The identification of an optimal I&M strategy for turbine support structures in offshore wind farms is a decision problem under uncertainty and risk (Farhan et al., 2021). This type of problem can be solved based on Bayesian decision theory (Raiffa and Schlaifer, 1961) and graphically represented by the generic decision tree shown in Fig. 1. Each branch of the decision tree corresponds to a realization of decisions represented by square nodes and random events or random variables represented by circular nodes. As an example, the lower branch in the decision tree in Fig. 1 corresponds to a realization of (a) a decision i concerning the inspection and monitoring regime, (b) the corresponding probabilistic inspection and monitoring outcomes Zi, (c) the decisions a concerning the maintenance actions, (d) the probabilistic parameters Y influencing the effect of the maintenance actions a, (e) the probabilistic parameters X influencing the system state, and (f) the probabilistic parameters W influencing the utility. Each of these realizations is associated with a utility U and an index representing the analysis type, depicted by the diamond-shaped leaves of the decision tree. According to utility theory (Von Neumann and Morgenstern, 1947), the maximization of the expected value of the utility U quantifies the optimality of decisions. From this it follows that the optimal decisions concerning inspection and monitoring as well as maintenance can be determined by maximizing the expected utility.

The utility U0 associated with the upper branch of the decision tree can be described by the following generic utility function: 1U0X,W=BX,W-CF(X,W), where BX,W and CF(X,W) describe the lifetime economic benefits from operating the wind farm and lifetime costs of structural failures as a function of X and W. Based on the utility function U0X,W and the prior probability distributions of X and W, the expected utility EU0 can be computed as described in Appendix A1. This quantity is primarily required as a reference value in the decision value analysis presented below in Sect. 2.2. In the literature, the evaluation of EU0 is referred to as system state analysis (SS-A) (Thöns and Kapoor, 2019).

Next, the utility U1 associated with the center branch of the decision tree can be generically expressed by the following utility function: 2U1a,Y,X,W=Ba,Y,X,W-CM(a,Y,X,W)-CF(a,Y,X,W), where Ba,Y,X,W, CM(a,Y,X,W), and CF(a,Y,X,W) are the lifetime economic benefits, maintenance costs, and failure costs as a function of a, Y, X, and W. The optimal maintenance actions a* are identified by maximizing the conditional expected utility EU1|a, which is computed based on the utility function U1a,Y,X,W and the prior probability distributions of Y, X, and W, as outlined in Appendix A2. Identifying a* in this way (i.e., without considering inspection and monitoring to inform decisions on maintenance actions) is referred to as prior decision analysis (Raiffa and Schlaifer, 1961) or predicted action decision analysis (PA-DA) (see also Thöns and Kapoor, 2019).

Finally, the utility U2 associated with the lower branch of the decision tree can be generically written as (see also Sorensen, 2009) 3U2i,Zi,a,Y,X,W=Bi,Zi,a,Y,X,W-CSHMi,Zi,a,Y,X,W-CIi,Zi,a,Y,X,W-CMi,Zi,a,Y,X,W-CFi,Zi,a,Y,X,W, where Bi,Zi,a,Y,X,W, CSHMi,Zi,a,Y,X,W, CIi,Zi,a,Y,X,W, CM(i,Zi,a,Y,X,W), and CF(i,Zi,a,Y,X,W) are the lifetime economic benefits, monitoring costs, inspection costs, maintenance costs, and failure costs as a function of i, Zi, a, Y, X, and W. The optimal information acquisition regime (or inspection and monitoring regime) i* is determined by maximizing the conditional expected utility EU2|i, which is determined based on U2i,Zi,a,Y,X,W and the probabilistic models of Zi, Y, X, and W as detailed in Appendix A3. As discussed in Appendix A3, this maximization jointly optimizes decisions on inspection and monitoring as well as maintenance actions based on (a) predicted information on the system condition and performance, (b) predicted maintenance actions, and (c) corresponding benefits and costs. This analysis is referred to as pre-posterior decision analysis (Raiffa and Schlaifer, 1961) or predicted information and predicted action decision analysis (PIPA-DA) (see also Thöns and Kapoor, 2019).

