491

1 INTRODUCTION

Offshore wind energy has emerged as a key player in

the global transition towards renewable energy [4].

Over the past decade, offshore wind installations have

grown rapidly, with global capacity surpassing

67.4 GW by 2024 [16], and projections indicate

continued expansion, particularly in regions such as

the North Sea and the East China Sea. This growth is

driven by the advantages of offshore wind over

onshore alternatives, higher and more consistent wind

speeds, reduced land use conflicts, and proximity to

densely populated coastal demand centers.

Despite its advantages, such as higher wind speeds and

reduced land use constraints, offshore wind farms face

significant operational challenges, particularly harsh

marine conditions, such as high winds, waves,

corrosion, and limited accessibility, creating a complex

and costly environment for maintenance activities. In

addition, accessing offshore turbines often requires

specialized vessels, helicopters, or remotely operated

vehicles, with substantial logistical planning. This not

only inflates costs but also introduces high operational

uncertainty, as weather windows narrow and delays

become frequent. The harsh marine environment and

unpredictable wind conditions complicate preventive

and corrective maintenance operations. Unscheduled

maintenance interventions can lead to prolonged

downtimes and high logistical costs, making efficient

maintenance scheduling essential [1].

Current maintenance strategies typically combine

preventive maintenance, which follows scheduled

intervals but can lead to unnecessary interventions;

corrective maintenance, which addresses failures as

they occur, often resulting in prolonged downtime and

high reactive costs; and condition-based and predictive

maintenance, which utilize sensor data and machine

Maintenance Activities Coordination for Offshore Wind

Farms Integrating Multivariate Stochastic Models

Y. Salgado Duarte & J. Szpytko

AGH University of Kraków, Kraków, Poland

ABSTRACT: The efficiency and reliability of offshore wind farms are significantly influenced by their

maintenance strategies. Effective maintenance coordination is crucial to minimize downtime and consequently

maximize reliability. This paper proposes a methodology for integrating multivariate stochastic models into the

maintenance scheduling process of offshore wind farms, enabling a more data-driven approach. The proposed

framework accounts for the temporal and spatial dependencies of wind variations to optimize maintenance

activities, ensuring minimal disruptions to energy production using the Expected Energy Not Supplied (EENS)

indicator. We validate our approach using case studies that demonstrate its effectiveness through a sensitivity

analysis. The results highlight more than 6% deviations in the EENS estimate for scenarios with the same

conditions, where the only difference is the wind speed simulation approach. The change in approach impacts

the overall allocation of maintenance activities to offshore wind farms.

http://www.transnav.eu

the International Journal

on Marine Navigation

and Safety of Sea Transportation

Volume 19

Number 2

June 2025

DOI: 10.12716/1001.19.02.19

492

learning to forecast failures, reducing unexpected

outages and improving reliability.

However, even the most advanced strategies often

rely on deterministic models that do not fully

incorporate the stochastic, spatio-temporal nature of

offshore wind conditions. This omission can

significantly undermine planning accuracy. For

example, variability in wind speed and direction, and

their correlation between spatially distributed

turbines, has a positive impact on energy production

and accessibility, although these are often

oversimplified in existing models [2].

As offshore wind farms scale in size and number,

the complexity of coordinating maintenance schedules

between multiple geographically dispersed units

increases. There is a pressing need for integrated

probabilistic decision-making frameworks that

account for environmental uncertainty, asset

conditions, and system-level constraints.

This paper addresses this need by proposing a

methodology that incorporates multivariate stochastic

wind modeling into the offshore maintenance

scheduling process. The approach accounts for both

temporal and spatial wind dependencies and

introduces more realism into the simulation of wind

behavior. Using the Expected Energy Not Supplied

(EENS) as a reliability indicator, the framework is

validated through sensitivity analysis and case studies.

Ultimately, the goal is to minimize downtime, enhance

system reliability, and enable smarter data-driven

maintenance planning. This paper proposes a revised

methodology for coordinating offshore wind farms'

maintenance activities that leverages multivariate

stochastic wind modeling to account for a more real-

driven approach, introducing new considerations into

the modeling process. The methodology, subject to

review, is taken from [3], and the study presented in

this article aims to address the following key research

questions.

How can multivariate stochastic wind models be

effectively integrated into maintenance decision-

making?

What are the potential impacts of integrating

multivariate stochastic wind models on offshore wind

farm maintenance scheduling?

How does the proposed approach compare with

existing maintenance optimization strategies in terms

of results?

The remainder of this paper is organized as follows.

Section 2 reviews the relevant literature on

maintenance planning and stochastic wind modeling,

and at the end of the section, the proposed

methodology and the new considerations addressed in

this contribution are presented. Section 3 presents the

parameterization, results and discussion of the

optimization solution for the case study, and Section 4

concludes with recommendations for future research.

2 MATERIALS AND METHODS

The maintenance of offshore wind farms has been

extensively studied in the literature, with a strong

emphasis on preventive and predictive maintenance

strategies [5]. First, this section reviews existing work

on maintenance planning for offshore wind farms and

highlights the role of stochastic wind modeling in

improving modeling and, consequently, decision

making. Then, knowing that the scale of offshore wind

farms is increasing daily and, certainly, the complexity

of coordinating maintenance schedules across multiple

geographically dispersed units increases, we present a

revised methodology with new considerations that

proposes an integrated probabilistic decision-making

framework that accounts for environmental

uncertainty, asset conditions, and system-level

constraints.

