# Subseasonal-to-Seasonal Forecasting for Wind Turbine Maintenance Scheduling

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Data

#### 2.2. Site Wind Speed Data

#### 2.3. Initial Transformations and Corrections

`gam`function from the

`mgcv`package in R, using its default numerical optimisation method. The basis dimension of the splines, k, was checked with

`gam.check`to ensure that it was sufficiently large. All available time points were used for training the GAM before predicting the corrected wind speed for all times in the hindcast and forecast sets. This correction is illustrated in Figure 2.

#### 2.4. Site-Specific Index Models

#### 2.4.1. Mean Wind Speed Index

#### 2.4.2. Variability Index

#### 2.4.3. Weather Window Indices

^{−1}to perform a given activity. Two obvious metrics to measure this on a weekly basis would be the number of hours in the week below the wind speed threshold, or the number of weather windows in the week. However, there are pitfalls to both of these. Counting the total number of ‘safe’ hours in the week does not tell us anything about the length of weather windows available, so that 10 h below the wind speed threshold could mean a continuous block of 10 h available, or 10 separate blocks of 1 h windows, which is much less useful for a task that takes several hours. If we instead report the number of weather windows of a certain minimum time length, a value of 1 window in a week could mean a single window of the minimum time duration, or that the wind speed is below the threshold for the entire week; the second of these scenarios would allow a lot more work to be done in practice than the first. The final metric used is therefore a combination of these: the number of hours in the week contained in a weather window of a certain minimum length. This way, every hour that is counted is valuable for work as it is guaranteed to be in a usefully long window and there is still a distinction between a single long and a single short weather window. This metric does not differentiate between one long and several shorter windows, but the minimum duration constraint ensures that each window counted is long enough for maintenance activities to be performed in it. Thus, the two parameters defining the weather windows (duration and wind speed threshold) are both determined by the maintenance activity of interest. All weather windows are counted, regardless of whether they fall within the usual working week (9–5 Monday–Friday) or not, as the decision to work at nights or weekends will depend on other factors, including the urgency of the job and availability of staff to work other hours. The number of useful hours metric ${H}_{T}^{(p,q)}$ is the number of hours in the week where wind speed is below p ms

^{−1}for q hours or more at a time and is modelled as

#### 2.5. Ensemble Model Output Statistics

**R**function

**fitDist**from the

**gamlss**package, which allows the fitting of a group of possible distributions with the optimal distribution having the lowest AIC score. In

**fitDist**, all distributions are fit with constant values for all parameters; this is then used to select a distribution that more closely matches the overall distribution of the target variable. For the mean wind speed and variability indices, the set of ‘realplus’ distributions were considered, as wind speeds (and their standard deviations) may only take real positive values. The gamma distribution was found to most closely fit the real distribution of weekly mean wind speeds and variabilities so this was used as the base parametric distribution to fit the EMOS correction model (a GAMLSS). The gamma distribution has two parameters, location $\mu $ and scale $\sigma $. First, we define two metrics of the K ensemble index forecasts ${x}_{k}$, their mean value $\overline{x}$ and mean difference $\overline{\Delta x}$ [32]. This value of mean difference takes into account the distance between each pair of ensemble members, and is less sensitive to outliers compared to measures such as standard deviation.

**gamlss**function and log link functions in Equations (11) and (12) to ensure positive values for ${\mu}_{t}$ and ${\sigma}_{t}$.

**R**are the location $\mu $, scale $\sigma $ and two parameters related to the amount of probability assigned at zero and one, $\nu $ and $\tau $. These are modelled as functions of the same ensemble mean ${\overline{x}}_{t}$ and ensemble mean difference $\Delta {\overline{x}}_{t}$ as in Equations (9) and (10):

**R**to those more commonly used to describe the beta distribution and the boundary probabilities, the final forecast distribution is given by

#### 2.6. Benchmark Climatology

## 3. Results

#### 3.1. Mean Wind Speed Forecasts Trained on MIDAS Data

#### 3.2. Mean Wind Speed Forecasts Trained on ERA5 Data

#### 3.3. Variability Indices

#### 3.4. Weather Window Indices and Economic Value of Forecasts

^{−1}in this work (Exact limits are tied to the specific job and equipment and are set out in the ‘authorised work procedure’. Interviews with site operations teams indicated that typical crane lift limits are between 5 and 8 ms${}^{-1}$). It is also assumed that a minimum of an 8 h window is needed to complete a maintenance task. As an indicative figure, in the 20 years of hindcast data at one site, 50.8% of time points were below the 7 ms

^{−1}threshold. Of the 5003 unique weather windows, 2582 of them lasted 8 h or longer.