From Eqs. (2) and (3) it can be seen that a model for quantifying the inspection and monitoring as well as maintenance costs is a key component of the utility functions required to measure the optimality of decisions concerning I&M of WT support structures in offshore wind farms. Such a cost model should be formulated as a function of the uncertain parameters influencing the I&M costs in order to capture the uncertainties in these costs in the decision-making.

The root node in the decision tree in Fig. 1 represents the basic decision concerning the implementation of an integrity management strategy (Thöns, 2018). This decision can be informed by a decision value (DV) analysis. Following Thöns and Kapoor (2019), three different DVs may be formulated based on the decision tree shown in Fig. 1. The first DV, VPA-DAPIPA-DA, is defined as the difference between the maximum expected utility resulting from the PIPA-DA, EU2|i*, and the maximum expected utility resulting the PA-DA, EU1|a*, i.e., 4VPA-DAPIPA-DA=EU2|i*-EU1|a*, where EU2|i* and EU1|a* are determined as described in Appendix A.

The second DV, namely the predicted value of information and actions VSS-APIPA-DA, is defined as the difference between the maximum expected utility resulting from the PIPA-DA, EU2|i*, and the expected utility resulting from the SS-A, EU0: 5VSS-APIPA-DA=EU2|i*-EU0.

The third DV, i.e., the predicted value of actions VSS-APA-DA, is the difference between the maximum expected utility resulting from PA-DA and the expected utility provided by the SS-A given as 6VSS-APA-DA=EU1|a*-EU0.

Essentially, an integrity management strategy should be implemented if the value of VPA-DAPIPA-DA, VSS-APIPA-DA, or VSS-APA-DA is positive.

Inspections are performed to obtain information on the condition of structural components. Concerning offshore wind turbine support structures, they look for (indicators of) deterioration (e.g., corrosion and/or fatigue cracks), which has an effect on the integrity of the structural systems. In this contribution, a probabilistic cost model is developed for I&M actions performed to detect and repair fatigue cracks in welded connections in steel support structures of wind turbines in offshore wind farms (e.g., monopiles, jackets). In such structural systems, the welded components can be located above and below water level. The components located above water are typically part of the turbine tower, transition piece, main access platform, and access systems, which can be inspected via rope access and getting closer to the structure, while in the areas of the transition piece and substructure below water, inspections are carried out by a diver or by utilizing a remotely operated vehicle (ROV). The location of the inspected welded component thus has an effect on the required personnel, vessels, equipment, and logistics.

Two types of inspection methods to identify fatigue damage in welded components located above and/or below water level are considered: visual inspection and electromagnetic (EM) inspection methods such as eddy current (EC), magnetic particle inspection (MPI), and alternating current field measurement (ACFM). Visual inspection is a coarse method capable of detecting only relatively large surface breaking defects in welds or fatigue failures of welded connections. It can be performed with the help of a camera mounted on an ROV or by naked-eye observation. In contrast, EM inspection methods detect smaller surface breaking defects in welds. They can also be applied by a diver below water.

After an inspection campaign, if any fatigue damage is detected, a subsequent maintenance action (e.g., a repair) is performed based on the inspection outcomes. Depending on the criticality of the identified fatigue damage, the maintenance campaign is launched in the same year or the following year.

During an inspection campaign, the length and depth of a detected surface breaking defect are measured to inform decisions on the repair methods. With regards to possible repairs for welded joints, we consider two methods. The first repair method is referred to as welding (Rodriguez-Sanchez et al., 2011). In this method, the welded joint is repaired by removing a surface crack through grinding and subsequent filling of the resulting groove with wet welding. This method is applied if the measured depth of the surface crack is greater than a defined percentage of the section thickness. Any surface crack with a measured depth less than the defined percentage of the section thickness may be repaired by grinding, which is the second repair method (Rodriguez-Sanchez et al., 2004).

There are several factors that influence the total cost of inspecting and maintaining support structures in an offshore wind farm. In this contribution, the influencing factors are identified, and their costs are estimated based on the existing scientific literature and interviews with I&M experts: a wind farm operator, engineering consultants, and operation and maintenance engineers – whose expertise does not necessarily cover all types of offshore wind farms. These experts are, however, able to provide sufficiently detailed information on established operational procedures and approximate estimates of I&M costs. The precise figures of I&M costs always depend on the actual wind farm type and the existing operational constraints. In this study, we consider the common case in which the wind farm is serviced by workboats operating from a nearby port base. Therefore, other forms of wind farm access such as helicopters are not considered in this contribution.