2.1 Review of the literature

Maintenance strategies have been proposed to

optimize offshore wind farm operations, mainly

preventive maintenance [7], predictive maintenance

[5], and condition-based maintenance [6]. As we know,

preventive maintenance schedules interventions based

on fixed time intervals, reducing unexpected failures,

but potentially leading to unnecessary service.

Predictive maintenance predicts the remaining useful

life of the equipment to anticipate potential failures,

enabling early detection of problems and timely

maintenance actions. Condition-based maintenance

uses real-time sensor data and predictive analytics to

schedule interventions based on actual component

conditions. Certainly, the most recent state-of-the-art in

this domain is the integration of these strategies. For

example, studies have shown that condition-based

maintenance significantly improves operational

efficiency by reducing unnecessary maintenance

actions and increasing turbine availability [8].

However, these approaches (preventive, predictive,

and condition-based) often neglect the stochastic

nature of offshore wind conditions, which can affect

accessibility and the feasibility of scheduled

maintenance operations.

Stochastic modeling of wind speed and turbulence

plays a crucial role in offshore wind energy forecasting

and decision making. Traditional wind forecasting

models rely on deterministic methods, which do not

capture the spatial-temporal uncertainties of offshore

wind patterns. On the contrary, stochastic models,

such as Markov processes, autoregressive integrated

moving average (ARIMA) models and Gaussian

mixture models, have been employed to provide

probabilistic wind forecasts [9].

Multivariate stochastic models extend these

approaches by considering the correlations between

different environmental variables, such as wind speed,

wind direction, and wave height. This improves

predictive accuracy and allows for more robust

maintenance planning [10]. For example, recent

research has shown that integrating these models into

offshore wind maintenance scheduling can improve

accessibility predictions and optimize vessel dispatch

planning [11].

Studies have explicitly incorporated multivariate

stochastic wind modeling into maintenance

coordination frameworks. Existing research focuses

primarily on single-variable wind speed forecasts,

neglecting interactions between wind, wave

conditions, and turbine reliability metrics. The need for

493

an integrated approach that captures these

dependencies is evident, particularly for offshore wind

farms, where accessibility constraints are highly

dependent on weather [12].

Recent advances in probabilistic optimization have

shown promise in addressing these challenges.

Methods such as Monte Carlo simulations, Bayesian

inference, and reinforcement learning have been

applied to improve maintenance decision making

under uncertainty [13]. However, there is still a gap in

integrating these approaches with multivariate

stochastic wind models to develop a fully adaptive and

proactive maintenance framework.

The reviewed literature highlights the interest in

optimizing offshore wind farm maintenance through

predictive and data-driven approaches. However,

there remains a lack of comprehensive frameworks

that integrate multivariate stochastic wind models into

maintenance coordination.

2.2 Contribution of this paper.

This paper aims to contribute by presenting a revised

methodology that uses multivariate probabilistic

modeling techniques to improve the maintenance

scheduling modeling process. As previously

highlighted, the methodology, subject to review, is

taken from [3]. However, in this section, we present

how the new consideration, essentially multivariate

stochastic wind modeling, is incorporated into the

existing methodology.

Figure 1 shows the revised flow diagram of the

optimization problem. Now, the wind energy and

thermal units are explicitly separated, although all the

generation units are aggregated at the system level. As

we can see in Figure 1, the heuristic optimization

algorithms propose a set of xi = Mi,1 starting

maintenance activity time for each generator unit

considered in the system, given the constraints of the

simulation window. The remaining parameters needed

to simulate the stochastic capacity of the units are

directly sourced from a SQL (Structured Query

Language) database.

Specifically, in the case of wind energy, we

introduce in this contribution a modeling change,

which aims to consider correlations in the wind speed

simulation. Essentially, considering correlations in

wind speed simulation means that the wind speed

characteristic in a region of the offshore wind farm is

homogeneous and that all wind turbines should

deliver a similar energy path. As we know, in the case

of wind energy, the primary source of energy is the

wind, so it is necessary first to simulate the wind speed

and then, using the characteristic function of the wind

turbine, translate the wind speed into energy.

In this research, we simulate the most likely wind

speed v with a Weibull model, estimating the shape

and scale parameters of the probability distribution

function with the mean and standard deviation of the

historical wind speed characteristics of the region.

Once the Weibull distribution is parameterized, the

inverse cumulative distribution function is used to

simulate the wind speed values v by generating

uniform random numbers u with (1).

( )

1/

lnvu





=−





(1)

where v is the wind speed, β and δ are the shape and

scale parameters of the Weibull probability

distribution function, and u is a uniform random

number between [0, 1].

As we can see, the number of random numbers u to

be generated depends on t = 1, 2, …, T, which is the

hourly simulation window.

Regardless of how the correlations can be

measured, we intend to assess the impact of correlated

wind speed simulations on the risk indicators of the

system, specifically assessing the disruptions to energy

production using the Expected Energy Not Supplied

(EENS) indicator, and eventually in the maintenance

scheduling activities to be conducted in the offshore

wind farms.

In this paper, we use the Gaussian elliptical copula

to simulate uniformly correlated samples U. In the

Gaussian copula, the correlation parameter ρ controls

the dispersion of random values. Certainly, values of ρ

close to one mean more correlated marginals and more

homogeneous wind speed simulations. In Section 3, we

illustrate the impact of highly correlated wind speed

simulations on the wind farm delivery energy path.

