is being conducted during timeperiod t to the maintenance capacity at that time period, which is denoted with M t . (cid:88) l ∈L (cid:88) i ∈G l α l,it ≤ M t , ∀ t ∈ T (11)11 .2.5. Energy Generation These constraints couple the energy generation decision variables, y , with component maintenancedecision variables, z and ν . Constraints (12) allow operational turbine i in location l to produceup to wind-induced production capacity at time t , namely p l,it , unless the turbine has an ongoingmaintenance. y l,it ≤ p l,it (1 − α l,it ) , ∀ l ∈ L , ∀ i ∈ G l , ∀ t ∈ T (12)Constraints (13) enforce that a failed wind turbine cannot produce electricity until it undergoescorrective maintenance. Once a turbine that started at a failed state is correctively maintained, itcan produce up to p l,it . Note that this decision considers the impact of turbine outage on productionloss. y l,it ≤ p l,it t − (cid:88) τ =1 α l,iτ , ∀ l ∈ L , ∀ i ∈ G lf , ∀ t ∈ T (13) The following constraints determine failure probabilities for each turbine i based on the conditionand maintenance status of each of its components. The binary variable u denotes whether or nota component underwent preventive or corrective maintenance before time period t ; u l,i,kt = 1 ifcomponent k of turbine i in wind farm l has been maintained before time period t . Constraints(14) and (15) establish this relationship between these variables and the maintenance schedulingvariables z and ν . t − (cid:88) τ =1 z l,i,kτ = u l,i,kt , ∀ l ∈ L , ∀ i ∈ G , ∀ k ∈ K io , ∀ t ∈ T (14) t − (cid:88) τ =1 ν l,i,kτ = u l,i,kt , ∀ l ∈ L , ∀ i ∈ G , ∀ k ∈ K if , ∀ t ∈ T (15)Failure probability of a turbine depends on the age and maintenance status of each of its components.Let H l,i define a set which contains all possible maintenance scenarios for each turbine i in wind farm l . If a turbine consists of n components, 2 n maintenance scenarios would be possible. For example,when a turbine consists of 2 components, the 4 possible maintenance scenarios are (i) no maintenanceis conducted on any of the components, (ii) maintenance is conducted only on component 1, (iii) maintenance is conducted only on component 2, and (iv) both components are maintained. Binaryvariable η is used to denote the maintenance status of a turbine, based on the maintenance status ofeach of its components. More specifically, η l,i,ht = 1 indicates that maintenance scenario h is realizedon turbine i of wind farm l in time period t . Constraints (16) establish this relationship.12 l,i,ht ≥ (cid:88) k ∈ K l,i,h u l,i,kt − (cid:88) k ∈ F l,i,h u l,i,kt − | K l,i,h | + 1 , ∀ l ∈ L , ∀ i ∈ G l , ∀ h ∈ H l,i , ∀ t ∈ T (16)where K l,i,h ( F l,i,h ) denotes the set of components of turbine i that have (not) been maintained undermaintenance scenario h . Recall that the continuous variable ρ l,it denotes the failure probability ofturbine i of wind farm l in period t . Constraints (17) enforce the relationship between this variableand the maintenance scenario variable η , and ensure that if maintenance scenario h is realized, thefailure probability will take the corresponding value, φ l,i,ht . ρ l,it ≥ φ l,i,ht η l,i,ht , ∀ l ∈ L , ∀ i ∈ G l , ∀ h ∈ H l,i , ∀ t ∈ T (17) In the formulation, component maintenance variables z and ν , turbine visit variable α , maintenancecrew visit variable x , maintenance scenario variable η , and cumulative maintenance variable u arebinary. On the other hand, failure probability variable ρ and energy generation variable y arecontinuous. Y is used to indicate the feasible region for the variable limits: z , ν , α , x , η , u , ρ , y ∈ Y . (18)
3. Reformulation and Solution Algorithm
The proposed mathematical model is computationally complex and not all instances can be solvedby a standard solver. Therefore the structural properties of the model is studied and a tailoredtechnique that enables solving instances of realistic sizes to optimality is developed.The solution method builds on a novel decomposition of the optimization model into a two-stageform. In the first stage the wind farm location- and turbine-level decisions are determined. At thesecond stage the decisions of the related wind turbine components are solved. This decompositionform is used to develop a master-slave type solution algorithm based on Benders’ and integer cuts -enabling a scalable integration of the proposed framework for wind farms.A restricted master problem