Accessibility is the main factor influencing the cost of I&M of turbine support structures in offshore wind farms. Depending on the location of the offshore wind farm, the type and location of the I&M activity (above water level (AW) and/or below water level (BW)), and the duration of the I&M activity, a certain type of vessel is employed. The choice of vessel for port-based operations could be a crew transfer vessel (CTV) or a service operation vessel (SOV). CTVs are usually used for frequent operations and are generally small aluminum catamarans employed to transfer personnel to and from offshore sites on a daily basis. CTVs do not have sufficient dynamic positioning redundancies to keep still during rough sea conditions. Their carrying capacity is usually 12 crew members who do 12 h shifts. SOVs are larger vessels designed and equipped to be present for a longer duration at the offshore wind farm for subsea or extensive I&M operations. These vessels have a capacity of around 40 technicians and can perform 24 h operations with multiple shifts (each shift is 12 h), which means that they come back to port only approximately once every 2 weeks (Martinez-Luengo and Shafiee, 2019). Table 1 shows the range of mobilization and demobilization costs as well as costs per shift for CTV and SOV.

The mobilization and demobilization costs of the vessels cover several aspects like the commuting time to the offshore wind farm, fuel consumption of the vessel, equipment and material costs, and project management costs, which account for logistics organization and reporting. The vessel cost per shift captures the number of people involved in the operation, the personnel costs, and the operational cost of the vessel during the I&M activity.

I&M also includes the additional effort for engineering the required repairs. This effort is associated with costs as summarized in Table 2. The engineering costs are usually dependent on the type of repair. In the case of grinding, the extra cost of engineering and preparation entails the design of the repair and laboratory tests among other factors. In the case of welding, the cost entails the design of the repair, chambers for underwater repair work if required, and laboratory tests. This additional cost is ideally incurred once during the service life of a wind farm because the type of hotspots and components is known; thus, if any repair is performed, the implementation of the repair has already been planned for the specific type of hotspot in the support structures.

The duration to complete an I&M activity is another factor that strongly influences the total I&M cost. It depends, for example, on the weather conditions, the experience of the personnel, the condition of the asset, and the existence of marine growth. The total time to complete an I&M activity usually entails transit time between WTs, the time required to complete the work package once stationed at a WT, and additional weather downtime due to unfavorable weather conditions. An increase in I&M activity time due to the aforementioned factors can lengthen the offshore time within a campaign. While this may not seem crucial, the time increase has an effect on other costs, such as the costs of the deployment of a vessel, personnel, and equipment. Table 3 shows the estimation of the time that each of the I&M activities takes for a single component in a support structure. A component is defined here as a hotspot in a welded connection (i.e., a certain section of a weld).

Weather downtime is mostly dependent on the type of vessel utilized and is given in Table 4. In the case of CTVs, the weather downtime is usually higher because they are small in size and lighter compared to SOVs and can easily lose position, especially if there are large waves and strong currents, while SOVs can safely withstand the harshest conditions even in winter.

The transit time between turbines also influences the inspection maintenance cost. It is estimated here based Martinez-Luengo and Shafiee (2019) and usually varies between 15 and 30 min.

Based on the discussion of the factors affecting I&M costs in Sect. 3.2, we develop a deterministic parametric cost model to describe the total cost of I&M of WT support structures in an offshore wind farm. This cost can be generally broken down into a campaign cost CC, engineering cost CE, and operational cost COp. In the case that inspection and maintenance are performed simultaneously (mixed I&M), the total cost CI&M is given as 7CI&M=CC+CE+COp.

In the usual case where inspections and maintenance are performed in separate campaigns, the total cost of inspection CI and the total cost of maintenance CM are given by 8CI=CC+CI,OpCM=CC+CE+CM,Op.

The campaign cost CC corresponds to the fixed one-time cost of initiating the I&M activities, which includes the cost of commuting to the wind farm and back, the cost of the equipment and materials required for the planned activities, fuel costs, and project management costs. These cost components are included in the mobilization and demobilization cost of the vessels (see Table 1). As discussed in Sect. 3.2, a CTV or SOV can be selected as a vessel for the planned I&M activities depending on their nature and extent. In the case of maintenance, an additional cost for planning and engineering repairs CE has to be considered. This cost depends on the chosen repair method (welding or grinding).