As we may know, copula functions try to capture

the dependence between marginals through the copula

parameter. Certainly, each density may have a

different operating space. In the case of the

Archimedean copulas, the parameter manages the

dispersion of the random numbers. For instance,

higher parameter values result in less dispersion of the

random values. When the parameter is close to one, the

random values are not correlated.

On the other hand, the Gaussian and t-copula are

elliptical copulas. In these cases, the correlation

parameter ρ controls the dispersion of the random

values. Values of ρ close to one, more correlated

marginals. Within the elliptical copulas, the t-copula

has two parameters; therefore, the t-copula offers

another feature, the tail dependency. This additional

parameter allows the t-copula to simulate values at the

corners of the distribution space. This flexibility is

given by the degrees of freedom υ. For instance, fixing

the correlation parameter and changing the degrees of

freedom to higher values, in a t-copula density, results

in a Gaussian copula. That said, t-copula is more

flexible than Gaussian.

We selected the Gaussian copula in this

investigation for two main reasons. First, elliptical

copulas can easily model high dimensions, since the

diagonal of the correlation matrix has the same

dimension as the number of components considered.

Archimedean copulas are practically implemented for

only two components. Second, since we do not have a

quantitative perception of the right value for the

degree of freedom υ in the t-copula and is certainly

more costly from the computation point of view, we

decided to use Gaussian over t-copula because we aim

to assess if correlated wind values impact the delivery

energy path and maintenance scheduling activities

allocation, and for that, Gaussian is sufficient.

494

Figure 1. Flow diagram to solve the optimization problem of maintenance scheduling.

Previously, with (1), uncorrelated random numbers

ui were generated for each unit i considered in the

system, to obtain the wind speed values vi for each

wind turbine. Now, we simulate a matrix with

dimension [I, T] of correlated marginals U[0,1](ρ), where

ρ is the correlation parameter controlling the

dispersion of random values, I is the number of

offshore wind turbines within the wind farm, and T is

the simulation window in hourly resolution. Then,

using the UI,T, and the parametrized Weibull

distribution, we obtain a matrix of VI,T, which is the

correlated wind speed simulations as shown in (2), and

then, with the power characteristics function of the

wind turbine, we translate the wind speed values into

energy. For wind turbines, the variable Pi(t|v)

represents the wind power of the i-th wind turbine at

the t-th time, as illustrated in Figure 1.

( )

1/

,,

ln

I T I T

VU







=−



(2)

At this point, we end the description of the

modeling changes. The following modeling steps

remain unchanged as presented in [3]. In Section 3, we

present the parameterized scenario used to assess the

impact of the modeling change introduced in this

paper.

3 RESULTS AND DISCUSSION

In this section, we describe scenario parameterization,

then we illustrate the experiment designed to measure

the deviations in the EENS estimations when

correlations in the wind speed are considered, and then

we provide the solution for the system maintenance

scheduling activities under highly correlated wind

speed conditions using two heuristic optimization

algorithms. The parameterization of each algorithm is

also presented, as well as a comparison between them.

The base scenario described in [3] is partially used

to validate the revised methodology. However, we

recap key information here to highlight what is

relevant for the analysis of this contribution. The IEEE

Reliability Test System (IEEE-RTS) suggested by the

Subcommittee on Applications of Probabilistic

Methods provides the basis for the scenario and is

particularly suitable for implementing maintenance

planning solutions [15]. Tables 1 and 2 show the

parameterization implemented in the scenario under

study. The original IEEE-RTS has 32 generators and an

overall capacity of 3405 MW. However, a modification

is introduced in the scenario to integrate offshore wind

farms. The wind turbine parameters assumed to make

up the offshore wind farms modeled in this

contribution are listed in Table 3. This modified

scenario replaced three 100 MW Oil/Stream generator

units with offshore wind farms. In particular, three

farms with 50 wind turbines of 2 MW, making a total

of 300 MW, which is the same capacity as the replaced

generator units.

In this modified scenario, even if the replacement

total nominal capacity is the same (300 MW), there is a

difference in the primary source of energy. Usually, in

the case of Oil/Stream generators, the primary source

of energy is always available. In the case of wind farms,

the primary source of energy is the wind and it is

subject to changes and variability. Therefore, the

contribution of wind energy will never be 100%; in

other words, even if we have 300 MW installed, in

practice, the potential energy to be used will certainly

be less. Selecting the same nominal capacity points out

this difference.

495

In this modified scenario, as we introduced three

offshore wind farms, the number of individual

generator units increased from 32 to 179. Figure 2

shows the connection schema of the wind farms with

the Power System, which is the connection to the grid.

Essentially, three wind farms are connected in parallel

and located in different locations, and the wind

characteristics are also related to the location.

Since we use a Monte Carlo method to estimate the

EENS, we assume that β = 5% (error criterion) and T =

8760 hours (simulation time window).

Table 1. Parameterization of Scenario (A)

No.

Units

Unit Capacity (MW) Ṗi

MTTF (hours)

MTTR (hours)

5

12

2940

60

4

20

450

50

6

50

1980

20

4

76

1960

40

4

155

960

40

3

197

950

50

1

350

1150

100

2

400

1100

150

2

3650

55

Table 2. Parameterization of Scenario (B)

No.