is formulated by incorporating decisions related to the wind farm loca-tion and the turbine levels, as well as an additional auxiliary variable Ψ to represent a lower boundof the optimal value from the subproblem, which includes decisions related to turbine components.13he subproblem is solved as a mixed integer programming model, and as a linear relaxation at dif-ferent time steps of the algorithm. Solutions from the first and second type of subproblems are usedto generate integer cuts, and Benders’ cuts, respectively. These cuts are iteratively incorporatedinto to the restricted master problem to obtain an optimal solution in a finite number of iterations.The restricted master problem is formulated as follows:max α , x , y (cid:88) l ∈L ,i ∈G ,t ∈T (cid:16) π t y l,it − V v,lT,t α l,it (cid:17) − (cid:88) l ∈L ,t ∈T V v,lF,t x lt − Ψ (19)s.t. (7) , (8) , (9) , (10) , (11) , (12) , (13)Ψ ≥ λ T α , ∀ λ ∈ Λ P (20)0 ≥ λ T α , ∀ λ ∈ Λ R (21)The objective function (19) maximizes the operational profit by explicitly accounting for the op-erational revenue and wind farm location- and turbine-level costs, while considering the auxiliaryvariable, Ψ, as a lower bound for all other costs. Note that these other costs, which are related tocomponent-level decisions, are explicitly accounted for in the subproblem. The master problem in-cludes constraints (7)-(13), which dictate the operational requirements and relationships among thewind farm location- and turbine-level decisions. Constraints (20) collectively denote the Benders’optimality cuts and the integer cuts. Constraints (21) denote the Benders’ feasibility cuts. Thelimits for the x , y , α variables are consistent with those given in (18).When the master problem is solved, the values for the α variables are used in generating thesubproblem, which solves for the second stage variables related to wind turbine components. Theformulation for the subproblem is given below:min z , ν , u , η , ρ (cid:88) l ∈L ,i ∈G o ,k ∈K io ,t ∈T C l,i,kt ok ,t z l,i,kt + (cid:88) l ∈L ,i ∈G o ,t ∈T C f,l,i ρ l,it (22)s.t. (3) , (4) , (14) , (15) , (16) , (17) z l,i,kt ≤ ¯ α l,it , ∀ l ∈ L , ∀ i ∈ G l , ∀ k ∈ K io , ∀ t ∈ T (23) ν l,i,kt ≤ ¯ α l,it , ∀ l ∈ L , ∀ i ∈ G l , ∀ k ∈ K if , ∀ t ∈ T (24)The objective function (22) minimizes the costs related to the second stage variables. The sub-problem includes constraints (3), (4), (14)-(17) of the original model, which ensure maintenancecoordination and manage the relationships between maintenance scenarios and failure probabilities.Constraints (23) and (24) enforce that the second stage decisions are consistent with the first stagedecision variables, whose values are denoted with the ¯ α vector. It is important to note that the sub-problem formulation is separable in turbines, which means that a separate model can be built andsolved for each turbine i ∈ G l in each location l ∈ L . This property allows solving a number of smalloptimization problems, rather than solving one big problem, which enables parallel implementation14nd further scalability.If a solution to the subproblem can be found, then the values of the dual variables corresponding tothis solution are computed. This dual vector is then added to the set of optimality cut coefficients,Λ P . These cuts will be later augmented by integer cuts as well. If, however, the subproblem has nofeasible solutions, this means that the dual of the subproblem is unbounded, and thus an extremeray is found and then added to the set of feasibility cut coefficients, Λ R .If the described procedure of iteratively solving master and subproblems terminates with integersecond stage variables, the algorithm stops since it reaches the optimal solution. If one or moresecond stage variables have non-integer value, then an integer cut of type [34] is added to penalizenon-integer values in the given solution. Without loss of generality, this cut is incorporated withinΨ ≥ λ T α , and the cut coefficient vector defining this integer cut is added to the set Λ P . The stepsof the described solution algorithm is summarized in Algorithm 1, and a flow diagram is given inFigure 2.The proposed solution algorithm converges to the optimal solution in a finite number of iterations.This is a straightforward result due to (i) finite convergence of Benders’ decomposition with integercuts, and (ii) the finite number of integer variables (denoted by α that couple