Moreover, CI,Op and CM,Op are the operational costs of inspection and maintenance, which are the costs of conducting the inspection or maintenance operation when the vessel is at the offshore wind farm. The total operational costs further depend on the time to complete the operation, the vessel type, and its shift pattern. The total time to complete the operation depends on the extent of the I&M activity and where it is carried out, i.e., above or below water. The operational cost of inspection CI,Op and maintenance CM,Op activity is given by 9CI,Op=tI,Optshift⋅CshiftCM,Op=tM,Optshift⋅Cshift, where Cshift is the cost of the vessel (CTV or SOV) per shift, tshift is the duration of a shift (in hours) (see Sect. 3.2), tI,Op is the total time (in hours) to complete the overall inspection operation, and tM,Op is the total time to complete the overall maintenance operation. The total operational time for I&M is estimated as 10tI,Op=∑i=1nI,WTnI,C,i⋅tI,C+nI,WT-1⋅ttransit⋅1+WDtM,Op=∑i=1nM,WTnM,C,i⋅tM,C+nM,WT-1⋅ttransit⋅1+WD, where nI,WT is the total number of WTs to be inspected in an offshore wind farm during an inspection campaign, nM,WT is the total number of WTs to be repaired in the offshore wind farm during a maintenance campaign, nI,C,i is the number of components to be inspected in the ith WT support structure in the offshore wind farm, nM,C,i is the number of components to be repaired in the ith WT support structure in the offshore wind farm, ttransit is the transit time between the different WTs, tI,C is the time to inspect a component above or below the water, and tM,C is the time to repair a component above or below water. Furthermore, WD is the weather-related downtime, which is defined here relative to the overall operation time. This parameter also depends on the vessel type.

The model developed in Sect. 4 provides deterministic estimates of the I&M costs. However, due to the uniqueness of each operation and the complexity of the individual activities involved, the different parameters governing Eqs. (7), (8), (9), and (10) – i.e., the campaign cost, the vessel costs, the engineering costs, the inspection duration, the repair duration, the transit time, and the weather downtime – are uncertain and their values are typically only known in terms of intervals, as estimated in Sect. 3.2. To capture these uncertainties, the parameters of the cost model are modeled as random variables. By probabilistically modeling the uncertainties in the parameters and propagating them through the deterministic cost model, a probabilistic description of the I&M costs is obtained.

Let W generically denote the vector of random variables influencing the total I&M costs. Based on Eqs. (8), (9), and (10), the probabilistic cost model of inspection CIW can now be written as 11CIW=CC+∑i=1nI,WTnI,C,i⋅tI,C+nI,WT-1⋅ttransit⋅1+WDtshift⋅Cshift.

Similarly, the probabilistic cost model of maintenance CMW is formulated as 12CMW=CC+CE+∑i=1nM,WTnM,C,i⋅tM,C+nM,WT-1⋅ttransit⋅1+WDtshift⋅Cshift.

Given the lack of empirical data on the uncertain parameters of the cost models W, their probabilistic models are – in a Bayesian sense – chosen based on the available expert knowledge. It should be emphasized up front, however, that these probabilistic models can be updated using Bayesian methods if data on the parameters W become available.

As a first step in the probabilistic model building, the parameters in W are assumed to be independent and their marginal distributions are assumed to follow a lognormal distribution. The first assumption is made as no information on the correlation of the different parameters is available. The second assumption is supported for the following reasons. First, each parameter of the cost models only takes non-negative values, and their statistical distribution is typically expected to be unimodal; i.e., one range of values in the distribution occurs more frequently than other ranges of values. A lognormal distribution is commonly chosen to probabilistically model such quantities as it is bounded by zero, has no upper limit, and is unimodal. Second, a lognormal distribution is skewed to the right with a long tail capturing rare extreme values of the cost model parameters. Third, the assumption that the parameters are lognormally distributed can be partially explained by the central limit theorem. Assuming that each parameter of the cost model itself derives from a multiplicative process, the sum of the logarithms of the factors in the underlying process approaches a normal distribution and their product approaches a lognormal distribution as the number of factors becomes large. For these reasons, a lognormal distribution is a plausible probabilistic model for the different cost model parameters (see also Moy et al., 2015).