Units

Mi,k (hours/year)

Maintenance (hours/year) Di,k

NMTi

5

672

168, 168

2

4

672

168, 168

2

6

672

168, 168

2

4

672, 672

168, 168, 168

3

4

672, 672, 672

168, 168, 168, 168

4

3

672, 672, 672

168, 168, 168, 168

4

1

672, 672, 672, 672

168, 168, 168, 168, 168

5

2

672, 672, 672, 672, 672

168, 168, 168, 168, 168, 168

6

150

672, 672, 672, 672, 672

168, 168, 168, 168, 168, 168

6

Table 3. Wind speed and wind turbine parameters

Parameter

Values Considered

µsw

19.52 km/h

σsw

10.99 km/h

Pr

2 MW

vci

15 km/h

vr

36 km/h

vco

80 km/h

Once the parameterization is known, we assess the

deviations in EENS when correlations in the wind

speed are considered within each wind farm. The

differences are clearly illustrated in Figures 3-6 and

discussed in this section.

Figure 3 shows the wind farm 1 delivery energy in

the first five days when the wind simulation is

assumed to be uncorrelated. As we can see, even if the

total capacity of wind farm 1 is 100 MW, there is no

peak value above 35 MW in the simulation. On the

other hand, Figure 4 shows the same wind farm 1, but

under the assumption that wind speed simulations are

highly correlated. Certainly, the outcome of wind farm

delivery is between these two extreme simulations

(uncorrelated and highly correlated) in a real scenario,

and the shape of how the energy is delivered changes

drastically.

In the system analyzed, we consider three wind

farms. Figure 5 shows the energy delivered in the first

five days if uncorrelated wind simulations are

assumed. As we can see, 300 MW of wind capacity only

delivers a peak value slightly above 90 MW. On the

other hand, Figure 6 shows the delivery energy when a

highly correlated wind speed is assumed in each wind

farm, but also assuming that each wind farm is in a

different location, which means that there should be no

correlation between wind farms. Again, the shape of

the delivery energy changes, and we can see peak

values reaching almost 250 MW.

Table 4. Variations in the correlation of wind speed.

Dev.

Wind Farm 1

Wind Farm 2

Wind Farm 3

EENS

6.23%

0.99

20,385

5.60%

0.90

20,265

4.93%

0.80

20,137

4.40%

0.70

20,035

2.79%

0.60

19,725

2.03%

0.50

19,579

1.73%

0.40

19,522

1.56%

0.30

19,490

1.18%

0.20

19,416

0.33%

0.10

19,253

0.00%

0.00

19,190

Figure 2. Power system and offshore wind farms connection schema.

496

Figure 3. One wind farm simulated delivery energy with no-correlated wind speed.

Figure 4. One wind farm simulated delivery energy with fully correlated wind speed.

Knowing that the shape of the delivery energy

changes, we first quantify the holistic impact using the

EENS indicator (MWh / year) and a predetermined

maintenance scheduling activity solution for this case

study, taken from [3]. Then we gradually introduce the

correlations as described in Table 4.

As we can see in Table 4, deviations above 6% are

observed in the EENS indicator. In fact, when an

uncorrelated wind speed is assumed in the simulation,

the dimension of the capacity of the system can be

overestimated, as Figure 7 shows. In this experiment,

we kept the random seeds unchanged to isolate the

actual impact of the correlations. Additionally, Table 5

shows deviations slightly higher than 3% in the annual

energy delivered by wind farms when correlations are

introduced. This result confirms that assuming

correlations in wind speed impacts not only the

individual energy delivery of each wind farm, but also

the overall behavior of the system, since the deviations

in the EENS indicator are higher than the deviations of

energy in the wind farms.

Table 5. Energy delivered (MWh/year)

Wind Farm

Uncorrelated

Correlated

Deviation

1

186,607

183,381

1.76%

2

185,665

190,449

-2.51%

3

185,814

191,879

-3.16%

Once presented with the impacts of the new

assumption, we solve the maintenance scheduling

problem under the new conditions, specifically, under

a highly correlated assumption (ρ = 0.99). For this aim,

we tested two heuristic optimization algorithms. In this

paper, we use MATLAB [14] in the proposed

implementation, and we use well-implemented

MathWorks algorithms for the solution, specifically

Particle Swarm Optimization (PSO) and Surrogate

Optimization. Independent of the algorithm strategy,

we assume that the optimization process ends when

the difference between two consecutive evaluations in

the objective function is less than 1.0E-06. Tables 6 and

7 show the maintenance scheduling activities solution

for each algorithm, and Figures 8 and 9 show the

convergence of the algorithms.

In the case of Surrogate, the algorithm required

8,950 evaluations in the objective function to achieve an

EENS value of 20,798 MWh/year. The PSO required

26,000 evaluations (260 iterations and populations of

100 candidates) to achieve 17,306 MWh/year. Although

Surrogate requires less execution time, the PSO can

find a better solution. Since we face a nondeterministic

polynomial time (NP) problem addressed with

heuristics, algorithms are prone to finding a local

minimum. Therefore, for this problem, PSO provides

the best solution.

497

Figure 5. Three wind farms simulated delivery energy without correlated wind speed.

Figure 6. Three wind farms simulated the delivery energy with a fully correlated wind speed.

As we can see, the PSO adapted the solution to

the new modeling conditions, since the EENS

achieved = 17,306 MWh/year is 18% lower than the

previous scheduling solution with the new

modeling conditions, EENS = 20,385 MWh/year.

Figure 10 illustrates the system capacity versus

system load under the new conditions, and Figure

11 shows the spatial distribution of the solution for

maintenance scheduling activities.

Table 6. Maintenance scheduling solutions for offshore wind farms

Unit

No.