restricted masterproblem with the subproblem). This result is demonstrated using two key observations. First, forany feasible solution to α ; one can obtain a Benders’ cut that imposes the cost of the continuousrelaxation of the subproblem, and an integer cut that imposes the exact cost of the subproblem.While imposing exact costs of the corresponding subproblems for a specific α , these cuts also generatea lower bound for any other feasible solution α (cid:48) . Second, due to the finite number of integer variables,one needs to execute a finite number of iterations before convergence is achieved. More specifically,if there are N many α variables, then a total number of 2 N Benders’ and 2 N integer cuts would besufficient in worst case scenario to recover the entire cost of the subproblems. For a detailed proofof finite convergence and optimality in a similar problem structure, the reader is referred to [35].To test the performance of the decomposition method, representative experiments for different casesare conducted. Table 1 shows analyses under different number of wind farms, wind turbines, andturbine components. All computational figures are average of 5 replication runs. In all consideredinstances, decomposition proves faster than the standard model. It is also noteworthy to mentionthat the decomposition method is able to provide solutions in reasonable times for instances whereno feasible solution is found by the standard model. This structural decomposition is especiallyimportant with the increasing number of turbines and wind farms to be operated. It is concluded thatthe proposed solution algorithm is beneficial and experiments can be continued by this technique.15 lgorithm 1 Solution Algorithm
IsInteger := F alse while
IsInteger = F alse do LB := −∞ , U B := ∞ while U B − LB > (cid:15) do Solve master problem (19)-(21); obtain solution vector (¯ x, ¯ y, ¯ α ) and objective valueSolve relaxed and MIP forms of the subproblem (22)-(24) using ¯ α values if subproblem returns a feasible solution then Update
U B by summing the current master and MIP subproblem objective valuesObtain dual vector λ ; add to Λ P Update LB based on relaxed subproblem objective value else if subproblem has no feasible solutions then Dual of the subproblem is unbounded; obtain dual extreme ray λ , add to Λ R end ifend whileif solution is integer then IsInteger = T rue else
Compute integer cut coefficients, add to Λ P . end ifend whileTable 1: Computational Performance: Full Model vs. Solution Algorithm for Problems with 4-and 8-Component Turbines |K| |L| , |G| Full Model SolutionAlgorithm ∗ Results indicate average computation time (seconds) across 5 replications. ∗ In 2 of the 5 replications, a feasible solution could not be found withinthe specified time limit of one hour. nitialize Λ P ← ∅ , Λ R ← ∅ Initialize UB ← ∞ , LB ← −∞ Solve master problem (19)-(21)Update UB Solve subproblems (22)-(24)
Feasible?Calculate feasibilitycut parameters
Update Λ R UB − LB ≤ (cid:15) ? Calculate optimalitycut parameters Update Λ P and LB Integer? Calculate integercut parameters
Update Λ P STOP
Optimal solution is found.No Yes Yes NoNoYes
Figure 2:
Flow diagram for Algorithm 1
4. Experimental Results
In this section experiments are performed to study the performance of different maintenance policesin a wide range of settings and conditions. The data used in the experiments can be introduced asfollows:
Degradation Data -
Rotating machinery degradation database from [36] is used to emulatethe degradation process in the wind turbine systems. Rotating machinery are run from brand newto failure, and their raw vibration spectra are captured continuously. These raw signals are thentranslated into degradation signals that are used to update the predictions on component remaininglife. For more information on the specifics of the database, we refer the reader to [36]. Rotatingmachinery degradation (e.g. bearing degradation) is an important contributor to turbine failure.However, turbines have many other distinct components, each with a different degradation processand a unique set of sensors for monitoring them (e.g. electrical or vibration based) [37, 13, 27]. Thefocus is to leverage on this existing literature, and build adaptive optimization models that can adaptto the resulting predictions on component remaining life, regardless of the underlying degradationprocesses and sensor requirements.