As a second step in the model building, the statistics of the different lognormal distributions are determined based on the lower and upper bounds of each cost model parameter specified in Sect. 3.2. These bounds represent the available expert knowledge on the ranges of the parameter values. Based on additional expert judgement, the lower and upper bounds are assumed to characterize the 1 % and 95 % quantile of the parameter values. Using this information, the different lognormal distributions are fitted as illustrated in Fig. 2. The resulting mean and coefficient of variation (CoV) of each probabilistic parameter of the cost models are summarized Table 5.

The probabilistic cost models for I&M of turbine support structures in the offshore wind farm defined in Eqs. (11) and (12) can be applied for different combinations of input parameters. The different combinations are defined by the vessel type, the inspection and repair methods, the number of inspected and/or repaired components above and/or below water level, and the number of inspected and/or repaired wind turbine support structures. For illustration, the probabilistic total I&M costs are in the following quantified at wind farm, wind turbine, and component level.

First, the total I&M costs are estimated at wind farm level. To this end, it is assumed that nI,C,BW=1 components of nI,WT=10 support structures are inspected below water. In addition, it is assumed that nM,C,BW=5 components of nM,WT=5 support structures are repaired below water. Inspections are performed using EM and visual inspection methods, while welding and grinding are applied as repair methods. Moreover, a CTV is utilized as a workboat in each scenario. For each considered scenario, the probabilistic distributions of the total I&M costs are determined using Monte Carlo (MC) simulations with nMC=106 samples of the corresponding model parameters. In the analysis, the model parameters are assumed to be statistically independent. The resulting empirical probability distributions of the total I&M costs are shown in Fig. 3.

Second, the total I&M costs are estimated at turbine level. In this case, it is assumed that 10 components of a support structure are inspected below water level, i.e., nI,WT=1, nI,C,BW=10, and 5 components in a support structure are repaired below water level, i.e., nM,WT=1, nM,C,BW=5. The assumptions regarding inspection and repair methods and the choice of vessel are the same as in the previous scenario considering I&M at wind farm level. The estimated empirical probability distributions of the total I&M costs together with their expected value and CoV are shown in Fig. 4.

Finally, the total I&M costs are estimated at the element level. In this scenario, nI,C,BW=1 components of nI,WT=1 turbines are inspected below water. The same is assumed for the maintenance campaign; i.e., only nM,C,BW=1 component of nM,WT=1 turbine support structure is repaired below water. The assumptions regarding inspection and repair methods and the choice of vessel are the same as in the scenario considering I&M at wind farm level. The empirical probability distributions of the total I&M costs are shown in Fig. 5.

From Figs. 3 to 5, it can be seen that propagating the uncertainties in the cost model parameters through the cost models defined in Eqs. (11) and (12) provides a probabilistic description of the total I&M costs. In each considered scenario, the total I&M costs exhibit an approximate lognormal distribution. The statistics of the total I&M costs shown in Figs. 3 to 5 are summarized in Table 6. Note that the coefficients of variation (CoVs) of the total I&M costs indicate that in each scenario certain parameters dominate the uncertainty in the total I&M costs. As an example, in Fig. 3a (wind-farm-level analysis considering EM inspection), the CoV of the total I&M costs is similar to the CoV of the vessel cost per shift. This finding is further substantiated by the variance-based sensitivity analysis in Sect. 5.2.1, where in Fig. 6 we observe that the vessel cost per shift has a sensitivity index close to 1 for the same inspection scenario.

As discussed in Sects. 3 and 4, the total I&M costs are influenced by numerous uncertain parameters W. To study the importance of each model parameter Wi, a variance-based sensitivity analysis is performed (Sobol, 1993), which quantifies the effect of Wi on the variance of the inspection and maintenance costs in terms of the following first-order measure: 13Vi=VarWiEW-i[C(W)|Wi], where CW can be the probabilistic model of the inspection costs defined in Eq. (11) or the probabilistic model of the maintenance costs defined in Eq. (12), EW-i[C(W)|Wi] is the expected value of the inspection or maintenance costs with respect to all parameters except Wi whose value is fixed, and VarWiEW-i[C(W)|Wi] is the variance of this average model. Normalizing Vi with the variance VarCW provides the first-order sensitivity index Si (Sobol, 1993): 14Si=ViVarCW=VarWiEW-i[C(W)|Wi]VarCW, in which VarCW is the variance of the inspection or maintenance costs. Si is evaluated here using an MC approach (Sobol, 2001).