## 4. Conclusions

- A comparison between forecasts generated with a complete measured time series and those using reanalysis data corrected with a limited history of site data. This bridges the gap between common methods for desk-based studies and those necessary to apply models to real-world sites.
- Determination of the skill of S2S forecasts across three different metrics that are relevant for maintenance planning.
- Implementation of a cost-loss model and investigation of the sensitivity of hiring decisions to electricity prices.

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

EA | East Atlantic |

EAWR | East Atlantic Western Russia |

ECMWF | European Centre for Medium-range Weather Forecasts |

EMOS | Ensemble Model Output Statistics |

GAM(LSS) | Generalised Additive Model (for Location, Scale and Shape) |

LASSO | Least Absolute Shrinkage and Selection Operator |

MIDAS | Met Office Integrated Data Archive System |

MJO | Madden–Julian Oscillation |

NAO | North Atlantic Oscillation |

NWP | Numerical Weather Prediction |

PCA | Principal Component Analysis |

S2S | Subseasonal-to-Seasonal |

S2S4E | Subseasonal-to-Seasonal Forecasting for Energy |

SCA | Scandinavian Pattern |

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**Figure 1.**Flowchart of data and modelling steps to produce the final site index forecasts. The Principal Component Analysis step identifies large-scale weather patterns in the grid of forecasts over a whole European and North Atlantic area; the ERA5 correction to site fits a Generalised Additive Model (GAM) to correct ERA5 wind speeds to more closely match the short history of observed site wind speeds. The site index model takes the S2S principal components as inputs and is trained to forecast a site-specific index given by the site data for the training set. The Ensemble Model Output Statistics step corrects any bias or dispersion problems with the ensemble of site index forecasts before forecasts are evaluated.

**Figure 3.**Principal components of 500 hPa geopotential height mapped back to geographical space, for January and July.

**Figure 4.**Weekly mean wind speed index forecast versus observed weekly mean forecast, for 2 weeks ahead forecasts at the MIDAS 4 station.

**Figure 5.**Time series of all individual ensemble member 2 weeks ahead index forecasts (light grey), overlaid with the actual weekly mean wind speeds (black). There is clear under-dispersion in the set of ensemble members.

**Figure 6.**Verification rank histogram of all 2 weeks ahead ensemble member index forecasts, showing the ‘U’ shape characteristic of under-dispersion.

**Figure 7.**Pinball skill score of weekly mean wind speed index, at MIDAS 1 and relative to climatology. The three index model configurations are with separate Principal Components, with extra weekly features (standard deviation, min and max) of the PCs inputs, and where all 100 m weather variables are transformed into PCs together. Error bars show 95% bootstrapped confidence intervals.

**Figure 8.**Normalised reliability diagram for mean wind speed index and the equivalent climatology forecast, at MIDAS 1. The forecast labelled ‘linear model’ is calculated with separate PCs and no extra engineered features. Horizons are in number of weeks.

**Figure 9.**Pinball skill score of weekly mean wind speed index relative to climatology. The forecasts labelled ‘WF 1’ are based on corrected ERA5 data for that wind farm. Error bars show 95% bootstrapped confidence intervals. MIDAS skill score is calculated relative to climatology of the MIDAS wind speed data and WF1 skill score is calculated relative to climatology of the corrected ERA5 data.

**Figure 10.**Reliability of weekly mean wind speed forecast at MIDAS 1. Intervals show 95% bootstrapped confidence bands.

**Figure 11.**Reliability of weekly mean wind speed forecast at WF1. Intervals show 95% bootstrapped confidence bands.

**Figure 12.**Pinball skill score of weekly mean wind speed index relative to climatology at WF 1, across the forecast distribution. The two week ahead forecast is shown here. Coloured band shows the 95% bootstrapped confidence interval.

**Figure 13.**Scatter plots of realisation probabilities for neighbouring sites. The solid black line shows $y=x$, i.e., perfect correlation in realisation probabilities, where the two sites are always under- or overforecast by the same proportion at the same times.

**Figure 14.**Pinball skill score of weekly variability (weekly standard deviation of hourly wind speeds) relative to climatology. The forecasts labelled ‘WF 1’ are based on corrected ERA5 data for that wind farm. Error bars show 95% bootstrapped confidence intervals.

**Figure 15.**Reliability of variability forecast at MIDAS 1. Intervals show 95% bootstrapped confidence bands.

**Figure 16.**Reliability of variability forecast at WF1. Intervals show 95% bootstrapped confidence bands.

**Figure 17.**(

**a**) Fan plot of 2 weeks ahead weather window index forecasts. (

**b**) Time series of realisation probabilities for the same forecasts as (

**a**).

**Figure 18.**Pinball skill score of weather window index relative to climatology for area 1. Error bars show 95% bootstrapped confidence intervals.

**Figure 19.**Reliability of weather window index forecast at WF1. Intervals show 95% bootstrapped confidence bands.

**Figure 20.**Relationship between number of hours in a weather window and weekly total lost energy at WF1. Both axes are normalised by their maximum (i.e., number of hours in a week and maximum weekly energy output).