Capacity

Start Maintenance

PSO

Surrogate

1

12

1727

'13-Mar-1900 23:00:00'

7053

'21-Oct-1900 21:00:00'

2

12

4007

'16-Jun-1900 23:00:00'

1392

'28-Feb-1900 00:00:00'

3

12

4674

'14-Jul-1900 18:00:00'

6344

'22-Sep-1900 08:00:00'

4

12

1117

'16-Feb-1900 13:00:00'

6296

'20-Sep-1900 08:00:00'

5

12

589

'25-Jan-1900 13:00:00'

0

'01-Jan-1900 00:00:00'

6

20

5562

'20-Aug-1900 18:00:00'

6214

'16-Sep-1900 22:00:00'

7

20

4999

'28-Jul-1900 07:00:00'

5406

'14-Aug-1900 06:00:00'

8

20

4548

'09-Jul-1900 12:00:00'

3954

'14-Jun-1900 18:00:00'

9

20

1190

'19-Feb-1900 14:00:00'

4970

'27-Jul-1900 02:00:00'

10

50

6091

'11-Sep-1900 19:00:00'

1549

'06-Mar-1900 13:00:00'

11

50

5958

'06-Sep-1900 06:00:00'

1287

'23-Feb-1900 15:00:00'

12

50

6600

'03-Oct-1900 00:00:00'

2496

'15-Apr-1900 00:00:00'

13

50

4432

'04-Jul-1900 16:00:00'

1436

'01-Mar-1900 20:00:00'

14

50

885

'06-Feb-1900 21:00:00'

5142

'03-Aug-1900 06:00:00'

15

50

5794

'30-Aug-1900 10:00:00'

4219

'25-Jun-1900 19:00:00'

16

76

614

'26-Jan-1900 14:00:00'

2980

'05-May-1900 04:00:00'

17

76

6216

'17-Sep-1900 00:00:00'

5656

'24-Aug-1900 16:00:00'

18

76

5531

'19-Aug-1900 11:00:00'

4924

'25-Jul-1900 04:00:00'

19

76

4648

'13-Jul-1900 16:00:00'

1667

'11-Mar-1900 11:00:00'

20

155

4295

'28-Jun-1900 23:00:00'

5039

'29-Jul-1900 23:00:00'

21

155

2340

'08-Apr-1900 12:00:00'

4697

'15-Jul-1900 17:00:00'

22

155

130

'06-Jan-1900 10:00:00'

63

'03-Jan-1900 15:00:00'

23

155

5575

'21-Aug-1900 07:00:00'

4574

'10-Jul-1900 14:00:00'

24

197

275

'12-Jan-1900 11:00:00'

5105

'01-Aug-1900 17:00:00'

25

197

3

'01-Jan-1900 03:00:00'

972

'10-Feb-1900 12:00:00'

26

197

4546

'09-Jul-1900 10:00:00'

4850

'22-Jul-1900 02:00:00'

27

350

1317

'24-Feb-1900 21:00:00'

530

'23-Jan-1900 02:00:00'

28

400

1840

'18-Mar-1900 16:00:00'

2749

'25-Apr-1900 13:00:00'

29

400

1642

'10-Mar-1900 10:00:00'

1689

'12-Mar-1900 09:00:00'

30

2

3880

'11-Jun-1900 16:00:00'

1294

'23-Feb-1900 22:00:00'

31

2

3589

'30-May-1900 13:00:00'

2115

'30-Mar-1900 03:00:00'

32

2

3085

'09-May-1900 13:00:00'

1531

'05-Mar-1900 19:00:00'

33

2

599

'25-Jan-1900 23:00:00'

736

'31-Jan-1900 16:00:00'

34

2

4388

'02-Jul-1900 20:00:00'

1519

'05-Mar-1900 07:00:00'

35

2

2122

'30-Mar-1900 10:00:00'

2420

'11-Apr-1900 20:00:00'

36

2

4389

'02-Jul-1900 21:00:00'

1800

'17-Mar-1900 00:00:00'

37

2

1902

'21-Mar-1900 06:00:00'

4141

'22-Jun-1900 13:00:00'

38

2

242

'11-Jan-1900 02:00:00'

1566

'07-Mar-1900 06:00:00'

39

2

3961

'15-Jun-1900 01:00:00'

1423

'01-Mar-1900 07:00:00'

40

2

1721

'13-Mar-1900 17:00:00'

3541

'28-May-1900 13:00:00'

41

2

1364

'26-Feb-1900 20:00:00'

2290

'06-Apr-1900 10:00:00'

42

2

3247

'16-May-1900 07:00:00'

61

'03-Jan-1900 13:00:00'

43

2

340

'15-Jan-1900 04:00:00'

2219

'03-Apr-1900 11:00:00'

44

2

7

'01-Jan-1900 07:00:00'

4118

'21-Jun-1900 14:00:00'

45

2

4378

'02-Jul-1900 10:00:00'

3994

'16-Jun-1900 10:00:00'

46

2

4182

'24-Jun-1900 06:00:00'

1593

'08-Mar-1900 09:00:00'

47

2

4278

'28-Jun-1900 06:00:00'

331

'14-Jan-1900 19:00:00'

48

2

3

'01-Jan-1900 03:00:00'

3441

'24-May-1900 09:00:00'

49

2

726

'31-Jan-1900 06:00:00'

1763

'15-Mar-1900 11:00:00'

50

2

228

'10-Jan-1900 12:00:00'

3167

'12-May-1900 23:00:00'

51

2

2241

'04-Apr-1900 09:00:00'

2153

'31-Mar-1900 17:00:00'