Weather Data -
Weather data from the KNMI North Sea Wind17tlas project [23] is used. This data set consists of hourly wind speed measurements at 8 differentheight levels (10, 20, 40, 60, 80, 100, 150 and 200 m) over a time period of January 2014 up toJanuary 2017. The hourly wind speed data is adapted for the presented maintenance optimizationmodel, where each time period corresponds to two days, by computing the average wind speed ofeach consecutive 2-day period. Measurements and parameters are based on an existing wind farm,GEMINI, in the North Sea, consisting of 150 Siemens SWT 4.0-130 turbines with a generatingcapacity of 4 MW and a rotor diameter of 130 meters. The power output is estimated from theKNMI wind speed measurements using the wind turbine power curve model [38] of SWT 4.0-130turbines (with cut-in, rated, and cut-off wind speeds of 5, 12, and 25 m/s, respectively).The rolling horizon based experimental framework, which is summarized in Figure 3, is formed bytwo phases. In the first phase the optimization model (given in Section 2) is solved with the tailoredsolution method (given in Section 3) to schedule the operations and maintenance actions over a 200-day planning horizon, denoted as 100 equal time periods of 2 days. Here, the dynamic maintenancecosts of operational turbine components are used. In the second phase, the chain of events aresimulated based on the rotating machinery degradation data for a sequential fixed number of timeperiods (16 days). For every day of this fixed period, the status of wind turbines is observed andrecorded: whether they undergo preventive or corrective maintenance, whether a failure occurs, andwhether the turbines have idle periods. For components that experience preventive maintenance, howlong the component had before failure is calculated – a term referred to as unused life . An unexpectedfailure occurs when an operational component was scheduled for a preventive maintenance but fails(i.e., is degraded beyond its failure threshold) before the scheduled preventive maintenance couldtake place. The wind turbines that have at least one failed component, and thus was in failed state atthe beginning of the planning horizon, remain idle until their failed components undergo correctivemaintenance. For each time period of the second phase, the metrics for revenues, costs, detailedmaintenance actions, and statistics such as idle days, availability of the wind turbines, failures,and unused life are recorded. At the end of the second phase, dynamic maintenance costs for eachcomponent in operational wind turbines are updated with the most recent sensor readings.
To demonstrate the effectiveness of the approach, a series of comparative studies that evaluate theperformance of the proposed model vs. benchmarks with different properties is conducted. In theexperiments, a large scale system composed of 3 wind farms with 100 turbines is considered. Eachturbine is composed of 4 components, amounting to 1200 components in total. First two sets ofbenchmarks (given in Sections 4.1.1 and 4.1.2) refer to sensor-driven approaches that have equalpredictive power to the proposed model in terms of predicting failure likelihoods. The final set ofbenchmarks (given in Sections 4.1.3 and 4.1.4) refer to more traditional maintenance policies thatdo not adapt to any sensor information. 18 nitialize t ← Phase I (Optimization) • Get sensor data for day t , update dynamic maintenance cost accordingly • Solve model for days t, t + 1 , . . . , t + 199 • Record maintenance decisions for days t, t + 1 , . . . , t + 15
Phase II (Simulation) • Implement maintenance decisions on days t, t + 1 , . . . , t + 15 • Observe events, such as preventive and corrective maintenance , failures • Record statistics, such as turbine availability, idle days, unused life, costs , and revenue , pertaning to days t, t + 1 , . . . , t + 15
End of simulation timehorizon reached?STOP
Calculate and report summary statistics for the entire simulation horizonYesNo t ← t + 16 Figure 3:
Flow diagram of the rolling horizon framework & SC-CBM ) These models maintain a single component per turbine visit. This is representative of a sensor-drivenmodel that does not use opportunistic cost across turbine components. In the first variant SC-CBM ,the following constraints are used to ensure that at most a single component is maintained at a time: (cid:88) k ∈K l,io z l,i,kt + (cid:88) k ∈K l,if ν l,i,kt ≤ , ∀ l ∈ L , ∀ i ∈ G l , ∀ t ∈ T (25)whereas in the second variant SC-CBM , this limit is restricted only to preventive maintenances(i.e. allowing single preventive and multiple corrective maintenances simultaneously). This variantrequires suppressing the second term in (25). This setting is most appropriate for cases where theturbine visit cost is negligible compared to the costs of component maintenance. & BC-CBOM ) In these models, maintenances on all operational components must be conducted in a single visitto a turbine. BC-CBOM enforces this rule to all the components, whereas BC-CBOM enforces19 igure 4: Component and Turbine Failure Probabilitiesbatching only for the components that are preventively maintained. For BC-CBOM , the followingconstraints are added: z l,i,kt = z l,i,k (cid:48) t , ∀ l ∈ L , ∀ i ∈ G l , ∀ k, k (cid:48) ∈ K l,io , ∀ t ∈ T (26)Likewise, the other variant BC-CBOM enforces additional constraints using ν l,i,kt for componentsin K l,if . This setting is important when turbine visit cost is significant and should be minimizedthrough aggressive batching. The PM model ensures that a component goes through preventive maintenance when its age iswithin a range that is optimized using the component failure times.To implement this policy, constraints (3) is replaced with ones that enforce PM limits, instead ofdynamic limits, ζ l,i,k .PM model does not use any insights from sensor-driven dynamic maintenance cost functions. There-fore, C l,i,kt ok ,t = 0 , ∀ l ∈ L , ∀ i ∈ G l , ∀ k ∈ K l,i . This policy is useful for understanding the conventionalpractice in wind farm maintenance. The RM model does not conduct any preventive maintenance actions. The rest of the model isidentical to the proposed model. To implement this policy, constraints (3) are removed from theMC-CBOM and z l,i,kt = 0 , ∀ l ∈ L , ∀ i ∈ G l , ∀ k ∈ K l,i , ∀ t ∈ T is added.A key advantage of the proposed MC-CBOM model over the conventional models is the explicit20odeling of the relationship between i) component outage risks, ii) turbine maintenance visit sched-ule, iii) component maintenance decisions, and iv) turbine failure risks. Figure 4 illustrates thedynamic progression of component and turbine failure probabilities in one of the runs as a functionof maintenance decisions. For this particular scenario, turbine i at wind farm l has maintenancevisits scheduled at time periods 11 and 17 (i.e. α l,i = α l,i = 1); maintaining components 1 & 4 inthe first visit ( z l,i, = z l,i, = 1), and 2 & 3 in the second visit ( z l,i, = z l,i, = 1). Note that bothturbine and component probabilities change as a function of turbine visit and component mainte-nance decisions. The costs associated with these maintenance decisions, in turn, are continuouslyupdated by sensor information. Table 2:
Performance of MC-CBOM Compared to Other Sensor-Based Maintenance Policies
Other Sensor-Based ModelsMC-CBOM SC-CBM SC-CBM BC-CBOM BC-CBOM Net Profit $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ Table 3:
Performance of MC-CBOM Compared to Reliability-Based Maintenance Policies
Reliability-Based ModelsMC-CBOM RM PM
Net Profit $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ ≥ .
03% better thanSC-CBM policies, and ≥ .
58% better than BC-CBOM policies. This is accomplished throughsignificant savings in maintenance costs (by ≥ .
30% and ≥ . .
138 components per turbine.The comparison of MC-CBOM with SC-CBM and BC-CBOM models indicate that access to degra-dation models and accurate prediction on component and turbine failure likelihoods does not nec-essarily provide improvements in fleet-level maintenance. While these predictions are important,their full potential can be recovered only if the complex interactions between different components,turbines and wind farm locations, are explicitly considered with the operations and maintenanceoptimization model.Note that MC-CBOM performs significantly better compared to the conventional (i.e. reliability-based) PM and RM models that do not use sensor information. MC-CBOM provides significantimprovements in profit, and reduces maintenance cost by 8 .
83% and 35 . The next set of experiments demonstrate the impact of market price on the performance of MC-CBOM. Table 4 presents the results for different price scenarios. Note that as market price increases,MC-CBOM increases the average batch size (i.e., the average number of components maintainedduring a turbine visit) to limit the downtime due to maintenance – by 5 .
88% between the scenarioswith market prices of 12 . $ /MWh and 50 $ /MWh. This results in an increase in availabilityand unused life – by 0 .
15% and 5 .
23% between 12 . $ /MWh and 50 $ /MWh, respectively. Thesechanges also impact component maintenance cost, which exhibits a small increase of 0 .