The SS-A determines the expected total lifetime cost EC0 assuming that no inspections and no maintenance actions are performed during the lifetime of the support structure. EC0 is equal to the expected total lifetime cost of system failure ECF (lifetime risk of failure), i.e., 15EC0=ECF=∑j=1mcF⋅γj⋅PrFj-PrFj-1, where cF=EUR2×107 is the failure cost, which is assumed here to be deterministic and equal to the investment cost of one wind turbine (Thöns et al., 2017); PrFj is the cumulative probability of failure up to the end of year j; PrFj-PrFj-1 is the probability of failure in year j; and γj is the discounting function, which discounts the failure cost cF occurring in year j back to the present. The discounting function is defined as γj=1/1+rj, wherein r=0.02 is the discount rate. The expected total lifetime cost EC0 of the case study related to the steel frame is EUR 7×105.

The PIPA-DA for jointly optimizing I&M is performed using the heuristic cost- and risk-informed approach proposed in Luque and Straub (2019) and Bismut and Straub (2021), which corresponds in essence to the normal form of the pre-posterior decision analysis. We adopt this approach as it is computationally tractable compared to the extensive form of analysis described in Appendix A3. In this approach, an I&M strategy, generically denoted by S, is defined by parameterized rules that specify what, when, and how to inspect and repair based on the available system information (i.e., inspection outcomes and corresponding repairs as well as the predicted system failure probability conditional on the inspection outcomes and previously performed repairs). In the current application, the parameterized rules are defined as follows (see also Bismut et al., 2017; Eichner et al., 2023; Schneider, 2019).

Inspection campaigns are performed at fixed intervals Δt.

From the above list of parameterized rules, it follows that the I&M strategy S is fully defined here by the parameters θ=Δt,pth,nI,C,η,aRT. To highlight the dependence of S on θ, we write Sθ in the following.

The utility function – or more precisely the cost function – underlying the current optimization is (generically) defined as 16C2Sθ,X,Y,Z,W=CISθ,X,Y,Z,W+CMSθ,X,Y,Z,W+CFSθ,X,Y,Z,W, where C2Sθ,X,Y,Z,W is the total lifetime cost, CISθ,X,Y,Z,W is the total lifetime inspection cost, CMSθ,X,Y,Z,W is the total lifetime maintenance cost, and CFSθ,X,Y,Z,W is the total lifetime cost of structural failure. These costs are defined as a function of the I&M strategy Sθ, the uncertain parameters X influencing the time-dependent system failure probability, the uncertain parameters Y influencing the effect of repairs, and the uncertain parameters of the I&M cost model W=W1T,…,WmTT, where Wj represents the uncertain parameters influencing the I&M costs in year j as defined in Table 5. The total I&M costs occurring in each year j are evaluated based on the parametric cost models described in Sect. 4. It is assumed here that the different Wj, j=1,…,m values are independent and identically distributed. Note that this assumption is not a limitation, as the model can be extended to account for dependent and non-identically distributed cost model parameters.

The optimal strategy Sθ* characterized by the optimal heuristic parameters θ* minimizes the expected value of the total lifetime cost EC2|Sθ=EX,Y,Z,Wc2Sθ,X,Y,Z,W. It follows that the optimal heuristic parameters θ* can be determined as 17θ*=arg⁡min⁡θEC2|Sθ.

EC2|Sθ is evaluated as described in Appendix B1. The optimal heuristic parameters θ* are identified here by conducting an exhaustive search across the following sets of parameter values: Δt∈4,8years, pth∈5×10-4,10-3, nI,C∈1,…,22, η=1, and aR=1mm. The estimated expected total lifetime cost EC2|Sθ as a function of θ is shown in Fig. 9.