**Figure 21.**Flowchart of crane hire decision and resulting costs, assuming that three turbines are broken. The week starts at time t = 0 and ends at time t = w. E is the expected cost of lost energy for one turbine for the whole week, given the weather window index forecast (see Equations (20)–(23)). C denotes the cost of crane hire for the week, J denotes the job time for repair of one turbine and T${}_{m}$ is a terminal cost representing the average energy loss in subsequent weeks before the next opportunity to repair the m unfixed turbines. x is the number of useful hours. Where one or more turbines are repaired, the times of repair are assumed to be divided evenly throughout the week.

**Figure 22.**Expected cost of hiring crane vs. expected cost of not hiring crane, for 2 weeks ahead forecasts of number of available hours in the week. Colour shows the q50 forecast value of normalised number of hours within a weather window.

**Figure 23.**Range of electricity prices at each site, where crane hire decision is sensitive to the forecast number of weather window hours, for either 2 or 3 turbines down. Marker shows the median price and whiskers show the minimum and maximum electricity prices where the hire decision is sensitive to the forecast.

**Figure 24.**Number of turbines fixed when index forecasts or climatology forecasts are used to make crane hiring decision, dependent on electricity price, at WF3. Error bars show 95% bootstrapped confidence intervals.

**Figure 25.**Cost of crane hire and lost energy when index forecasts or climatology forecasts are used to make hiring decision, for a range of electricity prices, at WF3. Error bars show 95% bootstrapped confidence intervals.

**Table 1.**Table of index model input features. gph = geopotential height; ws = wind speed. * 925 and 1000 hPa wind speed and mean sea level pressure are all included in one PC transform together.

Variable (s) | PCs Kept | Features | Index Models | ||
---|---|---|---|---|---|

Mean ws | Variability | Weather Window | |||

weekly mean | √ | √ | √ | ||

500 hPa gph | 20 | weekly sd | √ | ||

weekly min, max | √ | ||||

weekly mean | √ | √ | √ | ||

10 m ws | 40 | weekly sd | √ | ||

weekly min, max | √ | ||||

weekly mean | √ | √ | √ | ||

100 m variables * | 20 | weekly sd | √ | ||

weekly min, max | √ |

**Table 2.**Results of cost-loss analysis for a range of electricity prices at WF3, for the 2 weeks ahead forecasts. The cost per turbine includes both crane hire cost and the cost of lost energy before turbines were fixed. Number of turbines fixed is assuming ‘N turbines down’ needed repair for every week in the 2-year forecast period.

N Turbines Down | Electricity Price £/MWh | N Fixed (Index Model) | £/Turbine (Index Model) | N Fixed (Climatology) | £/Turbine (Climatlogy) |
---|---|---|---|---|---|

2 | 75 | 19 | 110,900 | 0 | - |

2 | 80 | 63 | 35,980 | 34 | 68,090 |

2 | 85 | 107 | 22,610 | 150 | 17,470 |

2 | 90 | 172 | 15,100 | 204 | 13,480 |

2 | 95 | 210 | 13,450 | 204 | 13,830 |

2 | 100 | 246 | 12,300 | 238 | 12,530 |

2 | 105 | 268 | 11,890 | 238 | 12,800 |

2 | 110 | 286 | 11,600 | 274 | 12,050 |

2 | 115 | 314 | 10,860 | 274 | 12,270 |

2 | 120 | 328 | 10,660 | 336 | 10,580 |

3 | 48 | 24 | 73,300 | 0 | - |

3 | 52 | 121 | 17,320 | 91 | 22,340 |

3 | 56 | 229 | 10,380 | 273 | 9330 |

3 | 60 | 337 | 8220 | 351 | 8000 |

3 | 64 | 381 | 7940 | 351 | 8200 |

3 | 68 | 426 | 7608 | 405 | 7850 |

3 | 72 | 477 | 7170 | 447 | 7520 |

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## Share and Cite

**MDPI and ACS Style**

Tawn, R.; Browell, J.; McMillan, D.
Subseasonal-to-Seasonal Forecasting for Wind Turbine Maintenance Scheduling. *Wind* **2022**, *2*, 260-287.
https://doi.org/10.3390/wind2020015

**AMA Style**

Tawn R, Browell J, McMillan D.
Subseasonal-to-Seasonal Forecasting for Wind Turbine Maintenance Scheduling. *Wind*. 2022; 2(2):260-287.
https://doi.org/10.3390/wind2020015

**Chicago/Turabian Style**

Tawn, Rosemary, Jethro Browell, and David McMillan.
2022. "Subseasonal-to-Seasonal Forecasting for Wind Turbine Maintenance Scheduling" *Wind* 2, no. 2: 260-287.
https://doi.org/10.3390/wind2020015