76% between12 . $ /MWh and 50 $ /MWh. This experiment shows that as turbine availability becomes moresignificant, MC-CBOM deviates slightly from its optimal maintenance policy to minimize turbineoutages. 22 able 4: Impact of Electricity Price on MC-CBOM
Market Price ( $ /MWh) 12.5 25 37.5 50 Component Maint. Costs $ $ $ $ In the last set of experiments, presented in Table 5, how the relative cost of different componentsimpact their respective maintenance schedules is studied. Components C1 through C4 denote thegearbox components, rotor, generator, and bearing, with associated preventive maintenance costs of $ k , $ k , $ k , and $ k (as in [11]).The key observation of this section is that while all components deviate from their component-specific optimal maintenance schedules, components that have less significance would be more liberalin adapting to turbine- and fleet-level requirements (deviations increase by 76% between C1 and C4).It is also observed that low cost components would initiate less turbine-level maintenances (by 27%between C1 and C4), and appear less often within turbine maintenances with single componentactions (by 13% between C1 and C4). This experiment demonstrates that the proposed model caneffectively incorporate how the importance of components play a significant role in determining thedegree of their interaction with turbine- and fleet-level decisions. Table 5:
Impact of Component Significance on MC-CBOM
Component
Average Deviation from Optimal CBM 4.66 5.92 6.12 8.21
An in-depth comparative performance analysis of the proposed MC-CBOM policy, as well as insightsinto how it adapts to varying conditions, is provided in Sections 4.1, 4.2, and 4.3. The MC-CBOMpolicy outperforms all considered benchmark policies thanks to its ability to consider and adaptto the complex interactions between different decision layers. By making use of real-time sensorinformation, it is able to significantly outperform traditional RM and PM policies, since these tendto either conduct maintenance too late, which results in low availability and production, or too early,which results in high maintenance costs. Additionally, it is observed that the flexibility of MC-CBOMin identifying which components/turbines are to be maintained during a crew visit, results in lower23aintenance expenditures and higher profits compared to more restrictive condition-based policies,such as SC-CBM and BC-CBOM.The MC-CBOM optimization model is adaptable to changing problem parameters, such as electricityprice, maintenance costs, turbine/location visit costs, fleet/crew size, and weather conditions. Asan example of how it adapts to changing conditions, Section 4.2 provides the outcome of MC-CBOM under varying electricity prices. Table 4 demonstrates that as electricity price increases,MC-CBOM adapts by pursuing a more aggressive maintenance strategy and therefore increasingoverall availability of the turbines.An important contribution of this study is to consider maintenance at a component, rather than at aturbine, level. This allows the MC-CBOM model to adapt to different component characteristics bydistinguishing between turbine components based on their maintenance costs and degradation levels.This feature of the proposed method is quantitatively demonstrated in Section 4.3. It is observedthat MC-CBOM prioritizes the components that are more “critical”, i.e., more expensive to main-tain, and schedules preventive maintenance on these critical components as close to their individualoptimal maintenance time as possible. This consideration of component criticality results in lowermaintenance expenditures than methodologies with only turbine-level maintenance schedules.In order to gain further insight into the impact of the MC-CBOM model, interesting directions forfuture work would be i) a comprehensive computational study that focuses primarily on testing theperformance of the proposed framework under a wide range of parameter values to highlight itspractical implications for the diverse needs of the practitioner community, and ii) a variant of theMC-CBOM focusing on shorter timescales (e.g., hours), which captures the interactions betweenmaintenance decisions and degradation due to short-term wind fluctuations.
5. Conclusion
A unified and scalable framework for integrating multi-component degradation models to larger scalewind farm systems is proposed in this paper. In doing so, a novel optimization model, MC-CBOM,is developed, that adapts to real-time sensor information while accounting for complex dependenciesacross components, turbines and wind farms. To ensure scalable deployment of the proposed model,a solution algorithm that exploits the structure of the reformulated optimization model is devisedto obtain the optimal O&M policy in an efficient manner.A simulation framework based on the rolling horizon methodology is used to extensively test theperformance of the proposed optimization model in terms of net profit, maintenance costs, numberof failures, unused life, and turbine availability. The proposed simulation framework builds on areal-world degradation database for rotating machinery and weather data from the KNMI North24ea Wind Atlas project [23] to create realistic test cases for testing. It is quantitatively demon-strated that the proposed framework provides significant cost and reliability improvements overexisting maintenance models. Furthermore, the ability of the proposed model to adapt to a widerange of operational and maintenance scenarios is illustrated. More specifically, it is observed that (i) the MC-CBOM model adapts to increasing electricity price by pursuing a more aggressive main-tenance strategy in order to ensure high turbine availability, and (ii) the multi-component natureof MC-CBOM is effective in prioritizing critical components over others in order to reduce overallmaintenance expenditure.
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