All strategies with nI,C=3,4,5,6 result in a similar expected total lifetime cost. This provides some flexibility to the decision-maker to choose a strategy based on their specific requirements regarding the inspection interval and structural reliability. Notably, both strategies with Δt=4 years exhibit similar expected costs for nI,C>7 regardless of the reliability threshold. When considering strategies with Δt=8, the reliability threshold has an impact on the expected total lifetime cost: a lower threshold results in more unscheduled inspections between regular inspections. In our current example the optimal strategy Sθ* is characterized by θ*=Δt=8,pth=1×10-3,nI,C=6,η=1,aR=1T.

Figure 10 shows the decomposed expected total lifetime cost E[C2|Sθ] as a function of the heuristic parameters θ=[Δt=8,pth=1×10-3,nI,C=1,…,22,η=1,aR=1]T. This cost is composed of the expected values of the failure cost, inspection campaign cost, inspection operation cost, repair campaign cost, repair operation cost, and engineering cost for repairs. As the inspection effort increases (i.e., more hotspots are inspected during each inspection campaign), the expected value of the system failure cost (i.e., the risk of structural failure) decreases, and – as expected – the expected values of the inspection and repair costs increase. This nicely illustrates the impact of the risk mitigation measures on the structural risk of failure. Note that the engineering cost for repairs is constant in this case study since it is incurred here only once at the beginning of the operational phase as repair solutions are engineered proactively before the frame is commissioned. Consequently, they could be neglected in the current optimization, as they only shift the expected total lifetime costs upwards by a fixed value.

To support the decision on whether one should implement an I&M strategy, the predicted value of information and actions is computed by the difference between the expected total lifetime cost EC0 and the expected total lifetime cost EC2|Sθ. By normalizing this difference with respect to EC0, the relative V‾SS-APIPA-DAθ is obtained (Farhan et al., 2021): 18V‾SS-APIPA-DAθ=EC0-EC2|SθEC0.

Figure 11 shows the V‾SS-APIPA-DA as a function of the parameters θ=[Δt=8,pth=1×10-3,nI,C=1,…,22,η=1,aR=1]T, where Δt=8 and pth=1×10-3 are the optimal inspection interval and reliability threshold. The dashed blue line corresponds to the expected total lifetime cost EC0 determined by the SS-A. The dashed–dotted blue line corresponds to the expected total lifetime cost EC2|Sθ. Notably, V‾SS-APIPA-DA is positive for nI,C=1,…,22. This result indicates that it is a rational decision to inspect and maintain the frame. As expected, the highest V‾SS-APIPA-DA is obtained when implementing the optimal strategy Sθ* with an optimal number of inspected hotspots in each inspection campaign nI,C=6.

The numerical example considers only a single turbine support structure. In this case, the cost models defined in Eqs. (11) and (12) can be expressed as linear functions of the number of inspected and repaired hotspots as follows: 19CI=CC+nI,C⋅CI,OpwithCI,Op=tI,C⋅1+WD⋅Cshifttshift and 20CM=CC+CE+nM,C⋅CM,OpwithCM,Op=tM,C⋅1+WD⋅Cshifttshift.

The expected lifetime cost EC2|Sθ can now be estimated based on the expected values of the parameters of the I&M cost model as described in Appendix B2. Subsequently, the optimal heuristic parameters θ* are identified based on Eq. (17) by again conducting an exhaustive search across the following sets of parameter values: Δt∈4,8years, pth∈5×10-4,10-3, nI,C∈1,…,22, η=1 and aR=1mm. The estimated expected total lifetime cost EC2|Sθ considering expected I&M costs is shown in Fig. 12.

It can be seen that the current analysis provides the same results as the analysis considering the probabilistic I&M costs (see also Fig. 9) and thus the same optimal strategy Sθ* with θ*=[Δt=8,pth=1×10-3,nI,C=6,η=1,aR=1]T. Consequently, the current analysis illustrates that the I&M costs can be considered deterministically as expected values in the DV analysis if they are included in the optimization on a linear basis.

Aligning with existing works, we utilize the probabilistic I&M cost model to derive deterministic or normalized cost ratios. In the literature, such a normalization is typically performed with respect to the failure cost or expected campaign cost due to their significant contribution to the overall lifetime costs. Applying the same methodology, we obtain the normalized cost models summarized in Tables 7 and 8. These models can subsequently be used to optimize I&M of offshore wind turbine support structures if the costs are included in the underlying models on a linear basis.