52

2

4064

'19-Jun-1900 08:00:00'

2353

'09-Apr-1900 01:00:00'

498

53

2

2583

'18-Apr-1900 15:00:00'

3018

'06-May-1900 18:00:00'

54

2

3732

'05-Jun-1900 12:00:00'

4

'01-Jan-1900 04:00:00'

55

2

2658

'21-Apr-1900 18:00:00'

3652

'02-Jun-1900 04:00:00'

56

2

4351

'01-Jul-1900 07:00:00'

3770

'07-Jun-1900 02:00:00'

57

2

2955

'04-May-1900 03:00:00'

3072

'09-May-1900 00:00:00'

58

2

83

'04-Jan-1900 11:00:00'

3164

'12-May-1900 20:00:00'

59

2

391

'17-Jan-1900 07:00:00'

3559

'29-May-1900 07:00:00'

60

2

930

'08-Feb-1900 18:00:00'

1816

'17-Mar-1900 16:00:00'

61

2

4364

'01-Jul-1900 20:00:00'

948

'09-Feb-1900 12:00:00'

62

2

3390

'22-May-1900 06:00:00'

574

'24-Jan-1900 22:00:00'

63

2

1600

'08-Mar-1900 16:00:00'

4341

'30-Jun-1900 21:00:00'

64

2

2213

'03-Apr-1900 05:00:00'

2561

'17-Apr-1900 17:00:00'

65

2

2253

'04-Apr-1900 21:00:00'

1010

'12-Feb-1900 02:00:00'

66

2

4

'01-Jan-1900 04:00:00'

678

'29-Jan-1900 06:00:00'

67

2

752

'01-Feb-1900 08:00:00'

579

'25-Jan-1900 03:00:00'

68

2

3008

'06-May-1900 08:00:00'

3060

'08-May-1900 12:00:00'

69

2

2232

'04-Apr-1900 00:00:00'

1287

'23-Feb-1900 15:00:00'

70

2

3665

'02-Jun-1900 17:00:00'

2890

'01-May-1900 10:00:00'

71

2

353

'15-Jan-1900 17:00:00'

322

'14-Jan-1900 10:00:00'

72

2

349

'15-Jan-1900 13:00:00'

291

'13-Jan-1900 03:00:00'

73

2

754

'01-Feb-1900 10:00:00'

1455

'02-Mar-1900 15:00:00'

74

2

3403

'22-May-1900 19:00:00'

3835

'09-Jun-1900 19:00:00'

75

2

4043

'18-Jun-1900 11:00:00'

2888

'01-May-1900 08:00:00'

76

2

1951

'23-Mar-1900 07:00:00'

997

'11-Feb-1900 13:00:00'

77

2

2325

'07-Apr-1900 21:00:00'

1056

'14-Feb-1900 00:00:00'

78

2

4085

'20-Jun-1900 05:00:00'

2241

'04-Apr-1900 09:00:00'

79

2

1663

'11-Mar-1900 07:00:00'

36

'02-Jan-1900 12:00:00'

80

2

2357

'09-Apr-1900 05:00:00'

1900

'21-Mar-1900 04:00:00'

81

2

601

'26-Jan-1900 01:00:00'

4392

'03-Jul-1900 00:00:00'

82

2

2627

'20-Apr-1900 11:00:00'

4327

'30-Jun-1900 07:00:00'

83

2

4379

'02-Jul-1900 11:00:00'

1035

'13-Feb-1900 03:00:00'

84

2

4364

'01-Jul-1900 20:00:00'

3722

'05-Jun-1900 02:00:00'

85

2

2004

'25-Mar-1900 12:00:00'

1002

'11-Feb-1900 18:00:00'

86

2

796

'03-Feb-1900 04:00:00'

4115

'21-Jun-1900 11:00:00'

87

2

1170

'18-Feb-1900 18:00:00'

1126

'16-Feb-1900 22:00:00'

88

2

3736

'05-Jun-1900 16:00:00'

3215

'14-May-1900 23:00:00'

89

2

1061

'14-Feb-1900 05:00:00'

2344

'08-Apr-1900 16:00:00'

90

2

4348

'01-Jul-1900 04:00:00'

1617

'09-Mar-1900 09:00:00'

91

2

4109

'21-Jun-1900 05:00:00'

159

'07-Jan-1900 15:00:00'

92

2

1545

'06-Mar-1900 09:00:00'

984

'11-Feb-1900 00:00:00'

93

2

60

'03-Jan-1900 12:00:00'

1933

'22-Mar-1900 13:00:00'

94

2

112

'05-Jan-1900 16:00:00'

1773

'15-Mar-1900 21:00:00'

95

2

4249

'27-Jun-1900 01:00:00'

2342

'08-Apr-1900 14:00:00'

96

2

1423

'01-Mar-1900 07:00:00'

3441

'24-May-1900 09:00:00'

97

2

126

'06-Jan-1900 06:00:00'

3616

'31-May-1900 16:00:00'

98

2

816

'04-Feb-1900 00:00:00'

2982

'05-May-1900 06:00:00'

99

2

233

'10-Jan-1900 17:00:00'

1694

'12-Mar-1900 14:00:00'

100

2

1229

'21-Feb-1900 05:00:00'

1659

'11-Mar-1900 03:00:00'

101

2

961

'10-Feb-1900 01:00:00'

632

'27-Jan-1900 08:00:00'

102

2

2851

'29-Apr-1900 19:00:00'

49

'03-Jan-1900 01:00:00'

103

2

4145

'22-Jun-1900 17:00:00'

3170

'13-May-1900 02:00:00'

104

2

207

'09-Jan-1900 15:00:00'

1227

'21-Feb-1900 03:00:00'

105

2

244

'11-Jan-1900 04:00:00'

1699

'12-Mar-1900 19:00:00'

106

2

1323

'25-Feb-1900 03:00:00'

20

'01-Jan-1900 20:00:00'

107

2

4174

'23-Jun-1900 22:00:00'

1779

'16-Mar-1900 03:00:00'

108

2

1683

'12-Mar-1900 03:00:00'

513

'22-Jan-1900 09:00:00'

109

2

4392

'03-Jul-1900 00:00:00'

3286

'17-May-1900 22:00:00'

110

2

565

'24-Jan-1900 13:00:00'

3653

'02-Jun-1900 05:00:00'

111

2

1783

'16-Mar-1900 07:00:00'

3676

'03-Jun-1900 04:00:00'

112

2

3555

'29-May-1900 03:00:00'

4040

'18-Jun-1900 08:00:00'

113

2

4309

'29-Jun-1900 13:00:00'

972

'10-Feb-1900 12:00:00'

114

2

2212

'03-Apr-1900 04:00:00'

2447

'12-Apr-1900 23:00:00'

115

2

4368

'02-Jul-1900 00:00:00'

2922

'02-May-1900 18:00:00'

116

2

2827

'28-Apr-1900 19:00:00'

0

'01-Jan-1900 00:00:00'

117

2

518

'22-Jan-1900 14:00:00'

3094

'09-May-1900 22:00:00'

118

2

4391

'02-Jul-1900 23:00:00'

2896

'01-May-1900 16:00:00'

119

2

3059

'08-May-1900 11:00:00'

1579

'07-Mar-1900 19:00:00'

120

2

2347

'08-Apr-1900 19:00:00'

4178

'24-Jun-1900 02:00:00'

121

2

873

'06-Feb-1900 09:00:00'

1476

'03-Mar-1900 12:00:00'

122

2

441

'19-Jan-1900 09:00:00'

25

'02-Jan-1900 01:00:00'

123

2

1876

'20-Mar-1900 04:00:00'

4356

'01-Jul-1900 12:00:00'

124

2

2066

'28-Mar-1900 02:00:00'

3851

'10-Jun-1900 11:00:00'

125

2

2572

'18-Apr-1900 04:00:00'

364

'16-Jan-1900 04:00:00'

126

2

2472

'14-Apr-1900 00:00:00'

2006

'25-Mar-1900 14:00:00'

127

2

4392

'03-Jul-1900 00:00:00'

3041

'07-May-1900 17:00:00'

128

2

1953

'23-Mar-1900 09:00:00'

3566

'29-May-1900 14:00:00'

129

2

967

'10-Feb-1900 07:00:00'

1514

'05-Mar-1900 02:00:00'

130

2

2021

'26-Mar-1900 05:00:00'

4216

'25-Jun-1900 16:00:00'

131

2

45

'02-Jan-1900 21:00:00'

2987

'05-May-1900 11:00:00'

132

2

1525

'05-Mar-1900 13:00:00'

2502

'15-Apr-1900 06:00:00'

133

2

1909

'21-Mar-1900 13:00:00'

757

'01-Feb-1900 13:00:00'

134

2

41

'02-Jan-1900 17:00:00'

3321

'19-May-1900 09:00:00'

135

2

4392

'03-Jul-1900 00:00:00'

1333

'25-Feb-1900 13:00:00'

136

2

3084

'09-May-1900 12:00:00'

4344

'01-Jul-1900 00:00:00'

137

2

570

'24-Jan-1900 18:00:00'

2328

'08-Apr-1900 00:00:00'

138

2

3065

'08-May-1900 17:00:00'

1103

'15-Feb-1900 23:00:00'

139

2

226

'10-Jan-1900 10:00:00'

0

'01-Jan-1900 00:00:00'

140

2

4388

'02-Jul-1900 20:00:00'

573

'24-Jan-1900 21:00:00'

141

2

1241

'21-Feb-1900 17:00:00'

4392

'03-Jul-1900 00:00:00'

142

2

4346

'01-Jul-1900 02:00:00'

1731

'14-Mar-1900 03:00:00'

143

2

345

'15-Jan-1900 09:00:00'

516

'22-Jan-1900 12:00:00'

144

2

3239

'15-May-1900 23:00:00'

3204

'14-May-1900 12:00:00'

145

2

2356

'09-Apr-1900 04:00:00'

4235

'26-Jun-1900 11:00:00'

146

2

418

'18-Jan-1900 10:00:00'

1320

'25-Feb-1900 00:00:00'

147

2

2376

'10-Apr-1900 00:00:00'

3596

'30-May-1900 20:00:00'

148

2

1

'01-Jan-1900 01:00:00'

4327

'30-Jun-1900 07:00:00'

149

2

3861

'10-Jun-1900 21:00:00'

29

'02-Jan-1900 05:00:00'

150

2

0

'01-Jan-1900 00:00:00'

3613

'31-May-1900 13:00:00'

151

2

'01-Jan-1900 02:00:00'

4354

'01-Jul-1900 10:00:00'

152

2

2288

'06-Apr-1900 08:00:00'

1621

'09-Mar-1900 13:00:00'

153

2

4183

'24-Jun-1900 07:00:00'

127

'06-Jan-1900 07:00:00'

154

2

304

'13-Jan-1900 16:00:00'

4097

'20-Jun-1900 17:00:00'

155

2

3052

'08-May-1900 04:00:00'

2257

'05-Apr-1900 01:00:00'

156

2

626

'27-Jan-1900 02:00:00'

0

'01-Jan-1900 00:00:00'

157

2

116

'05-Jan-1900 20:00:00'

1095

'15-Feb-1900 15:00:00'

158

2

524

'22-Jan-1900 20:00:00'

1380

'27-Feb-1900 12:00:00'

159

2

150

'07-Jan-1900 06:00:00'

2507

'15-Apr-1900 11:00:00'

160

2

1861

'19-Mar-1900 13:00:00'

3505

'27-May-1900 01:00:00'

161

2

1955

'23-Mar-1900 11:00:00'

416

'18-Jan-1900 08:00:00'

162

2

1868

'19-Mar-1900 20:00:00'

1881

'20-Mar-1900 09:00:00'

163

2

99

'05-Jan-1900 03:00:00'

1864

'19-Mar-1900 16:00:00'

164

2

4348

'01-Jul-1900 04:00:00'

2508

'15-Apr-1900 12:00:00'

165

2

2887

'01-May-1900 07:00:00'

3075

'09-May-1900 03:00:00'

166

2

4005

'16-Jun-1900 21:00:00'

72

'04-Jan-1900 00:00:00'

167

2

21

'01-Jan-1900 21:00:00'

2780

'26-Apr-1900 20:00:00'

168

2

495

'21-Jan-1900 15:00:00'

2387

'10-Apr-1900 11:00:00'

169

2

2936

'03-May-1900 08:00:00'

1972

'24-Mar-1900 04:00:00'

170

2

1171

'18-Feb-1900 19:00:00'

3668

'02-Jun-1900 20:00:00'

171

2

2984

'05-May-1900 08:00:00'

3336

'20-May-1900 00:00:00'

172

2

4377

'02-Jul-1900 09:00:00'

2398

'10-Apr-1900 22:00:00'

173

2

4386

'02-Jul-1900 18:00:00'

3395

'22-May-1900 11:00:00'

174

2

46

'02-Jan-1900 22:00:00'

1600

'08-Mar-1900 16:00:00'

175

2

3508

'27-May-1900 04:00:00'

4040

'18-Jun-1900 08:00:00'

176

2

120

'06-Jan-1900 00:00:00'

2137

'31-Mar-1900 01:00:00'

177

2

29

'02-Jan-1900 05:00:00'

2397

'10-Apr-1900 21:00:00'

178

2

3650

'02-Jun-1900 02:00:00'

3151

'12-May-1900 07:00:00'

179

2

746

'01-Feb-1900 02:00:00'

3975

'15-Jun-1900 15:00:00'

Table 7. Probabilistic impact assessment of maintenance

scheduling solutions considering wind farms.

Risk Indicator EENS (MWh/year)

PSO

Surrogate

17,306 [16,441; 18,171]

20,798 [19,758; 21,838]

499

Figure 7. Variations due to correlated wind speed assumptions.

Figure 8. Convergence of the surrogate algorithm.

Figure 9. Convergence of the PSO algorithm.

500

Figure 10. Generation capacity considering wind farms.

Figure 11. Maintenance scheduling schema.

4 CONCLUSIONS

This study introduced a revised methodology to

optimize the coordination of maintenance activities in

offshore wind farms within power systems. By

integrating multivariate stochastic wind simulations

into a maintenance scheduling framework, the

proposed model captures the spatial and temporal

dependencies inherent in offshore wind behavior, an

aspect oversimplified or overlooked in previous

results. Through extensive simulations based on IEEE-

RTS and a revised methodology drawn from previous

work, the results reveal that incorporating correlated

wind speed significantly affects the estimation of

Expected Energy Not Supplied (EENS). Specifically,

deviations greater than 6% in EENS were observed

under the same operational conditions when

comparing uncorrelated and highly correlated wind

scenarios. In addition, the new solution for

maintenance scheduling activities is 18% lower than

the previous solution under the new modeling

conditions. These findings confirm that the wind speed

correlation plays a critical role in system reliability and

should be taken into account in realistic maintenance

planning.

The use of copula-based modeling to simulate

correlated wind speeds, combined with heuristic

optimization algorithms (such as PSO and Surrogate

models), allowed for a nuanced and effective

scheduling of maintenance activities across dispersed

units. Among the heuristics tested, PSO achieved the

lowest EENS values, although further validation

through sensitivity analysis is needed to establish its

consistency across various scenarios. Certainly, this

modeling change introduced in this paper exposes

deviations (up to 6%) in the Expected Energy Not

Supplied (EENS) indicator under the same operational

conditions, purely due to changes in how wind speed

correlations are handled. Such deviations have a direct

impact on the allocation and timing of maintenance

activities, indicating that overlooking wind correlation

can lead to suboptimal or even misleading

maintenance strategies.

By incorporating realistic wind behavior and

spatial-temporal dependencies, the proposed

methodology enables more robust, data-driven, and

risk-aware planning of maintenance operations in

offshore environments.

In the case of directions for further research, we

plan to explore real operational data from actual

offshore wind farms to validate the evidence of

correlations in wind speed and introduce a dynamic

and real-time scheduling approach by using real-time

data assimilation and online adaptive optimization

algorithms to dynamically adjust maintenance

schedules.

501

ACKNOWLEDGMENT

The Polish Ministry of Science and Higher Education has

financially supported the work.

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